Effect of Dielectric Constant on Interaction Between Charged Macroions in Asymmetric Electrolyte

Abstract

The mean force between two highly like-charged macroions in the presence of monovalent counterions and added multivalent salt within solvents of varying dielectric constants was studied using Monte Carlo simulations. Without additional salt, the mean force is strongly repulsive at all macroion separations in solvents with a dielectric constant ${ϵ}_r$ ≥ 30. However, in solvents with ${ϵ}_r$≤ 30, macroions experience effective attraction, indicating that attractive interactions between highly charged macroions can occur even without multivalent salt in nonpolar solvents with low dielectric constants. The total multivalent counterion charge-to-total macroion charge ratio is defined as β which determines the amount of salt that is added to the system. At β = 0.075, the mean force becomes attractive at short separations in solvents with ${ϵ}_r$= 54 containing 1:3 salt, as well as in all solvents with 1:5 salt, while still exhibiting significant repulsion at longer separations. In contrast, for solvents with 1:3 salt and dielectric constants ${ϵ}_r$= 68 and ${ϵ}_r$= 78.4, the mean force turns attractive at a higher salt concentration, around β = 0.225. The shift in the mean force to an attractive state at short separations signifies charge inversion on the macroion surface when a sufficient amount of salt is present. At a stoichiometric ratio of multivalent counterions, long-range repulsion vanishes, and attraction becomes significant. However, with excess salt, the strength of the attractive mean force diminishes. Additionally, the attractive force at a given salt concentration increases as the dielectric constant decreases and is stronger in systems with 1:5 salt than in those with 1:3 salt.

1. Introduction

Charged colloids are widely present in diverse biological and technological applications. Examples include micelles formed by charged surfactants, latex particles, proteins, silica particles, and microemulsions created by water, oil, and charged surfactants. The physicochemical characteristics of these colloidal solutions are controlled by electrostatic interactions [1]. From a theoretical perspective, Derjaguin, Landau, Verwey, and Overbeek introduced what is now known as DLVO theory [2,3], which predicts purely repulsive electrostatic interactions between like-charged colloids. However, beginning in the mid-1980s, this foundational model was questioned following evidence that like-charged planar surfaces can attract under conditions of strong electrostatic coupling [4,5]. Similar attractive interactions have since been suggested for other geometries as well.

In order to comprehend the behavior of electrostatic interactions in colloidal suspensions, a variety of models have been used in theoretical and computational techniques [6,7,8,9,10,11,12]. The basic model is a helpful model that may be used to study the intercolloidal structure of colloidal solutions and the distribution of tiny ions close to the charged colloids [13]. This model treats the solvent as a dielectric medium and depicts charged colloids and tiny ions as hard, electrically charged spheres [14,15,16].

The Coulomb interactions in colloidal systems are primarily analyzed using the Poisson–Boltzmann (PB) mean field theory, where ions are treated as point charges and the solvent is represented as a dielectric medium. The results from PB theory for systems with weakly charged particles and low concentrations of monovalent salts align well with experimental observations. However, PB theory overlooks ion and electrostatic potential fluctuations. As a result, it has limitations and requires corrections, particularly when dealing with multivalent ions, highly charged colloids, polyelectrolytes, and other macroions [17,18].

In systems of charged colloids, the repulsive forces between particles are reduced by the presence of small ions, a process known as Coulomb screening. This mechanism is fundamental to understanding a wide range of disciplines, including polymer physics, nanofluidic, colloid science, and molecular biophysics. When multivalent counterions are present, attraction occurs between like-charged colloids [9], a phenomenon known as charge inversion or overcharging [19]. This effect arises from the correlations between counterions that emerge at short distances and high surface charge densities [14]. Charge inversion has been observed and explored through numerical simulations [9,10]. It happens when the effective charge of a surface exposed to a solution flips as a result of an excess of counterions accumulating near the surface. Charge inversion is significantly influenced by the valence Z of the ions [9,20]. Additionally, Besteman et al. [21] used atomic force spectroscopy to examine the effects of solvent dielectric constant and surface charge density on charge inversion. They discovered that the concentration of multivalent salt needed for charge inversion to occur can be decreased by decreasing the solvent’s dielectric constant and raising the surface charge density. Allahyarov et al.’s computer simulations of the electrolyte’s primitive model [22] showed that the concentration of salt has a significant impact on the efficient interaction between colloidal particles that are lightly charged in solvents with a low or moderate dielectric constant. Additionally, even at low concentrations of monovalent salt, the interaction between like-charged colloids is attractive, whereas when salt is absent, the interaction is repulsive.

