Specific ion effects via ion hydration: II. Double layer interaction

Transcription

1 Advances in Colloid and Interface Science 105 (003) Specific ion effects via ion hydration: II. Double layer interaction Eli Ruckenstein*, Marian Manciu Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 1460, USA Abstract A simple modified Poisson Boltzmann formalism, which accounts also for those interactions between electrolyte ions and colloidal particles not included in the mean potential, is used to calculate the force between two parallel plates. It is shown that the short-range interactions between ions and plates, such as those due to the change in the hydration free energy of a structure-makingybreaking ion that approaches the interface, affect the double layer interaction at large separations through the modification of the surface potential and surface charge density. While at short separations (below the range of the short-range ionhydration forces) the interaction can be attractive, at larger separations the interaction is always repulsive, as in the traditional theory. When the long-range van der Waals interactions between the ions and the system (ion-dispersion interactions) are accounted for in the modified Poisson Boltzmann approach, an attractive force between plates can be generated. At sufficiently large separations, this attraction can become even stronger than the traditional van der Waals attraction between plates of finite thickness, thus generating a dominant longrange double layer attraction. At small plate separations, the attraction generated by the ion-dispersion forces combined with the electrostatic repulsion due to the double layers overlap can lead to a variety of interactions, from a weak attraction (which is typically by at least one order of magnitude smaller than the traditional van der Waals attraction between plates) to a strong double layer repulsion (for sufficiently large surface charges). Both types of ion interactions (long-range van der Waals or short-range ionic hydration) strongly affect the magnitude of the double layer interaction, and can account for the specific ion effects observed experimentally. However, they do not affect qualitatively the traditional theory of the colloid stability, which predicts that the colloid is stable when there is a sufficiently large charge on the surface, and coagulates when the van der Waals interactions between two colloidal particles dominate. The only qualitative difference found when the ion-dispersion *Corresponding author. Tel.: q , ext. 14; fax: q address: feaeliru@acsu.buffalo.edu (E. Ruckenstein) /03/$ - see front matter 003 Elsevier B.V. All rights reserved. doi: /s ž x

2 178 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) interactions were incorporated into the traditional double layer theory was the emergence of a double layer attraction at very large separations, which, however, does not affect much the stability of colloids. 003 Elsevier B.V. All rights reserved. Keywords: Colloidal particles; Double layer interaction; Ion-hydration forces Contents 1. Introduction Ion distribution and interaction free energy. General formalism Interaction between parallel plates immersed in a gas of neutral particles Free energy of the system and the Gibbs adsorption equation Long-range interactions between neutral particles and plates Short-range interactions between neutral particles and plates The interaction between parallel plates immersed in an electrolyte Short-range interactions (ion-hydration forces) Long-range interactions (ion-dispersion forces) Conclusions References Introduction Langmuir demonstrated long ago w1x that the overlap of the ionic atmospheres generated near two charged parallel plates immersed in an electrolyte, calculated within the Poisson Boltzmann formalism w,3x, leads always to a repulsive force between the plates. This repulsion (the double layer force), combined with the van der Waals attraction, was later successfully employed to explain the stability of colloids (the DLVO theory) w4,5x. The original Poisson Boltzmann treatment (which assumes that the ions obey the Boltzmann equation and interact only via a mean potential, which satisfies the Poisson equation) is clearly a drastic simplification. However, it provided in many cases a good qualitative agreement with experimental data, and sometimes in a certain range of electrolyte concentrations even a good quantitative agreement. The limitations of the traditional Poisson Boltzmann treatment are numerous. A particularly important one refers to specific ion effects. The amphoteric latex particles coagulate when the concentration of CsNO3 is approximately 1 M, but remain stable at high ph values even when the concentration of KNO3 exceeds 3 M w6x. Nevertheless, in the traditional Poisson Boltzmann approach the double layer repulsions for Cs and K are the same. A number of attempts were made to include additional interactions in the formalism, some of them being briefly discussed in the first part of this article w7x. With these corrections, many of them dependent on unknown parameters, the modified Poisson Boltzmann approach regained its explanatory power, and in general most of the experiments could be accounted for

