Langmuir 2002, 18, 1457-1459
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Theories of Phase Separation in Colloidal Systems. Controversies and a Possible Resolution? Kenneth S. Schmitz Department of Chemistry, University of MissourisKansas City, Kansas City, Missouri 64110
Lutful Bari Bhuiyan* Laboratory of Theoretical Physics, Department of Physics, University of Puerto Rico, Box 23343, San Juan, Puerto Rico 00931-3343 Received October 22, 2001. In Final Form: January 4, 2002 The mechanisms of phase separation in colloidal systems as laid out in the volume-term theories, the Sogami-Ise model, and the Langmuir unipolar coacervation model are shown to have a common bond through standard thermodynamic relationships between the Gibbs free energy, the Helmholtz free energy, and the chemical potential. This leads to the interesting observation that the validity of the individual approaches is intrinsically tied to the validity of the generalized volume-term approach.
Introduction 1-5 have been
In recent years, volume-term (VT) models proposed to explain the two-state structure of colloidal suspensions readily observed in video microscopy techniques.6-17 Apart from maintaining an effective onecomponent view, a second interesting aspect of the current activity on the VT approach is that a phase separation may not be inconsistent with a “repulsive only” pair interaction between the macroions in the system. This stands in contrast to the model of Sogami and Ise (SI),18 which stipulates an effective pairwise interaction between macroions that exhibits a long-range attractive tail. The SI theory was based on the Gibbs free energy rather than the Helmholtz free energy, the latter of course being the basis of the venerable Derjaguin-Landau-VerweyOverbeek (DLVO) potential.19 In all this activity,1-18 the Langmuir unipolar coacervation (LUC) model for phase separation20 gets only a brief mention,3,5 perhaps primarily because of its dismissal as a viable model by Verwey and Overbeek (see for example, pp 195-199 of ref 19). (1) van Roij, R.; Hansen, J.-P. Phys. Rev. Lett. 1997, 79, 3082. (2) van Roij, R.; Hansen, J.-P. Prog. Colloid Polym. Sci. 1998, 110, 50. (3) van Roij, R.; Dijkstra, M.; Hansen, J.-P. Phys. Rev. E 1999, 59, 2010. (4) Denton, A. R. J. Phys.: Condens. Matter 1999, 11, 10061. (5) Warren, P. B. J. Chem. Phys. 2000, 112, 4683. (6) Carbajal-Tinoco, M. D.; Castro-Roma´n, F.; Arauz-Lara, J. L. Phys. Rev. E 1996, 53, 3745. (7) Larsen, A. E.; Grier, D. G. Phys. Rev. Lett. 1996, 76, 3862. (8) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (9) Grier, D. G. J. Phys.: Condens. Matter 2000, 12, A85. (10) Ito, K.; Ise, N. J. Chem. Phys. 1987, 86, 6502. (11) Ito, K.; Okumura, H.; Yoshida, H.; Ueno, Y.; Ise, N. Phys. Rev. B 1988, 38, 10852. (12) Yoshida, H.; Ito, K.; Ise, N. Phys. Rev. B 1991, 44, 435. (13) Yoshida, H.; Ito, K.; Ise, N. J. Chem. Soc., Faraday Trans. 1991, 87, 371. (14) Yoshida, H.; Yamanaka, J.; Koga, T.; Ise, N.; Hashimoto, T. Langmuir 1999, 15, 2684. (15) Dosho, S.; Ise, N.; Ito, K.; Iwai, S.; Kitano, H.; Matsuoka, H.; Nakamura, H.; Okumura, H.; Ono, T.; Sogami, I. S.; Ueno, Y.; Yoshida, H.; Yoshiyama, T. Langmuir 1993, 9, 394. (16) Ito, K.; Yoshida, H.; Ise, N. Science 1994, 263, 66. (17) Yoshida, H.; Ise, N.; Hashimoto, T. J. Chem. Phys. 1995, 103, 10146. (18) Sogami, I.; Ise, N. J. Chem. Phys. 1984, 81, 6320. (19) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (20) Langmuir, I. J. Chem. Phys. 1938, 6, 873.
