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Main Properties of Entangled Polymer Fluids Entangled polymer fluids are polymer melts and concentrated or semidilute ( above the concentration c* ) solutions. In these systems polymer coils strongly overlap with each other, and polymer chains are entangled. Specific properties of entangled polymer fluids: (i) Entangled polymer fluids normally have a very high viscosity ; (ii) Such fluids keep for a long time the memory about the history of flow ; (iii) the property of viscoelasticity: at fast (high frequency) external action the response is elastic, while for slow (low frequency) external action the response is viscous (i.e. flow starts). Viscosity of Fluids S z f, d Simplest setup for viscosity determination Newton-Stokes law: S d The proportionality coefficient is called the viscosity of the fluid. In the differential form the Newton-Stokes law can be written as follows f f d S dz where is imposed stress and z is the coordinate perpendicular to the plates. Viscosity is measured in poise; 1 poise 1 g cm sec . The viscosity of water at 200C is 10-2 poise, while viscosity of polymer melts can be of order 10101012 poise, or even higher. 2r Flow in capillary viscosimeter Viscosity is measured by viscosimeters. Most common are capillary viscosimeters. The method of measurements is then based on the Poiseille equation: r 4 Pt Q 8l where Q is the mass of the fluid flown through the capillary during the time t, P is the pressure difference at the ends of the capillary of length l and radius r. The Property of Viscoelasticity ln ln t ln t Step-wise stress starting at t =0 Reaction of the ordinary liquid ln * ln t Reaction of the polymer liquid • The reaction of the ordinary fluid to such a stress would be a normal flow (after some initial equilibration period), i.e. the shear angle will vary with time as t where is the viscosity of the fluid. • The typical reaction of polymer fluids is qualitatively different. At t << * the value of is practically constant E and t only at t >> * the flow starts . In other words for entangled polymer fluid at t << * we have the elastic response E where E is the effective Young modulus, while at t >> * we have t i.e. the response is viscous. This is just the property of viscoelasticity. By comparing these two relations, we have 1 or E E The described property of viscoelasticity is a general property of all entangled polymer fluids, as long as they are not crystalline, glassy or crosslinked. Therefore, as for the property of high elasticity, the general molecular explanation should be possible which is based on the fact of the chain structure of polymer molecules, without the explicit reference to the specific chemical nature of monomer units. Such explanation was developed by de Gennes, Doi and Edwards (1971-1979); it is called the theory of reptations. Theory of Reptations Let us consider one chain entangled with many others and let us “freeze” for a moment the conformations of the other chains. This gives rise to a certain “tube”: the given chain cannot escape in the directions perpendicular to the tube axis the only allowed type of motion is the snake-like diffusion along the tube axisreptations. If other chains are “defrozen” the competing mechanism appears:“tube renewal”, but it can be shown that reptations always give a dominant contribution. The neighboring chains forming the walls of the tube create restrictions for the motion of a given chain and are in this sense analogous to cross-links. But these are “quasi-cross-links” with finite lifetime: they “relax” after some time * , because the chain leaves the initial tube and has new neighbor after this time *. Thus, the property of visoelasticity has now molecular interpretation: at the t * polymer liquid behaves as network with “quasi-cross-links”, and the response is elastic; while at t * “quasi-cross-links” relax, and the response is viscous. What is the elastic modulus E for the network of “quasi-cross-links”? According to the classical theory of high elasticity E kT , where is the number of elastic chains per unit volume, 1 Na 3, N being the number of monomer units between two cross-links. It is normally assumed that N N e , where N e is some constant for a given polymer, most often in the interval 50 to 500; this is a phenomenological parameter reflecting that not each contact acts like cross-links. Two neighboring Two neighboring chains not forming a chains forming a “quasi-cross-link” “quasi-cross-link” Thus, E kT N e a 3 N.B. The