Langmuir 1993,9, 1442-1445
1442
On the Role of Solvation Forces in Colloidal Phase Transitions Laura J. Douglas,+Mark Lupkowski,tJ Travis L. Dodd,tJ and Frank van Swol*pt Department of Chemical Engineering, University of Illinois, Urbana, Illinois 61801 Received January 11,1993. In Final Form: February 16,1993
Phase transitions in colloidal systems characterized by increasing colloidal order with increasing temperature have been labeled entropy driven. Some colloidal phase transitions that may be identified as entropy driven are the swelling of clays, the crystallization of proteins, and the dehydration of silicotungstatecrystals. None of these transitions can be understood within the framework of classical colloidalforce (DLVO)theory. We present a new theoretical treatment of the colloidal force problem that describes the statistical mechanics of colloidal forces through a surface approach. A phase diagram that reproduces key features of entropy driven transitions is calculated consideringonly short-rangecolloidal forces in a system of parallel plate colloids and simple fluid. The role of entropy is further considered by quantifying transitions with Clapeyron equations. Colloidal systems are typically composed of particles (10-1OOO nm in diameter) interacting through a solvent medium. This categorization encompasses an enormous range of materials including polymers, ceramics, and biological materials. Forces between the colloids ultimately determine the aggregation state of a suspension. Furthermore, they are of key importance in many complex colloidal phenomena including clay swelling, protein crystallization, cell adhesion, protein folding, and the dehydration of inorganic colloidal crystals. In this paper, we present the first complete theoretical treatment of the contribution of colloidalforces to observed colloidalphase transitions. We restrict ourselves initially to short-range forces due to solvent structuring. These forces are oscillatory as has been demostrated in surface force apparatus1 and atomic force microscopy2 experiments. However, our theory is comprehensive in the sense that it may be extended in a straightforward way to include the many physical interactionspresent in collidal systems. As such, the formalism presented here should be viewed as the framework necessary to study and understand in a fundamental way the complex features (e.g. charge effects, surface roughness, and surface flexibility)present in many systems. We shall return to the extension of our theory in the discussion section. Putting aside the multiplicity of issues arising in these more complex cases, we recognize that clay swelling3p4 representa the ideal test case for our theory due to the parallel plate geometry of the clay species.6 The clay swelling experimenta we consider are those of Smalley et al.3 where a vermiculite clay is placed in a closed container with an aqueous solution of electrolyte. The inset of Figure 1shows a schematic of the experiment. A hydrostatic (or bulk) pressure, Pb, may be applied to the entire solution, and the spacing between the clay layers is monitored with X-ray scattering or neutron diffraction. The external t University of Illinois. t Presently at Southwest Research Institute, San Antonio, TX 70220. 1 Presentlvat Schoolof Chemical Eneineerine. -. CornellUniversitv. ", Ithaca, NY i4053. (1) Horn, R. G.;Iaraelachvili, J. N. J. Chem. Phys. 1981, 75,1400. (2) OShea, S.J.; Welland, M.E.; Rayment, T. Appl. Phys. Lett. 1992, 60.2368. (3) Smalley,M.V.;Thomas, R. K.; Braganza, L. F.; Matsuo, T.Clays Clay Miner. 1989,37,474. (4) Brag-, L. F.;Crawford, R. J.; Smalley, M.V.; Thomas, R. K. Clays Clay Mmer. 1990,38,90. (6) Bailey, S. W. In Crystal Structures of Clay Minerals and Their X-Ray Identification; Brindley G. W., Brown, G., Eds.; Spottiewoode Ballantyne: London, 1980, pp 90-99.
