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. 2021 Apr 5;6(14):9876–9886. doi: 10.1021/acsomega.1c00613

Multivalent Ion-Mediated Attraction between Like-Charged Colloidal Particles: Nonmonotonic Dependence on the Particle Charge

Cheng Lin †, Xiaowei Qiang †, Hai-Long Dong †, Jie Huo †,‡,*, Zhi-Jie Tan †,*
PMCID: PMC8047654  PMID: 33869968

Abstract

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Ion-mediated effective interactions are important for the structure and stability of charged particles such as colloids and nucleic acids. It has been known that the intrinsic electrostatic repulsion between like-charged particles can be modulated into effective attraction by multivalent ions. In this work, we examined the dependence of multivalent ion-mediated attraction between like-charged colloidal particles on the particle charge in a wide range by extensive Monte Carlo simulations. Our calculations show that for both divalent and trivalent salts, the effective attraction between like-charged colloidal particles becomes stronger with the increase of the particle charge, whereas it gradually becomes weakened when the particle charge exceeds a “critical” value. Correspondingly, as the particle charge is increased, the driving force for such effective attraction transits from an attractive electrostatic force to an attractive depletion force, and the attraction weakening by high particle charges is attributed to the transition of electrostatic force from attraction to repulsion. Our analyses suggest that the attractive depletion force and the repulsive electrostatic force at high particle charges result from the Coulomb depletion which suppresses the counterion condensation in the limited region between two like-charged colloidal particles. Moreover, our extensive calculations indicate that the “critical” particle charge decreases apparently for larger ions and smaller colloidal particles due to stronger Coulomb depletion and decreases slightly at higher salt concentrations due to the slightly enhanced Coulomb depletion in the intervening space between colloidal particles. Encouragingly, we derived an analytical formula for the “critical” particle charge based on the Lindemann melting law.

1. Introduction

Ion-mediated interactions between charged particles are critical to the structure assembly, complexation, and stability of colloids, nucleic acids, and proteins due to their polyelectrolyte nature.1−11 For over 2 decades, the effective interactions between like-charged particles have drawn rather considerable interest given that they are strongly coupled to many important processes in biology and physics such as colloid structure stability and nucleic acid structure assembly.12−25 Early on, classical models in the framework of mean-field theories such as the Poisson–Boltzmann theory have been widely employed to study the effective interactions between charged particles in electrolyte solutions.26−28 However, despite their great success, those mean-field-based approaches failed for polyelectrolyte systems under certain salt conditions such as at a very high monovalent salt concentration and in the presence of a multivalent salt. This is attributed to the mean-field approximation that inter-ion correlations are intrinsically ignored since such inter-ion correlations can play a fundamental role in ion-mediated interactions between charged particles.1,29−39

Counterions can bind to charged particles and consequently modulate effective interactions between charged particles. Beyond the mean-field descriptions, for oppositely charged particles, multivalent ions of a high concentration can modulate intrinsically Coulombic attractions into effective repulsions.39−41 In parallel, for like-charged particles, multivalent ions can modulate intrinsically Coulombic repulsions into effective attractions, which could drive the condensation or aggregation of charged particles such as nucleic acids,42−47 proteins,48,49 and colloids.50−62 The mechanisms for multivalent ion-mediated like-charge attractions are diverse and are attributed to counterion bridging,50,58,61−64 depletion force,52,63 and charge fluctuations of condensed counterions.55,65−68 Alternatively, condensed counterions could be modeled as one component plasma around oppositely charged particles (voids), and the effective like-charge attraction could be attributed to a competition among the ion–ion and void–void repulsions and the ion–void attraction.69 Recently, specific counterion configuration70 and depletion71 were reported to be able to cause the like-charge attractions at high monovalent salt concentrations.

Furthermore, it has been shown that effective like-charge attractions are strongly dependent on particle charge structures,15,72 competition between ions of different valences,73 dielectric constants,74,75 temperatures,76 ion sizes,77−82 and particle charge densities.83−85 For example, recent experiments and simulations show that ion-mediated interactions between nucleic acids are dependent strongly on their helical structures.44,47 Additionally, particles with a higher charge density would involve stronger ion–particle binding and consequently the multivalent ion-mediated like-charge attraction generally becomes stronger for higher particle charges.83,84 However, until now, the covered range of particle charge densities is rather limited in previous works, thus it is still unclear how the multivalent ion-mediated attraction between like-charged particles depends on the particle charge in a wide range.85

In this work, we employed Monte Carlo simulations to calculate the potentials of mean force (PMFs) between two like-charged colloidal particles with a wide range of particle charges in symmetrical divalent and trivalent salt solutions and to explore how multivalent ion-mediated like-charge attraction depends on the particle charge. Our calculations show that the effective like-charge attraction is nonmonotonically dependent on the particle charge: with the increase of the particle charge, the effective like-charge attraction becomes apparently stronger when the particle charge is lower than a “critical” value, whereas such attraction gradually becomes weakened when the particle charge exceeds the “critical” value. Furthermore, our calculations indicate that the “critical” value decreases slightly with the increase of the salt concentration and decreases apparently for larger ions and larger charged colloidal particles. Finally, we derived an analytical formula for the “critical” particle charge based on the Lindemann melting law.

