Phase Separation in Deionized Colloidal Systems: Extended Debye-Hu1 ckel Theory
Derek Y. C. Chan,*,† Per Linse,‡ and Simon N. Petris†
Particulate Fluids Processing Centre, Department of Mathematics & Statistics, The University of Melbourne, Parkville VIC 3010, Australia, and Physical Chemistry 1, Center of Chemistry and Chemical Engineering, Lund University, P.O. Box 124, S-22100 Lund, Sweden
Received November 8, 2000. In Final Form: May 3, 2001
Using an extension of the Debye-Hu¨ ckel theory for strong electrolytes, the thermodynamics, phase behavior, and effective pair colloidal potentials of deionized charged dispersions have been investigated. With the inclusion of colloid size effects, this model predicts the possibility of the existence of two critical points, one of which is thermodynamically metastable but can exhibit interesting behavior at high colloid charges. This analytic model also serves as a pedagogic demonstration that the phase transition is driven by cohesive Coulomb interactions between all charged species in the system and that this cohesion is not inconsistent with a repulsive effective pair potential between the colloidal particles.
Introduction actions based on the Deryaguin-Landau-Verwey-Over- beek4 (DLVO) picture, the colloidal dispersion is treated In a series of very careful experimental studies, Ise et 1 as a one-component fluid of colloidal particles interacting al. demonstrated that stable deionized aqueous disper- under a combination of attractive dispersion forces and sions of polystyrene latex colloids exhibit simultaneously repulsive electrical double layer forces. In particular, the regions of high and low colloid densities. When the density electrical double layer interaction between identically of the latex particles is carefully matched with that of the charged particles is repulsive for all separations. In this suspending aqueous medium (with a H2O/D2O mixture), situation, the purely repulsive double layer interaction large void regions of low particle concentration were due to Coulomb interaction between the colloidal particles observed to coexist with a phase of high particle concen- cannot, within the van der Waals picture, give rise to a tration. In both the high and low particle concentration vapor-liquid equilibrium. This is the dilemma that one regions, the colloidal particles remain as individual s faces in attempting to reconcile the observed phase particles there is no coagulation or flocculation between separation behavior with the well-tested DLVO theory. the particles. The observed coexistence of low and high We propose to study phase behavior in a deionized particle number density phases is reminiscent of a gas- colloidal dispersion or in a dispersion with added elec- liquid-phase equilibrium. trolyte, by going beyond the one-component paradigm, If the dispersion is regarded as a one-component system which only focuses on the separation dependent effective comprised of colloidal particles interacting via an effective pair potential between colloidal particles, as in the pair potential, then the familiar van der Waals picture of - - Sogami Ise approach. Instead, we consider the total free vapor liquid equilibrium will require the existence of an energy of the system that arises from Coulomb interactions attractive interaction between the particles. However, between all species in the system-between the colloidal even in the more concentrated region, the interparticle particles and the small ions as well as between the small separations are sufficiently large that attractive dispersion ions. By focusing on the free energy of the entire system, forces are totally negligible and therefore only Coulomb there is in fact no conflict between the picture of a purely interactions among the charged colloidal particles and repulsive effective interaction between any pair of colloidal their neutralizing counterions are important in such particles and the possible existence of vapor-liquid systems. Therefore, within the one-component paradigm equilibrium for the whole system. it must follow that the like-charged colloidal particles must In this paper, we use a simple extension of the Debye- have an attractive component in their effective pair Hu¨ ckel theory for strong electrolytes to provide a simple potential. This point of view led to the development of the and transparent illustration of how Coulomb interactions Sogami2 pair potential to explain the observed phase alone are sufficient to drive a vapor-liquid like phase separation. While the Sogami potential exhibits the transition in deionized colloidal systems and to resolve required attraction between like-charged particles at large the apparent paradox of the existence of phase separation separations, its theoretical foundation is not without behavior between like-charged colloidal particles that only controversy.3 In the traditional model for colloidal inter- experience mutual repulsion over all interparticle separa- tions. While this approach may have the limited utility * Author for correspondence. E-mail: D.Chan@ of being able to provide accurate quantitative predictions, ms.unimelb.edu.au. its analytic nature is still very instructive in helping to † The University of Melbourne. ‡ Lund University. illuminate two different ways of thinking about Coulomb (1) Ito, K.; Yoshida, H.; Ise, N. Science 1994, 263, 66. Yoshida, H.; interactions in colloidal systems. Furthermore, this ex- Ise, N.; Hashimoto, T. J. Chem. Phys. 1995, 103, 10146. Yoshida, H.; tended Debye-Hu¨ ckel theory of de-ionized colloidal Yamanaka, J.; Koga, T.; Ise, N.; Hashimoto, T. Langmuir 1998, 14, 569. systems predicts the existence of a critical point as well (2) Sogami, I.; Ise, N. J. Chem. Phys. 1984, 81, 6329. (3) Overbeek, J. Th. G. J. Chem. Phys. 1987, 87, 4406. Woodward, C. E. J. Chem. Phys. 1988, 89, 5140. Overbeek, J. Th. G. Mol. Phys. (4) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of 1993, 80, 685. Ettelaie, R. Langmuir 1993, 9, 1888. Lyophobic Colloids; Elsevier: Amsterdam, 1948.
