Dilute solution stress tensor

The presence of the fourth-rank tensor in (7.127) and its absence in (7.11) suggests that the stress optical rule should not apply for dilute solutions of rigid rods. Unfortunately, because of the difficulty of acquiring truly rigid rods, and the problems of making measurements of stress in dilute systems, there are no data available on dilute rigid rod solutions where the stress optical rule can be investigated on this class of polymer liquids. 

Extension of this theory can also be used for treating concentrated polymer solution response. In this case, the motion of, and drag on, a single bead is determined by the mean intermolecular force field. In either the dilute or concentrated solution cases, orientation distribution functions can be obtained that allow for the specification of the stress tensor field involved. For the concentrated spring-bead model, Bird et al. (46) point out that because of the proximity of the surrounding molecules (i.e., spring-beads), it is easier for the model molecule to move in the direction of the polymer chain backbone rather than perpendicular to it. In other words, the polymer finds itself executing a sort of a snake-like motion, called reptation (47), as shown in Fig. 3.8(b). 

As was demonstrated by Pyshnograi (1994), the last term in (6.7) can be written in symmetric form, if the continuum of Brownian particles is considered incompressible. In equation (6.7), the sum is evaluated over the particles in a given macromolecule. The monomolecular approximation ensures that the stress tensor of the system is the sum of the contributions of all the macromolecules. In this form, the expression for the stresses is valid for any dynamics of the chain. One can consider the system to be a dilute polymer solution or a concentrated solution and melt of polymers. In any case the system is considered as a suspension of interacting Brownian particles. 

The dilute polymer solution can be considered as a collection of non-interacting macromolecular coils suspended in a viscous liquid, the stress tensor of which is written as... 

Here the linear terms in respect to the coefficient of internal viscosity ipa have taken into account only. Averaging with respect to the velocity distribution has been assumed here. One ought to add the stresses (6.13) of carrier viscous liquid to stresses (6.14) to determine the stress tensor for the entire system, that is for the dilute solution of the polymer. 

Let us write down first of all the stress tensor for dilute solution (6.16) as a function of the velocity gradients. We can use expressions (2.41) for moments, in order to determine the stresses with accuracy within the first-order term with respect to velocity gradients... 

The set of constitutive equations for the dilute polymer solution consists of the definition of the stress tensor (6.16), which is expressed in terms of the second-order moments of co-ordinates, and the set of relaxation equations (2.39) for the moments. The usage of a special notation for the ratio, namely... 

The expressions for the stress tensor together with the equations for the moments considered as additional variables, the continuity equation, and the equation of motion constitute the basis of the dynamics of dilute polymer solutions. This system of equations may be used to investigate the flow of dilute solutions in various experimental situations. Certain simple cases were examined in order to demonstrate applicability of the expressions obtained to dilute solutions, to indicate the range of their applicability, and to specify the expressions for quantities which were introduced previously as phenomenological constants. 

The last equation can be compared with equations (9.1) and (9.3) for the stresses in dilute solutions. On can see that, when internal viscosity is neglected ipv = 0), there is a relation between the permittivity tensor and stress tensor in the form... 

Cooke BJ, Matheson AJ (1976) Dynamic viscosity of dilute polymer solutions at high frequencies of alternating shear stress. J Chem Soc Faraday Trans II 72(3) 679-685 Curtiss CF, Bird RB (1981a) A kinetic theory for polymer melts. I The equation for the single-link orientational distribution function. J Chem Phys 74 2016—2025 Curtiss CF, Bird RB (1981b) A kinetic theory for polymer melts. II The stress tensor and the rheological equation of state. J Chem Phys 74(3) 2026—2033 Daoud M, de Gennes PG (1979) Some remarks on the dynamics of polymer melts. J Polym Sci Polym Phys Ed 17 1971-1981... 

This hydrostatic approach also yields a formal closed formula for y in terms of the components of the stress tensor. When the stress tensor is expressed in terms of molecular variables, the resulting statistical mechanical formula for y provides a direct means for the calculation of surface tension. For example, it may be used directly to compute the surface tension of dilute ionic solutions (6). It also illustrates in molecular detail the iterative subtractive procedures that lead to the excess functions of the familiar phenomenological approach. 

Figure 6.17 Normalized intrinsic viscosity [r ]/[)7]o for a dilute solution of poly(y-benzyl-L-glutamate) (PBLG) = 208,000) in m-cresol. The line is a calculation for the rigid-dumbbell model, with the relaxation time t = lj6Dro adjusted to the value 10- sec to obtain a fit. The stress tensor for a suspension of rigid dumbbells is given by Eq. (6-36) with Cstr replaced by k T/Dro-(From Bird et al. 1987 data from Yang 1958, Dynamics of Polymeric Liquids, VoL 2, Copyright 1987. Reprinted by permission of John Wiley Sons, Inc.)...

The stress tensor for a semidilute solution of rods is given by Eq. (6-36), the formula for dilute solutions. However, if in a thought experiment one holds the shear rate fixed at a low value while increasing the concentration of rods from dilute to semidilute, the Brownian contribution to the stress will greatly increase, since the rotary diffusivity decreases according to (6-44). The viscous stress contribution, however, only increases in proportion to u. Thus, as Doi and Edwards (1986) argued, the ratio of viscous to Brownian stresses decreases as as the concentration increases in the semidilute regime. Hence, in the semidilute... 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

We confine our attention here to dilute solutions of several polymer speaes m a solvent. According to Sect. 7, the stress tensor is a sum of four contnbutions, the first three of which involve the singlet distribution function (s), whereas the fourth involves the doublet distribution fiinction(4) ... 

To demonstrate the point, let us fimt consider a dilute polymer solution. As shown in Section 4.5.2, the stress tensor is written as... 

For simplicity we proceed using eqn (9.50). The stress tensor is given in precisely the same manner as for dilute solutions (see eqn (8.123)). 

The simplest model for dilute polymer solutions is to idealize the polymer molecule as an elastic dumbbell consisting of two beads connected by a Hookean spring immersed in a viscous fluid (Fig. 2.1). The spring has an elastic constant Hq. Each bead is associated with a frictional factor C and a negligible mass. If the instantaneous locations of the two beads in space are riand r2, respectively, then the end-to-end vector, R = ri — ri, describes the overall orientation and the internal conformation of the polymer molecule. The polymer-contributed stress tensor can be related to the second-order moment of R. There are two expressions namely the Kramers expression and the Giesekus expression, respectively (Bird et al. 1987b) ... 

Although the UCM equation gives the polymer contribution to the stress in a dilute solution such as a Boger fluid, the solvent contribution to the stress cannot be neglected, and so the total stress tensor r in these solutions is the sum of the polymeric and solvent contributions... 

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