J. Phys. Chem. B 1998, 102, 5335-5340
5335
Nonequilibrium Generalization of Chemical Potential of Flowing Fluids M. Criado-Sancho,*,† D. Jou,‡,§ and J. Casas-Va´ zquez‡ Departamento de Ciencias y Te´ cnicas Fisicoquı´micas, Facultad de Ciencias, UNED, Senda del Rey s/n, 28040 Madrid, Spain, Departament de Fı´sica, UniVersitat Auto` noma de Barcelona, 08193 Bellaterra, Catalonia, Spain, and Institut d’Estudis Catalans, Carme 47, 08001 Barcelona, Catalonia, Spain ReceiVed: December 3, 1997; In Final Form: March 10, 1998
The contribution of a shear viscous pressure to the chemical potential of the components of a fluid mixture is considered. It is used to study the influence of the shear on the phase diagram (coexistence line, spinodal line, ...) of a polymer solution and on the shear-induced degradation effects.
1. Introduction In the context of equilibrium thermodynamics, the chemical potential plays one of the most important roles in the analysis of chemical systems,1 and in particular, it is the most useful tool to study the phase coexistence and the equilibrium conditions in chemical reactions. However, it is a well-known fact that some nonequilibrium effects, (heat flux, diffusion flux, shear viscous pressure, ...) imply modifications in the phase diagram of fluids and in the chemical rate constants. Such effects arise especially in polymer solutions, for which the application of a shear stress yields modifications of the viscosity and an enhancement of the turbidity, which seem to suggest a phase segregation,2-4 and furthermore, the modification of the molecular weight distribution (MWD) of the polymer because of the shear induced degradation has been verified experimentally.5-7 The influence of the shear in phase separation in fluids has been analyzed from different points of view. Some authors propose that the observed turbidity is due to enhanced monomer concentration fluctuations in the presence of the flow,8,9 rather than to a true phase separation. Other authors10-12 have suggested that there is a true shift of the coexistence line and the spinodal line, implying an actual phase separation, leading to turbidity that may be further enhanced by the dynamical effect of the flow on the concentration fluctuations. In principle, the shift due to purely thermodynamic effects should be manifested in other independet phenomena where the chemical potential plays a role, as for instance in chemical equilibrium. Concerning nonequilibrium effects in the chemical reaction rates, it is possible to find in the literature treatments that generalize the formalism of the equilibrium thermodynamics.13,14 Recently, Nettleton15,16 undertakes the problem by means of an extended thermodynamic model in which the fluxes play the role of independent variables. Such a model allows him to use the chemical potential in nonequilibrium situations, on the condition that some corrections are introduced in this quantity, whose explicit expression is determined from kinetic theory. When he deals with reactions among small molecules of simple geometry, he concludes that the nonequilibrium contributions are small. Of course, it seems reasonable to assume that such
effects will be more important when one takes into account more complex molecules (as for instance polymers), though their kinetic analysis is much more difficult; this difficulty outlines the importance of using a strictly thermodynamic model to deal with systems of complex geometry. Extended irreversible thermodynamics17,18 (which will be referred to as EIT from now on) adopts as a basic hypothesis the existence of a nonequilibrium entropy, which is an analytic function of the internal energy, the volume, the composition, the viscous pressure and the other fluxes acting in the system, and allows generalization of the definition of the Gibbs free energy in nonequilibrium situations. Thus, for systems in which there is neither heat nor diffusion flux but only a viscous pressure tensor PV, the Gibbs free energy is given by14,20,21
G ) G(eq) + VJ(PV12)2
(1)
with V the total volume of the system and J the steady-state compliance. This expression may be justified on a microscopic basis from kinetic theory17and from Hamiltonian formulation.21,22 To deal with a system with an appreciable extent of nonequilibrium effects, we will consider a solution with a polymer as a solute. In this case, the equilibrium contribution G(eq) may be expressed according to the Flory-Huggins model23,24 in such a way that (1) becomes V
V(P12) G ) n1 ln(1 - φ) + np ln φ + χ(1 - φ)Ωφ + ΩJ RT RT (2) 2
with n1 the number of moles of the solvent, V its molar volume, np the number of moles of the polymer, and χ the FloryHuggins interaction parameter, which depends on the temperature as
Θ 1 χ) +Ψ -1 2 T
(
)
(3)
where Θ is the theta temperature and Ψ a parameter that does not depend on T. Furthermore, the variable Ω is defined as
Ω ) n1 + mnp
(4)
†
UNED. Universitat Auto`noma de Barcelona. § Institut d’Estudis Catalans. ‡
where m is a new parameter (characteristic of the lattice model on which the Flory-Huggins approximation is based), which
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Criado-Sancho et al.
