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Polymer Dynamics

2017 School on Soft Matters and Biophysics

Instructor: Alexei P. Sokolov, e-mail: sokolov@utk.edu

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”.

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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

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INTRODUCTIONPolymers are actively used in many technologies

Traditional technologies:

Even better perspectives for use in novel technologies:

Light-weight materials

Materials for energy generation

Materials for energy storage

Polymers have huge potential for applications in Bio-medical field

“Smart” materials, stimuli-responsive, self-healing …

Example:Boeing-787 “Dreamliner” 80% by volume is plastic

Example: polymer solar cells

Example: Polymer-based Li battery

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Polymers

Polymers are long molecules

They are constructed by covalentlybonded structural units – monomers

Main difference between properties ofsmall molecules and polymers is relatedto 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

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Definitionso Monomer – repeat unit, structural block of the polymer chaino Oligomer – short chains, ~ 3-10 monomerso Polymers – long chains, usually hundreds of monomerso Degree of polymerization – number of monomers in the polymer chain

Molecular weight: Weight of the molecule M [g/mol]. Usually there is adistribution of M. Two main characteristics: Number average Mn and weightaverage Mw. Polydispersity is usually characterized by the ratio PDI=Mw/Mn.

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

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Structure of a polymer chainPolymer 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):

C C C

C

OCH3CO

CCOOCH3

H3C

CH3 C C C

C

COOCH3

C

COOCH3

H3C H3C

syndiotactic (sPMMA) isotactic (iPMMA)

Random tacticity is called atactic polymer (e.g. aPMMA)

Chain formed by different monomers - co-polymer. Can be random (randomposition of monomers A, B in the chain) or block co-polymers (blocks ofmonomers A connected to blocks of monomers B).Mixtures of two polymers called polymer blend.

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Chain Statistics

Freely joined chain assumes no correlations between directions of different bond vectors, <cosij>=0 for ij.

0R

22 nlR Freely rotating chain assume a constant angle

Distribution of the end-to-end distances :

2

22/3

2 23exp

23)(

RR

RRPN

Center of mass: ;

Mean-squared radius of gyration:

N

nnCM R

NR

1

1

N

nCMng RR

NR

1

22 )(1

n

i

n

jij

n

i

n

jji

n

jj

n

iin lrrrrR

1 1

2

1 111

2 cos

cos1cos122

nlRn

For polyethylene (PE) =68o: <Rn2> 2nl2

n

iin rR

1The end-to-end vector: ; its average value:

Here l is the bond length and is the bond angle.l

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Characteristic ratio, C, and Statistics for Real Chains

Traditional definition: ; here n is the number, l is thelength and m0 is the mass of bonds (or monomers) in a backbone; M is themolecular weight of the chain. C is a characteristic ratio, known for most of thepolymers.

2

0

222 lmMCnlClNR RR

Another important parameter is the Kuhn segment length defined as lK=<R2>/Rmax.In many cases, another simplification is assumed: Rmax=NRlK=nl0, i.e. Rmax is equalto completely extended chain. It leads to lK=Cl0. This approximation istraditionally 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, is defined not by C only, there is at least oneparameter more.

Bead and Spring model is a traditionalapproximation in Polymer Physics. Itpresents a chain as a line of beadsconnected by springs. Equilibrium lengthof springs is assumed to be b~lK. Anglesbetween the springs are free to change.

b

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Kuhn SegmentIn order to bring real chain to the freely jointed chain:

here N is the number of random steps (Kuhn segments) and b is the length of the random steps (Kuhn segment length).

222 NbnlCRn

The contour length of the chain L=Nb, so the Kuhn length

The assumption in many textbooks: L=Rmaxnl; e.g. for alltrans PE ~0.83. Then

LR

b n2

lCRR

b n

max

2

Parameters for known polymers

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Entanglement

According to chain statistics, R2M. It means that chain density, ~M/R3M-1/2,decreases with increase in molecular weight. We know, however, that polymershave 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 isan argument that Me depends on packing efficiency of the chain: if chain can packwell Me is large.

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

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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.

Neutron Scattering, S(Q,)Spin-Echo

Back-sc. Time-of-Flight

Light Scattering, Iij(Q,)

Photon – Correlation Spectroscopy

Interferometry

Raman spectroscopy

Inelastic X-ray Scattering, S(Q,) High-ResolutionIXS

Dielectric Spectroscopy, *()Traditional dielectric spectroscopy

Quasi-optics,TDS

IR-spectr.

Scattering techniques have an advantage due to additional variable – wave-vector Q

Frequency map of polymer dynamics

FREQUENCY, 10-2 1 102 104 106 108 1010 1012 1014 Hz

Structural relaxation, -processVibrations

Fast relaxationSecondary relaxations

Chain dynamics: Rouse and terminal relaxation

Mechanical relaxation G

In an isotropic medium, time-space pair correlation function: ),()0,0(),( trnntrG

where N

jj trrtrn ))((),( presents coordinate of all atoms at time t.

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 positionr1=0, GS(r,t) gives a probability to find the same particle around position r at time t, and Gd(r,t)gives a probability to find another particle around position r at time t.