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pressure (or stress), pest, applied directly to the clay booklet in Figure 1 is not present in the Smalley experiment. However,it may be found in similarexperimenta,M without discussion of ita distinctive role as a state variable. We include it here to expand the class of experimenta we address and to demonstrate the importance of the separation of these two types of sbesses @b and pea). Most fundamentally, pb is an osmotic stress in the sense that it controls solvent activity (or chemical potential, p ) while pest is a mechanical stress that has no effect on solvent activity. As the bulk pressure in the Smalley experiment is increased at constant temperature, the clay swells from an average separation of ( h ) = 1.94 nm (crystalline phase) to ( h ) = 12.0 nm (osmotic phase). A phase diagram, reproduced here in Figure 1, is then constructed by repetition of the experiment at different temperatures. Contrary to the more familiar molecular systems, the crystalline phase is the high temperature (or low pressure) phase! This phase diagram is then inverted with respect to the molecular systems, and reproducing this qualitative result will be essential for any theory. Finally, to indicate the generality of this type of swelling or, conversely,crystallizationand dehydrationphenomena, we point out that both the crystallization of DNA strands performed by Parsegian et al.9 and Leikin et al.10 and the dehydration of certain silicotungstate crystals by Spitayn and Kolli," are produced by increasing temperature. While it is most important to first understand the molecular physics that controls these transitions, it is also clearly valuable to demonstrate a complete connection with experimental analysis and observables. Analysis of these transitions has most often been approached through the familiar computation of entropy, enthalpy, and heat capacity changes (AS, AH, and AC,) using Clapeyron equations.3J We show here that proper Clapeyron equations may be derived in a straightforward manner once the appropriate free energy has been determined from a statistical mechanical analysis. Furthermore, we demonstrate that a Clapeyron style analysis is not as straight~~
(6) Rawel-Colom, J. A. Trans. Faraday SOC.1964,60, 190. (7) Prouty, M.S.; Schecter,A. N.; Pareegian, V. A. J. Mol. Biol. l98S, 184, 517. (8)Viani, B. E.; Low,P. F.; and Roth, C. B. J. Colloid Interface Sci. 1983,96,229. (9) Pareegian, V. A.; Rand, R. P.; Rau, D. C. In Physics of Complex and Supermolecular Fluids; Safran,S . A., Clark,N. A., Ede.;John Wiley & Sons: New York, 1987; pp 116-135. (10) Leikin, S.;Rau, D. C.; Pareegian, V. A. Phys. Reo. A 1991, 44, 5272. (11) Spitsyn, V. I.; Kolli, I. D. J. Znorg. Chem. 19S6, 1 , 2403.
0743-7463/93/2409-1442$04.00/00 1993 American Chemical Society
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Langmuir, Vol. 9, No. 6, 1993 1443
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T ("C) Figure 1. Phase diagram presented in Figure 6 of Smallsy et al.*for the swellingof n-butylammoniumvermiculite. The inset show a schematic of a model experiment with three colloidal state variables: p b ) , T, and p&.
forward as is generally assumed. The equations we derive are subtly different from those often applied to analyze experimental results and are furthermore dependent on specific experimental configuration. We turn now to our approach in modeling colloid and solvent species in these systems. At the most basic level of approximation, both colloid and fluid species may be represented by simple models. This is most appropriate for the developmentof a fundamentalcolloidalforce theory that addresses the type of phase transitions mentioned above. Features arising from specific solvent and colloid structures will alter phase diagrams in a quantitative way but are not likely to change qualitative features such as crystallizationwith increasing temperature. To illustrate this point, consider the liquid-vapor transition for a Lennard-Jones (U) solvent as compared to even a simple hydrocarbon (e.g. CsHs). In order to properly model the hydrocarbon, a model of vibrational and rotational states would be necessary. Nevertheless, like any complex molecule, C& will still vaporize with increasing temperature. Indeed once we have represented the phase diagram in the appropriate reduced variables, it transpires that all molecular fluids behave in essentially the same way. What we have learned from focusing on the general qualitative features of the phase diagram is that on a molecular level,liquid-vapor transitions are aconsequence of the existence, not the details, of attractive interactions. Here, we approach entropy-driven colloidal phase transitions in a similar vain and ask which features of the molecular interactions lead to ordered high temperature phases. Therefore, we consider smooth parallel plate colloids in a U solvent and concentrate on a proper description of the features of experiments in which these transitions areobserved. The two most important features of these experimenta are the constraints of chemical and mechanical equilibrium. It is clear that chemical equilibrium between solvent confined by colloids and solvent in the bulk must be maintained. Furthermore,the density (or spacing) of the colloids must be treated as the fundamental dependent variable of the system subject to constraints of mechanical equilibrium. More specifically,the model we consider consists of two smooth parallel plates of area A separated by a distance h in a closed reservoir of s 0 1 v e n t . l ~One ~ ~ of the plates (12)Ma&, J. J.; Tirrell, M.;Davis, H. T. J. Chem. Phys. 1986,86, 1888. (13) h h , S. G.;Everett, D. H.; Radke, C. J. Chem. Soc., Faraday Trons. 2 1973,69,1266.