3 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) by using suitable values for the unknown interaction parameters. However, both the simplicity and the predictability of the traditional treatment were lost, since in general, the choice of the parameter values was dependent on the particular problem investigated. In contrast, the specific ion effects seems to be rather well organized in a lyotropic series, first obtained empirically by Hofmeister from data on protein salting out w8x. The ions seems to obey rather well the lyotropic series sequence in many different experiments (an exhaustive compilation being provided in Ref. w9x), and the specific ionic effects relevant in biophysics are particularly strong for anions at moderate ionic strengths ( M w10x). It was also noted that different experiments might lead to slight alterations of the series w9x. An example is provided by the surface force experiments of Pashley w11x between crossed mica cylinders in various electrolytes, which indicated the occurrence of an additional repulsion in the vicinity of the surface in excess to the double layer interaction predicted by the traditional Poisson Boltzmann equation. Such an interaction was observed at low y4 y ionic strengths (;10 10 M) for NaCl, KCl and CsCl, but only at much larger y electrolyte concentrations (;6=10 M) for LiCl. Therefore, the specific ionic effect of Li seems to be quite different from those of Na, K and Cs. In contrast, the amphoteric latex colloids w6x were stable when the cation was K or Li, but coagulated for Cs. The relative regularity of the Hofmeister series suggested that the specific ion effects might be associated to the properties of the ions in the solvent, and it was noted long ago that the Hofmeister series is related to the hydration of ions w1x and to the ability of the ions to organize the neighboring water molecule w10x. More recently, it was proposed than the sequence of ions in the lyotropic series could be explained by the van der Waals interaction between the ions and the rest of the system w13x. When suitable constants for the van der Waals interactions were selected, the accounting for these ion-dispersion forces provided a quantitative agreement with the experiments regarding the surface tension of aqueous electrolytes w14,15x. More dramatic results were obtained when the ion-dispersion forces were accounted in the double layer interactions w16x. At low electrolyte concentrations (below 0.01 M), the specific ionic effects due to the van der Waals interactions of ions were negligible. However, under physiological conditions (c s0.1 M NaCl, E surface potential csf0.010 V) the double layer repulsion between parallel plates became attractive for separation distances smaller than approximately 0 A. The maximum interaction free energy due to the double layer repulsion calculated in Ref. w16x y5 was only approximately 5=10 ktya, which is one order of magnitude smaller than a typical van der Waals attraction between proteins at that separation distance. Therefore, the net interaction (namely, the double layer repulsion, which includes the van der Waals interactions of the electrolyte ions with the system, plus the van der Waals interactions between two parallel plates) is in general attractive and all the physiological colloids should coagulate. It should be noted that there is an extensive literature about the existence of a double layer attraction w17x, and there is no general agreement yet about its nature.

4 180 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) However, such an attraction occurs at large separation distances, and does not affect much the interactions at low separations, hence the coagulation of colloids. It is also well accepted that the classic Poisson Boltzmann treatment is inaccurate at concentrations higher than approximately 0.05 M, and in general either additional interactions or unknown fitting parameters had to be employed for quantitative agreement. One of the major successes of the inclusion of the ion-dispersion forces into the Poisson Boltzmann formalism was the explanation of the dependence of the surface tension of electrolytes on ionic strength w14,15x and ph w15x. In a previous article, it was shown that this dependence could be also explained by a simple accounting of the interactions between the ion and surface, through the changes in the ion hydration free energy between bulk and interface w7x. The latter treatment avoided some difficulties that occurred when only the van der Waals interactions of the ions with the system were taken into account (the accumulation of cations near the interface and the positive surface potential generated, in contrast to the accumulation of most anions and the negative surface potential provided by experiment and molecular simulations w18x). It also showed that the rather high values of the coefficients of the ion-dispersion interactions, obtained from the fit of the experimental data w14x, might have been caused by the fact that the ion-hydration interactions were neglected. In this article, a general formalism to calculate the interactions between colloidal particles, when additional interactions are included in the Poisson Boltzmann formalism is presented. The effects of ion-hydration and ion-dispersion interaction potentials on the force between parallel plates are investigated. While quantitative effects strongly dependent on the ion specificity are obtained, the calculations indicate that the DLVO theory remains essentially valid, namely that the colloids are stable if their surface is sufficiently charged. The values of the surface charge density and surface potential, which provide stability, are of course different if additional interactions are accounted in the Poisson Boltzmann formalism. It will be also shown that the ion-dispersion interactions might lead, at large separation distances, to an attraction between plates. This double layer attraction might become at large separations, even stronger than the traditional van der Waals attraction between parallel plates of finite thickness.. Ion distribution and interaction free energy: general formalism As in the previous article w7x, the equality of the electrochemical potential through the system leads to the following equation for the distribution of the ions of an uniunivalent electrolyte of concentration c : E B c sc expcy i E D i (y1) ecqdw E i kt F G (1) where i stands for the kind of ions (1 for anions and for cations), e is the protonic charge, k is the Boltzmann constant, T is the absolute temperature, DW is i

5 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) the change in the interaction energy of an ion i (not included in the mean potential c) with respect to the bulk. The potential c obeys the Poisson equation: =cs ecyc Ž 1. 0 () where is the vacuum permittivity and is the dielectric constant of the medium, 0 which here is assumed to be constant through the solvent. The distribution of electrolyte ions can be calculated, for suitable boundary conditions, from Eqs. (1) and (). The total free energy of the system is composed of several contributions: an electrostatic energy due to the distribution of charges, an energy due to the additional interactions of ions (not included in the mean potential), an entropic term due to the mobile ions, and a chemical free energy which is responsible for the formation of the double layer. The electrostatic energy is given by: 1 el 0 V F s (=c) dv, (3) and the entropic contribution can be calculated from w19x: B Bc E E C C i ent 8 i F i E F V D D ce G G i F sytdsskt c ln yc qc dv (4) where both integrals are calculated over the volume V of the solvent. The interaction free chemical energy (calculated with respect to the infinite separation of the components of the system) is given by w0x: B s C AD s( `) E DF s y c (s)dsfda (5) chem S G where s is the surface charge density, c the surface potential, the surface integral S is performed over the entire area A of the system (the interface between the solvent and the interacting surfaces), and the other integral is performed along the trajectory of the system from infinite separation distance between the surfaces to the final configuration of the system. To these components of the traditional double layer free energy one should add an energetic contribution due to the supplementary ion interactions: V Fints Ž c1dw1qcdw. dv. (6)

