9642
J. Phys. Chem. 1995, 99, 9642-9645
Energetics of Colloids: Do Oppositely Charged Particles Necessarily Attract Each Other? Nir Ben-Tal Department of Biochemistry and Molecular Biophysics and Center for Biomolecular Simulations, College of Physicians and Surgeons, Columbia University, 630 West 168th St., New York, New York 10032 Received: March 27, 1995@
The electrostatic free energy of two oppositely charged macroions, suspended in ionic solution is calculated for different distances between the macroions. This is done by using a lattice field theory formulation of the statistical mechanics of Coulomb gas particles of finite size interacting with a fixed charge distribution ( J . Chem. Phys. 1995, 102,4584). In many cases, it is found that, regardless of the size of the mobile ions, the minimum of the free energy is at a separation between the macroions corresponding to a noncontact configuration. Illustrative examples, which emphasize the source of the repulsion at near-touching configurations are presented and discussed.
The intriguing experimental study of oppositely charged colloidal particles,’ the development of novel theoretical approaches to these and a thorough examination of the need for a long-range attraction-repulsion rather than the commonly used repulsion-only potential between negatively charged macro ion^,^ all of which have emerged during the last couple of years, are drawing the attention of the scientific community to colloid sciences. Since the beginning of colloid sciences in the early 1800s the vast majority of systems of interacting macroions which have been studied (both experimentally and the~retically)~,’ have been composed of macroions with the same sign charge. For fluids and crystals of charged polystyrene spheres (“polyballs”), for example, the case with all polyballs negatively charged has received by far the most attention. These systems are relatively easy to prepare. In addition, like charges on all the polyballs keep these macroions from aggregating. Indeed, these systems can, under appropriate conditions (mainly associated with the ionic strength of the aqueous environment), order into stable crystalline arrays. Nevertheless, positively charged polyballs have been synthesized, and the possibility that under appropriate conditions a stable & polyball system can be prepared is currently under study.’ It certainly motivates more detailed studies of the interactions between oppositely charged macroions suspended in ionic solutions. Recently we used a lattice field theory (LlT) formulation of the statistical mechanics of interacting macroions to carry out a theoretical study of the effect of the dielectric constant on the energetics of oppositely charged p ~ l y b a l l s . ~ ~ Throughout our study, simple (mobile) ions were assumed to be infinitesimal. The most important (perhaps surprising) result of our study was that in many cases the minimum of the free energy was at a separation between the polyballs corresponding to a noncontact rather than a near-touching configuration. Very recently the LFT formalism has been generalized to treat mobile ions having a finite The present letter extends our theoretical studies of systems of two oppositely charged polyballs, having different charges and dielectric constants, with different ionic strengths for mobile ions which have finite sizes. The formalism will be briefly introduced, followed by results of our calculations. The results, which will be presented, emphasize the physical origin of the repulsive forces between oppositely charged polyballs. A full statistical mechanical treatment for the electrostatics @
Abstract published in Advance ACS Abstracts, June 1, 1995.
