Molecular coils

The static properties of an isolated chain constitute a good starting point to study polymer dynamics many of the features of the chain in a quiescent fluid could be extrapolated to the kinetics theories of molecular coil deformation. As a matter of fact, it has been pointed out that the equations of chain statistics and chain dynamics are intimately related through the simplest notions of graph theory [16]. 

The Rouse and Zimm models consider only minute deformations of the molecular coil in the presence of a constant velocity field. In the presence of a velocity gradient, each bead sample has a different fluid velocity resulting in different drag forces which must be incorporated into the system of equations (21). 

Most theories predict substantial stretching and alignment of the molecular coils at ... 

The above description refers to a Lagrangian frame of reference in which the movement of the particle is followed along its trajectory. Instead of having a steady flow, it is possible to modulate the flow, for example sinusoidally as a function of time. At sufficiently high frequency, the molecular coil deformation will be dephased from the strain rate and the flow becomes transient even with a stagnant flow geometry. Oscillatory flow birefringence has been measured in simple shear and corresponds to some kind of frequency analysis of the flow... 

The following qualitative picture emerges from these considerations in weak flow where the molecular coils are essentially undeformed, the polymer solution should behave approximately as a Newtonian fluid. In strong flow of a highly dilute polymer solution where the macroscopic velocity field can still be approximated by the Navier-Stokes equation, it should be expected, nevertheless, that in the immediate proximity of a chain, the fluid will be slowed down because of the energy intake to stretch the molecular coil thus, the local velocity field may deviate from the macroscopic description. In the general case of polymer flow,... 

From the weak dependence of ef on the surrounding medium viscosity, it was proposed that the activation energy for bond scission proceeds from the intramolecular friction between polymer segments rather than from the polymer-solvent interactions. Instead of the bulk viscosity, the rate of chain scission is now related to the internal viscosity of the molecular coil which is strain rate dependent and could reach a much higher value than r s during a fast transient deformation (Eqs. 17 and 18). This representation is similar to the large loops internal viscosity model proposed by de Gennes [38]. It fails, however, to predict the independence of the scission yield on solvent quality (if this proves to be correct). 

Under quiescent conditions, polymer solutions are divided into four categories depending on the average distance separating the centers of mass of the molecular coils the dilute, the semi-dilute (or semi-concentrated), the concentrated and the entangled state. 

A plausible assumption would be to suppose that the molecular coil starts to deform only if the fluid strain rate (s) is higher than the critical strain rate for the coil-to-stretch transition (ecs). From the strain rate distribution function (Fig. 59), it is possible to calculate the maximum strain (kmax) accumulated by the polymer coil in case of an affine deformation with the fluid element (efl = vsc/vcs v0/vcs). The values obtained at the onset of degradation at v0 35 m - s-1, actually go in a direction opposite to expectation. With the abrupt contraction configuration, kmax decreases from 19 with r0 = 0.0175 cm to 8.7 with r0 = 0.050 cm. Values of kmax are even lower with the conical nozzles (r0 = 0.025 cm), varying from 3.3 with the 14° inlet to a mere 1.6 with the 5° inlet. In any case, the values obtained are lower than the maximum stretch ratio for the 106 PS which is 40. It is then physically impossible for the chains to become fully stretched in this type of flow. 

In simple shear flow where vorticity and extensional rate are equal in magnitude (cf. Eq. (79), Sect. 4), the molecular coil rotates in the transverse velocity gradient and interacts successively for a limited time with the elongational and the compressional flow component during each turn. Because of the finite relaxation time (xz) of the chain, it is believed that the macromolecule can no more follow these alternative deformations and remains in a steady deformed state above some critical shear rate (y ) given by [193] (Fig. 65) ... 

From the standpoint of thermodynamics, the essential quantity which governs the course of a chemical reaction is the chemical potential of the system or, in the case of interest, the free energy storage within the molecular coil. This quantity, unfortunately, is difficult to evaluate in non-steady flow. At modest extension ratios (X < 4), the free energy storage of a freely-jointed bead-spring chain is... 

It is expected, however, that the Gaussian representation is inadequate in transient elongational flow, even if the chain is only weakly deformed. During a fast deformation, the presence of non-equilibrium effects, like internal viscosity , noncrossability and self-entanglements will stiffen the molecular coil which is now capable of storing a much larger amount of elastic energy than that predicted from Eq. (113). 

Kausch HH, Nguyen TQ (Feb. 1991) Molecular coils and their deformation, IUPAC Symposium Polymer 91, Melbourne 10-15... 

In dilute solution, the behavior of macromolecules is quite different to that of common low-molecular-weight molecules. For example, the shape of a macro-molecular coil is subject to permanent dynamic changes, and the coils are in a more or less swollen state when compared to their unperturbed (solid state)... 

The effect of branching is to increase the segment density within the molecular coil. Thus a branched molecule occupies a smaller volume and has a lower intrinsic viscosity than a similar linear molecule of the same molecular weight. The degree of branching is often characterized in terms of the branching factor [1] in Eq. (14), where the subscripts B and L, respectively, refer to branched and linear polymers of the same molecular weight ... 

This work examines the effect of long-chain branching on the low-shear concentrated solution viscosity of polybutadienes over a broad range of molecular weights and polydispersity. It will show that the reduction in molecular coil dimension arising from long-chain branching is more sensitively measured in concentrated than in dilute solutions for the polymers examined. 

For a semi-crystalline polymer the solidification model of Fischer [11] predicts that the chain dimensions are frozen in at Tm if crystallization occurs sufficiently fast so as to prevent the disentanglement of chains by unravelling and/or their segregation according to their mass. For a series of polyethylene samples Michler [12] has accomplished an instructive comparison between the dimensions of molecular coils and the thickness of crystalline lamellae (Fig. 4). 

Fig. 4 Influence of molecular weight Mw on (a) lamellar arrangement and (b) diffraction patterns of polyethylene (c) comparison of the diameter Dk of molecular coils and of the long period L of the lamellar structure (after Michler [12])...

Theoretical estimates of the quantity zv — 1 are in the range from 0.5 (non-draining Gaussian coil), to 1.11 (draining coil with excluded-volume interaction). A compilation of empirical values of K and of the power exponents for different polymers and different solvents may be found in the literature (Flory 1969, Tsvetkov et al. 1964). The empirical values of the exponent zv — 1 do not exceed 0.9, which indicates significant impermeability of the macro-molecular coil in a flow. We may note that once a relation of type (6.24)... 

In the simplest cases it can be reduced to the tensor of deformation of macro-molecular coils... 

For a polymer solution, p ss 10 2 Ps, p k, 1 g/cm3, and the size of macro-molecular coil is a k, 10-5 cm, which allow us to estimate the relaxation time r 10 1° s. Processes with relaxation times so small are not essential when compared to other relaxation processes in polymer solutions. 

We may note that it has been shown for the dumbbell (Altukhov 1986) (see Appendix F) that the combined allowance for the internal viscosity and the anisotropy of the hydrodynamic interaction leads to the appearance of a nonzero second difference between the normal stresses internal viscosity may be estimated, for example, from dynamic measurements, this effect may serve to estimate the anisotropy of the hydrodynamic interaction in a molecular coil. 

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