Anomalous Diffusion in Anisotropic Media Felix Kleinschmidt

Anomalous Diﬀusion in Anisotropic Media

Inauguraldissertation zur Erlangung der Doktorwürde der Fakultät fürChemie, Pharmazie und Geowissenschaften der Albert-Ludwigs-Universität Freiburg im Breisgau

vorgelegt von

Felix Kleinschmidt aus Freiburg im Breisgau

5. April 2005

”Il faut avoir ´etudi´ebeaucoup pour savoir peu.”

Montesquieu (1689-1755) Pens´ees, Nr. 1116 Œuvres compl`etes,Paris 1950

Vorsitzender des Promotionsausschusses: Prof. Dr. G. Schulz Dekan: Prof. Dr. H. Hillebrecht Referentin: Prof. Dr. C. Schmidt Korreferent: Prof. Dr. H. Finkelmann

Betreuerin der Arbeit: Prof. Dr. C. Schmidt

Tag der m¨undlichen Pr¨ufung:09.05.2005

Die vorliegende Arbeit entstand in der Zeit von April 2001 bis April 2005 am Institut f¨urMakromolekulare Chemie der Albert-Ludwigs-Universit¨at Freiburg im Breisgau.

Publications and Presentations

Publications

Kay Saalw¨achter, Felix Kleinschmidt and Jens-Uwe Sommer, Swelling Hetero- geneities in End-Linked Model Networks: A Combined Proton Multiple-Quantum NMR and Computer Simulation Study, Macromolecules 2004, 37, 8556-8568

Felix Kleinschmidt, Markus Hickl, Claudia Schmidt and Heino Finkelmann: Smec- tic Liquid Single Crystal Hydrogels (LSCH): Hygroelastic and 2H NMR Diﬀusion Measurements, in preparation

Felix Kleinschmidt and Claudia Schmidt: Multilamellar Vesicles from C10E3/D2O; NMR Line Shape and Pulsed Gradient Diﬀusion Measurements, in preparation

Scientiﬁc Talks

Felix Kleinschmidt, Patrick Becker, Laurence Noirez and Claudia Schmidt: Side- Chain Liquid Crystal Polymers under Shear Flow, Meeting of the TMR network ’Rheology of Liquid Crystals’, Lisbon / Portugal, 5-8/04/2001

Felix Kleinschmidt and Claudia Schmidt: Diﬀusion in Anisotropic Media, Meet- ing of the Collaborative Research Centre (SFB) 428, Waldau, 6-7/12/2001

Felix Kleinschmidt, Markus Hickl and Claudia Schmidt: PGSE-Diﬀusion Mea- surements in Anisotropic Hydrogels, Meeting ’Solid State NMR Methods and Applications in Material Sciences’, Oberjoch, 14-18/07/2002

Felix Kleinschmidt and Claudia Schmidt: Anisotropic Hydrogels: 2H PGSE Dif- fusion Measurements, Liquid Crystal Group Meeting, Paderborn, 11/2002 Felix Kleinschmidt and Claudia Schmidt: 2H NMR in Oriented Liquid Crystals, Meeting of the Collaborative Research Centre (SFB) 428, Waldau, 28-29/11/2002

Felix Kleinschmidt and Claudia Schmidt: 2H PGSE Diﬀusion Measurement in Oriented Liquid Crystals, ’NMR Spring Workshop’, Bayreuth, 14-17/02/2003

Felix Kleinschmidt and Claudia Schmidt: Structure of Multilamellar Vesicles, Meeting of the Collaborative Research Centre (SFB) 428, Waldau, 27-28/11/2003

Felix Kleinschmidt and Claudia Schmidt: Diﬀusion in Multilamellar Vesicles, ’NMR Spring Workshop’, Wandlitz (Berlin), 02/2004

Felix Kleinschmidt and Claudia Schmidt: Multilamellar Vesicles from C10E3/D2O; NMR Line Shape and Pulsed Gradient Diﬀusion Measurements, NMR - Group Meeting, Darmstadt, 26/7/2004

Felix Kleinschmidt and Claudia Schmidt: Multilamellar Vesicles from C10E3/D2O; NMR Line Shape and Pulsed Gradient Diﬀusion Measurements, ’32. Workshop on Liquid Crystals’ of the German Liquid Crystal Society, Halle, 24-26/03/2004

Poster Presentations

Claudia Schmidt, Felix Kleinschmidt and Daniel Burgemeister: Rheo-NMR and Diﬀusion Studies in Liquid Crystals, Meeting of the Magnetic Resonance Spec- troscopy Division of the German Chemical Society, Bremen, 24-27/09/2002

Felix Kleinschmidt and Kay Saalw¨achter: NMR Characterization and Self-diﬀusion Study of Partially Swollen Polymer Networks, Congress ’Macromolecular Collo- quium’, Freiburg, 27/02-1/03/2003

Felix Kleinschmidt, Markus Hickl, Heino Finkelmann and Claudia Schmidt: Smec- tic Liquid Single Crystal Hydrogels (LSCH): Hygroelastic and 2H NMR Diﬀusion Measurements, ’31. Workshop on Liquid Crystals’ of the German Liquid Crystal Society, Mainz, 19-21/03/2003

Felix Kleinschmidt and Claudia Schmidt: Multilamellar Vesicles from C10E3/D2O: NMR Line Shape and PFG Diﬀusion Measurements, ’GIDRM XXXIV Italian Na- tional Congress on Magnetic Resonance’, Porto Conte / Sardinia 21-24/09/2004

Table of Contents

1 Introduction 1 1.1 Motivation ...... 1 1.2 Restricted Diffusion in an Oriented Layered Environment . . . . . 2 1.3 Restricted Diffusion in Bent Lamellae ...... 4 1.4 Diffusion in Obstructed Geometries ...... 5

2 Theoretical Aspects 9 2.1 Diffusion ...... 9 2.1.1 Introduction to Translational Dynamics ...... 9 2.1.2 Fick´s Laws and Gaussian Diffusion ...... 9 2.1.3 Diffusion as a Step Process ...... 11 2.2 NMR Spectroscopy ...... 12 2.2.1 Zeeman Interaction and the Effect of HF-Pulses ...... 12 2.2.2 Quadrupolar Interaction ...... 14 2.2.3 Introduction to Gradient NMR ...... 16 2.3 Experiments ...... 17 2.3.1 The Pulsed Gradient Spin Echo (PGSE) ...... 17 2.3.2 The Stejskal-Tanner Equation: GPD and SGP Approximation ...... 20 2.3.3 Comparison of the GPD and SGP ...... 22 2.3.4 Notes for the Application ...... 22 2.3.5 The Pulsed Gradient Stimulated Echo (PGSTE) ...... 23 2.3.6 Relaxation ...... 24

i Table of Contents

3 Anisotropic Hydrogels 25 3.1 Hydrogel Sample Preparation ...... 25 3.2 Single Pulse Spectra ...... 30 3.3 Quadrupolar Splitting and Local Order ...... 31 3.4 Time Dependent Diﬀusion ...... 32 3.4.1 Estimation of Domain Sizes ...... 33 3.5 Temperature Dependent Diﬀusion ...... 37

3.5.1 Obstruction of D2O Diffusion ...... 37 3.5.2 Arrhenius Analysis ...... 42 3.5.3 Anisotropy of Diffusion ...... 45 3.6 Local Order ...... 47 3.6.1 The Orientational Distribution Function ...... 47 3.6.2 X-Ray Correlation Length ...... 51 3.6.3 Macroscopic Dye Diffusion ...... 53 3.7 Discussion: Hydrogels ...... 54

4 Multilamellar Vesicles 57 4.1 MLV Sample Preparation ...... 58 4.2 Polarizing Light Microscopy of MLV ...... 59 4.3 A Comment on the Interstitial Volume ...... 60 4.4 The Single Pulse Line Shape ...... 62

4.4.1 The Characteristic D2O Line Shape ...... 62 4.4.2 Theoretical Eﬀect of Polydispersity ...... 65 4.4.3 Estimation of the MLV Size from the Single Pulse Line Shape 66 4.4.4 A Surface Record Factor (SRF) for the MLVs ...... 73 4.5 Echo Line Shapes ...... 78 4.5.1 The Solid Echo ...... 78 4.5.2 Size Dependence of the Solid Echo ...... 82 4.5.3 The Stimulated Echo ...... 83 4.5.4 Relaxation ...... 85 4.5.5 Explanation for the Echo Line Shape ...... 88 ii Table of Contents

4.5.6 On Solid Echo Simulations ...... 95 4.6 Diﬀusion measurements ...... 96

4.6.1 Diffusion in the Oriented Lα Phase ...... 96 4.6.2 The PGSTE Echo Decay in MLVs ...... 99 4.7 Towards a Model for the PGSTE Echo Decay in MLVs ...... 105 4.7.1 Phenomenological Description of the Echo Decay on One Sphere ...... 106 4.7.2 Simulated Echo Decay for One Ideal MLV ...... 109 4.7.3 Effect of Finite Gradient Duration: ”Center of Mass”-Diffusion115 4.8 Discussion: MLV ...... 116

5 Swelling of PDMS Model Networks 119 5.1 Sample preparation ...... 120 5.2 Single Pulse Spectra ...... 121 5.3 Diﬀusion measurements ...... 122 5.3.1 On Temperature Stability ...... 125 5.3.2 Swelling Dependent Diﬀusion ...... 126 5.4 Discussion: PDMS Model Networks ...... 128

6 Summary 129

Appendices 135

A Hardware 135 A.1 The Spectrometer ...... 135 A.2 The Gradient System ...... 135 A.3 Temperature Control and Stability ...... 138 A.4 Gradient Calibration ...... 140 A.5 The Rotatable Sample Holder ...... 140

B Additional Sample Data 143 B.1 Solid Echo Spectra of MLV ...... 143

iii Table of Contents

B.2 PGSTE Diﬀusion Decay for MLV ...... 145

B.3 Arrhenius Plot for Diﬀusion in the Lα Phase ...... 146 B.4 PDMS Network Samples ...... 147

C Computer Programs 149 C.1 Numerical Calculations ...... 149 C.2 LabTalk Scripts ...... 165 C.3 Pulse Programs used in this Work ...... 178 C.3.1 Solid Echo ...... 178 C.3.2 Saturation Recovery ...... 179 C.3.3 PGSE ...... 179 C.3.4 PGSTE ...... 181

D NMR Calculations 185 D.1 Product Operators ...... 185 D.2 Solid Echo Calculation ...... 186

References 188

Chapter 1

Introduction

1.1 Motivation

Complex mesoscale structures formed by surfactant systems are of great academic interest [1–3]. The mesoscopic regime bridges the properties on the molecular level with macroscopic characteristics and features. Nuclear magnetic resonance (NMR) diffusion experiments provide the means to learn much about structure and dynamics of mesoscopic systems [4–8]. The technique of pulsed field gradients inserted in an NMR echo sequence, which was developed already in the 1960s by Stejskal and Tanner [9, 10], made it possible to detect a well resolved spectrum while measuring diffusion. The method is technologically demanding and its use was limited due to the necessity to construct a suitable NMR probe head. In the last years the technological progress made pulsed gradient equipment commercially available. The pulsed gradient diffusion NMR has access to timescales in the range of ten milliseconds up to seconds. The mean squared displacement due to translational diffusion therefore ranges from some µm up to several tens of µm. The method of deuterium NMR line shape analysis, being sensitive to rotational motions, is complementary to a certain degree, since rotational and translational motions are often coupled. Therefore rotational motion can be associated with a length scale if the curvature of the diffused path and the diffusion coefficient are known. This length scale ranges, in ’soft matter’ systems, from the nanoscopic regime

1 1. Introduction to several µm. The combination of both methods yields additional information on the translational displacement and on the dynamics of the molecule under investigation. The aim of this work was to implement the pulsed gradient spin echo method and to demonstrate its capabilities for gaining insight into structure and dynamics of ’soft matter’ systems. Three systems with different topological features were chosen. The first system is a lyotropic liquid crystalline (LC) anisotropic hydrogel [11], which consists of three-dimensionally crosslinked lamellar monolayers with a uniform, magnetically induced layer orientation. The local and mesoscopic characterization, with special focus on order phenomena of this novel material, was of special interest because of its projected application as a bifocal contact lens [12]. In the second system the complexity of the description of both deuterium line shapes and the translational diffusion is enhanced. Multilamellar vesicles, often referred to as ’onions’ [13–15], from a lyotropic liquid crystalline mixture of a low molar mass nonionic amphiphile and deuterated water exhibit strongly curved bilayers. Translational diffusion along the lamellae in this system is in- evitably associated with rotation and therefore a partial averaging of quadrupolar interaction. This leads to complex inhomogeneous line shapes and to a diffusive behavior, which is highly dependent on time or length scales, respectively. In the third system the focus is changed from restricted diffusion, which is subject to certain boundary conditions, to the different subject of obstruction. The fractal topology of swollen model networks provides the test field for recently developed theories on inhomogeneous network swelling, where diffusion time dependencies of apparent diffusion coefficients are expected [16–19].

1.2 Restricted Diﬀusion in an Oriented Layered Environment

Anisotropic Hydrogels. When looking at diﬀusion in liquid crystals the assumption of molecules diﬀusing in all three directions of space with the same

2 1. Introduction probability is no longer valid. The presence of boundaries or restrictions of any kind play an important role. For the Lα phase, for example, the diffusing water molecules are, in first approximation, not allowed to travel through the membranes and are confined between the layers. Nevertheless for very short distances the picture of free diffusion is still valid. If the majority of the molecules do not have time to travel to a barrier, their ensemble would not ’feel’ the boundary. For an experiment with a sufficiently good position determination the transition from free to restricted diffusion should be measurable [20]. This would mean that an experiment for diffusing water with a diffusion coefficient D of about 2 ·10−9 m2/s must have a resolution better than about one nanosecond to see free diffusion between two hydrophobic boundaries separated by typically about 20 A.˚ With the pulsed gradient experiment this is impossible. The temporal resolution of about 10 ms is not sufficient. The water molecule would have experienced a root mean squared (r.m.s) displacement of about 6 µm in this time. Therefore the NMR experiment is only sensitive to the long time behavior of the molecules, i. e., their behavior on a mesoscopic scale. The measured data contain information about the phase structure and texture of the sample, while information about microscopic properties is only indirectly accessible through theoretical modelling. The most important property of diffusion coefficients in such systems is their (potential) dependence on the diffusion time; or better say the travelled distance. This dependence can include information on defects in the lamellae (from the diffusion coefficient parallel to the layer normal, Dk) or on the lateral extension of a homogenous layer orientation (from the diffusion coefficient perpendicular to the layer normal, D⊥). But more impor- tantly, it yields information on characteristic length scales in the sample in the order of several tens of µm. Data analysis is not only possible for homogeneously ordered structures, but also for well defined layered systems with orientational distributions in two or three dimensions [21]. The anisotropic hydrogel investigated here is a model system with a homogeneously ordered layered structure. In this case the diffusion can still be described by the mathematics of isotropic diffusion. The difference is that the diffusion coefficient now becomes a second rank tensor and is no longer a simple number.

In the uniaxial environment of the Lα phase the diﬀusion tensor in the principal

3 1. Introduction axis system of the liquid crystalline phase (the z axis now lying parallel to the membrane layer normal) takes the following form:

  D⊥ 0 0   Db =  0 D⊥ 0  (1.1)   0 0 Dk

The notation ’parallel’ and ’perpendicular’ refer to the axis of symmetry of the layered structure. Field gradient NMR experiments measure diffusion only in the direction of the gradient. Therefore it is necessary to reconstruct the full diffusion tensor from different measurements, which can be quite simple if the axis of the field gradient can be aligned with any of the principal axes of the sample. Then no transformation of coordinate systems has to be performed and the desired part of the diffusion tensor can directly be extracted from the measurement.

1.3 Restricted Diﬀusion in Bent Lamellae

Multilamellar Vesicles. The theoretical description of the diffusive behavior of particles in a uniformly oriented layered structure is quite simple. The situation becomes more complicated, and therefore more interesting, when the layers get curved. For pulsed gradient diffusion NMR of D2O in a surfactant mesophase the radius of such a curvature must be in the micrometer range to produce significant effects. Furthermore such a curvature leads to effects of partial motional averaging of the deuterium line shape of D2O. Again the ability of the pulsed gradient method to acquire a well resolved spectrum together with the information on translational displacement is used. In surfactant mesophases, the deuterium line shapes yield information on the dynamics on a timescale in the same order of magnitude as the information on the displacement from the diffusion measurements. A system, which exhibits these features are the multilamellar vesicles (MLV). It makes them a suitable choice to demonstrate the accessability to simultaneous information on both the dynamics and the structure by the pulsed gradient experiment. MLVs are accessible by mechanical shear applied to a lamellar Lα phase.

4 1. Introduction

∗ The membranes in this Lα state are ordered in concentric spheres, resulting in an ’onion’-like structure (ﬁgure 1.1). The size and polydispersity of such MLVs that

Figure 1.1: An idealized MLV, consisting of many concentric spheres. were prepared under shear can be precisely controlled by the shear rate [13, 14]. This makes this method of preparation so unique and interesting. Since the procedure of preparation is simple, the shear induced MLVs are of special interest to the pharmaceutical industry. Drug encapsulation [15, 22] and the potential use as micro-reactors [23] are only two of the possibilities of this system. While the question of how the MLV structure is formed is still a matter of debate [24–27], this work focusses mainly on the question of what information on the structure of the metastable, but often long-lived, MLV state can be gained from deuterium NMR. Investigations by NMR on diﬀusive motion in spherical bilayers have been carried out on solid-supported bilayers and also on multilamellar vesicle systems by means of two-dimensional exchange [28–32] and relaxation [33] spectroscopy. Line shapes of vesicles have also been subject to theoretical treatment [34, 35]. The long term eﬀects on translational motions have not yet been investigated.

1.4 Diﬀusion in Obstructed Geometries

PDMS Model Networks. Not only regular barriers such as the soft membranes in liquid crystals or the hard walls in porous rocks or Swiss cheese [20,

5 1. Introduction

21, 36] are possible constraints to free diffusion. Also domains of different diffusivity act as such constraints to free movement and are interesting for diffusion measurements. In this case one speaks of obstructed diffusion to emphasize the statistical nature of these ’obstacles’. Such obstacles can be found in many materials, whose heterogeneous nature is not debated. On the other hand the detection of obstruction in materials which are claimed to be homogeneous would invoke the necessity to reform the classical models of these materials. One such system are polymer networks. When measurements are performed on such systems, it is necessary to define a quantity, which makes diffusion coefficients from various species comparable. A so called obstruction factor Q is defined as the ratio of the observed diffusion coefficient with the free diffusion coefficient of the observed molecule:

D Q = measured (1.2) Dfree

The classical model of polymer networks is based on several assumptions. No dangling chain ends are allowed and the segments between two crosslinks can be described by Gaussian statistics for polymer coils. Furthermore it is assumed that a deformation of the network acts equally on every part of the sample. This is called affine deformation. If the network is swollen this implies that it should be homogeneously permeable for diffusing solvent molecules (figure 1.2, left). In this case the diffusion coefficient of the solvent should only depend on the degree of swelling, resulting in faster diffusion at higher degrees of swelling. If, on the other hand, the swelling process results in an inhomogeneous network with regions of higher and lower crosslink density, which have different permeability for a solvent, then the diffusion coefficient depends strongly on the diffusion time [37]. For short diffusion times the molecules do not travel far enough to ’feel’ an effect of the density changes. An averaged diffusion coefficient of molecules in regions of different densities is then measured. For longer diffusion times the diffusing solvent molecules interact with the boundaries (figure 1.2, right). Their path is no longer free and the measured diffusion coefficient is reduced. In the long time limit the situation reaches a plateau. From the decay of the diffusion coefficient with the diffusion time (or better the corresponding length scale), an estimation

6 1. Introduction

freediffusion obstructeddiffusion

solventmolecule

Figure 1.2: Sketch of homogeneous, affine swelling (left) and inhomogeneous, non-affine swelling (right); red: possible pathways of diffusing solvent molecules; yellow: region of higher crosslink density. of the size of the inhomogeneities would be possible. Such heterogeneities were reported for swollen poly-N-isopropylacrylamid (PNIPAM) gels [38] and were also claimed to be found by pulsed gradient NMR in gellan gum hydrogels [39], where a depression of the apparent diffusion coefficient with diffusion time was observed. The heterogeneities were in the range of several microns. In another pulsed gradient work similar results were found but the apparent restriction effect was interpreted as cross-relaxation [40] and the picture of swelling heterogeneities was rejected. In this work the attempt is made to proof the existence of such micrometer size heterogeneities by means of diffusion NMR on mono- and bimodal poly(dimethylsiloxane) (PDMS) networks. These networks are well defined and the bimodal distribution of network chains makes it reasonable to anticipate detectable heterogeneities, therefore making it possible to test the suitability of the method.

Chapter 2

Theoretical Aspects

2.1 Diﬀusion

2.1.1 Introduction to Translational Dynamics

The motion of a molecule can be characterized by some time-dependent displacement ri(t). The conditional probability of finding any molecule (or scattering center) at a place r0 at time t, if it was initially at r at t = 0 is defined by the conditional probability P (r|r0, t). The problem is that this function correlates 0 the positions of ri(t) and rj(0) for i 6= j and for i = j. It is therefore sensitive to relative motions. There is another conditional probability that describes the 0 correlation only for i = j. This is the self -correlation function Ps(r|r , t). The possibility to measure Ps is very helpful since descriptions of self-diffusion are much simpler. At the microscopic level only two methods have the inherent ability to label the observed scattering centers: the pulsed gradient spin echo (PGSE) NMR and polarized neutron scattering [21].

2.1.2 Fick´s Laws and Gaussian Diﬀusion

In the PGSE experiment the motion of the particles is probed by two measurements at t = 0 and at a second time t. The recorded echo attenuation gives information on the displacement along the gradient axis (section 2.3.1) that has occurred between the two measurements. The attenuation can be related to the

9 2. Theoretical Aspects diffusion coefficient but it does not, at least directly, contain information about how the particle moved from its initial to its final position. On the macroscopic level diffusion against a concentration gradient is described by Fick´s first law, which relates the flux J(r, t) with the local concentration gradient ∇c(r, t): J(r, t) = −D∇c(r, t) (2.1)

The change of the concentration gradient with time is described by Fick´s second law: ∂c(r, t) = D∇2c(r, t) (2.2) ∂t

With the initial condition of c(r, 0) = δ(r − r0) the solution of equation (2.2) is the famous Gaussian diffusion profile [41, 42]. Equation (2.2) is valid if D is a scalar. If the diffusion becomes anisotropic, D has to be expressed as a tensor (see section 1.2) and D∇2 has to replaced by ∇D∇. In the case of self-diffusion there is no net concentration gradient, but the concentration can be replaced by the total probability P (r0, t) of finding a particle at the position r0 at the time t. This is given by Z P (r0, t) = ρ(r)P (r|r0, t)dr (2.3) where ρ(r) is the particle density and ρ(r)P (r|r0, t) is the probability of the particle moving from r to r0 during t. The integration accounts for all possible starting positions. Similarly to concentration, P (r0, t) gives the probability of finding a particle at a certain place at a certain time. Because the spatial derivatives in Fick´s laws refer to r0, the laws can be rewritten in terms of P (r|r0, t). The flux J then becomes the conditional probability flux and equation (2.2) becomes [21]:

∂P (r|r0, t) = D∇2 P (r|r0, t) (2.4) ∂t

For the case of isotropic three-dimensional diﬀusion P (r|r0, t) can be calculated as: (r0 − r)2 P (r|r0, t) = (4πDt)−3/2 exp − (2.5) 4Dt Equation (2.5) states that the angular distribution function of the particles is a

10 2.1. Diﬀusion

Gaussian. This free diffusion is therefore also called ’Gaussian diffusion’. Note that equation (2.5) does not depend on the initial position but on the net displacement r0 − r, which is referred to as the dynamic displacement R. P (r|r0, t) is often called the diffusion propagator. For the case of boundaries like reflect- ing walls or confining pores the evaluation of P (r|r0, t) is still possible but the mathematics is much more complicated. The diffusion propagator is the central function in self-diffusion studies, since in principle it contains all information on the movement in the observed sample.

2.1.3 Diﬀusion as a Step Process

One way to look at diﬀusion in gradient NMR is to picture it as a succession of discrete hops with the motion resolved in one dimension, the direction of the ﬁeld gradient, which will be referred to as the z-direction. Assuming that the motion in three spacial directions is independent, the three dimensions can be treated separately. The total displacement can then be obtained by adding quadratically according to Pythagoras’ theorem.

Let the mean time between two steps in one direction be τs and the root mean squared (r.m.s.) displacement in that direction be ξ. If the molecule has equal probability of jumping to the left or right, the travelled distance after n jumps at a time t = nτs is: n X Z(nτs) = ξai (2.6) i=1 with ai = ±1 as a randomly varying number. Z represents the z-axis displacement from the origin. The average mean squared displacement is therefore:

n n 2 X X 2 Z (nτs) = ξ aiaj (2.7) i=1 j=1 where the horizontal bar represents the ensemble average. Because ai varies randomly the average aiaj = 0 unless i = j. Therefore all cross terms vanish and

n n 2 X 2 2 2 X 2 Z (nτs) = ξ ai = ξ 1 = nξ . (2.8) i=1 i=1

11 2. Theoretical Aspects

With the definition of the self-diffusion coefficient according to Einstein and Smoluchowski [43, 44]: ξ2 D = (2.9) 2τ the average mean squared displacement in z-direction is

Z2(t) = 2Dt (2.10) or in n dimensions R2(t) = 2nDt. (2.11)

2.2 NMR Spectroscopy

2.2.1 Zeeman Interaction and the Eﬀect of HF-Pulses

The magnetic dipole moment µ of a spin is related to the spin operator I:

µ = γ~I (2.12) with the gyromagnetic ratio γ and Planck´s constant h divided by 2π. Two quantum numbers describe the eigenstate of an angular momentum operator: the spin quantum number I (with I2|ψi = I(I +1)|ψi and the magnetic quantum number m = [−I, −I + 1, ..., I] (with Iz|ψi = m|ψi). The interaction with an external magnetic ﬁeld B0 in direction of the z-axis is described by the Zeeman Hamilton operator

H0 = −µB0 = −ω0~Iz (2.13)

−1 −1 −1 with the Larmor frequency ω0 [rad s ], the gyromagnetic ratio γ [rad T s ] and the static magnetic field B0 [T]. The magnetic field lifts the degeneracy of the 2I + 1 eigenstates of Iz. Therefore a resonant electromagnetic alternating field can induce transitions between energy levels. Typical fields in NMR are up to several Tesla, leading to Larmor frequencies in the range of radio or high frequencies (hf). Resonant excitation is achieved by an external alternating field

B1 with:

B1(t) = 2B1,0 cos(ω1t) (2.14)

12 2.2. NMR Spectroscopy

which is oriented perpendicular (e. g. along the y-axis) to the static ﬁeld B0 with an amplitude of B1,0. This ﬁeld interacts with the magnetic moment of the nucleus as:

H1 = −µB1 = −2ω1~Iy cos(ω1t) (2.15) with ω1 = γB1. A convenient procedure is a transformation from the laboratory frame of reference into the frame of quantum mechanical interaction. This is done by the transformation into a rotating frame with angular velocity ω0. In this frame the static ﬁeld becomes zero and the Hamilton operator of the hf-ﬁeld becomes time independent1:

H1 = −ω1~Iy (2.16)

In a spin system in thermal equilibrium the spin density operator σ(0) is identical to Iz(parallel to B0). If a hf-pulse is now turned on, e. g. along the y-axis in the rotating frame, for a time t, the density operator is transformed into:

σ(t) = Iz cos(ω1t) + Ix sin(ω1t) (2.17)

This is equivalent to a rotation of the macroscopic magnetization M(t) = T r[σ(t)~µ] around the y-axis of the rotating frame. The ﬂip angle θ of the pulse is therefore deﬁned as follows:

θ = γB1t (2.18)

With the appropriate choice of the B1 field strength and the pulse length t, the magnetization can be flipped around an angle of e. g. 90◦ or 180◦. For a real system it is not sufficient to simply describe the magnetic interaction with just external fields. It becomes necessary to introduce interactions with much smaller internal fields, which can lead to shifts of resonance lines or splittings. The Hamiltonian of all interactions is the sum of the individual Hamil- tonians. Therefore it is possible to keep only those interactions in mind that are relevant for the problem and neglect the others. The different Hamiltonians are the Zeeman H0, the chemical shift HCS, the scalar HJ , the dipolar HD, the quadrupolar HQ and the hf-field H1 operators. In this work the chemical shift,

1 The part of B1 which rotates with 2ω1 in the opposite sense of the rotating frame can be neglected for B1 B0.

13 2. Theoretical Aspects scalar and dipolar Hamiltonians are neglected [45].

