Structure of Spherical Electric Double Layers Containing Mixed

Jul 23, 2010 - Theoretical Chemistry Section, Chemistry Group, Bhabha Atomic Research Centre, Mumbai 400 085, India ... E-mail: [email protected]. ...
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J. Phys. Chem. B 2010, 114, 10550–10557

Structure of Spherical Electric Double Layers Containing Mixed Electrolytes: A Systematic Study by Monte Carlo Simulations and Density Functional Theory Chandra N. Patra* Theoretical Chemistry Section, Chemistry Group, Bhabha Atomic Research Centre, Mumbai 400 085, India ReceiVed: May 11, 2010; ReVised Manuscript ReceiVed: July 9, 2010

The structure of spherical electric double layers in the presence of mixed electrolytes is studied using Monte Carlo simulation and density functional theory within the restricted primitive model. The macroion is modeled as an impenetrable charged hard sphere carrying a uniform surface charge density, surrounded by the small ions represented as charged hard spheres, and the solvent is taken as a dielectric continuum. The density functional theory uses a partially perturbative scheme, where the hard-sphere contribution to the one-particle correlation function is evaluated using weighted density approximation and the ionic interactions are calculated using a second-order functional Taylor expansion with respect to a bulk electrolyte. The Monte Carlo simulations have been performed in canonical ensemble. The system is studied at varying ionic concentrations, at different concentration ratios of mono- and multivalent counterions of mixed electrolytes, at different diameters of hard spheres, at different macroion radius, and at varying polyion surface charge densities. The theoretical predictions in terms of the density profiles and the mean electrostatic potential profiles are found to be in good agreement with the simulation results. This model study shows clear manipulations of ionic size and charge correlations in dictating a number of interesting phenomena relating to width of the diffuse layer and charge inversion under different parametric conditions. I. Introduction The distribution of the electrolyte around the macroparticles of different types such as micelles, colloids, electrode-electrolyte interface, etc. has been the key ingredient in establishing the structure and thermodynamics of these systems.1 The interfacial region between a charged surface along with the neutralizing diffuse layer consisting of smaller ions is often referred to as the electric double layer (EDL),2,3 which plays an important role in many aspects of interfacial phenomena. EDLs, although extensively studied in different geometries,4-7 are relatively less explored in spherical geometries.8-11 The spherical double layers (SDL) are the most popular model for colloidal solutions12 and are more generally used to represent polyelectrolyte solutions.13 Polyelectrolytes usually exist in spherical conformations because of electrostatic repulsions between their charged groups and thus a charged hard spherical model is sufficiently simple and convenient model to represent them. In the present decade, a considerable amount of work has been carried out by modeling polyelectrolytes as freely jointed chains of charged hard spheres to study their adsorption characteristics at charged interfaces.14 SDLs have also been utilized as a model to study technologically important systems like polymeric and oligomeric tethers in nanoparticles.15 The stability of colloidal solutions and their coagulation properties depend largely on various macroscopic conditions, which has direct relevance in many application areas such as dispersions of globular proteins, dendrimers, polyelectrolytes, aerosols, etc.16 There is a great deal of interest in mixtures of mono- and multivalent electrolytes because several macromolecular and biological phenomena occur just in the presence of mixed salts.17 The structure of SDL with multivalent ion solutions reveals the phenomenon of ionic layering, charge * Also at Homi Bhabha National Institute, Mumbai, India. E-mail: [email protected].

inversion, and charge reversal usually associated with higher valence of ions in concentrated solutions.18,19 The study of SDLs in solutions comprising of small ions with dissimilar valences and realistic sizes appears as an important issue toward the better understanding of phenomena related to the structure and dynamics of macromolecular solutions.20 There exist several levels of modeling of systems involving SDL, the simplest description being the so-called restricted primitive model (RPM),2 where the “colloid” particle (central ion with size ranging in nanometers) is considered as a hard sphere having uniform surface charge density and the small ions are represented by charged hard spheres of equal diameter and the medium is treated as uniform dielectric continuum. The inadequate treatment of the solvent has been recognized as one of the primary limitations of the theoretical treatment of SDL.21 A simple model such as solvent primitive model (SPM), which considers solvent molecules as the uncharged hard spheres in a dielectric continuum, offers a minimum alternative for the understanding of the charge density distribution10 by considering the steric role played by the solvent. However, the detailed structure of SDL can be possibly better understood using all-atom models with explicit water and ions through simulations.22 The present decade has seen a phenomenal growth in theoretical investigations of SDL by integral equation methods,23 modified PB approach,24 and density functional theories (DFT),9,25 revealing interesting phenomena of layering, charge inversion, and charge reversal, as well as overcharging under different parametric conditions. Several versions of DFT26 have been found to be accurate for the structural and thermodynamic properties of SDL9,25 when compared to Monte Carlo simulation results. Although extensive Monte Carlo simulation studies of planar and cylindrical EDLs are reported within the framework of RPM4,5,27,28 as well as SPM,29,30 for SDL the information is scanty.10,23,31 Similarly, a number of different versions of DFT have been applied quite successfully32,33 to planar and cylindrical

