Shear-induced shift of spinodal line in entangled polymer blends

Departamento de Ciencias y Te ́cnicas Fı ́sicoquı ́micas, Facultad de Ciencias, UNED, Senda del Rey, 9, 28040 Madrid, Spain Departament de Fı ́sica, Universitat Auto `noma de Barcelona, 08193 Bellaterra, Catalonia, Spain Institut d’Estudis Catalans, Carme 47, 08001 Barcelona, Catalonia, Spain Departamento de Polı ́meros, Instituto de Investigaciones en Materiales, UNAM, Apartado Postal 70-360, Me ́xico Distrito Federal, 04510, Mexico ~Received 23 April 2002; published 30 December 2002 !

Shear-induced effects in polymer solutions and blends are very interesting in thermodynamics, statistical mechanics, hydrodynamics, and engineering and, accordingly, much research is being carried out on them ͓1-3͔.Clarke and McLeish ͓4͔ recently studied the shear-flow effects on phase separation of entangled polymer blends.They considered a two-fluid model, taking into account the effects of mutual friction among the chains, of shear stress, and of the inhomogeneities in the chemical potential of the components.They assumed that the chemical potential keeps its localequilibrium form ͑i.e., that it does not depend on the flow͒, and the shear-flow effects were entirely attributed to the coupling between the divergence of the viscous pressure and the diffusion flux.On this basis, they showed the possibility of a rich variety of changes of the phase diagrams of flowing polymer blends.
Our aim here is to present an alternative study by keeping for the stress tensor the usual upper-convected Maxwell form and incorporating nonequilibrium contributions into the chemical potentials.Such an approach has been used in the description of shear-induced diffusion and of phase separation in polymer solutions ͓1,2,5-7͔.A comparison of the two approaches may be useful for discussion of the role of dynamical and thermodynamical effects in this active field of research.
First of all, we summarize the results by Clarke and McLeish ͓4͔, who use a two-fluid description proposed by Doi and Onuki ͓8͔.The main result of this approach is the following expression for the evolution of the volume fraction A of one of the polymers, say, polymer A: where v is the volume average velocity, M the mobility tensor, the viscous stress tensor, and ␣ a parameter depending on the ratio of entanglements in both polymers.Then the viscous stress is coupled to the diffusion flux J, which is given by In ͓4͔, this coupling plays an essential role in the shift of the spinodal line, whereas A and B are the localequilibrium chemical potentials of polymers A and B, respectively.Thus, the shear-induced effects are attributed to a purely dynamical origin.
As the memory function for the stress relaxation of the blend, Clarke and McLeish take where i , G i , and i are, respectively, the volume fraction, plateau modulus, and relaxation time of polymer i.This nonlinear mixing rule, following from the model of double reptation ͓9,10͔, is the simplest one to describe the details of coupled stress relaxation in polymer blends.It yields for the steady viscous stresses the following expressions: where ␥ ˙is the shear rate, and X( A ) and Y ( A ) are polynomials defined in order to have more compact expressions and whose explicit forms are given in Eqs.͑A2͒ and ͑A3͒ in the Appendix.
The diagonal components of the stress tensor are assumed to have the form ͓4͔ assume.For instance, if an upper-convected Maxwell model ͓11͔ is used to describe the viscous stress tensor, it follows that xx ϭN 1 , yy ϭ zz ϭ0.͑7͒ Although the stress tensor can only be determined within an isotropic constant, the options ͑6͒ and ͑7͒ do not differ by a constant, but a variable, namely, Ϫ 1 3 N 1 , and thus they are not equivalent.
The different choices ͑6͒ and ͑7͒ lead to very different results concerning the flow effects on the diffusion flux.Indeed, consider for instance the y component of J, which according to Eq. ͑2͒ is given by

