Anomalous Diffusion in Anisotropic Media Inauguraldissertation zur Erlangung der Doktorw¨urde der Fakult¨at f¨urChemie, Pharmazie und Geowissenschaften der Albert-Ludwigs-Universit¨at 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 iv 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 gra- dients 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.