It is shown in the present communication that both the SI and LUC ideas for phase separation are contained within the general formalism of the VT theories, and as such inconsistencies in any would also be reflected in others. It is not necessary for our purposes to delve into the details of the VT approach, but rather we will outline only the salient features as these are well-established thermodynamic expressions, the underlying assumption being the equality of the chemical potential of all the species present and the equality of the pressure across a phase boundary. The relevant variables for the phase separation are the mean number densities of the added electrolyte, n0s , and the mean number density of the macroions (and hence the mean number density of the counterions released by the macroions), n0p. The screening parameter plays an absolutely crucial role in the VT approach,
κVT )
x
( ||)
4πλBZs2 2n0s +
Zp 0 n Zs p
(1)
where λB ) βqe2/ is the Bjerrum length, qe is the magnitude of the electron charge, and is the bulk dielectric constant. Also, β ) 1/kBT with kB the Boltzmann constant and T the absolute temperature. For simplicity, the added electrolyte is taken to be a symmetric valency Zs:Zs salt and shares the same absolute value of charge as the counterion, viz., |Zc| ) |Zs|, and 2n0s ) n+0 + n-0. The picture of the macroion phase separation that ensues is that the microions respond to a spinodal instability in the system and phase separate, whereupon the macroions, by virtue of being married to their parent ion clouds in the form of the electrical double layer, follow their ion clouds in the phase separation process. The Gibbs free energy for either phase is calculated in the VT approach from the general thermodynamic relationship,
j s + npµ j p ) ns G h ) nsµ
( ) ∂A h ∂ns
np
( )
+ np
∂A h ∂np
(2)
ns
where A is the Helmholtz free energy and the bar (over the function) denotes a reduced function per unit volume, viz., βG/V ) G h . Likewise, the reduced pressure takes on the operational form
10.1021/la011577c CCC: $22.00 © 2002 American Chemical Society Published on Web 02/08/2002
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Langmuir, Vol. 18, No. 5, 2002
βP ) ns
( ) ∂A h ∂ns
+ np
np
Letters
( ) ∂A h ∂np
-A h
(3)
ns
Let us now examine the explicit electrical contributions to the Helmholtz free energy arising from the macroionmacroion, microion-microion, and macroion-microion interactions. The contribution from the interaction between the macroions, as given by Warren,5 is
A h elec pp ≈
n0p 2 2
∫
Zp2λB exp(-κVTr) gpp(r) ) r n0p 2 2 exp(-κVTr) Z λ (4) 2 p B r
d3r
〈
〉
where gpp(r) is the macroion-macroion pair distribution function. The second equality above indicates that this is a “preaveraged” quantity identified with the traditional DLVO screened Coulomb interaction. Thus, eq 4 may be recast as
A h elec pp )
A h elec DLVO
(5)
The second contribution to the Helmholtz free energy comes from the microion-microion interaction (the Debye-Hu¨ckel contribution21)
A h elec ss ) -
κVT3 12π
(6)
with the final contribution of interest being due to the microion-macroion interaction,5
A h elec ps ) -
2κVTZp2λBn0p 3
(7)
It is now a simple matter to show from eqs 2-7 that the electrostatic Gibbs free energy and pressure for these three components are
G h elec DLVO
( (
G h elec ss ) ns
) )
(8)
κVTr elec A h DLVO 2
(9)
( ) ( ) ()
3 κVT3 ∂A h elec ∂A h elec ss ss 3 κVT + np ))∂ns ∂np 2 12π 8π (10)
G h elec ps ) ns
( )
κVT3 κVT3 κVT3 - )8π 12π 24π
(11)
∂A h elec ∂A h elec 3 elec ps ps + np ) A h ∂ns ∂np 2 ps
(12)
βPelec ss ) -
( ) ( ) ()
and
A h elec ps ) βPelec ps 2
n0j
j
n0j
) kBT
(π)1/2 (λB 3 -
κ3
)
24π
∑j Zj2n0j
κ2 ) 4πλB
The basic equations used thus far are to be found in the papers on the VT model for phase separation. Within the (21) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976.