2. Results and Discussion

In this section, first, we calculate the PMFs between two like-charged colloidal particles with a wide range of particle charges in 2:2 and 3:3 salt solutions (see Figure 1), and the attractive PMFs mediated by multivalent ions exhibit a nonmonotonic dependence on the particle charge for both 2:2 and 3:3 salts around a “critical” particle charge. Afterward, we analyze the contributions of electrostatic force and depletion force as well as counterion distributions to explore the microscopic mechanism for such nonmonotonic dependence on the particle charge. Additionally, we examine the effects of salt concentration, ion size, and colloidal particle size on the nonmonotonic dependence of multivalent ion-mediated like-charge attraction on the particle charge. Finally, we derive an analytical formula for the “critical” particle charge based on the Lindemann melting law.

Figure 1.

Figure 1

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Schematic representation of the model system with two like-charged colloidal particles in a symmetrical multivalent (2:2 or 3:3) salt solution and x is the separation between the centers of the two colloidal particles. Here, the big spheres represent the colloidal particles with negative charge Z and the small spheres the salt ions.

2.1. Effective Like-Charge Attraction is Nonmonotonically Dependent on the Particle Charge

As shown in Figure 2a,b, the PMFs between two like-charged colloidal particles are attractive and strongly dependent on the particle charge |Z| and the ion valence; see the complete PMFs for different |Z|’s in Figures S1a,b in the Supporting Information. For the 0.01 M 2:2 salt, as the particle charge |Z| is increased from 6 e to 54 e, the PMF changes gradually from a weak effective repulsion to a strong attraction, with the minimums of the PMFs at the separation of x ∼ 22 Å. For a 0.1 mM 3:3 salt, the attractive PMF continuously becomes stronger when the particle charge |Z| is increased until |Z| ∼ 90 e, with the minimums of the PMFs at the separation of x ∼ 21 Å. Such an enhanced multivalent ion-mediated like-charge attraction by higher particle charges is in accordance with previous works.52,54,83 The comparison between a 2:2 salt and a 3:3 salt indicates that trivalent counterions bind more tightly to charged colloidal particles and can cause a stronger effective attraction with a lower PMF minimum and a slightly closer equilibrium separation between like-charged colloidal particles.

Figure 2.

Figure 2

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(a,b) PMFs between like-charged colloidal particles in 0.01 M 2:2 (a) and 0.1 mM 3:3 (b) salt solutions from the MC simulations. (c) Dependence of the like-charge attraction strength on the colloidal particle charge in 0.01 M 2:2 salt (black) and 0.1 mM 3:3 salt (red) solutions. Lines in (c) are guide to the eye.

It is also shown in Figure 2a,b that when the particle charge |Z| exceeds a high “critical” value |Z|c, the PMFs between like-charged colloidal particles gradually become less attractive with the increase of |Z| for both 2:2 and 3:3 salts, which is beyond the prior expectation based on the results of low/medium particle charges and previous works.34 To show such a phenomenon more directly, we utilized the minimum depth ΔGmin of the PMF to describe the strength of effective attractions between like-charged colloidal particles. As clearly shown in Figure 2c, with the increase of |Z|, negative ΔGmin decreases monotonically until a “critical” value and increases when |Z| exceeds the “critical” value. Namely, there is a nonmonotonic dependence of multivalent ion-mediated like-charge attraction on the particle charge for both 2:2 and 3:3 salts, and the “critical” values of particle charges are |Z|c ∼ 54 e for the 0.01 M 2:2 salt and |Z|c ∼ 90 e for the 0.1 mM 3:3 salt, respectively.

2.2. Driving Forces for Effective Like-Charge Attraction over a Wide Range of Colloidal Particle Charges

To understand the above-shown nonmonotonic dependence of the multivalent ion-mediated attraction between like-charged colloidal particles on particle charge, we first analyzed the driving forces for the effective like-charge attractions over a wide range of particle charges for 2:2 and 3:3 salts by calculating the electrostatic forces and depletion forces according to eqs 7 and 8; see Figures 3 and S2 in the Supporting Information. Afterward, we plot the minimum values of the electrostatic force and the depletion force (denoted Fmin) versus the particle charge in Figure 3c,f, and the separations for the minimum value are chosen where the electrostatic forces and the depletion forces appear to be most attractive.