6 18 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) Consequently, the interaction free energy of the system (with respect to infinite separation distance between components) is given by: DFsDF qdf qdf qdf (7) el ent chem int 3. Interaction between parallel plates immersed in a gas of neutral particles 3.1. Free energy of the system and the Gibbs adsorption equation The formalism described in Section can be applied to systems of any geometry. To simplify the calculations, it will be applied here to the interactions between two parallel plates. Let us first assume that two large parallel plates are immersed into a reservoir containing a neutral gas of concentration c N. If the gas interacts with the inner side of the plates (the interactions with the other sides are neglected), the free energy of the system (the plates and the adsorbed gas between them) per unit area is given by: d F N(d)sF int(d)qf ent (d)s c(x)dw(x)dx yd d B Bc(x) E E qkt Cc(x) lnc Fyc(x)qcNFdx yd D D cn G G d B B DW(x) EE scn kt C1yexpCy FFdx (8) D D kt GG yd where d is the distance between plates, DW(x) is the change in the energy of a particle located at a distance x from the middle between the plates with respect to that at infinite distance and the gas particles are assumed to obey a Boltzmann distribution. The same result can be obtained using a Gibbs adsorption procedure, introduced by van der Waals and Makor w1x and generalized by Ash et al. wx. The positive or negative surface excess adsorption of gas particles, G N, between plates is related, via the Gibbs adsorption equation, to the change in the surface tension g: B C dgsyg dm sy Ž c(x)yc. dx dm d N N N N D 0 G B db B DW(x) E E E NC C C F F F N D 0 D D kt G G G E F syc exp y y1 dx dm (9) where mn is the chemical potential of the neutral particles, which is constant through the system and can be related to the concentration of the neutral particles in the

7 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) reservoir, for a perfect gas, through: dmnsdžkt lnž c N... (10) Integrating Eqs. (9) and (10) for a concentration of particles varying from 0 to c, one obtains: N D D G G N G (c) DW(x) gž cn. yg(0)sykt dcsyc kt exp y y1 dx (11) c db B E E N N C C F F 0 c 0 kt and, as expected, the force between plates w1x, g PŽ d. sy d (1) is identical to that obtained from the derivative of the free energy, Eq. (8), with respect to the separation distance d. For a neutral gas, the adsorption (which is proportional to the free energy of the system, see Eq. (8)) can be calculated easily. For charged particles one should account in the Gibbs adsorption equation for the adsorption of all particles of the system (including those responsible for the charging of the surface w3x). Therefore, one should first identify the mechanism of formation of the double layer. In this case, m c ( ) G gž ce. yg(0)sykt Gidm i (11a) 8 y` i where i is taken over all the ionic species present in the system and Gi is the adsorption of ion i. The result is incomplete when one accounts only for the distributions of electrolyte ions between the plates. Indeed, for an uni-univalent electrolyte of concentration c E, the adsorption G9 of the two species of ions between two plates with constant surface charges s, is given at low surface potentials by: d d C C F F C F ydd D G G D G yd G9Ž ce. B Bec(x) E E B e E s cosh y1 dx( (c(x)) dx c kt kt E Bx E l Bd E cosh C F sinhc Fqd B D G esl E d l B esl E D l G ( C F dxs C F (13) DkT G BdE DkT G BdE 0 yd 0 sinh C F sinh C F DlG DlG where l is the Debye length. Since ( y (d))(g9(c )yc )-0 for any c, using E E E

8 184 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) Eqs. (9) (1) would predict an attraction between the plates (P-0), which is incorrect. In what follows, the direct calculation of the free interaction energy by adding the energetic and entropic contributions (Eq. (7)) will be employed. 3.. Long-range interactions between neutral particles and plates To have an idea about how important are the dispersion interactions of electrolyte ions, we will consider in this section the force between plates generated by a neutral gas, whose particles interact with the plates by long-range dispersion forces. An interaction of the type w16x: B B DW vdw(x)s q (14) 3 3 (dyx) (dqx) will be assumed, where d is the separation distance between plates. For a molecule of gas, B is negative (the particles are attracted by the plates). However, if the particles are immersed in a medium, it is possible to have positive B values, depending on the Hamaker constants of the plates, particles and medium. Assuming a cut-off distance D for the interaction, the free energy per unit area (Eq. (8)) becomes, in the linear approximation (which is accurate if B is small or if the particle is sufficiently far from any interface): d BDW (x) E vdw 4d(dyD) N N C F N ydd kt G D (dyd) F (d)(c kt dxsbc (15) and the corresponding interaction free energy, per unit area, is: Bc N F(d)yF(`)sy (16) (dyd) This interaction, due to the interaction between particles and plates, has the same functional form at large separations (d4d) as the well-known van der Waals attraction between thick parallel plates w4x: F A H vdw,platessy 1p(d) (17) where AH is the Hamaker constant. When the particles are attracted by the plates through dispersion forces (B-0), the net interaction between plates is repulsive, while when the interactions between particles and plates are repulsive (B)0), the plates are attracted to each other. The magnitude of the interaction generated by the dispersion forces between the neutral particles and the plates (Eq. (16)) is proportional to the concentration of

