1. Last Lecture:
The radius-of-gyration of a polymer, Rg, is 1/6 of its root-mean-square end-to-end distance <R2>1/2.
Excluded volume effects cause polymer molecules to swell (in a self-avoiding walk).
Elastic (entropic) effects cause a polymer molecule to coil up.
Polymer-solvent interactions, described by the c-parameter, can favour tight polymer coiling into a globule (large c) or swelling (low c).
Thus there is a competition between three effects!

2. Last Lecture:
When c = 1/2, excluded volume effects are exactly balanced by polymer/solvent interactions. Elastic effects lead to a random coil: <R2>1/2 ~ aN1/2
When c < 1/2, excluded volume effects are dominant over polymer/solvent interactions. In competition with elastic effects, they lead to a swollen coil: <R2>1/2 ~ aN3/5
When c > 1/2, polymer/solvent interactions are dominant over excluded volume effects. Along with elastic effects, they lead to polymer coiling: a globule results.
A rubber (or elastomer) is created from a network of crosslinked polymer molecules.
The shear modulus, G, for a rubber with n crosslinks per unit volume is G = nkT. With a bulk density of r and an average molar mass, Mx, between crosslinks:

3. Polymer Reptation, Viscosity and Diffusion 3SCMP
23 March, 2006
Lecture 10
See Jones’ Soft Condensed Matter, Chapt. 5

4. Response of Soft Matter to a Constant Shear Stress: Viscoelasticity
Soft matter, such as a polymer melt or solution, usually exhibits a combination of viscous and elastic behaviour.
Viscous response
Elastic response t
The shear strain, and hence the shear modulus, both change over time: gs(t) and

5. Response of Soft Matter to a Constant Shear Stress: Viscoelasticity
Slope: t
t is the relaxation time t
We see that 1/Go  (1/h)t
An alternative expression for viscosity is thus h  Got

6. Examples of Viscoelasticity

7. Viscoelasticity of Soft Matter
With a constant shear stress, ss, the shear modulus G can change over time:
G(t) is also called the “stress relaxation modulus”. t
G(t) can also be determined by applying a constant strain, gs, and observing stress relaxation over time: s

8. Example of Viscoelasticity

9. Relaxation Modulus for Polymer Melts
Elastic t
Viscous flow
Gedde, Polymer Physics, p. 103

10. Experimental Shear Relaxation Moduli
Poly(styrene) GP
High N
Low N
~ 1/t
G.Strobl, The Physics of Polymers, p. 223

11. Relaxation Modulus for Polymer Melts
At very short times, G is high. The polymer has a glassy response.
The glassy response is determined by the intramolecular bonding.
G then decreases until it reaches a “plateau modulus”, GP. The value of GP is independent of N for a given polymer: GP ~ N0.
After a time, known as the terminal relaxation time, tT, viscous flow starts (G decreases with time).
Experimentally, it is found that tT is longer for polymers with a higher N. Specifically, tT ~ N3.4
Previously, we said that the relaxation time is related to ratio of h to G at the transition between elastic and viscous behaviour. Thus: tT~h/GP

12. Viscosity of Polymer Meltsho
Extrapolation to low shear rates gives us a value of the “zero-shear-rate viscosity”, ho.
Shear thinning behaviour
For comparison: h for water is 10-3 Pa s at room temperature.
Poly(butylene terephthalate) at 285 ºC

13. h ~ tTGP Scaling of Viscosity: ho ~ N3.4 ho ~ N3.4 N0 ~ N3.4 Why? 3.4
Data shifted for clarity!
Viscosity is shear-strain rate dependent. Usually measure in the limit of a low shear rate: ho
h ~ tTGP
3.4
ho ~ N3.4 N0 ~ N3.4
Universal behaviour for linear polymer melts
Applies for higher N: N>NC
Why?
G.Strobl, The Physics of Polymers, p. 221

14. Concept of “Chain” Entanglements
If the molecules are sufficient long (N >100 - corresponding to the entanglement mol. wt., Me), they will entangle with each other.
Each molecule is confined within a dynamic “tube”.
Tube
G.Strobl, The Physics of Polymers, p. 283

15. Network of Entanglements
There is a direct analogy between chemical crosslinks in rubbers and “physical” crosslinks that are created by the entanglements.
The physical entanglements can support stress (for short periods up to a time tT), creating a “transient” network.

16. Plateau Modulus for Polymer Melts
• Recall that the elastic shear modulus of a network depends on molecular weight between crosslinks, Mc. In a polymer melt, GP therefore depends on Me.
• That is, GP ~ N0 (where N are the number of repeat units in the molecule).
• Just as for a polymer network:
• It makes sense that Me is independent of N - consistent with measurements of GP versus N.

17. Entanglement Molecular Weights, Me, for Various Polymers
Me (g/mole)
Poly(ethylene) 1,250
Poly(butadiene) 1,700
Poly(vinyl acetate) 6,900
Poly(dimethyl siloxane) 8,100
Poly(styrene) 19,000

18. An Analogy!
There are obvious similarities between a collection of snakes and the entangled polymer chains in a melt.
The source of continual motion on the molecular level is thermal energy, of course.

19. Reptation Theory Polymer molecules “dis-entangle” after a time tT.x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
Polymer molecules “dis-entangle” after a time tT.
Chain entanglements create restraints to other chains, defining a “tube” through which they must travel.
The process by which a polymer chain moves through its tube formed by entanglements is called “reptation”.
Reptation (from the Latin reptare: to crawl) is a snake-like diffusive motion that is driven by thermal motion.
Models of reptation consider each repeat unit of the chain as diffusing through a tube with a drag coefficient, xseg.
The tube is considered to be a viscous medium surrounding each segment.
For a polymer consisting of N units: xpol = Nxseg.

