Density Functional Analysis of Like-Charged Attraction between Two

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Density Functional Analysis of Like-Charged Attraction between Two Similarly Charged Cylinder Polyelectrolytes Shiqi Zhou School of Physics and Electronics, Central South University, Changsha, Hunan 410083, China S Supporting Information *

ABSTRACT: A systematic theoretical investigation is performed for electrostatic potential of mean force (EPMF) between two similarly charged rods (modeling DNA) immersed in a primitive model electrolyte solution. Two scientific anomalies are disclosed: (i) although a like-charge attraction (LCA) generally becomes stronger with bulk electrolyte concentration, the opposite effect unexpectedly occurs if the two rod surfaces involved are sufficiently charged and (2) contrary to what is often asserted, that the presence of multivalent counterion is necessary to induce the LCA, it is found that the univalent counterion induces the LCA solely only if bulk electrolyte concentration rises sufficiently and the rod surface charge quantities are high. On the basis of the system energetics calculated first by a classical density functional theory in three-dimensional space, a hydrogenbonding style mechanism is advanced to reveal the origin of the LCA, and by appealing to fairly common-sense concepts such as bond energy, bond length, number of hydrogen bonds formed, and counterion single-layer saturation adsorption capacity, the present mechanism successfully explains the scientific anomalies and effects of counterion and co-ion diameters in eliciting the LCA first investigated in this work. To add weight to the hydrogen-bonding style mechanism, a theoretical investigation is further performed regarding the effects of the rod surface charge density, co-ion valence, relative permittivity of the medium, temperature, nonelectrostatic interion interactions, and rod diameter in modifying the EPMF, and several novel phenomena are first confirmed, which is self-consistently explained by the present hydrogen-bonding style mechanism.

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INTRODUCTION

proven that the standard mean-field PB theory will always predict repulsion between like-charged macroions, regardless of their shape.4 However, the presence of multivalent salt ions can induce an effective attraction between like-charged macroions, and this was confirmed by a large number of computer simulations5 and observed experimentally.6 The like-charged attraction (LCA) is used to elucidate experimentally observed multivalent counterion-induced condensation of DNA, a stiff, highly and negatively charged polyelectrolyte. Notorious examples include the formation of dense packages of DNA molecules (DNA condensates),7 aggregates of colloidal particles,8 and the 2D superlattice of the rodlike tobacco mosaic virus (TMV) formed in an aqueous electrolyte solution.9 The LCA is believed to be important for the

Aqueous solutions of charged macroions play an important role in everyday life and have various industrial, biotechnological, and medical applications. Electrostatic interactions (EI) necessarily are critical to most of these applications since a huge class of soft-matter and biological systems are water soluble and, thus, bear electric charges in aqueous solutions, and it is indicated that the EI offers the basis of a novel particletrapping method for manipulation of particle systems at the nanometer scale1 and is believed to be the dominant mechanism of a precise and large-scale placement of nanoparticles through a scheme named “electrostatic funneling”.2 The presence in solution of smaller ions influences the effective electrostatic interactions between charged macro-ions. In the weak coupling limit, the effective interactions between like-charged surfaces are described satisfactorily by a widely used Poisson−Boltzmann (PB) theory, which predicts that likecharged polyelectrolytes always repel,3 and it can be rigorously © 2013 American Chemical Society

Received: December 24, 2012 Revised: September 3, 2013 Published: September 10, 2013 12490