9 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) Fig. 1. The force between parallel plates for a neutral system, generated by ion-dispersion forces at c E s y50 3 y M, for B1sBs31=10 J m (curve ()); B1sBs15=10 J m (curve (3)); B1sBsy y50 3 1=10 J m (curve (4)) are compared to the traditional van der Waals attraction between plates for y0 A s10 J (curve (1)). H particles in the reservoir (c ) and is typically much smaller than the traditional van N der Waals attraction between plates. The two interactions (Eqs. (16) and (17)) become comparable at large separation distances for: A H c N(. 1pB (18) Since the Hamaker constant for proteins in a physiological solution is of the y0 order of 10 J, Eq. (18) provides a concentration of approximately M even for y50 3 y the large value of Bs31=10 J m corresponding to Cl ions w14x. This corresponds to an ideal gas pressure of approximately 50 atm. Consequently, the long-range repulsion or attraction between plates due to the van der Waals interactions between a gas of neutral particles and the plates is expected to be negligible in most cases. The interactions between neutral plates immersed in an electrolyte can be calculated from Eqs. (11) and (1), providing that the dispersion forces are the same for both kinds of ions and hence the potential vanishes everywhere. In Fig. 1,

10 186 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) the forces (per unit area) between parallel plates, calculated numerically for c 1 s 0.1 M, Ts300 K, a cut-off distance Ds A and various values B1sB (curves ( 4)) are compared with the traditional van der Waals attraction due to the y0 polarizability of the plates, for a typical Hamaker constant of AHs1=10 J. The attraction generated by the ion-dispersion forces at this concentration is in general much smaller than the traditional van der Waals attraction (curve 1). When B 'B -0 (curve 4), the interaction generated by ion-dispersion forces is a 1 repulsion. In general, B /B and a double layer is generated by the asymmetric distribution 1 of the electrolyte ions even if there is no charge on the plate surface. A treatment based on the modified Poisson Boltzmann formalism will be presented in Section Short-range interactions between neutral particles and plates In a previous article w7x, it was suggested that the changes in ion hydration between bulk and the vicinity of the surface might be approximated by a simple step function. In what follows the interaction between parallel plates will be investigated for an interaction between particles and plates of the type: DW(j)sW 0-j-d (19a) 1 1 DW(j)s0 d -j-` (19b) 1 where j is the distance to one of the plates and W can be negative (corresponding 1 to an attraction between particles and plates) or positive (repulsion). It is also assumed that the interactions between particles and plates are additive (Fig. ): DWsDW(j)qDW(dyj), (19c) d being the separation between plates. Due to the short-range of this interaction, the force between plates vanishes for d)d 1. For (d1y)-d-d 1, the free energy per unit area of the system (Eq. (8)) is given by: d B B DW(x) EE F(d)scN kt C1yexpCy FFdx D D kt GG yd W sc kt d yd 1yexp y dyd 1yexp w B B EE B B EEz 1 1 N x Ž 1. C C FF Ž 1. C C FF y D D kt GG D DkT GG~ w B B E E B B E B EEz W1 W1 W1 scnktx dcexpc Fy1F qd1cexpc Fyexp C FF (0) y D DkT G G D DkT G D kt GG~ W

11 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) Fig.. An additive step-like interfacial interaction between neutral particles and plates. There is no force between plates for d)d 1, attraction when (d1y)-d-d1 and either attraction or repulsion for 0-d- (d1y). and the force per unit area between plates by F(d) B BW E E 1 d1 N C C F F 1 D D G G P(d)sy syc kt exp y1-0 -d-d (1) (d) kt

12 188 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) is always attractive in this region (P-0), regardless of whether the particles are attracted toward (W1-0) or repelled (W 1)0) by the plates. For the range 0-d-(d1y), the force per unit area is given by: B BW E E 1 d1 P(d)scNkTCexpC Fy1F 0-d- () D DkT G G which leads to a net repulsion between plates if the neutral particles are attracted by the plates and to an attraction when the neutral particles are repelled by the plates. In summary, short-ranged step-like interactions for ions generate only short-range interactions between plates, which are always attractive for (d1y)-d-d1 and become either attractive or repulsive, depending on the sign of W 1, for 0-d-(d1y ). 4. The interaction between parallel plates immersed in an electrolyte 4.1. Short-range interactions (ion-hydration forces) The change in the ion hydration energy between the bulk and the water air interface for structure-making ions is so large compared to kt, that these ions practically do not approach the interface. The change is not so steep when a solid interface is immersed in water, particularly when the surface has sites which can bound structure-making ions. In what follows, it will be assumed that the anions are structure breaking and hence interact with the surface via an attractive potential of the type (W )0, see Fig. 3a): 1 DW (j)syw 0-j-d (3a) DW (j)s0 d -j-` (3b) 1 1 and the cations are structure making and interact with the surface via a repulsive potential (W )0): DW (j)sw 0-j-d (4a) DW (j)s0 d -j-` (4b) The modified Poisson Boltzmann equation for a uni-univalent electrolyte confined between parallel plates is: d c ec DW ec DW ec s exp y exp yexp y exp y (5) d x kt kt kt kt B B E B E B E B EE E 1 C C F C F C F C FF 0 D D G D G D G D GG