2.2.2 Quadrupolar Interaction

1 Nuclei with a spin I > 2 have a non-spherical charge distribution and exhibit therefore an electric quadrupole moment eQ. This can interact with the local electric ﬁeld gradient (EFG), arising from the electron shell. The Hamilton operator of the quadrupolar interaction is:

eQ H = I~QˆI~ = I~Vˆ I~ (2.19) Q ~ 2I(2I − 1) where the quadrupole coupling tensor Qˆ can be written in terms of the electric ﬁeld gradient tensor Vˆ at the nuclear site. In its principal axis system (PAS) the quadrupolar Hamiltonian for the axially symmetric case can be written as:

2 2 HQ = ~ωQ(3Iz − I ) (2.20) where the quadrupolar coupling constant

3e2qQ ωQ = (2.21) 4~ controls the magnitude of the interaction. Fast molecular motions on the NMR time scale average out the quadrupolar interactions and the time averaged quadrupolar Hamiltonian becomes zero. This is the case in an isotropic system (figure 2.1, center). Here only the Zeeman splitting occurs and the spectrum exhibits only one line. If the molecular motions (e. g. diffusion) have correlation times slower than the NMR time scale, which is given by the inverse of the width of the spectrum (e. g. the splitting), the quadrupolar interaction does not vanish and the energy level difference is changed. This leads to a splitting in the NMR spectrum (figure 2.1, right side). The two resonant lines are then separated by a frequency difference ∆ν. The magnitude of the observed splitting reflects not only quadrupolar averaging due to fast reorientational motions, which result in a reduction of the splitting. It has also an angular dependence, which opens the possibility to gain informa-

14 2.2. NMR Spectroscopy

w w +w 0 0 Q

w w- w 0 0 Q

B0 = 0 HQ = 0 HQ 6= 0

Figure 2.1: Energy levels with time averaged, HQ = 0, (only Zeeman) and

quadrupolar interaction, HQ 6= 0, for spin I = 1; ω0/2π is the Larmor frequency.

tion of the relative orientation of the molecules with respect to the magnetic field. The splitting ∆ν is dependent on the angle θ between the principal axis of the electric field gradient tensor and the axis of the external static magnetic field B0. 1 2 It scales with the second Legendre polynomial 2 (3 cos θ − 1). This is valid only for the case of a uniaxial symmetric field gradient tensor (Vxx = Vyy, Vzz = eq). For the case of the deuterium-oxygen bond in water, the principal axis of the EFG is parallel to the bond and the above simplification is valid. The quadrupolar coupling constant, and thereby the splitting ∆ν, is directly proportional to a local order parameter S. For lyotropic mesophases the coupling between the ethylene oxide units and the water is normally stronger at lower temperatures and lower content of water. Water is in fast exchange between a fraction of ’free’ water between layers and a fraction of water, which is strongly coupled to the ethylene oxide units. This can be seen in the absence of an isotropic line in the spectrum. With increasing amount of water, the fraction of ’free’

15 2. Theoretical Aspects water increases and the average degree of local order, the splitting, decreases. The observed splitting is furthermore a weighted average over all degrees of local order in that part of the sample, which is covered by diﬀusion of the water molecules on the NMR time scale.

2.2.3 Introduction to Gradient NMR

All field gradient NMR techniques rely on the fact that the Larmor frequency of the spins in an inhomogeneous magnetic field depends on their absolute position in space. For the case of a homogeneous magnetic field B0 the Larmor equation holds (see section 2.2.1):

ω0 = γB0 (2.22)

−1 If the magnetic ﬁeld B0 is assumed to have a constant gradient ~g [T m ] over the volume of the sample in z-direction, the Larmor frequency of the individual spins is then related to the z-coordinate of their position ~r.

ωeff (~r) = ω0 + γ~g~r (2.23) where the gradient ~g is deﬁned by the grad of the ﬁeld component parallel to

B0: ∂B ∂B ∂B ~g = ∇~ B = z~i + z~j + z ~k (2.24) 0 ∂x ∂y ∂z with the unit vectors ~i,~j and ~k in the laboratory frame. For translational motion the time dependent z-coordinate of a particle is thus projected into a time-dependent Larmor frequency. In such a gradient ﬁeld, the transverse magnetization will decay primarily owing to two mechanisms. First due to spin-spin-relaxation which is partly reversible in an echo experiment and second due to spatial movement. Moving spins will exhibit a time-dependent angular velocity, gaining a phase factor Φ in their evolution during a period t which depends on the space-time trajectory of the moving particle:

Z t 0 0 0 Φ(t) = γB0t + γ ~g(t )~r(t )dt (2.25) 0

16 2.3. Experiments where the first term on the right side corresponds to the phase shift due to the static field and the second term represents the phase shift due to the effect of gradients. The degree of dephasing is therefore proportional to the type of nucleus and to the strength and duration of the gradient.

2.3 Experiments

2.3.1 The Pulsed Gradient Spin Echo (PGSE)

The basis for accurately measuring diffusion is the ability of precisely labelling the position of a spin with a well defined magnetic field gradient (see appendix A.2). There are two possibilities for magnetic gradients and both have advantages and drawbacks. The strongest gradients are static (∼ 200 T/m) and diffusion coefficients down to 10−15 m2/s can be measured, but this advantage is greatly reduced by the fast decaying signal in the inhomogeneous field, which results in broad, unspecific spectra. Pulsed gradients as parts of echo sequences overcome this problem. Since no gradient is present during data acquisition, the recorded spectra are highly resolved and a diffusion analysis for different spectral components is possible. Pulsed gradients have large hardware requirements since the gradient amplifiers have to be very stable upon very fast switching (' 100 µs) of large currents (' 40 A). Pulsed gradients are limited in strength (up to about 20–30 T/m) and have to overcome the problem of ringdown delays of the gradient resonant circuit, since fast switching of strong gradients produces large ringdown effects. By using computer controlled shaped gradients these problems can be limited. The best compromise between power and precision is a ramped trapezoidal gradient. Gradients are always applied in a certain direction. Since most of the experiments in this work were performed with the gradient parallel to the magnetic field

B0, the gradient will be assumed as lying along the z-axis, if it is not otherwise mentioned. Ramped trapezoidal gradients [46] of duration δ and strength g were always used, for details on the gradient see appendix A.2. The simplest approach for measuring diﬀusion with pulsed ﬁeld gradients is to use an echo pulse sequence (see Appendix D.2) with gradient pulses inserted in the transverse evolution pe-

17 2. Theoretical Aspects riods. This method goes back to Stejskal and Tanner [9, 10] who were the first to modify the Hahn echo experiment. The quadrupolar PGSE experiment is built up by a basic solid echo [47] and two gradient pulses inserted in the evolution delays τ (figure 2.2). The first gradient pulse induces a spatially dependent phase jump, which is shown in the left column of figure 2.3. The relative phase change of the spins is depicted by the small arrows.

After the period t = τ1, see equation (2.25), the spins i (with their z- coordinate zi) have experienced a phase shift (ﬁgure 2.3, left column) due to the Larmor precession and the ﬁrst gradient of:

Z δ Φi(τ1) = γB0τ1 + γg zi(t)dt (2.26) 0

At the end of the period τ1 a pulse is applied that inverts all phases (ﬁgure 2.3, center column). This pulse is either a π-pulse for protons or a π/2-pulse for spin-1 nuclei (e. g. 2H). Directly after the echo pulse a second gradient pulse of exactly the same duration and strength as the ﬁrst is applied, leading to a total phase shift of:

Z δ Z τ1+δ 0 0 Φi(τ1 + τ2) = γB0τ1 + γg zi(t)dt − γB0τ2 + γg zi(t )dt 0 τ1 | {z } | {z } first period τ1 second period τ2 (2.27) Z δ Z τ1+δ 0 0 = γg zi(t)dt − zi(t )dt 0 τ1

The ﬁrst gradient pulse builds up a magnetization helix. The second one un- winds it. The spins that did not change their z-position (black dots) are fully refocussed. But those spins that moved during the evolution intervals τ do not experience their appropriate unwinding and are not fully refocussed (ﬁgure 2.3, right column). Since the phases are averaged over all spins in the sample the echo is not phase shifted but attenuated. If all delays in the pulse sequence are kept constant, relaxation can be removed by dividing through the signal S(2τ)g=0 in the absence of gradients and the echo amplitude depends solely on translational

18 2.3. Experiments

æ p ö æ p ö ç ÷ ç ÷ 2 2 è ø y è ø x t1 t2 d

Figure 2.2: PGSE pulse sequence with a solid echo with gradient magnitude g, duration δ and separation ∆.

Figure 2.3: Phase evolution during gradients, black labelled molecules do not change z-position, colored molecules change z-axis position, the ﬁrst column corresponds to the ﬁrst gradient pulse, the second to free movement and the third to the refocussing gradient pulse (see also text).

19 2. Theoretical Aspects

diﬀusion. The signal of the echo, occurring at τ2 = τ1 is then given by:

Z +∞ iφ S(2τ) = S(2τ)g=0 P (φ, 2τ)e dφ (2.28) −∞

P (φ, 2τ) is the (relative) phase distribution function. For the pulsed gradient experiment, P (φ, 2τ) is normally written as P (φ, ∆), since it is the separation of the gradients which constitute the ’active’ part of the sequence. For an evaluation of diffusion coefficients it is therefore necessary to find an expression for P (φ, ∆) for the PGSE experiment.

2.3.2 The Stejskal-Tanner Equation: GPD and SGP Approximation

As mentioned in the previous section, in an actual application it is necessary to derive an expression for P (φ, ∆). As discussed by Callaghan [21], exact solutions have so far been found for only two cases, namely for free diffusion and for free diffusion superimposed on flow. Therefore, if more complicated boundary conditions come into play, one is forced to use different approximations. There are two major approaches. One, the Gaussian phase distribution (GPD), goes back to suggestions of Douglass and McCall [48], who developed the method of phase accumulation (see equation 2.25). In this framework P (φ, ∆) can be derived from equation (2.27). zi(t) is described by the one-dimensional diffusion equation, which is a Gaussian for the case of free diffusion. The (one-dimensional) positional probability density along the z-direction is, analogous to equation (2.5):

z2 P (z |z0, ∆) = (4πD∆)−1/2 exp − i (2.29) i i 4D∆

As discussed by Neuman [49], one assumes that the phases have a Gaussian distribution. The probability density of the integral of a variable, which has a

Gaussian probability density (zi), is a Gaussian [50]. Therefore one obtains:

φ2 P (φ , ∆) = (2πhφ2i )−1/2 exp − i (2.30) i av 2hφ2i

20 2.3. Experiments where hφ2i is the mean squared phase change after 2τ, which is given by the average of the square of equation (2.27). A lengthy calculation [42] leads to:

hφ2i = 2γ2g2δ2(∆ − δ/3)D (2.31)

With the assumption of a Gaussian phase distribution from equation (2.30), equation (2.28) yields: hφ2i S(∆) = S(∆) exp − (2.32) g=0 2 Inserting equation (2.31) results in the famous Stejskal-Tanner equation for the intensity of the echo signal:

δ S(δ, ∆, g) = S(δ, ∆, g = 0) exp −γ2g2δ2D ∆ − (2.33) 3

The diffusion time ∆ is modified by the term δ/3 [51], which accounts for the finite gradient duration. This modification of equation (2.33) is still only valid in the limit of short gradient pulses compared to the diffusion time (δ ∆).

The other approach is the short gradient pulse (SGP) limit. No assumptions are made for the distribution of phases. Instead the gradient duration is set equal to zero, δ → 0, while the product gδ is kept constant. This corresponds to the physical situation where there is no motion of the spins during the gradient. For this case the intensity of the spin-echo can be shown to be [9, 21]:

ZZ S(δ, ∆, g) = ρ(r)P (r|r0, ∆) exp [iγdδ(r − r0)] drdr0 (2.34) with the conditional probability P (r|r0, ∆) as given in section 2.1.1. To utilize equation (2.34) the knowledge of P (r|r0, ∆) for the particular case of interest is required. In the case of free diﬀusion the Gaussian form of P (r|r0, ∆) is taken and one obtains also equation (2.33).

21 2. Theoretical Aspects

2.3.3 Comparison of the GPD and SGP

Both the GPD and the SGP approximation are only valid within certain limits, namely that no diffusion occurs during δ for the SGP and the assumed Gaussian behavior during δ for the GPD. This means that the SGP approximation is good for e. g. viscous or solid samples and the GPD approximation for very short gradients in samples without close boundaries. Both assumptions have their limits in the case of restricted diffusion, for example in the case of diffusion in the MLVs. The assumptions of GPD are most obviously violated due to the special boundary conditions and the effect of movement in this case during the finite gradient duration will be discussed in more detail in section 4.7.3.

2.3.4 Notes for the Application

The absolute echo intensity in equation (2.33) is still dependent on spin-lattice

(T1) relaxation and/or on spin-spin (T2) relaxation. In the pulsed gradient experiment all delays are kept constant and the echo attenuation can be made independent of relaxation by division of the data by the echo intensity without gradient, S(∆)g=0. The applicability of equation (2.33) on experimental data is checked by plotting the normalized echo amplitude as function of the square of the gradient strength g, which is the only parameter that is varied within one experiment. If such a plot is linear the above equation can be applied to derive a diffusion coefficient. If necessary an offset can be added to equation (2.33), if there are immobile fractions in the sample which do not decay with increasing gradient. Many ways of data presentation are possible. In the present work echo decays are presented in the following manners:

• Echo intensity as function of the gradient magnitude g, given in [G/cm]. The diffusion coefficient is derived by fitting the data according to equation (2.33).

• Logarithm of the echo intensity as function of the gradient magnitude squared times a parameter k, which is deﬁned as:

k = γ2δ2(∆ − δ/3) (2.35)

22 2.3. Experiments

g2 · k is given in [s/m2]. The slope of this curve therefore yields directly the diffusion coefficient. For free Gaussian diffusion the slope is constant, independent of ∆.

• Echo intensity as function of a parameter q that is deﬁned as:

q = γgδ (2.36)

This q is the analogue to the scattering vector and is given in units of reciprocal length, mostly [mm−1]. This was used in some plots in section 4.6.2.

2.3.5 The Pulsed Gradient Stimulated Echo (PGSTE)

Most experiments in this work were carried out with the stimulated version of the pulsed gradient echo [52–54] as it is shown in ﬁgure 2.4. Here the gradients are, like in the PGSE, introduced into the transverse evolution delays τ1 of the sequence. But the use of a third pulse to store the magnetization along z during τ2, where it is subject only to T1 and not to T2 relaxation, greatly extends the range for diﬀusion time dependent measurements, since in liquid crystalline samples T1 is usually longer than T2. Another advantage is the possibility to independently

Figure 2.4: PGSTE pulse sequence analogous to the PGSE sequence, including a spoiler gradient during z-storage. vary an evolution delay and the diﬀusion time. This was especially important for the MLV investigations, where the inhomogeneous line shape is strongly depen-

23 2. Theoretical Aspects

dent on the transverse evolution delays τ1. Measurements with diﬀerent diﬀusion times would in this case not have been possible with the PGSE experiment alone.

A lower limit for the evolution delay τ1 is given by the ﬁnite gradient duration δ, which is in the millisecond range.

2.3.6 Relaxation

NMR relaxation theory considers the behavior of a spin in the presence of a randomly ﬂuctuating time dependent perturbation of the Zeeman Hamiltonian. In the case of 2H NMR spectroscopy, the perturbing quadrupolar interaction depends on the orientation of the principal axes of the electric ﬁeld gradient

(EFG) tensor that is associated with a particular D2O time averaged orientation relative to the static magnetic ﬁeld B0. Relaxation is governed by the dynamics 2 of the system. The T1 relaxation processes in H NMR are sensitive to spectral densities Jn(nω)(n = 1, 2) and therefore to correlation times on the order of the inverse of the Larmor frequency, which means about 2 · 10−9s. This corresponds mainly to internal motions of the molecules. T2 relaxation is also aﬀected by J(0) (n = 0), which means that it is sensitive also to slow motions with correlation −1 times τ ω0 .

T2 relaxation was measured with a solid echo sequence (see appendix C.3).

The evolution delays τ were increased and T2 was obtained from ﬁtting the normalized (S(0) ' 1) echo intensity S(2τ) to the following formula:

2τ S(2τ) = S(0) · exp − (2.37) T2

T1 relaxation was measured with the saturation recovery pulse sequence (see appendix C.3). The delay τ (parameter ’vd’ in the pulse program) after the initial pulse train was varied. T1 was obtained from ﬁtting the echo intensity, normalized to the intensity at the longest delay (S(τmax) ' 1), according to:

τ S(τ) = S(∞) 1 − exp − (2.38) T1

24 Chapter 3

Anisotropic Hydrogels

3.1 Hydrogel Sample Preparation

Anisotropic hydrogels were prepared in cooperation with M. Hickl [11, 12, 55]. In his PhD work M. Hickl synthesized a mesogen [2,5-bis(4-monomethyl hexa ethylene glycole benzophenonoxy)-benzoic acid-2-(2-methacrylate)-ethylester], which exhibits a lyotropic lamellar Lα phase at room temperature. He also synthesized a crosslinker [2,5-bis-(4-methacrylic acid tetra ethylene glycole benzophenonoxy)- benzoic acid-2-(2-methacrylate)-ethylester], which was expected not to disturb the mesophase, because the length of the rigid core part and the hydrophilic part are as long as those of the mesogen. The chemical structures of mesogen and crosslinker can be seen in ﬁgure 3.1. Figure 3.2 shows the phase diagram for the monomer (mesogen) and the corresponding polymer [55]. The phase diagram of the linear polymer is of course not directly comparable to that of the network, but the phase transition from the lamellar to the isotropic phase occurs in a similar temperature range. It occurred at about 327 K for one of the networks (Gel7, table 3.1). The phase transition was unfortunately not reversible due to loss of water upon heating and very slow equilibration upon cooling. But it was possible to rehydrate the dry gel with an excess of water. It was of course impossible to exactly control the water content with this procedure and therefore the order (splitting) of the rehydrated hydrogel was about 50 % lower. Network samples for NMR measurements were prepared from a mixture of

25 3. Anisotropic Hydrogels

Figure 3.1: Similarity in the chemical structure between mesogen and crosslinker.

mesogen/crosslinker/D2O and a small amount of photoinitiator which was ho- mogenised in a vibrating mill with a small magnetic stir bar for several minutes. The mixture was then injected with a small syringe into a UV permeable quartz glass sphere (figure 3.3). The sample filled the volume but not the small capillary. The diameter of the sphere was about 4 mm and the inner diameter of the capillary about 0.5 mm. The sphere was then sealed with glue in the capillary and additionally a plug of molten glass. The sample was then macroscopically orientated in the magnetic field by either heating or cooling from the isotropic

(L1 or L2) into the lamellar Lα phase as sketched by the arrows in figure 3.2. Due to their positive anisotropy of the diamagnetic susceptibility (∆χ > 0) the mesogens align parallel to the external field B0. The orientation of the director (layer normal) was chosen in the direction perpendicular to the capillary for samples 1 to 4 to enable a controlled sample rotation and thereby give the possibility to measure diffusion in different directions with only one gradient. Samples 5 to 7 were oriented with the director parallel to the capillary to ensure a known sample orientation during measurements with gradients in different directions. The definition of ’parallel’ and ’perpendicular’ as used e. g. in the diffusion experiments

26 3.1. Hydrogel Sample Preparation

Figure 3.2: Phase diagram of the monomer mixture (left) and the linear polymer (not the network !) (right). All measurements were carried out in the temperature range from 303 to 320 K (gray bars). The arrows symbolize the temperature pathways during the preparation procedures.

sample glue glass

Figure 3.3: Sketch of the hydrogel sample. always refers to the axis of symmetry, i. e., the layer normal or director. The sphere was removed from the magnet and the mixture was polymerized (and crosslinked) by UV irradiation for 180 seconds. Because of the high viscosity in the Lα phase no signiﬁcant change in orientation was expected during this procedure.

27 3. Anisotropic Hydrogels

Table 3.1: Composition of synthesized hydrogels (wt.%.)

sample name Gel5 Gel 1,2,3,4,6,7 mesogen 61 28

D2O 33 66 crosslinker 5 5 initiator 1 1

In order to keep the temperature in the desired range of the Lα phase (about 20 ◦C) a gas flow of cooled nitrogen was blown over the sample and the temperature in the UV irradiation chamber was measured by a Pt100. The most unwanted effect of sample heating during irradiation would induce micro- or sub- microscopic defects in the phase or even destroy the sample by phase separation. The success of the preparation procedure was controlled by comparison of the line shapes before and after UV irradiation. Hydrogels with two different amounts of water were synthesized as summarized in table 3.1. The hydrogels Gel5 and Gel7 were used in the diffusion experiments presented in sections 3.4 and 3.5, while the line shape analysis in section 3.6.1 was performed on Gel1. Data from Gel2 to Gel6 are not presented, since they did not show any difference. In figure 3.4 the spectra before and after UV irradiation of a typical hydrogel with 66 wt.% D2O (Gel2) are compared. The crosslinked sample has a smaller splitting and broader peaks, which are slightly asymmetric, with less steeper slopes towards the middle of the spectrum. These observations indicate a less ordered sample and a broader orientational director distribution. The small isotropic peak in the center is about 2 % of the total intensity.

28 3.1. Hydrogel Sample Preparation

-400 -200 0 200 400 Frequency [Hz]

Figure 3.4: Spectrum of the oriented phase of Gel2 before (—) and after (··· ) crosslinking. The small additional peaks to the left of all peaks are due to susceptibility eﬀects and did not disappear on shimming.

29 3. Anisotropic Hydrogels

3.2 Single Pulse Spectra

Samples with two different chemical compositions both within the Lα phase were investigated (see figure 3.2). In figure 3.5 the D2O splitting of Gel5, the one with lower content of D2O (33 %), and Gel7, with a higher content of D2O (66 %), are shown at 303 K. The splitting of Gel5 is 850 Hz and the full width at half maximum (FWHM) is 50 Hz. Gel7 has a splitting of 700 Hz and a FWHM of 150 Hz. The smaller quadrupolar splitting shows that Gel7 has a lower order parameter (see section 2.2.2) and the higher FWHM indicates a broader distribution of director orientations or larger director fluctuations. This was expected to some extent, since the viscosity is expected to be lower in the sample with higher content of water, which would broaden the orientational distribution due to local heating.

-2000 -1500 -1000 -500 0 500 1000 1500 2000 - 2 0 0 0 - 1 5 0 0 - 1 0 0 0 - 5 0 0 0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 Frequency [Hz] F r e q u e n c y [ H z ]

Figure 3.5: Deuterium splitting of Gel5 (33 wt.% D2O, left) and Gel7

(66 wt.% D2O, right) at 303 K.

Proton measurements were also performed on the hydrogels. In figure 3.6 the proton spectrum of Gel5 (left) and Gel7 (right) and two component Lorentzian fits are plotted. For Gel5 a narrow (FWHM of 170 Hz) and a broad peak (1650 Hz) are observed. The narrow peak contributes with 38 % to the spectrum. Diffusion measurements showed that it corresponds to mobile proton bearing components in the sample. It is not clear whether the narrow peak is residual uncrosslinked mesogen or H2O. The magnitude of the diffusion coefficient measured for protons, which is lower than DD2O, suggests that both contribute to the narrow peak. The

30 3.3. Quadrupolar Splitting and Local Order measured diffusion coefficient is a weighted sum of the two components, which leads to an apparent diffusion coefficient much lower than the expected value for

H2O. Nevertheless, with regard to anomalous diﬀusion, the observed molecule (or mixture) can be used to calculate domain sizes (section 3.4). The spectrum of Gel7 shows less obvious features, but analogous ﬁtting resulted in a ’narrow’ peak with 20 % spectral contribution (700 Hz) and a broad peak (2100 Hz).

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 Frequency [Hz] Frequency [Hz]

Figure 3.6: Proton spectrum of Gel5 (left) and Gel7 (right) with two component Lorentzian ﬁts (dashed lines) at 303 K.

3.3 Quadrupolar Splitting and Local Order

In figure 3.7 the splittings of Gel5 and Gel7 are plotted as a function of the temperature. The quadrupolar splitting is related to a local order parameter of the phase, which reflects the average orientational order on the length scale of the diffusing water, i. e., several microns. A smaller degree of order and hence a smaller splitting is expected for the hydrogel with higher content of water, Gel7 (see section 2.2.2). The splitting decreases with temperature, which indicates a lower degree of local order due to membrane undulations or increased generation of holes or other defects. It should be emphasized that there is no discontinuity in the splittings at 312 – 314 K, where the monomeric mixtures separate into two phases, in contrast to the measurements of the diffusion coefficients (see section 3.5).

31 3. Anisotropic Hydrogels

8 5 0 2 Φ r e g i o n i n t h e G e l 5 u n c r o s s l i n k e d s y s t e m G e l 7 8 0 0 ] z 7 5 0 H [

g n i t t i

l 7 0 0 p s

r a l o

p 6 5 0 u r d a u q 6 0 0

5 5 0

3 0 4 3 0 6 3 0 8 3 1 0 3 1 2 3 1 4 3 1 6 3 1 8 3 2 0 T [ K ]

Figure 3.7: Quadrupolar splitting of Gel5 (33 wt.% D2O) and Gel7

(66 wt.% D2O).

3.4 Time Dependent Diﬀusion

Diffusion was measured for each composition in the direction parallel to the director (layer normal) and perpendicular to it, resulting in diffusion coefficients

Dk and D⊥, respectively [56]. The echo decays gained from PGSE- and PGSTE- experiments of Gel5 are presented in ﬁgures 3.8 and 3.9 (top) in a semi-logarithmic plot. The linearity of the natural logarithm of the normalized echo amplitude

S/S0 against the square of the gradient amplitude g signifies that the echo decay has a Gaussian shape (see equation 2.33). The diffusion time ∆ was varied from experiment to experiment while the gradient duration δ and the maximum gradient amplitude gmax were chosen adequately to reach an echo intensity of zero at gmax. Since all gradient parameters and the pulse sequence were varied independently, technical artifacts can be excluded. The diffusion coefficients, fitted according to equation 2.33, are also plotted against the diffusion time in figure

32 3.4. Time Dependent Diﬀusion

3.8 and 3.9 (bottom). No tendency with varying ∆ is observed but the scattering of the data is bigger than their individual fitting error. Because of this a new statistical error for the mean value is taken. This mean value is also plotted at the hypothetical diffusion time of zero. PGSE and PGSTE diffusion experiments with the proton signal of the hydrogels were also performed. Since T1, which limits the experimental timescale of 1 the diffusion experiment in the stimulated echo, is longer for H than for D2O, ∆ could be raised in the 1H experiment to a maximum of one second. In figure 3.10 the initial 1H-echo decays of Gel5, parallel (top) and perpendicular (bottom) to the layer normal, are shown for different diffusion times ∆. The data are taken from integration over the whole spectra, but the tails from immobile components, which show up at strong gradients are not shown. There is no dependence on ∆ up to one second. The diffusion coefficients were derived from fitting the raw data with equation 2.33, including an offset, which accounts for the immobile ’crosslinked’ protons from the gel. Again the mean value from the different diffusion times was taken. The results for proton and deuterium measurements for both gels are summarized in table 3.2.

3.4.1 Estimation of Domain Sizes

Based on the time dependent diffusion data the minimal lateral extension of a ’perfectly’ ordered domain can be estimated. Assuming that equation 2.11 (see section 2.1.3) is valid, the minimal lateral and longitudinal expansion of domains, which is proportional to the square root of the mean squared displacements, can 2 be calculated with hz i = 2D∆max. The results are summarized in table 3.2. With a layer distance of 63 A˚ (see section 3.6.2) this means that homogeneity parallel to the layer normal is given at least over 1160 layers for Gel5 and at least 1900 layers for Gel7. It should be emphasized that this does not mean that Gel7 is ordered more perfectly, it just gives the lower limit of homogeneity. The fewer layers of Gel5 are owed to the lower diffusion coefficient Dk, which results from less defective membranes. This gives a hint that overall Gel5 is more perfect, as it is already expected from the measurements of the splitting.

33 3. Anisotropic Hydrogels

[ms] 0 30.4 50.2 90.2 209.6 -1 ) [a.u.] ) 0

-2 ln(S/S

-3

0 1x1010 2x1010 3x1010 4x1010 g2 * k [s/m2] 5.0x10-11

4.5x10-11

-11 /s]

2 4.0x10 [m D

3.5x10-11

3.0x10-11 0.00 0.05 0.10 0.15 0.20 0.25 [s]

Figure 3.8: 2H-PGSTE echo decay (semi-logarithmic scale) for the gradi-

ent axis parallel to the layer normal and Dk against the diﬀusion time ∆ for Gel5 at 303 K.

34 3.4. Time Dependent Diﬀusion

[ms] 0 33.7 52.9 72.9 92.1 -1 121.2 161.2 210.9 ) [a.u.] ) 0

ln(S/S -2

-3 0 1x109 2x109 3x109 4x109 g2 * k [s/m2] 6.5x10-10

6.0x10-10

5.5x10-10 /s] 2

[m 5.0x10-10 D

4.5x10-10

4.0x10-10 0.00 0.05 0.10 0.15 0.20 0.25 [s]

Figure 3.9: 2H-PGSTE echo decay (semi-logarithmic scale) for the gradi-

ent axis perpendicular to the layer normal and D⊥ against the diﬀusion time ∆ for Gel5 at 303 K. 35 3. Anisotropic Hydrogels

0 [ms] D 10-11 [m2/s]

29.4 2.684 (0.012) 59.4 2.731 (0.012) -1 109.4 2.751 (0.013) 208.6 2.735 (0.020) 507 2.710 (0.038) 1006.6 2.700 (0.037) ) [a.u.] )

0 -2 ln(S/S

-3

-4 0.0 5.0x1010 1.0x1011 1.5x1011 g2 * k [s/m2] 0 [ms] D 10-10 [m2/s] 29.4 3.564 (0.009) 59.4 3.586 (0.005) -1 108.6 3.641 (0.006) 207.8 3.679 (0.008) 507 3.597 (0.017) 1006.8 3.483 (0.023) ) [a.u.] )

0 -2 ln(S/S

-3

-4 0.0 4.0x109 8.0x109 1.2x1010 g2 * k [s/m2]

Figure 3.10: 1H-PGSTE echo decay (semi-logarithmic scale) for the gradient axis parallel (top) and perpendicular (bottom) to the layer normal Gel5 at 303 K.

36 3.5. Temperature Dependent Diﬀusion

Table 3.2: Average diffusion coefficients Dav at 303 K and minimal ex- 2 tensions of monodomains from diffusion data; ∆max( H) = 200 ms, 1 ∆max( H) = 1 s.

