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Last Lecture:

Last Lecture:. The root-mean-squared end-to-end distance, < R 2 > 1/2 , of a freely-jointed polymer molecule is N 1/2 a , when there are N repeat units, each of length a . The radius-of-gyration of a polymer, R g , is 1/6 of its root-mean-square end-to-end distance < R 2 > 1/2 .

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Last Lecture:

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  1. Last Lecture: • The root-mean-squared end-to-end distance, <R2>1/2, of a freely-jointed polymer molecule is N1/2a, when there are N repeat units, each of length a. • The radius-of-gyration of a polymer, Rg, is 1/6 of its root-mean-square end-to-end distance <R2>1/2. • Polymer coiling is favoured by entropy.The elastic free energy of a polymer coil is given as • Thinner lamellar layers in a diblock copolymer will increase the interfacial energy and are not favourable. Thicker layers require chain stretch and likewise are not favourable! A compromise in the lamellar thickness, d, is reached as:

  2. Last Lecture: • Elastic (entropic) effects cause a polymer molecule to coil up. • Excluded volume effects cause polymer molecules to swell (in a self-avoiding walk). • Polymer-solvent interactions, described by thec-parameter, also have an effect. • Thus there is a competition between three effects! • When c= 1/2, excluded volume effects are exactly balanced by polymer/solvent interactions. Elastic effects (from an entropic spring) lead to a random coil: <R2>1/2 ~ aN1/2 • When c< 1/2, excluded volume effects dominate 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. They lead to polymer coiling: a globule results.

  3. PH3-SM (PHY3032) HE3 Soft Matter Lecture 10 Polymer Elasticity, Reptation, Viscosity and Diffusion 13 December, 2010 See Jones’ Soft Condensed Matter, Chapt. 5

  4. Rubber Elasticity A rubber (or elastomer) can be created by linking together linear polymer molecules into a 3-D network. Chemical bonds between polymer molecules are called “crosslinks”. Sulphur can crosslink natural rubber. To observe “stretchiness”, the temperature should be > Tg for the polymer.

  5. We define an extension ratio, l, as the dimension after a deformation divided by the initial dimension: Bulk: Strand: l lo Affine Deformation With an affine deformation, the macroscopic change in dimension is mirrored at the molecular level.

  6. z z R = lxxo + lyyo + lzzo Ro = xo+ yo+ zo Single Strand Ro R y y x x Transformation with Affine Deformation z Bulk: y x If non-compressible (volume conserved): lxlylz=1 R2 = x2+y2+z2

  7. Initially: Finding DS: Entropy Change in Deforming a Strand The entropy change when a single strand is deformed, DS, can be calculated from the difference between the entropy of the deformed coil and the unperturbed coil: DS = S(R) - S(Ro) = S(lxxo, lyyo, lzzo) - S(xo, yo, zo) We recall our expression for the entropy of a polymer coil with end-to-end distance, R:

  8. But, if the conformation of the coil is initially random, then <xo2>=<yo2>=<zo2>, so: For a random coil, <R2>=Na2, and alsoR2 = x2+y2+z2 = 3x2, so we see: Substituting: This simplifies to: Entropy Change in Polymer Deformation

  9. For a one-dimensional stretch in the x-direction, we can say thatlx = l. Incompressibility then implies Thus, for a one-dimensional deformation oflx=l: The corresponding change in free energy: (F = U - ST) will be DFfor Bulk Deformation If there are n strands per unit volume, then DS per unit volume for bulk deformation: If the rubber is incompressible (volume is constant), thenlxlylz =1.

  10. In Lecture 3, we saw thatsT = Ye. The strain,e, for a 1-D tensile deformation is Substituting inL/Lo = e + 1: A Force for Rubber Deformation At the macro-scale, if the initial length isLo, thenl = L/Lo. Realising thatDFbulkis an energy of deformation (per unit volume), then dF/deTis the force, F (per unit area, A) for the deformation, i.e. the tensile stress,sT.

  11. Young’s and Shear Modulus for Rubber This is an equation of state, relating together F, L and T. In the limit of small strain,sT 3nkTe, and the Young’s modulus is thus Y = 3nkT. The Young’s modulus can be related to the shear modulus, G, by a factor of 3 to find a very simple result: G = nkT This result tells us something quite fundamental. The elasticity of a rubber does not depend on the chemical make-up of the polymer nor on how it is crosslinked. Gdoes depend on the crosslink density. To make a higher modulus, more crosslinks should be added so that the lengths of the segments become shorter.

