Polymer Dynamics

2017 School on Soft Matters and Biophysics

Instructor: Alexei P. Sokolov, e-mail: [email protected]

Text: Instructor's notes with supplemental reading from current texts and journal articles.

Please, have the notes printed out before each lecture.

Books (optional): M.Doi, S.F.Edwards, “The Theory of Polymer Dynamics”. Y.Grosberg, A.Khokhlov, “Statistical Physics of Macromolecules”. J.Higgins, H.Benoit, “Polymers and Neutron Scattering”.

1 Course Content:

I. Introduction

II. Experimental methods for analysis of molecular motions

III. Vibrations

IV. Fast and Secondary Relaxations

V. Segmental Dynamics and Glass Transition

VI. Chain Dynamics and Viscoelastic Properties

VII. Rubber Elasticity

VIII. Concluding Remarks

2 INTRODUCTION Polymers are actively used in many technologies

Traditional technologies:

Even better perspectives for use in novel technologies: Example: Light-weight materials Boeing-787 “Dreamliner” 80% by volume is plastic

Materials for energy generation Example: polymer solar cells

Materials for energy storage Example: Polymer-based Li battery

Polymers have huge potential for applications in Bio-medical field

“Smart” materials, stimuli-responsive, self-healing … 3 Polymers

Polymers are long molecules

They are constructed by covalently bonded structural units – monomers

Main difference between properties of small molecules and polymers is related to the chain connectivity

Advantages of Polymeric Based Materials:  Easy processing, relatively cheap manufacturing  Light weight (contain mostly light elements like C, H, O)  Unique viscoelastic properties (e.g. rubber elasticity)  Extremely broad tunability of macroscopic properties 4 Definitions o Monomer – repeat unit, structural block of the polymer chain o Oligomer – short chains, ~ 3-10 monomers o Polymers – long chains, usually hundreds of monomers o Degree of polymerization – number of monomers in the polymer chain

Molecular weight: Weight of the molecule M [g/mol]. Usually there is a

distribution of M. Two main characteristics: Number average Mn and weight average Mw. Polydispersity is usually characterized by the ratio PDI=Mw/Mn.

Properties of a polymer (Tg, viscoelastic properties) can be changed by variation of molecular weight, without change of monomer chemistry. Properties can be also changed by changing macromolecular architecture: • Linear chain • Branched chain • Cyclic molecule •Star • Dendrimer

5 Structure of a polymer chain Polymer chain has a backbone (the main atoms of the chain, e.g. C-C-, or Si-O-Si)

and side groups (e.g. CH3, phenyl ring, etc.)

Tacticity – the way monomers are connected also affect the polymer properties. Example poly(methyl methacrylate) (PMMA):

OCH3 H3CH3C C CO CH3 C C C C C C C Random tacticity is called C C CO CO atactic polymer (e.g. aPMMA) CO OCH H3C OCH3 3 OCH3 syndiotactic (sPMMA) isotactic (iPMMA) Chain formed by different monomers - co-polymer.Canberandom (random position of monomers A, B in the chain) or block co-polymers (blocks of monomers A connected to blocks of monomers B). Mixtures of two polymers called polymer blend. 6 n Chain Statistics The end-to-end vector: R n   r i ; its average value: R  0 i1  n  n  n n n n R2  r  r   rr  l 2 cos n  i  j   i j  ij  i1  j1  i11j i11j Here l is the bond length and  is the bond angle. N 1 l Center of mass: RCM   Rn ; N n1 N 2 1 2 Mean-squared radius of gyration: Rg   (Rn  RCM ) N n1 Freely joined chain assumes no correlations between directions of different bond vectors, =0 for ij. R2  nl 2 Freely rotating chain assume a constant angle  1 cos R2  nl 2 n 1 cos o 2 2 For polyethylene (PE) =68 :  2nl Distribution of the end-to-end distances : 3 / 2      3   3R2  P (R)  exp  N  2   2   2 R   2 R      7 Characteristic ratio, C, and Statistics for Real Chains

2 2 2 M 2 Traditional definition:R  N RlR  Cnl  C l ; here n is the number, l is the m0 length and m0 is the mass of bonds (or monomers) in a backbone; M is the molecular weight of the chain. C is a characteristic ratio, known for most of the polymers. 2 Another important parameter is the Kuhn segment length defined as lK=/Rmax. In many cases, another simplification is assumed: Rmax=NRlK=nl0,i.e.Rmax is equal to completely extended chain. It leads to lK=Cl0. This approximation is traditionally used for characterization of real chains and bead size (Kuhn segment).

However, C characterizes just the coil size and not the bead (segment) length. It means that the bead size, lR,isdefinednotbyC only, there is at least one parameter more.