Figure 3.

Figure 3

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(a,b) Electrostatic forces between like-charged colloidal particles immersed in the 0.01 M 2:2 salt solution (a) and in the 0.1 mM 3:3 salt solution (b). (c,d) Depletion forces between like-charged colloidal particles immersed in the 0.01 M 2:2 salt solution (c) and in the 0.1 mM 3:3 salt solution (d). (e,f) Minimum values of the electrostatic force and depletion force over separation x versus the particle charge in the 0.01 M 2:2 salt solution (e) and in the 0.1 mM 3:3 salt solution (f). Full and open symbols represent the electrostatic force and the depletion force, respectively.

It is shown that there are roughly three regimes of colloidal particle charges associated with the electrostatic and depletion forces. At a low particle charge (up to |Z| ∼ 18 e for the 2:2 salt and up to |Z| ∼ 12 e for the 3:3 salt), the electrostatic forces are obviously attractive and almost dominate the overall PMFs compared with the repulsive or weakly attractive depletion force. The effective like-charge attraction in this range of particle charge was attributed to the obviously attractive electrostatic force caused by the accumulation and bridging effect of counterions in the region between two like-charged colloidal particles.13,84 As the particle charge rises up to |Z| ∼ 54 e for the 2:2 salt and |Z| ∼ 90 e for the 3:3 salt, the attractive electrostatic force between colloidal particles is not very sensitive to the particle charge and does not play a dominating role in determining the overall PMFs any longer compared to the strongly attractive depletion forces. The attractive depletion force is attributed to the Coulomb depletion and the resultant imbalance of counterions inside and outside the charged colloidal particles, which is a combined effect of the Coulomb attraction between colloidal particles and counterions and the strong Coulomb repulsion between condensed counterions.52 As the particle charge exceeds |Z| ∼ 54 e for the 2:2 salt and |Z| ∼ 90 e for the 3:3 salt, the attractive electrostatic forces apparently increase to repulsive ones and become more repulsive for higher particle charges, while the depletion forces become slightly more attractive. Consequently, the attractive PMFs become weakened in this particle charge range; see Figure 3c,f.

Therefore, based on the above analyses, we can draw the following conclusions: (i) according to the components of driving forces, the like-charge attraction mediated by multivalent ions for a wide range of colloidal particle charges can be roughly divided into three particle-charge regimes, that is, the regime with an attractive electrostatic force and a repulsive/weakly attractive depletion force, that with an attractive electrostatic force and an attractive depletion force, and that with a repulsive electrostatic force and an attractive depletion force; (ii) the nonmonotonic dependence of multivalent ion-mediated like-charge attraction on the particle charge is mainly attributed to the transition of the electrostatic force from attraction to repulsion when the particle charge exceeds the “critical” value. The transition of electrostatic force from attraction to repulsion at a high particle charge is explicitly shown above and the mechanism will be discussed below.

2.3. Distribution of Ions around Colloidal Particles Is Responsible for Driving Forces

To understand the nonmonotonic dependence of multivalent ion-mediated like-charge attractions on the particle charge at the microscopic level, we analyzed the distribution of the counterions condensed on the surface of colloidal particles since the electrostatic and depletion forces are both determined by the condensed counterions around colloidal particles. As shown in the top portion of Figure 4a, the distribution of counterions near the particle surface depends mainly on the polar angle θ with respect to the X-axis at close separation. Here, we used the charge density σc(θ) of condensed counterions within a condensation shell of a thickness of ΔR = rion + 2 Å at the particle surface to analyze the electrostatic and depletion forces and the slight change of ΔR does not visibly affect our following analyses; see Figure S3 in the Supporting Information. Figure 4 shows the two-dimensional landscapes of counterion distributions and the charge densities σc(θ) of condensed counterions around two colloidal particles with different particle charges |Z| for the 2:2 salt and at the 3:3 salt where the inter-particle separation was taken as a typical value of x = 2R + 2rion + 1 Å.

Figure 4.

Figure 4

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(a) Two-dimensional distribution of counterions around colloidal particles with different charges in the 0.01 M 2:2 salt solution. (b) Charge density σc(θ) of the condensed counterion around one colloidal particle with different charges in the 0.01 M 2:2 salt solution (black) and in the 0.1 mM 2:2 salt (red) solution. From bottom to top: |Z| = 12 e to |Z| = 78 e. (c) Two-dimensional distribution of counterions around colloidal particles with different charges in the 0.1 mM 3:3 salt solution (black) and in the 0.01 mM 3:3 salt (red) solution. (d) Angular density σc(θ) of the condensed counterions around one colloidal particle with different charges in the 0.1 mM 3:3 salt solution. From bottom to top: |Z| = 6 e to |Z| = 114 e. Here, the particle–particle separation x was taken as a typical value x = 2R + 2rion + 1 Å for all panels; see the main text.