of systems containing charged macromolecules in salt solutions, carried out recently, yielded a closed formula for calculating the total electrostatic energy of such systems for each distance between the macrom~lecules.~ In addition to the Coulombic interaction, a Yukawa repulsive potential was added to account for the ions’ “self-volume”. Following a Bom-Oppenhimerlike approach, each mobile ion (which is able to move fast due to its relatively small-though not infinitesimal-size) moves in the electrostatic field caused by the macromolecules and in the repulsive field caused by the presence of its neighboring ions, and adjusts itself instantaneously to each orientation of the (relatively big, hence, slow) macromolecules. The Helmholtz free energy, A, of a particular macromolecular configuration is calculated as
where n+ (n-) is the total number of positive (negative) mobile ions, y+ ( y - ) is the activity coefficient of the positive (negative) mobile ions, and S({&},{&}) is a function of the electrostatic, &, and the repulsive Yukawa, &, fields, respectively. A meanfield (Poisson-Boltzman) approximation for the “action” function, S, is given by the fields, {&} and {&}, which minimize its value:4d
r
where a = ca/4z/3e2,ti = a/45cpe2 ( E is the dielectric constant of water, a is the lattice step, /3 = (kBT)-’, and e is the proton charge), and K’ and C are parameters which determine a Yukawa repulsive potential: K’ 3 Ka ( K is the characteristic repulsion length) and C is the characteristic repulsion strength. Ql is the charge on the macromolecules, and is a single-particle potential which is zero everywhere but large and positive on the macromolecules in order to prevent mobile ions from penetrating into the macromolecules. Ax%and &, are finite difference matrix representations of the Laplacian operator for spatially constant and spatially varying dielectrics, respectively.*
0022-3654/95/2099-9642$09.00/0 0 1995 American Chemical Society
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J. Phys. Chem., Vol. 99, No. 24, 1995 9643
The fields {&} and {&} are than determined by the conditions asla& = 0 and asla& = 0, or more specifically
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-v;
-10 J
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(3b)
which are Poisson-Boltzmann type equations. The activity coefficients yi, are determined from the prescribed number of mobile ions, n+, by the thermodynamic relation: r
(4)
Equations 3a and 3b were solved to a preset accuracy by a simple annealing technique which was described in refs 4b-d. The activity coefficients were adjusted every few iterations until self-consistent electrostatic and Yukawa repulsive fields were obtained. These fields and the self-consistent activity coefficients were used to calculate the mean-field approximation for the electrostatic contribution for the Helmholtz free energy via eqs 1 and 2. Higher order “loop corrections” for the mean field level (Le., fluctuations) can also be calculated. However, in polyball systems tested to date, these have tumed out to be negligibly mall.^^,^ We studied systems composed of two spherical polyballs (having different charge distributions and dielectric constants), in ionic solutions of different strengths. We assume that the relatively short range of the electrostatic forces in ion solutions enable us to extend our conclusions to system of many interacting polyballs. This presumption was verified earlier.& The two polyballs were placed on a 363 cubic lattice characterized by a spacing of a = 0.05 pm. Each polyball was constructed from elemental “polycubes” (one lattice spacing on a side). The results presented below are for polyballs consisting of 33 polycubes arranged as follows: There is a central 3 x 3 x 3 cube (the center of the polyball is the central lattice point of this cube); appended to the center of each face of the 3 x 3 x 3 cube is one additional polycube. Except for the case described in Figure 3 (which is presented in order to emphasize a theoretical point) charge is distributed evenly over the six appended polycubes. Although this arrangement is only a rough approximation for a (spherical) polyball, it is reasonable to assign to it an effective radius of 2.5 lattice spacings (0.125 pm). The polyballs were placed on the lattice with different distances between them. The Helmholtz free energy for each distance was calculated. Figures 1-3 present electrostatic “energy curves’’ (Le., electrostatic free energy, A , versus interpolyball distances, R, measured between the centers of the polyballs) for three (different) systems of two oppositely charged polyballs. In each of the three cases the total number of anions was taken as 20 OOO and the total number of cations was set, in accord with the total charge of the two polyballs, to preserve global electroneutrality. The dielectric outside the polyballs was chosen as 80.0 to fit aqueous solutions, since water is the most commonly used solvent for polyballs systems. A temperature of 300 K was
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Figure 1. Energy curves for a realistic model system. Helmholtz free energy, A (in units of KT), vs interpolyball separation, R (in units of micrometers) for a two-polyball system. Polyballs have charges of +40 and -1500 on their surfaces. The radius of each polyball is 0.125 pm. Simple ion concentrations are given in text. The dielectric constant, E , is 80.0 for the solvent and 1.0 for the polyballs. The dotted line with squares and the solid line with circles mark the results obtained for ions with infinitesimal and finite diameters, respectively.