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compaction of DNA inside viral capsids10 and the DNA biological function by influencing DNA folding, packaging,11 pairing,12 and interactions with other biological macromolecules.13,14 The approximations introduced by the PB theory include omitting finite ion size and ion−ion correlation effects, which prohibit the PB theory application to the nonweak coupling region (i.e., when multivalent counterions are present, the macroions are highly charged) or at high Bjerrum lengths. However, it is precisely in the nonweak coupling region where LCA occurs, and this is exactly the reason why the PB theory fails to predict the LCA. As a result, something beyond the PB theory is needed for a satisfactory determination of the electrostatic properties, a highly nontrivial many-body problem involving long-range interactions. It is known that some specifically designed theoretical approaches are available, which can predict the LCA. However, external concepts such as counterion condensation are introduced into these schemes for the aim of simplification, and this makes the resulting theories deviate somewhat from the first principle feature and undesirably contain nonelectrostatic components by the modeling constraint that the counterions fall into two explicit populations, condensed and uncondensed.15 All in all, counterion condensation is a physical reality, which is observed in simulations and in countless experiments.16 Moreover, the approximations involved only allow for reliable applications of the resulting theories for a certain interval of parameters, and one such example is a strong coupling (SC) theory. It is confirmed that strong deviations between simulations and the SC theory predictions for the electrostatic potential of mean force (EPMF) between two rods occur at all temperatures, if two relevant geometrical parameters are not sufficiently large;17 in fact, for most colloid systems and relevant ranges of coupling parameters, the geometrical conditions are not met.17 On the other hand, a long-range correlation of the electrostatic force makes the computer simulation in high-dimensional spaces a very timeconsuming activity, and this makes it hardly realistic (additionally because of a need for a much larger simulation box) to simulate the relevant EI over a wide range of parameters, as evidenced by a scarcity of reports of systematical colloid EI knowledge. Consequently, probable scientific anomalies and systematical exploration about the EPMF may be mostly out of sight of the current computer simulation practices. Finally, the main instruments in intermolecular and surface force measurements are adapted to large mica sheets with a radius of curvature standing of ∼1 cm,3 and it is difficult to offer valuable information about nanoscale colloids, for which the Derjaguin approximation18 is clearly inappropriate. Consequently, the EPMF between two small colloids remain less understood; however, they are relevant to polymers, biological molecules, and other nanoscale colloids. The state above of theoretical progresses is not conducive to exploring the EI of small colloids over a wide range of parameter space and is going against the disclosure of the LCA mechanism with a global validity. As a result, in spite of the spectacular progresses achieved on the LCA during the past two decades,19−23 the mechanism capable of explaining all phenomena about the LCA has not yet taken shape. In contrast, the two effects omitted in the PB theory can coherently be retrieved in a classical density functional theory (DFT) framework.24 Success of the classical DFT in dealing with the electrical double layer formed around the charged

colloid,25−27 particularly the two effects missing in the PB theory, critically depends on availability of a bulk second-order direct correlation function (DCF) expression for electrolyte solutions,28 whose utilization in a functional perturbation expansion approximation and satisfactorily accounts for the correlation effects;25−27,29 as for the finite ion size effect, its accurate estimation is guaranteed due to the development of sophisticated density functional approximation for hard sphere correlation.24,30 However, it is too bad that the classical DFT is not analytically solvable, even under the linearization approximation as the PB theory, and thus, any attempt to go beyond the Derjaguin approximation for the EPMF between curved surfaces inevitably involves a numerical solution of the classical DFT for charged systems in two- and threedimensional spaces, which is not something easily dealt with as demonstrated by a complete lack of relevant literature. The aim of this work is two-fold. One is to find novel phenomena related to the LCA, whose existence is not observed previously in experiments, theories, and simulations; second, the present observations and previous ones are combined to formulate a globally effective mechanism explanation for the LCA. To achieve this goal, we first perform 3D classical DFT calculations (displayed graphically in Figure 1) for two parallel charged planes decorated with adsorbed 3D charged spheres to represent the counterions. This calculation is performed primarily as an illustration of the present globally effective mechanism explanation for the LCA and to provide deterministic evidence and dispel the conjecture character of our mechanism explanation. Then, the rest of the paper is devoted to two-dimensional calculations of the EPMF between two infinitely long, hard, parallel, and negatively charged cylinder rods immersed in a primitive model (PM) electrolyte solution, which can be regarded as a simple DNA model. As far as the author knows, although the PB theory is solved for arbitrarily complicated geometries, previous numerical implementation of the classical DFT for the charged systems is limited to one-dimensional space, and the present article for the 2D and 3D spaces is presented for the first time. With a prerequisite that validity of the classical DFT for charged systems is demonstrated by a large amount of work24−27 and there is no need to introduce additional approximations to numerical solutions, the numerical results can be informative over a reasonable range of parameters and systematic and reliable phenomenological exploration of the LCA is guaranteed.

