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Langmuir 1996, 12, 2605-2607
Stability of Suspensions of Charged Colloids Alfred Delville Centre de Recherche sur la Matie` re Divise´ e, CNRS, 1B rue de la Fe´ rollerie, 45071 Orle´ ans Cedex 02, France Received July 5, 1995
Introduction Colloids resulting from the dispersion of charged particles are used in many industrial applications. They have been the subject of numerous experimental1-10 and theoretical11-17 studies. A central question concerns the relevance of statistical physics, which has successfully modeled molecular systems, to explain the behavior of colloids implicating much larger objects. In this context, suspensions of monodisperse spherical particles (silica, latex) are useful reference systems, since no influence of the shape of the particle18-20 may interfere with interparticle interactions. The phase diagram of colloidal suspensions presents a variety of transitions (gas-liquid, liquid-solid, sol-gel) generally well understood17-23 on the basis of the primitive model.24 An effective interparticle potential is often used in the framework of the ‘one-component plasma’10,12,16 resulting from a balance between three interactions among the charged particles and their neutralizing counterions: Coulombian energy, excluded volume effect, and perfect gas entropy.17,25 While the first contribution is always attractive, the other two are repulsive and lead generally to a net repulsion among the particles17,26 neutralized by monovalent counterions.27,28
Recent direct observation of spontaneous segregation1-3 of dilute aqueous suspensions of charged large latex particles seemed inconsistent with this generally accepted repulsion among charged particles. A direct interpretation of this segregation requires attraction between the particles.29 However, because of the large interparticle separation, no evident source of attractive potential can be identified. The present work suggests that salt transfer between dense and empty microdomains of segregated suspension reduces the effective interparticle repulsion and can explain the segregation of particles which remain in a repulsive regime. We numerically solve the PoissonBoltzmann (PB) equation to quantify the influence of this ionic exchange on the decrease of the Helmoltz free energy of the segregated suspension. We further elucidate the influence of ionic strength and particle concentration on the coexistence between dense and empty microdomains. Methods The central approximation25 identifies the mean force potential with the electrostatic potential. In the framework of the cell model,30 the local concentration ci(r) at a distance r from the central particle is described by Boltzmann’s law
ci(r) ) Mi exp(-eiΨ(r)/kT)
(1)
where the multiplicative constant Mi is related to the choice of the additive constant of the electrostatic potential Ψ. Because of the spherical symmetry of the latex particle, the PoissonBoltzmann (PB) equation reduces to
d2Ψ
2dΨ
∑
eiMi exp(-eiΨ(r)/kT)
)-
+ dr2
(1) 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. (2) Yoshida, H.; Ise, N.; Hashimoto, T. Langmuir 1995, 11, 2853. (3) Yoshida, H.; Ise, N.; Hashimoto, T. J. Chem. Phys. 1995, 103, 10146. (4) Ito, K.; Ieki, T.; Ise, N. Langmuir 1992, 8, 2952. (5) Bradbury, A.; Goodwin, J. W.; Hughes, R. W. Langmuir 1992, 8, 2863. (6) Einarson, M. B.; Berg, J. C. J. Colloid Interface Sci. 1993, 155, 165. (7) van Megen, W.; Underwood, S. M. Nature 1993, 362, 616. (8) Imhof, A.; van Blaaderen, A.; Maret, G.; Mellema, J.; Dhont, J. K. G. J. Chem. Phys. 1994, 100, 2170. (9) Bonnet-Gonnet, C.; Belloni, L.; Cabane, B. Langmuir 1994, 10, 4012. (10) Palberg, Th.; Kottal, J.; Bitzer, F.; Simon, R.; Wu¨rth, M.; Leiderer, P. J. Colloid Interface Sci. 1995, 169, 85. (11) Sogami, I.; Ise, N. J. Chem. Phys. 1984, 81, 6320. (12) Groot, R. D. J. Chem. Phys. 1991, 94, 5083. (13) Sogami, I. S.; Shinohara, T.; Smalley, M. V. Mol. Phys. 1992, 76, 1. (14) Lo¨wen, H.; Madden, P. A.; Hansen, J. P. Phys. Rev. Lett. 1992, 68, 1081. (15) Snook, I. K.; Hayter, J. B. Langmuir 1992, 8, 2880. (16) Fushiki, M. J. Chem. Phys. 1992, 97, 6700. (17) Delville, A. Langmuir 1994, 10, 395. (18) Onsager, L. Ann. N.Y. Acad. Sci. 1949, 51, 627. (19) Buinning, P. A.; Philipse, A. P.; Lekkerkerker, H. N. W. Langmuir 1994, 10, 2106. (20) Mourchid, A.; Delville, A.; Lambard, J.; Le´colier, E.; Levitz, P. Langmuir 1995, 11, 1942. (21) Ottewill, R. H. Ber. Bunsen.