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9644 J. Phys. Chem., Vol. 99, No. 24, 1995
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Legend finite size ions lnflnllesinmi ions
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Figur 3. Electrostatic repulsion due to dielectric pressure. Energy curves; Helmholtz free energy, A , vs interpolyball separation, R, for a
two-polyball system. Polyballs have charges of +500 and -500-each located at the point on the surface of the polyball which faces the other polyball. The radius of each polyball is 0.125 pm. Simple ion concentrations are given in text. The dielectric constant, E , is 80.0 for the solvent and 1.0 for the polyballs. The dotted line with squares and the solid line with circles mark the results obtained for ions with infinitesimal and finite diameters, respectively. can see a convincing energy minimum when the polyballs are located at a noncontact configuration. The minimum appears both for infinitesimal ions (dotted line with squares), at interpolyballs distance of 0.35 pm, and for very big ions of 0.05 pm in diameter (solid line with circles), at an interpolyballs distance of 0.40 pm. In the following, we explain why opposite charges can sometimes repel each other at short distances; hence stabilize the polyballs at a noncontact configuration. The common electrostatic intuition of “opposites attracts” would lead to an energy curve which is an increasing function of the interpolyballs distance. Such a trivial electrostatic energy curve is, indeed, observed for many systems including, for example, a system of two equal but oppositely charged polyballs in a uniform dielectrics. The (trivial) electrostatic attraction is also observed for the system presented in Figure 1 when the polyballs are far apart from each other such that their “ion atmospheres” (the ionic cloud around each charged polyball) hardly polarizes one another. When the polyballs are close enough to each other, their ion atmospheres mutually polarize each other, such that the electrostatic energy of the “near-touching configuration” increases. The result is a stable energy minimum for a noncontact configuration. Similar energy minima have also been observed for other systems including (i) A repressor operator interaction? (ii) two oppositely charged polyballs, polystars, and parallel plates,4c and (iii) for binding of basic peptides to negatively charged lipid bilayers. l o Two different mechanisms can cause the “effective repulsion” between the polyballs: (a) the “osmotic pressure”, or the exclusion of counterions from the surface of the charged polyballs because of the proximity of the macroions, and (b) the “dielectric pressure”, or Born repulsion which results from having a dielectric boundary. In the following two sections we explain and manifest these two effects. Electrostatic Repulsion due to Osmotic Pressure. The charge on the positive polyball is more than an order of magnitude smaller than the charge on the negative polyball in the case which is presented in Figure 1. Thus one could treat the source terms on the positive polyball as a small perturbation to the electrostatic field created by the negative polyball. Placing the positive polyball in close proximity to the negative one reduces the total energy of the system due to the electrostatic
attraction between the charges on their surfaces. There is, however, a simultaneous increase in the energy of the system caused by the fact that mobile cations, which surround the negatively charged polyball, are pushed away from the space occupied by the positive polyball. If the total charge on the positive polyball is less than the sum of charges of all the cations that could have been in this space, than the energy of a neartouching configuration will exceed that of a noncontact configuration. The result will be a stabilization of two oppositely charged polyballs in a noncontact configuration. The stabilization obtained via this mechanism depends on the ability of the mobile ions to form a dense enough ionic cloud, hence, the connection to osmotic pressure. The results presented in Figure 2, which were obtained for a system of two polyballs with total surface charges of -1500 and +25 proton units, a total number of anions and cations of 20 000 and 21 475, respectively, in a uniform dielectric of 80.0, manifest a stabilization of a noncontact configuration via the “osmotic pressure” mechanism. Looking at the figure one can see that for infinitestimal ions (dotted line with squares)-when no repulsive force is there to prevent the simple ions from accumulating around the negative polyball-the energy curve has a minimum at an interpolyball distance of 0.30 pm. This minimum diminishes for simple ions with a finite diameter (solid line with circles) which is big enough to prevent the ions from forming a dense atmosphere. A more direct support for the explanation given above comes from looking at the densities of positive and negative infinitestimal ions near the negative polyball for each interpolyball distance. For a configuration with the positive polyball at infinity, the net charge of mobile ions in those 33 grid points near the negative polyball, which would be taken by the positive polyball in a near-touching ( R = 0.25 pm) configuration, is about 27 proton charge units. This