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Overcharging of Nanoparticles in Electrolyte Solutions Sathyajith Ravindran and Jianzhong Wu* Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521 Received March 11, 2004. In Final Form: May 22, 2004 Monte Carlo simulations are performed to investigate the effects of salt concentration, valence and size of small ions, surface charge density, and Bjerrum length on the overcharging of isolated spherical nanoparticles within the framework of a primitive model. It is found that charge inversion is most probable in solutions containing multivalent counterions at high salt concentrations. The maximum strength of overcharging occurs near the nanoparticle surface where counterions and coions have identical local concentrations. The simulation results also suggest that both counterion size and electrostatic correlations play major roles for the occurrence of overcharging.
I. Introduction Charge inversion or overcharging is an electrostatic phenomenon frequently encountered in colloids or polyelectrolyte solutions in which a macroion adsorbs an excess amount of counterions such that the overall charge of the macroion along with its surrounding smaller ions has a sign opposite to its bare charge. This nonintuitive phenomenon was first reported a long time ago by Bungenberg de Jong1 and more extensively by Strauss et al.2 observing the reversal of the electrophoretic mobilities of highly charged macroscopic colloidal aggregates or polysoaps. The charge inversion phenomenon is not supported by the classical electrostatic theories based on the PoissonBoltzmann (PB) equation, which ignores counterion size and the correlation of small ion distributions. According to the PB theory, counterions can only partially neutralize a macro charge by electrostatic screening. There had been erroneous speculations that the reversal of charges was probably due to some binding forces between counterions and the macroion other than the Coulombic interactions.2 The phenomenon of charge inversion regains substantial research interests in recent years for both scientific and technical considerations. On the theoretical side, the limitations of the traditional PB approaches for representing various electrostatic phenomena in colloids and biological systems become ever-increasingly evident in comparison to the results from molecular simulations and experiments. More advanced statistical-mechanical theories are required for quantitative and sometimes even qualitative interpretation of novel electrostatic behavior such as attraction between similar charges and charge inversion. Theoretical efforts to capture the overcharging phenomenon beyond the PB-like approaches were recently reviewed in two excellent accounts from different perspectives but both emphasizing the correlations among counterions in the vicinity of the macroion surface.3,4 One concludes that the correlation of counterions at the surface * To whom correspondence should be addressed. E-mail: jwu@ engr.ucr.edu. (1) Bungenberg de Jong, M. G. In Colloid Science; Kruyt, H. R., Ed.; Elsevier: New York, 1949; p 335. (2) Strauss, U. P.; Gershfeld, N. L.; Spiera, H. J. Am. Chem. Soc. 1954, 76, 5909-5911. (3) Grosberg, A. Y.; Nguyen, T. T.; Shklovskii, B. I. Rev. Mod. Phys. 2002, 74, 329-345. (4) Quesada-Perez, M.; Gonzalez-Tovar, E.; Martin-Molina, A.; Lozada-Cassou, M.; Hidalgo-Alvarez, R. ChemPhysChem 2003, 4, 235248.