10.1021/jp1042975  2010 American Chemical Society Published on Web 07/23/2010

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EDLs, whereas the same is lacking for spherical EDLs. Although a few models of CDL containing mixed electrolytes propose34-36 that multivalent small ions form strongly correlated layers leading to overcharging at the polyion surface, the behavior of SDL in presence of mixed electrolytes has not been addressed so far. In a number of works, the systematic investigation of the structure of SDL in the framework of RPM and SPM through density functional theory and Monte Carlo simulations has been reported quite recently.9,10,25 The presence of multivalent small ions has been shown to influence the characteristics of the double layer (layering, charge inversion, etc.) phenomenally.31,37 Therefore, it would be wise to study such systems where the mixture of mono- and multivalent small ions is added in the electrolyte solution. The density functional formalism and MC simulation method are similar to the already reported studies.10 The theory is partially perturbative, where the hard-sphere contribution is approximated through a weighted density approximation (WDA) using the Denton and Ashcroft recipe38 and the residual Coulombic interaction is calculated perturbatively around a uniform fluid. The second-order direct correlation function required as input is taken from mean spherical approximation (MSA).39 The interplay between the charge correlations and excluded volume effects has been reflected in the density and the mean electrostatic potential profiles. The theoretical predictions are found to be in quantitative agreement with the Monte Carlo simulation data under various macroscopic parameters such as concentration of electrolytes, ionic valences, ionic diamters, macroion radius, and charge density on the macroparticle surface. The rest of the paper is organized as follows: the respective model and the theoretical formalism are described in section II, the numerical results are presented in section III, and some conclusions are offered in section IV. II. Theoretical Formulation A. Molecular Model. The SDL model considered here consists of an isolated charged macroparticle dispersed in a RPM electrolyte solution. The macroparticle is a charged hard sphere of radius R with a uniform surface charge density Q given as

where r is the radial distance of the ion from the macroion surface. Unless otherwise stated, the diameter of the small ions is taken as 0.425 nm, the typical value used in most EDL simulations. B. Density Functional Theory. In DFT for the present system, the grand potential Ω can be expressed as an exact functional of the singlet density profiles, {FR(r)}, viz.

Ω[{FR}] ) F[{FR}] +

∑ ∫ dr FR(r)[UMR(r) - µR] R

(4) where, F[{FR}] is the Helmholtz free energy functional and µR is the chemical potential of ion R. The Helmholtz free energy F[{FR}] for the present system can be decomposed into its respective contributions as40

F[{FR}] ) kBT 1 2

∑ ∫ dr FR(r)[ln(FR(r)λ3R) - 1] + R

∑ ∑ zRzβ ∫ ∫ dr1 dr2 R

β

FR(r1)Fβ(r2) + Fhs ex[{FR}] + |r1 - r2 | Felex[{FR}] (5)

where β0 ) (kBT)-1 represents the inverse temperature, with kB as the Boltzmann constant, and λR is the thermal de Broglie wavelength corresponding to the Rth component. In eq 5, the first term represents the ideal-gas free energy functional Fid, C , and the second term gives the direct Coulomb contribution Fex hs el and Fex , correspond whereas the third and the fourth terms, Fex to the hard-sphere interactions and the electrostatic interactions, respectively. At equilibrium, the grand potential functional attains minimum value with respect to variations in the density profiles, leading to the final expression of ionic density distribution along the radial direction r as

FR(r) ) F0R exp{-β0zRψ(r) + c(1)hs R (r;[{FR}]) Q)

Ze 4πR2

(1)

where e is the electronic charge and Z is the valence of the macroparticle. The electrolyte solution consists of small ions (R) taken as charged hard spheres of equal diameter σ and the solvent is treated as a uniform dielectric continuum with the dielectric constant of water  ) 78.5 at temperature T ) 298 K. The pair potential between the small ions is given as

uRβ(r) ) ∞, )

rσ r

(2)

where zR is the valence of ion R and r is the interionic distance. Similarly, the macroion-ion potential is defined as