The effect of the dielectric constant on the zeta potential of highly charged colloidal particles in solvents ranging from polar (e.g., water) to nonpolar, with multivalent salts present, was studied through primitive Monte Carlo (MC) simulations [23]. Results indicate that the zeta potential (ξ) decreases with a lower dielectric constant and declines further as salinity and salt valency increase. To build on these findings, we extended our study by employing Monte Carlo (MC) simulations to examine how the medium’s dielectric constant influences charge inversion between two highly charged colloids in asymmetric electrolyte solutions. Here, we extended our study by applying Monte Carlo (MC) simulation to investigate the dependence of charge inversion between highly charged colloids in asymmetric electrolyte solutions on the dielectric constant of the medium.

2. Model and Method

2.1. Model

The primitive model, in which macroions are represented as hard spheres with a radius of R_M = 20 Å and a charge of Z_M = −60, was used to analyze systems of asymmetric electrolytes. Counterions are modeled as charged hard spheres with the radius R_I = 2 Å and the valences Z_I = +1, +2, and +3. The solvent is treated as a continuous medium characterized by a dielectric constant ${ϵ}_r$. Added salt consists of small ions, also with a radius of 2 Å, including multivalent cations with valencies +3 and +5 and monovalent anions with Z_a = −1. The addition of multivalent salts can alter the effective dielectric constant of the medium. However, we treat it as a fixed parameter to maintain analytical simplicity and focus on core electrostatic effects. While this is a simplification, it captures the main trends and is standard in mean-field models. Here the ratio of the total charge of the additional cation charge to the total charge of the macroion, $\beta$ = ${\displaystyle \frac{Z_c{\rho }_c}{Z_M{\rho }_M}}$ where ${\rho }_c$ and ${\rho }_M$ are the number densities of the added cations and macroions, respectively, determines the fixed amount of salt that is added.

In this work, we employed a cylindrical cell geometry to model an isolated macroion [11]. Ewald summation is not applied, as the system is not periodic. While this avoids artificial periodic image effects, it also means the treatment of long-range interactions differs from that in periodic cubic simulations with Ewald sums. Thus, the two approaches are not strictly equivalent [18,24]. A simulation cell with a radius of R_cyl = 80 Å and a length of L_cyl = 398 Å is defined. Two macroions (N_M = 2) are symmetrically positioned along the $C_{\infty }$ symmetry axis within this cell. N_I = |Z_M/Z_I|N_M is used to determine the proper amount of counterions to introduce in order to maintain charge neutrality. The total potential energy $U_T$ of the cylindrical system, which is electrostatically isolated from its surroundings, can be found using the following formula:

$$U_T=U_{hs}+U_{elec}+U_{ext}$$

(1)

where $U_{hs}$, representing the hard-sphere repulsion, is defined by

$$U_{hs}=\sum _{i<j}{u_{ij}}^{hs}\left(r_{ij}\right)$$

(2)

with

$${u_{ij}}^{hs}\left(r_{ij}\right)=\left\{\begin{array}{c}\infty , r_{ij}<(R_i+R_j)\\ 0, r_{ij}\ge (R_i+R_j)\end{array}\right.$$

(3)

where r_ij is the center-to-center distance between the particles i and j, where i and j stand for a macroion, a monovalent counterion, a multivalent counterion, or a coion. This expression defines the hard-sphere (hs) interaction as a purely repulsive potential, with an infinite energy penalty for any overlap between particles and zero interaction otherwise. As a result, the associated mean force is undefined (infinite) at contact and zero elsewhere. In our numerical implementation, this contribution was handled by ensuring that particle overlaps were strictly avoided and by computing the mean force only for separations strictly greater than the hard-core diameter, where the force was finite. The definition of the electrostatic interaction $U_{elec}$ is

$$U_{elec}=\sum _{i<j}{u_{ij}}^{elec}\left(r_{ij}\right)$$

(4)

with

$${u_{ij}}^{elec}\left(r_{ij}\right)={\displaystyle \frac{Z_i{Z}_j e^2}{4\pi {ϵ}_0{ϵ}_{r }{r}_{ij}}}$$