-Ges. Phys. Chem. 1985, 89, 517. (22) Dubois, M.; Zemb, Th.; Belloni, L.; Delville, A.; Levitz, P.; Setton, R. J. Chem. Phys. 1992, 96, 2278. (23) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1985. (24) Carley, D. D. J. Chem. Phys. 1967, 46, 3783. (25) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: New York, 1986. (26) Linse, P.; Jo¨nsson, B. J. Chem. Phys. 1983, 78, 3167. (27) Kjellander, R.; A° kesson, T.; Jo¨nsson, B.; Marcelja, S. J. Chem. Phys. 1992, 97, 1424. (28) Valleau, J. P.; Ivkov, R.; Torrie, G. M. J. Chem. Phys. 1991, 95, 520.
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r dr
0r
i
(2)
which is solved numerically.17,31 In the absence of salt, the only ionic species are the counterions neutralizing the charged particles. The complete characterization of the electrostatic potential requires a priori knowledge of the potential or of its derivative at the two cell boundaries corresponding respectively to the contact with the particle (r ) ap) and to the cell upper limit (r ) R). Both methods are useful,32,33 but the assignment of a specific value to the electrostatic potential or field must be done on physical grounds. While a constant value of the potential may be imposed to conductors, the only constant electrical property of dielectrics is their total charge (Q), which remains unaltered in the absence of specific binding of ions to surface sites of the particle.34 Gauss’s law then determines the boundary conditions
(dΨ/dr)r)ap ) -
Q 4πap20r
(3)
and
(dΨ/dr)r)R ) 0 because of the electroneutrality of the cell. In the presence of salt, a supplementary relation among M1, M2, and the reference potential is required. In a closed system, the number of salt molecules is constant. As a consequence, the (29) Tata, B. V. R.; Arora, A. K.; Valsakumar, M. C. Phys. Rev. E 1993, 47, 3404. (30) Rice, S. A.; Nagasawa, M. Polyelectrolyte Solutions; Academic Press: New York, 1961. (31) Delville, A.; Herwats, L.; Laszlo, P. New J. Chem. 1984, 8, 557. (32) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyotropic Colloids; Elsevier: Amsterdam, 1948. (33) McCormack, D.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 169, 177. (34) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: New York, 1981.
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Langmuir, Vol. 12, No. 10, 1996
Notes
average number of co-ions (noted 2) in the cell satisfies the relationship
N2 )
c2°4π(R3 - ap3) 4πM2 ) 3 3
∫r
R 2
ap
exp(-e2Ψ/kT) dr (4)
where c2° is the mean co-ion or salt concentration. The latex particles2 are modeled by uniformly charged hard spheres with a radius ap ) 3500 Å and a net charge Qp ) 3.69 × 105 electrons. The salt concentration (c2°) varies between 0.1 and 2 µM, and the radius of the solvated sodium counterions is 2.13 Å.35
Results and Discussion (A) Equation of State of Colloidal Suspensions. Within the framework of statistical physics, the pressure of colloidal suspensions of charged particles may be derived from the virial equation25 Nλ Nµ
P)
∑λ FλkT + /6∑λ Fλ∑µ Fµ∑ ∑∫rijFijgij drij i)1 j)1 1
(5)
where the indices λ and µ are relative to the different elements (particle, counterion, or co-ion), Fλ is their average density, Nλ is their total number, and gij is the distribution function of the (i,j) pair of elements. For charged particles, the entropic contribution from particle translation is negligible (FpkT ) 2.3 × 10-4 Pa at density φ ) 0.01). In the context of the “primitive model”,24 the second term of eq 5 contains two contributions: a long-range electrostatic interaction and a short-range hard-core repulsion which leads to the contact pressure25
∑i Fi∑j Fjσij3gij(σij)
Pcontact ) (2π/3)kT
(6)