−10 2 Dav · 10 [m /s] minimal domain size [µm] parallel (k) perpendicular (⊥) longitudinal (k) lateral (⊥) gel5 2H 0.40 ± 0.01 5.35 ± 0.04 4.0 14.6 gel7 2H 0.84 ± 0.02 7.09 ± 0.07 5.8 16.8 gel5 1H 0.27 ± 0.01 3.61 ± 0.05 7.3 26.9 gel7 1H 0.72 ± 0.02 4.20 ± 0.12 12.0 29.0

3.5 Temperature Dependent Diﬀusion

Temperature dependent diffusion measurements were performed. The samples were heated by moderate air flow. Artifacts arising from potential convection (see appendix A.3) could be excluded, since the sample is a gel and convection is highly improbable. Furthermore no ∆-dependence was ever seen at any temperature in the investigated range. The average diffusion coefficients for different diffusion times for the gels are plotted in figures 3.11. There is a change of temperature dependence in the diffusion coefficients Dk at around 312K for Gel5 and at 314 K for Gel7. In the temperature range of 312 – 314 K the monomeric mixtures are already phase separated. In figure 3.11 this is not so easy to see, but in section 3.5.1 it becomes clearer that this change corresponds to an increased permeability of the bilayers. The increased permeability could lead to the idea of the formation of holes or cracks in the membrane. In the following this idea is discussed in more detail.

3.5.1 Obstruction of D2O Diﬀusion

A more general way to discuss diffusion is by means of the obstruction factor Q of equation (1.4). A smaller value of Q means stronger obstruction. The temperature dependence of the free diffusion coefficient of D2O and H2O were measured by Mills et al. [57]. They found a linear dependency of D on the

37 3. Anisotropic Hydrogels

1.8x10-10 Gel5 Gel7 1.6x10-10

1.4x10-10

1.2x10-10 /s] 2 1.0x10-10 [m D 8.0x10-11

6.0x10-11

4.0x10-11

304 308 312 316 320 324 328 T [K] 1.6x10-9 Gel5 Gel7

1.4x10-9

1.2x10-9 /s] 2 1.0x10-9 [m D

8.0x10-10

6.0x10-10

304 308 312 316 320 324 328 T [K]

Figure 3.11: Diﬀusion coeﬃcients of D2O parallel (top) and perpendicular (bottom) to the layer normal against temperature for Gel5.

38 3.5. Temperature Dependent Diﬀusion absolute temperature in the range from 280 to 320 K with:

2 −9 DD2O [m /s] = (0.049 · T [K] − 12.7) · 10 (3.1)

2 −9 DH2O [m /s] = (0.054 · T [K] − 13.8) · 10 (3.2)

Using these relationships for Dfree and the measured values of D, temperature dependent obstruction factors were calculated. In figure 3.12 Qk and Q⊥ (orientation relative to the layer normal) are plotted against the temperature. Qk of Gel5 shows a step from 0.018 to 0.026 at around 312 K, a similar step can be estimated for Gel7 at around 314 K, although the unfortunate lack of data in this region is obvious. The hinderance of the water in these regions is suddenly reduced. Qk above and below the step is constant. The temperature dependence of Q⊥ shows only a slight increase and no significant step. The values of Q for Gel7 are higher (see table 3.3), as expected for the gel with higher content of water, where the lamellae are expected to be less perfectly organized [58]. The ideal lamellar phase would have obstruction factors Qk of zero, corresponding to impenetrable lamellae and a Q⊥ of one for supposed free diffusion along the lamellae. In table 3.3 the values of the obstruction factors at 303 K are given for the two gels of different composition. The measured Q values indicate that, on the one hand, diffusion across the layers is possible and, on the other hand, that the motion along the layers is not free. The diffusion in the mesophase is therefore obstructed to a certain degree. The existence of defects in the lamellae like holes or cracks can be deduced from Qk [59].

Table 3.3: Obstruction factors Q of Gel5 and Gel7 at 303 K with 2 −9 2 1 −9 2 Dfree( H) = 2.165 ·10 m /s and Dfree( H) = 2.582 ·10 m /s.

Qk Q⊥ Gel5 2H 0.018 0.248 Gel7 2H 0.039 0.328 Gel5 1H 0.011 0.139 Gel7 1H 0.018 0.159

39 3. Anisotropic Hydrogels

0.055 2 region in the uncrosslinked system 0.050

0.045

0.040 Gel 5

0.035 Gel 7 Q 0.030

0.025

0.020

0.015 304 308 312 316 320 324 328 T [K] 0.480 2 region in the uncrosslinked system

0.440

0.400

Gel 5 0.360 Gel 7 Q

0.320

0.280

0.240 304 308 312 316 320 324 328 T [K]

Figure 3.12: Temperature dependent D2O obstruction factors Qk (top)

and Q⊥ (bottom), the dotted lines are sigmoidal and linear ﬁts, serving as a guide to the eye.

40 3.5. Temperature Dependent Diﬀusion

Such a breaking up of the lamellae could be explained by the elastic restoring force from the network. In an isotropic environment it is normally expected to be in a spherical, Gaussian, coil conformation. Swelling experiments [11] showed a slightly oblate coil conformation in the isotropic state of the network, which is then the equilibrium conformation with the lowest entropy. In the liquid crystalline swollen sample the coil is forced into a more oblate conformation due to the coupling of the mesophase to the network. But elastic forces tend to restore its more spherical shape. (Nevertheless, the mesophase is stabilized by the network due to the motional constraints on the mesogens.) It is a reasonable assumption that above a certain temperature the opposing elastic forces of the network exceed the coupling forces of the mesophase. This phenomenon leads to a partial disruption of the mesophase. It occurs at a temperature, where the corresponding uncrosslinked low molecular weight mixture is already phase separated. This indicates the higher thermal stability of the mesophase in the hydrogel compared to the low molecular weight liquid crystal. The network also seems to prevent the complete isotropization. Whether the step in Qk is caused by a formation of holes, cracks or increased defects of any other kind cannot be decided by NMR diﬀusometry.

The Two-Site Model

The water in the system is bound to the EO units to a certain degree, usually about 1.5 water molecules per EO unit are assumed [60–62]. Since the NMR spectra do not show any central isotropic line, water is in the limit of fast exchange, i. e., the molecules change fast, compared to the NMR time scale, between bound (ordered) and freely moving water. The diffusion along the lamellae can therefore be modelled by a simple two-site model. The measured diffusion coefficient Dexp is then a weighted sum of the different populations x of the water states [63, 64].

Dexp = xfreeDfree + xboundDbound (3.3)

−9 2 Free D2O has a diffusion coefficient of Dfree = 2.165 · 10 m /s at 303 K according to the literature [57]. PGSE measurements yielded a diffusion coeffi- −10 2 cient perpendicular to the layer normal of D⊥ = 5.35 ·10 m /s for Gel5 and

41 3. Anisotropic Hydrogels

−10 2 D⊥ = 7.09 ·10 m /s for Gel7 (see table 3.2). With the assumption of 1.5 bound water molecules per EO unit, Gel 5 contains about 66 mol-% of bound water (xfree = 0.34) and Gel 7 about 17 mol-% of bound water (xfree = 0.83). But equation (3.3) is not solvable with these fractions, even with the assumption that the bound water does not move at all (Dbound = 0). The measured diffusion coefficients are too small, which means that the mobility of the water must depend on at least one more factor. The main factor is the fluctuating curvature of the membranes, the undulations. These undulations must have low amplitudes and small periods so that their time average (on the diffusion time scale) refers to only one average director orientation. If the periods of these layer fluctuations were larger than about one µm, strongly curved surfaces would induce a ∆-dependence of the diffusion coefficients and one could no longer speak of a local domain.

The reduced diffusion coefficient D⊥ can also be made plausible by simple geometric considerations. The layer repeat distance was determined to 63 A˚ [11]. The mesogen in its all trans conformation has a hydrophobic core of 18 A˚ and two EO-tails of 22 A˚ each. When an all trans conformation of the EO-chains is assumed, this results in a structure where water is always in close contact to the EO-units and its diffusion is hindered, even though the all trans conformation is not the most probable one. This means that even the ’free’ water is simply not free in this case.

3.5.2 Arrhenius Analysis

If a thermally activated process for liquid diffusion is assumed, then the diffusion coefficient is given by the Arrhenius equation (3.4) and activation energies and frequency factors for the diffusive motion (2.1.3) can be extracted from the temperature dependence of the diffusion coefficients:

E D = D · exp − a (3.4) 0 RT

Linear fitting of the natural logarithm of the diffusion coefficient plotted against the inverse of the temperature (over the whole temperature range), therefore yields the activation energy Ea and the frequency factor D0. The fit was applied

42 3.5. Temperature Dependent Diffusion in the temperature range before and after the jump in the diffusion coefficient (figure 3.13). For both hydrogels a similar activation energy was obtained below and above the jump and the average value was taken. The fact that a similar activation energy is obtained, is not surprising, since the same dependence on temperature became already obvious for Qk above and below the step. Frequency factors for each gel differed only about 10 % and also the average value is reported. The results obtained by 1H and 2H diffusion measurements are summarized in table 3.4.

-22.4 Gel 5 Gel 7

-22.8

-23.2 ln D ln

-23.6

-24.0 3.05 3.10 3.15 3.20 3.25 3.30 1000 / T [K]

Figure 3.13: Linearized Arrhenius plots for D2O-diffusion in Gel5 and Gel7, solid lines are fits according to equation 3.4 below and above the respective diffusivity jumps.

A surprising observation is that the activation energy for D2O-diffusion through the lamellae is the same as for diffusion along the lamellae; the slightly lower activation energy for Gel7 through the layers could be explained by the overall lower degree of order as can be seen from the smaller splitting. Nevertheless a larger activation energy for the diffusive step through the hydrophobic membrane would

43 3. Anisotropic Hydrogels

Table 3.4: Arrhenius activation energies Ea and frequency factors D0 for the two gels in the directions parallel and perpendicular to the layer normal (director).

a −6 2 Ea [kJ/mol] D0 · 10 [m /s] k ⊥ k ⊥ gel5 2H 23.1 23.0 0.23 5.03 gel7 2H 18.3 22.5 0.12 5.56 gel5 1H 55.2 22.3 8094 2.43 gel7 1H 55.2 21.5 5110 2.14

a Due to the very few data used for fitting, the error is of the same order of magnitude. have been expected. The main difference is the frequency factor which differs by more than one order of magnitude. Activations energies below and above the jumps were the same. The absolute values of the activation energies are slightly higher than for free D2O, which is 18.8 kJ/mol [57]). They are comparable to activation energies reported for transport through aquaporins (water channels) in lipid bilayers of leaves (≤ 21 kJ/mol) [65, 66]. Diffusion across a lipid bilayer without channels has significantly higher activation energy (40 – 60 kJ/mol) [67]. Arrhenius analysis only seems to be sensitive to the diffusive behavior on the length scale of the individual layers and not on the length scale of the texture. This would be an explanation for the similar activation energies parallel and perpendicular to the layers. Only the diffusion within one layer is responsible for

Ea and the activation energy is approximately the same as for free D2O. The low activation energy for diffusion across the lamellae confirms the existence of holes or cracks. Water primarily does not diffuse through the hydrophobic part directly. The frequency factors represent somehow the vibrational mobility of each water molecule. The higher value for diffusion perpendicular to the layer normal can be made plausible by the classical picture of water that does not have to cross a hydrophobic membrane. Yet the error of the frequency factors is of about the same magnitude as their values due to the very few data points that where used

44 3.5. Temperature Dependent Diffusion for fitting. Therefore one should not overinterpret and especially not compare them in terms of different mobilities. The analysis of 1H diffusion resulted in strange values. The activation energy parallel to the layers is comparable to D2O but the frequency factors parallel to the layer normal are far too high. Although the corresponding semi-logarithmic plots (not shown) are linear, it must be assumed that an Arrhenius analysis is not applicable. The fact that the observed apparent diffusion coefficients result from different components, namely hydrophobic mesogens and water, makes this reasonable.

3.5.3 Anisotropy of Diﬀusion

The degree of anisotropy of diffusion can be defined by the ratio of the diffusion coefficients perpendicular and parallel to the layer normal, D⊥/Dk. This ratio is plotted in figure 3.14. At low temperature the anisotropy stays approximately constant at a value of 13 – 14 for Gel5 and of 9 – 10 for Gel7. The anisotropy decreases above about 312 K for Gel5 and at slightly higher temperatures for Gel7. This corresponds to the temperatures of the jump in the obstruction factors Qk (figure 3.12, top). The further decrease of anisotropy is associated with the increased permeability of the lamellae with increasing temperature. The same explanations as for obstruction in section 3.5.1 are possible. The increased solubil- ity in the hydrophobic phase and the formation of holes decrease the obstruction through the lamellae and thereby the anisotropy.

45 3. Anisotropic Hydrogels

G e l 5 1 4 G e l 7

1 3

1 2  1 1  D

⊥

D 1 0

7 3 0 2 3 0 4 3 0 6 3 0 8 3 1 0 3 1 2 3 1 4 3 1 6 3 1 8 3 2 0 3 2 2 T [ K ]

Figure 3.14: Anisotropy of D2O diﬀusion in Gel5 and Gel7 as deﬁned in the text.

46 3.6. Local Order

3.6 Local Order

3.6.1 The Orientational Distribution Function

Information on the degree of order can be obtained from an angle dependent line shape analysis. The width and symmetry of the deuterium line shape F(t) depends not only on the natural line width (relaxation) but also on the orientational distribution function of the local director. The estimation of the orientational distribution from NMR line shapes is a typical ill-posed inverse problem. A Tikhonov regularization procedure employing the self-consistent method to ﬁnd the regularization parameter has proven particularly reliable [68, 69]. The special problem for the solid echo has the form:

Z π/2 F (t) = K(t, β)p(β)dβ (3.5) 0 and Z π/2 Fθ(t) = Kθ(t, β)p(β)dβ (3.6) 0 where K(t, β) is the so called kernel function, describing the NMR signal, which is weighted by the orientational distribution function p(β). θ is the angle that the global average director has with respect to the magnetic field, which is in this case identical to the orientation of the sample with respect to the magnetic field. β is the angle that describes the variation of the local director field with respect to the sample axis, see figure 3.15. The kernels K(t, β) and Kθ(t, β) depend on five parameters of the sample and the experimental procedure: quadrupolar frequency

ωQ, transverse relaxation time T2, pulse duration p1, resonance frequency ωRF and pulse separation τ. The parameters ωRF , p1 and τ are set while ωQ, T2 and an additional time offset for the FID are subjected to the fitting procedure. Thus p(β) gives the weighting factor for the deviation of the local director from the average one. Winterhalter et al. [70] have written a program to obtain p(β) by Tikhonov regularization from two FIDs, one recorded with the director parallel to the magnetic field and another at an angle θ. The analysis was performed for Gel1 for 14 different angles θ. The distribution functions that differ somewhat for each

47 3. Anisotropic Hydrogels

p( )

Figure 3.15: Deﬁnition of the angles in the Tikhonov regularization procedure: θ is the tilt angle of the sample with respect to the magnetic ﬁeld

B0 and given by the experiment; β is the angle between the local director and the sample axis. The orientational distribution function p(β) is determined. Its rotational symmetry is indicated by the cone. value of θ are plotted in figure 3.16 to allow an estimation of the quality of the result. The variations of the results are due to imperfections of the model, e. g. a single transverse relaxation time T2 is fitted for both FIDs, although T2 is angular dependent. Yet the width of the orientational distribution function is about the same of all plots. The average over the orientational distribution functions is plotted in figure 3.17. The maximum differences of the individual results from the average were taken as the error bars. The full width at half maximum of the averaged p(β) in figure 3.17 corresponds to an angle of 13.5◦. A domain order parameter S can be obtained from X-ray investigations [11]. The azimuthal width of the small angle reflexes (see figure 3.18) thereby reflects the order. According to Lovell and Mitchell [71] the domain order parameter S

48 3.6. Local Order

3.0 3.0 3.0

2.5 2.5 2.5

2.0 2.0 2.0

1.5 1.5 1.5

1.0 1.0 1.0 40 0.5 2020°0.5 30 30°0.5 40°

0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

3.0 3.0 3.0

2.5 2.5 2.5

2.0 2.0 2.0

1.5 1.5 1.5

1.0 1.0 1.0 65 b 80 0.5 5555° 0.5 65°0.5 80°

0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

3.0 3.0 3.0

2.5 2.5 2.5

2.0 2.0 2.0

1.5 1.5 1.5

1.0 1.0 1.0 100 0.5 100° 0.5 110110° 0.5 120120°

0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

3.0 3.0 3.0

2.5 2.5 2.5

2.0 2.0 2.0

1.5 1.5 1.5

1.0 1.0 1.0

orientationaldistributionfunctionp( )[a.u.]

0.5 130130°0.5 140°140 0.5 150

0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

3.0 3.0

2.5 2.5

2.0 2.0

1.5 1.5

1.0 1.0 160 0.5 160° 0.5 165

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

b [rad]

Figure 3.16: Orientational distribution functions p(β) from the regularization analysis of the two FIDs at θ = 0 and θ as indicated next to each graph.

49 3. Anisotropic Hydrogels

1.5 averaged Regularization SAXS

1.0 ) [a.u.] )

p( 0.5

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 [rad]

Figure 3.17: Orientational distribution function p(β) averaged over the 14 diﬀerent angles θ. For comparison the azimuthal X-ray scan is plotted. is calculated according to:

R π/2 I(q, β) sin β 1 (3 cos2 θ − 1)dβ S = 0 2 (3.7) R π/2 0 I(q, β) sin βdβ

I(q, β) is the scattering intensity depending on the azimuthal angle β and the scattering vector q. The order parameter S reflects the distribution of the local director of the domains in the beam. The distribution of X-ray scattering intensity as function of the azimuthal angle β is broader than the NMR orientational distribution function. Both are plotted in figure 3.17. The NMR orientational distribution function reflects a similar order parameter but on a larger length scale. Equation 3.7 can be used to derive a domain order parameter SNMR by replacing I(q, β) by p(β). The order parameter was calculated for each of the distributions in figure 3.16 and the statistical average was taken as SNMR. The analysis according to Lovell [71] yielded order parameters SSAXS = 0.65 and

50 3.6. Local Order

SNMR = 0.89(±0.02) at 297 K. This indicates a well ordered structure averaged over the NMR time scale. This time scale τ, given by the inverse splitting, can be converted to a length scale, using hx2i = 2Dτ, of about 5 µm. The X-ray order parameter on the other hand indicates signiﬁcant defects on smaller scales. X-ray gives a snapshot of the sample, since X-ray data are summed up over time whereas for NMR data averaging over time is given.

3.6.2 X-Ray Correlation Length

The radial intensity scan of the x-ray diffraction pattern of the swollen hydrogel (equilibrium degree of swelling) is given in figure 3.18. Fitting of the two peaks with a Lorentz function yields the repeat distance of the layers [11] (63 A)˚ and the smectic correlation length ξs, which is proportional to the full width at half maximum (FWHM). This correlation length gives the number of stacked layers that make up a domain. The fit in figure (3.18) fails in the central part due to the beam stop, but the FWHM is not affected. The radial intensity profile was fitted according to [72]: a1 I(q) = 2 (3.8) (q − q0) + a2 where a1 is a fitting parameter proportional to the area under the peaks, q0 is the peak position and a2 is proportional to the FWHM. a2 was determined as 2.3(±0.3) · 10−4 A˚−1. The smectic correlation length, given by:

2π ξs = √ (3.9) a2 is calculated to 420(±8) A.˚ This corresponds to about 6.7 layers with the layer distance of 63(±0.5) A.˚ The correlation length gives the probability to ﬁnd another layer at a distance from a certain layer. In this case the probability that the seventh layer is where it should ideally be is reduced to 1/e (36 %).

51 3. Anisotropic Hydrogels

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

q [Å-1]

Figure 3.18: Radial intensity proﬁle of the hydrogel at equilibrium degree of swelling, solid line are two Lorentzian ﬁts with equation (3.8).

52 3.6. Local Order

3.6.3 Macroscopic Dye Diﬀusion

Another experiment showing the macroscopic anisotropy of diffusion is the direct visual observation of the diffusion of a dye. Hickl demonstrated this by putting a dye crystal (methylene blue) in the middle of a hydrogel at its equilibrium degree of swelling. The experiment was followed by taking pictures with a digital camera [11]. A quantitative analysis as it was tried by Hickl is questionable, since the boundary conditions of the experiment, which are needed in the model, are very complex. Nevertheless the anisotropy is evident as shown in figure 3.19. The anisotropy can be determined by this method just by measuring the lateral (horizontal) and axial (vertical) dimension of the spot. The axial ration is 2.3 : 1. The ratio of the diffusion coefficients parallel and perpendicular is then calculated by the square of this axial ratio, since it is the mean squared displacement, which is proportional to the diffusion coefficient. The anisotropy D⊥/Dk is 5.3 : 1.

Figure 3.19: Dye diﬀusion after several minutes. The crystal was placed in the middle of the dye spot. The layer normal in this picture is vertical with a little tilt to the left.

53 3. Anisotropic Hydrogels

3.7 Discussion: Hydrogels

Anisotropic hydrogels were investigated in order to learn about their microscopic properties and structure. In particular the anisotropic behavior and the order on different length scales was of interest. The mesoscopic behavior was investigated by means of NMR diffusometry. The anisotropy of the diffusion coefficient was found to be in the order of 10 – 14 at a length scale in the order of several tens of µm. This is rather low, when compared to low molecular weight systems [73]. The low anisotropy could be made plausible by the interpretation of temperature dependent diffusion data, which showed the existence of defects in the lamellae. Arrhenius activation energies in the range from 300 to 340 K were found to be in agreement with the literature on biological membrane systems. The temperature dependent obstruction factors showed an anomalous step at around 312 – 314 K which was interpreted as the occurrence of further defects and cracks due to restoring elastic forces from the network, which become dominating in the coupled system of mesophase and network above this temperature. It occurred at a temperature range where the corresponding low molecular weight mixture was already phase separated. Diffusion NMR could also proof that the minimal extension of homogeneously ordered domains is rather big with a lateral extension of 30 µm × 30 µm and a longitudinal (parallel to the layer normal) extension of 10 µm. The X-ray analysis showed a structure that is locally, on the length scale of 0.1 µm, not well ordered. But the ordering on intermediate scales is better as can be seen from lineshape analysis (several µm) and diffusion NMR (several 10 µm). Macroscopic dye diffusion again showed a decrease of order at larger scales. The anisotropy of the diffusion goes down to about 5.3 on the scale of several mm. Concluding, one can say that the hydrogel is a mesoscopically well ordered structure with defects on small scales as seen from a nonzero obstruction factor

Qk and X-ray scattering. On large scales the order is lower due to macroscopic defects in the texture. The analysis of NMR line shapes and diﬀusometry data for the two gels, which diﬀer in their position in the phase diagram, showed an overall lower degree of order for the gel with higher content of water.

54 3.7. Discussion: Hydrogels

In future research on these hydrogels one should be aware of the problem associated with the crosslinking process. In the present work the UV-irradiation dissipated a lot of energy, which could be the reason for local disorder (X-ray) and defects in the lamellae.

Chapter 4

Multilamellar Vesicles

In 1993 Roux and coworkers found an interesting substructure of the lamellar Lα phase [13]. They investigated a quaternary mixture of sodium dodecyl sulfate (SDS), water, pentanol and dodecane. This system forms a lamellar phase and by the application of a shear force to this phase, a metastable state of so called ∗ multilamellar vesicles (MLV) can be reached [13, 74]. The membranes in this Lα state are ordered in concentric spheres, resulting in an ’onion’-like structure (see section 1.3). While the process of formation is still a matter of debate [24–27], the diameter, which is in the µm range, and the polydispersity of the MLVs can be controlled by the applied shear rate. With higher shear rate the diameter of the onions becomes smaller and they become more monodisperse. At low shear rates below 2 – 3 s−1 the size distribution of the MLVs is broader and their average diameter is larger. Freeze fracture measurements demonstrated the coexistence of large and small MLVs at such low shear rates [75]. They also show that the MLV state is a close packed structure, where every MLV touches its next neighbors [76]. It should be emphasized that there is no interstitial or gusset volume as it would be expected for close packed structures of hard beads. It is rather the concept of the truly spherical structures that has to be given up in favor of a more ﬂexible structure of a convex polyhedron that has to adapt to its nearest neighbors. This will be discussed in more detail in section 4.3. MLVs are usually stable after interruption of shear for several hours and even days or longer. Depending on the temperature and the shear rate of preparation the long-term stability of the MLVs can vary, a higher preparation shear rate thereby leading to more stable

57 4. Multilamellar Vesicles

MLVs. Closed aggregated structures play an important role in many biological, pharmacological or industrial processes, only to mention drug encapsulation as a big ﬁeld of interest [15, 22, 23, 77].

As an exemplary system the binary system of C10E3/D2O, which exhibits at

298 K a broad lamellar Lα phase from about 5 to 75 wt.% C10E3 was chosen [78, 79]. The excellent knowledge of the phase behavior and structure formation under shear ﬂow of this system makes it an appropriate choice to study the possibilities of structure elucidation by means of NMR. A comparison of NMR line shapes is made in section 4.4.3 with the similar system C12E4/D2O [79].

4.1 MLV Sample Preparation

Multilamellar vesicles were prepared by a ’standard procedure’, which was developed by Daniel Burgemeister [27]. An amount of 40 wt.% triethylene glycol mono n-decyl ether (purchased from Nikko Chemicals; trade name: BD-3SY) in D2O was mixed in an Eppendorf tube and shaken in a vibrating mill with a small magnetic stir bar for several minutes to get a homogeneous sample. The mixture was transferred to the home built [27] rheo-NMR probe head, equipped with a Couette cell with a gap of 500 µm. The sample was heated to the isotropic phase (318 K) outside of the magnet. A constant shear rateγ ˙ was applied and the sample was kept for 20 – 30 minutes at 318 K. It was then cooled to room temperature at a very slow rate (0.2 K/min) and sheared simultaneously. During the cooling process the Lα phase was reached and MLVs of shear rate dependent size were obtained. The sample was then quickly transferred by a spatula into a 4 mm NMR sample tube by smearing it on the upper part of the tube and then ’centrifuging’ (by flinging with the hand) it to the bottom. No big shearing forces are expected in this procedure, maybe only a slight compression of the MLVs. A ’heating’ of the sample by handling it was carefully avoided. After the transfer a tightly locking teflon spacer with a pinhole in the center (to let the air escape) was inserted in the tube to seal the sample from the air and then the whole tube was sealed by melting the glass (figure 4.1). The success of the procedure was then controlled by comparison of the NMR lineshape with the previous results of in situ NMR measurements by D. Burgemeister under shear. The initial line

58 4.2. Polarizing Light Microscopy of MLV shape was regularly checked and taken as a reference for the sample stability. The MLVs were usually stable for about one to two days, depending on their size.

teflon MLVsample air

5mm spacer

Figure 4.1: Sketch of a sealed MLV sample.

4.2 Polarizing Light Microscopy of MLV

Two samples, that were prepared following the standard procedure described in the previous section, were investigated by light microscopy under crossed polarizers. The samples were sheared at 10 and 2 s−1, respectively, and then smeared on a microscope slide. Spherical structures like the MLVs should display a texture of maltese crosses. The parts of the vesicles, where the lamellae are aligned parallel to one of the polarizers are dark and those areas where the lamellae are aligned at 45◦ to both polarizers are bright. This is shown for both samples in ﬁgure 4.2. Two homogeneous samples are seen, showing the typical texture with a modula- tion of the refractive index on the micrometer scale as expected [14, 80, 81]. Four of these spots make up one MLV, which is sketched in some exemplary circles. The circles have a diameter in the order of 5 µm for the sample sheared at 10 s−1. The sample that was prepared at 2 s−1 shows bigger MLVs and is more polydisperse. The circles range from 7.8 up to 12 µm, yet the majority is around 7.8 µm. Both is in good agreement with the data obtained by light scattering [82]. The maltese crosses that can be seen can be taken as a proof that the preparation procedure leads to the expected result. It should be mentioned that ﬁgure 4.2 does not show an MLV monolayer but a rather thick sample, where some MLVs on top of each other might darken the image in some places.

59 4. Multilamellar Vesicles

Preparationat2s-1 Preparationat10s

5 m m 5 mm

Figure 4.2: MLV sample prepared at 10 (left) and 2 (right) s−1 under crossed polarizers. One MLV consists of four spots, which results in an MLV-diameter of about 5 µm at 10 s−1. The sample at 2 s−1 is more polydisperse and the radii vary from 7.8 (majority) up to 12 µm (only a few)

4.3 A Comment on the Interstitial Volume

When the MLV state is discussed the question of what is between the MLVs, in the so called gusset, is often posed. This comes mainly from the somehow misleading description of an MLV as an ’onion’. This picture covers well the multilamellarity of the structure but on the other hand invokes the association of hard spheres. Such an idea leads to an image of a closed packed structure of beads with a certain fraction of free volume between the MLVs. According to NMR this picture is not true. Yet it is not fully clear what is in between. Numerous electron transmission microscopy (TEM) freeze fracture pictures exist of MLVs. They all show that each MLV is more or less directly connected to all of its neighbors as a consequence of the ﬂuid properties of the system [76]. One could of course argue that the freeze fracture technique changes the system and can not be trusted to yield an image of the unperturbed sample. Assuming that the onions remain spherical, there are mainly two possibilities for the volume between the idealized hard onions. One possibility is unoriented

Lα phase, the other one is isotropic solution of smaller micelles or MLVs. From the NMR single pulse spectra both possibilities can be rejected. First, the vol-

60 4.3. A Comment on the Interstitial Volume ume fraction between close packed beads is about 25 %. This means that both unoriented Lα phase or isotropic solution should significantly contribute to the spectrum either as a Pake spectrum or as a small isotropic central line. Neither is observed. From the homogeneous appearance of the broad peak (although it is in fact inhomogeneous, as will be discussed in section 4.5.1) follows that the sample is dynamically uniform. The dynamics inside the MLV-onions and ’outside’ is the same. The easiest way to account for this, is to change the picture of rigid objects to more flexible structures, where each of the four MLVs surrounding a gusset form equal bulges/convexities to fill the void. The phase should then be regarded more as ’foam’-like, where especially the gusset is of undefined shape. Energetically this leads to only one singularity in the very center of the gusset, besides those in the center of each MLV. From the NMR point of view, this is the easiest way to describe the situation.