(5)

where e is the elementary charge, ${ϵ}_0$ is the vacuum permittivity, ${ϵ}_r$ is the solvent’s relative permittivity, and Z_i and Z_j are the charges of particles i and j, respectively. In Equation (1), the confinement potential energy $U_{ext}$ is provided by

$$U_{ext}=\sum _i{u}_i^{ext}\left(r_i\right)$$

(6)

with

$$u_i^{ext}\left(r_i\right)=\left\{\begin{array}{l}0, \sqrt{(x_i^2+y_i^2)} \le R_{cyl} and \left|z_i\right|\le L_{cyl}/2\\ \infty , other wise \end{array}\right.$$

(7)

All systems maintain the macroion number density ${\rho }_M=2.5\times {10}^{-7} {\AA }^{-3}$, which corresponds to the macroion volume fraction ${\phi }_M=0.0084$. T = 298 K is chosen as the temperature.

The distance between two unit charges where their Coulomb interaction equals the thermal energy is defined by the Bjerrum length, $l_B=e^2/{4\pi ϵ}_0{ϵ}_{r}kT$, where k is Boltzmann constant. The dielectric constants and matching Bjerrum lengths for the solvents investigated in this investigation are shown in

Table 1. Lists solvents’ Bjerrum lengths and dielectric constants at 25 °C.

$\mathbf{Dielecrtiric} \mathbf{Constant} \left({\mathit{ϵ}}_{\mathit{r}}\right)$	78.4	68	54	40	30	20	15	10
$l_B$	7.1	8.23	10.36	14.0	18.657	27.986	37.315	55.973

.

2.2. Method and Simulation Settings

Metropolis Monte Carlo (MC) simulations within the canonical ensemble (constant temperature, volume, and particle count) were used to solve the model. Equations (1)–(8) were used to obtain the mean force. The two macroions were initially positioned symmetrically along the cylindrical cell’s $C_{\infty }$ axis, with z = 0 at its center. The fixed separations were usually between 42 Å and 80 Å. Initially, tiny ions were dispersed at random throughout the cylindrical cell, and their locations were subjected to trial displacement. Production runs, which usually had 10⁶ trail moves per particle, were carried out following the equilibration phase. The integrated Monte Carlo/molecular dynamics/Brownian dynamics simulation program MOLSIM (Version 6.3.5) [25] was used for all of the simulations.

2.3. Mean Force and Potential of Mean Force

The effective interactions between two macroions have been analyzed before, using the cylindrical cell technique [11,26]. The effective interaction is widely thought to be only marginally impacted by the particular form of the cylindrical cell, even though it is influenced by the cell size due to counterion entropy [27]. When projected onto the interparticle vector r, the mean force F(r) acting on macroion M is represented as

$$F\left(r\right)=\sum _{i\ne M}^N\langle -{\nabla }_{r_{Mi}}{U}_{Mi}(r_{Mi})\rangle$$

(8)

$〈\dots 〉$ indicating an ensemble average over the particle’s positions, where F(r) > 0 indicates a repulsive mean force and F(r) < 0 indicates an attractive mean force. The associated mean force potential (pmf) is defined as U^pmf (r), where

$$U^{pmf}\left(r\right)\equiv -{\int }_{\infty }^{r}F\left(r^{\prime }\right)d{r}^{\prime }$$

(9)

F(r) and $U^{pmf}(r)$ both become closer to 0 when r becomes large enough. The force is a mean force rather than an effective one because the macroions and their counterions are restricted within a cell, approximating the effect of the surrounding electrolyte [28]. There are several ways to express the mean force operating on a single macroion based on the broad concepts of regional equilibrium [26]. In one of them, also being numerical advantageous, F(r) is decomposed according to [11,26]

$$F\left(r\right)=F_{ideal} \left(r\right)+F_{elec} \left(r\right)+F_{hs }\left(r\right)$$

(10)

The first term F_ideal(r) is given by

$$F_{ideal} \left(r\right)=kT\sum _i^{small ions} [{\rho }_i\left(Z=0\right)-{\rho }_i(Z=l_{cyl}/2)]A_{cross }$$

(11)

F_elec(r) denotes the difference in the transfer of linear moments across the planes Z = 0 and Z = L_cyl/2 with ${\rho }_i(Z=Z^{\prime })$ being the number density of species i in the plane $Z=Z^{\prime }$ averaged over the cross-section area A_cross of the cylinder, as long as r $\ll$ L_cyl/2, F_ideal(r) is dominated by the ${\rho }_i(Z=0)$ term.