of a highly charged macroion, which resembles the cohesive energy of a highly correlated two-dimensional fluid, leads to the attraction of extra counterions to the surface.3 Conversely, the other approach, based on the integral equation theory of electrolyte solutions, is focused on the spatial correlations between small ions at the charged surface and beyond.4 In recent years, charge inversion was also extensively studied using computer simulations at planar charged surfaces,5,6 cylindrical rods,7-9 and spheres.10-13 Effects of macroion geometry on the overcharging were also investigated using a novel random sampling technique.14 Simulation results indicate that the correlation and the excluded volume effects of counterions are the prime reasons of charge inversion. On the other hand, charge inversion is also of current interest from a technical point of view because this phenomenon has been widely utilized for the delivery of therapeutic biomacromolecules,15,16 immobilization of biosensors,17,18 and layer-by-layer fabrication of materials.19 In addition, there has been extensive experimental work on the complexation of DNA with positively charged liposomes20,21 and with cationic protein aggregates22,23 and on the (5) Denton, A. M.; Lowen, H. Thin Solid Films 1998, 330, 7-13. (6) Greberg, H.; Kjellander, R. J. Chem. Phys. 1998, 108, 29402953. (7) Deserno, M.; Jimenez-Angeles, F.; Holm, C.; Lozada-Cassou, M. J. Phys. Chem. B 2001, 105, 10983-10991. (8) Montoro, J. C. G.; Abascal, J. L. F. J. Chem. Phys. 1995, 103, 8273-8284. (9) Das, T.; Bratko, D.; Bhuiyan, L. B.; Outhwaite, C. W. J. Chem. Phys. 1997, 107, 9197-9207. (10) Messina, R.; Holm, C.; Kremer, K. Comput. Phys. Commun. 2002, 147, 282-285. (11) Messina, R.; Gonzalez-Tovar, E.; Lozada-Cassou, M.; Holm, C. Europhys. Lett. 2002, 60, 383-389. (12) Tanaka, M.; Grosberg, A. Y. J. Chem. Phys. 2001, 115, 567574. (13) Terao, T.; Nakayama, T. Phys. Rev. E 2001, 63, 041401. (14) Mukherjee, A. K.; Schmitz, K. S.; Bhuiyan, L. B. Langmuir 2003, 19, 9600-9612. (15) Trimaille, T.; Pichot, C.; Delair, T. Colloids Surf., A 2003, 221, 39-48. (16) Dinsmore, A. D.; Hsu, M. F.; Nikolaides, M. G.; Marquez, M.; Bausch, A. R.; Weitz, D. A. Science 2002, 298, 1006-1009. (17) Nguyen, Q. T.; Ping, Z. H.; Nguyen, T.; Rigal, P. J. Membr. Sci. 2003, 213, 85-95. (18) Rossi, S.; Lorenzo-Ferreira, C.; Battistoni, J.; Elaissari, A.; Pichot, C.; Delair, T. Colloid Polym. Sci. 2004, 282, 215-222. (19) Decher, G. Science 1997, 277, 1232-1237. (20) Radler, J. O.; Koltover, I.; Salditt, T.; Safinya, C. R. Science 1997, 275, 810-814. (21) Koltover, I.; Salditt, T.; Radler, J. O.; Safinya, C. R. Science 1998, 281, 78-81.
10.1021/la0493619 CCC: $27.50 © 2004 American Chemical Society Published on Web 07/24/2004
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overscreening of clay surfaces.24 Charge inversion is also frequently encountered in the adsorptions of multivalent small ions and polyelectrolytes on oppositely charged micelles, latex, and proteins.25-27 Although most practical applications of charge inversion are concerned with the adsorption of polyeletrolytes or highly charged biomacromolecules and the phenomena are necessarily more complicated, the physics underlying the charge reversal phenomena resembles that in relatively simple systems involving structureless small ions and charged surfaces.28 All previous theoretical investigations on charge inversion have been primarily focused on macroions bearing a uniform surface charge that is independent of solution conditions. However, realistic colloidal particles and biomacromolecules have discrete charges at the surface, and the charge density is sensitive to solution pH and local salt conditions. Using molecular dynamics simulations, Allahyarov et al. recently reported that the discrete nature of macroion charges may lead to a striking nonmonotonic variation of the osmotic second virial coefficient with salt concentration. Such nonmonotonic behavior is not captured within a smeared charge model.29 These simulation results provide a convincing explanation to some long-standing experimental observations pertaining to the effect of salt concentration on the osmotic second virial coefficients, solubility, and cloud point temperature of protein solutions. In this work, Monte Carlo simulations are applied to investigating the effects of charge discreteness at the surface of isolated nanoparticles on the distributions of neutralizing small ions and charge inversion. In considering that the charge of each ionizable group at the nanoparticle surface varies with the solution conditions and it is influenced by the local environment of small ions, we allow the discrete charges to migrate at the nanoparticle surface. The migrating discrete charges are strongly correlated not only within the nanoparticle surface but also with the neutralizing counterions. Such