͑8͒
In the steady state, one has ‫ץ‬ xy /‫ץ‬xϭ0 and ‫ץ‬ zy /‫ץ‬zϭ0, and one is left with If Eq. ͑6͒ is assumed, yy ϭϪ 1 3 N 1 and the coupling term contributes to J y , whereas if Eq. ͑7͒ is considered, the coupling term does not contribute to the diffusion flux.This is the main difference between the approach in ͓4͔ and the present one.In ͓4͔, A and B do not depend on the flow and all the shear-induced effects are attributed to the coupling term.In the approach proposed in ͓1,7,12,13͔ for dilute polymer solutions and applied here to polymer blends, A and B depend on the flow and the coupling term vanishes ͑for the plane Couette flow studied in this paper, but not in some other flows such as the cone-and-plate one ͓14,15͔͒.Thus, from now on we will use Eq.͑7͒ and a nonequilibrium contribution for .
Anyway, we will follow the standard procedure ͓4,8,12͔ of defining an effective diffusion coefficient D eff by rewriting Eq. ͑1͒ for the space Fourier transform of the perturbations ␦ A in the volume fraction of polymer A in the form

͑10͒
where q is the wave vector.For wave vectors in the y and z directions ͑when the velocity is in the x direction and the velocity gradient in the y direction͒, the effective diffusion coefficient is where is the Flory-Huggins interaction parameter, which is a function of the temperature of the form

͑12͒
where ⌿ and ⌰ are parameters that depend on the system, c its value on the quiescent spinodal, and the interfacial energy.The spinodal line indicating the onset of instability of the homogeneous phase is given by the condition D eff ϭ0.Indeed, for positive values of D eff , inhomogeneities will tend to disappear, whereas negative values of D eff will enhance inhomogeneities ͓8,12,13͔.Thus, ⌬ c describes the ␥ ˙-dependent shift in the spinodal line.When ⌬ c Ͻ0, shearinduced demixing occurs, i.e., phase separation is enhanced, whereas ⌬ c Ͼ0 corresponds to shear-induced mixing, which contributes to the stability of the one-phase system.In the approach by Clarke and McLeish ͓4͔, ⌬ c is given by where XЈ and Y Ј denote the derivatives with respect to A .
The results ͑13͒ have been taken as the basis for the analysis of some experiments by assuming an effective diffusion coefficient of the form D eff ϭD eq Ϫa␥ ˙2, where D eq is the diffusion coefficient at equilibrium and a is taken as a parameter to be identified by fitting the experimental data ͓16͔.
In contrast with ͓4͔, we include in the chemical potential contributions from the flow, which may be expressed in terms of the viscous stress in a macroscopic approach, or of the macromolecular configuration tensor in a more microscopic approach ͓1,2,5-7,17͔.In particular, in extended irreversible thermodynamics the nonequilibrium contribution of the viscous stress tensor to the Gibbs free energy G is given by ͓1,2,18 -22,15͔ V being the total volume, J ˜the steady-state compliance ͑i.e., the ratio of the viscoelastic relaxation time to the shear vis-cosity͒, n i the number of moles of species i per unit volume, N i the number of monomers of polymer i (iϭA,B), and v 0 the molar volume of the monomer, which in this simple illustration is supposed the same for A and B, although the result could be straightforwardly generalized to different molar volumes.This contribution is supported by several microscopic arguments ͓1͔ and, in summary, it takes into account the contribution to the Gibbs free energy of the orientation and stretching of the macromolecules due to the flow.From Eq. ͑14͒, one may derive the nonequilibrium contributions to the chemical potentials, T being the temperature, p the pressure, and ␦ i j the Kronecker delta.
The explicit form of the steady-state compliance J ˜for polymer blends may be obtained from the general formula ͓11͔

͑16͒
and from the fact that the memory function G(t) in Eq. ͑3͒ may be explicitly rewritten as where C Ϫ1 ϭ2( A Ϫ1 ϩ B Ϫ1 ).By combining Eqs.͑16͒ and ͑17͒, it follows that Therefore, the following expressions for the nonequilibrium contribution ͑15͒ to the chemical potentials are obtained: where Z( A ) and W( A ) are polynomials given in the Appendix ͓Eqs.͑A4͒ and ͑A10͒, respectively͔.Therefore, the total shift of the spinodal line becomes in our model