As shown in Figure 4, for low particle charges (|Z| < ∼30 e for the 2:2 salt and |Z| < ∼18 e for the 3:3 salt), there are apparently condensed bridging ions between two like-charged colloidal particles for both 2:2 and 3:3 salts, reflected by the high density of counterions and apparent peak in σc(θ) between two charged colloidal particles, that is, around |θ| ∼ 0. With the increase of the particle charge (up to ∼54 e for the 2:2 salt and up to ∼66 e for the 3:3 salt), the condensed bridging ions gradually becomes less apparent, and simultaneously the ion depletion zone begins to appear apparently, reflected by the relatively decreased peak height and the apparent valley depth in σc(θ) between two charged colloidal particles. When the particle charge becomes very high (|Z| > 66 e for the 2:2 salt and |Z| > 90 e for the 3:3 salt), the bridging ions nearly disappear and ion depletion becomes very apparent, as shown by the crystal-like charge density σc(θ) with almost a disappeared peak and a very apparent valley. The apparent crystal-like structure suggested that the Coulomb depletion can not only affect the depletion force but also affect the electrostatic force.

The electrostatic and depletion forces are strongly associated with the peak height of σc(θ) for bridging ions and the valley depth for ion depletion, respectively.14 At a low particle charge, there is an apparent peak of bridging ions and no valley (depletion zone), and correspondingly, the electrostatic force is attractive and the depletion force is repulsive. As the particle charge is increased, more bridging ions are necessary to be condensed to counteract the repulsion between like-charged colloidal particles to induce an effective like-charge attraction. However, for a very high particle charge, bridging ions cannot increase continuously and even decrease relatively to condensed counterions outside with the increase of the particle charge due to the Coulomb depletion (repulsion) between condensed counterions and the valley of σc(θ) in the limited region between two colloidal particles. Consequently, with the increase of the particle charge, the electrostatic force becomes more attractive for low particle charges while it can become repulsive at very high particle charges. Simultaneously, the depletion force is repulsive at a low particle charge |Z| and become (more) attractive with the increase of |Z| due to the more apparent valley of σc(θ) and the consequent stronger collisions from condensed counterions outside than those inside the two colloidal particles. It is also shown in Figure 4 that the more apparent peak height and valley depth of σc(θ) at high particle charges for the 0.1 mM 3:3 salt than that for the 0.01 M 2:2 salt suggests that the attractive electrostatic and depletion forces induced by trivalent counterions is generally stronger than those induced by divalent ones, and therefore the effective like-charge attractions for the 3:3 salt are generally stronger than those for the 2:2 salt.

To characterize the strength of Coulomb depletion, we introduce a reduced distance d/2rion between neighboring condensed counterions relative to the diameter of the counterions. Here, d can be measured by the distance between the first peak and the second peak of the charge densities σc(θ) shown in Figure 4b,d, and a smaller d/2rion corresponds to a stronger Coulomb depletion. As shown in Figure 5a,b, with the increase of particle charge |Z|, the reduced distance d/2rion decreases and approaches ∼1 at an extremely high particle charge, where d/2rion ∼ 1 means the minimum distance between counterions before overlapping. This indicates that the strength of Coulomb depletion increases with the increase of the particle charge, that is, the crystal-like structure becomes more apparent at a higher particle charge. Interestingly, when the reduced distances d/2rion decrease to a value slightly smaller than 1.5, |Z|’s are close to the “critical” values |Z|c for both the 2:2 salt and the 3:3 salt though the “critical” particle charge |Z|c for the 2:2 salt is visibly lower than that for the 3:3 salt.

Figure 5.

Figure 5

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Reduced distance d/2rion between adjacent condensed counterions on the colloidal particle surface. (a) For the case of 0.01 M 2:2 salt, colloidal particle radii R = 9 Å and ion radii rion = 2 Å; (b) For the case of 0.1 mM 3:3 salt, colloidal particle radii R = 9 Å and ion radii rion = 2 Å; (c) For the case of 0.1 mM 3:3 salt, colloidal particle radii R = 9 Å and ion radii rion = 2.5 Å; (d) For the case of 0.1 mM 3:3 salt, colloidal particle radii R = 8 Å and ion radii rion = 2 Å. See also Figures 4 and S5a in the Supporting Information.

Based on the above analyses, we can understand the mechanism for the nonmonotonic colloidal particle-charge dependence of the like-charge attraction as follows. When the particle charge becomes very high, the counterion condensation in the limited region between two particles is suppressed severely due to the Coulomb depletion; thus, the attractive electrostatic force will become less attractive and even repulsive since the suppressed counterion condensation and bridging effect between two particles cannot compensate for the increased Coulomb repulsion between like-charged particles with higher charges. Consequently, the like-charge attraction dominated by the attractive depletion force becomes weakened by the repulsive electrostatic force for a very high particle charge. Furthermore, we found that the “critical” particle charge |Z|c corresponds to the reduced distance between adjacent condensed counterions d/2rion ∼ <1.5 for both the 2:2 and 3:3 salts.