model of reptations is valid only for N N e . The picture of a tube d The chain is a sequence of subcoils, each containing N e links and having the size d N e1 2.aThe width of the tube is d N e1 2 a . The length of the tube is N Na d 1 2 Na L Ne Ne The diffusion coefficient corresponding to reptation along the tube is Dt kT , where is the corresponding friction coefficient; N0 ( 0 - friction coefficient for one monomer unit) Dt kT 0 N On the other hand, 2 Dt *, thus 2 N 2 a 2 0 N N 3 0 a 2 N 3 0 Dt N e kT N e kT Ne * where 0 0 a 2 kT 10 12 sec is the characteristic microscopic time. N3 0 Ne * Note the very strong N -dependence. For N 105 N e 102, this gives * 10 sec - macroscopic relaxation time. This is the reason for slow relaxation and high viscosity in polymer liquids. Thus 3 3 kT N N E * 0 2 0 3 Nea Ne Ne where 0 kT 0 a 3 is a characteristic viscosity for low molecular liquid 0 1 poise Therefore, the reptation theory gives for the viscosity ~ N 3 . This relationship is close to the best experimental fit ~ N 3..4 The Method of Gel-Electrophoresis Macromolecules containing charged monomer units are called polyelectrolytes. Charged monomer units appear as a result of dissociation reaction: neutral monomer unit charged counter + monomer unit ion Most important polyelectrolytes are biological macromolecules: DNA and proteins. As an application of the reptation theory, let us consider the gel-electrophoresis of polyelectrolytes, having in mind mainly DNA molecules. This method is used in practice for the separa- tion of DNA fragments of different length and composition (for DNA sequencing). - + E Gel is swollen polymer network. The negatively charged DNA molecules (total charge Q) move through the gel in external electric field E . The drift velocity depends on the length of the chains the separation of the DNA molecules of different lengths is achieved. In the absence of the gel: F QE and F , where F is the electric force acting on the DNA molecule and is the friction coefficient. Thus, QE . But Q ~ L and ~ L (different parts of DNA molecule exhibit independent friction), therefore is independent of L and DNA molecules cannot be separated in the solution. In the gel DNA molecules are in the effective “tubes” and move via reptations. Let us divide the moE lecule in small fragments and count only the forces acting S i along the tube. QRE QS i Q Ft E S i E i L L i L The drift velocity along the tube t Ft , where 0 N, 0 being the friction coefficient for one monomer unit. Thus, QRE t L 0 N We have Q ~ L, R ~ N 1 / 2 (in weak field), 1/ 2 N ~ L , therefore t is proportional to 1 L (or 1 N 1 2), shorter chains move faster, and the separation of DNA fragments of different length is possible. However, in stronger fields DNA molecules stretch, R becomes ~ N, and the resolution of gel-electrophoresis vanishes. To deal with this problem the direction of the field is turned at 900 after each interval *( * being the time of renewal of initial tube via reptations). E1 E2 Then the chain does not have time to stretch during one cycle. In this way it is possible to keep a good resolution of the method of DNA gel-electrophoresis even in strong enough field. Gel Permeation Chromatography This method is also based on the idea of the separation of chains of different length in the process of their motion through microporous (gel) medium chromatographic column. The differences from gel-electrophoresis: • The driving force for the motion is pressure gradient (due to the pumping of polymer solution through chromatographic column), not electric field. Therefore, the method can be applied to all polymers, not only to charged ones. •The gel medium is normally solid microporous material, rather than swollen soft gel. Contrary to gel-electrophoresis, the size of the largest pores is usually much higher than the coil size (although the pore sizes exhibit very wide distribution). • Contrary to gel electrophoresis, in the normal exclusion regime of gel permeation chromatography (no specific interactions of polymers with the column) larger chains move faster. Explanation: shorter chains can penetrate even in small pores of microporous system, while long chains move only through largest pores. Therefore, the “effective way” for the long chains is shorter. There is another regime, called adsorption chromatography, when polymer chains are attracted to the walls of microporous system of the column and “stick” to them. In this case, the “sticking energy” for longer chains is higher, and they move slower.