10.1021/la001560e CCC: $20.00 © xxxx American Chemical Society Published on Web 00/00/0000 PAGE EST: 8.9 B Langmuir Chan et al. as an interesting metastable phase behavior that can be The interaction potential between two ionic species at long-lived at high colloid charges. There is the possibility a distance r between their centers is taken to be made up that the vapor-liquid like phase separation observed in of a hard-sphere (hs) part and a Coulomb (coul) part uRâ(r) hs coul deionized latex systems corresponds to this long-lived ) uRâ (r) + uRâ (r) where R and â denote either a colloidal metastable state. particle (0) or a counterion (1). The hard-sphere and the The present work also provides a clear analytic dem- Coulomb interactions are given by onstration that while there are different equivalent hs ) g + methods of calculating the thermodynamic properties of uRâ (r) 0, r (RR Râ) Coulomb systems, those approaches that use an effective ) ∞ < + pair potential between the colloids will involve cancel- , r (RR Râ) (1) lations between terms of similar magnitude that arise for instance from one-body and two-body terms, and therefore z z e2 coul ) R â > these should be evaluated in a consistent manner. uRâ (r) , r 0 (2) The study of the statistical thermodynamics of systems r involving long-ranged Coulomb interactions can be quite with e denoting the protonic charge. This is the specifica- subtle, and sometimes well-established approximations tion of our primitive asymmetric electrolyte model for the that are valid in treating systems with short-ranged deionized colloidal dispersion. potentials can lead to incorrect conclusions. We therefore The configurational part of the Hamiltonian of this also attempt a comparison between the present work and system is a sum of hard-sphere interactions, Hhs, and the work of van Roij et al. and of Warren, who took similar Coulomb interactions, Hcoul, where the latter has the form approaches to study the phase separation problem. The general physical content invoked in explaining the phase coul ) coul separation phenomenon in these papers is very similar to H ∑∑uRâ (rij) (3) R>âi∈R that discussed here. However, the technical details of each j∈â approach are quite different in each case, so that in spec- j*i ial known limits, there are significant quantitative dif- ferences between the results obtained via different ap- The (excess) electrostatic internal energy, Ecoul, arising proaches. from the Coulomb interactions is given in terms of the Very recently, computer simulation studies of the pair distribution functions, gRâ(r), between various species structure and thermodynamics of charged colloidal sys- Ecoul 1 tems using a model of discrete small ions and discrete ) ∫ coul charged colloidal particles have been performed.5,6 While ∑nRnâ uRâ (r) gRâ(r)dr (4) V 2 R it is not yet feasible at present to carry out simulations ,â for system parameters that are within the range where 2 2πe ∞ phase separation has been observed in experimental ) - ∑nRzRnâzâ∫ + rhRâ(r)dr systems, the simulation results clearly indicate the Ra Râ R,â existence of a Coulomb-interaction-driven phase separa- πe2 tion. + 2 ∑nRzRnâzâ(RR Râ) (5) R,â Colloidal System as an Asymmetric Electrolyte Taken individually, each integral in eq 4 diverges because Our starting point for considering phase behavior of of the long-ranged nature of the Coulomb potential. But ) - Coulomb dispersions is to calculate the Helmholtz free using the total correlation function hRâ(r) gRâ(r) 1 and energy and the pressure of the system, and hence we first the bulk electroneutrality condition, the divergences in summarize the necessary standard formalism.9 For a eq 4 cancel to give finite integrals in eq 5. The second term )- deionized colloidal dispersion stabilized only by Coulomb in eq 5 follows from the exact condition hRâ(r) 1 which < + interactions, we model the colloidal particles as hard must hold for r (RR Râ) and arises from the combined spheres of radius R0 and valence z0 that are neutralized effects of size differences and Coulombic interactions by counterions of radius R1 and of valence z1. Later we between the charged species. will study in detail the limits z0 ) Z . 1, z1 )-1, and R1 The configurational part of the Helmholtz free energy ) 0. We assume N0 colloidal particles and N1 counterions F can accordingly be separated as a sum of the hard- in a volume V and that the amount of residual electrolyte sphere free energy Fhs and the (excess) electrostatic free or added salt is negligible. The number densities of the energy Fcoul. By employing a coupling constant or charging coul coul colloids n0 ) N0/V and of the counterions n1 ) N1/V that integration, F can be evaluated from E using the neutralize the colloidal charge are related by the bulk formally exact relation electroneutrality condition: n0z0 + n1z1 ) 0. The aqueous 1 dλ 1 dλ solvent is treated as a dielectric continuum with a relative Fcoul ) ∫ 〈Hcoul〉 ) ∫ Ecoul(λ) (6) dielectric permittivity . 0 λ λ 0 λ