61 4. Multilamellar Vesicles

4.4 The Single Pulse Line Shape

4.4.1 The Characteristic D2O Line Shape

The characteristic lineshapes of MLV samples prepared at two different shear rates are presented in figure 4.3. A broad isotropic line is observed in the single pulse experiment. This line shape was previously identified as the one of the MLV structure by comparison of light scattering and NMR experiments for the system 2 C12E4/D2O [80]. The typical H-NMR line widths, taken as the full width at half maximum (∆νves), are in the range of 150 to 350 Hz while the typical line width of pure D2O in the geometry of the Couette shear cell is only about 10 Hz. The

FWHM is related to the outer radius, Rmax, of the MLV multispheres, although there exists no well deﬁned relationship so far. An attempt to establish such a relationship is undertaken in section 4.4.3.

1.0 10 s-1 2 s-1 0.8

0.6

0.4

normalized intensity [a.u.] intensity normalized 0.2

0.0

-1500 -1000 -500 0 500 1000 1500 Frequency [Hz]

Figure 4.3: Characteristic single pulse spectra of MLV structures, prepared at shear rates of 2 and 10 s−1.

62 4.4. The Single Pulse Line Shape

Generally the line shape is governed by the rotational motion of D2O on the surface of the spherically bent layers. The motion of the water on the spheres leads to a change of the orientation θ of the director relative to the magnetic ﬁeld

B0 on the time scale of the experiment. This motion is defined by the rotational diffusion coefficient Drot, which depends on the translational diffusion coefficient

Dtrans. While Dtrans is the same for all molecules, Drot on the other hand changes with the radius R of the sphere, according to the following equation:

D D = trans (4.1) rot R2

However the time dependence of θ may be due to both isotropic reorientation of the entire object, in this case the whole sphere, and diffusion of the molecule around the surface of this object. Both diffusional processes can be associated with diffusion coefficients Dtumbling and Drot, respectively. The precise description of the rotational diffusion must therefore be written as [35, 83]:

D 3kT D = D + D = trans + (4.2) rot tumbling R2 8πηR3

For a radius of R = 2 µm, a viscosity of η = 7 Pa s [82], T = 298 K and −9 2 D = 10 m /s, the effect of the second term Dtumbling is below 10 % and was therefore neglected in the following considerations. Two extreme cases of large and small rotational diffusion coefficients can be considered for a given time scale. One extreme situation are fast moving molecules on small spheres. In this case the molecules travel around the sphere for even several times on the experimental time scale. The angular dependent NMR signal in this case is averaged over all angles and a narrow isotropic line is observed. The other situation is a molecule, which is fixed on the sphere, or at least does not significantly change its orientation during the experiment. Here a Pake spectrum would be expected, corresponding to the case of a polydomain, since the director is oriented in every direction with equal probability. The lowering of the rotational diffusion coefficient can be realized experimentally by observing slower molecules (low Dtrans) on large spheres [75]. This is the case for the deuterated surfactant, C10E3-d2, where a diffusion coefficient parallel to the lamellar layers of

63 4. Multilamellar Vesicles

−12 2 Dtrans = 8.6 · 10 m /s was measured in the Lα phase. As it is shown in ﬁgure 4.4, the expected Pake spectrum is observed in the single pulse experiment. It should be noted that such a Pake spectrum is only observed when Drot is low enough even on the inner spheres of the MLVs, although their impact on the ﬁnal spectrum is relatively low due to surface weighting.

- 2 0 0 0 0 - 1 5 0 0 0 - 1 0 0 0 0 - 5 0 0 0 0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 F r e q u e n c y [ H z ]

Figure 4.4: Single pulse spectrum of C10E3-d2 in the MLV state, prepared −1 at 10 s , the isotropic peak in the middle arises from residual D2O and corresponds to the spectrum in ﬁgure 4.3.

Table 4.1 shows the solid angle corresponding to the surface area 2 hx i = 4DτNMR, that is visited by the water and the surfactant, respectively, for the assumption of surface diﬀusion on a sphere, calculated for a radius R =

2.5 µm. The translational diﬀusion coeﬃcients have been measured in the Lα phase at 298 K. The NMR time scale is given by the inverse of the splitting of the NMR signal, which is 1.6 kHz for D2O and 25 kHz for C10E3-d2, leading to

τNMR(D2O) = 625 µs and τNMR(C10E3-d2) = 40 µs. Additionally calculated in table 4.1 are the solid angle Ω [sr] = hx2i/R2 in steradians (sr) and the opening

64 4.4. The Single Pulse Line Shape angle α of the cone that makes up the solid angle. α is deﬁned by the solid angle according to Ω = 2π [1 − cos(α/2)].

Table 4.1: Calculations of rotational diﬀusion on a sphere with a radius R = 2.5 µm. hx2i: surface area covered by the molecule on the sphere

after the respective τNMR. Ω: corresponding solid angle Ω after τNMR. α: opening angle of the cone that makes up the solid angle.

D2OC10E3 2 −9 −12 Dtrans [m /s] 1.0 · 10 8.6 · 10 −1 Drot [s ] 160 1.38

τNMR [µs] 625 40 x2 [µm2] 2.5 1.37 10−3 Ω [sr] 0.4 2.2 10−4 α [rad] 0.717 0.017 α [◦] 41 0.97

During the typical time of the NMR experiment water diﬀuses (averages) over 41◦, while the surfactant only covers a small angle. This is in good agreement with the observation of a signiﬁcantly broadened line for water, where the quadrupolar interaction is not completely averaged and the Pake spectrum for the surfactant.

In order to obtain a value of Drot for D2O large enough to observe a Pake pattern, a radius of more than 100 µm would be necessary. The formation of MLVs of such a size was never observed and their stability is highly questionable.

4.4.2 Theoretical Eﬀect of Polydispersity

Douliez et al. [34] performed computer simulations of 2H line shapes of MLV structures. They simulated the transition from Pake patterns at large radii to isotropic lines at small radii. At intermediate MLV sizes single broad peaks were found. They also simulated the eﬀect of polydispersity on a system of unilamellar vesicles (ULVs). A sharp isotropic line was observed when the radii of ULV were highly polydisperse. This sharp line occurred also in MLVs, when the number of inner bilayers was increased in the simulations, which means that eﬀectively

65 4. Multilamellar Vesicles a polydisperse structure of small and large MLVs is simulated. In our case such a sharp isotropic line was never observed. This can be taken as a hint that polydispersity is not signiﬁcant in our system, although it can not be excluded.

4.4.3 Estimation of the MLV Size from the Single Pulse Line Shape

The MLVs investigated in this work are all prepared according to a standard procedure as described in section 4.1. A direct ’on-line’ control of the radius of the MLVs is not possible. For comparing NMR data on the other hand it is crucial to know something about the size if not even the polydispersity of a speciﬁc sample. The only knowledge about the MLV size is available by taking the respective MLV radius determined by light scattering at the same shear rate. Optical microscopy of the samples only provides a rough estimate of the size and distribution. In the following an attempt is made to establish a relationship between NMR line shape and MLV size. Based on detailed data from Nettesheim [82] and M¨uller[80] a simultaneous analysis of two systems is performed in order to show the generality of the calculations.

Simultaneous analysis of C12E4 and C10E3

Measurements of the MLV size were performed on two system: C12E4 and C10E3, both containing 40 wt.% D2O. Nettesheim et al. [82] measured the size of the MLVs in both systems as a function of shear rate by the method of rheo-light scattering. The radii found in the polarizing microscope in section 4.2 are in good agreement with their data. M¨ulleret al. performed measurements of the

NMR line width of D2O (∆νves) as a function of the shear rate for the system

C12E4/D2O. Up to now there is no direct relationship between the radius of the

MLV, RMLV , and the line width. A plot of both data (NMR and LS) on top of each other suggests that there could be a direct relationship (ﬁgure 4.5). This is especially valid for C12E4, the data of ∆νves for C10E3/D2O scatter much more. A reason for this could be that they were acquired in many diﬀerent experiments and not subsequently in one experiment.

66 4.4. The Single Pulse Line Shape

400 C E 4.5 12 4

4.0 R from Rheo-light scattering, 350 MLV F. Nettesheim and S. Müller 3.5 from Rheo-NMR, S. Müller VES 300 3.0 m] [Hz] [ 2.5 250 VES MLV R 2.0 200 1.5

1.0 150

0 5 10 15 20

shear rate [s-1] 10 C E 10 3 9 R 500 MLV

8 VES

7 400

6 m] [Hz] VES [ 5 300 MLV R 4

3 200 2

1 100 0 5 10 15 20 25 30

shear rate [s-1]

Figure 4.5: Comparison between rheo-light scattering data and rheo-

NMR data, the superposition suggests a direct relationship between ∆νves

and RMLV .

67 4. Multilamellar Vesicles

The ﬁrst experiments on shear induced MLVs by Diat et al. [13] suggested a scaling of the MLV radius in dependence on the shear rate of the form R ∼ γ˙ −0.5. This square root dependence was not found in this work, but the assumption of a scaling relationship was adopted for the analysis. For the relation of the radius versus the shear rate, scaling laws of

−0.60 (±0.01) RMLV (C12E4) = 6.5 (±1) · γ˙ −0.82 (±0.08) RMLV (C10E3) = 18.2 (±4) · γ˙ (4.3) could be ﬁtted for the two system (see ﬁgure 4.6). The assumption that a similar scaling relation is also valid for the FWHM versus the shear rate resulted in the scaling laws:

−0.43 (±0.02) ∆νves(C12E4) = 500 (±20) · γ˙ −0.31 (±0.04) ∆νves(C10E3) = 430 (±40) · γ˙ (4.4)

Both ﬁts are presented in ﬁgure 4.7. From these two independent measurements a direct relationship between the MLV radius and the FWHM can be established. Combining equations (4.3) and (4.4) resulted in:

−3 1.38 (±0.06) RMLV (C12E4) = 1.1 (±0.1) · 10 · (∆νves) −6 2.64 (±0.73) RMLV (C10E3) = 2.3 (±0.3) · 10 · (∆νves) (4.5)

The calculated radii of the MLVs at the measured shear rates according to equations (4.5) are shown in ﬁgure 4.8 for both samples. In addition, the points where both radius and FWHM were measured at the same shear rate are plotted. They agree quite well with the calculated data although the error is very large. It should be kept in mind that these equations were derived assuming scaling relations. In particular it has to be emphasized that the above equations can only be valid in a certain range of ∆νves. The minimal ∆νves, for which equations (4.5) are valid is determined by the line width of a system with complete averaging of the quadrupolar interaction. Therefore there is lower limit for the radii of the

MLVs that can be determined by NMR. This minimal ∆νves, resulting mostly

68 4.4. The Single Pulse Line Shape from hardware line broadening, was not accounted for in the analysis for the sake of simplicity of the ﬁtting function. The data in ﬁgure 4.5 seem to level out at about 150 Hz for both samples. This would then lead to a value of 1.2 µm for the smallest vesicle radius that can be determined by NMR. Moreover the error for C10E3/D2O is at least 1 µm and therefore the NMR line width can give only an estimated value of the radius.

Comparing the two samples (ﬁgure 4.8) it can be observed that ∆νves at a given radius is smaller for C10E3/D2O than for C12E4/D2O. Since the line width is determined by the dynamics of the water molecules it can be concluded that the motion of D2O in the C10E3/D2O sample is faster. This can be made reasonable by the ratio of D2O per ethyleneoxide (EO) unit. With the usual assumption of 1.5 D2O-molecules bound per EO-unit the fraction of free and thereby mobile

D2O is higher in the C10E3/D2O-sample.

69 4. Multilamellar Vesicles

5 C E 4 12 4

2 m] [ MLV R 1

10 100

shear rate [s-1] 8 7 C E 10 3 6

3 m] [ MLV R 2

1 5 10 15 20 25 30 35

shear rate [s-1]

Figure 4.6: Scaling ﬁts of the MLV size as function of the shear rate,

leading to equations (4.3). The data for C12E4 are taken from Nettesheim 70 and M¨uller. 4.4. The Single Pulse Line Shape

400 C E 12 4 350

300

250 [Hz] VES

200

150 2 4 6 8 10 12 14 16 18

shear rate [s-1] 500 C E 10 3

400

300 [Hz] VES

200

1 10

shear rate [s-1]

Figure 4.7: Scaling ﬁts of the ∆νves as function of the shear rate, leading to equations (4.4). The data for C12E4 are taken from Nettesheim and M¨uller.

71 4. Multilamellar Vesicles

400 C E 12 4 calculated 350 measured

300

[Hz] 250 ves

200

150

0 2 4 6 8 R [ m] MLV 400 calculated C E 10 3 measured 350

300

[Hz] 250 VES

200

150

0 2 4 6 8 R [ m] MLV

Figure 4.8: Dependence of the ∆νves on the radius for C12E4, solid squares are calculated from equation (4.5), open circles are measured data from Nettesheim and M¨uller. 72 4.4. The Single Pulse Line Shape

4.4.4 A Surface Record Factor (SRF) for the MLVs

In order to check if the above considerations are consistent to at least some degree, a theoretical approach to the problem of the surface area covered by diffusion on the sphere is attempted. The NMR line width depends primarily on the solid angle, over which the quadrupolar interaction is averaged. It should be possible to define a parameter that can display this averaging and which should then be directly related to the ∆νves of the MLV spectra. A convenient parameter should be zero, when the motion is averaged out, that is, when the MLVs are small and the diffusion is fast, and it should assume a constant maximum value when no averaging takes place. This parameter will be called the ’Surface Record Fac- tor’, SRF. In analogy to an order parameter, that is defined for thermodynamic equilibrium and therefore independent of time, this parameter is in our case defined as the integral over all angles over the second Legendre polynomial times the angular distribution function P (θ), obtained for diffusion on a sphere for a limited time. The time limit is the characteristic time of the NMR experiment (see section 4.4.1). Z π 1 SRF = P (θ) (3 cos2 θ − 1)dθ (4.6) 0 2

The Angular Distribution Function

A description for a diffusing particle on a sphere is based on the angular distribution function [84], which gives the probability of finding a particle at a point r0 after a time t, thus having covered an angle θ, assuming that is has started at r, as can be seen in figure 4.9. For convenience r is chosen to point to the north pole. The diffusion on a sphere is described by the three-dimensional diffusion equation ∂ P (r0, t|r, t ) = ∇Dˆ∇P (r0, t|r, t ) (4.7) ∂t 0 0 with the anisotropic diffusion tensor Dˆ. Since the system exhibits spherical sym- 0 metry, the conditional probability function P (r , t|r, t0), depending on the positions, is usually expressed as a function P (θ, t|θ0, t0) of the polar angle θ, which is covered by the diffusing particle on the unit sphere. P (θ, t|θ0, t0) is the probability that a particle starting from the north pole at t0 has an angle with the

73 4. Multilamellar Vesicles

z-displacement

r r’ q

Figure 4.9: Displacement on a unit sphere of a particle starting at the north pole r and ending at r0, having passed an angle θ.

magnetic field (being parallel to the polar axis) after a time t. The magnitude of the angle depends on the diffusion coefficient, the radius and the diffusion time. Changing the conditional probability function to spherical coordinates and inserting the Laplace operator in spherical coordinates yields:

∂ 1 ∂ ∂ 1 ∂ P (θ, t|θ , t ) = (sin θ) + P (θ, t|θ , t ) (4.8) ∂t 0 0 sin θ ∂θ ∂θ sin2 θ ∂φ 0 0 with the condition that the radius is constant. The solution of this diﬀusion equation, the angular distribution function for a sphere of radius R, has the form [84]: ∞ 1 X Dt P (θ, t) = (2l + 1) P (cos θ) · exp −l(l + 1) sin θ (4.9) 2 l R2 n=1 where Pl are the Legendre polynomials of order l. The angular distribution function depends on the magnitude of the factor Dt/R2 and one of the three parameters D, t or R can be chosen to be varied. In the long time limit the distribution reaches the orientational distribution of vectors normal to the surface 2 1 of a sphere P (θ, t R /D0) = 2 sin θ.

In figure 4.10 Ps(θ, t) is plotted for a sphere of radius R = 2.6 µm (which is comparable to MLVs prepared at a shear rate ofγ ˙ ≈ 10 s−1) and a diffusion coefficient of 1 · 10−9 m2/s for different diffusion times t. The long time limit of the angular distribution function in this case is reached for diffusion times around 20 ms, which can be easily reached in a typical pulsed field gradient experiment.

74 4.4. The Single Pulse Line Shape

1.2 t [ms]

1.0 1 3 7 0.8 20

,t) 0.6 P(

0.4

0.2

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 [rad]

Figure 4.10: Angular distribution function for diﬀusion on a sphere.

The surface record factor SRF, as defined in equation (4.6) can now be calculated using P (θ, t) given by equation (4.9). In figure 4.11 a plot of the SRF as a function of the radius for one sphere with radius R is plotted for different times t. The different averaging times show the behavior of the SRF for different radii. When t is long, complete averaging over the sphere, that is SRF = 0, is found up to larger radii. On the other hand a short t leads to a fast increase of the SRF since then the molecules on the sphere can not travel over the whole surface even for small radii. The set of curves in figure 4.11 can all be scaled to collapse on one master curve (say the one at 1 ms), when the radii are divided by the square root of t. There is a threshold Rcrit below which the SRF remains zero. This has an experimental analogy as the line width also approaches a constant minimal value when complete motional averaging over the whole sphere is reached. This minimal line width then depends mainly on the spectral resolution, which is determined by hardware and the viscosity of the sample (about 150 Hz in our case). Furthermore the SRF reaches a maximum of one. In this case the particles

75 4. Multilamellar Vesicles on the sphere are stationary and in the experiment a Pake spectrum would be visible. An analysis of ∆νves for this situation is no longer possible, since ∆νves does not account for the shape of the spectrum. The analysis is only valid, as long as the spectrum is a broad single line.

1.0 1 ms 2 ms 3 ms 0.8 5 ms 7 ms 10 ms 0.6 15 ms 20 ms

0.4 Surface record factor SRF factor record Surface 0.2

0.0 0 2 4 6 8 10 R [ m] one sphere

Figure 4.11: Surface record factor SRF for diﬀusion on one sphere, calculated with equation (4.6) for diﬀerent experimental time scales t. D was kept constant at 10−9 m2/s and the summation over the Legendre

polynomials Pl(cos θ) in equation (4.6) was taken from l = 0 to 100.

When the SRF in figure 4.11 is compared to the line width dependence on the radius in figures 4.8 a principal analogy is visible. It was not possible to superimpose the simulated data (figure 4.11) for a single sphere (which would then be considered the outermost sphere of the MLV, which dominates the spectra) on the calculated data (figures 4.8). Nevertheless the shape of the simulated curves resembles those of the experiment and the most prominent qualitative features like the minimal radius Rcrit and the upper limit are present. Concluding, it has to be stated that the analysis with the simple model of one sphere explains the

76 4.4. The Single Pulse Line Shape experimental data in a qualitative way, but is not yet quantitatively satisfying. But the agreement of the data in ﬁgures 4.5 looks promising enough to recommend a further reﬁnement of the description of the above relationships, taking into account the summation over all spheres in an MLV.

77 4. Multilamellar Vesicles

4.5 Echo Line Shapes

4.5.1 The Solid Echo

The observations and calculations on the single pulse spectra in section 4.4.1 lead to the idea that the broad line of D2O is essentially a superposition of motionally narrowed Pake spectra. The hypothesis is that each signal is a sum from the contributions of the spheres with different radii. The dynamics of the system, namely the diffusion coefficient of D2O, is assumed to be independent of the radius. The only difference within one MLV then are the radii of the spheres. D2O on an inner sphere with a small radius is travelling over the whole surface of the sphere on the experimental timescale, which is much longer than the correlation time for quadrupolar interaction. The correlation time for quadrupolar interaction τc for molecules diffusing on the surface of a sphere of radius R with a diffusion constant D is given by [83, 85, 86] 1:

R2 τ = (4.10) c 6D

For a small sphere such a motion leads to complete averaging of the quadrupolar interaction and a narrow isotropic line. D2O on an outer sphere with a large radius, on the other hand, does not diﬀuse over the whole sphere and keeps a residual quadrupolar interaction. The extreme of this situation is the Pake spectrum, which is observed for the slowly diﬀusing surfactant molecules in section

4.4.1. To construct the D2O-line shape of the full MLV, the line shapes from each of the spheres are weighted according to their surface area with factors 4πR2. Be- cause of this weighting the outermost spheres have the greatest impact on the ﬁnal spectrum. Some typical experimental spectra and their dependence on the evolution delay τ are shown in ﬁgure 4.12. At short echo delay times a broad line equivalent to the single pulse experiment is obtained. When the evolution delay τ is increased the line shape changes and a doublet with a central peak is observed. The occurrence of the doublet gets more and more pronounced with longer evolution delays. But the spectra at long delays have intensities that are

1The factor 6 is given in the literature, although a factor of 4 should be more appropriate for diﬀusion on a surface.

78 4.5. Echo Line Shapes

2 = 0.3 ms 2 = 6 ms 4x108 6x107 I = 100 % I = 12 %

2x108 3x107

0 0 -2000 -1000 0 1000 2000 -2000 -1000 0 1000 2000

1x107 2 = 12 ms 3x106 2 = 18 ms

Intensity [a.u.] Intensity I = 2 % I = 0.6 %

5x106 2x106

0 0 -2000 -1000 0 1000 2000 -2000 -1000 0 1000 2000

Frequency [Hz]

Figure 4.12: Solid echo line shapes at different evolution delays 2 τ at 298 K. The integrated intensities I are given relative to the first spectrum. MLVs prepared at 10 s−1. largely reduced by about two orders of magnitude, compared to the spectrum at 2τ = 0.3 ms, which can also be seen from the increased noise level (figure 4.12, lower right). This can be explained by the slow-motion effect [87, 88], which predicts a strong signal reduction in the solid echo for motions on the time scale τNMR of the inverse of the splitting ∆ν, which is of the same order as the inverse line width of the MLV single pulse spectrum. That means, if the orientation of the quadrupolar tensor of the water molecule changes significantly during the evolution delay of the solid echo, the dephasing is irreversible and the refocussing −1 is incomplete. For the D2O signal in MLVs, prepared at 10 s , the line width is 200 Hz, which corresponds to a time scale in the range of milliseconds. This is of the order of τc according to equation (4.10). Of course, τc varies for the different shells of an MLV and a distribution of correlation times is assumed. Nevertheless,

79 4. Multilamellar Vesicles the observed reduction of the echo intensity (by a factor depending on 2τ) can be explained by the slow motion of the diffusing molecules. According to the simple calculations in section 4.4.1, the orientation of the water molecule changes during the evolution delay. This leads to a complex solid echo line shape due to motions on the intermediate scale [89]. The signal reduction is especially strong for the water molecules that are moving on the spheres of intermediate size. Following the considerations made above, the explanation is that the observed splitting comes from the 90◦ singularity of a Pake spectrum, produced from the outermost spheres. The reason that only the 90◦ orientation remains, is mainly geometric since the relative fraction of angles is proportional to sin θ. A second reason is orientation dependent transverse relaxation. This will be discussed in more detail in section 4.5.4. The signal from the spheres in the middle of the MLVs with intermediate radii is distributed over the whole spectral range. Figure 4.13 shows the solid echo from figure 4.12, lower right, sketched together with a Pake spectrum. The 90◦ horns of the Pake pattern are scaled to match the measured splitting. The missing intensity in the measured splitting in the middle and at the shoulders of the Pake spectrum can be taken as a hint that the line shape is strongly influenced by intermediate dynamics. The observed splitting is only slightly reduced in comparison to the magnetically ◦ oriented Lα phase, where the layer normal is also oriented at 90 with respect to the external field. The origin of the central line after long evolution delays 2τ in the spectra is yet more complicated. On the one hand the central peak comes from the relatively fast motions on the inner spheres, where the quadrupolar interactions are averaged out on the experimental time scale. On the other hand the splitting of deuterium follows the second Legendre polynomial and is therefore zero at the magic angle. This means that a second contribution to the central line could originate from particles on the outer spheres that are oriented at 54.7◦. Never- theless they would have to stay at this orientation during the whole sequence. In section 4.5.5 an attempt is made to explain the origin of the center line in more detail.

80 4.5. Echo Line Shapes

2 = 18 ms Pake spectrum

L -phase

-2000 -1000 0 1000 2000 Frequency [Hz]

Figure 4.13: Solid echo spectrum at 2τ = 18 ms, shown together with a schematic Pake spectrum (dotted line). The splitting in the solid echo spectrum (∆ν = 750 Hz) matches the splitting in the magnetically oriented Lα phase (∆ν = 775 Hz), which is shown below.

81 4. Multilamellar Vesicles

4.5.2 Size Dependence of the Solid Echo

According to the above considerations the solid echo line shape should depend on the solid angle that is covered by the molecules during the evolution delays. This angle is a function of radius and time. The same angle is covered during a short observation time at small radii or during a long observation time at large radii. This means that the solid echo line shapes from MLVs prepared at low shear rates (large radii) should be similar to those prepared at higher shear rates (smaller radii), if the evolution delays are chosen shorter. In figure 4.14 three solid echo spectra at different evolution delays 2τ from MLVs prepared at three different shear rates are presented. The line shapes look all very similar if the delays are chosen longer for those MLVs prepares at lower shear rates, corresponding to larger radii (see section 4.4.3). No simple relationship between shear rate, i. e. MLV size, and echo time was found.

shear rate [s-1] 2 [ms] 10 8 5 16 2 30

-2000 -1000 0 1000 2000 Frequency [Hz]

Figure 4.14: Solid echo spectra for MLVs sheared at 10, 5 and 2 s−1 with evolution delays of 5, 16 and 30 ms, respectively.

82 4.5. Echo Line Shapes

Additional solid echo spectra with diﬀerent evolution delays for MLVs sheared at 5 and 2 s−1 are given in appendix B.1.

4.5.3 The Stimulated Echo

Since all measurements of diffusion were carried out with a stimulated echo sequence (STE), a knowledge of the STE line shapes of the echo on the respective evolution delays is essential. Mainly the effect of the z-storage period, denoted τ2 in section 2.3.5, on the Fourier transforms of the stimulated echoes is of interest. Two sets of data are plotted in figures 4.15 and 4.16. Each with one transverse evolution time τ1 of 4 and 8 ms, respectively, and with four different z-storage periods τ2 up to 30 ms.

8.0x107 5 ms 10 ms 20 ms 35 ms

4.0x107 Intensity [a.u.] Intensity

0.0

-1000 -500 0 500 1000 Frequency [Hz]

Figure 4.15: Stimulated echo line shape for diﬀerent exchange times τ2 −1 and an evolution time τ1 of 4 ms, MLVs prepared at 10 s , Rmax = 2.6 µm.

There are two main features. First, the intensity of the doublet with the large splitting seems to depend solely on the evolution delay τ1 between the

83 4. Multilamellar Vesicles

4.0x107

0.5 ms 5.5 ms 15.5 ms 30.5 ms

2.0x107 Intensity [a.u.] Intensity

0.0

-1000 -500 0 500 1000 Frequency [Hz]

Figure 4.16: Stimulated echo line shapes for diﬀerent exchange times τ2 −1 and an evolution time τ1 of 8 ms, MLVs prepared at 10 s , Rmax = 2.6 µm.

first two pulses. It does not change when the exchange time, τ2, is changed. Second, the central peak is reduced with increasing exchange time. The longer the exchange time τ2 is, the more of the central peak vanishes. The effect is even more pronounced when the evolution delay τ1 is chosen longer (8 ms). In this case the central line vanishes almost completely at exchange times of about 30 ms (figure 4.16).

84 4.5. Echo Line Shapes

4.5.4 Relaxation

The NMR time scale τNMR for D2O in 40 wt.% C10E3/D2O is 625 µs (section

4.4.1). The correlation time for quadrupolar interaction τc for molecules diﬀusing 2 on the surface was given in equation (4.10) to τc = R /6D. −9 2 Inserting D = 10 m /s and R from 0.2 to 3 µm results in values for τc from 6 µs up to 1.5 ms. The echo pulse spacing 2τ in the solid echo ranges from 40 µs up to 18 ms. This means that the correlation time for quadrupolar interaction, τc, is in the order of the NMR and the echo time scale. Neither complete averaging (fast limit) over the motions nor the static limit case is given.

Spin-Spin Relaxation

Transverse relaxation was measured with the solid echo sequence. In ﬁgure 4.17 the decay with increasing evolution delay 2τ is plotted for three diﬀerent frequencies in the spectrum of MLVs, that were prepared at 10 s−1. Initially the decays are identical for all positions in the spectrum. The line shape does not change here, it corresponds to a broad single line in the one pulse spectrum (left inset). But at longer delay times the decay of the splitting (± 400 Hz) is slower than the decay at zero frequency, which is monoexponential. The data at very low intensities, below 5 · 10−5, are attributed to (absolute) noise and are not shown. Because of phase instabilities during the experiment a power spectrum was used for the analysis.

Fitting the initial decay according to equation (2.37) yields T2 = 2.8 ms while the second decay at long times above 15 ms for the splitting decays with

T2 = 18 ms. The intermediate decay of the spectral dip at ± 240 Hz seems to be a simple superposition of the other two. A similar echo decay was also found for −1 MLVs prepared at shear rates of 2 and 5 s with a fast T2 of the initial decay of around 2 ms and a slower T2 of 5 ms at longer times. Although the second decay is faster for these two, the general feature that the splitting decays more slowly than the peak at zero frequency is preserved.

85 4. Multilamellar Vesicles

100

10-1 I = 100% 0

-2000 -1500 -1000 -500 0 500 1000 1500 2000 Frequency [Hz] 10-2

-400 +400 ) [a.u.] ) -3 10 -240 +240 I = 5% S(2 S(2 -2000 -1500 -1000 -500 0 500 1000 1500 2000 Frequency [Hz] 400 Hz 10-4 240 Hz 0 Hz

10-5 0 4 8 12 16 20 24 28

2 [ms]

Figure 4.17: Solid echo T2 decay as function of the evolution delays 2τ at three diﬀerent frequencies. The frequencies are marked in the inset, I gives the relative integrated intensity of the two spectra. MLVs prepared at 10 s−1.