UMR(r) ) ∞,

r < R + σ/2 4πR2QzRe ) , r > R + σ/2 r

(3)

0 (1)el (1)el 0 c(1)hs R ([{FR}]) + cR (r;[{FR}]) - cR ([{FR}])} (6)

where ψ(r) is the mean electrostatic potential of the diffused double layer due to the macroion surface charge density and the ionic distributions, given by

ψ(r) )

4πe 

(

2

∫r∞ ∑ zRFR(r′) r′ - r′r R

)

dr′

(7)

(1)el and c(1)hs R (r;[{FR}]) and cR (r;[{FR}]) represent, respectively, the hard-sphere and the electrical contributions to the first-order correlation function. Although eq 6 is formally exact, due to lack of knowledge and c(1)el for the nonuniform SDL system, suitable of c(1)hs R R approximations have to be introduced. In general, two categories of approximations are used, viz., functional perturbative approaches41 and nonperturbative weighted density approaches (WDA),38,42,43 although the mixed forms of the two with different modifications are also plenty.44 The weighted density approximation and its variants have been applied to EDL quite successfully using fully nonperturbative40 as well as partially perturbative6,9,10 approaches. How-

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Figure 1. Ion density profiles around a spherical macroion of R ) 1.5 nm and Q ) 0.102 C m-2 having 1 M bulk concentration of NaCl with added MgCl2 with different [Mg2+]:[Na+] concentration ratios as (a) pure NaCl, (b) 1:8, (c) 1:2, and (d) pure MgCl2. Symbols are simulation results and solid and dashed curves represent DFT and PB predictions, respectively. Empty symbols represent Na+, symbols with cross represent Mg2+, and solid symbols correspond to Cl-.

ever, most of these approaches provide quantitative results when compared to Monte carlo simulation data on density and potential profiles. Hence, in the present work, we resort to using WDA of Denton and Ashcroft (DA)38 to calculate c(1)hs R (x;[{FR}]), where the same has been approximated with that of uniform fluid counterpart, c˜, at a weighted density, FjR(r), i.e.

c(1)hs ˜ (1)hs(F¯ (r)) R (r;[{FR}]) ) c

(8)

are randomly inserted inside the box and made to move through a small distance randomly. The total potential energy of the system, i.e., macroion-ion, UMR, and ion-ion Coulombic interaction, URβ, decides the selection criteria for any move. The final equilibrated density distributions are obtained after total moves of 8 × 108 and final average is done over 5 blocks each having 8 × 108 moves. The acceptance ratio is kept between 0.2 and 0.4 for proper sampling. III. Results and Discussion

where

F¯ (r) )

∫ dr′ ω

hs

(|r - r′|;

F¯ (r))[

∑ FR(r′)]

(9)

R

with whs(r) the weight function obtained through the DA prescription.38 However, the electrical contribution is obtained through perturbation around the bulk fluid given as (1)el 0 c(1)el R (r;[{FR}]) - cR ([{FR}]) ) 2

∑ c˜Rβ(2)el(|r - r′|;

{F0R})(Fβ(r′) - Fβ0 ) (10)

β)1

(2)el taken from the analytical expression within the mean with c˜Rβ spherical approximation (MSA).39 C. Monte Carlo Simulations. Canonical Monte Carlo (CMC) simulations (N,V,T) have been performed using standard Metropolis sampling procedure45 for spherical double layers using ions as charged hard spheres of mixed valences, all of equal diameters. The simulation cell is a cubic box with the macroion fixed at the center of the box. The periodic boundary conditions are employed in all directions. The length of the box is considered sufficiently large to neglect macroion-macroion interactions and to obtain stable density profiles. The number of ions for each species (NR) is adjusted to satisfy the electroneutrality condition ΣRNRzR + Ze ) 0. The small ions

We present a systematic study of the structure of SDL consisting of a spherical macroion surrounded by the electrolyte containing the mixture of mono- and multivalent co-ions. The density profiles of ions in SDL formed from the mixed salt, i.e., 1:2:1 (NaCl/MgCl2) having 1 M 1:1 (NaCl) electrolyte along with varying concentration of 2:1 (MgCl2) salt are shown in Figure 1a-d as obtained from both DFT and MC simulations. The accumulation of Cl- counterions and depletion of both Na+ and Mg2+ co-ions at the macroparticle surface are the obvious representations of attractive interactions between the macroion charges and the counterions and repulsions of the co-ions. The addition of small amount of divalent co-ions [Mg2+] causes the decrease of layering, although the depletion in co-ions remains constant. As can be seen in Figure 1b-d, with increase in concentration of the divalent co-ion, the system tends toward charge inversion, which is completely absent in the nonlinear PB theory. This is due to the interplay of electrostatic interaction from the macroion, the interionic correlation, and the hard-core exclusion by the macroion and the small ions. For higher-valent co-ions, the positive charge on the macroion is effectively screened by large number of counterions, which leads to a point where the co-ion density crosses the counterion density, leading to charge inversion. As expected, higher effective screening of the charge on the macroion also amounts to the damping of the density profiles with increasing valence of the co-ion. However, the concentration of the divalent co-ions does not affect the width of the inversion layer [cf. Figure 1b-d].