coul t coul 2f 2 (5) Lobaskin, V.; Linse, P. J. Chem. Phys. 1999, 111, 4300. Linse P.; where E (λ) E (e λe ). Lobaskin, V. Phys. Rev. Lett. 1999, 83, 4208. Linse, P.; Lobaskin, V. J. The pair distribution functions, gRâ(r), obey the local Chem. Phys. 2000, 112, 3917. electroneutrality condition about an ionic species of (6) Linse, P. J. Chem. Phys. 2000, 112, 4359. (7) van Roij, R. M.; Hansen, J.-P. Phys. Rev. Lett. 1997, 79, 3082. van valence zR Roij, R.; Dijkstra, M.; Hansen, J.-P. Phys. Rev. E 1999, 59, 2010. The first paper in the series contained omissions that were corrected in the )- ∫ second paper. zR ∑nâzâ gRâ(r) dr (local electroneutrality) (8) van Roij, R, M.; Evans, R. J. Phys. Condens. Matter 1999, 11, â 10047. (7) (9) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976. This condition expresses the requirement that the total Phase Separation in Deionized Colloidal Systems Langmuir C net charge surrounding a particular particle of valence zR Substitution of eqs 2 and 10 into eq 5 gives the Coulomb must cancel the charge on that particle exactly. This contribution to the internal energy per unit volume condition provides a check on approximate expressions coul 2 for gR (r). Also as a consequence of the long-ranged 1/r E 2πe â )- {n 2z 4 + 2(n z 2)(n z 2) + n 2z 4} form of the Coulomb potential, the Stillinger-Lovett V 2 0 0 0 0 1 1 1 1 10 kTκD second moment condition should also hold for the gRâ(r): kTκ 3 3kT )- D )- 2 (DHLL) (12) ∑nRzRnâzâ∫r gRâ(r)dr 8π 2 2πe R,â | | . | | (Stillinger-Lovett) (8) When z0 z1 , the dominant contribution is from the first (colloid-colloid) term. It follows from eq 6 that the Up till now, all the results are completely general and Coulomb contribution to the free energy per unit volume we are in the position to investigate various approximate is methods of calculating the free energy of a system of coul kTκ 3λ3/2 deionized colloidal dispersions and, from that, study the F ) ∫1(- D ) dλ phase behavior of such systems. In general, one only has V 0 8π λ available approximate expressions for gR (r), so the â kTκ 3 accuracy of any approximation will depend on the method )- D ∼ n 3/2Z3 (DHLL) (13) that one uses to compute the relevant thermodynamic 12π 0 quantities. Equations 4 and 6 provide one such method, but equally, one can calculate the pressure directly using and the Coulomb contribution to the pressure is the virial equation9 or one may choose a free energy density coul 3 7 ∂F kTκD functional formalism or some other approach. The very Pcoul t - )- (DHLL) (14) general local electroneutrality and Stillinger-Lovett ∂V 24π conditions, both of which are consequences of the Coulomb potential, provide simple checks on the internal consis- From eq 12 we see that the Coulomb contributions due to tency of the approximate pair distribution functions colloid-colloid, colloid-counterion, and counterion-coun- gRâ(r). terion interactions are always negative. In other words, the bare Coulomb repulsions between the colloid-colloid species and between the counterion-counterion species - 1 Point-Ion Debye Huckel Theory are outweighed by the bare Coulomb attraction between the colloid-counterion species with the net effect that the Thermodynamics. While the thermodynamics of an - Coulomb interaction provides a cohesive effect that will electrolyte described by the Debye Hu¨ ckel theory is well- tend to collapse the ionic system; this cohesive effect known, its analytic nature means that one can readily counter-balances the ever-present entropic effects. make explicit the subtle cancellations that arise when For a deionized colloidal dispersion with monovalent trying to describe a colloidal system in terms of effective counterions, modeled as a Z:1 electrolyte with z0 ) Z (>0) pair potentials when it is treated as an asymmetric 2 and z1 )-1, the inverse Debye screening length κD ) electrolyte. For point colloids and point ions (R0 ) 0 ) R1), 2 4e n0Z(Z + 1)/(kT) is dominated by the highly charged the potential, φâ(r), at r given that there is a point charge (Z . 