20. Experimental Evidence for Reptation
Fluorescently-stained DNA molecule
Chain follows the path of the front
Initial state
Stretched
Chu et al., Science (1994) 264, p. 819.

21. Development of Reptation Scaling Theory
Pierre de Gennes (Paris) developed the concept of polymer reptation and derived scaling relationships.
Sir Sam Edwards (Cambridge) devised tube models and predictions of the shear relaxation modulus.
In 1991, de Gennes was awarded the Nobel Prize for Physics.

22. Polymer Diffusion along a Tube
In our discussion of colloids, we defined an Einstein diffusion coefficient as:
If we consider the drag on a polymer molecule, we can express D for the diffusion of the molecule in a tube created by an entangled network as:
Hence, the rate of tube diffusion is inversely related to the length of the molecules.

23. Tube Relaxation Time, ttube
The polymer terminal relaxation time, tT, must be comparable to the time required for a polymer to diffuse out of its confining tube, ttube.
The length of the tube must be comparable to the entire length of the polymer molecule (contour length): Na
By definition, a diffusion coefficient, D, is proportional to the square of the distance travelled (x2) divided by the time of travel, t.
For a polymer escaping its tube:
Comparing to our previous Einstein definition:
We thus can derive a scaling relationship for ttube:

24. Scaling Prediction for Viscosity
We see that
We have also found that GP ~ N0
Recalling that h ~ Gt
Then:
But, recall that experiments find h ~ N3.4

25. Polymer Self-Diffusion
Time = 0
Time = t X
Reptation theory can also describe the self-diffusion of polymers, which is the movement of the centre-of-mass of a molecule by a distance x in a matrix of the same type of molecules.
In a time ttube, the molecule will diffuse the distance of its entire length. But, its centre-of-mass will move a distance on the order of its r.m.s. end-to-end distance, R. R
In a polymer melt: <R2>1/2 ~ aN1/2

26. Polymer Self-Diffusion CoefficientX
A self-diffusion coefficient, Dself, can then be defined as:
But we have derived this scaling relationship:
Substituting, we find:
Larger molecules are predicted to diffuse much more slowly than smaller molecules.

27. Testing of Scaling Relation: D ~N -2
Experimentally, D ~ N-2.3 -2
Data for poly(butadiene)
Jones, Soft Condensed Matter, p. 92
M=Nmo

28. “Failure” of Simple Reptation Theory
Reptation theory predicts h ~ N3, but experimentally it varies as N3.4. It predicts Dself ~ N-2, but it is found to vary as N-2.3.
One reason for this slight disagreement between theory and experiment is attributed to “constraint release”.
The constraining tube around a molecule is made up of other entangled molecules that are moving. The tube has a finite lifetime.
A second reason for disagreement is attributed to “contour length fluctuations” that are caused by Brownian motion of the molecule making its end-to-end distance change continuously over time.
Improved theory is getting even better results!

29. Relevance of Polymer Self-Diffusion
When welding two polymer surfaces together, such as in a manufacturing process, it is important to know the time and temperature dependence of D. R
Good adhesion is obtained when the molecules travel a distance comparable to R, such that they entangle with other molecules.

30. Application of Theory: Electrophoresis
DNA is a long chain molecule consisting of four different types of repeat units.
DNA can be reacted with certain enzymes to break specific bonds along its “backbone”, creating segments of various sizes.
Under an applied electric field, the segments will diffuse into a gel (crosslinked molecules in a solvent) in a process known as gel electrophoresis.
Reptation theory predicts that shorter chains will diffuse faster than longer chains.
Measuring the diffusion distances in a known time enables the determination of N for each segment and hence the position of the bonds sensitive to the enzyme.

31. Application of Theory: Electrophoresis
From Giant Molecules

32. Relaxation Modulus for Polymer Melts
Viscous flow
Gedde, Polymer Physics, p. 103

33. Problem Set 5
1. A polymer with a molecular weight of 5 x 104 g mole-1 is rubbery at a temperature of 420 K. At this temperature, it has a shear modulus of 200 kPa and a density of 1.06 x 103 kg m-3. What can you conclude about the polymer architecture? How would you predict the modulus to change if the molecular weight is (i) doubled or (ii) decreased by a factor 5?
2. Two batches of poly(styrene) with a narrow molecular weight distribution are prepared. If the viscosity in a melt of batch A is twice that in a melt of batch B, what is the predicted ratio of the self-diffusion coefficient of batch A over that of batch B? Assume that the reptation model is applicable.
3. The viscosity h for a melt of poly(styrene) with a molecular weight of 2 x 104 g mole-1 is given as X. (This molecular weight is greater than the entanglement molecular weight for poly(styrene)). (i) According to the reptation theory, what is h for poly(styrene) with a molecular weight of 2 x 105 g mole-1. (ii) Assuming that poly(styrene) molecules exist as ideal random coils, what is the ratio of the root-mean-square end-to-end distance for the two molecular weights?
4. The plateau shear modulus (GP) of an entangled polymer melt of poly(butadiene) is 1.15 MPa. The density of a poly(butadiene) melt is 900 kg m-3, and the molecular mass of its repeat unit is 54 g mole-1.
(i) Calculate the molecular mass between physical entanglements.
(ii) The viscosity (in Pa s) of the melt can be expressed as a function of the degree of polymerisation, N, and temperature, T (in degrees Kelvin), as:
Explain why h has this functional form.
(iii) Estimate the self-diffusion coefficient of poly(butadiene) in the melt at a temperature of 298 K when it has a molecular mass of 105 g mole-1.