Structure of Spherical Electric Double Layers

Figure 2. Mean electrostatic potential profiles around a spherical macroion of R ) 1.5 nm and Q ) 0.102 C m-2 having 1 M bulk concentration of NaCl with added MgCl2 with different [Mg2+]:[Na+] concentration ratios as pure NaCl (9), 1:8 (2), 1:4 ( b), and 1:2 ([). Symbols are simulation results and lines represent DFT predictions.

The mean electrostatic potential (MEP) profiles for systems of varying [Mg2+]:[Na+] ratios are shown in Figure 2. The figure indicates that the contact value of MEP at the polyion surface becomes less on increasing the amount of divalent co-ions (Mg2+) keeping the monovalent co-ion (Na+) concentration constant. This merely corroborates the findings on the density profiles as shown above (cf. Figure 1). The MEP profile decays faster and touches the zero line at much shorter distance from the surface in presence of higher amount of MgCl2. The sign of MEP profile changes from positive to negative (indication of the charge inversion) before coming back to zero line for [Mg2+]:[Na+] ) 1:4 system, which does not happen in case of 1:16 ratio. The depth of the inversion layer becomes larger in case of higher [Mg2+]:[Na+] ratio. In order to see the implications of stronger ion-ion correlations, the bulk concentration of NaCl is varied from 0.01 to 2 M with added MgCl2 at each concentration, while keeping 2:1 (MgCl2) and 1:1 (NaCl) salt concentration ratio as 1:2 (i.e., [Mg2+]:[Na+] ) 1:2). This is depicted in Figure 3, where strong oscillations in density profiles of counterions and co-ions become amply clear at higher concentration. The counterion as

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Figure 4. Mean electrostatic potential profiles around a spherical macroion of R ) 1.5 nm and Q ) 0.102 C m-2 for NaCl/MgCl2 salt with [Mg2+]:[Na+] ) 1:2, having bulk concentration as 0.01 M (9), 0.1 M (2), 1 M (b), and 2 M ([). Symbols are simulation results and solid and dashed curves represent DFT and PB predictions, respectively.

well as the co-ion density profiles at the interface increase. It must be noted that the density profiles shown in figures are normalized with respect to the bulk density and are not the absolute density values. Higher electrolyte concentrations should lead to higher accumulation of ions at the surface; thus, macroion charge is screened more sharply, which results in a narrow diffuse layer. The double layer also becomes structured at high electrolyte concentrations because of volume exclusions due to small ions. At higher electrolyte concentration, charge inversion is quite significant, where the co-ion density profile crosses that of the counterion. At 2 M concentration, even the co-ion density shows a marked increase within the Stern layer. The narrowing down of the diffuse layer as well as the appearance of charge inversion at a higher electrolyte concentration is quite clear in the potential profiles as depicted in Figure 4, which shows that, for lower electrolyte concentration, the damping of density profiles occurs at a longer distance from the interface. Once the inversion sets in, the depth of the inversion layer increases with increasing concentration. The DFT predictions are found to be in quantitative agreement with the simulation results. As

Figure 3. Ion density profiles around a spherical macroion of R ) 1.5 nm and Q ) 0.102 C m-2 for NaCl/MgCl2 salt with [Mg2+]:[Na+] ) 1:2, having bulk concentration as (a) 0.01 M, (b) 0.1 M, (c) 1 M, and (d) 2 M. The key is the same as in Figure 1.

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Figure 5. Ion density profiles for 1 M NaCl/MgCl2 salt with [Mg2+]:[Na+] ) 1:2, around a spherical macroion of R ) 1.5 nm at varying surface charge densities: (a) Q ) 0.102 C m-2, (b) Q ) 0.204 C m-2, (c) Q ) 0.306 C m-2, and (d) Q ) 0.408 C m-2. The key is the same as in Figure 1.