F_elec(r) is given by

$$F_{elec}\left(r\right)=\underset{i<j}{\overset{N}{{\sum }^{\prime }}}\langle -{\nabla }_{r_{ij }}{U}_{ij}^{elec} \left(r_{ij}\right)\rangle$$

(12)

F_elec(r) represents the average force across the plane Z = 0 originating from the electrostatic interaction among the charged species and the prime that the summation only include pairs of species located on different sides of the plane Z = 0. The electrostatic force can be calculated directly as it is a continuous function.

The third term is given by

$$F_{hs}\left(r\right)=\underset{i<j}{\overset{N}{{\sum }^{\prime }}}\langle -{\nabla }_{r_{ij }}{U}_{ij}^{hs} \left(r_{ij}\right)\rangle$$

(13)

F_hs(r) represents the averaged force across the plane Z = 0 arising from hard-sphere contacts due to hard-core overlaps at the midplane. Only particle pairs situated on sides of the plane Z = 0 should be taken into consideration, according to the prime in the summation of Equations (12) and (13). However, due to the discontinuous nature of hard-sphere interactions, direct sampling of the average force is not straightforward. Following the approach proposed by Wu et al., F_hs(r) is calculated by performing a virtual displacement $∆s$= 0.1 Å of counterions near the plane Z = 0 toward the plane Z = 0 and checking for hard-sphere overlaps [29].

The hard-sphere contribution F_hs(r) is always positive and becomes significant at high concentrations of small ions. This term vanishes when the small ions are treated as pointlike particles [26]. Previously Linse et al. calculated the potential of mean force between two macroions in the primitive model in a cylindrical cell [11,27]. In those studies, the decomposition of the mean force between macroions varied depending on the microscopic model employed. Notably, in the study performed in 2002 [27] where counterions were modeled as point charges, the mean force was expressed as

$$F\left(r\right)=F_{ideal} \left(r\right)+F_{elec} \left(r\right)$$

(14)

since excluded-volume (hard-sphere) interactions were absent in that model. However, in later work performed in 2005 [11], finite-sized counterions with a radius of 2 Å were used. In that case, steric (hard-sphere) interactions between particles were explicitly included, and the mean force took the same form as Eq. 10. In our present work, we follow the latter approach: counterions are assigned a finite size with a radius of 2 Å, and hard-sphere interactions are explicitly treated in the simulations. For this reason, we include the $F_{hs }\left(r\right)$ term in the total force decomposition. This ensures that all relevant physical interactions—electrostatic, ideal, and steric—are captured consistently in the force calculations.

3. Results and Discussion

3.1. Salt-Free Solvents

For a 60:1 system in different solvents with dielectric constants given in Table 1, Figure 1a shows the mean force between two like-charged macroions and their counterions trapped in a cylindrical cell as a function of macroion spacing r. The potential of mean force pmf, promoted by integrating the mean force using Equation (10), is shown in Figure 1b.