is fixed while the other is free to move in response to the state variables. The thermodynamic state of the surrounding bulk fluid is fully specified by the system temperature, T,and chemical potential, p. The force per unit area on the outside of the mobile plate, F d A = Pb + ped, is due to bulk fluid and mechanical contributions. At equilibrium, the plates are separated by h,. The solvation force arises fromsolventstructure in the confined fluid192J2 and is defined per unit area as fa = p - pb(T,p) where p is the pressure between and normal to the parallel plates. Mechanical equilibrium clearly requires fs = pea. This solvation force is identical with the swelling pressure6bJ6and structural disjoining pressure16 introduced by others. The statistical mechanics for this parallel plate model is an extension of an analysisof the surface forces apparatus presented by Lupkomki and van Sw0l.l' In their generalized ensemble, the volume of the confined fluid may vary in response to all three state variables, p , T, and p h . The partition function of this ensemble is T = SO"dhe-@tUm+O) where the force on the mobile plate is written as the gradient of an energy (Fed = -VU&, Q(h;p,T) = -plIn Z is the grand potential, and Z is the grand canonical partition function of an open system with fixed plate separation. For our model colloids, Ued @b + p e t ) V&t, where V&t = Ah is the volume in the slit. Furthermore, a surface free energy, Qa = Q + PbV&, may be introduced reducing the generalized partition function to
A normalized probability distribution for h can then be written P(h) = T-le-@A@&+m/A), and an average plate separation may be defined as ( h ) = Jo"&hP(h). In the thermodynamic limit ( A O D ) , the probability distribution becomes a delta function and h, is found where P(h) is maximized or, equivalently, where p e d + Q8(h;p,T)/Ais minimized. In this paper, without loss of generality, the colloids are taken to be noninteracting. Inclusion of direct interactions, q5pp, is mathematicallytrivial, and results of such interactions will be discussed presently. It has been shown with molecular dynamics that solvent structuringin confined spaces leads to spacial oscillations in W(h;p,T) out of phase from those of f8(h).12 This is implicit in the definition of solvation force, fs = -(d(Qa/ A ) / d h ) ~ , .From our identification of p e d + Qs(h;p,T)/A as the relevant free energy, we now observe that these oscillations in Q8 provide the basis for transitions between phases characterized by different h,. The contribution of a mechanical stress, p e d , is linear in h, and ita magnitude simply changes the relative depths of the local minima of Q8. Direct colloid interactions would play the same role as pea although they may have different functional forms. Since the key to studying these transitions is then the calculation of P ( h ; p , T ) , this problem is ideally suited to the nonlocal density functional theory18J9 approach taken here. We can at this stage identify how these phase transitions must fit into the classical picture of colloidal forces. In
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(14) Evans, R.; Marini Bettolo Marconi, U. J. Chem. Phys. 1987,86, 7138. (16) Low, P.F. hngmuir 1987,3,18. (16) Derjnguin, B. V.;Churaev, N. V. J. Colloid Interface Sci. 1974, 49, 249. (17) Lupkowski, M.;van Swol, F. In Dynamics in Small Confining Systems;Drake,J. M.,Klafter, J.,Kopelman,R.,Ede.;MaterialeRaearch Socity: Pittsburgh, PA, ISSO, pp 19-22. (18) van Swol, F.;Hendereon, J. R. P h y . Reu. A 1989,40,2667. (19) van Swol, F.;Henderson, J. R. Phys. Rev. A lSSl,4S, 2932.
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1444 Langmuir, Vol. 9, No.6, 1993
classical theory (known as DLVO theorym),only electrostatic and van der Waals forces are considered. These , are most forces lead to interaction potentials, ~ D L V O that properly thought of as contributions to the free energy. The minima in potential diagrams, ~ D L V Ovs h, then represent potential equilibrium states for the colloidal suspension. There are typically two minima present in these diagrams. One is the primaryvan der Waals mimium (at short range) and the other is a secondary minimum representing a dispersed state for the colloids. The two forces embodied in DLVO theory describe colloidal stability well for certain cases, but they cannot describe even qualitatively the phenomena observed in clay swelling, protein crystallization, or the dehydration of silicotungstate crystals. In each of these cases the details of the forces at short-range and, in particular, their dependence on p and T ,are crucial to a complete understanding of transitions and interactions. Our theory, which begins with these dependencies, results in a detailed description of the primary (short-range) minimum of DLVO theory. When combined with electrostatics and direct (e.g. van der Waals) interactions, it will represent a major extension of DLVO theory. In our treatment, the primary minimum is properly sensitive to all the state variables (most importantly solvent activity) demostrated in experiments. Phase transitions may be further quantified for comparison with experiment by derivingClapeyron equations. The route we take is similar to the capillary condensation work of Evans and Marini Bettolo Marconi14 where equations are derived in terms of the excess adsorption, r,and the excess entropy, s per unit area. These variables are not easily accessible through experimental methods. However, a connection can be made with macroscopic changes in entropy and volume, written Ast.,t,AVt.,t, p d AV&t, by defining r = Johq[p(z)-Pbl dz q d s = Johqo[u(z) - &,I dz, where p is a number density, u is an entropy density, and the subscript b refers to the bulk. If total system changes are equated with the sum of changes in the slit and reservoir fluids as ASt.,t = Asm + A&t and AVht = AVrw+ AV&,Clapeyron equations may be written in terms of microscopic variables as
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Figure 2. Surface free energies calculated at p d = 0.01bIe = -7.87), p d O.l&/c = -4.85), p d = 0.5b/€ = -2.92), and p d 0.7(Jc = -0.67) all at keTlc = 1.5 and p&oYe = 0.