(14) (15)
where the summation extends over all species present. The first term on the right-hand side of eq 14 is the ideal pressure of the colloidal system. For our purposes, the second term is more interesting as it is formally identical to the electrostatic pressure given by eq 11. However, there is an important, subtle difference in that the definition (1) excludes the mean macroion number density whereas the definition (15) includes this density. A plausible reason as to why the macroion number density n0p appears in κ may be seen from the fact that Langmuir treated the “dressed” micelles as point ions and therefore contributed Zp2n0p to the evaluation of κ. His argument was that at
sufficiently large concentrations κ is proportional to xn0p. He then wrote eq 14 in the form
(16)
A somewhat similar argument was put forth by van Roij and Hansen1 as the “essence” of the term responsible for the spinodal instability. In the VT theories, however, the macroion concentration enters into the expression for κ indirectly through the counterion contribution as given in eq 1. We next examine the criticism of Verwey and Overbeek regarding the n0p dependence of κ on pages 197 and 198 of their monumental work,19 “The double layer theory, however, shows that this thickness is determined by the electrolyte concentration in the sol medium, far from any particle, and is, therefore, independent of the sol concentration.” This is clearly suggestive of the notion that the screening parameter in the DLVO potential be calculated only on the basis of added electrolyte. That is, κDLVO is, by definition, of the form
κDLVO ) x4πλBZs2n0s (13)
)
∑j Zj2n0j )3/2
P ≈ 3n0p - 2(n0p)3/2
κVTr elec ) 2A h DLVO 2
βPelec DLVO ) 1 -
(∑ (∑
P ) kBT
j
(n0p)2 2
framework of this formalism, one obtains in the case of the (electrical) macroion-macroion pair interaction the conversion factor of [2 - (κVT/2)] between the Gibbs and the Helmholtz free energies in accordance with eq 8. This is the same conversion factor first derived by Sogami and Ise18 and later identified by a different means with the electrostatic pressure contribution given by eq 9.22-24 We now turn our attention to the electrostatic pressure h elec (derived from A ss ) given by eq 11. Note that this quantity was discussed by McQuarrie21 (cf. the chapter on the Debye-Hu¨ckel theory) and is simply the Debye-Hu¨ckel (limiting law) expression for the osmotic pressure. This term can now be related to the osmotic pressure in the Langmuir model (cf. eq 15 of ref 20),
(17)
A seeming dilemma now arises with regard to the VT theories vis-a`-vis the LUC model and the VerweyOverbeek criticisms. Warren5 has given a detailed de(22) Schmitz, K. S. Langmuir 1996, 12, 3828. (23) Schmitz, K. S. Phys. Chem. Chem. Phys. 1999, 1, 2109. (24) Schmitz, K. S.; Bhuiyan, L. B. Phys. Rev. E 2000, 63, 011503-1.