Intermediate scattering function is a space-Fourier transform of G(r,t): V

rdiqrtrGtqI 3)exp(),(),(

Definition from quantum mechanics: i j

jii j

ji trriqtiqriqrtqI )}]()0({exp[)](exp[)]0(exp[),(

Time-Fourier transform of I(q,t) gives dynamic structure factor:

dttqItiqS ),()exp(21),(

There are coherent and incoherent S(q,):

VSinc rdtqritrGdtqS 3)(exp),(

21),(

Vdcoh rdtqritrGdtqS 3)(exp),(

21),(

These equations were first introduced by van Hove [van Hove, Phys.Rev. 95, 249 (1954)].

What is Measured in Scattering: Time-Space Correlation Function

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Separation of translation, rotation and vibration)()()()( tutbtctr iiii ci(t) – center of mass position;

bi(t) – rotation about the center of mass;ui(t) – distance of the particle from its average position.

i j

jijiji tuuiqtbbiqtcciqtqS )}]()0({exp[)}]()0({exp[)}]()0({exp[),(

If we assume that all 3 vectors are independent:

Thus translation, rotation and vibration can be considered separately.

i j

VIBROTTR tqstqstqstqS ),(),(),(),(

Examples:Diffusion: Dttxx 2)]([ ; D is a diffusion coefficient; )exp(),( 2DtqtqsTR

)exp()exp(),( 2 tNDtqNtqSTRinc

22)exp()exp(

2),(

NdtttiNtSTR

inc

An exponential decay for S(q,t) A Lorentzian for S(q,)The decay rate (or width) q2

0.0

0.2

0.4

0.6

0.8

1.0

q1 > q2q2

q1

S(q,

t)

t 0

q1

q2

S(q,)

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Rotation: Let’s assume that there are two equal positions and molecule makes rotational jumps between r1 and r2 positions. In isotropic case:

qdqdqA

qdqdqA

rrdtqAqANtqS rotinc

sin1)2/1()(;sin1)2/1()(

;)/2exp()()(),(

10

1210

In the frequency domain:

2210 4

21)()()(),(

qAqANqS rotinc

0

S(q,)

0.0

0.2

0.4

0.6

0.8

1.0

qd~2

qA1

A0

S(q,

t)

Vibrations: For a vibration of atoms in isotropic media:

)()(2

exp),(

cos)0(1))(exp()()0(exp))(exp(),(

00

22

02

2222

ququqS

tuqqutquququtqS

vibinc

vibinc

0

S(q,)

-0+0

103 102 101 100 10-1

10-6

10-7

10-8

10-9

10-10

10-11

10-12

10-13

10-14

10-2 10-1 100 101 102

106

107

108

109

1010

1011

1012

1013

Freq

uenc

y [H

z]

Q [nm-1]

NeutronScattering

InelasticX-Ray

Ligh

t Sca

tterin

g

Tim

e [s

]

Length [nm]

Length- and time- scales accessible for different scattering techniques

Scattering cross-section

106 108 1010 1012 [Hz]

10-4 0.01 1 100 E [meV]Triple-axis

TOFNSE

Back-ScatteringSpectrometers

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 volumesor 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

RamanInterferometryPhoton Correlation Spectroscopy

10-2 102 106 1010 [Hz]1014

Spectrometers

Inelastic X-ray Scattering and X-PCSAdvantages:-scatter on charge density, i.e. nearly directly onatoms;-wavelength is comparable with interatomic distances;-good statistics.

Disadvantages:-bad frequency resolution, at present ~300 GHz;-high energy of photons, leads to samplesdegradation;-Only a few spectrometers are available in theworld

A comparison of inelastic X-ray and neutronscattering [Sokolov, et al. PRE 60, R2464(1999)] clearlydemonstrates much higher (~105) count rate inthe case of X-ray.

0

100

200

300

400

500

320 K

X-rays DHOIn

tens

ity [

cts/s

]

-10 -8 -6 -4 -2 0 2 4 6 8 100

50

100

150

200 140 K

E [meV]

0

20

40

60

80

100

Intensity neutrons [cts/h]

Neutrons

0

10

20

30

40

50

The absolute intensities were scaled at the first maximumQ=1.5A-1. The comparison demonstrates a reasonable agreement.

Brillouin X-ray and neutron scattering in PB

X-Ray Photon Correlation Spectroscopy(X-PCS) is analogous to usual PCS. It couldbe very important technique, but highenergy of photons destroys samples too fast.

Complex permittivity

00)exp()(11)( dttiti

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 amolecule in time and space: (0)

(t)

)(cos/)()0()( 2 ttt

where (t) is the angle between dipole vector at t=0 and t=t.

- - - - - - - -t<0 , t=0 , t=∞ ,

0i

i 0i

i

0E

0E

0EE

0EE

10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10

10 -1

10 0

10 1

''

'

''

'

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

1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 6 1 0 7 1 0 8 1 0 9 1 0 1 0

1 0 - 1

1 0 0

1 0 1

1 0 2

1 0 3

1 0 4

'

''

C o n d u c t i v i t yE P

Study of Dynamics with NMRNMR 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 iscalled spin-lattice relaxation, characterized by a time T1. The “lattice” means other molecules thatprovide exchange of energy via molecular motion.

Each spin has also precession about the direction of the magnetic field. The precession inducesmagnetic 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 thespin state. The spin-spin relaxation is characterized by T2.

Gradient-field NMR is an effective technique for measurements of molecular diffusion. It is basedon 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 foranalysis 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().