13 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) Fig. 3. (a) A typical ion-hydration potential: the anions are attracted toward the interface (W1s1 kt, d s6 A) while the cations are repelled by the surface (W s kt,d s3 A). 1 (b) The potential and the corresponding ion distributions, for W s1 kt,d s6 A,W s kt,d s3 A,ds15 A, 1 1 ces0.1 M, css y0.010 V, s80, Ts300 K. which has to satisfy two boundary conditions, one due to the symmetry: dc Z ZZ dx xs0 Z s0 (5a)

14 190 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) and the other providing the surface charge density, the surface potential or a relation between them. Since the selected DW (j) is independent of j on each of the intervals 0-j-d, i d -j-d, d -j-d (Fig. 3a), the solution in the linear approximation (valid at 1 1 low potentials) in the interval j is of the type: y B E B E j j j j j j j C F jq1 C F j j D G D G kt A yd AqD j A qd j c(j)sy qc exp qc exp y (6) e AqD l l where Ajsexp(y(DW 1) jy(kt)), Djsexp(y(DW ) jy(kt)), ls6(( 0kT)y(e c E)) is the Debye length and in each interval j, C j, Cjq1 are constants to be determined from the boundary conditions and the continuity of the electric potential and electric field. A typical result for the potential and the corresponding ion distribution is presented in Fig. 3b, for: W s1 kt,d s6 A,W s kt,d s3 A,ds A, ces 0.1 M, cssy0.010 V, s80, Ts300 K. Once the potential has been determined, the interaction free energy of the system can be calculated using Eq. (7). For separation distances d)d,the Poisson Boltzmann equation is obeyed within 1 d -j-dyd.the system is completely equivalent with a traditional double layer, 1 1 but formed between the surfaces S9(jsd ) and not between the real surfaces S(js 1 0). The corresponding surface potential is the potential at S9, c 'c(jsd ), and S 1 the corresponding surface charge density, s9, is given by: y d 1 dc 0 Z 0 dj jsd 1 s9ssq r(j)djsy. (7) Z Therefore, the short-range interactions due to ion hydration do not modify qualitatively the double layer interaction, which remains repulsive at distances larger than the range of the interfacial interaction of the ions. The magnitude of the repulsion is, however, modified, since the relation between the true surface charge density s and the apparent surface charge density s9 depends on DWi and hence on the specificity of the ions. To illustrate this, the apparent charge density s9 is calculated in Fig. 4a and b as a function of the separation distance, for d 'd s5 A 1 and constant surface charge density. The apparent charge density is proportional to the derivative of the electric potential at the distance jsd 1'd. When the surface is negatively charged (ssy Cym, Fig. 4a), both the accumulation of anions (DW1-0) and the cation depletion (DW )0) lead to an increase of the charge in the vicinity of the surface, therefore to an increase in the double layer repulsion. When the surface is positively charged (ss0.010 Cym, Fig. 4b), the ion hydration interaction can lead to a decrease in the apparent surface charge density (curve (4)), which can even change sign. The apparent charge density is plotted in Fig. 4c as a function of electrolyte concentration, for ds0 A and ssq0.010 Cym. Depending on the potentials of

15 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) ion hydrations, s9, which is positive at sufficiently low concentrations, can become negative when the ionic strength is sufficiently high. The electrolyte concentration which correspond to s9s0 and hence, roughly, to coagulation depends strongly, via W and W, on the type of electrolyte. Therefore, short-range interfacial interactions 1 of ions, of the order of kt, can account for specific ion effects in the double layer interactions. The force between neutral plates (ss0) immersed in an electrolyte of concentration c s0.1 M, calculated from the derivative of the free energy of the system (Eq. E (7)) is plotted in Fig. 5a. The double layer, which is generated by the different distributions of the positive and negative ions of the electrolyte alone (it vanishes when DW sdw ), is always repulsive for d)d 'd, and can be either attractive 1 1 or repulsive for d-d. 1 Fig. 4. (a) The apparent surface charge density s9 for an interaction at constant surface charge density (ssy0.01 Cym, ces0.1 M) as a function of the separation between plates. The long-range double layer interaction is controlled by s9, which in turn depends on the short-range potential of interaction between ions and surface. (b) The apparent surface charge density s9 for an interaction at a positive constant surface charge density (ssq0.01 Cym, ces0.1 M) as a function of the separation between plates. The apparent surface charge density can be decreased or even change its sign when the shortrange interaction between ions and surface is accounted for. (c) The apparent surface charge density s9 for ssq0.01 Cym and ds0 A as a function of the electrolyte concentration. While the double layer interaction is always repulsive at sufficiently large concentrations, the electrolyte concentration corresponding to s9s0 (hence, to a vanishing double layer repulsion) depends strongly on the short-range interactions between ions and surface.