Figure 6. Ion density profiles for 1 M NaCl/MgCl2 salt with [Mg2+]:[Na+] ) 1:2, around a spherical macroion of Q ) 0.102 C m-2 at diifferent macroion radii: (a) R ) 0.5 nm, (b) R ) 1 nm, (c) R ) 1.5 nm, and (d) R ) 6 nm. The key is the same as in Figure 1.

expected, the nonlinear PB theory does not show any such effects due to the neglect of ionic correlations. The effect of variation in the surface charge density on diffuse layer characteristics is studied by varying Q for 1 M 1:2:1 (NaCl/ MgCl2) electrolyte, where the [Mg2+]:[Na+] ratio is kept at 1:2. Figure 5 depicts the density profiles of small ions at four different surface charge densities, viz., at Q ) 0.102, 0.204,, 0.306, and 0.408 C m-2, respectively. The layering of Clincreases at the surface while the depletion of both Na+ and Mg2+ enhances as Q increases. The charge inversion becomes quite stronger in the case of high surface charge density because of stronger electrostatic correlations. The depleted co-ions (Na+ and Mg2+) from the first layer get accumulated in the second layer leading to stronger charge inversions. The depth of the inversion layer increases with increasing surface charge on the macroion, although the layer width remains constant.

To study the charge correlations in SDL, the radius of the macroion in the SDL is varied from 0.5 to 6 nm. Figure 6 depicts the ionic density profiles of 1 M NaCl/MgCl2 mixed electrolyte with [Mg2+]:[Na+] ratio as 1:2 at Q ) 0.102 C m-2 at four different macroion radii, viz., R ) 0.5, 1, 1.5, and 6 nm. For larger macroions, there is a considerable enhancement of counterions and depletion of co-ions at the interface because of substantial increase of value of absolute charge (Z) on the macroion. The concentration profile of counterions decays faster and the co-ion profile grows sharply. It is evident from the graphs that the diffuse layer width goes on decreasing with increase in the radius of the macroion. Also, the extent of charge inversion and depth of the inversion layer increases, although its width remains constant. This trend is also expected from planar double layer which resembles the larger sized macroion case quite well.

Structure of Spherical Electric Double Layers

Figure 7. Ion density (a) and mean electrostatic potential (b) profiles for 1 M NaCl/AlCl3 salt with [Al3+]:[Na+] ) 1:2, around a spherical macroion of R ) 1.5 nm and Q ) 0.102 C m-2. Symbols are simulation results and lines represent theoretical predictions. For density profiles (a), empty symbols represent Na+, symbols with cross represent Al3+, and solid symbols correspond to Cl-.

DFT and Monte Carlo simulation results quantitatively agree with each other. To have an idea of the effect of valence on the behavior of double layer characteristics, the trivalent co-ion (Al3+) is added instead of Mg2+. Thus, Figure 7 shows the density and the potential profiles of 1:3:1 (NaCl/AlCl3) electrolyte with [Al3+]:[Na+] ratio as 1:2 and bulk concentration of 1:1 salt (NaCl) is 1 M. Although the density variations of the counterion (Cl-) and the co-ions (Na+ and Al3+) are not substantial [cf. Figure 1c], the mean electrostatic potential profiles show strong charge inversion in the presence of higher valent co-ion (cf. Figure 2). This is because of the presence of a large amount of counterions which could effectively screen the surface charge on the macroion. This also indicates the decrease of Al3+ density profiles compared to divalent co-ion Mg2+ [cf. Figure 1c]. To investigate the effect of ionic size on the structure of SDL, we present in Figure 8 the local concentration profiles for 1 M 1:2:1 (NaCl/MgCl2) electrolyte, where the [Mg2+]:[Na+] ratio is kept at 1:2, at four different ionic diameters, viz., 0.3, 0.4, 0.5, and 0.6 nm. The counterion density profiles decay monotonically for 0.3 nm diameter while the system shows strong structures in the density profiles for 0.6 nm size. This is a clear indication that larger sized small ions experience more volume exclusion effects contributing strong charge inversion phenomena. That the double layer becomes thinner and more structured with increasing small ion diameter can also be evident in Figure 9 which depicts the mean electrostatic potential profiles for the same systems. The potential profiles also indicate the macroion overcharging; that is, more counterions accumulate in the vicinity of the macroion than is necessary to screen the surface charge. The DFT and MC results again show remarkable consistency at all ionic diameters. The charge layering and the oscillations observed in the ionic density profiles can be better quantified through the integrated charge distribution function P(r) defined as