͑21͒
when we consider the y direction ͑where xy ϭconst) and ͑22͒ in the z direction ͑along which ␥ ˙ϭconst).The parameter is defined as the ratio of the number of monomers, N A and N B , in the polymeric chains A and B, i.e., ϭN B /N A .Note that, as well as Eqs.͑13a͒ and ͑13b͒, the results ͑21͒ and ͑22͒ point out that in the presence of the flow one should consider an effective diffusion coefficient of the form D eff ϭD eq ϪaЈ␥ ˙2, with aЈ a coefficient different from the one obtained in the Clarke and McLeish theory, but yielding similar qualitative results.
In Fig. 1 we compare our results ͑21͒ with ͑13a͒ obtained in ͓4͔.In this figure is plotted the border of the instability behavior, namely, ⌬ c ϭ0, in a diagram of log 10 vs , for Gϭ1.In the Clarke and McLeish results, such a border is a single line, whereas in our model two different lines arise.The zones where ⌬ c Ͻ0 correspond to shear-induced demixing, whereas where ⌬ c Ͼ0 the shear induces mixing and it enhances stability.The physical differences between the predictions of Clarke and Mc Leish and ours are directly seen: in the former, the nonequilibrium and equilibrium spinodal lines cross over each at a single point, whereas in the latter there are crossings at two points, depending on the range of and A considered.It is also noted that, although appears in Eq. ͑21͒, calculations reveal only a minor influence on ⌬ c (q y ) in our results ͑21͒, whereas it does not even appear in Eq. ͑13a͒.The limit when Ӷ1 would correspond to an entangled polymer solution of chains A in solvent B; in this case, the shear flow induces mixing and enhances stability.When increases, i.e., when the viscoelastic effects of the solvent B increase, a demixing behavior appears.Following Clarke and McLeish ͓4͔, the parameter ␣ can be written as where N i,ent is the degree of polymerization of an entanglement segment in polymer i.The last equation shows that ␣ takes values near zero when A and B have equal degrees of polymerization and its entanglement behavior is similar ( ϭ1).In this situation, the ⌬ c proposed in ͓4͔ vanishes as follows from Eqs. ͑13a͒ and ͑13b͒.
FIG. 1. Borders of instability in the y direction when Gϭ1.The continuous curve is calculated from Eq. ͑21͒ when ϭ1 and the dashed curve corresponds to the results ͓4͔ calculated from Eq. ͑13a͒.
In Fig. 2 we compare the results for ⌬ c (q z ) in our approach, namely, Eq. ͑22͒, and in that of Clarke and McLeish Eq. ͑13b͒ for Gϭ1.In contrast with the method for obtaining ⌬ c (q y ), where the gradient of A Ϫ B is carried out at constant xy , in the analysis of ⌬ c (q z ) this gradient is performed at constant shear rate ␥ ˙.Thus, the ''spinodal line'' is no longer univocal, but depends on the direction of observation.One difference of ⌬ c (q z ) with respect to ⌬ c (q y ) is that the former is sensitive to the ratio (ϵN B /N A ) whereas the second one remains practically unchanged under modifications of .
When a T-A diagram is considered ͑Figs.3 and 4͒, in the zones where ⌬ c Ͻ0 the spinodal line is shifted toward higher temperatures ͓according to the relation ͑12͒ between and T͔, whereas where ⌬ c Ͼ0 it is shifted in the opposite direction.In order to apply the previous results to a real system, we consider one of the isotopic blends of poly͑dim-ethylsiloxane͒ ͑PDMS͒ studied by Beaucage et al. ͓23͔.In this reference, neutron scattering experiments are reported which yield the values of the Flory-Huggins parameter and its dependence on temperature.The system that we have chosen is the blend in which the degree of polymerization of hydrogenous PDMS is 964 and that of deuterated PDMS is 957 ͑whose density is of the order of 9.7ϫ10 Ϫ3 kg m Ϫ3 ) for which ⌿ϭ0.5013 and ⌰ϭ1.17K. Taking into account the values of the quoted degrees of polymerization, one has Ϸ1 and N A ϭ957.From the rheological quantities reported by Migler ͓24͔ we have estimated G A ϭ1.5ϫ10 4 Pa and A ϭ2.0ϫ10 Ϫ3 s.Note that in Fig. 3, both the Clarke and McLeish ͑CM͒ model and the extended irreversible thermodynamics ͑EIT͒ model predict an increase of the critical temperature of the same order.However, the shift of the value of the critical concentration is opposite in the two models: it is positive in the CM model and negative in the EIT model.In contrast, in the situation reported in Fig. 4, EIT predicts a shear-induced increase in critical temperature whereas the CM model predicts a decrease; note, furthermore, that the shift predicted by the CM model is more sensitive to the value of the shear rate than is the shift predicted by EIT.Experimental analysis of these spinodal lines could allow one to establish which of these two theoretical models is more suitable.FIG. 3. Spinodal lines predicted for the isotopic blend described in the text.The dotted curve corresponds to the Flory-Huggins model without interfacial contributions.The other curves are associated with fluctuations in the y direction when the system is submitted to a shear stress xy ϭ250 Pa.The continuous curve and the dashed curve are calculated from Eqs. ͑21͒ and ͑13a͒, respectively.The calculations have been carried out using Gϭ1, log 10 ϭϪ0.5, and *ϭ0.5.FIG. 4. Spinodal lines predicted for the isotopic blend described in the text.The dotted curve corresponds to the Flory-Huggins model without interfacial contributions.The other curves are associated with fluctuations in the z direction taking Gϭ1, log 10 ϭϪ0.5, and ϭ1 when the system is submitted to a constant shear rate.The continuous curves are calculated from Eq. ͑22͒ considering two shear rates ␥ ˙ϭ10 ͑curve 1͒ and 5 s Ϫ1 ͑curve 2͒.All dashed curves correspond to Eq. ͑13b͒: *ϭ0.9 and ␥ ˙ϭ5 s Ϫ1 ͑curve 3͒, *ϭ0.5 and ␥ ˙ϭ5 s Ϫ1 ͑curve 4͒, and *ϭ0.5 and ␥ ˙ϭ10 s Ϫ1 ͑curve 5͒.
In summary, we have pointed out here two different descriptions of the shear-induced shift of the spinodal line, due to the coupling between J and "• and the nonequilibrium contribution to the chemical potentials .Whereas in the plane Couette flow studied here the coupling term disappears when the upper-convected Maxwell model is used, this is not so in other flow geometries, such as for instance the coneand-plate model.In such situations, the two contributions should be added, as was done, for instance, in ͓13͔.Indeed, in addition to the relative importance of the latter contributions to the steady-state nonequilibrium spinodal line outlined here, other good motivations to consider them arise in the analysis of time-dependent phenomena, such as, for instance, the rate of shear-induced separation.Indeed, in ͓14͔ it was shown ͑in a polymer dilute solution in a cone-and-plate experiment͒ that by keeping only the coupling between viscous shear and diffusion flux ͑i.e., by ignoring the nonequilibrium contributions to the chemical potential͒ one predicts a shear-induced polymer separation which is two to three orders of magnitude slower than the one experimentally observed.Instead, inclusion of nonequilibrium contributions such as Eq.͑14͒ yields the correct order of magnitude for this separation time, since the separation is accelerated because of a thermodynamical instability due to the nonequilibrium terms of the chemical potential ͓13͔.The analysis of similar experiments in polymer blends would be of much interest in clarifying the physical relevance of the several kinds of nonequilibrium contribution to the effective diffusion coefficient.