The figures show that the mean force and pmf are strongly repulsive and increase monotonically for monovalent counterions in solvents with dielectric constants of 78.4, 68, 54, and 40. A vertical shift is used to make sure the potential of mean force pmf approaches zero as r→∞ since the mean force stays positive at the largest separation, r = 80 Å. The effective long-range repulsion is essentially nonexistent in a solvent with a dielectric constant of 30, and the mean force approaches attractiveness between r = 45 Å and 48 Å. At r = 45 Å, the pmf correspondingly displays a shallow minimum. The mean force for solvents with ${ϵ}_r$ < 30 is only nonzero at short separations; it is repulsive for r < 44 Å and attractive between 44 and 55 Å. The macroion–macroion surface distance equal to the counterion’s diameter coincides with the 44 Å spacing. The pmf minimum is roughly −5 kT, −10 kT, and −20 kT for solvents with dielectric constants of 20, 15, and 10, respectively. In these instances, monovalent counterions operate as a mediator between the macroions, facilitating their effective attraction to one another. This implies that the attractive interaction between highly charged colloids may take place in a fluid with a low dielectric constant and without salt. In contrast, E. Allahyarov et al. [22] found that, in the absence of salt, weakly charged colloids interacted repulsively with monovalent counterions in liquids with moderate and low dielectric constants. The charged surface’s counterion–counterion coupling parameter, Γ, was less than two in their system. And correlation-induced attraction is known to occur at $\Gamma >{\Gamma }^{*}$ $\approx$ 2 [10,16]. Here, the intensity of electrostatic interactions is described by $\Gamma =Z_I^2{l}_b/a_Z$, which can be raised by decreasing the solvent’s dielectric constant. The average distance between two near by counterions on a surface with a surface charge density of σ is denoted by $a_Z={\left[Z_I/\left(\sigma /e\right)\right]}^{1/2}.$ The coupling parameter defines the degree of ordering in the counterion system and thereby the correlation attraction. The attraction is short-ranged and important only at h $\le$ $a_Z$, where h is the distance between the charged surfaces [16].

Table 2. The Coulomb coupling parameter (Γ) between monovalent, divalent, trivalent, and pentavalent counterions on a charged surface is evaluated for all solvents.

$\mathbf{Dielecrtiric} \mathbf{Constant} \left({\mathit{ϵ}}_{\mathit{r}}\right)$	78.4	68	54	40	30	20	15	10
$l_B$ (Å)	7.1	8.2	10.4	14.0	18.6	28.0	37.3	56
${\Gamma }_1$	0.8	0.9	1.1	1.5	2.0	3.0	4.1	6.1
${\Gamma }_2$	2.3	2.5	3.1	4.2	5.6	8.5	11.6	17.2
${\Gamma }_3$	4.1	4.7	5.7	7.8	10.4	15.6	21.3	31.7
${\Gamma }_5$	8.9	10.1	12.3	16.8	22.4	33.5	45.8	68.2

displays the determined coupling parameters for each system between the counterions on the charged surface. While the solvent with ${ϵ}_r$ = 30 corresponds to $\Gamma$ = 2.04 $\approx {\Gamma }^{*}$, the solvents with dielectric constants of 20, 15, and 10 correspond to Γ = 3.058, 4.078, and 6.117, respectively, which are more than ${\Gamma }^{*}$ $\approx$ 2, while the solvents with dielectric constants of 78.4, 68, 54, and 40 have values of Γ < 2.

As the solvent’s dielectric constant drops, an attractive force shows that charge inversion has taken place on the macroion’s surface. This is seen in solvents with ${ϵ}_r$ = 20, 15, and 10.

The distribution of counterions surrounding the macroion in a 60:1 system with various solvents is shown in Figure 2. Figure 2h unequivocally demonstrates how the strong polarity of the aqueous solvent, which has a high dielectric constant of 78.4 at 298 K, influences the separation of counterions from the macroion surface while maintaining the stability of the colloidal system. As the solvent polarity decreases—reflected by lower dielectric constants when moving from aqueous to less polar solvents—the dissociation of counterions from the macroion surface is reduced, resulting in their closer association with the macroion, as illustrated in Figure 2. Figure 2a shows that the counterions are compacted and firmly adhered to the macroion’s surface in a solvent with a dielectric constant of 10.

By computing the macroion–counterion radial distribution functions (rdfs), which characterize the relative densities of counterions at a distance r from the macroion, the distribution of monovalent counterions close to the macroions was investigated. Systems of uncorrelated particles are represented by the horizontal dotted line, where rdfs equals unity in the bulk. We used a spherical cell with one macroion and an equal number of monovalent counterions in our investigation. The findings shed light on how counterions are distributed around a particular macroion that is distinct from other macroions. Figure 3 displays the macroion–counterion rdfs in various solvents. The accumulation of the counterions close to the macroion is indicated by the peak in the figure at the hard-sphere contact separation r = R_M + 2R_I = 22 Å for the macroion–counterion rdfs. The contact value is 633 at ${ϵ}_r$ = 10. The contact value drops monotonically as the solvent’s dielectric constant rises, reaching a value of 120 at ${ϵ}_r$ = 78.4.