allows one to define the volume fraction as
φ ) mnp/Ω
(5)
At first sight, m may be identified as the polymerization index, and according to a strict interpretation of the lattice model, its value should coincide with the ratio of the molar volumes of the polymer and the solvent. It is obvious that in practice there is not such a coincidence, and this leaves open the way to several options to determine the value of m in the literature.3,4,25 It is possible to generalize the chemical potential of component j for a system out of equilibrium from (1) by means of
µj )
( ) ∂G ∂nj
(6)
n T,p,P12
which, when one uses (2) for the Gibbs free energy, yields for the chemical potential of the solvent and of the polymer the following expressions
( )
µ1 1 ) ln(1 - φ) + 1 - φ + χφ2 + RT m
[()
V(PV12)2 ∂J Ω RT ∂φ
dn(β) dn(R) j j + )0 dt dt
(9)
and the rate of variation of G associated with the material flux is given by (R)
dnj dG (β) ) (µ(R) j - µj ) dt dt
The simplest constitutive equation for the latter flux compatible with a definite decrease of G is
dn(R) j (β) ) -L(µ(R) j - µj ) dt
]
µp ) ln φ + (1 - m)(1 - φ) + χm(1 - φ)2 + RT V(PV12)2 ∂J ∂φ Ω + mJ (8) RT ∂φ PV12∂np
[()
]
The definition of the chemical potential for nonequilibrium situations opens the possibility to generalize the analysis of phase diagrams and of reacting systems, which is usually carried out in the framework of equilibrium thermodynamics. In sections 2 and 3 it is shown how to carry out such generalization for any kind of system for which the corresponding chemical potentials are available. In the following sections, we make use of the previous results for the analysis of a particular system, in which the nonequilibrium effects are relevant. 2. Phase Diagram under Nonequilibrium Effects According to EIT, the entropy out of equilibrium is a convex function of the whole set of variables, and this condition allows us to use the extremal character of G (and consequently that of the chemical potential) as a stability criterion. Therefore, when two phases coexist, their respective limits of stability are determined by those compositions for which the value of the total chemical potential has an extremal value. The limit of stability corresponds to the spinodal line in the plane T-φ, built from the condition (∂µ1/∂φ)PV12 ) 0, and the critical temperature corresponding to the maximum of the spinodal line is specified by the further condition (∂2µ1/∂φ2)PV12 ) 0. In equilibrium thermodynamics, the extremal character of the Gibbs function is usually used as criterion of stability, and the equilibrium condition in any chemical system is obtained from the minimization of G at constant temperature and pressure. Such arguments cannot be blindly extrapolated to nonequilibrium situations, where all conclusions must be derived from a dynamical point of view. In this order of ideas, for a closed system whose composition differs from a phase (R) to another (β), the mass conservation is expressed by the balance equation of any component j, namely
(11)
with L a positive phenomenological coefficient. This is a usual argument in linear irreversible thermodynamics. Thus, the stationary condition implies the generalization for a system under flow of the usual equilibrium condition (β) (j ) 1, 2, ...) µ(R) j ) µj
∂φ + J (7) ∂n PV12 1
(10)
(12)
where the only restriction of the previously defined nonequilibrium chemical potential is used. 3. Chemical Processes under Nonequilibrium Hydrodynamic Situations It is well-known20 that the variation of the Gibbs function during a chemical process at constant temperature and pressure is given by
dG dξ )A dt dt
(13)
where ξ is the degree of advancement of the reaction and A is the affinity, defined as A ) -∑iνiµi, with νi the stoichiometric coefficients of the process being considered. The constitutive equation relating the reaction rate with the affinity is given by