2.4. Salt Concentration Effect

To examine the effect of the salt concentration on the nonmonotonic dependence of the like-charge attraction between colloidal particles on particle charge, we made additional calculations of the PMFs for different 2:2 and 3:3 salt concentrations. As shown in Figure 6a,b, when the salt concentration decreases, the “critical” particle charge |Z|c increases slightly and the negative ΔGmin increases for both 2:2 and 3:3 salts. At a lower salt concentration, the less negative ΔGmin corresponds to the weaker effective like-charge attraction, which is attributed to the higher entropy penalty for counterion condensation and the resultant weaker counterion condensation.

Figure 6.

Figure 6

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Nonmonotonic dependence of the like-charge attraction strength on the colloidal particle charge for the 2:2 salt solution (a) and for the 3:3 salt solution (b).

In order to understand the higher “critical” particle charge |Z|c at a lower salt concentration, we calculated the charge density σc(θ) of condensed counterions around the colloidal particles since the suppression of counterion condensation in the limited region between two particles is mainly responsible for the appearance of the “critical” particle charge as discussed above. As shown in Figure 4b,d, a higher concentration leads to a lower relative peak height of bridging ions because there is no noticeable change of the ion condensation in the intervening region between two colloidal particles (i.e., region of |θ| < 0.15π), while the counterion condensation in the outer region becomes slightly stronger. This suggests that the suppression of counterion condensation in the intervening region at a higher salt concentration is more severe due to the stronger Coulomb depletion, and consequently the electrostatic force begins to become repulsive at a lower |Z| for the higher salt. Therefore, the “critical” particle charge |Z|c for the nonmonotonic dependence of effective like-charge attraction on the particle charge decreases slightly with the increase of the 2:2 or 3:3 salt concentration. However, such effect of 2:2 and 3:3 salt concentrations is still very slight, and this is because the multivalent ions can interact very strongly with charged colloidal particles and the condensation of multivalent ions is only weakly dependent on the salt concentration.86

2.5. Ion Size Effect

To examine the ion size effect for the dependence of the like-charge attraction between colloidal particles on particle charge, we made another series of calculations on the PMFs between two like-charged colloidal particles at the 0.1 mM 3:3 salt with larger ion radii (rion = 2.5 Å). As shown in Figures 7a,b and S1 (c) in the Supporting Information, the PMFs for rion = 2.5 Å are less attractive than for rion = 2 Å and the “critical” particle charge (|Z|c ∼ 60 e) for rion = 2.5 Å is apparently lower than that (|Z|c ∼90 e) for rion = 2 Å. To understand the ion size effect, we plot the minimum values of electrostatic force and depletion force as functions of particle charge |Z| in Figures 7c and S4a–c in the Supporting Information. As shown in Figures 7 and S4a–c in the Supporting Information, the effective attraction weakening at a high charge is obviously attributed to the repulsive electrostatic force at a high particle charge, given that the attractive depletion force always becomes stronger with the increase of the particle charge. Furthermore, Figure 7c shows that the electrostatic force for rion = 2.5 Å becomes repulsive at a lower particle charge and is more repulsive than that for rion = 2 Å at a high particle charge, while the depletion force for rion = 2.5 Å is less attractive than that for rion = 2 Å at a high particle charge. Thus, compared with the case of rion = 2 Å, the PMFs for rion = 2.5 Å are less attractive and the enhanced trend of the attractive depletion force by a higher particle charge for rion = 2.5 Å could be counteracted by a repulsive electrostatic force at a lower particle charge and consequently, the effective like-charge attraction becomes weakened at a lower “critical” particle charge for larger ions.

Figure 7.

Figure 7

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(a) PMFs between like-charged colloidal particles with radii R = 9 Å in the 0.1 mM 3:3 salt solution with radii rion = 2.5 Å from the MC simulations. (b) Dependence of the like-charge attraction strength on the particle charge in the 0.1 mM 3:3 salt solution: R = 9 Å and rion = 2.5 Å (black); R = 9 Å and rion = 2 Å (red). (c) Minimum values of the electrostatic force and the depletion force versus the particle charge in the 0.1 mM 3:3 salt solution: R = 9 Å and rion = 2.5 Å (black); R = 9 Å and rion = 2 Å (red). (d) PMFs between like-charged colloidal particles with radii R = 8 Å in the 0.1 mM 3:3 salt solution with radii rion = 2 Å from the MC simulations. (e) Dependence of the like-charge attraction strength on the particle charge in the 0.1 mM 3:3 salt solution: R = 8 Å and rion = 2 Å (black); R = 9 Å and rion = 2 Å (red). (f) Minimum values of the electrostatic force and depletion force over separation x versus the particle charge in the 0.1 mM 3:3 salt solution: R = 8 Å and rion = 2 Å (black); R = 9 Å and rion = 2 Å (red). Lines in (b,e) are guide to the eye.