correlations may result in a drastic effect on the small ion distributions and, thereby, charge inversion. II. Model and Simulation Method We consider a negatively charged nanoparticle dispersed in a primitive model electrolyte solution as shown schematically in Figure 1. Both counterions and coions in the solution are represented by hard spheres embedded with fixed central charges, and the solvent is represented by a dielectric continuum. The diameter of the charged nanoparticle is assumed to be σN ) 20 Å in all simulations performed in this work, and the charges are represented by electronic units distributed on a spherical shell right underneath the nanoparticle surface. The diameter of each unit charge is assigned as σu ) 1 Å. The unit charges on the nanaparticle surface are treated as mobile, and their distribution is correlated with the local densities of small ions. The electrostatic interactions between unit charges (22) Grunstein, M. Nature 1997, 389, 349-352. (23) Luger, K.; Mader, A. W.; Richmond, R. K.; Sargent, D. F.; Richmond, T. J. Nature 1997, 389, 251-260. (24) Kekicheff, P.; Marcelja, S.; Senden, T. J.; Shubin, V. E. J. Chem. Phys. 1993, 99, 6098-6113. (25) Radeva, T. Physical chemistry of polyelectrolytes; Marcel Dekker: New York, 2001. (26) Ladam, G.; Schaad, P.; Voegel, J. C.; Schaaf, P.; Decher, G.; Cuisinier, F. Langmuir 2000, 16, 1249-1255. (27) Paton-Morales, P.; Talens-Alesson, F. I. Langmuir 2002, 18, 8295-8301. (28) Netz, R. R.; Andelman, D. Phys. Rep. 2003, 380, 1-95. (29) Allahyarov, E.; Lowen, H.; Louis, A. A.; Hansen, J. P. Europhys. Lett. 2002, 57, 731-737.
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Figure 1. (a) Nanoparticle with migrating charges at the surface. (b) Cubic simulation box with one nanoparticle fixed at the center immersed in an electrolyte solution within the primitive model.
are accounted for explicitly. Because these unit charges are strongly repulsive to each other, a small variation of σu will not significantly alter the results as long as the size of the unit charge remains much smaller than the nanoparticle itself, and the excluded volume effect does not affect the charge distributions on the nanoparticle surface. Table 1 lists the solution and model parameters for various simulations performed in this work. In this Table, C stands for the molar concentration of small ions; N+ and N- are, respectively, the numbers of cations and coions employed in the simulation; L is the length of the cubic simulation box; σM is the diameter of microions (i.e., counterions and coions); ZN is the valence of the nanoparticle; and κ-1 is the Debye screening length. The Bjerrum length, lB, represents the separation between two unit charges where the pair electrostatic potential is equal to the thermal energy kT. All Monte Carlo simulations were performed in the NVT ensemble where the number of particles, volume, and the temperature of the system are constant. The cubic simulation box (Figure 1) contains a single nanoparticle fixed at the center and small counterions and coions. The condition of electrostatic neutrality is satisfied for all systems. To compensate the charge of the central nanoparticle, the overall charge density of cations is necessarily larger than that of coions. Periodic boundary conditions
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Table 1. Parameters for Various Simulation Runs Performed To Calculate the Radial Distributions of Small Ions around a Nanoparticle with Migrating Charges on the Surface run salt A B C D E F G H I J K L M N
2:2 2:2 2:2 2:2 2:2 2:2 2:2 2:2 2:2 2:2 2:2 1:2 1:1 2:1
N+
N-
C (M) lB (Å) κ-1 (Å) L (Å) σM (Å)
740 740 740 740 220 220 220 220 220 220 220 220 220 220
730 730 730 730 215 210 205 210 210 210 210 100 200 420
2.4 7.14 0.71 7.14 1.23 7.14 0.154 7.14 0.71 7.14 0.71 7.14 0.71 7.14 0.71 7.14 0.71 7.14 0.71 5 0.71 10 0.71 7.14 0.71 7.14 0.71 7.14
0.98 1.81 1.37 3.88 1.81 1.81 1.81 1.81 1.81 2.16 1.53 2.08 3.61 2.08
80 120 100 200 80 80 80 80 80 80 80 80 80 80
4 4 4 4 4 4 4 2 6 4 4 4 4 4
ZN -20 -20 -20 -20 -10 -20 -30 -20 -20 -20 -20 -20 -20 -20
are applied to each direction of the simulation cell. To mimic the dispersion of an isolated nanoparticle in a bulk solution, we choose the simulation box at least one order of magnitude larger than the Debye screening length such that the correlation between nanoparticles in different image boxes is assumed negligible. The interaction potential between a pair of migrating unit charges at the nanoparticle surface is given by
{
e2 r g σu φuu(r) ) 4π0r ∞ otherwise
(1)
where r is the center-to-center distance, e ) -1.602 × 10-19 C is the electronic charge, 0 ) 8.854 × 10-12 C2/(J m) is the permittivity of free space, and the dielectric constant is assumed the same as that of the solvent (otherwise there will be image charges). A similar expression is applied for interactions between microions in the solution. Because the discrete unit charges are free to move on the nanoparticle surface, their distribution is strongly correlated with the local density of small ions. The total potential between the fixed nanoparticle and a microion of charge qM and diameter σM is given by