Lr A dξ ) Lr 1 - exp A ≈ dt RT RT
[
( )]
(14)
where Lr is a phenomenological coefficient and where it is seen that the equilibrium is reached for affinity equal to zero. In the textbooks on Chemical Thermodynamics,1 the relationship between the equilibrium chemical potential and the activity of the component considered is given by µj ) µ0j (T) + RT ln aj. In the framework of EIT, (1) and (6) show that out of equilibrium it is possible to generalize the expression for the chemical potential as
µj ) µ0j (T) + RT ln aj + µ(ne) j
(15)
Alternatively, one could define an activity out of equilibrium eq ne as RT ln ane a(ne) j j ) RT ln aj + µj . As a consequence, in the steady state the following relation is satisfied
∏j aVj ) Kλ j
(16)
where K is the equilibrium constant without the flow and λ is given by
Chemical Potential of Flowing Fluids
[
λ ) exp -
1
J. Phys. Chem. B, Vol. 102, No. 27, 1998 5337
∑j νjµ(ne) j
RT
]
(17)
4. Application to a Polymer Solution The application of a shear flow to a polymer solution modifies its phase diagram (for instance, the transition from one phase to two phases is modified under shear conditions) and its molecular mass distribution (in a polydisperse system). A wide bibliography on experimental observations of both effects may be found in ref 20. As the experimental basis for our analysis, we start from the results of the previous sections and of the experimental information in ref 2 for the system formed by polystyrene with a molecular weight Mw ) 1.8 × 103 kg mol-1 and polydispersivity index (r ) Mw/Mn ) 1.3) solved in dioctyl phthalate (solvent density 900 kg m-3) under shearing stress when PV12 remains constant. For this solution (Θ ) 288 K, Ψ ) 1.48), the critical temperature at rest is 285 K and the parameter m is estimated by fitting the critical point predicted by the FloryHuggins model and the experimental one. In the present paper we derive the coexistence curves for the system polystyrene-dioctyl phthalate, as well as the modification induced by the shear on the molecular mass distribution function of the polymer. The computations are carried out by using for J the functional relation obtained by fitting the experimental data of ref 2, which was previously used to derive the spinodal curves14,19 of this system; it turns out that J is given by
J)
[
]
0 b0 b0 b3φ-b 0 (PV12)-2φb0 exp b4(PV12)1/2 + (PV12)b2 - pφp 2 b1 pφ
0
(18)
The values of the parameters are b0 ) 4, b1 ) 7.85, b2 ) 0.319, b3 )14.7, b4 ) 0.164, and φ0 ) 0.042, and where we have introduced the function
p ) b1(PV12)-b2
(19)
Expression (18) has some similarities with that used by RangelNafaile et al. in ref 2 (this was partly our motivation for using it), but it improves the quality of the fitting with the available data. However, it is not easy to be interpreted in molecular terms, and it has a complicated nonlinear (in fact, nonanalytic) behavior. This expression for J, of course, is not necessarily unique: other expressions could probably be found that would also fit satisfactorily the experimental data. It is therefore reasonable to ask up to what point our results, obtained by using (18), are reliable. We discuss this point in detail in the concluding remarks. From (7), (8), and (18), the chemical potential of the solvent and of the polymer are given by
V(PV12)2 µ1 1 ) ln(1 - φ) + 1 - φ + χφ2 + J(1 - F) (20) RT m RT
(
)
µp ) ln φ + (1 - m)(1 - φ) + χm(1 - φ)2 + RT V(PV12)2 1 - φ J F + 1 m (21) RT φ
(
where F is defined as
)
Figure 1. Coexistence curves of polystyrene in dioctyl phatalate calculated from (24) for the Flory-Huggins model (FH) and for several values of PV12. The dashed curves correspond to the spinodal curves taken from ref 19.