Similar to the analyses in the above subsections, the disappearance of bridging ions due to the strong Coulomb depletion in the intervening space between two colloidal particles is responsible for the weakening of like-charge attraction at a high particle charge; see Figure S5a in the Supporting Information. It indicates that the Coulomb depletion for larger ions is stronger than that for the smaller ones, causing a lower “critical” particle charge |Z|c for the case of larger ions; see Figure 5c. It is worth noting that the reduced distance d/2rion at the “critical” particle charge |Z|c for rion = 2.5 Å is also slightly smaller than ∼1.5.

2.6. Effect of Particle Size

We have also made the calculations for smaller like-charged colloidal particles with R = 8 Å to examine the effect of particle size for the dependence of the like-charge attraction between colloidal particles on particle charge |Z|. As shown in Figures 7c,d and S1d in the Supporting Information, the multivalent ion-mediated attraction becomes weakened at a lower particle charge (|Z|c ∼ 80 e) compared with that for the charged colloidal particle with radii R = 9 Å (|Z|c ∼ 90 e). Such a lower |Z|c for smaller colloidal particles is attributed to the relations between the repulsive electrostatic force and the particle charge. As shown in Figures S4d–f in the Supporting Information, the electrostatic force at a high particle charge for R = 8 Å becomes repulsive at a lower particle charge and becomes rapidly more repulsive with |Z| than that for R = 9 Å, while the depletion force for R = 8 Å is only slightly more attractive than that for R = 9 Å. Thus, the attractive depletion force for R = 8 Å can be counteracted by a strong repulsive electrostatic force at a lower |Z|, and consequently, the nonmonotonic dependence appears at a lower “critical” |Z|c for smaller charged colloidal particles.

Similar to the proceeding analyses, the disappearance of the bridging ions’ peak in σc(θ) due to the strong Coulomb depletion in the intervening space between two colloidal particles is responsible for the weakening of like-charge attraction at a high particle charge; see Figure S5b in the Supporting Information. The more apparent Coulomb depletion for a smaller particle size is reasonable. First, counterions are more difficult to condense to serve as bridging ions in the more limited region between two smaller particles. Second, compared to larger colloidal particles, the smaller ones will attract more counterions to condense on the smaller surface area, and thus the average distance is reduced and the resultant Coulomb repulsion between condensed counterions enhanced. As shown in Figure 5b,d, the reduced distance d/2rion between condensed counterions for the smaller particles is obviously smaller than that for the larger ones. This indicates that the Coulomb depletion for the smaller colloidal particle is stronger than that for the larger ones, which causes a more repulsive electrostatic force and a lower “critical” particle charge |Z|c for the smaller colloidal particles. Furthermore, we found that the reduced distance d/2rion at |Z|c for R = 8 Å and rion = 2 Å is close to a value slightly smaller than 1.5, which is consistant with the above discussed cases.

2.7. Analytical Formula for the “Critical” Particle Charge

Based on the above extensive MC simulations and analyses, the nonmonotonic dependence of like-charge attraction between colloidal particles on the particle charge is attributed to the Coulomb depletion of condensed counterions at a high particle charge, which suppresses the counterion condensation in the intervening space between two like-charged colloidal particles. In the following, we will derive an analytical formula for the “critical” particle charge. As shown above, the distance d between two adjacent condensed counterions can be used to measure the strength of Coulomb depletion of condensed counterions, and d is given by63,87

2.7. 1

where q is the charge of the counterions. d is derived based on the assumption that the charged colloidal particles were fully neutralized63,87,88 and such an assumption is obviously valid for our multivalent ion-particle systems; see Figure S3 in the Supporting Information. As shown in Figure 5, d from eq 1 agrees well with the values from the MC simulations. Inspired by the formation of the crystal-like structure of condensed counterions around the “critical” particle charge |Z|c and following previous works,88−90 we used the Lindemann melting law to characterize the strong Coulomb depletion of condensed counterions at |Z|c. Thus, the distance between adjacent condensed counterions dc at the “critical” |Z|c satisfies

2.7. 2

where Δr is the fluctuation displacement of condensed counterions around its equilibrium position. Here, d > dc and d < dc correspond to the liquid-like state and crystal-like state of condensed counterions, respectively. According to the Lindemann melting law, Δr = 0.15dc corresponds to the value of the solid–liquid transition.86,88 Then, eq 2 gives

2.7. 3

As shown in Figure 8a, eq 3 agrees well with the extensive MC simulations over system parameters including ion valence, ion size, ion concentration, and colloidal particle size. Furthermore, the combination of eqs 1 and 3 gives the “critical” particle charge |Z|c

2.7. 4

Figure 8.