  12. Experiments on Rubber Elasticity Rubbers are elastic over a large range ofl! Strain hardening region: Chain segments are fully stretched! Treloar, Physics of Rubber Elasticity (1975)

  13. Alternative Equation for a Rubber’s G  For a rubber with a known density, r, in which the average molecular mass of a strand is Mx (m.m. between crosslinks), we can write: strand Looking at the units makes this equation easier to understand: Substituting forn: We have shown that G = nkT, where n is the number of strands per unit volume.

  14. Network formed by H-bonding of small molecules Blue = ditopic (able to associate with two others) Red = tritopic (able to associate with three others) H-bonds can re-form when surfaces are brought into contact. For a video, see: http://news.bbc.co.uk/1/hi/sci/tech/7254939.stm P. Cordier et al., Nature (2008) 451, 977

  15. G(t) can also be determined by applying a constant strain,gs, and observing stress relaxation over time: s t 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”.

  16. Example of Viscoelasticity High molecular weight polymer dissolved in water. Elastic recovery under high strain rates, and viscous flow under lower strain rates.

  17. tT Relaxation Modulus for Polymer Melts Elastic tT= terminal relaxation time Viscous flow Gedde, Polymer Physics, p. 103

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

  19. 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 thattTis longer for polymers with a higher N. Specifically,tT ~ N3.4 • Previously in Lecture 3, we said that in the Maxwell model, the relaxation time is related to ratio ofhto G at the transition between elastic and viscous behaviour. That is:tT~h/GP

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

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

  22. 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.

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

  24. 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 timetTube),creating a “transient” network.

  25. Plateau Modulus for Polymer Melts • Recall that the elastic shear modulus of a network depends on molecular weight between crosslinks, Mx. In a polymer melt, GP therefore depends on the molecular weight between entanglements, Me. • That is, GP ~ N0 (where N is the number of repeat units in the molecule). • Using an equation for the polymer melt that is analogous to a crosslinked network: • It makes sense that Me is independent of N - consistent with experimental measurements of GP versus t for various values of M.

  26. 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 Me corresponds to the Nc that is seen in the viscosity data.

  27. Reptation Theory 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,tTube. • 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.

  28. 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.

  29. 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.

  30. 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: Polymer Diffusion along a Tube In our discussion of colloids, we defined an Einstein diffusion coefficient as: Hence, the rate of 1-D tube diffusion is inversely related to the length of the molecules.

  31. Tube Relaxation Time,ttube For a polymer escaping its tube: Comparing to our previous Einstein definition: We thus can derive a scaling relationship forttube: 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.

  32. Then: Scaling Prediction for Viscosity We can think oftTas the average time required for chains to escape the confinement of their tube,ttube. We see thatwhich is comparable to experiments in whichtT ~ N3.4 We have also found that GP ~ N0 Recalling thath ~ GtT But, recall that experiments findh ~ N3.4. Agreement is not too bad!

  33. R 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 timettube, 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. In a polymer melt: <R2>1/2 ~ aN1/2

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

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

  36. “Failure” of Simple Reptation Theory • Reptation theory predictsh~N3, but experimentally it varies as N3.4. • Theory 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!

  37. 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.

  38. Application of Theory: Electrophoresis From Giant Molecules

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

  40. Stages of Interdiffusion at Polymer/Polymer Interfaces Chain extension across the interface:likely failure by chain “pull-out” Chain entanglement across the interface: possible failure by chain scission (i.e. breaking) Interfacial wetting:weak adhesion from van der Waals attraction

  41. Environmental SEM Immediate film formation upon drying! Tg of latex  5 °C; Hydrated film Example of Good Coalescence Bar = 0.5mm • Particles can be deformed without being coalesced. (Coalescence means that the boundaries between particles no longer exist!) J.L. Keddie et al., Macromolecules (1995) 28, 2673-82.

  42. Strength Development with Increasing Diffusion Distance s Full strength is achieved when d is approximately the radius of gyration of the polymer,Rg. Rg d K.D. Kim et al, Macromolecules (1994) 27, 6841

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

  44. Problem Set 6 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.

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