Spin-Lattice Relaxation

Spin-lattice relaxation was measured with the saturation recovery pulse sequence (section 2.3.6 and appendix C.3). For all samples the magnetization recovery shows a single exponential asymptotic increase according to equation (2.38), as it can be seen, for instance, in ﬁgure 4.18 for MLVs prepared at 10 s−1. The relaxation rates did not vary signiﬁcantly with the MLV size.

A second measurement of T1 was performed with the stimulated echo sequence. When the short delay τ1 in figure 2.3.5 was chosen very short (20 µs), the line shape is the same as in the saturation recovery. The echo intensity relaxes with a slightly different T1 of 250 ± 4 ms, as shown in figure 4.19. It also

86 4.5. Echo Line Shapes

1.0

0.8

-1 0.6

-2 )) )/S(

) [a.u.] ) -3 0.4 ln(1 - S( - ln(1 -4 ln S( ln

-5 0 200 400 600 800 1000 0.2 [ms]

0.0 0 500 1000 1500 2000 2500 3000 [ms]

Figure 4.18: Saturation recovery T1 measurement. A single exponential

ﬁt yields a T1 of 230 ± 8 ms. The data are scaled to the value at τ = 3 s. The inset shows the monoexponential behavior. MLVs were prepared at 10 s−1.

shows perfect monoexponential behavior. A different T1 is obtained from the echo intensity in the stimulated echo, when longer delays τ1 of 4 ms are chosen and gradients are included in the sequence. These are the data from the diffusion experiments, but now analyzed in a different manner, by plotting the echo intensity as function of the delays. This can be done for different gradient strengths. The intensity decay for a gradient strength of zero is also plotted in figure 4.19.

The T1 relaxation is much faster at longer evolution delays τ1, which originates from the fast decay of the central line as shown in figure 4.15. By choosing τ1 longer, a selection of a different subensemble of the sample is made, which relaxes differently. It increases further when gradients are included in the delay (the regular PGSTE experiment). The dependence of T1 on the gradient strength is shown in figure 4.20.

87 4. Multilamellar Vesicles

1.0 stimulated echo w/o gradients ( = 20 s ) 1 T = 250 ± 4 ms 1 0.8 stimulated echo w/o gradients ( = 4 ms ) 1 T = 53 ± 7 ms 1

0.6 [a.u.] 0 0.4 S/S

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 [ms]

Figure 4.19: Stimulated echo T1 measurement. Decay of the echo inten-

sity in the stimulated echo at τ1 = 20 µs and τ1 = 4 ms both without gradients in the transverse evolution delays. MLVs were prepared at 10 s−1.

4.5.5 Explanation for the Echo Line Shape

In the following the various possible explanations for the echo line shape in the solid echo and the stimulated echo are discussed in a qualitative manner. Starting from purely geometric considerations and discussing the eﬀect of relaxation, the impact of the respective factor on the line shape is considered, with particular emphasis on the origin of the central line.

Geometric Consideration

As already mentioned in section 4.5.1, the central line in the solid echo could in principle originate from contributions of particles moving on inner spheres and on outer spheres around the magic angle. In order to estimate the relative impact on the spectrum from the particles on the inner and on the outer spheres, one

88 4.5. Echo Line Shapes

40 [ms] 1 T

0 200 400 600 800 1000 g [G/cm]

Figure 4.20: T1 relaxation time in the stimulated echo with τ1 = 4 ms as function of the gradient strength g. MLVs were prepared at 10 s−1. can calculate the relative surface areas of the two fractions. The inner surface is simply calculated as the sum over the surfaces of spheres with radii Rin up to a certain radius R1, according to:

R1 X 2 Sin = 4πRin (4.11) Rin=0

The surface on the outer spheres is chosen as a ring on these spheres. It is constructed from the cross plane of a cone with an opening angle of 41◦ (taken from the calculations in section 4.4.1), that is rotated around the polar axis at the magic angle of 54.7◦, with the sphere. By choosing this opening angle a certain confidence level of ∆θ around the magic angle is introduced. The particles in this confidence level are all treated as equally contributing to the spectrum. Both surfaces on the northern and southern hemisphere contribute equally, as it is drawn in figure 4.21.

89 4. Multilamellar Vesicles 54.7° h

41°

Figure 4.21: Contributions to the central line in the solid echo spectra. Both particles in the center of the MLV (light grey) and at the magic angle on the outer spheres (dark grey) play a role.

The surface of both rings is calculated according to:

R Xmax Sout = 2 · 2 · h π Rout (4.12)

Rout=R2 with the height of ◦ h = Rout · sin(41 ) (4.13)

The ratio for only two spheres, one inner and one outer, for this speciﬁc opening angle of 41◦ is then calculated to:

Rmax , R1 Sout X 2 X 2 = 0.656 Rout Rin (4.14) Sin Rout=R2 Rin=0

An exemplary calculation for an MLV with a radius Rmax = 3µm consisting of 100 equally spaced spheres was carried out. The inner surface was calculated from zero up to a certain radius R1 while the outer surface around the magic angle was calculated between a certain radius R2 > R1 and Rmax. The surfaces from spheres with radii between R1 and R2 are completely left out. Some exemplary data for freely chosen radii R1 and R2 are summarized in table 4.2. The main

90 4.5. Echo Line Shapes result from this calculation is that the contribution from the particles at the magic angle on the outer spheres is signiﬁcant. As the case may be, it could even be dominating the spectrum at zero frequency from the geometric point of view.

Table 4.2: Ratios between outer and inner surface of an exemplary MLV

(100 spheres, Rmax = 3 µm) radius of radius of inner surface outer surface surface ratio inner MLV outer MLV

0 - R1 [µm] Sin [a.u.] R2 - Rmax [µm] Sout [a.u.] Sout/Sin 0 - 1 12.3 2.8 - 3 38.9 3.16 0 - 1 12.3 2.5 - 3 85.2 6.92 0 - 0.75 5.0 2.8 - 3 38.9 7.78 0 - 0.5 1.6 2.8 - 3 38.9 24.32

On Angular Dependent Spin-Spin Relaxation

The transverse relaxation time T2 in the MLV system is in principle strongly orientation dependent [28, 90–93]. When T2 is measured by the 1D spin echo method, the main orientation dependence of the relaxation rate can be described by the empiric relation [33, 86]:

−1 2 2 T2 = A + B sin θ cos θ (4.15) with A and B constant and θ the angle between the local director and the static

field. The degree of orientational dependence of T2 thereby depends on the degree of averaging over the θ-dependent part on the experimental time scale τNMR, √ during which the molecule diffuses a distance 4DτNMR (see section 4.4.1). For particles with a large rotational diffusion coefficient, equation (4.15) is able to describe the orientational dependence of T2 only for short time scales (say up to several ten µs) due to lateral diffusion [33]. For longer time scales as in the echo experiment, significant averaging over θ takes place and therefore removes the angular dependence. For the actual situation in the MLV it is therefore the case

91 4. Multilamellar Vesicles

that averaging over θ removes the angular dependence of T2 for the inner part, causes a vanishing signal due to slow motion in the middle part. Therefore an angular dependence of T2 is only present in the outer part. Equation (4.15) states that relaxation is slow at 0 and 90◦ and fast around 45◦ for the largest radii. To estimate the contribution from diﬀerent parts of the MLV with diﬀerent orientations on the spectrum, the orientation dependence of

T2 has to be weighted further with sin θ, which leads to vanishing contribution at 0◦ and leaves the 90◦ horns, as observed. The fast relaxation around 45◦ puts the impact of the outer spheres in a diﬀerent perspective. Although there are many particles at the outer spheres, they do not signiﬁcantly contribute to the spectrum at longer transverse evolution delays.

On Angular Dependent Spin-Lattice Relaxation

In oriented lipid bilayer systems no strong angular dependence of T1 was observed [94]. The same observation was made for unoriented powder-like samples of phospholipid bilayers [93], in which case this was attributed to rapid lateral diﬀusion, which averaged over all angular dependencies of T1. If T1 is independent of the director orientation, the spectrum S(ω) should have the same shape for all values of τ in the saturation recovery experiment [95]. This was indeed the case for all investigated MLV samples. It can be tested by use of the second moment of the spectrum, M2, which is deﬁned as:

Z ∞ −1 2 M2 = A S(ω)(ω − ω0) dω (4.16) −∞ where A is the area of the spectrum, ω0 is the Larmor frequency and S(ω) is the line shape [93]. If T1 is angular independent, then M2 should be constant for all values of τ. As it can be seen in figure 4.22 M2 varies insignificantly with τ, allowing an error of about 2 %. This means that the rapidly diffusing molecules do not show any angular dependency and that T1 relaxation must be attributed mainly to fast local motions and not to changes of the orientation due to diffusion.

92 4.5. Echo Line Shapes

1.0

0.8

0.6 2 M 0.4

0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 [s]

Figure 4.22: Second moment of T1 weighted spectra from MLVs, prepared at 10 s−1. No signiﬁcant variation with τ is observed.

The Stimulated Echo Line Shape

With the conclusion that angular dependencies do not play a role for T1 relaxation, the observed behavior of the central line in the stimulated echo must be explained in a diﬀerent manner. Three time periods are important for the ﬁnal echo: the two transverse evolution periods, denoted τ1(1) for the period between the excitation pulse and the second pulse and τ1(2) for the period between the third pulse end the stimulated echo maximum, where the acquisition is started.

The third time is the mixing time τ2 between the second and third pulses (see also the picture in 2.4). From the change of the line shape it might be concluded that a positional change has taken place during the mixing time, denoted τ2. The picture would then be as follows: the ﬁrst evolution time τ1(1) selects the central and outer spheres of the MLV, the medium spheres vanish due to the slow motion eﬀect (section 4.5.1) and do no longer contribute to any further echo. The same takes

93 4. Multilamellar Vesicles

place during the second evolution time τ1(2) (after the third pulse and before the echo acquisition). This means that only those molecules that were in the center or the outer spheres at the excitation pulse and stayed there for the whole sequence contribute to the stimulated echo. But now during the sequence, and especially during the mixing period τ2 diffusion through the layers might take place. This can happen in both radial directions. First, molecules from the center can move to the spheres with medium radius and vice versa. Both fractions do not contribute to the echo, since the ones originating from medium spheres were not stored with the second pulse and those from the center vanish during the second evolution time τ1(2). Second, molecules from the outer spheres can also travel to the medium positions (and vice versa) and vanish during the evolution periods, but they can also travel to the outer spheres of an adjacent MLV, where they contribute to the echo. This means that diffusion through the layers taking place during the sequence leads to a signal reduction of the central line. On the other hand this means that diffusion through the layers must play a significant role in the effort to find a description of the translational dynamics of water in the MLVs, since the timescales of the change in the lineshape and the translational diffusion measurements are comparable. PFG diffusion measurements (see section 4.6.1) −12 2 lead to a diffusion coefficient Dk < 10 m /s for diffusion through the layers. For a diffusion time of 30 ms the associated length is 240 nm. This corresponds to about 40 layers or 10 % of the radius of MLVs, that were prepared at 10 s−1. In other words it is not likely that the decay of the central line can be attributed solely to exchange between different shells. Therefore a different explanation for the line shape has to be taken into account instead. The strong reduction of the signal as function of the evolution delay τ1, see figure 4.19, gives rise to the assumption that the center line originates from only a very small fraction of the MLVs. The signal intensity of the solid echo at τ = 4 ms is only about 10 – 15 % of the single pulse intensity.

This means that motions with correlation times in the order of τ1 are present. In the usual calculation of the loss of signal intensity due to exchange in the stimulated echo the frequency is kept constant during the short evolution periods τ1 [87]. The signal intensity due to exchange is then only dominated by hcos(ω1τ1(1)) cos(ω2τ1(2))i, where the frequency changes only during τ2 from ω1

94 4.5. Echo Line Shapes

to ω2 [96–98]. The averaging is performed over all angles. In the MLVs this is not true as it can be seen in the solid echo. The rotational motion during both periods τ1 is relevant in addition to the motion during τ2. This means that the intensity at a certain frequency in the stimulated echo spectrum depends on the decay of at least two diﬀerent correlation functions. Furthermore these correlation functions are depending on complex distributions of correlation times over the radius of the MLVs. This makes it diﬃcult it to further comment or predict the stimulated echo line shape without proper simulations.

4.5.6 On Solid Echo Simulations

As mentioned in the previous section, it is necessary to simulate the echo line shapes in order to gain further information. Westlund presented an approach for heavy water line shapes and relaxation in bilayer systems [99]. The model derives a line shape function that describes the orientational dependence of D2O signal for studies of structure and dynamics of lipid-water interfaces in a phosphatidyl- choline bilayer system. The model comprises two chemically interchanging fractions of water namely, ’free’, and ’bound’. There are four molecular parameters characterizing the ’bound’ water of the lipid water interface, namely, (1) the fraction of water P d molecules bound to lipid molecules, (2) the local water order parameter S0 , dD (3) the order parameter S0 which describes the averaged ’bound’ water, and

(4) an eﬀective correlation time τc, characterizing water translational diﬀusion at the interface. This model allows for analyzing quadrupole splittings, spin-lattice relaxation rates, and water powder line shapes. Thus, dynamics as well as structural information about the water molecules residing in the water lipid interface may be extracted [100, 101]. An adaption of this model to the MLV solid echo line shape analysis was started in a collaboration [102]. This would then be a test to what extent the above considerations are correct. First results of the simulations were able to reproduce the broad single line of the one pulse experiment but not the solid echo line shapes. This will be subject of further work.

95 4. Multilamellar Vesicles

4.6 Diﬀusion measurements

4.6.1 Diﬀusion in the Oriented Lα Phase

D2O-diﬀusion was measured in the magnetically oriented Lα phase of 40 wt-%

C10E3/D2O. The oriented lamellae were obtained by cooling from the isotropic phase. Very slow cooling (∼ 0.2 K/min) and annealing in the two-phase region is necessary to obtain a homogeneous orientation. Since C10E3 orients with its long molecular axis perpendicular to the magnetic field, a two dimensional powder distribution of layer normals perpendicular to B0 results. Measuring diffusion along the lamellae is therefore easily possible by applying a z-gradient and fitting of the resulting echo decay with equation (2.33), which yields D⊥. Measurements of the coefficients of diffusion across the lamellae, Dk, on the other hand is only possible in an indirect way. Diffusion measurements in any direction perpendicular to B0 can be analyzed by the diffusion equation for two-dimensional powder distributions [21, 103]:

Z 2π 2 S(g, δ, ∆) = Sg=0 exp(−D⊥k) exp(−[Dk − D⊥] sin θ)dθ (4.17) 0

Fitting of the experimental echo decay curves yielded Dk and D⊥ simultaneously.

The results for Dk using equation (4.17) with both Dk and D⊥ as adjustable parameters are, within the experimental error, the same as with D⊥ taken from the separate diffusion experiment, and only Dk as the fitting parameter. Tem- perature dependent measurements of the diffusion were performed between 298 – 321 K. Measurements with the gradient perpendicular to the magnetic field yielded simultaneously the diffusion coefficients Dk and D⊥ (figures 4.24). At temperatures below 306 K the fit with both parameters varying freely became −12 2 unstable, as Dk was below 10 m /s. Below this temperature the best result was obtained by setting Dk equal to zero. An Arrhenius analysis (see appendix B.3) yielded activation energies of 16 kJ/mol perpendicular to the layer normal, which is comparable to the value of 18.8 kJ/mol for free water [57]. The Arrhenius activation energy parallel to the layer normal was 157 kJ/mol, which indicates that diffusion through the lamellae

96 4.6. Diﬀusion measurements

0 D D and D

-1 ln S [a.u.] S ln -2

-3 0.0 5.0x108 1.0x109 1.5x109 2.0x109 2.5x109 3.0x109

g2 * k [s/m2]

Figure 4.23: Example for the PGSTE echo decay in the Lα phase with

gradients in z- and x-direction. D⊥ (z-gradient) is obtained from the

slope. Dk (x-gradient) has to be derived with equation (4.17). is strongly hindered. The lamellae appear to be free of defects, as it is also seen from the anisotropy of the diﬀusion. The anisotropy, deﬁned as the ratio D⊥/Dk varies from 500 ± 200 at 306 K to 70 ± 5 at 318 K. The large error at low temperatures is due to the low value of D⊥.

The Two-Site Model

The repeat distance of the Lα phase as determined by x-ray diﬀraction is 65 A˚ [27].

In the all trans conformation C10E3 has an overall length of 24 A,˚ about 12 A˚ for the hydrophilic and hydrophobic part each. This results in a minimal gap of 17 A˚ between the ethylene oxide (EO) units (assuming all trans conformation) that can be occupied by free water. With the assumption of 1.5 bound water molecules per EO unit, the mixture of 40 wt.% C10E3/D2O contains about 20 mol-% of bound water, thus xfree = 0.8. But equation 3.3 from section 3.5.1,

97 4. Multilamellar Vesicles

1.6x10-9

1.5x10-9

1.4x10-9 /s] 2 [m

D 1.3x10-9

1.2x10-9

304 306 308 310 312 314 316 318 T [K] 2.0x10-11

1.5x10-11

1.0x10-11 /s] 2 [m D

5.0x10-12

0.0

304 306 308 310 312 314 316 318 T [K]

Figure 4.24: Temperature dependent diﬀusion coeﬃcients parallel and

perpendicular to the layer normal for 40 wt.% C10E3/D2O.

98 4.6. Diffusion measurements where an apparent average diffusion coefficient was calculated for the assumption of free and bound water, is not solvable with this molar fraction, like it was the case for the hydrogels. The measured diffusion along the layers is too slow. With the assumption that the bound water does not move at all, the maximal fraction of free water would have to be 56 mol-%. Again undulations or membrane defects are possible explanations. Since the lamellae are defect-free, undulations are a reasonable explanation for the low diffusivity along the lamellae. Assum- −12 ing that xfree = 0.8 and that the bound water diffuses more slowly than 10 m2/s (at 298 K) the theoretical diffusion coefficient, according to the two-site −9 2 model, is about Dfree = 1.5 · 10 m /s. The measured diffusion coefficient D⊥ is 1.05 ·10−9 m2/s, which means that diffusion is slowed down by about 30 % compared to bulk water. It should be mentioned that the simple two-site model does not include water, which interacts weakly with the EO units. This could also explain the reduced diffusivity along the membrane.

4.6.2 The PGSTE Echo Decay in MLVs

Translational diffusion in MLVs was measured with the pulsed gradient stimulated echo sequence (PGSTE). The sequence was chosen in order to have the ability to independently vary two delays. During the delay between the first two pulses T2 relaxation takes place. The duration of this delay governs the line shape of the final spectrum. Included in this delay is also the encoding gradient pulse. The second, usually longer, delay between the second and the third pulse controls the diffusion time ∆. It is not possible to perform such experiments with the PGSE sequence, since the line shape changes in this case with the diffusion time. Measurements were performed on MLV samples prepared at different shear rates and with different diffusion times. All measurements were performed with the PGSTE sequence and the short evolution delay was mostly chosen to be minimal, the limit given by the gradient duration necessary to appropriately attenuate the signal. This resulted in a line shape where the splitting already occurs as it can be seen in figure 4.25. This is important because it has to be kept in mind for the analysis of the echo decays, since only certain regions (radii) of the MLVs (see section 4.5.1)

99 4. Multilamellar Vesicles contribute to the signal. A typical set of echo decay curves is given in ﬁgure 4.26.

-2000 -1000 0 1000 2000 Frequency [Hz]

Figure 4.25: Spectrum of a typical PGSTE diﬀusion measurements, echo decays were obtained by integration over the whole spectral width.

The logarithm of the echo intensity (from integrating over the whole spectrum) is plotted as a function of g2k with k = γ2δ2(∆ − δ/3). The parameters δ and ∆ are kept constant in the experiments. In such a diagram a Gaussian diffusion behavior leads to a straight line, independent of ∆. Faster diffusion would be seen as a steeper decay. As an example the diffusion echo decay perpendicular to the layer normal in the Lα phase is plotted for ∆ = 202 ms. For the aligned

Lα phase the echo decay curves with different diffusion times match within the experimental error, indicating Gaussian diffusion behavior. The PGSTE echo decay in the MLV system is completely different. The data depend strongly on the diffusion time. Furthermore the graphs are curved. This already indicates anomalous, hindered diffusion. On the one hand this is not too surprising, since the known structure of the samples lets one expect such an effect. Nevertheless it is not evident that this effect is visible in our experiment. This becomes clearer when the diffusion time scale is extended and when averaging effects begin to

100 4.6. Diﬀusion measurements

1 [ms] ------0.8 10.1 15.1 25.1 0.6 40.1

) L , , 0.4 S(g,

0.2 0.0 5.0x109 1.0x1010 1.5x1010 2.0x1010 g² k [s/m²]

Figure 4.26: PGSTE echo decay for MLVs prepared at 10 s−1 for different diffusion times ∆. play a role, as it can be seen in the following. In figure 4.27 (top) the data set of another sample, also prepared at 10 s−1, is given. Here the range of diffusion times was extended to the T1-limit. For long diffusion times the echo decays become less and less curved and eventually seem to reach a limit (here from about 100 ms on), where a more or less straight line is measured. This second data set is presented in a different way in figure 4.27 (bottom). Here the axes are both linear and the intensity is given as a function of q = γgδ. In such a representation Gaussian diffusion looks like a Gaussian curve. An example is the curve of the Lα phase. The echo decay of the Lα phase in this plot is a function of the diffusion time. The longer the diffusion time is the steeper is the echo decay; therefore a long diffusion time of ∆ = 202 ms was chosen in order to avoid a superposition of the curves. The differences between the echo decays for the MLVs are more pronounced here, especially at long diffusion times. In figure 4.27 (top) the long time limit seemed already reached at about 100 ms while in figure 4.27 (bottom) it becomes

101 4. Multilamellar Vesicles

1 [ms]

10.1 55.1 15.1 105.1 25.1 305.1 40.1 L ) S(g,

0.1

0 1x1010 2x1010 3x1010 4x1010

g2 k [m2/s] [ms] 1.0 10.1 55.1 15.1 105.1 25.1 305.1 0.8 40.1 L (202)

0.6 S(q) 0.4

0.2

0.0 0 200 400 600 800

q [mm-1]

Figure 4.27: PGSTE echo decay for MLVs prepared at 10 s−1 for different diffusion times ∆. (a) Plotted against g2k, such that the slope yields directly the diffusion coefficient. (b) Plotted against the ’scattering vector’ q, the solid line is a Gaussian fit to the data for ∆ = 305.1 ms.

102 4.6. Diffusion measurements clear that this limit is not reached until at least 300 ms. Nevertheless from about 100 ms on the echo decay starts to develop a Gaussian shape. At longer times ∆ the averaging over the texture of the MLV state becomes more pronounced. In the long time limit the diffusion does no longer reflect the more local properties of the curved bilayer spheres, instead the isotropic nature of the whole sample becomes visible in the Gaussian shape of the echo attenuation curve. A Gaussian fit to the curve at 300 ms yields an apparent diffusion coefficient of 1.26 (± 0.04) ·10−10 m2/s.

Size Dependence

MLVs of different sizes were prepared according to the standard procedure. Ap- plication of higher shear rates during preparation led to smaller MLVs. In figure 4.28 the PGSTE echo decays for three different shear rates of 2, 5 and 10 s−1 are plotted as a function of q, each for a diffusion time of ∆ = 10 ms. The size of the MLVs is 2.5 µm for the shear rate of 10 s−1, 4 µm for 5 s−1 and 6 to 8 µm for 2 s−1 (due to polydispersity). The shape of the curves is similar in all three cases, whereas the echo decay is faster for the smaller MLVs. Additional time dependent diffusion data for MLVs, prepared at 2 and 5 s−1 are given in appendix B.2.

103 4. Multilamellar Vesicles

-1 1.0 shear rate [s ] 2 5 0.9 10

0.8

0.7 S(q)

0.6

0.5

0.4 0 200 400 600 800 1000

q [mm-1]

Figure 4.28: PGSTE echo decay for shear rates of 2, 5 and 10 s−1 at a diﬀusion time of 10 ms.

104 4.6. PGSTE Echo Decay Model

4.7 Towards a Model for the PGSTE Echo Decay in MLVs

A quantitative description of the translational diffusion of water in a system consisting of multilamellar vesicles is very complicated. There are numerous possible ways for the displacements of the diffusing water molecules with different restrictions and varying probabilities for the respective steps. First, the molecules diffuse along the bilayers on the surface of each sphere of one MLV. Second, they can pass through the lamellae although at a very low rate (see the diffusion in the pure Lα phase) and, third, they can jump from one vesicle to another. From pulsed gradient NMR all that is accessible is the mean squared displacement hz2i along the axis of the gradient as a function of the diffusion time. hz2i depends on the constant diffusion coefficient, the diffusion time, an experimental parameter, and the particular structure. It is the structure that sets the boundary conditions for the diffusing particle and defines the possible pathways along which it may travel. This is the part where modelling is necessary in order to learn about the particular system. It is necessary to develop all these steps with respect to the particular experiment. It should again be emphasized that only projections (on the gradient direction) of the mean square displacement are observable in gradient NMR as it is sketched in figure 4.29. The principal steps for the development of such a model are the following:

• Diﬀusion on a spherical surface of radius R

• Diﬀusion in one MLV (without jumps between spheres, i. e. Dk = 0), approximated as concentric multispheres

• Jumps between diﬀerent concentric spheres of radii R < Rmax ; D⊥ Dk

• Jumps between vesicles

• Inclusion of deviations from ideal geometry

Furthermore it has to be taken into account that the measured echo attenuation might further depend on relaxation. The PGSTE design ensures that relaxation does not play a role during the course of one experiment since all delays are kept

105 4. Multilamellar Vesicles

Z 2

Figure 4.29: Example for a possible pathway (···) and the net displacement (→) along a preferred axis (the gradient direction) of a water molecule in the MLV phase. constant. But when the diffusion time, and therewith a delay, is changed between experiments, relaxation effects might play a role in the echo attenuation. Namely if loss of quadrupolar (or dipolar) correlation due to motion would contribute to the decay of the signal, the situation becomes very complicated [104]. It is therefore necessary to check if the echo intensities really depend only on the translational displacement. The movement of the MLVs as a whole can be safely neglected, since the viscosity is very high [74, 82]. In the following an attempt is made to first describe the problems associated with the geometry and the particular pulsed gradient experiment and then to discuss a simple model for translational diffusion in the spherical geometry.

4.7.1 Phenomenological Description of the Echo Decay on One Sphere

The considerations are started with a visualization of the speciﬁc problem of pro- jecting the diﬀusion path onto the gradient axis. First, only those displacements

106 4.7. PGSTE Echo Decay Model that have a component in the direction of the gradient axis contribute to the echo attenuation. The movements perpendicular to the gradient axis (along a latitude of a sphere) do not lead to a phase change. For convenience in the following considerations the MLV is defined in such a way that the axis from the center to the north pole is chosen to be parallel to the gradient axis, the z-axis. The first step for the model is to go from the 1-dimensional visualization for diffusion on a sphere (see figure 4.10) into two dimensions. Starting from the north pole (θ = 0) the molecule will have a displacement according to equation (4.9) in all directions of the azimuthal angle φ. In figure 4.30 this function is plotted. It is constructed from a simple rotation of equation (4.9) around the z-axis. The sphere as the basic plane is projected on a planar surface. In this case the movement along θ for any φ contributes equally to the echo attenuation.

Gradientaxis (a) (b) j q

Figure 4.30: Two dimensional angular distribution function for diﬀusion on a sphere. The diﬀusing particle starting from the north pole (θ = 0). (a) Top view onto the north pole / the gradient axis. (b) Qualitative 3-D sketch.

The other extreme situation is a particle starting from the equator. Displace- ment along the equator is not seen, since no phase change is connected with this movement. A particle diﬀusing only towards the north pole would have a probability equal to the angular distribution function from equation (4.9) but with the

107 4. Multilamellar Vesicles angle θ shifted by 90◦. The two-dimensional distribution function for the particle starting from the equator must therefore be a combination of figure 4.30 with a cut through the equator as it is sketched in figure 4.31. This cut reflects the fact that the signal is not attenuated, when the particle moves along the equator.

Figure 4.31: Two-dimensional distribution function for diﬀusion on a sphere measured in a PFG experiment. The diﬀusing particle starting from a point on the equator. (a) Side view on the equator of the sphere. (b) Qualitative 3-D sketch.

The situation for one sphere as a whole can therefore be described as follows: molecules start to diffuse at positions all over the sphere. The encoding gradient then cuts the sphere to slices of equal height, labelling them with the well defined phase shift (see section 2.3.1). This means that all molecules within one slice are treated the same. The effect on the echo intensity depends on the relative amount of molecules in each slide. This amount is given in first approximation by the surface area S on a sphere with radius R (see section 4.5.1), which is:

S = 2πRh (4.18)

The thickness of the slice is given by h and is directly proportional to the gradient

108 4.7. PGSTE Echo Decay Model strength g. For the two situations given in ﬁgure 4.32 this means that molecules start at either the pole cap or from within an equatorial band. The probability of displacement in each direction on the sphere is equal, yet the molecules on the equator have a higher probability to stay within their bands. For molecules on the pole cap it is more likely to change their z-coordinate and, therefore, their signal is more strongly attenuated in the pulsed gradient experiment. Recapitulating, this means that the closer the initial position of the molecules is towards the poles, the more their signal is attenuated. Two conclusions from these considerations

Figure 4.32: Surface areas on a sphere. The surface area depends only on h and not on the position. can be drawn concerning probabilities. First, a longer diffusion time leads to a stronger echo attenuation, since it is more likely that eventually all molecules have left their initial z-coordinate. Second, at a given diffusion time spheres with larger radii yield less attenuated signals than spheres with smaller radii. This is because the curvature at the pole caps is smaller for larger radii. This explains the observed behavior, namely that the echo is more strongly attenuated for longer diffusion times and diffusion on smaller MLVs, in a qualitative way.