For reference, the equivalent mean forces with different counterion valences but without added salt are shown. Figure 4 shows the emergence of attractive interactions with divalent (Γ = 2.3, 2.5, 3.1, and 4.2) and trivalent (Γ = 4.1, 4.7, 5.7, and 7.8) counterions in solvents with dielectric constants of 78.4, 68, 54, and 40. On the other hand, the typical image of a like-charged macroion displaying effective repulsion is seen for monovalent counterions. As seen in Table 2, attraction between monovalent and multivalent counterions happens in solvents with low dielectric constants ${ϵ}_r$ < 30 and no salt because the coupling parameters are bigger than Γ* ≈ 2. Where the minimum is becoming deeper than with divalent counterions, the attraction is strongest with trivalent counterions and weakest with monovalent ones. Because a higher coupling parameter indicates a stronger electrostatic connection, the attraction rises as the dielectric constant falls.

3.2. Solvents with the Presence of Added Salt

The electrostatic interactions between macroions can be modulated by adding salt to the solution, as the presence of additional counterions and coions enhances screening and reduces repulsion [9,10,12]. Under varying concentrations of 1:3 and 1:5 salts, respectively, Figure 5 and Figure 6 show the mean force between two like-charged macroions in a 60:1 system as a function of their separation distance r. For solvents with dielectric constants of 78.4, 68, and 54, the findings are shown. The three solvents’ mean force curves show notable variations with salt concentration. The repulsive mean force first becomes weaker across all separations, with the greatest decrease seen at short distances, as the concentration of simple salt rises. Then, the mean force changes to an attractive interaction. Later, the attractive force’s strength and range increase, while the long-range repulsion disappears. Finally, the attractive force’s magnitude decreases. Interestingly, a considerable initial decrease in repulsion can be induced with a very modest amount of salt. Additionally, the commencement of attraction at short separation happens as early as β = 0.075 in a solvent with ${ϵ}_r$ = 54 that contains 1:3 salt and in all solvents with 5:1 salt. On the other hand, attraction appears at greater salt concentrations (β > 0.075) for solvents with a 1:3 salt content and dielectric constants of ${ϵ}_r$ = 68 and 78.4. In accordance with Linse et al. [26], the maximum attraction is seen close to β = 1, although it diminishes at β = 6.25.

The 60:1 systems with dielectric constants of ${ϵ}_r$ = 78.5, 68, and 54 have mean forces that are quite similar to those of the 60:1 systems at β = 1 for the 1:3 salt (Figure 5a–c). This extra electrolyte has very little effect on screening the attractive mean force since the 60:1 systems at β = 1 can be thought of as the result of adding a 1:1 electrolyte to the 60:3 systems. In contrast to solvents with dielectric values of 68 and 78.4, the decrease in repulsive force is more noticeable in the solvent with a dielectric constant of 54. Likewise, because the coupling parameters in 1:5 salt systems are higher than those in 1:3 salt systems, the reduction proceeds more quickly in the former than in the latter. In particular, for dielectric constants of 78.4, 68, and 54, the coupling parameter values for solvents containing 1:3 salt are 4.1, 4.7, and 5.7, respectively, whereas for 1:5 salt, they are 8.9, 10.1, and 12.3 (Table 2).

When there is enough salt present, the repulsive mean force changes to an attractive one at short separations (around 42 Å < r < 50 Å), indicating charge inversion on the macroion surface in solvent. In solvents with ${ϵ}_r$ = 54 that contain 1:3 salt and in all solvents that include 1:5 salt, this inversion takes place at β = 0.075. On the other hand, charge inversion is seen at a larger concentration, approximately β ≈ 0.225, for liquids with 1:3 salt and dielectric constants of ${ϵ}_r$ = 68 and 78.4. These findings imply that when the dielectric constant falls, the attractive force at a given concentration rises. Because of the greater coupling parameter (Table 2), Figure 7 illustrates a stronger attraction in 1:5 salt systems compared to 1:3.