1) colloidal particles. The explicit expressions for the (zâe) at the origin satisfies the Poisson equation: total pressure, P, which in the present model will include an entropic contribution from colloids and counterions 4πe 4πzâe ∇2 )- - g treated as an ideal gas mixture, is φâ(r) ∑nγzγgγâ(r) δ(r), r 0 (9) γ P ) Pideal + Pcoul kT 4πe2 3/2 which together with the linearized mean-field approxi- ) n (Z + 1)kT - ( ) [n Z(Z + 1)]3/2 (15) 0 0 mation, gRâ(r) = 1 - zReφâ(r)/kT, gives the approximate 24π kT total correlation function The simple treatment provided by the point-ion Debye- 2 Hu¨ ckel theory demonstrates the existence of a cohesive zRz e exp(-κ r) )- â D > contribution to the free energy due to Coulomb interactions hRâ(r) , r 0 (DHLL) (10) kT r and therefore signals the possibility of a vapor-liquid phase equilibrium. Indeed, this is precisely the point of where κD is the inverse Debye screening length of the view adopted by Langmuir11 in considering phase behavior asymmetric electrolyte in strongly interacting dispersions and by Michaeli et al.12 in considering phase separation in polyelectrolytes. 4πe2 4πe2 κ 2 ) (n z 2) + (n z 2) However, this simple instructive treatment cannot as D kT 0 0 kT 1 1 yet predict a phase transition. This is evident from the f t 2 + 2 pressure in eq 15, where at low densities (n0 0) the ideal κ0 κ1 (11) gas term dominates as expected but at high densities the 3/2 cohesion due to Coulomb interactions (∼n0 ) overwhelms The inverse Debye screening length has contributions from the ideal gas term. This offers no great surprise, as the both the colloidal particles, κ0, and the counterions, κ1. point-ion Debye-Hu¨ ckel theory is only expected to be valid The total correlation functions, eq 10, satisfy the local at low concentrations. In a later section, we will provide electroneutrality condition, eq 7, as well as the Stillinger- an attempt to remedy this shortcoming. Lovett condition, eq 8. (11) Langmuir, I. J. Chem. Phys. 1938, 6, 873. (10) Stillinger, F. H.; Lovell, R. J. Chem. Phys. 1968, 49, 1991. Carnie, (12) Michaeli, I.; Overbeek, J. Th. G.; Voorn, M. J. J. Polym. Sci. S. L.; Chan, D. Y. C. Mol. Phys. 1984, 51, 1047. 1957, 23, 443. D Langmuir Chan et al.
We can understand the origin of this cohesion in more .... Within the point-ion Debye-Hu¨ ckel theory, the detail by considering the forms of the pair distribution expansion ends exactly after two terms: functions within the point-ion model. Equation 12 showed | | . | | coul ) + that when z0 z1 , the dominant term in the cohesive Heff N0W1 ∑W2(rij) (17) contribution to the internal energy is from the colloid- i
coul 3 instead of retaining both terms in eq 22. F F1 F2 kTkD ) + )- (28) In the point-ion limit, their excess electrostatic free V V V 12π energy, obtained as the sum of eqs 29 and 30 according to eq 20, becomes8 (again in our notation) which is exactly the point-ion Debye-Hu¨ ckel result (eq 13) that we expect to recover. Notice how the terms in the coul 2 F kTκ0 κ1 3/2 5/2 two-body free energy F2 that arise from the effective pair )- ∼ n0 Z potential W2(r) cancel out all three terms in the one-body V 8π or volume term F1 to give the point-ion Debye-Hu¨ ckel (van Roij, Dijkstra, Hansen, Evans) (31) result. While this calculation is for the idealized case of point which differs from the point-ion Debye-Hu¨ ckel limiting ions, the analytic result suggests that computing ther- law (eq 13) by a factor (2/3)Z1/2, a difference that can be modynamic quantities using eqs 4 and 6 is likely to be significant when Z . 1, as in typical colloidal systems.1 more robust numerically. On the other hand, using the The reason for this difference is thus traced to the way effective one-component approach will require the evalu- in which the free energy contributions F1 and F2 were ation of one- and two-body and possibly higher body calculated. potentials between the colloids as well as the calculation of the contributions to the free energy from the effective (14) Levin, Y.; Barbosa, M. C.; Tamashiro, M. N. Europhys. Lett. - 1988, 41, 123. Tamashiro, M. N.; Levin, Y.; Barbosa, M. C. Physica A colloid colloid potentials. Indeed, in the effective one- 1988, 258, 341. Tamashiro, M. N.; Levin, Y.; Barbosa, M. C. Eur. Phys. component approach for point ions, while the final answer J. B 1988, 1, 337. F Langmuir Chan et al.