F ) b0[1 - (φ/φ0)p]
(22)
Therefore, the spinodal curve is given by the points in T-φ that satisfy the equation
( )
( )
∂ µ1 1 ) -(1 - φ)-1 + 1 + 2χφ + ∂φ RT m V(PV12)2 -1 J[φ F(1 - F) - F′] ) 0 (23) RT where F′ denotes the derivative of F with respect to φ. The composition of the coexisting phases are given from condition (12), which requires to solve the set of equations
{
µ1(φ(R)) ) µ1(φ(β)) µp(φ(R)) ) µp(φ(β))
(24)
where µ1 and µp correspond to (20) and (21), respectively. In Figure 1 are plotted the coexistence curves of polystyrene in dyoctil-phatalate for several values of the applied shear stress PV12, and they are compared with the spinodal curves. Regarding both curves, it is convenient to note that the composition of the phases that coexists in the steady state is information simpler to obtain than that corresponding to the limits of stability indicated by the spinodal lines. According to some authors8,9 the turbidity produced by the application of the shear could have an explanation not purely thermodynamic, but it would be a purely dynamic effect associated with the enhancement of density fluctuations. Our point of view is that both effects do not exclude each other,12 and therefore the coexistence curves are a source of information about the degree of relevance of the thermodynamic contribution. This hypothesis about the superposition of thermodynamic and dynamic effects agrees with the fact that the values for the critical temperature predicted by our model based on EIT14,19 (see Figure 2) are lower than the experimental values reported by Rangel-Nafaile.2 5. Degradation of Polymers under Shear Flow as a Chemical Reaction Up to now, we have implicitly assumed that we were dealing with a monodisperse polymer. However, in the practice it is often not so, because the system is composed of the solvent
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Criado-Sancho et al.
Figure 2. Critical temperatures predicted by our model for several values of PV12.
and of a mixture of polymeric chains of different lengths, distributed according to certain molecular weight distribution (MWD). If we consider that the polydispersivity is a chemical equilibrium among segments of different lengths,20,26 the MWD in a steady state is the result of breaking and recombination processes, which may be identified with the reversible reaction
Pj a Pi + Pj-i
(25)
where Pk indicates a macromolecule composed of k monomers. If N(j) is the probability of finding a macromolecule composed of j monomers and we assume that all species Pk have an ideal behavior, the equations (16) take the following particular form (eq)
N (j) N(j) λ-1 ) (eq) N(i)N(j - i) N (i)N(eq)(j - i) ji
(26)
where the quantity (17) referred to reaction 25 is denoted as λji. To determine the explicit form of λji, it is necessary to define the chemical potential for macromolecules with a given degree of polymerization; thus, eq 21 is generalized as
µj µ(eq) j ) + Ξ(φ, PV12)jN(j) (j ) 2, 3, ...) RT RT
(27)
is the equilibrium contribution and where the where µ(eq) j function
Ξ(φ, PV12) )
V(PV12)2 1 - φ F+1 RT φ
(
)
(28)
has been used, so that
λji ) exp[-Ξ[(j - i)N(j - i) + iN(i) - jN(j)]]
(29)
The value for the polydispersivity previously reported in this system allows us to consider for this system a Schultz distribution,23 i.e.
Figure 3. Continuous curve representing the difference between the probability N(j) of finding a segment of length j when the system is submitted to PV12 ) 400 N m-2 and the value of such a probability in a rest state N(eq)(j).