φMN(r) )
{
eqM
1
∑ 4π R r ∞
0
r g σMN ) (σM + σN)/2
Ri
(2)
otherwise
where r is the center-to-center distance between the nanoparticle and a microion and rRi is the distance between a unit charge R and a microion i. Because all charges are explicitly considered, no screening is applied to the interactions between microions or between a microion and the charges located on the nanoparticle surface. The Ewald sum method is used to take into account the long-range electrostatic interactions introduced as a result of the periodic boundary conditions. The system is allowed to reach equilibrium by making trial moves of microions and the unit charges at the nanoparticle surface following the standard Metropolis algorithm. In the course of simulations, a microion or a unit charge of the nanoparticle is randomly chosen, and it is subjected to an infinitesimally small displacement. In the case of a microion, a small linear displacement is given along all three coordinates. When a unit charge is chosen, it is displaced by a small angle along the nanoparticle surface calculated using Euler’s method.30 In this technique we generate three angles R, β, and γ uniformly distributed over 0 to 2π using
a random number generator. Then the unit charge on the nanoparticle is rotated with respect to the center of the nanoparticle. The move is accepted or rejected on the basis of the difference in energy of the original and the trial microstates. If the energy of a trial configuration is lower than that of the original configuration, the Monte Carlo move is always accepted; otherwise, a random number is generated and the move is accepted only if it is larger than the ratio of the Boltzmann factors of the trial and the original configurations. The number of microions in each simulation varies from approximately 400 up to 1500 particles. By monitoring the total energy of the system, we find that the system attains equilibrium after approximately 5 × 105 simulation steps per particle. To calculate the radial distributions of the counterions and coions, we divide the simulation box into approximately 250 concentric shells of equal thickness. The numbers of counterions and coions in each shell are counted during the simulations, and they are averaged over 107 Monte Carlo moves. The average densities of counterions and coions are calculated by dividing these numbers with the shell volume. III. Results and Discussion The radial distributions of small ions near each fixed nanoparticle were calculated at various concentrations of the electrolyte, valence and size of small ions, the surface charge, and the Bjerrum length. The key quantity reflecting charge inversion is the integrated charge distribution function defined as4
P(r) ) -ZN +
∑i ∫σ
r MN
ziFi(r)4πr2 dr
(3)
where ZN is the number of unit charges of the nanoparticle and zi stands for the valence of a counterion or coion. Even though the system is overall electrically neutral, the local charge depends on the densities of the counterions and coions. At the surface of the fixed nanoparticle, P(r ) σMN) is equal to the total number of unit charges for the particle, and the absolute value of P(r) decreases as the surface charge is neutralized by adsorbing counterions and approaches 0 far from the surface (bulk). Charge inversion occurs when P(r) changes the sign from negative to positive at some intermediate separations. Figure 2 shows the integrated charge distribution function P(r) for run A. In this case, the Debye screening length is κ-1 ) 0.98 Å, much smaller than the size of microions (σM ) 4 Å). According to a classical electrostatic theory, the electrostatic interactions should be insignificant for this case. However, that is not true as shown by Monte Carlo simulations. While P(r) approaches 0 when r f ∞ as expected, it overshoots the neutralization line (or isoelectric line) at r/σM ≈ 3.25 and reaches a maximum at r/σM ≈ 3.7. At larger separations, the integrated charge profile exhibits an oscillatory behavior indicating the existence of alternating counterion- and coion-dominated adsorption layers. In particular, the overall charge within the shaded region is opposite to the bare charge of the nanoparticle, indicating a charge inversion. The charge inversion does not occur exactly at the nanoparticle surface as speculated before. Instead, it shows a maximum reversal charge at a position where the counterions and coions have the same number density. This is most clearly depicted in Figure 3, which presents the radial density (30) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: New York, 1989.