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Figure 3. Phase diagram of parallel plate colloids in a LennardJones solvent in the plane p&Vc = 0.
These equations highlight the importance of identifying the correct regime of operation for any particular exper-
iment. With all the appropriate tools in hand we can now consider phase transitions in our model system. In the clay swelling experiment of Smalley et al.3 no mechanical stress is applied to the clay so pea = 0. The colloidal state variables are T and p , and eq 5 may then be used (as is assumed by Smalley et al.3) to find entropy changes for the transition. In this case U,, = PbV&t and @(h;p,T)/Aalone is minimized to find h,. Figure 2 shows how QB varies with h for various p all at T* = kBT/e = 1.5. The solvent is a 12-6Leonard-Jones (U) fluid interacting through a 9-3W potential with the walls. All potentials are cut and shifted, and the fluid completely wets the wall. The constants k g , u, and e are introduced in the figure to nondimensionalize all the variables. They are the Boltzmann constant, the LJ diameter of the fluid particles,and the depth of the W energywell, respectively. As the bulk fluid becomes more dense (increasing p ) , a swelling transition between hlu = 1.79 and hlu = 2.77 occurs at p"/e = -2.88. This transition is clearly first order with a discontinuous change in h. Repeating this experiment over a wide21temperature range, 0.70 < T* < 1.75, leads to the complete phase diagram presented in Figure 3. As observed for real colloidal systems, the crystalline phase is the high temperature (lowpressure)phase. Thismost simplecasestudy showsthat the state of a colloidalsuspension can be altered by solvent state variables ( p and 7')in the correct manner
(20) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elaevier: Amsterdam, 1948.
(21) Nota that the triple point and critical temperatures of the cut and shifted U fluid are Tt* = 0.64 and To*= 1.11, respectively.
(3) (4)
or, applying standard thermodynamic transformations, to eqs 2 and 3, they may be written in terms of macroscopic variables as (5)
(7)
Letters
Langmuir, Vol. 9, No. 6,1993 1445
0
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Figure 4. Entropy change per unit surface area, A S d A along coexistence at ph = 0.
if the analysis includes a granular solvent in the place of the DLVO continuous medium. Further analysis of this phase transition may now proceed throughthe applicationof eq 7. Resulting changes in entropy are shown in Figure 4. Two regimes may be identified from this plot. At low T, the bulk is a gas or dilute fluid at coexistence, and AS,$ varies rapidly with temperature. At high T, the bulk is a dense fluid, and AS, is small and nearly constant. Therefore, entropy effects may not always dominate the free energy minimization which ultimately determines the equilibrium state of the system. The me of such terminology as entropy driven is then probably best avoided for related phenomena. While there are two regimes present for ASht, we observe that in all cases, ASbt -C 0. An increase in overall order is therefore coincident with the loss of order in the large particles as the temperature decreases. Extension of the phase diagram into three dimensions by inclusion of the nonzero mechanical stress is now
straightforward. For p e a > 0, the depth of the first minimum in the free energy curve will be increased relative to the second.22 A higherpb(p,T) is then requiredto induce a transition at any given T. Details of the P b - p e a and p e a - T projections of phase space will be discussed elsewhere. Furthermore, extension of the model to nonzero plate-plate interactions, #pp(h),is possible by assuming some functional form for such direct interactions. Additional flexibility in the global minimum of the relevant free energy, now written am/ A +p e a+ #pp, may be achieved in this manner. It can be shown that with an exponentially decaying repulsion, completely separated colloids may be observed. A complete understanding of such states would require a description of electrostatic and van der Waals interactions. In summary, we have presented a new method for studying colloidal forces and phase transitions. This method is based on a granular solvent which accurately represents the effects of an oscillatory solvation force and most importantly, is properly sensitive to solvent state variables ( p and T). Although charges have not been discussed in any detail, we note that these can be included in the density functional theory in a straightforward manner. Doing so will lead to a completetheory of colloidal interactions in the sense that both the short-range structural forces, with their key dependence on solvent activity,and the long range electrostaticand van der Waals forces will be captured. However, as we have demonstrated,the underlying physics of so-called entropy driven phase transitions can be fully understood by analyzing a simplified model of parallel plates in a neutral fluid. Acknowledgment. This work was supported through an NSF Fellowship for L.J.D. and a Hauser Scholarship for T.L.D. (22) For p& always.
< 0 and in the approximations of this model, h, =