where M0 is the monomer molecular weight, Γ is the gamma function, and the parameters y and h satisfy the relations
y)
h+1 Mw
h)
1 r-1
(31)
When one combines (26), (29), and (30), one arrives at the following results
[
]
Γ(h(eq)) N(j) j ) N(i)N(j - i) (y(eq)M )h(eq) i(j - i)
[
0
h(eq)-1
λ-1 ji
(yM0)h {(j - i)h exp[-yM0(j - i)] + Γ(h)
λji ) exp -Ξ
(32)
]
ih exp[-yM0i] - jh exp[-yM0j]} (33) Equations (32) and (33) allow one to determine how are modified the parameters of a Schultz distribution under the action of a shear flow with a given value of PV12. To do that, one may use the algorithm described in the Appendix, which is a modification of the model used26 to study the degradation in the system polystyrene-transdecalin. In Figure 3 is plotted with a continuous curve the difference between the probability N(j) of finding a segment of length j when the system is submitted to shear flow and the value of such probability in a rest state N(eq)(j). In the same figure is plotted with a discontinuous curve the equilibrium distribution of segments. This shows that, for PV12 ) 400 N m-2, the modification of the MWD is of the order of 1%. Furthermore, a comparison of both curves yields the conclusion that the application of shear implies a reduction in the number of long chains (negative values of the continuous curve), the chains with a length of the order of 20 000 segments being those that are most affected by the degradation. In contrast, the positive value of N - N(eq) for lengths below 15000 segments shows the increase in the number of short chains: this increase is maximum for macromolecules of the order of 7000 segments, whereas the macromolecules of the order of 15 000 segments are almost unaffected by the degradation. 6. Concluding Remarks
(yM0)h h-1 N(j) ) j exp(-yM0j) Γ(h)
(30)
Here we have shown how the chemical potential of the components of a fluid mixture is modified in the presence of a
Chemical Potential of Flowing Fluids
J. Phys. Chem. B, Vol. 102, No. 27, 1998 5339
shear viscous pressure. In contrast to what happens in gases or in solutions of small molecules, the nonequilibrium effects are rather large in polymer solutions. We have obtained the coexistence line in the phase diagram for polystyrene in dioctyl phthalate for several values of the applied shear viscous pressure and have compared them with the corresponding spinodal curves, which were obtained in previous works. Furthermore, we have shown the effects of shear-induced degradation on the MWD of the polymer: it turns out that the depletion is maximum for macromolecules of length of the order of 20 000 segments and the increase is maximum for shorter molecules of length of 7000 segments, whereas the macromolecules of 15 000 segments are almost unaffected. In the previous analysis by Nettleton15,16 on the nonequilibrium modifications of the chemical potential, the nonequilibrium effects considered turned out to be very small. This is not surprising, because he was dealing with simple gases. Thus, that analysis could leave the impression that the inclusion of nonequilibrium effects in the chemical potential was merely academic. Here, we have shown that when applied to polymer solutions, where the relaxation times are much longer than in simple gases, the effects turn out to be appreciable in situations of practical interest. As we have commented in the discussion on (18), it could be asked whether our results depend strongly on that particular form for the steady-state compliance J. The reliability of our results is reinforced from the fact that they are qualitatively confirmed by using simpler theoretical models for J, which are, however, not applicable to the system under study. Indeed, for dilute polymer solutions where the Rouse-Zimm model is applicable, the steady-state compliance is given by
(
)
ηs CM J) 1cRT η
Figure 4. Diagram showing how the MWD out of equilibrium is obtained.
2
(34)
where c is the polymer concentration expressed in terms of mass per unit volume, η and ηs are the solution and solvent viscosities, M is the molecular mass of polymer, and C is a parameter whose value depends on the model used (0.4 for the Rouse model and 0.206 for the Zimm model). The prefactor agrees with the predictions of the scaling arguments according to which J is proportional to c-1 in that regime. If one uses only the prefactor and repeats the analysis given in this paper, one finds a downward shift of the spinodal line under shear, instead of the observed upward shift.25 This fact has led some authors to conclude that the thermodynamic scheme is inadequate to describe the shear-induced shift of the spinodal line. However, this criticism is not well-founded: indeed, if one uses the full expression (34) instead of only the prefactor and if one takes into account the concentration dependence of η, one finds an upward shift, in agreement with observations. Therefore, one must indeed be careful with the description of J, and one must go beyond simple scaling arguments. In our case, (18) fits well the available experimental data for J, and it predicts results that agree, at least qualitatively, with the observations. Therefore, the present results deserve to be seriously taken into consideration. Acknowledgment. This work has been partially supported by the Direccio´n General de Investigacio´n Cientı´fica y Te´cnica of the Spanish Ministry of Education and Science under Grant PB94-0718.
Appendix We adopt as starting point the following recurrence relation26
N(j) )
1
L
∑Λ(i, j) N(i) N(j - i)
L i)1
(j ) 2, 3, ...) (A1)
where L ) E(j/2) is the integer part of j/2 and the explicit expression of Λ(i, j) is derived from (26). In this way, it is generated a discrete distribution N(j) for which the sums
Sn )
∑j jnN(j)
(n ) 0, 1, ...)