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Figure 4. Effect of salt concentration on charge inversion. The inset represents Pmax as a function of salt concentration (runs A-D).
Figure 2. Integrated charge distribution for run A. The shaded area indicates that the overall charge of the nanoparticle along with the neutralizing counterions has a sign opposite to the bare charge of the nanoparticle.
Figure 5. Overcharging in electrolyte solutions of different valences (runs E, L, M, and N).
Figure 3. Density distributions of the counterions (solid line) and coions (dotted line) for run A. Here, the density profiles are in the units of 1/lB3.
distributions of the counterion (squares) and the coion (triangles). At the position where the counterions and coions have the same number density, the overall charge of the nanoparticle and its surrounding small ions is opposite to the nanoparticle bare charge. As the distance is further increased, the additional adsorption of coions diminishes the charge inversion. Figure 3 also indicates that near the nanoparticle surface, the distributions of small ions are significantly asymmetric. Different from the predictions of the PB theory, the distributions of both counterions and coions clearly exhibit nonmonotonic behavior. The two density profiles cross at r/σM ≈ 3.7, corresponding to the position of maximum charge reversal as shown in Figure 2. At larger separations, the local concentration of coions becomes larger than that for the
counterion. As a result, the integrated charge distribution declines. Figure 4 shows the effect of electrolyte concentration on the integrated charge distributions (runs A-D). The integrated charge densities exhibit oscillatory behavior at high salt concentrations but become monotonic when the concentration is below 0.154 M. In this case, no charge inversion is observed. Figure 4 indicates that charge inversion is more probable at high salt concentrations. The inset of Figure 4 presents the maximum value of the integrated charge distribution as a function of salt concentration. It shows that the magnitude of charge inversion increases monotonically with the concentration of the electrolyte. However, the layer thickness decreases with the solution concentration; that is, the first peak of the integrated charge distribution becomes sharper as the salt concentration increases. Previous investigations point toward electrostatic correlations and the excluded volume effects as the main causes of charge inversion.6 To investigate these two different aspects separately, we simulate the charge distributions near the nanoparticle at different valences and sizes of small ions and at different Bjerrum lengths. Figure 5 presents the integrated charge distribution function P(r) for a nanoparticle immersed in 1:1, 1:2, 2:1, and 2:2 electrolytes (runs E, L, M, and N). All these electrolyte solutions have the same concentration. We find that there is essentially no charge inversion in the 1:1 and the 1:2 electrolytes where the counterions are monovalent. However, in solutions containing divalent counterions as in the 2:1 and the 2:2 electrolytes, significant charge inversion appears. Figure 5 suggests that charge inversion is directly related to the strong
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Figure 7. Integrated charge distributions for different microion sizes (runs E, H, I).