In their density functional formulation, vRDH neglected coul 2 4 2 E 2π(n0Z) e Z 2Z the ion-ion correlation term in the one-body potential as )- + + 1 - V 2 (1 + 2κ R 1 + κ R ) well as omitted the important ring diagram contributions κDkT D D 9 to the electrostatic free energy from the two-body term, 2π(n ZeR)2 so their result will not reduce to the Debye-Hu¨ ckel 0 (33) limiting law expected in the point-ion and point-colloid limit or at low densities. In their calculation that included effects of finite colloid size and added salt, the free energy in which all terms are negative, with the colloid-colloid contribution from the colloid-colloid effective pair po- term being dominant when Z . 1. The last term in eq 33, tential is calculated using a variational approach based which is exact, arises from the contributions from the on a hard-sphere reference system. Since some of the domains r < 2R and r < R in the colloid-colloid and interesting phase behavior founded by vRDH occurs at colloid-counterion integrals, respectively. very low colloid densities, it seem important to be able to The excess electrostatic free energy can again be found recover known low-density limiting behavior. by a coupling constant integration over λ. Substitution of 15 1/2 2 Warren also studied phase separation of a colloidal eq 33 into eq 6 and remembering that κD f λ κD and e dispersion under only Coulombic interactions. The physi- f λe2 leads to cal model and initial assumptions are very similar to that of vRDH and that in this paper. Warren also started the coul 2 4 F 4π(n0Z) e analysis with expressions for the internal energy as given )- [Z2f(2κ R) + 2Zf(κ R) + 1] - V 2 D D by eq 4. However, all the approximate pair distribution 3 κDkT functions used by Warren to calculate internal energies 2π(n ZeR)2 do not satisfy the local electroneutrality condition (eq 7). 0 (34) Also, like vRDH, he also used a variational method to calculate the contribution of the effective pair potential to the colloid-colloid free energy, but he did not give a where f(x) t (3/x3)(log(1 + x) - x + x2/2), and in the point- detailed justification of the form of the pair potential he colloid limit R f 0, eq 34 reduces to the point-ion Debye- used. As a consequence, it is not possible to make a more Hu¨ ckel result of eq 13, since f(0) ) 1. detailed comparison with the work of Warren. The Coulomb contribution to the pressure is For finite colloid size, the exact term that arises from the combined effects of size differences and Coulombic 2 4 2π(n0Z) e interactions between the charged speciessthe second term Pcoul )- [Z2p(2κ R) + 2Zp(κ R) + 1] - 2 D D in eq 5sis not present in the treatments of vRDH or 3 κDkT Warren. 2 2π(n0ZeR) Extended Debye-Hu1 ckel TheorysFinite Colloid t coul + coul Size Pcc Pex (35) - Thermodynamics. The Debye Hu¨ ckel theory for where p(x) t (3/x3)((1 + x) - 1/(1 + x) - 2 log(1 + x)). In electrolytes can be extended to include hard-sphere radii eq 35, we have identified for later discussion the contri- RR > 0. Due to the hard-sphere repulsion, gRâ(r) ) 0 (or coul )- < + bution from charge correlations Pcc that arises from equivalently hRâ(r) 1) for r (RR Râ). An ap- the three integrals involving correlation functions in eq proximation for the total correlation functions hRâ(r) for coul 33 and the exact term Pex , which arises from the size r > (RR+ Râ) is to retain the Debye-Hu¨ ckel form hRâ(r) ≈ - > + difference between the charged species. In the point-colloid zReφâ(r)/kT, for r (RR Râ), where φâ(r) is the potential limit R f 0, eq 35 reduces to the point-ion Debye-Hu¨ ckel at r