value is (slightly) higher than the total charge of the positive polyball. Therefore the value of the electrostatic free energy observed for near-touching configuration is somewhat higher than the one obtained for noncontact configuration. The net charge of mobile ions in those 33 grid points which would be taken by the positive polyball in a configuration which corresponds to R = 0.30 p m is only 15 proton charge units, which is smaller than the total charge of the positive polyball. This configuration is, therefore, the one which is energetically favorable. Another test for the explanation given above is obtained by looking at the electrostatic energy curve of a similar system with the positive polyball having a charge of 30 proton units, a charge which is somewhat larger than the net charge of mobile ions in those 33 grid points near the negative polyball which would be taken by the positive polyball in a near-touching ( R = 0.25 pm) configuration. Indeed, the energy-curve for this system is a monotonically increasing function of the interpolyball distance (Le., the energy minimum is at the trivial neartouching configuration) for both infinitestimal and finite size mobile ions. Electrostatic Repulsion due to Dielectric Pressure. The effect of the “dielectric pressure” to stabilize equal but oppositely charged polyball “mutants” in a noncontact configuration, which has been studied in ref 4c,is manifested in Figure 3. The system which was studied is composed of two polyballs similar to those studied before, except that the total charge assigned to each of them was submitted to that lattice point which is facing the other polyball. The positive (negative) polyball had a charge of +500 (-500), and a total of 20 000 cations and 20 OOO anions were added. The dielectric was set to 1.0 at the interior of each polyball and 80.0 elsewhere. This, unrealistic, system is
J. Phys. Chem., Vol. 99,No. 24, 1995 9645
Letters presented here since, due to its simplicity, the net effect of having a dielectric boundary is clearly understood. The electrostatic energy curves in Figure 3 have minima at a noncontact configuration (Le., an interpolyballs distance of 0.30 pm). The minima are observed both for infinitesimal mobile ions (dotted line with squares) and for simple ions with finite diameter (solid line with circles). The energy curve of a similar system with a uniform dielectric of 80.0 is a monotonically increasing function of the interpolyballs distance (i.e., the energy minimum is at the trivial near-touching configuration). Looking at Figure 3, one can see the attraction between the oppositely charged polyballs at large distances. One can also see a pattern, which is observed in Figures 1 and 2 as well; since the ion atmosphere is less perfect for ions with finite diameter than it is for infinitesimal ions, both the range and the strength of the attraction are larger for the former. At a near-touching configuration, however, the opposite behavior is observed. The charge on each polyball, which is exposed to water with their high dielectric at all other configurations, is exposed to the low dielectric of the other polyball. The result is an increase in the energy of the near-touching configuration (compared to a near-touching configuration) due to Born repulsion," which causes a stabilization of a neartouching polyball configuration. Note that, since the dielectric pressure effect results from having a dielectric boundary, this effect is not related to properties of the mobile ions. Indeed, the same energy difference between near-touching, R = 0.25 pm, and noncontact, R = 0.30 pm, configurations is observed for infinitesimal and finite size ions. Now that the physical principles which can lead to electrostatically stable noncontact configuration for oppositely charged polyballs are clear, we can try to make a more quantitative statement regarding the stability of such state. To do that we need to evaluate changes in other components of the free energy as a function of the interpolyball distance. The van der Waals forces and hydrophobicity, which were not accounted for in this study, should affect the system. These forces, which are expected to cause an attraction between the polyballs at short interpolyball distances, can in principle be calculated separately and added later to the electrostatic repulsion. These calculations require an atomic description of the polyballs which is not available at the moment. However, on the basis of the interactions of basic peptides with negatively charged membranes, which are currently under theoretical and experimental study,I0 we believe that the behavior of polyballs is likely to be govemed by electrostatic energy. For the system described in Figure 1 we calculated huge electrostatic energy differences (660KT = 400 kcal/mol for infinitesimal ions, and 770KT = 456 kcal/mol for ions having a diameter of 0.05 pm) between near-touching and noncontact configurations. To sum up, the electrostatic energy difference (few hundreds of kcaumol) between near-touching and noncontact configura-
tions of the two oppositely charged polyballs presented in Figure 1 is expected to dominate the energetics of the system. This huge energy difference, which results from both the osmotic and the dielectric pressures, stabilizes the noncontact configuration.