Figure 6. Effects of Bjerrum length on charge inversion. lB ) 7.14 Å corresponds to an aqueous system at an ambient condition (runs E, J, and K).
interactions between the counterions as shown for the electrostatic attraction between two similarly charged nanoparticles.31-33 In addition to stronger electrostatic interactions, the entropic penalty associated with localization of multivalent counterions is much smaller than that for monovalent counterions. This is because the number of multivalent ions required to neutralize the charged particle is smaller than that for the monovalent counterions while the entropy loss for the immobilization of each ion depends only on density. Even though Figure 5 shows no sign of overcharging when the solution contains only monovalent counterions, a recent density functional theory calculation predicts the appearance of overcharging even in monovalent electrolyte solutions as long as the salt concentration is sufficiently high.34 The integral equation and molecular dynamics study of Deserno et al. also indicated the possibility of overcharging caused by large monovalent ions.7 We use the Bjerrum length, defined by lB ) e2/(4π0kT), as an input in Monte Carlo simulations to control the strength of electrostatic interactions. In real experiments, this parameter can be controlled by changing the temperature or the dielectric constant of the solvent. The change of Bjerrum length investigated in this work corresponds to that due to the latter approach. A larger Bjerrum length (or a smaller electrostatic constant) means stronger electrostatic interactions among nanoparticles and surrounding small ions. Figure 6 shows the charge inversion for systems with three different Bjerrum lengths (runs E, J, and K), approximately corresponding to those for the ionic species dispersed in a solvent of N-propylpropanamide (lB ) 4.74 Å), water (lB ) 7.14 Å), and dimethyl sulfate (lB ) 10.19 Å) at ambient conditions. Figure 6 clearly indicates that a higher Bjerrum length introduces stronger charge inversion. Both Figures 5 and 6 suggest that charge inversion is most likely in systems with strong electrostatic interactions. (31) Wu, J. Z.; Bratko, D.; Prausnitz, J. M. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 15169-15172. (32) Wu, J. Z.; Bratko, D.; Blanch, H. W.; Prausnitz, J. M. J. Chem. Phys. 1999, 111, 7084-7094. (33) Linse, P.; Lobaskin, V. Phys. Rev. Lett. 1999, 83, 4208-4211. (34) Yu, Y. X.; Wu, J. Z.; Gao, G. H. J. Chem. Phys. 2004, 120, 72237233.
Figure 8. Effect of surface charge density of nanoparticles on charge inversion (runs E-G).
The effect of excluded volume on overcharging may be studied by changing the size of small ions while keeping all other conditions fixed. Figure 7 shows the integrated charge distributions in solutions containing microions of different sizes (runs E, H, and I). We observed that charge inversion is more significant for microions of larger size. This implies that charge inversion at higher salt concentrations as shown in Figure 4 is likely attributed to the excluded volume effects. Figure 7 also indicates that larger counterions are significantly more efficient in shielding the charge effects of nanoparticles. Qualitatively, the two major mechanisms behind the overcharging, that is, the electrostatic correlations and the excluded volume effect as shown in Figures 5-7, are consistent with a previous investigation by Greberg and Kjellander for the charge inversion in a planar electric double layer.6 Finally, Figure 8 presents charge inversion for nanoparticles with different surface charge densities (runs E-G). Here, the charge densities are expressed in terms of ZN/σN2. It shows that the charge inversion becomes more significant as the surface charge of the nanoparticle increases. This can be explained by the fact that an increase in nanoparticle charge attracts more counterions near the surface, introducing stronger electrostatic and excluded-volume correlations.
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IV. Conclusions Monte Carlo simulations have been applied to studying the ionic density profiles and integrated charge distribution functions near nanoparticles bearing mobile unit charges at a variety of solution conditions. In general, the results are qualitatively similar to those corresponding to a smeared charged model or discrete but fixed charged model. We find that charge inversion is most likely in electrolyte solutions of high ionic strength. Upon the increase of salt concentration, the magnitude of inversed charges almost linearly increases while the thickness of the first adsorption layer declines. The charge inversion phenomenon can be attributed to two main factors, namely, the strength of electrostatic potential and the
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size of counterions. At similar solutions conditions, charge inversion occurs in solutions containing multivalent counterions and the magnitude of reversal charge increases with the counterion size. As for the interactions between similarly charged nanoparticles, the counterions play a dominant role in overcharging. Charge inversion becomes more likely as the Bjerrum length or the charge density of the nanoparticle surface increases. Acknowledgment. This project was financially supported by a grant from the University of California Energy Institute. LA0493619