given that there is a point charge (zâe) at the origin result of eq 14, since p(0) ) 1. ∇2 - 2 ) > + and satisfies φâ(r) κD φâ(r) 0,r (RR Râ). In other Critical Points and Phase Diagrams. We will now words, the size effect is taken into account by assuming investigate the phase behavior of the extended Debye- that the point charge model remains valid up to the + Hu¨ ckel theory by examining the critical points of the distance of closest approach (RR Râ) between species. equation of state. The state of the Z:1 colloidal system can This gives be characterized by the following three nondimensional h (r) )-1, r < (R + R ) variables: the colloid charge Z, the colloid volume fraction Râ R â φ, and the coupling strength or inverse temperature Γ 6 z z e2 defined by )- R â × kT(1 + κ (RR + R )) 3 2 D â t 4πR t e - - + φ n0 Γ (36) exp[ κD(r (RR Râ))] 3 kTR , r > (R + R ) (32) r R â and all other nondimensional parameters can be con- 2 These correlation functions satisfy the local electroneu- structed from these, for example, (κDR) ) 3φΓZ(Z + 1). trality condition (eq 7) but not the second moment To analyze the critical behavior of the system, we use condition (eq 8). Equation 32 for the colloid-counterion the Carnahan-Starling formula9 for the contribution to correlation function is identical to that in the treatment the pressure, Phs, from the reference system of uncharged of vRDH. hard sphere colloids and uncharged point ions: Insertion of z0 ) Z, z1 )-1, R0 ) R, and R1 ) 0 into eq 32 gives the excess electrostatic internal energy from eq + + 2 - 3 hs t t 1 φ φ φ + Z 5 P n0kTΦ(φ,Z) n0kT( ) (1 - φ)3 1 - φ (15) Warren, P. B. J. Chem. Phys. 2000, 112, 4683. (37) Phase Separation in Deionized Colloidal Systems Langmuir G
All thermodynamic functions can now be written in terms 1 4πR3 1 κ LcrR = φLcr t n Lcr = of the three nondimensional parameters (Z, φ, and Γ): D 2 3 0 384Z Electrostatic Internal Energy e2 32 4πR3PLcr (16 log(2) - 11) coul ZΓκ R 2 ΓLcr t = = E )- D Z + 2Z + - Lcr Z Lcr × 7 ( + + 1) kT R 3kT 3 2 N0kT 2(Z + 1) 1 2κDR 1 κDR (44) 3 2 /2φZ Γ (38) These analytical results demonstrate qualitatively the possibility of the existence of a vapor-liquid-phase Helmholtz Free Energy transition in a deionized dispersion driven by Coulomb interactions and ideal gas entropy. For Z . 1, this critical ZΓκ R F ) - D × point occurs at a very low volume fraction. If the critical Θ(φ,Z) temperature is to be around room temperature and the N0kT 3(Z + 1) relative dielectric permittivity is to be the aqueous value 2 + + - 3 2 [Z f(2κDR) 2Zf(κDR) 1] /2φZ Γ (39) of 78, the present extended Debye-Hu¨ ckel theory with finite-sized colloids would predict a colloidal charge Z ∼ where 2200 for a colloidal radius of 50 nm. Experimental colloidal charges are estimated to be around this order of magni- 2 1 tude.1 Θ(φ,Z) t (Z + 1) log(φ) + + - 1 - φ - φ 2 The above estimates for the low-density critical values (1 ) - - have the same form as that obtained earlier for a Debye Z log(1 φ) Hu¨ ckel treatment of symmetric electrolyte with equal size ions.16 For this restrictive primitive model electrolyte, the Pressure coul third term (Pex ) in eq 40 for the pressure that originates 3 from size differences between the charged species is 4πR P φZΓκDR ) φΦ(φ,Z) - × identically zero. Indeed, the study of phase transitions in 3kT 6(Z + 1) symmetric electrolytes is covered very extensively in the [Z2p(2κ R) + 2Zp(κ R) + 1] - 3/ φ2Z2Γ literature, and the general consensus is that while the D D 2 Debye-Hu¨ ckel theory is able to provide instructive 17 3 hs 3 coul 3 coul analytical results, it has only qualitative accuracy 4πR P 4πR Pcc 4πR Pex t + + (40) because of the inherent linearization assumptions. 