Acknowledgment. I would like to thank Professor Rob D. Coalson for helpful discussions while the work on this project has been carried out, and for his critical remarks on the manuscript, to Professor Barry Honig for being supportive throughout this project, and to Dr. Carey Bagdassarian and Dr. Richard Friedman for their critical remarks on the manuscript. The work was financially supported by NSF Grant MCB9304127, and the National Center for Research Resources division of the Biomedical Technology Program at the NIH, through a Research Resource grant (P41 RR06892) at Columbia University. References and Notes (1) Asher, S. A,, private communication. (2) For variational approaches to the electrostatic free energy of colloids see, for example: (a) Reiner, E. S.; Radke, C. J. J . Chem. Soc., Trans. 1990, 86, 3901. (b) Reiner, E. S.; Radke, C. J. AIChE J . 1991, 37, 805. (3) A class of approaches are derived from density functional theory of the classical Coulomb gas. See, for example: (a) Stevens, Robbins, M. 0. Europhys. Lett. 1990, 12, 81. (b) Lowen, H.; Madden, P. A,: Hansen, J.-P. Phys. Rev. Lett. 1992,68, 1081. (c) Lowen, H.; Hansen, J.-P.; Madden, P. A. J . Chem. Phys. 1993, 98, 3275. (d) Lowen, H.; Kramposthuber, G. Europhys. Lett. 1993, 23, 673. (4)For a lattice field theory formulation see: (a) Coalson, R. D.; Duncan, A. J . Chem. Phys. 1992, 97,5653. (b) Walsh, A. M.; Coalson, R. D. J . Chem. Phys. 1994, 100, 1559. (c) Ben-Tal, N.: Coalson, R. D. J . Chem. Phys. 1994, 101,5148. (d) Coalson, R. D.; Walsh, A. M.; Duncan, A.; Ben-Tal, N. J . Chem. Phys. 1995, 102, 4584. ( 5 ) (a) Sogami, I. S.: Shinohara, T.; Smalley, M. V. Mol. Phys. 1992, 76, 1. (b) Ito, K.; Yoshida, H.; Ise, N. Science 1994, 263, 66. (c) Ise, N.; Matsuoka, H. Macromolecules 1994, 27, 5218. (d) Ise, N.: Smalley, M. V. Phys. Rev. 5 1994, 50, 16722. (e) Yananaka, J.; Ise, N.: Miroyuki, H.: Yamaguch, T. Phys. Rev. E. 1995, 51, 1276. (f) Ray, J.: Manning, G. S. Langmuir 1994, 10, 2450. (6) See, for example, the following textbooks: (a) Venvey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. This book contains references to other work from the same period. (b) Alexander, A. E.; Johnson, P., Colloid Science: Clarendon Press: Oxford, 1949. (c) Physical Chemistry of Surfaces, 5th ed.: Adamson, A. W.; John Wiley & Sons: New York, 1990; Chapter XIII, section 5. (7) For modem overviews see: (a) Asher, S. A.: Flaugh, P. L.: Washingter, G. Appl. Spectrosc. 1986, 1, 26. (b) Thirumalai, D. J . Phys. Chem. 1989, 93, 5673. (c) Sood, A. K. In Solid State Physics: Ehrenreich, H., Turnbull, D., Eds.; Academic: New York, 1991; Vol. 45, p 1. (8) Expressions for the Laplacian matrix for the cases of spatially constant and spatially varying dielectrics are given in eq 7 of ref 4a and in eq 5 of ref 4c, respectively. (9) Zacharias, M.; Luty, B. A,; Davis, M. E.; McCammon, J. A. Biophys. J . 1992, 63, 1280. (10) Ben-Tal, N.; Honig, B.; Bagdassarian, C.; Peitzsch, R. M.: Denisov, G.; McLaughlin, S., manuscript in preparation. (11) See, for example, Chapter 2 o f Bockris, J. O'M.; Reddy, A. K. N. Modern Electrochemistry: Plenum and Rosetta edition, 1973. JP9508618