4.7.2 Simulated Echo Decay for One Ideal MLV

The phenomenological description of the echo decay can only serve to make the measured data plausible. But in order to extract any information on the structure of the MLV state, a model must include the mathematical description of the assumed properties of the system. As a ﬁrst step the approximation of the MLVs

109 4. Multilamellar Vesicles as concentric multispheres is suitable. The NMR signal could then be constructed from the description of diﬀusion on one sphere [105]. In order to calculate the NMR signal in a pulsed gradient experiment, it is necessary to evaluate the conditional probability function (equation (2.3), section 2.1.2). According to Yosida [106], the conditional probability for diﬀusion on a sphere is given by:

∞ k 0 0 X X m m 0 0 P (t, θ, φ, θ , φ ) = exp(−k(k + 1)t)Yk (θ, φ)Yk (θ , φ ) (4.19) k=0 m=−k

m where Yk (θ, φ) are the spherical harmonics. This gives the probability that a particle starting at (θ, φ) moves to (θ0, φ0) during t. In the PGSTE experiment the φ-dependent part is not relevant. Including the radius and the diffusion coefficient eventually leads to the PGSTE echo decay Ssph(q, ∆) for the diffusion of particles on the surface of one sphere of radius R, calculated with the short gradient pulse approximation (see section 2.3.2):

∞ X Dtrans S (q, ∆) = (2l + 1) j2(q) exp(−l(l + 1) · ∆) (4.20) sph l R2 l=0

The jl(q) are the spherical Bessel functions of order l, which depend on the phase encoding by the gradient: q = γgδ. ∆ is the diffusion time and has to be corrected for finite gradient pulse length. For integer l the spherical Bessel functions are defined by: r π j (q) = J (q) (4.21) l 2q l+(1/2) with the normal Bessel functions Jl+(1/2)(q). Bessel functions of the first kind

Jl(q) are deﬁned as the solution to the Bessel diﬀerential equation:

d2y dy q2 + q + (q2 − l2)y = 0 (4.22) dq2 dq

This equation was first defined by Friedrich Wilhelm Bessel (1784-1846) in 1817 for the problem of determining the motion of three bodies moving under mutual gravitation. Bessel functions appear as coefficients in the series expansion of the indirect perturbation of a planet, that is the motion caused by the motion of the sun caused by the perturbing body. The Bessel functions can be evaluated by

110 4.7. PGSTE Echo Decay Model numerical methods (Appendix C.1). Equation (4.20) gives the echo decay for diffusion on one sphere of radius R as a function of the gradient parameters and the diffusion time. In the simulations the parameter D∆/R2 was varied. Since S(q, ∆) depends only on the ratio D∆/R2, the same effect can be reached by either varying D, ∆ or R. It should be emphasized that a low value of D∆/R2 corresponds to large values of R. The limit of equation (4.20) for very long times ∆, which corresponds to an equal dis- th 2 tribution over the sphere, is the square of the Bessel function of 0 order, j0 (q). This equality is obtained in the simulation (appendix C.1). In figure 4.33 a plot of equation (4.20), normalized to one, is given for different values of D∆/R2. The signal is decaying faster for spheres with a higher value of D∆/R2. The ordinate is in units of reciprocal length and the number was chosen to be identified with the q from the experimental data.

D 1.0 R2

10-3 -5 0.8 10 2.5 10-6 10-6 0.6 ) [a.u.] ) ,

, 0.4 S(g,

0.2

0.0 0 200 400 600 800 1000 q [a.u.]

Figure 4.33: Echo decay simulation following equation (4.20) for four different values of D∆/R2.

In order to simulate the echo decay of a full MLV the sum over all radii from the inner core to the outer shell was taken. Surface weighting with 4πR2

111 4. Multilamellar Vesicles was applied to account for the diﬀerent amount of molecules in each sphere that contribute to the full echo decay. This leads to the full equation for the PGSTE echo decay in an ideal MLV:

Rmax ∞ X X Dtrans S (q, ∆) = 4πR2 (2l + 1) j2(q) exp −l(l + 1) · ∆ (4.23) MLV l R2 R=Rmin l=0

In figure 4.34 the curves resulting from equation (4.20) and equation (4.23) are compared. The sum over the radii in equation (4.23) was taken from R ' 0 (D∆/R2 is large) to an upper limit (D∆/R2 is small). This upper limit of R is taken into equation (4.20) to simulate the echo decay of the corresponding outer radius of the MLV. This was done for two different maximal radii. The echo decays coincide of course at low and very high values of g, since all curves are normalized and since the asymptotic limit is zero for high values of q. The intermediate shape is different: the simulated decay of the MLV according to equation (4.23) is faster. The comparison with experiment is not so straightforward. The measured echo decays arise from the integration over a spectrum, that is affected by loss of correlation due to motion during τ1 and τ2 and does not contain the intensity from all shells (see figure 4.25). According to the hypothesis in section 4.5.1 the signal from intermediate shells contributes the least. Therefore additional weighting factors in the first sum over R in equation (4.23) must be added. But it is not possible to define these weighting factors before the line shape is fully understood. In figure 4.35 some experimental data from MLV (prepared at 10 s−1) are compared with the simulations. The dash-dotted line (− · −) corresponds to an MLV with a maximum radius R1,max that is too small to describe the data. The dotted line (··· ) is another simulation according to equation (4.23) but with a bigger radius R2,max > R1,max. This curve decays too slowly. Both lines are summed up from the same minimal radius. It is conceivable that a sum of both curves in figure 4.35, which leaves some intermediate shells out could lead to a curve that resembles the data. There is unfortunately a big discrepancy in all the above considerations. In- serting the correct values for the diffusion coefficient and the diffusion time in

112 4.7. PGSTE Echo Decay Model

D R2 1.0 10-6 R max,1 10-3-10-6 MLV 0.8 1 10-5 R max,2 10-3-10-5 MLV 0.6 2 ) [a.u.] ) ,

, 0.4 S(g,

0.2

0.0 0 200 400 600 800 1000 q [a.u.]

Figure 4.34: Echo decay simulation following equation (4.23). Compar- ison of the surface weighted sum of many radii (i. e. an MLV) and the

corresponding decay of the maximal outer sphere Rmax for two different minimal values of D∆/R2. equation (4.23), as it was done in the simulations, makes it necessary to choose the radii of the MLVs in the range of millimeters to get a decay on the order of the measured data. This is about three orders of magnitude too large. Al- ternatively the apparent diffusion coefficient D would have to be of the order of 10−15 m2/s, which is 6 orders of magnitude smaller than the measured value of D(D2O). A solution to that problem could not be found so far. This could lead to the idea that in fact the model is wrong, but nevertheless the principal agreement between the shape of the simulated curves and the experimental data is observed. The simulations showed a faster echo decay with less curvature for a larger value of D∆/R2, that means for long diffusion times ∆, which corresponds to the experiment as it is presented in figure 4.27 (bottom) and also for smaller outer radii of the MLVs as the experiment in figure 4.28 shows. So the observed experimental trends are mirrored by the simulation. This is on the other hand

113 4. Multilamellar Vesicles

1.0 D R2

-3 -6 0.8 10 -10 10-3-10-5 experimental =25 ms 0.6 ) [a.u.] ) ,

, 0.4 S(g,

0.2

0.0 0 200 400 600 800 1000 g [G/cm]

Figure 4.35: Comparison of simulations according to equation (4.23) with exemplary experimental data from MLVs, prepared at 10 s−1. quite surprising since, although the calculations are complicated, the model itself is very simple. It only satisfies the first two points of the listing at the beginning of section 4.7 and one cannot expect that this should already describe reality. Especially the effect of diffusion through the layers is not included, which is of at least some relevance, as discussed in section 4.5.

114 4.7. PGSTE Echo Decay Model

4.7.3 Eﬀect of Finite Gradient Duration: ”Center of Mass”- Diﬀusion

There is a significant drawback of all pulsed field gradient methods. They all rely on a finite gradient encoding time in the real experiment. During this time evolution changes the line shape, so that diffusion measurements on the one pulse line shape are impossible. On the other hand the diffusion along the lamellae is fast compared to the gradient duration. During a typical gradient duration δ of 1.5 ms the water is travelling a distance in the order of: √ x = 2 · D · ∆ = p2 · 1.0 10−9 m2/s · 1.5 10−3 s = 1.7 µm (4.24)

This means that the orientation of the molecules is already averaged over a significant part of the MLV. Assuming a typical radius of about 2.5 µm it follows that even on the outermost sphere an average over about 40◦ occurs. From that follows that all further considerations towards a model for the PGSTE echo decay have to deal with the so called ”center of mass” diffusion [107–109]. The phase encoding no longer depends on the single point-like water molecule diffusing around the sphere, but on the center of mass of the volume, in which the particle diffuses during gradient encoding. This volume might be a complicated object. In first approximation it is something like a cap that covers the surface of the sphere to a certain degree, depending on the initial distance of the particle from the center of the sphere and the diffusion coefficient. With the assumption that the particle diffuses also through the layers to a certain degree, the above cap gets an additional spatial extension in the radial direction and becomes a lens-like object.

115 4. Multilamellar Vesicles

4.8 Discussion: MLV

The dynamics of D2O in the multilamellar vesicle phase of C10E3/D2O was investigated by means of 2H NMR line shape analysis and pulsed gradient diffusion NMR. It could be shown that a relationship between MLV size and the single pulse line shape exists. Such a relationship was deduced from simplified scaling relations and was described by the introduction of a theoretical factor that related the dynamic element of surface coverage by the diffusing D2O, represented by the line width, with the static property of the radius. Furthermore the inhomogeneity of the single pulse line shape could be shown by solid and stimulated echo measurements, where a strong dependence of the echo line shape on the transverse evolution delays was observed. At long evolution delays a splitting occurred which was identified with the remaining 90◦ singularity of a motionally averaged Pake spectrum. Line shapes from stimulated echo experiments showed that diffusion through the layers plays a minor role. Instead the stimulated echo line shape is dominated by rotational motions with correlation times in the order of the delays τ1 and τ2 in the pulse sequence. Therefore the line shape originates from a complex superposition of correlation functions that depend on both delays. Further analysis including simulations will be necessary. Pulsed gradient diffusion measurements showed a strong dependence of the echo decay on the diffusion time. This anomalous diffusion behavior was attributed to the restricted diffusion along the spherically bent lamellae in the MLVs. A theoretical model for diffusion in an idealized MLV structure was tested for the ability to describe the measured echo decay. Although a principal agreement between the shape of the experimental echo decay and the model could be observed, the simulation parameters differed by several orders of magnitude from the experiment. The differences were mainly attributed to three factors, which were not taken into account in the derivation of the echo decay curve: first, the incomplete description of the dependence of the line shape on transverse evolution, second, the invalidity of the model for finite gradient duration, since it was calculated in the short gradient approximation, and third, the diffusion through the layers.

116 4.8. Discussion: MLV

This means that a correct model for the complete description of the PGSTE echo decays further has to take into account the ’center of mass’ diffusion due to finite gradient duration, the different weighting of different spheres in the MLVs due to the finite transverse evolution delay that changes the one pulse spectrum, and in the end the overall picture of the MLVs as concentric multispheres. This means that before one could reasonably proceed with a refinement of the model for the echo decay, it is absolutely necessary to take the knowledge of the line shape into account.

117

Chapter 5

Swelling of PDMS Model Networks

As mentioned in the introduction (section 1.4) new theories about heterogeneities in swollen polymer networks have been proposed [16–19]. There are different ways to proof the existence of such heterogeneities by means of NMR diffusometry. They all rely on deviations from Gaussian diffusion behavior. First this could be a direct deviation from the Gaussian shape of the echo decay in a single pulsed gradient experiment. Second, anomalous diffusion could show up as a dependence of an apparent diffusion coefficient on the diffusion time. The diffusion coefficient decreases with increasing diffusion time, reaching eventually a constant value for long times. In this case it is necessary to change the degree of swelling and to ’adjust’ thereby the heterogeneities. ’Adjusting’ means that the diffusivity contrast between regions of different crosslinking density is maximized. The contrast should be more pronounced at lower degrees of swelling. A third way to observe obstructed diffusion is to try to compare networks with different chains lengths and also mixtures of chain lengths. By choosing deliberately a heterogeneous chemical composition, such bimodal networks provide a test for the applicability of the method.

119 5. Swelling of PDMS Model Networks

5.1 Sample preparation

Experiments are carried out on mono- and bimodal poly(dimethylsiloxane) (PDMS) networks prepared from linear hydroxyl-terminated precursor polymers (Gelest, Inc.) using tetraethoxysilane as crosslinker [110]. The synthesis and characterization of the samples is described in [111], and key properties are summarized in Table 5.1. The samples of dry network used in this work were provided by the group of A. Vidal in Mulhouse [111], who also removed the sol.

Table 5.1: Investigated mono- and bimodal PDMS model networks com-

posed of long (Mn = 47 200 g/mol, PD = 1.64) and short (Mn = 5200 g/mol, PD = 1.86) chains.

a b wt.% sol Qeq net47k 7 2.9 net5k 4.2 2.4 netb30c 5 2.2

a Determined by swelling experiments in toluene. b The equilibrium degree of swelling in octane. c The bimodal network, netb30, consists of 70 wt.% long and 30 wt.% short chains.

The networks are characterized by a very good conversion of precursor chain ends, yielding very low sol contents and low amounts of dangling chains. In this work, the samples of approximately (1 × 1 × 3) mm3 were cut from dry PDMS networks with a scalpel and swollen in octane, which was chosen due to its low volatility. Diﬀerent protocols were tested, in order to gently swell the networks and not to introduce microscopic cracks by rapid swelling. The protocols were direct swelling in liquid octane, swelling in octane vapor, and gradual swelling in diﬀerent mixtures of ethanol and octane. The gradual swelling method relies on the fact, that ethanol is a bad solvent for PDMS. The networks were successively put into mixtures of ethanol/octane with ratios of 3/1, 1/1, 1/3 and pure octane in the end. The samples were swollen for about 12 – 16 hours in each solution in order to let the system equilibrate. Both methods of vapor and gradual swelling were found to work equally well, yet the equilibrium degree of swelling, that could be reached with the vapor method was lower. In order to

120 5.2. Single Pulse Spectra

reach a speciﬁc degree of swelling, Qw (deﬁned as the mass ratio mswollen/mdry =

1 + moctane/mpdms), which should not be mixed up with the obstruction factor Q, the swollen samples were placed on a balance at room temperature and the octane was allowed to evaporate. When the desired weight was reached after several minutes, the samples were transferred into 4 mm magic-angle spinning rotors and closed with KelF caps, thus establishing a swelling equilibrium with the solvent vapor pressure. The tightness of the caps is checked regularly by measuring potential weight loss. No loss of weight was ever observed for several weeks. It was carefully avoided that the swollen samples exceeded the volume of the rotors, thereby compressing the networks. The degree of volume swelling, denoted as

QV , is in ﬁrst approximation linearly related to Qw − 1 = moctane/mpdms through simple additivity of volumes: QV = 1+(ρpdms/ρoctane)(Qw −1) = 1+1.4(Qw −1).

5.2 Single Pulse Spectra

Figure 5.1 shows a typical proton spectrum of the highly swollen sample f2 (see appendix B.4) with QNMR = 2.92. The two narrow peaks originate from the mobile octane protons, the broader peak to the right originates from the network.

The network peak disappears at long evolution delays due to its faster T2 relaxation. The degree of swelling can be determined by deconvolution of the peaks and is calculated from the integral intensities I as follows:

m I pdms = 3 · 0.65 · pdms (5.1) moctane NMR Ioctane with Mw(pdms)/Mw(octane) = 0.65 and moctane QNMR = 1 + (5.2) mpdms NMR

Both Qw, determined during the sample preparation by weighing, and QNMR were usually in good (± 10 %) agreement.

121 5. Swelling of PDMS Model Networks

- 4 0 0 0 - 2 0 0 0 0 2 0 0 0 4 0 0 0 F r e q u e n c y [ H z ]

Figure 5.1: Typical single pulse spectrum of a swollen PDMS network.

Sample f2 (QNMR = 2.92.)

5.3 Diﬀusion measurements

In order to measure the effect of network inhomogeneities, diffusion time dependent pulsed field gradient (PFG) measurements were performed on various samples with different degrees of swelling and different chemical composition (see appendix B.4 and table 5.1). The apparent diffusion coefficients Dapp were obtained by fitting the integral intensity of the whole spectrum with equation 2.33, including an offset which accounts for the signal originating from the immobile network protons. The diffusion coefficient is called ’apparent’, because of the possible dependence on the diffusion time. An example is given in figure 5.2. No deviation from Gaussian diffusion, in particular no weaker attenuation at stronger gradients, was observed. It should be mentioned, that such a deviation could also arise from the immobile network protons. But when the delays in the pulse sequence where longer than about 5 – 8 ms, the network protons were completely

122 5.3. Diffusion measurements relaxed and did not show up in the echo signal. The fact that the echo decay could be fitted with a Gaussian curve means that there is no direct evidence for obstructed diffusion. The non-linear least square fitting error of Dapp was usually of the order of 2 – 3 %.

1.0

0.8

0.6 0 S S 0.4

0.2

0.0 0 10 20 30 40 50 60 70 80 g [G/cm]

Figure 5.2: Exemplary fit of the normalized experimental data for the sample net47k(4) for a diffusion time ∆ = 28 ms. The solid line is a fit according to equation 2.33, including an offset.

Although no direct evidence for obstructed diffusion was observed, a dependency of the apparent diffusion coefficient on the diffusion time would reveal such obstruction effects. Selected results of PGSE and PGSTE experiments for three different degrees of swelling of the network net47k are presented in figure 5.3. The diffusion coefficients are plotted as a function of the corresponding length √ scale D∆. The error bars are within the width of the points. For the PGSE experiments the error increases at longer diffusion times due to a worse signal to noise ratio. Samples net47k(4) and net47k(7) were prepared at approximately the same

123 5. Swelling of PDMS Model Networks

degree of swelling (Qw ' 2), the difference is that net47k(4) was swollen with normal (protonated) octane while net47k(7) was swollen with (deuterated) octane- d18. The degree of swelling of net47k(3) was a little lower, Qw = 1.32. In the PGSE-experiment, the sample net47k(4) and net47k(3) show a decrease of the apparent diffusion coefficient Dapp with the diffused distance. From such an observation one could conclude that indeed diffusion is obstructed. The values of Dapp for net47k(3) are overall lower, which is reasonable for the sample with the lower degree of swelling. The observed behavior in the PGSTE-experiment is yet completely different. With this experiment, which differs only in the z-storage period, no dependence on the diffused distance is observed. Even at very long diffusion times no depression in Dapp is observed. Dapp stays constant, the statistical scattering of the data in the PGSTE experiments at different diffusion times being of the order of 5 – 8 %. The constant plateau value of Dapp in the PGSTE experiment is comparable to the value that is obtained in the PGSE-experiment at very short diffusion times (data not shown for sample net47k(3)). This result is most irritating, since two different results are obtained, when the PFG diffusion measurement is performed with the two different methods of PGSE and PGSTE. An even more irritating result is that the sample net47k(7) with the deuterated solvent showed an increase of Dapp with the diffusion time. An acceleration of the diffusing particle has no physical meaning. On the other hand Dapp stays also constant in the PGSTE experiment at approximately the value of Dapp of sample net47k(4). This is reasonable since the two samples are swollen to about the same degree. No explanation is found for the strange behavior in the PGSE experiments. √ This indicates that the measured dependence of Dapp on D∆ in the PGSE experiment is an artifact. The origin of this artifact is yet unclear. The PGSTE experiment was therefore regarded to be more reliable in determining a possible ∆ dependence of Dapp. All other investigations on the different samples concerning this ∆ dependence were therefore carried out with PGSTE. It was checked for all diffusion measurements on the samples if this ∆ dependence occurred, but the observed Dapp remained always constant. The statistical aver-

124 5.3. Diﬀusion measurements

net47k(3) PGSE Q = 1.32 NMR 3x10-9 net47k(4) PGSE Q = 1.96 net47k(4) PGSTE NMR net47k(7) (2H) PGSE Q = 2.12 net47k(7) (2H) PGSTE NMR

2x10-9 /s] 2 [m app D

1x10-9

0 0 2x10-5 4x10-5 6x10-5 D [m]

Figure 5.3: Diffusion coefficients as a function of the corresponding length scale for three degrees of swelling of the network net47k. Different methods lead to different results!

age of Dapp over the different diffusion times was taken as the the representative value of Dapp for each sample. Because the statistical scattering was found to be larger by a factor of typically 10 compared to the individual error from the fitting of each diffusion measurement, the variance of the averaged Dapp was taken as the error.

5.3.1 On Temperature Stability

The strange effects in the PGSE experiments in figure 5.3 led to the question, if the observed dependence on the diffusion time might instead be caused by a temperature variation during the course of the experiment. The suspicion was that

125 5. Swelling of PDMS Model Networks sample heating due to energy transfer by the pulsed gradients could take place. This was tested by measuring the diffusion coefficient of pure octane as function of the diffusion time. The experimental parameters of gradient strength, recycle delay and method of temperature control were varied. Temperature control via airflow resulted in the worst data, possibly due to convective flow. Therefore only the gradient cooling water was used for establishing reliable temperature conditions. The temperature of the cooling water itself is dependent on the power in the gradient coil. A recycle delay of 2 seconds between two transients with three gradient pulses (including one dummy pulse) was found to be necessary. The gradient duration was chosen not to exceed 1.5 ms. With these conditions gradients with any desired strength could be used with the ’micro5’ system. Nevertheless a slight dependence of the diffusion coefficient of pure octane on the diffusion time was observed. A 10 % decrease of Dapp with increasing ∆ (up to 160 ms) was found, which is much lower than what was observed for the network net47k(4). For this sample the value at long ∆ (300 ms) was only 45 % of the value at short ∆.

5.3.2 Swelling Dependent Diﬀusion

There was no experimental proof for obstructed diffusion, neither in a single PFG experiment nor on the same sample at various diffusion times, where a true decrease of Dapp was never observed. However, the expected heterogeneities could become evident in different dependencies of Dapp on Qw when different samples are compared. In particular, differences could show up at low degrees of swelling Qw, where percolation structures of differently swollen regions could lead to a large diffusivity contrast between these regions. Therefore the three different samples were investigated at various degrees of swelling. The results are depicted in figure 5.4. A difference is not observable, given the accuracy of the data. Therefore one can conclude that the diffusivity contrast between differently swollen regions is either too low or that the length scale associated with the heterogeneity is simply too small to be detected by the rapid diffusion of the solvent molecules. The length scale, on which heterogeneities can be excluded, can roughly be estimated

126 5.3. Diﬀusion measurements

0 . 8 o c t a n e - d 1 8 0 . 7

0 . 6 0 D / 0 . 5 D

r o t c

a 0 . 4 f

n o i t

c 0 . 3 u r t

s n e t 4 7 k b

o 0 . 2 n e t 5 k n e t b 3 0 0 . 1 l i n e a r P D M S 1 1 0 k

0 . 0 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 2 . 8 3 . 0 d e g r e e o f s w e l l i n g Q N M R

Figure 5.4: Obstruction factors determined by PFG diffusion measure- −9 2 ments at 297 K. Dfree for octane was measured as 2.0 · 10 m /s. The error bars indicate the variation in D for different diffusion times ∆. from the length scales in figure 5.3. It ranges from about 4 to 60 µm. In other words, on these length scales no heterogeneities are detected by diffusion NMR. If any heterogeneities are smaller and the diffusion already averages over them or if they are on a larger scale can not be definitely deduced. However, large heterogeneities should have been visible in a dependence of the apparent diffusion coefficient on the diffused distance (which is up to several ten µm). Furthermore large heterogeneities (clusters of dense and loose network chains) should lead to a superposition of two Gaussian functions with different diffusion coefficients in the echo decay. Therefore it is most probable that the contrast between differently swollen regions is too low for octane. Further experiments with larger molecules, that would interact more strongly with the percolation structure as they would be repelled stronger from the ’walls’ of the

127 5. Swelling of PDMS Model Networks less swollen regions, might answer that question. A good approach would be the use of rigid spherical proteins of known diameter.

5.4 Discussion: PDMS Model Networks

Various PDMS endlinked model networks were investigated by pulsed gradient diffusion NMR in order to detect anomalous diffusional behavior. Networks swollen with a good solvent to different degrees of swelling were proposed. Diffu- sion time dependent measurements were carried out, monitoring especially careful variations of the temperature, which could be misinterpreted as anomalous diffusion. Suitable conditions to avoid any effect due to temperature were established. The attempt to find a proof of inhomogeneous swelling of networks by means of diffusion NMR was not successful so far. Neither the shape of the echo decay nor diffusion time dependencies nor variation of the degree of swelling showed a significant effect. Even in the deliberately inhomogeneously synthesized bimodal networks, netb30, where heterogeneities were suspected to be detectable most easily, no evidence was found. In fact, all the networks behaved more or less the same. This should however not leave the impression that the method in general is not suitable. It is most likely that it is a problem of length scales. The diffusing solvent molecule is very tiny and might therefore not interact strongly enough with the supposed percolation structure. The diffusion method should indeed be sensitive to the heterogeneities, if the relative sizes between the diffusing particle and the size of the heterogeneities were chosen properly.

128 Chapter 6

Summary

The pulsed gradient spin echo method was established and proven valuable for the investigation of structure and dynamics in ’soft matter’ systems. Three systems with different topological features were investigated by pulsed gradient diffusion NMR and complementary deuterium line shape analysis. Restricted diffusion was studied in two lyotropic lamellar systems. The first consisting of crosslinked flat lamellae and the second consisting of spherically bent lamellae. Obstruction effects on diffusion were investigated in the fractal topology of swollen networks.

Anisotropic Hydrogels. The structure and ordering of anisotropic hydrogels was investigated on the microscopic and mesoscopic length scale. The X-ray correlation length of about seven layers shows a rather low order on very small scales. The director distribution on mesoscopic scales, which was investigated by deuterium line shape analysis, was found to have its maximum parallel to the magnetic field, as expected. The distribution has a full width at half maximum of 13◦. Diffusion NMR proves the existence of defects in the lamellae, but gives also an estimate of the minimal extension of homogeneous domains of the order of 10 µm parallel to the layer normal and 30 µm perpendicular to it. Arrhenius activation energies for diffusion in the range from 300 to 340 K were found to be in agreement with the literature on biological membrane systems. The temperature dependent obstruction factors for diffusion parallel to the layer normal showed an anomalous step at around 312 – 314 K. This decrease in obstruction was interpreted as the generation of further defects and cracks due to elastic forces

129 6. Summary from the network, which tend to restore the original shape of the coil and become dominating in the coupled system of mesophase and polymer network above this temperature range. The change of obstruction occurs at a temperature range where the corresponding low molecular weight mixture is already phase separated. The anisotropy of diffusion was found to decrease with increasing length scale. It changes from about 10 : 1 on the mesoscopic level, deduced from diffusion NMR and lineshape analysis, down to 5.3 : 1 on the macroscopic level, as it was observed by dye diffusion. The comparison of two different hydrogels with different content of water showed higher order and less structural defects in the system with lower content of water.

Multilamellar Vesicles. The dynamics of D2O in the multilamellar vesicle phase of C10E3/D2O was investigated for MLVs of three different radii. It could be shown that a relation between MLV size and the D2O single pulse line shape exists. Such a relationship was deduced from simple empirical scaling relations and was explained by the introduction of a theoretical factor. This factor relates the dynamic element of the surface area visited by the diffusing D2O, represented by the line width, with the static property of the vesicle radius. Furthermore the inhomogeneity of the single pulse line shape could be shown by solid and stimulated echo measurements which showed a strong dependence of the echo line shape on the transverse evolution delays. At long evolution delays a splitting occurred which was identified as the remaining 90◦ singularity of the Pake spectrum of disordered flat lamellae. Line shapes from stimulated echo experiments showed that diffusion through the layers plays a minor role. It also shows that the stimulated echo selects only a small subensemble of spins from the MLVs. The stimulated echo line shape is dominated by rotational motions with correlation times on the order of the delays τ1 and τ2 in the stimulated echo pulse sequence. Therefore the line shape originates from a complex superposition of correlation functions that depend on both delays. Pulsed gradient diffusion measurements showed a strong dependence of the echo decay on the diffusion time. This anomalous diffusion behavior was attributed to the restricted diffusion along the spherically bent lamellae in the MLVs. A theoretical model for diffusion in an idealized MLV structure was

130 tested for the ability to describe the measured echo decay. Although a principal agreement between the shape of the experimental echo decay and the model could be observed, the simulation parameters differed by several orders of magnitude from the experiment. The differences were mainly attributed to three factors, which were not taken into account in the derivation of the echo decay curve: first the incomplete description of the dependence of the line shape on relaxation in the stimulated echo, which selects only a small subensemble of spins from the MLVs, second the validity of the model for finite gradient duration, since it was calculated in the short gradient approximation, and third the diffusion through the layers. It was shown that a quantitative description of the diffusional echo decay in the PGSTE experiment cannot be given without simulation of the echo line shape.

PDMS Model Networks. PDMS model networks were investigated by pulsed gradient diffusion NMR in order to detect anomalous diffusional behavior. This was done as a test for recently developed theories on inhomogeneous network swelling. Two networks with different chain lengths between the crosslinks were swollen with octane, which is a good solvent for PDMS, to different degrees. One of them was constituted from shorter chains (Mn = 5200 g/mol, PD = 1.86), another one from longer chains (Mn = 47 200 g/mol, PD = 1.64). A third bimodal network consisting of 30 wt.% long chains and 70 wt.% short chains was investigated. The netpoint density of all three networks was similar. Diffusion time dependent measurements were carried out on all samples, monitoring especially careful variations of the temperature, which could pretend anomalous diffusion. Suitable conditions to avoid any effect due to temperature were established. No evidence for inhomogeneous swelling of the networks by means of diffusion NMR was found. Neither a non-Gaussian echo decay shape nor diffusion time dependencies nor variation of the degree of swelling showed a significant effect from which obstructed diffusion could have been deduced. Even in the deliberately inhomogeneously synthesized bimodal networks, where heterogeneities were suspected to be detectable most easily, no such evidence was found. This means that the small solvent molecule octane did not experience any restrictions on its mobility in the swollen samples. The dependence of the diffusion coefficient on

131 6. Summary the degree of swelling followed an expected asymptotic increase as the degree of swelling increases. It cannot be decided by octane diﬀusion whether averaging over heterogeneities is recorded or truly homogeneous swelling takes place.