BeadandSpringmodelis a traditional approximation in Polymer Physics. It presents a chain as a line of beads connected by springs. Equilibrium length of springs is assumed to be b~lK. Angles b between the springs are free to change. 8 Kuhn Segment 2 2 2 In order to bring real chain to the freely jointed chain: Rn  Cnl  Nb here N is the number of random steps (Kuhn segments) and b is the length of the random steps (Kuhn segment length). R2 The contour length of the chain L=Nb, so the Kuhn length b  n L The assumption in many textbooks: L=R nl;e.g.forall max R2 trans PE ~0.83. Then n b   Cl Rmax

Parameters for known polymers

9 Entanglement

According to chain statistics, R2M. It means that chain density, ~M/R3M-1/2, decreases with increase in molecular weight. We know, however, that polymers have normal density. It happens because different chains interpenetrate each other.

As a result, they become entangled above particular molecular weight Me.

Me is important polymer parameter. It depends on polymer chemistry and rigidity. For example, Me~15,000 in polystyrene, and Me~2,000 in polybutadiene. There is an argument that Me depends on packing efficiency of the chain: if chain can pack well Me is large.

Entanglement results in very interesting polymer-specific properties, such as rubber-like elasticity even in a non-crosslinked state, extremely high viscosity and very long relaxation times.

10 Different Types of Molecular Motion

I. Vibrations Oscillations around fixed positions (Center of mass does not move). II. Translation motion Change of the position (Center of mass shifts), Diffusion is one of the examples.

III. Rotation

IV. Conformational jumps. Example – transitions of cyclohexylene ring.

The last three (II-IV) can be considered as a relaxation motion.

Usually one can separate contributions from these motions because in most cases they occur on different time scales. However, these motions can be coupled and one should consider them all together or use some approximations. 11 Frequency map of polymer dynamics Chain dynamics: Rouse and terminal relaxation Structural relaxation, -process Vibrations Secondary relaxations Fast relaxation FREQUENCY,  10-2 1102 104 106 108 1010 1012 1014 Hz Mechanical relaxation G

Dielectric Spectroscopy, *() Quasi-optics,TDS

Traditional dielectric spectroscopy IR-spectr.

Light Scattering, Iij(Q,) Interferometry

Photon – Correlation Spectroscopy Raman spectroscopy

Spin-Echo Neutron Scattering, S(Q,) Back-sc. Time-of-Flight

Inelastic X-ray Scattering, S(Q,) High-Resolution IXS

Scattering techniques have an advantage due to additional variable – wave-vector Q What is Measured in Scattering: Time-Space Correlation Function

In an isotropic medium, time-space pair correlation function: G(r,t)  n(0,0)n(r,t) N where n(r,t)   (r  rj (t)) presents coordinate of all atoms at time t. j

G(r,t) has self and distinct part, G(r,t)=GS(r,t)+Gd(r,t). If at time t1=0 a particle was at position r1=0, GS(r,t) gives a probability to find the same particle around position r at timet,andGd(r,t) gives a probability to find another particle around position r at time t.

3 Intermediate scattering function is a space-Fourier transform of G(r,t): I(q,t)  G(r,t)exp(iqr)d r V

Definition from quantum mechanics: I(q,t)   exp[iqri (0)]exp[iqrj (t)]   exp[iq{ri (0)  rj (t)}] ij ij  1  Time-Fourier transform of I(q,t) gives dynamic structure factor: S(q, )   exp(it)I(q,t)dt 2     1 3 Sinc (q, )    dt GS (r,t)expi(qr t) d r There are coherent and incoherent S(q,): 2  V  1 3 Scoh (q, )   dt Gd (r,t)expi(qr t) d r 2  V

These equations were first introduced by van Hove [van Hove, Phys.Rev. 95, 249 (1954)]. Separation of translation, rotation and vibration c (t) – center of mass position; ri (t)  ci (t)  bi (t)  ui (t) i bi(t) – rotation about the center of mass; ui(t) – distance of the particle from its average position.

S(q,t)   exp[iq{ci (0)  c j (t)}]exp[iq{bi (0)  bj (t)}]exp[iq{ui (0)  u j (t)}] ij If we assume that all 3 vectors are independent: S(q,t)  sTR (q,t)  s ROT (q,t)  sVIB (q,t) ij Thus translation, rotation and vibration can be considered separately. Examples: TR 2 Diffusion: [x  x(t)]2  Dt ; D is a diffusion coefficient; s (q,t)  exp(q Dt)

TR 2 TR N N  Sinc (q,t)  N exp(q Dt)  N exp(t) Sinc ( ,t)  exp(i t)exp( t )dt  2  2   2 An exponential decay for S(q,t) A Lorentzian for S(q,) 1.0 q > q S(q,) The decay rate (or width) q2 0.8 1 2 q 2 0.6 q 2

0.4 S(q,t) q 1

0.2 q 1

0.0 0 t  15 Rotation: Let’s assume that there are two equal positions and molecule makes rotational jumps between r and r positions. In isotropic case: 1.0 1 2 A 0 S rot (q,t)  NA (q)  A (q)exp(2t / );d  r  r 0.8 inc 0 1 2 1 0.6

0.4  sin qd   sin qd  S(q,t) A (q)  (1/ 2) 1 ; A (q)  (1/ 2) 1 0   1   0.2 qd~2 A  qd   qd  1 0.0  S(q,) q

In the frequency domain:   rot  1 2  Sinc (q, )  N A0 (q) ( )  A1(q)  4   2 2   0 