3kT 3kT 3kT Extended Debye-Hu¨ ckel Theory High-Density Critical Point (Z . 1). At higher densities for which κDR . 1, the Chemical Potential second term for the pressure in eq 42 is of order (κDR) ∼ φ1/2Γ1/2Z, and so it is small compared to the third term. 3 . µ t 1 dF ) 1 4πR P + F Hence for Z 1 we can neglect the second term in eq 42 (41) to give kT kT dN0 φ 3kT N0kT 3 Equation 41 is a standard thermodynamic relation for 4πR P = Z - 3 2 2 φ /2φ Z Γ (45) the chemical potential µ of a single colloid particle plus 3kT (1 - φ) its neutralizing counterions. Critical points can be located by solving (∂P/∂V) ) 0 ) and we find a second critical point characterized ap- (∂2P/∂V2). For Z . 1, we can retain only the leading term proximately by in Z in P coul in eq 40 and write the pressure as cc 1/2 Hcr = 3Z Hcr = 1 κD R φ 4πR3P 1 1 2 3 = φΦ(φ,Z) - (1 + 2κ R) - - 3kT 48 ( D 1 + 2κ R D 9 4πR3PHcr Z ΓHcr = = (46) + - 3 2 2 Hcr 2 log(1 2κDR)) /2φ Z Γ (42) 4Z 3kT 8
There are two density regimes of interest: κ R e 1 and This high-density critical point occurs at a volume fraction D Hcr κ R . 1 that we can examine. When κ R e 1, the second φ of around 33%, and it is determined by the hard- D D hs term on the RHS of eq 42 is of order (κ R)3 ∼ φ3/2Γ3/2Z3 and sphere contribution P and the exact Coulomb contribu- D coul is therefore the larger of the two negative terms when Z tion Pex due to the size-difference effects between the . 1. So by retaining only the first two leading terms in colloids and the ions. eq 42 and approximating Φ(φ,Z) by 1, we have The general form of the pressure isotherm is shown in Figure 1 for a colloid charge Z ) 100. Note the location 3 of the two critical points at very different volume frac- 4πR P = - 1 + - 1 - tions: one at around φ ∼ 10-4 (eq 44) and one at around φZ ((1 2κDR) + - 3kT 48 1 2κDR φ ∼ 10 1 (eq 46) separated by a region of near ideal gas ∼ -3 ) + behavior at φ 10 .AtZ 10, only the high-density 2 log(1 2κDR)) (43) critical point is present (see Figure 2). In fact, in this model, the system only exhibits two critical points provided Z g The critical parameters can now be estimated by solving (∂P/∂V) ) 0 ) (∂2P/∂V2) to give approximate expressions (16) McGahay, V.; Tomozawa, M. J. Chem. Phys. 1992, 97, 2609. for the low density critical parameters. (17) Fisher, M. E.; Levin, Y. Phys. Rev. Lett. 1993, 71, 3826. Fisher, - M. E.; Levin, Y. J. Stat. Phys. 1994, 75, 1. Levin, Y.; Fisher, M. E. Extended Debye Hu¨ ckel Theory Low-Density Critical Physica A 1996, 225, 164. Point (Z . 1) (18) Blum, L.; Høye, J. S. J. Chem. Phys. 1977, 81, 1311. H Langmuir Chan et al.
Figure 4. Schematic illustration of the pressure isotherm with Figure 1. Pressure (P) scaled by the approximate low-density two critical points. critical pressure PLcr (see eq 44) as a function of the colloid volume fraction æ at colloid charge Z ) 100. From bottom to top, the isotherms are at Γ ) (e2/kTR) ) 32/(ZT*) for T* ) 0.8, 0.9, 1.2, 2, 4, 9, and 15. The pressure is plotted on an arcsinh scale.
Figure 2. Pressure (P) scaled by the approximate low-density critical pressure PLcr (see eq 44) as a function of the colloid volume fraction φ at colloid charge Z ) 10. From bottom to top, the isotherms are at Γ ) (e2/kTR) ) 32/(ZT*) for T* ) 0.8, 1.1, 1.25, 1.5, 2.5, 5, 8, 10, and 15. The pressure is plotted on an arcsinh scale.
Figure 3. Scaled temperature ΓHcr/Γ ) 9/(4ZΓ) ) 9kTR/(4Ze2) versus volume fraction φ phase diagram corresponding to colloid charges of Z ) 10-1000.