Letters
Langmuir, Vol. 18, No. 5, 2002 1459
scription of the relationship between the macroion, its associated ion cloud that forms the double layer, and the inclusion of macroions in the screening parameter in a manner employed by Langmuir. If indeed the counterions surrounding the macroion must remain in the vicinity of the macroion such that they are the reason for the separation of the suspension into “dense” and “sparse” regions, then these counterions cannot be included in the calculation of κ for they are not “far from any particle” in the spirit of Verwey and Overbeek. It would therefore appear that arguments against the LUC model likewise affect the VT theories and the SI model for phase separation in colloidal systems. A possible resolution is suggested in the following argument. It is conceivable that Verwey and Overbeek may have overlooked an aspect of the LUC model that is intrinsically tied to the distributions of the macroions and microions in the system. Let us consider, for example, the Poisson equation for a system of macroions and microions. This is an open thermodynamic system, identical to the one used in the VT theories1-5 and in line with the chemical potential expressions for equilibrium between two phases (see for example ref 3). The Poisson equation reads
∇2φ(r) ) )-
4πqe
∑Zini(r)
4πqe [Zpnp(r) + Zcnc(r) + Z+n+(r) + Z-n-(r)] (18)
As mentioned earlier following eq 1, we assume a symmetric Zs:Zs added electrolyte and |Zc| ) |Zs|. Assuming a Boltzmann distribution for the microions, viz.,
nc(r) ) n0c exp(-βqeZc φ(r)) n+(r) ) n0+ exp(-βqeZ+ φ(r))
(19)
n-(r) ) n0- exp(-βqeZ- φ(r)) and expanding these expressions up to the second term and using the salt electroneutrality,
Z+n0+ + Z-n0- ) 0
(20)
we get
∇2 φ(r) ) -
4πqe [Zpnp(r) + Zcn0c ] + κ2 φ(r) (21)
where κ is defined by eq 1 and n0c represents the average number density of the counterions. In writing eq 21, we have also used the fact that for the symmetric valency added salt n0+ + n0- ) 2n0s . The simple reason the macroions are not included directly in the calculation of κ is that np(r) does not exhibit a Boltzmann distribution. In view of the fact that crystalline structures were accepted to have negative total energies, Langmuir apparently used that model as a starting point for his argument that the total energy of a colloidal system is negative. In modern-day computer simulations, the col-
loidal system is taken to have a net negative energy even when all of the particles move.25,26 The volume-term theories have drawn attention to the holistic nature of phase separation in macroion systems: that the microions as well as the macroions must be considered and they need to be treated on an equal footing. We have tried to develop arguments in this letter to the effect that the Sogami-Ise and Langmuir equations responsible for phase separation in their respective theories are contained within the general mathematical framework of the volume-term theories. The physics of the VT and SI premises is that both employ a preaveraged pair interaction between the macroions, while the VT and LUC theories both include the Debye-Hu¨ckel expression for the osmotic pressure attributed to the microions. In the VT theories,1-3 the mechanics of phase separation is independent of the nature of the macroion-macroion interaction. The retention of the DLVO picture is thus understandable. The connection between the VT and the SI indicates that this interaction indeed has a (microionmediated) long-range attractive component. The common threads running through the VT, SI, and LUC models are the fundamental thermodynamic expressions linking the Gibbs and Helmholtz free energies, the osmotic pressure, and the chemical potential, the logical implication being that any argument directed at the validity of one would also, albeit indirectly, apply to the others. A basic premise of this discussion has been the DebyeHu¨ckel linearization model, certainly in suggesting the connection between the VT and the LUC models. Literature abounds on the inadequacies of such linearization even for simple electrolytes at moderate concentrations and/or in the presence of multivalent ions. The situation in polyelectrolyte solutions is no different. As Denton4 points out, “First of all, the assumption of linear response of the counterions is strictly valid only for dilute suspensions of weakly charged macroions.” The LUC itself is probably valid only under such circumstances. However, even when nonlinear effects are included the conclusions remain somewhat ambiguous. In a very recent paper,27 von Gru¨nberg, van Roij, and Klein have shown that a gas-liquid phase coexistence in a spherical colloidal suspension for monovalent microions predicted by volume-term theory occurs within a linearized PB equation but not in the nonlinear PB equation. In contrast, Denton28 has found that the inclusion of nonlinear terms actually enhances the occurrence of spinodal instability. In general, though, one may question therefore the employment of a preaveraged pair potential between macroions, such as the DLVO potential, in a phase defined as a dense population of macroions. Nonlinear effects will surely come into play, especially when the ion clouds of the macroions show extensive overlap. Acknowledgment. We thank Patrick Warren, Rene van Roij, and Alan Denton for the many stimulating discussions of their work. L.B.B. acknowledges a grant from FIPI, University of Puerto Rico. LA011577C (25) Delville, A. Langmuir 1994, 10, 395. (26) Linse, P. J. Chem. Phys. 2000, 113, 4359. (27) Von Gru¨nberg, H. H.; van Roij, R.; Klein, G. Europhys. Lett. 2001, 55, 580. (28) Denton, A. Private communication.
