Confined polymers More complex systems-3

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Initial considerations Reptation: snake-like thermal motion of very long, linear, entangled macromolecules in a melt or concentrated polymer solutions Theories based on diffusion of single chain molecules predict reptation time proportional to the cube of the contour length These phenomena may change in confinement because of the finite space available and MW dependence of the chain dynamics

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.

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

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

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) expression for the entropy of a polymer coil with end-to-end distance, R: Initially: Finding DS:

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

DF for 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), then lxlylz =1. For a one-dimensional stretch in the x-direction, we can say that lx = l. Incompressibility then implies Thus, for a one-dimensional deformation of lx = l: The corresponding change in free energy: (F = U - ST) will be

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

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. G does depend on the crosslink density. To make a higher modulus, more crosslinks should be added so that the lengths of the segments become shorter.

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. G does depend on the crosslink density. To make a higher modulus, more crosslinks should be added so that the lengths of the segments become shorter.

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

Alternative Equation for a Rubber’s G We have shown that G = nkT, where n is the number of strands per unit volume.  strand 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: Looking at the units makes this equation easier to understand: Substituting for n:

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

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

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

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

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

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 In the Maxwell model, the relaxation time is related to ratio of h to G at the transition between elastic and viscous behaviour. That is: tT~h/GP

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: h for water is 10-3 Pa s at room temperature. Poly(butylene terephthalate) at 285 ºC From Gedde, Polymer Physics

Scaling of Viscosity: ho ~ N3.4 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 G.Strobl, The Physics of Polymers, p. 221

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. Tube G.Strobl, The Physics of Polymers, p. 283

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.

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 tTube), creating a “transient” network.

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.

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.

Reptation Theory Polymer molecules “dis-entangle” after a time, tTube. 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.

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

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.

Polymer Diffusion along a Tube Einstein diffusion coefficient : 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 1-D tube diffusion is inversely related to the length of the molecules.

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:

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

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

Polymer Self-Diffusion Coefficient X 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.

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

“Failure” of Simple Reptation Theory Reptation theory predicts h ~ 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!

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.

scanning electron micrograph of PS nanorods MW: 5.91 x105 g/mol prepared with alumina membranes with cylindrical nanoscopic pores Pore diameter (15 nm) < radius of gyration of the polymer (~22 nm) The configuration of the polymer is perturbed with an entropic penalty due to chain deformation The capillary force is enough to draw the polymer into the pore

measurement of the capillary rise small angle X-Ray scattering(SAXS)

reduction of the integrated intensity as PS fills the pores is proportional to the fraction of unfilled pores

Flux of PS into the pores flux can be determined from the SAXS experiments and compared to that expected from the bulk viscosity of the polymer flux depends weakly on the MW

flux depends weakly on the MW flux is inversely proportional to N1.5 and inversely proportional to viscosity,  h ~ N1.5 In bulk polymers, h ~ N3.4 for MW > Me (entanglement MW) The MW observed in this case is much less than that of the bulk PS  confinement reduces viscosity, enhancing PS flow for N > Ne

glass transition region is broader than that of the bulk and shows little dependence on the MW onset temperature varied only in ~10o observed reduction in viscosity is not due to the glass transition because the experiments were done well above the end point of the transition

Small angle neutron scattering yields information about chain conformation data for the confined polymer follows the behavior of an unperturbed chain confined polymer bulk

radius of gyration Rg in the direction of the pore axis is similar to bulk since the chain is unperturbed, and the radius of the pore is < 2Rg,  # of chains per unit volume must be smaller than in bulk as the entangled polymer enters the pore, the # of chains per unit volume decreases, and the interpenetration of polymer chains decreases

evolution of conformation after flowing into the pore, chains are not stretched in the direction of the flow, but due to confinement they are compressed in a direction perpendicular to the flow diffusivity of a single unperturbed chain is proportional to N-2.4 based on this assumption, it can be shown that the flux F ~ N-1.4 and therefore the viscosity h ~ N1.4 in agreement with the results

Application of Theory: Electrophoresis From Giant Molecules

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.

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

Example of Good Coalescence Immediate film formation upon drying! Hydrated film Tg of latex  5 °C; Environmental SEM Bar = 0.5 mm • 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.

Example of Good Coalescence Immediate film formation upon drying! Hydrated film Tg of latex  5 °C; Environmental SEM Bar = 0.5 mm • 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.

Strength Development with Increasing Diffusion Distance 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

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