16 19 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) Fig. 4 (Continued).

17 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) Fig. 5. (a) The force per unit area between neutral parallel plates immersed in an electrolyte with c E s 0.1 M. The (asymmetric) short-range interactions between ions and interface lead to an asymmetric distribution of electrolyte ions, which in turn generates a long-range double layer repulsion. In the vicinity of the surface (d-10 A), the force can become attractive. (b) The force per unit area between nega- tively charged parallel plates immersed in an electrolyte with c s0.1 M. The interactions between ions E and surfaces increase in this case the apparent surface charge density, hence increase the repulsion at large separation distances. The force can become attractive for d-10 A.

18 194 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) A more realistic calculation is presented in Fig. 5b, which implies that there is a surface charge density generated by the dissociation of surface groups. Assuming that the cations can be bound to N negative sites per unit area, the dissociation equilibrium provides the expression: q q (hn)c hc h B ec W E E E S KDs s s ceexpcy y F (8) (1yh)N (1yh) 1yh D kt kt G q where KD is the dissociation constant, ce denotes the cation concentration in the liquid in the vicinity of the surface and h is the fraction of dissociated sites. The surface charge density is therefore given by: en ssyehnsy c B ec W E E S 1q expcy y F K D kt ktg D (9) The ion hydration interactions lead to an accumulation of anions and a depletion of cations near the interface, thus increasing the magnitude of the apparent surface charge density s9. Consequently, the double layer repulsion is enhanced at large distances. When the surface charge density is positive at low electrolyte concentrations (e.g. when the charge is generated by the dissociation of basic surface groups), the ion hydration can decrease the magnitude of the repulsion at large distances, but cannot transform it there into an attraction. The apparent charge density, which is ultimately responsible for the double layer interaction, depends strongly on the hydration potential of ions. Although the double layer interaction is always repulsive, for some electrolytes s9 can become sufficiently large to prevent coagulation, while for other electrolytes sufficiently low to allow coagulation. Therefore, the ion hydration interaction can provide a suitable explanation for the specific ion effects observed in double layer interactions. 4.. Long-range interactions (ion-dispersion forces) In this section, the effect of the ion-dispersion forces on the double layer interactions will be investigated, by assuming that they provide the only correction to the traditional Poisson Boltzmann equation. When the dispersion interaction coefficients are different for anions and cations, the different distributions of the two kinds of ions generate a potential even when the plates are neutral. The potential is obtained from the solution of the correspondingly modified Poisson Boltzmann equation (Eqs. (1) and ()) and the interaction free energy can be calculated via the numerical integration of Eq. (7). The force between neutral plates immersed in an uni-univalent electrolyte of concentration ces0.1 M, when B 1/B, is presented in Fig. 6(a), for a cut-off distance Ds A. While the force was always attractive when B1sB )0 (Fig. 1), the double layer generated by the asymmetric distributions of electrolyte ions, when

19 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) B 1/B, can dominate the interaction (which becomes repulsive) at intermediate y50 3 y50 3 separations (B1s31=10 J m, Bs1=10 J m, curve (1) in Fig. 6a). If one y50 3 type of the ions (the cation) is attracted by the interface (Bsy1=10 J m, curve ()), the repulsion induced by the ion-dispersion forces becomes, as expected, even stronger. Fig. 6. (a) The force between neutral plates immersed in an electrolyte with ces0.1 M, due to the iondispersion forces. At intermediate separations, the double layer generated by the asymmetric distribution of electrolyte ions leads to a net repulsion, although both types of ions are repelled by the interface y50 3 y50 3 (B1s31=10 J m, Bs1=10 J m ). When the ions of one kind are attracted by the interface y50 3 (B1sy1=10 J m ), the repulsion increases. The interactions at large separation distances (inset) are attractive. (b) The force between negatively charged plates immersed in an electrolyte with ces M, (1) Ns3=10 sitesym, KDs0.1 M, B1sBs0; () Ns3=10 sitesym, KDs0.1 M, B1s y50 3 y50 3 y50 3 y =10 J m and Bsy1=10 J m ; (3) ss0, B1s31=10 J m and Bsy1=10 J m ; 17 y50 3 y50 3 (4) Ns1=10 sitesym, KDs0.1 M, B1s31=10 J m and Bsy1=10 J m ; the double layer attraction (inset) occurs at large separations, even in the presence of large surface charges. (c) The force between parallel plates, for a moderate surface charge generated by the dissociation of surface 17 groups, for Ns1=10 sitesym, KDs0.1 M, s80, Ts300 K and various sets of ion-dispersion y50 3 y50 3 y50 3 coefficients: (1) B1s0, Bs0; () B1s15=10 J m, Bs3=10 J m ; (3) B1s15=10 J m, y50 3 y50 3 y50 3 y50 3 Bs1=10 J m ; (4) B1s15=10 J m, Bsy1=10 J m ; (5) B1s15=10 J m, Bsy y50 3 y50 3 y50 3 y50 3 y50 3=10 J m ; (6) B1s31=10 J m, Bs3=10 J m ; (7) B1s31=10 J m, Bs1=10 3 y50 3 y50 3 y50 3 y50 3 Jm;(8) B1s31=10 J m, Bsy1=10 J m ; (9) B1s31=10 J m, Bsy3=10 J m ; 17 In the inset, the calculations are repeated for a strong double layer (Ns=10 sitesym, K s1 M). D

20 196 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) Fig. 6 (Continued).