Vibrations: For  a vibration of atoms in isotropic media:

S vib (q, t)  exp(  (qu ) 2 ) exp qu (0)qu (t)  exp(  (qu ) 2 )1  q 2u 2 (0) cos t inc  2 2   vib 2  u q   S(q,) S inc (q, )  exp  qu  0  ( 0  )  ( 0   )   2 

- + 0 0

0  16 Length [nm] 103 102 101 100 10-1 -14 1013 10 Inelastic -13 1012 10 X-Ray -12 1011 10 -11 1010 10 Neutron -10 109 10 Scattering -9 108 10 -8 107 10 Time [s] Light Scattering Frequency [Hz] -7 106 10 10-6 10-2 10-1 100 101 102 Q [nm-1]

Length- and time- scales accessible for different scattering techniques

Scattering cross-section

Spectrometers Back-Scattering TOF NSE Triple-axis 10-4 0.01 1 100 E [meV] 106 108 1010 1012  [Hz] Light Scattering Spectroscopy

Advantages of the light scattering are: -Needs relatively inexpensive experimental setup; -broad frequency range, three techniques cover total 10-3 –1014 Hz; -good statistics combined with high frequency resolution; -small probing spot ~1-10 m allows measurements of very small volumes or of high lateral resolution; -polarization of light provides additional information on type of motion; -in most cases the spectra do not require corrections.

Disadvantages of the light scattering: -does not measure atomic motion directly, measures fluctuation of (r,t); -wavelength of light, ~300-1000 nm, is much larger than characteristic interatomic distances; -in many cases requires samples with good optical quality; -very sensitive to impurities. Light Scattering Intensity

Spectrometers Interferometry Photon Correlation Spectroscopy Raman

10-2 102 106 1010 1014  [Hz] Inelastic X-ray Scattering and X-PCS Brillouin X-ray and neutron scattering in PB Advantages: 500 -scatter on charge density, i.e. nearly directly on 320 K 100 400 atoms; 80 -wavelength is comparable with interatomic distances; 300 -good statistics. 60 200 40 Disadvantages: Intensity neutrons[cts/h] -bad frequency resolution, at present ~300 GHz; 100 20 -high energy of photons, leads to samples 0 0 Neutrons 50 degradation; 200 140 K X-rays

-Only a few spectrometers are available in the Intensity [cts/s] DHO 40 world 150 A comparison of inelastic X-ray and neutron 30 100 scattering [Sokolov, et al. PRE 60, R2464(1999)] clearly 20 5 demonstrates much higher (~10 ) count rate in 50 the case of X-ray. 10 0 0 -10-8-6-4-20246810 E [meV] X-Ray Photon Correlation Spectroscopy The absolute intensities were scaled at the first maximum (X-PCS) is analogous to usual PCS. It could Q=1.5A-1. be very important technique, but high The comparison demonstrates a reasonable agreement. energy of photons destroys samples too fast.

       ( )  1  Complex permittivity   1 i (t)exp(it)dt 0   0 where 0 and  are the limiting low and high frequency values of the ; (t) is a macroscopic relaxation function that can be expressed as the time- autocorrelation function for the reorientational motions of dipole vector  of a molecule in time and space: (0)  (t) 2   (t)  (0) (t) /  cos (t) 

where (t) is the angle between dipole vector at t=0 and t=t. 1  10 ' E 0  '' 100 ''  ------'    t=∞ ,    t<0 ,E  0 t=0 , E  E 0 E  E 0

   0 -1   i  0  i 10 i i 100 101 102 103 104 105 106 107 108 109 1010 

However, many samples have ionic conductivity , that affects ’’() spectrum, and electrode polarization (EP) effect that contributes to ’() spectrum.

104 Conductivity 103 EP 2

 10 ''

101  ' 

100

10-1 100 101 102 103 104 105 106 107 108 109 1010 

Study of Dynamics with NMR NMR measures motions of nuclear spin. This method provides direct measurements of atomic motion. However, the method provides very local information only.

An equilibrium population of spins in a magnetic field can be disturbed by resonant radio- frequency. The process by which the spin system attains equilibrium from a non-equilibrium state is called spin-lattice relaxation, characterized by a time T1. The “lattice” means other molecules that provide exchange of energy via molecular motion.

Each spin has also precession about the direction of the magnetic field. The precession induces magnetic field that interacts with spins of other nucleus precessing at the same frequency. This spin- spin interactions do not change the overall energy of the spin system, but shortens life-time of the spin state. The spin-spin relaxation is characterized by T2.

Gradient-field NMR is an effective technique for measurements of molecular diffusion. It is based on analysis of the NMR signal in the magnetic field with gradient along some direction.

Multi-dimensional NMR (sequence of many resonance radio-frequencies) is a very effective tool for analysis of molecular motion, dynamic heterogeneity.

Recently fast filed-cycling NMR has been applied to analysis of polymer dynamics [Herrmann, et al.

Macromolecules 42, 5236 (2009)]. In that case, one measures T1() by changing the magnetic field H0. This allows to calculate the susceptibility function ”() in a broad frequency range from T1().