132 Appendices

133

Appendix A

Hardware

A.1 The Spectrometer

Experiments were performed on a Bruker Avance 500 solid-state NMR spectrome- 2 ter (B0 = 11.7 T, H resonance at 76.773 Hz), equipped with a commercial Bruker static double-resonance probe (2H 90◦ pulses of 7 µs length) including the wiring for three orthogonal gradients. Two different gradient tops, ’diff30’ and ’micro5’, could be attached to the probe. The ’diff30’ is equipped with a single z-gradient and the ’micro5’ with three orthogonal gradients. Data acquisition was done with Bruker XWin-NMR 3.0 software. All further data treatment (phasing, etc.) was done with PV-Wave.

A.2 The Gradient System

The Bruker gradient coil systems ’diff30’ and ’micro5’ both were used with the same amplifier system BAFPA-40. This provides a maximum current output of 40 A, which causes gradient strengths of about 1180 G/cm for the ’diff30’ and about 190 G/cm for the ’micro5’. The crucial parameters for diffusion measurements are the gradient stability, reproducibility and effective rise times. The stability was usually good, when the gradient amplifiers were running permanently. Also reproducibility was very good. In all PFG pulse programs an additional dummy gradient pulse was applied before each transient with the same timing as the two

135 A. Hardware encoding/decoding gradients to establish a pseudo equilibrium state. The minimal rise times of amplifiers were checked with a Tektronix TDS3032B 300 MHz digital oscilloscope. The characteristic rise time was chosen to be the time inter- val between 10 and 90 % of the amplifier setting. Results are plotted in figure A.1. The rise times were all below 70 µs even for 100 % amplifier power. Below 20 %, however, the response is nonlinear and therefore it was avoided in the PFG experiments to use gradients below that value. Instead the gradient duration was adapted. This fast switching of gradients results in strong overshoots upon rising

60 s]

30 risetime 10-90 % [ % 10-90 risetime

10 0 20 40 60 80 100 gradient magnitude [%]

Figure A.1: Ampliﬁer rise time between 10 and 90 % of the gradient magnitude. A magnitude 100 % corresponds to 40 ampere output power. and large ringdown eﬀects, when the gradient is switched of [112]. Furthermore the quick switching of very large currents (up to 40 ampere in less than 100 µs) resulted in mechanical responses of the probe head and sample. Therefore a smoother way to apply the gradient was chosen. A ramp of ten equally separated steps was applied where the gradient strength was successively increased. A time parameter ’p17’, the ramp time, had to be set and the ten steps were equally

136 A. Hardware separated during this time. The same ramp was applied when the gradient was switched off. With this method the electric overshoots and mechanical responses could be minimized. Different ramps with up to 100 steps were tested but yielded no improvement. The ramp however increases the total duration of the gradient needed to reach the same phasing strength gδ as without a ramp. In figure A.2 the increase of a ramped gradient is plotted for three ramp times p17. If this

100 gradient rise time [ s] 800 500 80 100

40 gradient magnitude [%] magnitude gradient

0 0 200 400 600 800 1000 time [ s]

Figure A.2: Ramped rise of a trapezoidal gradient for three different ramp times p17. A ramp time of 700 µs was usually chosen as default, since it was suitable for all gradient strengths. time was chosen too short (100 µs), the amplifier could not respond to it and no ramp is present. The optimal time to create a ramp was adjusted if the experiment needed short evolution delays. As a standard parameter which fitted well for most applications the ramp time was usually set to 700 µs. The phasing strength gδ for such a trapezoidal gradient is calculated by simply multiplying the gradient strength with the sum of the rise time (p17) and the time during which the gradient is at its desired power (p18). By this calculation the trapezoidal shape is approximated by a rectangular shape. The gradient magnitude during

137 A. Hardware the time where the gradient decreases is added to the the magnitude during the rise time, as sketched in ﬁgure A.3. The gradient decrease time has nevertheless to be taken into account for the calculation of the diﬀusion time.

p17 p18

Figure A.3: Sketch of the trapezoidal gradient. The gradient duration δ is approximated by (p18 + p17).

A.3 Temperature Control and Stability

Temperature control during the diffusion experiments was a crucial task since the diffusion coefficient is usually very sensitive to temperature variations. The temperature of the sample was measured by a chemical shift thermometer: ethylene glycole. Ethylene glycole has two resonances and the chemical shift difference ∆ν is temperature sensitive in the range from 273 to 416 K [113]. The temperature was controlled by a Bruker Eurotherm BVT 3000 temperature controller with a constant air flow of 670 l/h, the temperature stability was about 0.2 K. In order to obtain comparable results for the diffusion measurements the gradient cooling water was switched on and set to 23◦C at the cooling unit. This setup leads to about 298 K without airflow. The chemical shift difference of ethylene glycole is

138 A. Hardware related to the sample temperature by:

Tsample [K] = 466.5 − 102 · ∆ν [ppm] (A.1)

The thermocouple element from the temperature control unit contains traces of iron and is therefore about 10 – 15 cm away from the sample. The diﬀerence between true temperature of the sample Tsample and the temperature Tset, that is reported by the thermocouple, is therefore quite large. The relation was found to be linear in good approximation. A ﬁt yielded:

Tsample [K] = 127.6 (±0.7) + 0.576 (±0.002) Tset [K] (A.2)

Temperature control by air flow on the other hand has a severe drawback for diffusion measurements, since the samples are heated from below and in systems with a low viscosity (like water) this could lead to convection in the sample. Furthermore samples, whose height is in the order of the rf-coil (∼ 8 mm), experience a temperature gradient across the sample. In order to prevent artifacts which would lead to false interpretation of the diffusion data, the temperature was usually controlled only by the gradient cooling water with an accuracy of about 0.5 K. This setup on the other hand has two severe drawbacks. One is that temperature dependent measurements are not recommendable, since the cooling unit equilibrates very slow, it needs about 3 hours to establish constant temperatures. Furthermore the temperature range is limited to between 10 and 40 ◦C. The other drawback is that recycle delays have to be chosen very long to prevent a heating of the sample via the gradient coil. The high currents in the gradient coils significantly heat up the gradient system, e. g., up to 3 K during the cause of an experiment of half an hour, when the recycle delay was set to 1 s and maximum gradient power was reached. To prevent this heating a recycle delay of at least 2 s is necessary, which of course leads to long experiment times. For these reasons the appropriate method of temperature control was chosen for each sample separately. The temperature dependent measurements of the hydrogels and the Lα phase of C10E3/D2O were performed with air flow, since the samples were small (no temperature gradient) and the viscosity was assumed

139 A. Hardware to be high enough to prevent convection. For the measurements of the MLV and the PDMS networks, temperature was controlled only by the gradient cooling water.

A.4 Gradient Calibration

It is necessary to calibrate gradients very accurately. In the formulas used to calculate diffusion coefficients, the gradient magnitude enters quadratically. The gradients were usually calibrated by measuring diffusion of pure water and comparing the obtained diffusion coefficient Dmess with literature data Dlit. The variable of interest is called the Gradient Calibration Constant (GCC), which relates gradient field strength with the output power of the amplifier. The output power is controlled in the pulse programs as the percentage of maximum power (i. e. GPZ1). The field strength is the given by:

g [G/cm] = GCC [G/cm] · output power [%] (A.3)

The correct GCCnew for the data analysis is determined from the diffusion measurement where the diffusion coefficient Dmess, that was determined with some arbitrary GCCold is inserted into: r Dmess GCCnew = GCCold (A.4) Dlit

The exact knowledge of the real temperature of the sample is necessary (see appendix A.3) to insert the correct literature value of Dlit that were taken from Mills [57]. Gradient calibration constants were regularly checked, but no severe variations were observed (GCC = 118.2 ± 1.5 [G/cm]).

A.5 The Rotatable Sample Holder

There are two possibilities to measure orientation dependent diffusion coefficients. One is to set the gradient axis in the direction of interest, the other is to change the orientation of the sample. The first method, which was mostly used in this work,

140 A. Hardware is by far more convenient, but expensive hardware is necessary. In particular one gradient coil, including ampliﬁers and control hardware, is needed for each direction. This hardware was not available in the beginning of this work and therefore it was necessary to develop a possibility to rotate the sample with respect to the gradient. This rotation had to take place not only inside the gradient coil system (inner diameter of about 15 mm), but also inside the high frequency rf-coil, which had an inner diameter of about 5 mm. Furthermore, since the gradient system was cooled with water it was necessary to build a connection to the outside of the probe head in order not to disassemble the probe head each time the orientation should be changed. The limited space posed a challenge to the construction. In ﬁgure A.4 a sketch of the rotatable sample holder is given.

outeraxisofrotation upperaxisofrotation

samplewithcapillary rubbero-ring

rf-coil

loweraxisofrotation teflonholder innerdiameterofthegradient

Figure A.4: Sketch of the rotatable sample holder. Side (left) and top view (middle). To the right is a picture of the tool in a transparent tube with the same inner diameter as the gradient coil system.

The sample had to be prepared in a small glass sphere, to which a capillary was attached (see section 3.1). This capillary was inserted into a hole in the lower axis of rotation, where it ﬁtted exactly . The possibility to control the rotation

141 A. Hardware from the outside was given by two rubber o-rings that were spanned between the two inner axes and the outer axis of rotation (see figure A.4). A slight strain of the rubber occurred when rotation was started. But when the direction of rotation was kept, this strain was negligible and a relatively precise control of the angle (better than 3 – 5◦) of rotation was possible. This was proven by rotation of a lamellar hydrogel, which was prepared with its axis of symmetry oriented perpendicular to the capillary. The observed quadrupolar splitting followed perfectly the second Legendre polynomial. Because of the vicinity to the rf-coil, the material used for the whole tool was glass-fiber enforced teflon.

142 Appendix B

Additional Sample Data

B.1 Solid Echo Spectra of MLV

Evolution time dependent echo spectra for MLV from 40 wt-% C10E3/D2O prepared at shear rates of 2 and 5 s−1 are shown in ﬁgures B.1 and B.2 at a temperature of 298 K. Plots are analogous to those in section 4.5.1.

4x108 3x107

0.3 ms 6 ms 3x108 2x107

2x108

1x107 1x108

0 0 -2000 -1000 0 1000 2000 -2000 -1000 0 1000 2000 6 6 4x10 1x10

intensity [a.u.] intensity 12 ms 18 ms 3x106

2x106 5x105

1x106

0 0

-2000 -1000 0 1000 2000 -2000 -1000 0 1000 2000

Frequency [Hz]

Figure B.1: Solid echo line shapes at diﬀerent evolution delays 2 τ, MLV prepared at 2 s−1.

143 B. Additional Sample Data

1.2x108 3x106

0.08 ms 8 ms

8.0x107 2x106

4.0x107 1x106

0.0 0 -2000 -1000 0 1000 2000 -2000 -1000 0 1000 2000 5 1.5x10

Intensity [a.u.] Intensity 26 ms 1.2x10 12 ms

1.0x105

8.0x105

5.0x104 4.0x105

0.0 0.0 -2000 -1000 0 1000 2000 -2000 -1000 0 1000 2000

Frequency [Hz]

Figure B.2: Solid echo line shapes at diﬀerent evolution delays 2 τ, MLV prepared at 5 s−1.

144 B. Additional Sample Data

B.2 PGSTE Diﬀusion Decay for MLV

0.0 [ms] 10.1 -0.2 15.1 25.1 -0.4 40.1

-0.6 ln(S) [a.u.] ln(S) -0.8

-1.0

-1.2 0.0 5.0x109 1.0x1010 1.5x1010 g2 * k [s/m2]

0.0 [ms] 9.8 24.8 -0.5 39.8 54.8

-1.0 ln(S) [a.u.] ln(S)

-1.5

-2.0 0 2x1010 4x1010 6x1010 g2 * k [s/m2]

Figure B.3: PGSTE echo decay for MLV prepared at 2 and 5 s−1 for different diffusion times ∆. Plotted against g2k, such that the slope yields directly the diffusion coefficient.

145 B. Additional Sample Data

B.3 Arrhenius Plot for Diﬀusion in the Lα Phase

-23

-24

-25 2- region ln D ln

-26

-27

3.12 3.16 3.20 3.24 3.28

1000 / T [K-1]

-20.3

2- region

-20.4 ln D ln

-20.5

3.12 3.16 3.20 3.24 3.28

1000 / T [K-1]

Figure B.4: Arrhenius plots for diﬀusion in the Lα phase, parallel (top) and perpendicular (bottom) to the layer normal. The solid lines are ﬁts yielding activation energies of 157 kJ/mol (parallel) and 16 kJ/mol (perpendicular).

146 B. Additional Sample Data

B.4 PDMS Network Samples

Diﬀusion measurements were performed on a variety of PDMS network samples. The samples were swollen with diﬀerent methods as described in section 5.1. In tables B.1, B.2 and B.3 important parameters of the samples are summarized.

The respective degrees of swelling Q determined by weighing (Qw) on a balance and by integration of NMR signals (QNMR) are given. Additionally the method of preparation is speciﬁed.

Table B.1: Samples prepared from the network net47k sample f1 f2 f3 f4 f5 f7 f11 f12

Qw 3.01 3.26 1.36 1.89 2.21 2.12 1.53 1.61

QNMR 2.73 2.92 1.32 1.96 2.2 2.12 1.53 1.61 method vapor gradient gradient gradient vapor vapor vapor vapor

Table B.2: Samples prepared from the network net5k sample x1 x2 x3 x4 x5 x6 x7 x8

Qw 1.94 2.53 1.56 2.6 1.68 1.49 1.1 1.29

QNMR 1.68 2.11 1.42 2.37 1.63 1.49 1.17 1.31 method gradient gradient gradient vapor vapor vapor vapor vapor

Table B.3: Samples prepared from the network net30b sample fx1 fx2 fx3 fx4 fx5 fx6 fx7 fx8

Qw 2.22 1.07 1.3 1.78 2.32 1.28 1.53 1.97

QNMR 2.2 1.09 1.33 1.76 2.11 1.32 1.67 1.97 method vapor vapor vapor vapor gradient gradient vapor vapor

147

Appendix C

Computer Programs

C.1 Numerical Calculations

Numerical simulations of the PGSTE echo decay curves were carried out in C++, using the commercial compiler Microsoft Visual C++ r 6.0.

The PGSTE Echo Decay for One Sphere

The spherical Bessel functions jl(q) in equation 4.23 were calculated according to the procedures in ’Numerical Recipes in C, The Art of Scientiﬁc Computing, Second Edition’ [114] taken from the on-line edition [115]. In the following the code for the calculation of

∞ X Dtrans S(q, ∆) = (2l + 1) j2(q) exp(−l(l + 1) · ∆) l R2 l=0 is printed. The output of the program is the signal intensity as function of some q-values for the abscissa for a number of radii R. The signal intensities S for each R are all consecutively stored in the same ﬁle. They have to be rearranged and summed up by a diﬀerent program. This sum over R was done using Origin with the LabTalk script from section C.2.

#include #include #include

149 C. Computer Programs

#include #include #include #include #include #include #include #include #include

#define EPS 1.0e-10 #define FPMIN 1.0e-30 #define MAXIT 10000 #define XMIN 2.0 #define PI 3.141592653589793 #define RTPIO2 1.2533141

/*error messages from num. rec.*/ void nrerror(char error_text[]){ fprintf(stderr, "Numerical Recipes run-time error. . .\n"); fprintf(stderr, "%s\n", error_text); fprintf(stderr, ". . .now exiting to system. . .\n"); exit(1); }

/*bessjy from num. rec.*/ void bessjy(double x, double xnu, double *rj, double *ry, double *rjp, double *ryp) { void beschb( double x, double *gam1, double *gam2, double *gampl, double *gammi); int i,isign,l,nl; double a,b,br,bi,c,cr,ci,d,del,del1,den,di,dlr,dli,dr,e,f,fact,fact2,fact3,ff,gam, gam1,gam2,gammi,gampl,h,p,pimu,pimu2,q,r,rjl,rjl1,rjmu,rjp1,rjpl,rjtemp,ry1,rymu, rymup,rytemp,sum,sum1, temp,w,x2,xi,xi2,xmu,xmu2; if (x <= 0.0 || xnu < 0.0) nrerror("bad arguments in bessjy");

150 C. Computer Programs nl=(x < XMIN ? (int)(xnu+0.5) : IMAX(0,(int)(xnu-x+1.5))); xmu=xnu-nl; xmu2=xmu*xmu; xi=1.0/x; xi2=2.0*xi; w=xi2/PI; isign=1; h=xnu*xi; if (h < FPMIN) h=FPMIN; b=xi2*xnu; d=0.0; c=h; for (i=1;i<=MAXIT;i++) { b += xi2; d=b-d; if (fabs(d) < FPMIN) d=FPMIN; c=b-1.0/c; if (fabs(c) < FPMIN) c=FPMIN; d=1.0/d; del=c*d; h=del*h; if (d < 0.0) isign = -isign; if (fabs(del-1.0) < EPS) break; } if (i > MAXIT) nrerror("x too large in bessjy; try asymptotic expansion"); rjl=isign*FPMIN; rjpl=h*rjl; rjl1=rjl; rjp1=rjpl; fact=xnu*xi; for (l=nl;l>=1;l--) { rjtemp=fact*rjl+rjpl; fact -= xi; rjpl=fact*rjtemp-rjl; rjl=rjtemp;

151 C. Computer Programs

} if (rjl == 0.0) rjl=EPS; f=rjpl/rjl; if (x < XMIN) { x2=0.5*x; pimu=PI*xmu; fact = (fabs(pimu) < EPS ? 1.0 : pimu/sin(pimu)); d = -log(x2); e=xmu*d; fact2 = (fabs(e) < EPS ? 1.0 : sinh(e)/e); beschb(xmu,&gam1,&gam2,&gampl,&gammi); ff=2.0/PI*fact*(gam1*cosh(e)+gam2*fact2*d); e=exp(e); p=e/(gampl*PI); q=1.0/(e*PI*gammi); pimu2=0.5*pimu; fact3 = (fabs(pimu2) < EPS ? 1.0 : sin(pimu2)/pimu2); r=PI*pimu2*fact3*fact3; c=1.0; d = -x2*x2; sum=ff+r*q; sum1=p; for (i=1;i<=MAXIT;i++) { ff=(i*ff+p+q)/(i*i-xmu2); c *= (d/i); p /= (i-xmu); q /= (i+xmu); del=c*(ff+r*q); sum += del; del1=c*p-i*del; sum1 += del1; if (fabs(del) < (1.0+fabs(sum))*EPS) break; } if (i > MAXIT) nrerror("bessy series failed to converge"); rymu = -sum;

152 C. Computer Programs ry1 = -sum1*xi2; rymup=xmu*xi*rymu-ry1; rjmu=w/(rymup-f*rymu); } else { a=0.25-xmu2; p = -0.5*xi; q=1.0; br=2.0*x; bi=2.0; fact=a*xi/(p*p+q*q); cr=br+q*fact; ci=bi+p*fact; den=br*br+bi*bi; dr=br/den; di = -bi/den; dlr=cr*dr-ci*di; dli=cr*di+ci*dr; temp=p*dlr-q*dli; q=p*dli+q*dlr; p=temp; for (i=2;i<=MAXIT;i++) { a += 2*(i-1); bi += 2.0; dr=a*dr+br; di=a*di+bi; if (fabs(dr)+fabs(di) < FPMIN) dr=FPMIN; fact=a/(cr*cr+ci*ci); cr=br+cr*fact; ci=bi-ci*fact; if (fabs(cr)+fabs(ci) < FPMIN) cr=FPMIN; den=dr*dr+di*di; dr /= den; di /= -den; dlr=cr*dr-ci*di; dli=cr*di+ci*dr;

153 C. Computer Programs temp=p*dlr-q*dli; q=p*dli+q*dlr; p=temp; if (fabs(dlr-1.0)+fabs(dli) < EPS) break; } if (i > MAXIT) nrerror("cf2 failed in bessjy"); gam=(p-f)/q; rjmu=sqrt(w/((p-f)*gam+q)); rjmu=SIGN(rjmu,rjl); rymu=rjmu*gam; rymup=rymu*(p+q/gam); ry1=xmu*xi*rymu-rymup; } fact=rjmu/rjl; *rj=rjl1*fact; *rjp=rjp1*fact; for (i=1;i<=nl;i++) { rytemp=(xmu+i)*xi2*ry1-rymu; rymu=ry1; ry1=rytemp; } *ry=rymu; *ryp=xnu*xi*rymu-ry1; };

/*Beschb from num. rec.*/ #define NUSE1 7 #define NUSE2 8 void beschb( double x, double *gam1, double *gam2, double *gampl, double *gammi) { double chebev( double a, double b, double c[], int m, double x); double xx; static double c1[] = {-1.142022680371168e0,6.5165112670737e-3, 3.087090173086e-4,-3.4706269649e-6, 6.9437664e-9,3.67795e-11,-1.356e-13};

154 C. Computer Programs static double c2[] = {1.843740587300905e0,-7.68528408447867e-2, 1.2719271366546e-3,-4.9717367042e-6, -3.31261198e-8,2.423096e-10,-1.702e-13,-1.49e-15}; xx=8.0*x*x-1.0; *gam1=chebev(-1.0,1.0,c1,NUSE1,xx); *gam2=chebev(-1.0,1.0,c2,NUSE2,xx); *gampl= *gam2-x*(*gam1); *gammi= *gam2+x*(*gam1); }

/*chebev from num. rec.*/ double chebev( double a, double b, double c[], int m, double x) { void nrerror(char error_text[]); double d=0.0,dd=0.0,sv,y,y2; int j; if ((x-a)*(x-b) > 0.0) nrerror("x not in range in routine chebev"); y2=2.0*(y=(2.0*x-a-b)/(b-a)); for (j=m-1;j>=1;j--) { sv=d; d=y2*d-dd+c[j]; dd=sv; } return y*d-dd+0.5*c[0]; }

/*spherical bessel functions from num. rec.*/ void sphbes( double n, double x, double *sj, double *sy, double *sjp, double *syp) { void bessjy( double x, double xnu, double *rj, double *ry, double *rjp, double *ryp); void nrerror(char error_text[]); double factor,order,rj,rjp,ry,ryp; if (n < 0 || x <= 0.0) nrerror("bad arguments in sphbes"); order=n+0.5;

155 C. Computer Programs bessjy(x,order,&rj,&ry,&rjp,&ryp); factor=RTPIO2/sqrt(x); *sj=factor*rj; *sy=factor*ry; *sjp=factor*rjp-(*sj)/(2.0*x); *syp=factor*ryp-(*sy)/(2.0*x); }

#define XMAX 500 //x-axis maximum value #define POINTS XMAX //number of points #define DT 1e-9 //Diffusion coefficient #define BD 1e-2 // big Delta in STE //#define Rad 5e-4 // vesicle radius [micro m] #define RMIN 100 #define RMAX 100 #define RINC 100

/*MAIN */ int main() { int c, n, r, tr, s; double x, sj, sy, sjp, syp, Rad; FILE * pFile; double sum[POINTS]; double axis[POINTS]; double mlv[POINTS]; for (c=0;c<=POINTS;c++){sum[c]=0;}; for (c=0;c<=POINTS;c++){mlv[c]=0;}; sj=0.0; for(s=RMIN;s<=RMAX;s=s+RINC){ printf("Radius=%i um \n",s); for (c=0;c<=POINTS;c++){sum[c]=0;};

156 C. Computer Programs

Rad=s*1e-6; //spherical Bessel functions of order n for (n=0;n<=XMAX;n++){ for (r=1;r<=POINTS;r++){ x=(r-1.0+1e-10)/POINTS*XMAX; sphbes(n, x, &sj, &sy, &sjp, &syp);

// empiric ’overfloat error: -1.#IND00’ correction if (n>=110) tr=3+0.4*(n-100)+0.00009*(n-100)*(n-100); else tr=1; if (n == 0) tr=0; if (r<=tr){sj=0.0;}; sum[r-1]=sum[r-1]+sj*sj*(2*n+1)*exp(-n*(n+1)*DT/(Rad*Rad)*BD); axis[r-1]=x; }; }; for (c=0;c

157 C. Computer Programs

1.0

0.8

0.6

0.4

0.2

0.0 0 2 4 6 8 10 12 14 X

Figure C.1: Test of the MLV echo decay program. The program gives the curve for ∆ → ∞ (—) and the square over the spherical Bessel function th 2 of 0 order j0 (x)(), which coincide as predicted.

A test of the program is the long time limit in ∆ of one sphere. The curve th 2 should be equal to the square over the spherical Bessel function of 0 order j0 (x). This result was obtained as it can be seen in ﬁgure C.1.

The Angular Distribution Function

The angular distribution function for diﬀusion on a sphere was calculated according to equation (4.9) for a sum over the Legendre polynomials up to n = 100:

100 1 X Dt P (θ, t) = (2l + 1) P (cos θ) · exp −l(l + 1) sin θ 2 l R2 n=1

The program yields the angular distribution function for one radius.

158 C. Computer Programs

/* RAD_sphere.c */

#include #include #include #include #include #include #include #include #include #include #define PI 3.141592653589793

/*Legendre polynomial of order l,m for values of x*/ float plgndr(int l, int m, float x) { void nrerror(char error_text[]); float fact,pll,pmm,pmmp1,somx2; int i,ll; if (m < 0 || m > l || fabs(x) > 1.0) nrerror("Bad arguments in routine plgndr"); pmm=1.0; if (m > 0) { somx2=sqrt((1.0-x)*(1.0+x)); fact=1.0;

159 C. Computer Programs for (i=1;i<=m;i++) { pmm *= -fact*somx2; fact += 2.0; } } if (l == m) return pmm; else { pmmp1=x*(2*m+1)*pmm; if (l == (m+1)) return pmmp1; else { for (ll=m+2;ll<=l;ll++) { pll=(x*(2*ll-1)*pmmp1-(ll+m-1)*pmm)/(ll-m); pmm=pmmp1; pmmp1=pll; } return pll; } } }

#define m 0 #define D 1e-9 #define R 2e-6 #define t 0.0001

/*MAIN */ float main() { float x,Legendre,sum; int c,l,k; FILE * pFile; float axis[100]; float rad[100]; for (k=0;k<=100;k++){rad[k]=0;};

160 C. Computer Programs

Legendre=0.0; sum=0.0; for (l=0;l<=100;l++){ for (c=0;c<=100;c++){ x=cos(c/100.0*PI); Legendre=plgndr(l, m, x)*sin(c/100.0*PI); rad[c]=rad[c]+0.5*(2*l+1)*Legendre*exp(-l*(l+1)*D*t/(R*R)); axis[c]=c/100.0*PI; }; }; axis[0]=0; for (c=0;c<100;c++){ //printf("%f \n",rad[c]);}; pFile = fopen ("RAD.txt","a"); fprintf (pFile, "%f \t",axis[c]); fprintf (pFile, "%f \n",rad[c]/l); fclose (pFile); }; return 0; };

The Surface Record Factor

The surface record factor according to equation (4.6) was calculated as function of the radius. Z π 1 SRF = P (θ, R) (3 cos2 θ − 1)dθ 0 2 The program yields the SRF for the specific parameters, like diffusion coefficient and diffusion time, which have to be set in the program.

/* SRF_MLV.C */

#include #include #include #include #include

161 C. Computer Programs

#include #include #include #include #include #define PI 3.141592653589793

/*error messages*/ void nrerror(char error_text[]){ fprintf(stderr, "Numerical Recipes run-time error. . .\n"); fprintf(stderr, "%s\n", error_text); fprintf(stderr, ". . .now exiting to system. . .\n"); exit(1); } float plgndr(int l, int m, float x) { void nrerror(char error_text[]); float fact,pll,pmm,pmmp1,somx2; int i,ll; if (m < 0 || m > l || fabs(x) > 1.0) nrerror("Bad arguments in routine plgndr"); pmm=1.0; if (m > 0) { somx2=sqrt((1.0-x)*(1.0+x)); fact=1.0; for (i=1;i<=m;i++) { pmm *= -fact*somx2; fact += 2.0; } } if (l == m) return pmm; else { pmmp1=x*(2*m+1)*pmm;

162 C. Computer Programs if (l == (m+1)) return pmmp1; else { for (ll=m+2;ll<=l;ll++) { pll=(x*(2*ll-1)*pmmp1-(ll+m-1)*pmm)/(ll-m); pmm=pmmp1; pmmp1=pll; } return pll; } } } //define relevant paramters #define m 0 #define D 1.0e-9 //diffusion coefficient #define t 0.001 //diffusion time #define Rmax 10.0 //max. x-axis #define Points 100 //number of points

/*MAIN */ float main() { float sum,x,Legendre,Radius,rmax; int R,c,l,k;

FILE * pFile; float axis[100]; float rad[100]; float P2[100]; float order[100]; float s[100]; float smlv[100]; float area[100]; float sphere[100]; Legendre=0.0;

163 C. Computer Programs sum=0.0; for (k=0;k<=100;k++){P2[k]=0.0;}; // second Legendre; for (c=0;c<=100;c++){ x=cos(c/100.0*PI); P2[c]=plgndr(2, 0, x); }; for (k=0;k<=100;k++){order[k]=0.0;}; for (k=0;k<=100;k++){s[k]=0.0;}; //SRF(R); for (R=1;R<=100;R++){ for (k=0;k<=100;k++){rad[k]=0;}; Radius=(R-1)/100.0*1e-6*Rmax; for (l=0;l<=100;l++){ for (c=0;c<=100;c++){ x=cos(c/100.0*PI); Legendre=plgndr(l, m, x)*sin(c/100.0*PI); rad[c]=rad[c]+0.5*(2*l+1)*Legendre*exp(-l*(l+1)*D*t/(Radius*Radius)); axis[c]=c/100.0*PI; }; }; for (k=0;k<=100;k++){order[k]=0.0;}; for (c=0;c<=100;c++){order[c]=order[c]+rad[c]*P2[c];}; for (c=0;c<=100;c++){s[R]=s[R]+order[c];}; sphere[R-1]=Radius*1e6; }; axis[0]=0; sphere[0]=0.0;s[1]=0.0; // S(Rmax) for (c=0;c<=100;c++){area[c]=sphere[c]*sphere[c]*PI;}; for (c=0;c<100;c++){smlv[c]=0.0;}; for (c=0;c<100;c++){ for (k=0;k<=c;k++){

164 C. Computer Programs smlv[c]=smlv[c]+s[k]/(10*PI)*area[k]; }; smlv[c]=smlv[c]/100; printf ("%f \t",smlv[c]); printf ("%f \n",s[c]/(10*PI)); }; for (c=0;c<100;c++){ pFile = fopen ("SRF.txt","a"); fprintf (pFile, "%f \t",sphere[c]); fprintf (pFile, "%f \n",s[c]/(10*PI)); fclose (pFile);}; return 0; };

C.2 LabTalk Scripts

All data analysis was performed with Microcal Originr 6.0 and 7.0. For many standard procedures in the data analysis, like diﬀusion experiments, rearranging simulation output data, regrouping many plots etc., the same operations had to be performed over and over again. Therefore LabTalk Scripts were programmed that greatly simplify these routine operations. The following scripts were programmed:

• A plot routine for diﬀusion data for the three methods of PGSE, PGSTE and PGSTE including a spoiler gradient.