APPENDIX
In order to obtain the explicit form of the polynomials quoted in the previous sections, the following parameters are introduced: which allow us to write for X( A ) and Y ( A ) appearing in Eqs.͑4͒ and ͑5͒, The polynomial Z( A ) in Eqs.͑19͒ and ͑20͒ is found to be with coefficients given by Analogously, we can write for W( A ) appearing in Eq. ͑19͒, W͑ A ͒ϭw 0 ϩw 1 A ϩw 2 A 2 ϩw 3 A 3 , ͑A10͒ with the following coefficients:

FIG. 2 .
FIG.2.Borders of instability in the z direction when Gϭ1.All the curves have been calculated by means of Eq. ͑22͒ using different values of the parameter , which are indicated on the respective curves.For the considered value of G, Eq. ͑13b͒ always predicts ⌬ c Ͻ0, as was pointed out in Ref.͓4͔.
We acknowledge financial support from the Spanish Ministry of Science and Technology under Grant No. BFM2000-0351-C03-01 and from the Direccio ´General de Recerca of the Generalitat of Catalonia under Grant No. 2001 SGR 00186.Also, one of us ͑L.F.C.͒ acknowledges the support of the Direccio ´n General de Asuntos de Personal Acade ´mico of the UNAM ͑Me ´xico͒ under Grant No. IN-119200.