J. Phys. Chem. B, Vol. 114, No. 32, 2010 10555 which represents the total charge of the macroion and its surrounding ionic clouds within the radius r. It should finally converge to zero at bulk limit to satisfy the electroneutrality condition. Figure 10 shows the integrated charge distribution function P(r) profiles of SDL under different parametrical conditions. The presence of multivalent co-ions (Figure 10, panel a for Mg2+ and panel b for Al3+) leads to efficient charge inversion due to strong charge correlations. Stronger charge correlations is also observed for systems with high surface charge densities which show distinct appearance of charge inversion phenomena (Figure 10c). The larger sized ions (macroion and smal ions) show the charge inversion due to larger volume exclusion leading to effective reduction of diffuse double layer width (Figure 10, panels d and e). In all cases, the nonlinear PB theory is unable to predict the charge inversion due to neglect of ionic correlations. The theory and the simulation results match satisfactorily in all these profiles. The diffuse layer characteristics of a SDL is better understood in terms of zeta potential ζ, which is the value of the mean electrostatic potential at the closest approach to the charged surface, i.e., ζ ) ψ(R + σ/2). The zeta potentials of the SDL formed from the mixed salt, i.e., 1:2:1 (NaCl/MgCl2) having 1 M 1:1 (NaCl) electrolyte along with varying concentrations of 2:1 (MgCl2) salt, is shown in Figure 11 as obtained from both DFT and MC simulations. The magnitude of ζ decreases with increase in concentration of the divalent ions, indicating the narrowing down of the double layer as was ascribed in the density profiles (cf. Figure 1). The variations of bulk concentration as well as small ion diameters are shown in Figure 12, where the system of 1:2:1 (NaCl/MgCl2) electrolyte, with [Mg2+]:[Na+] ratio as 1:2 is chosen at 0.01, 0.1, and 1 M with three small ionic diamaters as 0.3, 0.4, and 0.5 nm. It is quite clear from the graphs that ζ potential decreases with increasing salt concentration for all ionic diameters. Because of overscreening of the macroion charge more effectively, the system with larger sized small ions gives rise to lower ζ potential values. IV. Concluding Remarks The systematics of the structure of spherical electric double layers formed from the mixed electrolytes is studied using density functional theory and Monte Carlo simulations. The present work differs from all the earlier works in the sense that the effect of multivalent ions are seen gradually in the density as well as mean electrostatic potential profiles. The spherical double layer consists of a charged macroparticle in the presence of charged hard spheres of ions within the restricted primitive model in a continuum dielectric solvent. The density functional approach is partially perturbative where hard-sphere contribution is calculated through weighted density approximations and the residual Coulombic contribution is treated as a perturbation around the bulk fluid. The system is studied for varying bulk electrolyte concentrations, concentration ratios of mono- and multivalent co-ions of mixed electrolytes, diameters of components, radii of macroions, and their surface charge densities. The density profiles and the mean electrostatic potential profiles as obtained through theoretical predictions are compared with the Monte Carlo simulation results and found to be in overall agreement in most cases. The interplay between charge correlations and excluded volume effects shows important implications on the double layer properties such as charge inversions and zeta potentials. Thus, the addition of divalent co-ions (Mg2+) to the 1:1 (NaCl) electrolyte system causes the decrease of layering of counterions (Cl-). The phenomena of charge inversion are seen to become

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Figure 8. Ion density profiles for 1 M NaCl/MgCl2 salt with [Mg2+]:[Na+] ) 1:2, around a spherical macroion of R ) 1.5 nm and Q ) 0.102 C m-2 with small ion diameters as (a) σ ) 0.3 nm, (b) σ ) 0.4 nm, (c) σ ) 0.5 nm, and (d) σ ) 0.6 nm. The key is the same as in Figure 1.

Figure 9. Mean electrostatic potential profiles for 1 M NaCl/MgCl2 salt with [Mg2+]:[Na+] ) 1:2, around a spherical macroion of R ) 1.5 nm and Q ) 0.102 C m-2 with small ion diameters as σ ) 0.3 nm (9), σ ) 0.4 nm (2), σ ) 0.5 nm (b), and σ ) 0.6 nm ([). The key is the same as in Figure 2.