where σij is the contact separation of the (i,j) pair. For charged colloids, the virial equation (eq 5) is governed by three leading terms: (i) the entropic repulsion resulting from ion translation; (ii) the electrostatic attraction between polyions and their neutralizing counterions, partially reduced by the polyion-polyion and ionion electrostatic repulsions; and (iii) the contact repulsion between polyions and their condensed counterions (eq 6). In the context of the cell model,30 the PoissonBoltzmann approximation estimates the pressure from the sum of the ionic activities17,23,26,36,37
∑ci(R)
P ) kT
(7)
Monte Carlo simulations validate this approximation for monovalent counterions,17,26 since correlation forces are then negligible.27,28 Figure 1 shows the equation of state of an homogeneous suspension of latex particles deduced from a numerical solution of the PB equation (eqs 1-4) describing a closed system. The total pressure increases with the particle concentration and the ionic strength. This last result differs from the behavior observed for colloidal suspensions of charged particles in the open condition (dialysis experiment), for which an increase of the ionic strength reduces the swelling pressure.17,22,23 Salt exclusion (the so-called Donnan equilibrium)22,38 is then responsible for the decrease of the effective interparticle repulsion. (35) Cooker, H. J. Phys. Chem. 1976, 80, 2084. (36) Marcus, R. A. J. Chem. Phys. 1955, 23, 1057. (37) Langmuir, I. J. Chem. Phys. 1938, 6, 873. (38) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: London, 1991.
Figure 1. Equation of state of dilute latex suspensions calculated from a Poisson-Boltzmann treatment of the ion distribution at ionic strength I ) 0.1 (s), 1 (-‚-), 1.5 (‚‚‚), and 2 (-‚‚‚-) µM.
(B) Salt Transfer within Segregated Suspensions. We interpret the reported segregation1-3 of dilute suspensions (volume fraction φ ∼ 0.01) of latex particles as due to the occurrence of an ionic transfer from dense to empty coexisting microdomains. Since P ) -(∂F/∂V)T,N, the relative Helmoltz free energy of a suspension may be estimated from the integral
dφ ∫∞V∆P dV ) Vp∫0φ∆P φ2
∆F(N,V,T) ) -
(8)
where Vp is the volume of the particle. Two limiting models of segregated suspensions are considered: we treat the central empty cavity of ionic strength IE either as a pure electrolyte solution or as a colloidal suspension with a particule volume fraction φE determined by the number of particles located in the first shell at the boundary between dense and empty microdomains. These two models are somewhat complementary: while the first approach is mainly valid in the limit of large segregation (VE/VT f 1), the second gives an exact description of weakly segregated suspensions (VE/VT f 0). We then search for a possible coexistence between dense and empty microdomains, satisfying the constraints of conservation of the number of particles and salt molecules. We further impose the equality of the pressure between the empty and dense microdomains. From the equations of state of closed colloidal suspensions (Figure 1) we predict a decrease of the salt concentration in the dense domain, since the pressure is an increasing function of both particle density and ionic strength. However, the three constraints can generally not be satisfied without a pressure increment ∆P of the segregated suspension referred to the initial homogeneous suspension
P(φ,I) + ∆P ) P(φD,ID) ) P(φE,IE)
(9)
The sign of this increment ∆P is crucial, since it determines the spontaneity of the segregation (cf. eq 8). Figure 2 shows the prediction resulting from the first model, which treats voids as a solution of a simple electrolyte. It predicts a large region of coexistence between dense and empty domains but overestimates the stability of segregated suspensions. The diffuse layers from the particles located at the boundary between dense and empty domains contribute also to the pressure within the void. This contribution is partially included in the
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Notes
Figure 2. Variation of the Helmoltz free energy of a segregated suspension of latex (total particle volume fraction φ ) 0.01 and ionic strength I ) 1.5 µM) as a function of the void volume fraction. The empty cavity is modeled as a pure electrolyte solution (see text).