Figure 8

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(a) Minimum values of the PMFs between like-charged colloidal particles as a function of the distance d/2 rion for system parameters; see eq 1 for d in the main text. (b) “Critical” particle charge |Z|c (MC) from the MC simulations versus |Z|c (eq 4) from eq 4. Due to the spare data in the MC simulations, |Z|c (MC) were obtained by the interpolation based on the MC data. Red symbols: 0.1 mM 3:3 salt, R = 9 Å and rion = 2.5 Å; black symbols: 0.1 mM 3:3 salt, R = 8 Å and rion = 2 Å; olive symbols: 0.1 mM 3:3 salt, R = 9 Å and rion = 2 Å; blue symbols: 0.01 mM 3:3 salt, R = 9 Å and rion = 2 Å; cyan symbols: 0.01 M 2:2 salt, R = 9 Å and rion = 2 Å; purple symbols: 0.001 M 2:2 salt, R = 9 Å and rion = 2 Å; brown symbols: 0.1 mM 2:2 salt, R = 9 Å and rion = 2 Å.

As shown in Figure 8b, |Z|c’s from eq 4 are nearly in quantitative accordance with those from the MC simulations including system parameters.

Therefore, we can generalize the weakening of multivalent ion-mediated attraction between two like-charged colloidal particles at a high particle charge as follows: when particle charge |Z| becomes very high and exceeds the “critical” value |Z|c, the condensed counterions transit from the liquid state to the crystal one due to the very strong Coulomb depletion. This would lead to the repulsive electrostatic force, which counteracts the attractive depletion force and consequently results in the effective attraction weakening. If the condensed counterions are modeled as one component plasma,1 such weakening of multivalent ion-mediated attraction would correspond to the phase transition from the liquid state to the crystal one for one component plasma.

3. Conclusions

In summary, in the present work, we made extensive Monte Carlo simulations for calculating the PMF between like-charged colloidal particles with a wide range of particle charges for both divalent and trivalent salts of different concentrations. Through extensive calculations and detailed analyses, we have reached the following major conclusions:

  • (1)

    The multivalent ion-mediated attraction between like-charged colloidal particles is nonmonotonically dependent on particle charge: with the increase of particle charge, the effective attraction is apparently strengthened when the particle charge is lower than a “critical” value, while such attraction would become gradually weakened when the particle charge exceeds the “critical” value. The “critical” particle charge for the 3:3 salt is higher than that for the 2:2 salt.

  • (2)

    The driving forces for effective attraction between like-charged colloidal particles mediated by divalent/trivalent ions could be divided into three particle-charge regimes: (i) a low particle charge regime with an attractive electrostatic force and a repulsive/weakly attractive depletion force; (ii) a medium particle charge regime with an attractive electrostatic force and an attractive depletion force; (iii) a high particle charge regime with a repulsive electrostatic force and an attractive depletion force.

  • (3)

    The nonmonotonic dependence of the multivalent ion-mediated like-charge attraction on particle charge is attributed to the unexpected repulsive electrostatic force induced by Coulomb depletion, which suppresses the counterion condensation in the limited region between colloidal particles.

  • (4)

    The “critical” particle charge decreases apparently for larger ions and smaller colloidal particles due to the stronger Coulomb depletion effects for larger ions and smaller colloidal particles and decreases slightly at a higher salt concentration due to the slightly enhanced Coulomb depletion of counterions between colloidal particles.

  • (5)

    We derived an analytical formula for the “critical” particle charge based on the Lindemann melting law and such a formula agrees well with our extensive MC simulations including system parameters.

In spite of the above major conclusions, our model system involves some important simplifications. First, the charges of colloidal particles were placed at the centers of the respective particles and consequently the discreteness of the particle charges at the surfaces was ignored. Such simplification may affect counterion distributions in the very close vicinity of colloidal particles. Second, the solvent was modeled as a continuous dielectric medium, and the dielectric boundary between particles and solvent was ignored. In fact, colloidal particles generally have a lower dielectric constant than the solvent outside,91−93 and counterions at the particle surface would experience the repulsion from the induced ion image charges, which would disfavor the counterion condensation.93−95 Such an effect may be partially compensated by the enhanced electrostatic attraction between counterions and particles with a low dielectric constant.86 Third, for simplicity, divalent and trivalent salts were modeled as symmetrical 2:2 and 3:3 salts with equal ion sizes rather than realistic salts. Although these simplifications were often used in previous model systems for ion–particle interactions, more detailed and accurate treatments are still required to be involved in future works, including discrete charge distributions, a discontinuous dielectric boundary effect, and more realistic (mixed) salts. Nevertheless, our finding and analyses can be very helpful for understanding the ion-mediated effective interactions between charged colloidal particles and the assembly of charged colloidal particles.