P(r) ) Z +

∫0r dr′ ∑ zRFR(r′)

(11)

R

stronger in cases where either the ionic correlations or the excluded volume effects are enhanced. The ionic correlations predominate in the system by increasing [Mg2+]:[Na+] concentration ratio and the bulk ionic concentration, and by increasing the radii of the macroion as well as its surface charge density, while the excluded volume effects are manipulated by increasing the diameters of small ions. The effect of increasing the ionic correlations leads to increased electrostatic attractions between the macroion and the counterion, which in turn causes more effective screening of the surface potential, thus resulting in stronger charge inversions. The excluded volume effects due to small ions pushes all small ions from the bulk toward the surface, thus neutralizing the effective charge on the macroion, thereby causing stronger charge inversions. The mean electrostatic potential profiles corroborate the density profiles in predicting the layering and charge inversion phenomena. The zeta potential values give clear indications about the thickness of the double layer. The theory predicts the density profiles and MEP profiles that are in quantitative agreement with the simulation results under

Figure 10. Integrated charge distribution function profiles for 1 M mixed salt around a spherical macroion under different parametrical conditions: (a) NaCl/MgCl2 salt, [Mg2+]:[Na+] ) 1:2, R ) 1.5 nm, Q ) 0.102 C m-2, σ ) 0.425 nm; (b) same as (a), except NaCl/AlCl3 salt, [Al3+]:[Na+] ) 1:2; (c) same as (a), except σ ) 0.6 nm; (d) same as (a), except Q ) 0.408 C m-2; (e) same as (a), except R ) 60 nm. Symbols are simulation results and solid and dashed curves represent DFT and PB predictions, respectively.

all parametric conditions studied here. The lack of accuracy in certain instances is due to the input taken from the mean spherical approximation that calculates the coupling between hard sphere and Coulombic interactions. The major defect of the present model is the use of of a continuum dielectric solvent, thereby neglecting all the effects coming out of the solvent as an individual component. A good solvent model that accounts for the appropriate relative permittivity is the need of the hour. Addition of neutral solute in the system of mixed electrolytes

Structure of Spherical Electric Double Layers

Figure 11. Zeta potentials around a spherical macroion of R ) 1.5 nm and Q ) 0.102 C m-2 having 1 M bulk concentration of NaCl with added MgCl2 with different [Mg2+]:[Na+] concentration ratios as 1:16 (9), 1:8 (2), 1:4 (b), and 1:2 ([). Symbols are simulation results and lines represent theoretical predictions.

Figure 12. Zeta potentials around a spherical macroion of R ) 1.5 nm and Q ) 0.102 C m-2 for NaCl/MgCl2 salt with [Mg2+]:[Na+] ) 1:2, having bulk concentrations as (a) 0.01 M, (b) 0.1 M, and (c) 1 M, with small ion diameters as σ ) 0.3 nm (9), σ ) 0.4 nm (b), and σ ) 0.5 nm (2). Symbols are simulation results and lines represent theoretical predictions.

to mimic the macromolecular crowding interactions will be an important area of study as this could provide rational explanation of many important phenomena in biophysical systems.46 Since the solution of the MSA equation also exists for size-asymmetric electrolytes,47,48 it would be interesting to study the effect of multivalent counterions on the potential drop and static structure of colloidal solutions formed from such electrolytes giving rise to charge reversal and overcharging.49 At present, work along these directions are in progress in our laboratory and will be reported in the near future. Acknowledgment. The author gratefully acknowledges Swapan K. Ghosh for useful discussions. It is a pleasure to thank Tulsi Mukherjee for his kind interest and constant encouragement. References and Notes (1) Berg, J. C. An Introduction to Interfaces and Colloids: The Bridge to Nanoscience; World Scientific: Singapore, 2009. (2) Carnie, S. L.; Torrie, G. M. AdV. Chem. Phys. 1984, 56, 141. (3) Hansen, J. P.; Lo¨wen, H. Annu. ReV. Phys. Chem. 2000, 51, 209.