Figure 3. Variation of the Helmoltz free energy of a suspension of latex (total particle volume fraction φ ) 0.01 and 0.5, ionic strength I ) 0.8 and 1.5 µM) as a function of the void volume fraction. The empty cavity is modeled as a suspension of latex particles (see text).
second model, which restricts the extent of segregation (Figure 3) and the order of magnitude of the decrease of the Helmoltz free energy. We also determine the influence of ionic strength and particule density on the stability of segregated suspensions: both parameters drastically restrict void formation, in agreement with experimental evidence.3 From the results shown in Figure 3, we estimate the order of magnitude of the variation of the Helmoltz free energy (eq 8) resulting from the segregation of the suspension (∆F ≈ -0.8 kT per particle), for a total ionic strength I ) 1.5 µM and particle volume fraction φ ) 0.01. (C) Limitations. The estimate of the Helmoltz freeenergy difference suffers from a major drawback: it treats coexisting dense and empty microdomains of the segregated suspension as independent regions. However charged particles located at the boundary between dense and empty microdomains must show some lack of symmetry of their cloud of counterions. In addition to the previously discussed salt transfer from the dense domain to the voids, one also expects a partial transfer of neutralizing counterions. Because of the low salt concentration necessary to observe void formation, one cannot rule out a priori this “void contamination”. Monte Carlo simulations of ion distribution around mobile charged particles are now under progress to quantify the influence
Langmuir, Vol. 12, No. 10, 1996 2607
of both phenomena on the stability of voids and elucidate the mechanism of void formation. The Poisson-Boltzmann approximation neglects the long-range correlation of the Coulomb potential. Monte Carlo simulations have already shown the occurrence of a net interparticle attraction due to ionic correlation.27,28 However, the attraction is sensitive only for divalent counterions and small interparticle separation. Calculations performed on dilute suspension of particles neutralized by monovalent counterions do not show such effects,17,26 fully justifying the use of the PB approach. One may also question the validity of the primitive model24 which uses smooth spherical ions and particles to describe the interactions between solvated sodium counterions and latex particles. The inclusion of some roughness of the particle will probably not modify its longrange electrostatic interactions but must introduce an angular modulation39 of the contribution of short-range electrostatic and contact interactions between a surface site of the particle and one of a neighboring counterion. For the pressure calculation, this angular modulation is expressed by the scalar product rijFij of the virial equation (eq 5). Another problem with dilute suspensions of latex particles is the possible contamination of the suspension due to impurities released by latex particles or ionic exchangers. Smaller fragments of polyelectrolyte may be responsible for the suspension segregation, confining the latex particles in dense microdomains through their own diffuse layer. The possible occurrence of such a depletion mechanism is certainly reduced by using carefully washed samples. In any case, chemical analysis of solvent isolated from the suspension by centrifugation or elution through a large-pore membrane should answer this most important question. Other interpretations of the spontaneous segregation of dilute latex suspensions may be suggested. The occurrence of a long-range attraction29 among the particles also explains the phenomenon, but the interparticle separation is too large to permit strong Van der Waals attraction.23,38 The dense microdomains contain only a gaslike suspension of highly mobile latex particles,2 excluding any trapping of the particles in the minimum of an attractive-repulsive potential (like the DLVO potential). Conclusions We suggest a new interpretation of the spontaneous segregation of dilute suspension of large latex particles,1-3 without invoking long-range interparticle attraction. The transfer of ions between dense and empty microdomains is the driving force responsible for this segregation. A Poisson-Boltzmann treatment of the ionic distribution around latex particles under dilute conditions was used to quantify the contribution of this ionic exchange to the relative stability of segregated latex suspensions. Although generally present, the ionic transfer is not taken into account by classical simple models of charged colloids which use, in the framework on the one-component plasma, an empirical Yukawa potential10,12,15,16 to describe the interparticle interactions. Acknowledgment. It is a pleasure to acknowledge Drs. R. Setton and P. Levitz (CRMD) for helpful discussions and IFP (Rueil) for the purchase of the minicomputers (HP 9000) used for the PB calculations. LA950972Y (39) Nezbeda, I.; Reddy, M. R.; Smith, W. R. Mol. Phys. 1990, 71, 915.