4. Model and Method

In this work, we investigated effective interactions between like-charged colloidal particles in symmetrical multivalent salt solutions by canonical ensemble MC simulations based on a primitive model in which salt ions were considered as small charged spheres and the solvent was modeled as a continuum medium with a dielectric constant ε.54 In our simulations, two large like-charged colloidal particles were immersed in a rectangular box and the particles were symmetrically located on two sides of the plane at x = 0. For simplicity, interactions between charged colloidal particles and ions are composed of Coulombic interactions and hard-core interactions

4. 5

here, ai and qi stand for the radius and charge of sphere i (small ions and large colloidal particles), and rij is the center-to-center distance between spheres i and j. ε0 and ε are the vacuum permittivity and the dielectric constant of the solvent, respectively. In practice, to diminish the boundary effect, the sizes of simulation boxes were always kept larger than two colloidal particles by at least 6 times the Debye–Hückel length, and the calculated results are stable as tested against different box sizes. In the model system, we fixed the radii of two colloidal particles R to 9 Å, the radii of salt ions rion to 2 Å, and the temperature to 298.15 K (room temperature) in all the MC simulations. Our simulation systems are illustrated in Figure 1. The charge Z of colloidal particles ranges from −6 to −78 e for the 2:2 salt and ranges from −6 to −114 e for the 3:3 salt, respectively. Furthermore, we made the calculations for a different colloidal particle size (R = 8 Å) and a different ion size (rion = 2.5 Å) to examine the colloidal particle size effect and ion size effect. Additionally, we made the calculations for different concentrations of the 2:2 salt and the 3:3 salt to examine the salt concentration effect.

In our simulations, the Metropolis algorithm96 was employed to generate the distributions of ions at equilibrium. Each MC simulation starts from an initial configuration with fixed colloidal particles in the X-axis with separation x and randomly distributed ions, and the probability to accept a trial move of an ion is given by p = min[1,exp(−ΔU/kBT)], where ΔU is the interaction energy change associated with the trial move of the ion in the simulation box. kB is the Boltzmann constant, and T is the absolute temperature in Kelvin. Six million configurations were collected to calculate the average mean force after the pre-equilibrium process.

The total mean force acting on colloidal particle i along the reaction coordinate (the X-axis) is composed of two terms

4. 6

Here, Felei(x) is the electrostatic force between colloidal particle i and all other charged objects, and Fhsi(x) is the depletion force (hard-sphere collision force) between colloidal particle i and the counterions in contact with it. The electrostatic force Felei(x) is expressed as40,52,54,97

4. 7

where θ is the polar angle formed with the reaction coordinate and the angular brackets···denote the ensemble average. x is the separation between two colloidal particles and rij is the distance between colloidal particle i and ion j. Zi and qj are charges of colloidal particle i and ion j, respectively. The depletion force Fhsi(x) based on the contact theorem in spherical geometry is given by

4. 8

where n(θ) is the number of counterions located in a spherical shell with polar angle θ and thin thickness ΔR at the surface of colloidal particle i. In practice, ΔR = 0.02 Å was used in our calculations for the calculation accuracy and efficiency and the choice of ΔR around the value does not have a visible influence on our calculated results. The PMFs were calculated through integrating the ensemble-averaged mean forces along the reaction coordination x to describe the effective interactions between like-charged colloidal particles in multivalent salt solutions.40,52,54,97 Thus, the PMF ΔG(x) can be calculated numerically as40,54

4. 9

where the outer reference separation was taken as xref = 40 Å in practice.

Acknowledgments

We are grateful to Prof. Shi-Jie Chen (Univ. Missouri) and Prof. Xiangyun Qiu (George Washington Univ.) for valuable discussions. This work was supported by the National Natural Science Foundation of China grant nos. (11774272 and 12075171). Parts of numerical calculations in this work were performed on the supercomputing system in the Supercomputing Center of Wuhan University.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.1c00613.

  • PMFs between like-charged colloidal particles with a wide range of particle charges in 2:2 and 3:3 salt solutions; electrostatic forces and the depletion forces between like-charged colloidal particles in 2:2 and 3:3 salt solutions; definition of the charge fraction of the counterions condensed on the surface of colloidal particles; and condensed charge density σc(θ) of counterions under different conditions for different particle charges |Z| (PDF)

The authors declare no competing financial interest.

Supplementary Material

ao1c00613_si_001.pdf (1MB, pdf)

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