A variety of macroions are of importance in chemistry and biology, ranging from the charged surface of mica or charged solid particles to charged lipid membranes, colloids, DNA, actin, and even cells and viruses. Multivalent metal ions, charged micelles, dendrimers, and short or long polyelectrolytes including DNA can play the role of the screening ions [30]. The observed transition from repulsive to attractive mean forces between like-charged macroions, modulated by salt valency, concentration, and solvent polarity, has significant implications for a wide range of systems. In colloidal science, these results offer insights into controlling stability and aggregation in suspensions through dielectric tuning or selective salt addition similarly to the study performed by Lobaskin et al. [10]. They demonstrated that adding a simple monovalent salt gradually reduces the effective macroion charge and decreases the stability of the solution. Similarly, small amounts of multivalent salt have a comparable effect. When the concentration of multivalent salt exceeds the macroion’s isoelectric point, it leads to charge inversion and electrostatically driven aggregation of macroions.

In biological systems, the findings are relevant to processes such as DNA condensation, protein aggregation, and virus assembly, where multivalent ions play a crucial role [30,31]. Moreover, the ability to engineer short-range attractions while suppressing long-range repulsion can be harnessed in the design of self-assembled nanostructures and stimuli-responsive drug delivery systems [32].

4. Conclusions

Monte Carlo simulations using a cylindrical cell model containing two highly charged macroions were conducted to determine the mean force between them under varying salt concentrations and solvent dielectric constants. In the absence of additional salt, the repulsive mean force observed with monovalent counterions progressively weakens and eventually becomes purely attractive as the counterions are replaced by trivalent and then pentavalent ions in solvents with the dielectric constant ${ϵ}_r$ $\ge$ 30. In contrast, in solvents with ${ϵ}_r$ < 30, an attractive mean force is present even with monovalent counterions, regardless of salt absence. With added salt, a similar but gradual transition occurs when trivalent and pentavalent counterions are introduced into a monovalent counterion system, up to a stoichiometric ratio of multivalent counterions. Beyond this point, the attractive mean force starts to diminish. The introduction of small amounts of multivalent salt progressively reduces the effective charge of the macroions, resulting in a decrease in the stability of the solution.

In solvents with polarity lower than that of water, the transition from a repulsive to an attractive mean force takes place at an extremely low multivalent counterion-to-macroion charge ratio (β = 0.075). Furthermore, at a fixed salt concentration, the mean force increases as solvent polarity decreases and is stronger in systems with pentavalent counterions than in those with trivalent ones due to the higher Coulomb coupling parameter. Moreover, the addition of small quantities of multivalent salt gradually lowers the effective charge of the macroion, resulting in reduced solution stability. Ultimately, this study further supports the development of attractive forces between like-charged colloids, driven by multivalent counterions, across various solutions. The observed inversion in the potential of mean force corresponds to the overcharging of the macroion by multivalent counterions. This charge inversion also manifests as a reversal of electrophoretic mobility, where the macroion moves opposite to the direction expected from its bare charge. For the negatively charged macroions, the effective charge becomes positive once the adsorbed counterion layer overcompensates the bare charge [33].

While monovalent counterions (e.g., H⁺, Na⁺) are more common in physical systems, multivalent ions such as +3 and +5 are used to probe strong electrostatic effects. Our results show that replacing them with monovalent ions at the same bulk concentration significantly weakens ion–macroion attraction, reduces counterion condensation, and eliminates charge inversion. The pmf becomes shallower or purely repulsive, and screening becomes less effective, consistently with previous studies [30].

In the current model, the solvent is treated as a continuous medium. However, when the solvent molecule size becomes comparable to the macroion size, this approximation may no longer hold. Future work may benefit from employing explicit solvent models to capture molecular-level interactions more accurately. The dielectric constant of the solvent is also a key parameter: a lower ${ϵ}_r$ enhances ion–macroion attraction, promotes counterion condensation, and can lead to charge inversion. It also reduces screening efficiency, resulting in longer-range interactions and stronger effective forces. These effects impact observable properties such as colloidal stability, phase behavior, and electrophoretic mobility, underscoring the importance of solvent polarity in tuning electrostatic phenomena in soft matter systems.

In this work, we have presented a simplified model to capture the dominant electrostatic interactions in colloidal systems with multivalent electrolytes. We acknowledge that image charge effects were not included in the current calculations. While such interactions can be significant near dielectric interfaces, our aim was to isolate the primary electrostatic contributions within a tractable framework. Incorporating image forces would be a valuable extension in future, more detailed models.