(A2)
allow one to determine the normalization constant and the number average molecular weight
Mn )
S1 M S0 0
(A3)
The former distribution does not necessarily coincide with Schultz’s distribution given by (30), which will now be denoted as N(S)(j), and for which are defined the sums analogous to (A2)
S(S) n )
∑j jnN(S)(j)
(n ) 0, 1, ...)
(A4)
Furthermore, recalling (31) and the definition of polydispersivity index, it is concluded that the weight-averaged molecular mass satisfies the following relations
5340 J. Phys. Chem. B, Vol. 102, No. 27, 1998
Mw )
h+1 Mn h
h ) yMw - 1
Criado-Sancho et al.
(A5) (A6)
If one formulates the hypothesis that for values of PV12 considered the MWD is still a distribution of the Schultz’s form and only its parameters are modified, the new parameters h and y are obtained by means of the algortithm sketched in Figure 4. References and Notes (1) Kirkwood, J. G.; Oppenheim, I. Chemical Thermodynamics; McGraw-Hill: New York, 1961. (2) Rangel-Nafaile, C.; Metzner, A.; Wissbrun, K. Macromolecules 1984, 17, 1187. (3) Wolf, B. A. Macromolecules 1984, 17, 615. (4) Kra¨mer, H.; Schenck, H.; Wolf, B. A. Makromol. Chem. 1988, 189, 1613, 1627. (5) Bueche, F. J. Appl. Polym. Sci. 1960, 4, 101. (6) Wolf, B. A. AdV. Polym. Sci 1988, 85, 1. (7) Nguyen, T. Q.; Kausch, H. H. Makromol. Chem. 1989, 190, 1389; AdV. Polym. Sci. 1992, 10, 73. (8) Helfand, E.; Fredrickson, G. H. Phys. ReV. Lett. 1989, 62, 2468. (9) Milner, S. T. Phys. ReV. Lett. 1991, 66, 1477.
(10) Onuki, A. Phys. ReV. Lett. 1989, 62, 2472. (11) Onuki, A. J. Phys.: Condens. Matter 1997, 9, 6119. (12) Criado-Sancho, M.; Casas-Va´zquez, J.; Jou, D. Phys. ReV. E 1997, 56, 1887. (13) Garcı´a-Colı´n, L. S.; De la Selva, S. M. T.; Pin˜a, E. J. Phys. Chem. 1986, 90, 953. (14) Lebon, G.; Casas-Va´zquez, J.; Jou, D.; Criado-Sancho, M J. Chem. Phys. 1993, 98, 7434. (15) Nettleton, R. E. J. Phys. Chem. 1996, 100, 11005. (16) Nettleton, R. E. Z. Phys. Chem. 1996, 196, 177. (17) Jou, D.; Casas-Va´zquez, J.; Lebon, G. Extended IrreVersible Thermodynamics, 2nd ed.; Springer: Berlin, 1996. (18) Salamon P., Sieniutycz, S., Eds.; Extended Thermodynamic Systems; Taylor and Francis: New York, 1992. (19) Criado-Sancho, M.; Jou, D.; Casas-Va´zquez, J. Macromolecules 1991, 24, 2834. (20) Jou, D.; Casas-Va´zquez, J.; Criado-Sancho, M. AdV. Polym. Sci. 1995, 120, 207. (21) Grmela, M. Physics Lett. A 1987, 120, 276. (22) Grmela, M. Phys ReV. E 1993, 48, 919. (23) Kurata, M., Thermodynamics of Polymer Solutions; Harwood Academic Publishers: Chur, Switzerland, 1982. (24) Doi, M. Introduction to Polymer Physics, Clarendon Press: Oxford, 1996. (25) Criado-Sancho, M.; Casas-Va´zquez, J.; Jou, D. Polymer 1995, 36, 4107. (26) Criado-Sancho, M.; Jou, D.; Casas-Va´zquez, J. J. Non-Equilib. Thermodyn. 1994, 19, 137.