• A script to calculate the derivative the echo decay, yielding directly the diﬀusion coeﬃcient.

Plot Routine for the Diﬀusion: DiﬀPlot

The script calls first for a 0.dat0 ASCII file, containing solely the echo intensity in a single column. It then calls for a ’.txt’ ASCII file containing the parameters (delays, gradient strengths etc.) of the measurement in the format created with the XWin-NMR command: xwp lp. The type of experiment has to be specified in order to read in the correct values from the parameter file.

165 C. Computer Programs

# DiffPlot wks.template(origin); getfile *.dat;open -w %A;%D=%A;%T=%B; #parameter einlesen; win -t data; getfile *.txt;work -e Ascii;open -w %A; #default experiment; #%B=diff_se; #%B=diff_ste; %B=diff_ste_spoil;

%N=col(B)[3]$; %M=col(B)[4]$; getstring (pgse, pgste, ...) (%B) Type of experiment; #Experiment festlegen; if ("%B"=="diff_se") {exptype=1}; else {exptype=0} if ("%B"=="diff_ste") {exptype=2}; if ("%B"=="diff_ste_spoil") {exptype=3}; switch (exptype) { case 1: pgse; case 2: pgste; case 3: pgste_spoil; case 0: type -c "not yet"; break; }; }; #********Hahn/Solid echo type*********************************** define pgse{ #Kernsorte bestimmen; %F=%[%A,’.’]_B[33]$; if ("%F"=="2H") {nuc=1}; else {nuc=0}; if ("%F"=="1H") {nuc=2};

166 C. Computer Programs switch (nuc) {case 1: Gamma=4106.5211; break; #Hz/G case 2: Gamma=26751.5255; break; #Hz/G default: type " nucleus not identified"; break; }; D2=col(B)[25]; D9=col(B)[26]; D11=col(B)[28]; P2=col(B)[35]; ramp=col(B)[52]; on=col(B)[53]; GPZ1=col(B)[49]; TD1=col(B)[57];

BD= (ramp*2e-6+on*1e-6+D2+D9+D11+P2*1e-6); #s LD=(ramp+on)*1e-6; #s BDrel=(BD-LD/3); #s k=BDrel*(Gamma*LD)^2*1e4; GCC=19.000; type -b "little Delta \t$(LD*1e3) ms \nbig Delta \t\t$(BD*1e3) ms \nk \t\t$(round(k,1)) \nD2 \t\t$(D2*1e3) ms \nD9 \t\t$(round(D9*1e3,1)) ms \nD11 \t\t$(D11*1e3) ms; \nP2 \t\t$(P2) us \nP17 \t\t$(ramp) us \nP18 \t\t$(on) us \nmax. gradient \t$(GPZ1) % \ngradient steps \t$(TD1)"; getnumber (grad. calib. const.) GCC; win -ca; # calculations; wks.addcol(g); wks.addcol(g^2k); wks.col1.name$=area; wks.col2.name$=lnArea; for (ii=1;ii

167 C. Computer Programs

#mark -m %[%D,’.’]_area -b 1 -e 1; col(area)=col(area)/col(area)[2]; col(lnArea)=ln(col(area)); wks.col1.label$=$(round(BDrel*1000,1)); wks.col2.label$=$(round(BDrel*1000,1)); wks.col3.label$=G/cm; wks.col4.label$=s/m^2; wks.labels(); wks.col1.digits=5; wks.col4.digitmode=2; wks.col4.digits=5; # plotten; # ln(area) vs. g2 *k; wks.col1.type=1; wks.col4.type=4; worksheet -s 2 0; worksheet -p 201; window -r %H lnDiff%[%D,5:6]; legend; layer.X.from=0; layer.X.majorTicks=5; layer.X.labelSubtype=2; label -xb g\+(2) * k [s/m\+(2)]; label -yl ln(area) [a.u.]; #ln-plot linear ? lin=-1; getyesno (continue ?) lin [fitting possible ?]; if (lin<=0) {return}; # Area vs. g; window -a %[%D,’.’]; wks.col4.type=1; wks.col3.type=4; wks.colSel(2,0); worksheet -s 1 0; worksheet -p 201;

168 C. Computer Programs legend; layer.X.from=0; layer.X.to=GCC/10*GPZ1*1.3; layer.Y.from=0; label -xb g [G]; label -yl echo intensity [a.u.]; label -s -sa -q 1 \g(d) $(ld*1000) ms \n\g(D) $(bd*1000) ms \n\g(D) \-(rel) $(bdrel*1000) ms \n \nG\-(max) $(gpz1) % \nGCC $(gcc) \nTD1 $(td1) \nGamma $(Gamma) (%F) \nk $(k); window -r %H %[%D,5:6]; label -d 5000 500 -s %N %M $(@D,D0); window -a %[%D,’.’]; wks.colsel(1,0); window -a %[%D,5:6]; }; #************stimulated echo w/o spoiler gradient***************************** define pgste{ #Kernsorte bestimmen; %F=%[%A,’.’]_B[34]$; if ("%F"=="2H") {nuc=1}; else {nuc=0}; if ("%F"=="1H") {nuc=2}; switch (nuc) {case 1: Gamma=4106.5211; break; #Hz/G case 2: Gamma=26751.5255; break; #Hz/G default: type " nucleus not identified"; break; }; D2=col(B)[25]; D5=col(B)[26]; D9=col(B)[27]; D11=col(B)[29]; P1=col(B)[35]; ramp=col(B)[52]; on=col(B)[53];

169 C. Computer Programs

GPZ1=col(B)[49]; TD1=col(B)[57];

BD= (ramp*2e-6+on*1e-6+D2+D5+D9+D11+P1*2e-6); #s LD=(ramp+on)*1e-6; #s BDrel=(BD-LD/3); #s k=BDrel*(Gamma*LD)^2*1e4; GCC=118.02; type -b "little Delta \t$(LD*1e3) ms \nbig Delta \t\t$(BD*1e3) ms \nk \t\t$(round(k,1)) \nD2 \t\t$(D2*1e3) ms \nD5 \t\t$(round(d5*1e3,1)) ms \nD9 \t\t$(round(D9*1e3,1)) ms \nD11 \t\t$(D11*1e3) ms; \nP1 \t\t$(P1) us \nP17 \t\t$(ramp) us \nP18 \t\t$(on) us \nmax. gradient \t$(GPZ1) % \ngradient steps \t$(TD1)"; getnumber (grad. calib. const.) GCC; win -ca; # calculations; wks.addcol(g); wks.addcol(g^2k); wks.addcol(q); wks.col1.name$=area; wks.col2.name$=lnArea; for (ii=1;ii

170 C. Computer Programs wks.col1.digits=5; wks.col4.digitmode=2; wks.col4.digits=5; # plotten; # ln(area) vs. g2 *k; wks.col1.type=1; wks.col4.type=4; worksheet -s 2 0; worksheet -p 201; window -r %H lnDiff%[%D,5:6]; legend; layer.X.from=0; layer.X.majorTicks=5; layer.X.labelSubtype=2; label -xb g\+(2) * k [s/m\+(2)]; label -yl ln(area) [a.u.]; #ln-plot linear ? lin=-1; getyesno (continue ?) lin [fitting possible ?]; if (lin<=0) {return}; # Area vs. g; window -a %[%D,’.’]; wks.col4.type=1; wks.col3.type=4; wks.colSel(2,0); worksheet -s 1 0; worksheet -p 201; legend; layer.X.from=0; layer.X.to=GCC/10*GPZ1*1.3; layer.Y.from=0; label -xb g [G]; label -yl echo intensity [a.u.]; label -s -sa -q 1 \g(d) $(ld*1000) ms \n\g(D)

171 C. Computer Programs

$(bd*1000) ms \n\g(D) \-(rel) $(bdrel*1000) ms \n \nG\-(max) $(gpz1) % \nGCC $(gcc) \nTD1 $(td1) \nGamma $(Gamma) (%F) \nk $(k); window -r %H %[%D,5:6]; label -d 1800 300 -s %[%T,#6,\b] $(@D,D0); window -a %[%D,’.’]; wks.colsel(1,0); window -a %[%D,5:6]; }; #***********stimulated echo with spoiler gradient*************** define pgste_spoil{ #Kernsorte bestimmen; %F=%[%A,’.’]_B[34]$; if ("%F"=="2H") {nuc=1}; else {nuc=0} if ("%F"=="1H") {nuc=2}; switch (nuc) {case 1: Gamma=4106.5211; break; #Hz/G case 2: Gamma=26751.5255; break; #Hz/G default: type " nucleus not identified"; break; }; D2=col(B)[25]; D5=col(B)[26]; D9=col(B)[27]; D11=col(B)[29]; P1=col(B)[35]; ramp=col(B)[64]; on=col(B)[65]; spoil=col(B)[66]; GPZ1=col(B)[58]; TD1=col(B)[70]; BD= (ramp*4e-6+on*1e-6+spoil*1e-6+D2+D9+D5+2*D11+P1*2e-6); #s LD=(ramp+on)*1e-6; #s BDrel=(BD-LD/3); #s k=BDrel*(Gamma*LD)^2*1e4;

172 C. Computer Programs

GCC=118.02; type -b "little Delta \t$(LD*1e3) ms \nbig Delta \t\t$(BD*1e3) ms \nk \t\t$(round(k,1)) \nD2 \t\t$(D2*1e3) ms \nD5 \t\t$(round(d5*1e3,1)) ms \nD9 \t\t$(round(D9*1e3,1)) ms \nD11 \t\t$(D11*1e3) ms; \nP1 \t\t$(P1) us \nP17 \t\t$(ramp) us \nP18 \t\t$(on) us \nP19 \t\t$(spoil) \nmax. gradient \t$(GPZ1) % \ngradient steps \t$(TD1)"; getnumber (grad. calib. const.) GCC; win -ca; # calculations; wks.addcol(g); wks.addcol(g^2k); wks.col1.name$=area; wks.col2.name$=lnArea; for (ii=1;ii

173 C. Computer Programs legend; layer.X.from=0; layer.X.majorTicks=5; layer.X.labelSubtype=2; label -xb g\+(2) * k [s/m\+(2)]; label -yl ln(area) [a.u.]; #ln-plot linear ? lin=-1; getyesno (continue ?) lin [fitting possible ?]; if (lin<=0) {return}; # Area vs. g; window -a %[%D,’.’]; wks.col4.type=1; wks.col3.type=4; wks.colSel(2,0); worksheet -s 1 0; worksheet -p 201; legend; layer.X.from=0; layer.X.to=GCC/10*GPZ1*1.3; layer.Y.from=0; label -xb g [G]; label -yl echo intensity [a.u.]; label -s -sa -q 1 \g(d) $(ld*1000) ms \n\g(D) $(bd*1000) ms \n\g(D) \-(rel) $(bdrel*1000) ms \n \nG\-(max) $(gpz1) % \nGCC $(gcc) \nTD1 $(td1) \nGamma $(Gamma) (%F) \nk $(k); window -r %H %[%D,5:6]; label -d 1800 300 -s %[%T,#6,\b] $(@D,D0); window -a %[%D,’.’]; wks.colsel(1,0); window -a %[%D,5:6];

174 C. Computer Programs

Derivative of the Echo Decay: derivative

The script has to be executed directly after the plot routine DiffPlot. It differ- 2 entiates the plot of ln(S/S0) (ln(area) in Origin) vs g k, yielding the diffusion coefficient. It first smoothes the decay function, then differentiates it by calcu- lating the slope of the linear fit of the 3 or 5 points around each point. Both the smoothing parameters and the number of points considered for the local derivative can be manually adjusted. The program then plots the diffusion coefficient as a function of g2k.

# derivative window -a %[%D,’.’]; if (wks.col5.type==1) {getyesno (overright data ?) overright}; if (overright==1) { delete %[%D,’.’]_smooth; delete %[%D,’.’]_deriv; }; if (break.abort==1) {break}; if (overright==0) {break}; #worksheet -v deriv; wks.addcol(deriv); wks.addcol(smooth); curve.data$=%[%D,’.’]_area; curve.result$=%[%D,’.’]_smooth; sl=2;sr=2;pd=2; getnumber (smoothleft) sl (smoothright) sr (polyDeg) pd [smooth paramters]; curve.smoothleftpts=sl; curve.smoothrightpts=sr; curve.polyDeg=pd; curve.sgsmooth(); sp=5;ddg=1; getnumber (smoothpoints) sp (derivative degree) ddg [derivative data]; wks.col3.type=2; wks.col4.type=4; col(smooth)=ln(col(smooth));

175 C. Computer Programs curve.data$=%[%D,’.’]_smooth; curve.result$=%[%D,’.’]_deriv; curve.i1=-1; curve.smoothPts=sp; curve.derivdeg=ddg; curve.deriv(); wks.col5.digits=2; mark -m %[%D,’.’]_deriv -b 1 -e 3; mark -m %[%D,’.’]_deriv -b (td1-3) -e td1; wks.colSel(1,0); col(deriv)=-col(deriv); worksheet -s 5 0; worksheet -p 202 ORIGIN %[%D,’.’]deriv; layer.X.from=0; layer.X.majorTicks=5; layer.X.labelSubtype=2; label -xb g\+(2) * k [s/m\+(2)]; label -yl D [m\+(2)/s];

The PGSTE Echo Decay for one MLV: Summation over R

A LabTalk script was necessary to rearrange the data from the C-program MLV_ste.c to obtain the echo decay for an MLV. MLV_ste.c writes the echo decay curves for several radii successively in the same ASCII file, all normalized to one. The LabTalk script must be applied in the following manner: first the ASCII file must be imported, creating a worksheet named ’ste’. Then the first part of the following script has to be executed, which creates several worksheets: ’umsort’, ’weight’ and ’sum’. The columns in the ’umsort’ worksheet then have to be enumerates to the right, creating column names A1, A2, A3, ... . Then the parameters ’RMIN’, ’RMAX’, ’RINC’, and ’XMAX’ have to be adjusted to the correct values (from MLV_ste.c) and the rest of the script can be executed. It creates the individual echo decay curves for each radius in worksheet ’umsort’ and applies surface weighting. The normalized sum over these echo decays is given in the worksheet ’sum’ as function of the gradient strength (’XMAX’).

176 C. Computer Programs

#MLVsum wks.template(origin); win -r %H weight; wks.template(origin); win -r %H sum; wks.template(origin); win -r %H umsort; for (jj=1;jj<=80;jj+1){wks.addcol()}; wks.col1.type=1;

#first execute until here !!!!!!!!!!

#MLV sort, input equvalent paramters from MLV_ste.c RMIN=10; RMAX=100; RINC=10; XMAX=200;

POINTS=XMAX; SPHERES=(RMAX-RMIN)/RINC+1; for (jj=1;jj<=SPHERES;jj+1){ for (ii=1;ii<=POINTS;ii+1){umsort_A$(jj)[$(ii)]=ste_A[$(ii)];}; mark -d ste_A -b 1 -e POINTS; window -a umsort; wks.col$(jj).label$=$(RMIN+(jj-1)*RINC); }; wks.labels(); #RADsphere-weighted; for (jj=1;jj<=SPHERES;jj+1){weight_A[jj]=RMIN+(jj-1)*RINC}; for (jj=1;jj<=SPHERES;jj+1){weight_B[jj]=weight_A[jj]^2*4*PI}; for (jj=1;jj<=SPHERES;jj+1){weight_B[jj]=weight_B[jj]/weight_B[$(SPHERES)]}; for (jj=1;jj<=SPHERES;jj+1){umsort_A$(jj)=weight_B[jj]*umsort_A$(jj)}; #sum all up for (ii=1;ii<=POINTS;ii+1){sum_B[ii]=0}; for (jj=1;jj<=SPHERES;jj+1){

177 C. Computer Programs for (ii=1;ii<=POINTS;ii+1){sum_B[ii]=sum_B[ii]+umsort_A$(jj)[$(ii)]}; }; #x-axis for (ii=1;ii<=POINTS;ii+1){sum_A[ii]=(ii-1+1e-10)/POINTS*XMAX}; #normalize; sum_B=sum_B/sum_B[1];

C.3 Pulse Programs used in this Work

The following pulse programs (Avance spectrometer) were used in this work. The most important parameters are explained after each pulse program. Common to all are: ze: set the buﬀer to zero ; p1: 90◦ hard pulse ; phx: pulse phase (x being pulse number, x=31 for receiver phase) go=y: starts acquisition and jumps to pointer y

C.3.1 Solid Echo

#include 1 ze 2 d1 p1 ph1 d2 p1 ph2 d3 go=2 ph31 30m wr #0 if #0 zd lo to 2 times td1 exit ph1= 0 2 1 3 0 2 1 3 ph2= 3 1 0 2 1 3 2 0

178 C. Computer Programs ph31= 0 2 1 3 0 2 1 3 5 d2, d3: transverse evolution delays τ ; d3 has to be set as d2 minus the receiver dead time

C.3.2 Saturation Recovery

#include 1 ze 2 d1 3 p1 ph0 d2 lo to 3 times 4 vd p1 ph1 go=2 ph31 30m wr #0 if #0 ivd zd lo to 2 times td1 exit ph0= 0 ph1= 0 3 2 1 ph31= 0 3 2 1

d2: saturation pulse train spacing - has to be longer than T2 to avoid echo eﬀects; vd: variable delay to allow T1 relaxation

C.3.3 PGSE

#include #include ze 10u

179 C. Computer Programs

5m pl1:f1 ;set rf power level start, d1 d11 UNBLKGRAMP if (l3) { dummy, p17:gp1*diff_ramp ;trapezoidal gradient pulse p18:gp2*diff_ramp ;trapezoidal gradient pulse p17:gp3*diff_ramp ;trapezoidal gradient pulse d2 ;gradient stabilization time d9 BLKGRAMP ;tau d11 UNBLKGRAMP ;unblank gradient amplifier lo to dummy times l13 } p1:f1 ph1 ;p1 :90 degree pulse p17:gp1*diff_ramp ;trapezoidal gradient pulse p18:gp2*diff_ramp ;trapezoidal gradient pulse p17:gp3*diff_ramp ;trapezoidal gradient pulse d2 ;gradient stabilization time d9 BLKGRAMP ;tau d11 UNBLKGRAMP ;unblank gradient amplifier p2:f1 ph2 ;180 degree pulse p17:gp1*diff_ramp ;trapezoidal gradient pulse p18:gp2*diff_ramp ;trapezoidal gradient pulse p17:gp3*diff_ramp ;trapezoidal gradient pulse d2 ;gradient stabilization time d10 BLKGRAMP ;tau go=start ph31 100u wr #0 if #0 zd igrad diff_ramp lo to start times td1 ;td1 = number of gradient steps 1s rf #0 ;reset file pointer lo to start times l1 ;l1 = Number of repetitions exit

180 C. Computer Programs ph1=0 ph2=0 1 2 3 ph31=0 2

The parameters can be conveniently set with the Bruker Tcl/Tk diff-script, but it is highly recommendable to check the values directly and maybe set them to even numbers. The Tcl/Tk diff-script has to be run afterwards! It creates files, which are necessary (e. g. for the diff_ramp parameter). d2: evolution delay τ ; d11: delay to unblank the gradient amplifier - should be set at least to 200 µs ; l3: set to one if dummy gradients are desired - which is recommended ; l13: number of dummy gradients - one is enough ; p17: ramp time for the ten steps of the trapezoidal gradient - 700 µs is enough for 100 % gradient strength ; p18: gradient duration ; d2: gradient stabilization time - about 200 µs ; d9: delay to blank the gradient amplifier - about 200 µs ; d10: gradient blanking before acquisition - set d10 equal d9 minus receiver dead time ; gp1-3: gradient pulses - the maximum magnitude is controlled by gpz1-3 (in %) ; diff_ramp: a counter taken from a file, which is created by the Bruker Tcl/Tk diff-script

C.3.4 PGSTE

#include #include ze 10u

5m pl1:f1 ;set rf power level start, d1 d11 BLKGRAMP if (l3) { dummy, p17:gp1*diff_ramp ;trapezoidal gradient pulse p18:gp2*diff_ramp ;trapezoidal gradient pulse p17:gp3*diff_ramp ;trapezoidal gradient pulse d2 ;gradient stabilization time d9 BLKGRAMP ;tau

181 C. Computer Programs

if (l11) { d11 UNBLKGRAMP ;unblank gradient amplifier p17:gp4 ;trapezoidal gradient pulse p19:gp5 ;trapezoidal gradient pulse p17:gp6 ;trapezoidal gradient pulse d2 d5 BLKGRAMP ;long tau d11 UNBLKGRAMP ;unblank gradient amplifier } lo to dummy times l13 } p1:f1 ph1 ;90 degree pulse p17:gp1*diff_ramp ;trapezoidal gradient pulse p18:gp2*diff_ramp ;trapezoidal gradient pulse p17:gp3*diff_ramp ;trapezoidal gradient pulse d2 ;gradient stabilization time d9 setnmr0^34^32^33 ;tau p1:f1 ph2 ;90 degree pulse if (l11) { d11 UNBLKGRAMP ;unblank gradient amplifier p17:gp4 ;trapezoidal gradient pulse p19:gp5 ;trapezoidal gradient pulse p17:gp6 ;trapezoidal gradient pulse d2 d5 BLKGRAMP ;long tau d11 UNBLKGRAMP ;unblank gradient amplifier } p1:f1 ph3 ;90 degree pulse p17:gp1*diff_ramp ;trapezoidal gradient pulse p18:gp2*diff_ramp ;trapezoidal gradient pulse p17:gp3*diff_ramp ;trapezoidal gradient pulse d2 ;gradient stabilization time d10 BLKGRAMP ;tau go=start ph31 100u wr #0 if #0 zd igrad diff_ramp

182 C. Computer Programs

lo to start times td1 ;td1 = number of gradient steps 1s rf #0 ;reset file pointer lo to start times l1 ;l1 = Number of repetitions exit ph0=0 ph1=0 0 2 2 1 1 3 3 2 2 0 0 3 3 1 1 ph2=0 2 0 2 1 3 1 3 2 0 2 0 3 1 3 1 ph3=0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 ph31=0 2 2 0 1 3 3 1 2 0 0 2 3 1 1 3 d5: longitudinal evolution delay ; p19: spoiler gradient duration - can be made very short, since two times p17 is set anyway for the spoiler ; l11: set to one if a spoiler gradient is desired ; l13: number of dummy gradients (including spoiler)

183

Appendix D

NMR Calculations

D.1 Product Operators

Product operators used in the calculations of the pulse sequences. The notation was taken from Spiess and Schmidt-Rohr [87].

Table D.1: Cartesian Spin Operators and Spherical Spin Operators

q 2 1 2 −→ Tb10 = Ibz Tb20 = {3Ib − I } √ 3 3 z √ Tb11 = −(Ibx + iIby)/ 2 Tb2±1 = (Tb10Tb1±1 + Tb1±1Tb10)/ 2 √ 2 Tb1−1 = −(Ibx − iIby)/ 2 Tb2±2 = (Tb1±1)

q 2 Table D.2: Eﬀect of the Quadrupolar Coupling ωQ 3 Tb20 on Spherical Tensor Operators q q ρ exp(−i 2 T ω t) ρ exp(i 2 T ω t) b 3 b20 Q b 3 b20 Q Ibz Ibz

Tb20 Tb20

(Tb22 ± Tb2−2)(Tb22 ± Tb2−2)

Ibx Ibx cos(ωQt) + i(Tb21 + Tb2−1) sin(ωQt)

(Tb21 + Tb2−1)(Tb21 + Tb2−1) cos(ωQt) + iIbx sin(ωQt)

Iby Iby cos(ωQt) + (Tb21 − Tb2−1) sin(ωQt)

(Tb21 − Tb2−1)(Tb21 − Tb2−1) cos(ωQt) − Iby sin(ωQt)

185 D. NMR Calculations

Table D.3: Eﬀect of Rotation (rf) Pulses on Spherical Tensor Operators

ρb exp(−iχIbx) ρb exp(iχIbx) exp(−iχIby) ρb exp(iχIby) Ibz Ibz cos χ − Iby sin χ Ibz cos χ + Ibx sin χ

Ibx Ibx Ibx cos χ − Ibz sin χ

Iby Iby cos χ + Ibz sin χ Iby 1 2 1 2 Tb20 Tb20 2 (3 cos χ − 1) Tb20 2 (3 cos χ − 1) q 3 q 3 −i 8 (Tb21 + Tb2−1) sin(2χ) − 8 (Tb21 − Tb2−1) sin(2χ) q 3 2 q 3 2 − 8 (Tb22 + Tb2−2) sin χ + 8 (Tb22 + Tb2−2) sin χ q 3 (Tb21 + Tb2−1) −i 8 Tb20 sin(2χ)(Tb21 + Tb2−1) cos χ − (Tb22 − Tb2−2) sin χ

+(Tb21 + Tb2−1) cos(2χ) 1 −i 2 (Tb22 + Tb2−2) sin(2χ) q 3 (Tb21 − Tb2−1)(Tb21 − Tb2−1) cos χ 2 Tb20 sin(2χ)

−i(Tb22 − Tb2−2) sin χ +(Tb21 − Tb2−1) cos(2χ) 1 − 2 (Tb22 + Tb2−2) sin(2χ) q 3 2 q 3 2 (Tb22 + Tb2−2) − 2 Tb20 sin χ 2 Tb20 sin χ 1 1 −i 2 (Tb21 + Tb2−1) sin(2χ) + 2 (Tb21 − Tb2−1) sin(2χ) 1 2 1 2 + 2 (Tb22 + Tb2−2)(cos χ + 1) + 2 (Tb22 + Tb2−2)(cos χ + 1) (Tb22 − Tb2−2)(Tb22 − Tb2−2) cos χ (Tb22 − Tb2−2) cos χ + (Tb21 + Tb2−1) sin χ

−i(Tb21 − Tb2−1) sin χ

D.2 Solid Echo Calculation

We start with the density operator σ(0) = Iz. The ﬁrst pulse, which is a y- π pulse will transform it into σ(1) = Iz cos θ + Ix sin θ. If θ = ( 2 ) the resulting density operator is σ(1) = Ix. Between the ﬁrst and the second pulse quadrupolar evolution takes place and the density operator evolves into:

σ(2) = Ibx cos(ωQτ1) + i(Tb21 + Tb2−1) sin(ωQτ1) (D.1) corresponding to the initial FID, neglecting relaxation. Then a second pulse is applied in direction perpendicular to the ﬁrst one, namely in x-direction. With

186 D. NMR Calculations

π ◦ Figure D.1: The solid echo pulse sequence with two ( 2 )-pulses 90 phase shifted.

π a ﬂip angle 2 this leads to:

σ(3) = Ibx cos(ωQτ1) − i(Tb21 + Tb2−1) sin(ωQτ1) (D.2)

Then the second evolution period t2 takes place:

σ(4) = Ibx cos(ωQ(τ1 − τ2)) + i(Tb21 + Tb2−1) sin(ωQ(τ1 − τ2)) (D.3)

which leads to an echo at τ2 = τ1.

187

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200 Danksagung

Den folgenden Personen m¨ochte ich gerne danken, da sie alle, teils mehr teils weniger, dazu begetragen habe, diese Arbeit zu einem sch¨onenAbschluss zu bringen:

Kerstin, meiner Frau, daf¨ur, dass sie einfach immer da und in der Lage war, aufkommende Zweifel sofort wegzuargumentierten.

Meinen Eltern Maria und Erich f¨ureinfach alles, was seit Juli ’74 passiert ist.

Ein spezieller Dank an Prof. Claudia Schmidt fürdie vielen Möglichkeiten, Vorträge zu halten und Tagungen zu besuchen. Insbesondere aber fürauch fürdie gewährten Freiheiten, die ständige Diskussionsbereitschaft und die angenehme Art im Um- gang.

Prof. Heino Finkelmann fürgroßzügiggewährte finanzielle Unterstützung. Ein Vorbild dafür,dass Forschung frei von materiellen Sorgen sein muss.

Burkhard Geil f¨urseine Hilfe bei der Erstellung der Formel des Modells der Diﬀusion in den MLV.

Alfred Hasenhindl, der täglich wieder beweist, dass Geduld und Planung meist die besseren Lösungenergeben fürseine große Hilfe bei NMR-technischen Dingen.

Meinen Mitarbeitern Rouven Streller und Johanna Becker f¨urIhr reges Interesse und Durchhalteverm¨ogen am NMR.

Allen Rennradfahrern, die so oft bereit waren, alles stehen und liegen zu lassen, um eine Abendrunde ¨uber den Kandel zu drehen. Insbesondere aber Kerstin, Michael, Felix, Willi, Olli, Simon, Dani, ...

All den anderen Kollegen, Mitarbeitern und Freunden der letzten Jahre im Ar- beitskreis fürdie angenehme und natürliche Arbeitsatmosphäre und die nicht zuletzt fürdie häufigeBereitschaft zu spontanen Pausen.