21 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) The maximum repulsion generated by the asymmetric distribution of ions is however, small when the surface charge is zero (approximately one order of magnitude smaller than a typical van der Waals attraction) and therefore the neutral physiological colloids will coagulate, in agreement with the DLVO theory. More interesting are the effects of ion-dispersion forces when the plates are charged. Let us first assume that a small negative surface charge is generated by the dissociation of surface groups (Eqs. (8) and (9), with ces0.1 M, Ns 16 3=10 sitesym and KDs0.1 M). In the absence of the van der Waals interactions of ions (B1sBs0), a weak double layer repulsion is provided by the traditional theory (curve (1), Fig. 6b). By accounting in the modified Poisson Boltzmann y50 3 equation for the van der Waals interactions of ions with B1s31=10 J m, Bs y50 3 y1=10 J m and Ds A, but assuming ss0, curve (3) was obtained. When both the van der Waals interaction and the surface charge density were accounted for (curve ()), the interaction became attractive at most distances. However, if the surface charge density is sufficiently large (curve (4) in Fig. 6b, 17 with Ns1=10 sitesym ) then the interaction in the vicinity of the surface is repulsive. Therefore, a strong double layer interaction is not modified much at small separations by ion-dispersion forces. The ion-dispersion forces (as the ion-hydration forces, discussed previously) might constitute a suitable candidate to account for the ion specific effects within the Poisson Boltzmann formalism, provided that suitable values for the interaction parameters are chosen. In Fig. 6c, the force between plates was calculated for a moderate double layer, generated by the dissociation of surface groups (Ns 17 1=10 sitesym,kds0.1 M), and for ces0.1 M, s80 and Ts300 K. In the absence of ion dispersion interactions (B1sBs0), the surface potential is approximately y0.00 V. When the ion dispersion interactions were included, the repulsion in the vicinity of the surface has always decreased. The decrease of the repulsion seems to be particularly affected by B 1, which typically has larger magnitudes than B w14,15x y50 3 (B1s15=10 J m for the curves () (5) and B1s y50 3 y50 31=10 J m for the curves (6) (9)). The change of B from Bs1=10 to y50 3 Bs3=10 J m (curves () (3) and (6) (7)) had a much smaller effect. The specific ionic effects of cations become more important when B-0, because the accumulation of positive charges at the surface compensates for the negative charge generated through the dissociation and this attenuates the double layer. In the inset of Fig. 6c, the calculations were repeated for the same values of the 17 parameters, but a stronger double layer (Ns=10 sitesym, KDs1.0 M). The ion-dispersion forces have in this case only a minor effect, and the interaction can be well approximated by the traditional Poisson Boltzmann approach, with slightly modified parameters (density of sites, dissociation constant). There is, however, a range in which the ion-dispersion forces generate qualitatively new results. The double layer between two plates immersed in an electrolyte decays exponentially with the distance and becomes vanishingly small at large separations. The interactions generated by the van der Waals interactions of the ions with the system decay slower and become dominant at those separations. In the inset of Fig.

22 198 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) b, the double layer force between plates becomes attractive at distances larger than approximately 40 A, despite the relatively large surface charge density, which generates a repulsion in the vicinity of the surface. The equivalent Hamaker constant of such an attraction f1pc E(B1qB ) is small, for usual electrolyte concentrations (ce-1 M), when compared to the y0 Hamaker constants of proteins (AHf10 J). However, if the plates have a finite thickness l, their van der Waals interactions at large separations is given by w4x: B E AH 1 1 AHl 1 vdw,plates C F 4 4 D G F sy q y fy Ay (30) 1p (d) (dql) (dql) p(d) (d) At very large separations (d4l), the attraction induced by the van der Waals interactions of ions, which decays only with the second power of the distance (Eq. (16)), becomes much larger than the traditional van der Waals attraction between plates, and its magnitude is proportional to the electrolyte concentration. 5. Conclusions The traditional Poisson Boltzmann treatment of the electrolyte ions, while simple and relatively accurate, is incomplete, since the interactions are treated only via a mean field. It predicts that the double layer interactions depend only on the valence of the electrolyte ions, whereas it is well documented that different electrolytes of the same valence lead to qualitatively different results. It was recently suggested that the van der Waals interactions of the ions with the system can explain the ion specific effects w13x, and it was shown that by accounting for this interactions one can successfully predict the dependence of surface tension of aqueous solutions of electrolytes on their concentration w14,15x. In the first part of this article w7x, it was shown that the ion hydration interaction can also explain the surface tension behaviour. In this article, the effect of the interfacial interaction of ions (ion-hydration and ion-dispersion forces) on the double layer interaction was investigated. When only short-ranged interactions were taken into account (such as the ion hydration interactions), the magnitude of the repulsion between plates at large separations became dependent on the magnitude of an apparent charge density, which is the sum between the real surface charge density and the charge due to the accumulationy depletion of electrolyte ions in the vicinity of the interface. The accounting of the ion hydration interactions can increase or decrease the apparent surface charge density, hence increases or decreases the interaction, which remained, however, always repulsive at large separations. The long-range double layer interactions are strongly dependent on the short-range interfacial interaction of ions. Consequently, specific ion effects can be accounted for in this manner. When the ion-dispersion forces are included in the Poisson Boltzmann treatment, an attraction was obtained at most distances, in the absence of a surface charge density. The magnitude of the attraction was in general smaller than the traditional