11. From the magnitude of the pressure, only the high- density critical point around φ ∼ 10-1 is thermodynami- - cally stable and examples of the temperature volume Figure 5. Free energy (F/FLcr), pressure (P/PLcr), and chemical fraction phase diagram for a range of colloid charges Z potential (µ/kT) versus volume fraction φ at the scaled tem- are given in Figure 3. For the case Z ) 10, the present perature of 1/Γ ) (kTR/e2) ) 0.95(1/ΓLcr) ) 0.95 (Z/32) near the Lcr theory predicts a critical point at φc ) 0.24 and Γc ) 0.32, metastable critical point for a colloid charge of Z ) 100. F is and for Z ) 20, φc ) 0.27 and Γc ) 0.14. These modest the free energy (eq 39) evaluated at the state point given by eq colloid charges are certainly accessible by simulation 44. Horizontal tie lines and the common tangent construction studies.6 between the equilibrium densities are shown. The critical point at low volume fraction is in the separate into the metastable points C and D, and will metastable region of the pressure isotherm. This general finally attain thermodynamic equilibrium at the equi- feature of the pressure isotherm is illustrated schemati- librium state points A and B. To obtain a simple measure cally in Figure 4. The portion of the isotherm between of the “stability” of the metastable states C and D, we points A and B is in a thermodynamically metastable consider the variation of the chemical potential in the region. At the composition S, the system may phase neighborhood of the metastable low-density critical point. Phase Separation in Deionized Colloidal Systems PAGE EST: 8.9 Langmuir I
Debye-Hu¨ ckel model gave a scaled critical temperature 2 1 (kTcR/e ) ) /32, which is within about 10% of the best estimate from Monte Carlo simulations.17 However, the predicted critical density is too low by over a factor of 5. This would suggest that the extended Debye-Hu¨ ckel model may also be reasonable in estimating the critical temperature of the Z:1 system. Hopefully, the results obtained herein will stimulate further theoretical and simulation studies. Conclusion Using an extension of the Debye-Hu¨ ckel theory of an asymmetric electrolyte, we have demonstrated how a vapor-liquid phase separation can be driven by Coulomb Figure 6. Temperature-volume fraction phase boundaries at the low- and high-density critical points. The colloid charge Z interactions between the charged species because the ) 100. Each curve is scaled with respect to its own critical overall Coulomb interaction is cohesive. We also showed values. that this cohesive contribution to the free energy due to Coulomb interactions is consistent with a purely repulsive In Figure 5 we show the variation of the free energy, effective pair interaction potential between the colloidal pressure, and chemical potential with volume fraction for particles in the system within the primitive model. While a system with colloid charge Z ) 100 at a reduced the DLVO paradigm of regarding a colloidal system as a temperature of (1/Γ) ) 0.95(Z/32). The usual common one-component system of colloidal particles interacting tangent construction for the free energy (Figure 5a) and with pair potentials has been extremely successful in the Maxwell equal area construction for the pressure understanding colloidal stability, the phase separation (Figure 5b) are indicated by the dotted lines. We observe phenomenon requires consideration of the thermodynamic that there is a large barrier in the chemical potential state of the whole system. In low salt or deionized systems associated with the pressure maximum (point K in Figure where Coulomb interactions dominate, a consideration of 4). We note from Figure 5c that, at Z ) 100, the chemical the colloid-colloid pair interaction alone will not provide potential barrier is of the order 100kT and, at Z ) 50 (not sufficient insight into the phase behavior of the system; shown), the chemical potential barrier still has a mag- instead it is necessary to consider additional contributions nitude of about 30kT. While considerations of the lifetime from the small ions. of the metastable states are beyond a simple equilibrium As with other studies,7,13,15 our consideration of the study undertaken here, this observation would suggest point-ion, point-colloid Debye-Hu¨ ckel model indicates that, at large colloid charges, the low-density critical point that, in using the effective pair potential paradigm to offers the possibility of the existence of long-lived meta- calculate thermodynamic properties, there will be sig- stable states that may be manifested as the vapor-liquid nificant cancellations between contributions from the one- like phase separation observed in deionized latex disper- and two-body terms because both contribute to the same sions.1 Finally, we observe that the variation of the free order in the colloid density. More generally, with finite energy between the two metastable states is only around colloid size, higher body terms will appear although their 2-3%, which gives an indication of the numerical precision contributions to the thermodynamics may remain small. needed if this metastable region is to be explored by Nonetheless, it is important to ensure that all types of simulation studies. terms are calculated in a consistent manner. On the other In Figure 6, we compare the characteristics of the hand, a direct approach to calculate the internal energy temperaturesvolume fraction coexistence phase bound- in terms of distribution functions does not involve such aries about the low-density and high-density critical points cancellations and may provide a numerically more stable for a colloid charge Z ) 100. The results have been scaled route to system thermodynamics. with respect to their own critical values for comparison. So far, previous theories7,13,15 of such colloidal systems The phase boundaries corresponding to the low-density are based on variants of the linear Debye-Hu¨ ckel theory. (metastable) critical point have a very large density However, it is possible to go beyond the linear Debye- difference between the metastable states which is remi- Hu¨ ckel theory by exploiting the idea of effective pair niscent of the low particle densities observed within the potentials between colloidal particles to include nonlinear void regions in deionized dispersions1. effects in the Coulomb interaction. This will be presented Clearly, the reliability of the above results obtained in the next paper in this series. from the extended (linear) Debye-Hu¨ ckel model is difficult Acknowledgment. This research is funded in part by to gauge without further work. We can gain some insight the Australian Research Council and by the Swedish by comparing with earlier studies of the critical behavior Research Council (NFR), and their support is gratefully of the restricted primitive model (RPM) electrolytesa acknowledged. symmetric electrolyte in which the anions and cations have equal hard sphere radii. For this system, the extended LA001560E