J. Phys. Chem. B, Vol. 114, No. 32, 2010 10557 (4) Torrie, G. M.; Valleau, J. P. J. Chem. Phys. 1980, 73, 5807; J. Phys. Chem. 1982, 86, 3251. (5) Mills, P.; Anderson, C. F.; Record, M. T., Jr. J. Phys. Chem. 1985, 89, 3984. (6) Mier-y-Teran, L.; Suh, S. H.; White, H. S.; Davis, H. T. J. Chem. Phys. 1990, 92, 5087. (7) Patra, C. N.; Ghosh, S. K. J. Chem. Phys. 2002, 117, 8938. (8) Gonza´lez-Tovar, E.; Jimenez-Angeles, F.; Messina, R.; LozadaCassou, M. J. Chem. Phys. 2004, 120, 9782. (9) Yu, Y.-X.; Wu, J.; Gao, G.-H. J. Chem. Phys. 2004, 120, 7223. (10) Patra, C. N. J. Phys. Chem. B 2009, 113, 13980. (11) Guerrero-Garcia, G. I.; Gonzalez-Tovar, E.; Lozada-Cassou, M.; de J. Guevara-Rodriguez, F. J. Chem. Phys. 2005, 123, 034703. (12) Schmitz, K. S. Macroions in Solution and Colloidal Suspension; VCH: New York, 1993. (13) Hara, M. Polyelectrolytes: Science and Technology; Dekker: New York, 1993. (14) Yethiraj, A. J. Phys. Chem. B 2009, 113, 1539. (15) Smagala, T. G.; Patrykiejew, A.; Pizio, O.; Fawcett, W. R.; Sokolowski, S. J. Chem. Phys. 2008, 128, 024907. (16) Highlights in Colloid Science; Platikanov, D., Exerowa, D., Eds.; Wiley-VCH: Weinheim, Germany, 2008. (17) Gelbart, W. M.; Bruinsma, R. F.; Pincus, P. A.; Parsegian, V. A. Phys. Today 2000, 53, 38. (18) Gonza´lez-Tovar, E.; Lozada-Cassou, M. J. Phys. Chem. 1989, 93, 3761. (19) Bhuiyan, L. B.; Vlachy, V.; Outhwaite, C. W. Int. ReV. Phys. Chem. 2002, 21, 1. (20) Khokhlov, A. R.; Grosberg, A. Y. Statistical Physics of Macromolecules; AIP: New York, 1994. (21) Minton, A. P. Curr. Biol. 2006, 16, R269. (22) Mittal, J.; Best, R. B. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 20233. (23) Degre`ve, L.; Lozada-Cassou, M.; Sa´nchez, E.; Gonza´lez-Tovar, E. J. Chem. Phys. 1993, 98, 8905. (24) Bhuiyan, L. B.; Outhwaite, C. W. Condens. Matter Phys. 2005, 8, 287. (25) Goel, T.; Patra, C. N. J. Chem. Phys. 2007, 127, 034502. (26) Evans, R. In Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Dekker: New York, 1992. (27) Boda, D.; Fawcett, W. R.; Henderson, D.; Sokolowski, S. J. Chem. Phys. 2002, 116, 7170. (28) Goel, T.; Patra, C. N.; Ghosh, S. K.; Mukherjee, T. J. Chem. Phys. 2008, 129, 154906. (29) Zhang, L.; Davis, H. T.; White, H. S. J. Chem. Phys. 1993, 98, 5793. (30) Goel, T.; Patra, C. N.; Ghosh, S. K.; Mukherjee, T. J. Chem. Phys. 2008, 129, 154707. (31) Degre`ve, L.; Lozada-Cassou, M. Mol. Phys. 1995, 86, 759. (32) Gillespie, D.; Nonner, W.; Eisenberg, R. S. J. Phys.: Condens. Matter 2002, 14, 12129. (33) Reszko-Zygmunt, J.; Sokolowski, S.; Henderson, D.; Boda, D. J. Chem. Phys. 2005, 122, 084504. (34) Patra, C. N.; Yethiraj, A. J. Phys. Chem. B 1999, 103, 6080. (35) Ni, H.; Anderson, C. F.; Record, M. T., Jr. J. Phys. Chem. 1985, 89, 3984. (36) Patra, C. N.; Yethiraj, A. Biophys. J. 2000, 78, 699. (37) Terao, T.; Nakayama, T. Phys. ReV. E 2001, 63, 041401. (38) Denton, A. R.; Ashcroft, N. W. Phys. ReV. A 1991, 44, 8242. (39) Waisman, E.; Lebowitz, J. L. J. Chem. Phys. 1972, 56, 3086; 1972, 56, 3093. (40) Patra, C. N.; Ghosh, S. K. Phys. ReV. E 1993, 47, 4088. (41) Denton, A. R.; Ashcroft, N. W. Phys. ReV. A 1989, 39, 426. (42) Tarazona, P. Phys. ReV. A 1985, 31, 2672. (43) Rosenfeld, Y. Phys. ReV. Lett. 1989, 63, 980. (44) Choudhury, N.; Patra, C. N.; Ghosh, S. K. J. Phys.: Condens. Matter 2002, 14, 11955. (45) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1953, 21, 1087. (46) Ellis, R. J. Trends Biochem. Sci. 2001, 26, 597. (47) Hiroike, K. Mol. Phys. 1977, 33, 1195. (48) Li, Z.; Wu, J. Phys. ReV. E 2004, 70, 031109. (49) Guerrero-Garcia, G. I.; Gonza´lez-Tovar, E.; Chavez-Paez, M. M. J. Chem. Phys. 2010, 132, 054903.

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