23 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) van der Waals attraction between plates. When a small (negative) charge was added on the surface, the attraction was even increased. However, when a large charge density was present on the interface, the interaction became repulsive (and strong) at small separations. When either ion hydration interaction or ion-dispersion forces were included in the treatment, the results were qualitatively identical to the traditional DLVO theory, which roughly predicts that strongly charged colloidal particles are stable, and weakly charged particles coagulate. Significant quantitative differences, which can account for specific ion effects, could be introduced by either mechanism, when suitable interaction parameters were selected. There are some qualitative difficulties when the specific ion effects are explained via the dispersion forces of the ions. Particularly the anions, for which the dispersion coefficients B1 are large, affect the double layer interactions. However, experiments on colloid stability w6x or colloidal forces w11x revealed strong specific ion effects especially for cations. Furthermore, the ions which affect most strongly the solvating properties of the proteins are those from their vicinity, since they perturb mostly the structure of water near the proteins. However, the van der Waals interactions of ions predict that the cations remain in the vicinity of an interface, and the anions are strongly repelled, while Hofmeister concluded that anions are mainly responsible for the salting out of proteins. Accounting for ion hydration is more appealing qualitatively. The structuremaking ions (mostly cations) are repelled by the surface, which therefore becomes more negatively charged. Consequently, the cations can generate via this mechanism long-range specific effects. Furthermore, there are the chaotropes (structure-breaking) anions, which are expelled from the bulk water toward the interface, therefore they modify more strongly the structure of water near the proteins. The structuremaking cations prefer the bulk of the electrolyte, hence, they are expected to affect less the solvation properties of the proteins. It is noteworthy that the ion dispersion interactions induce an attraction between plates, which prevails at sufficiently large separations, regardless of their surface charge density. At separations much larger than the thickness of the plates, this attraction can even exceed the traditional van der Waals attraction between plates. References w1x I. Langmuir, J. Chem. Phys. 6 (1938) 893. wx G. Gouy, J. Phys. Radium. 9 (1910) 457. w3x D.L. Chapman, Phyl. Mag. 5 (1913) 475. w4x B.V. Deryagin, L. Landau, Acta. Physicochim. URSS 14 (1941) 633. w5x E.J. Verwey, J.Th.G. Overbeek, Theory of Stability of Lyophobic Colloids, Elsevier, Amsterdam, w6x T.W. Healy, A. Homola, R.O. James, R.J. Hunter, Farady Discuss. Chem. Soc. 65 (1978) 156. w7x M. Manciu, E. Ruckenstein, Adv. Colloid Inter. Sci. 105 (003) 63. w8x F. Hofmeister, Naunin-Schmiedebergs Archiv fur Experimentelle Pathologie und Pharmakologie (Leipzig) 4 (1888) 47. w9x K.D. Collinns, M.W. Washabaugh, Q. Rev. Biophys. 18 (1985) 33.

24 00 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (003) w10x M.G. Cacace, E.M. Landau, J.J. Ramsden, Q. Rev. Biophys. 30 (1997) 41. w11x R.M. Pashley, J. Colloid. Interf. Sci. 83 (1981) 531. w1x N.K. Adam, The Physics and Chemistry of Surfaces, Oxford University Press, London, w13x B.W. Ninham, V. Yaminsky, Langmuir 13 (1997) 097. w14x M. Bostrom, D.M.R. Williams, B.W. Ninham, Langmuir 17 (001) w15x K.A. Karraker, C.J. Radke, Adv. Colloid. Interf. Sci. 96 (00) 31. w16x M. Bostrom, D.M.R. Williams, B.W. Ninham, Phys. Rev. Lett. 87 (001) w17x N. Ise, T. Okubo, M. Sugimura, K. Ito, H.J. Nolte, J. Chem. Phys. 78 (1983) w18x P. Jungwirth, D.J. Tobias, J. Phys. Chem. 105 (001) w19x J.Th.G. Overbeek, Colloid. Surf. 51 (1990) 61. w0x M. Manciu, E. Ruckenstein, Langmuir 19 (003) w1x E.L. Makor, J.H. van der Waals, J. Colloid. Sci. 7 (195) 535. wx S.G. Ash, D.H. Everett, C.J. Radke, J. Chem. Soc. Faraday Trans. 69 (1973) 156. w3x E. Ruckenstein, J. Colloid. Interf. Sci. 8 (1981) 490. w4x R.J. Hunter, Foundations of Colloid Science, Oxford Science Publications, 1987.