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RESULTS AND DISCUSSION In the present investigation, the PM is used for the electrolyte solution, where the solvent is implicitly described by its dielectric constant, and the salt ions consist of hard spheres with a centered point charge, such that their electrostatic interactions are as described in Methods. For simplicity, the rod and solution have the same dielectric constant. Our calculation results are presented in Figures 1−9 in the text and Figures S1−S4 of the Supporting Information. To help in illustrating our view underlying the present new mechanism explanation for the LCA, we first consider two negatively charged planar walls of infinite extent immersed in a 2:2 type electrolyte water solution. A quantitative relevance in the EPMF between two planar geometries and two cylindrical ones is described by a well-known Derjaguin approximation (DA),31 which allows one to determine the force between two gently curved colloidal particles in a close proximity, based on 12491

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the contrary, the symmetrical permutation causes a rising of the system’s excess grand potential and gives a repulsive EPMF over the entire range of the surface separation. Rising or dropping of the system energy, we think, depends on a balance between direct attraction between the positively charged HS inlaid at the left planar wall, for example, and the negative surface charge on the right one and direct repulsion between two charged HS inlaid at the two face-to-face surfaces. For the cross permutation of the tightly adsorbed counterions, strength of the direct attraction outstrips that of the direct repulsion, since the former operates at shorter separation than the latter because of a dislocated distance between two tightly adsorbed counterions on the two face-to-face surfaces, and the attractive EPMF results, as illustrated in Figure 1a. Conversely, in the case of symmetrical configuration, the direct repulsion outstrips the direct attraction in strength, and thus, rising of the system excess grand potential results in contrast to the dropping in the case of asymmetrical configuration as shown in Figure 1b. In the name of the EPMF, the symmetrical and asymmetrical configurations correspond to a repulsive and an attractive EPMF, respectively. The attractive EPMF between electrically neutral bodies due to the cross permutation mechanism is reminiscent of hydrogen bond attraction occurring between two neutral molecules such as water, and we will try to explain the LCA and our new observations by making an analogy between them. In the case of two cylinder rods, we think that one of the two negatively charged rods takes the role of the oxygen atom (hydrogen donor), and it adsorbs counterions around itself; the adsorbed counterions play the role of hydrogen atom of the same water molecule and tend to be directly attracted to the surface domain of another of the two face-to-face cylinder rods if this surface domain is not covered by counterions. Consequently, the empty surface domain serves as the oxygen atom (hydrogen acceptor) of another water molecule involved in the hydrogen bonding. If the hydrogen bonding-like mechanism is a leading one in the generation of the LCA, it should explain equally well all LCA phenomena as reported previously and in this work. Below, we will try to show that this is true. Figure 5 shows that the nearest surface separation Hmin,i, at which the EPMF is the most attractive, actually does not * ). This change with alteration of the co-ion’s diameter (danion observation convinces one that the counterion dominates the LCA, and it is adsorbed onto two face-to-face surfaces of the two cylinder rods, but arranged alternately. The cross permutation is energetically favorable as explained above and also supported by the observation that Hmin,i is less than about 1.25 (high surface charge case) or 1.5 (low surface charge case) times the counterion diameter (d*cation) because otherwise Hmin,i will exceed 2 times the dcation * if the symmetrical configuration is operating. As the counterion serves as the role of the hydrogen atom involved in the hydrogen bonding, the higher the valence of the counterion, the more stable the hydrogen bonding formed, so the hydrogen bonding-like mechanism can easily explain why it is a multivalent counterion rather than a univalent counterion that can usually cause the LCA, as observed experimentally and in simulation. One notes that the hydrogen-bonding style mechanism clearly accords with the observation shown in Figure 2, which indicates that with the growth of rod surface charge density strength |σ*| the LCA becomes more appreciable, and one can attribute the positive correlation to two aspects. First, rod surface of high |σ*| will adsorb more counterions onto itself,

Figure 1. EPMF between two planar walls of infinite extent and the relevant excess grand potential vs surface separation h*. The two planar walls are periodically adorned with a hard sphere of unitreduced diameter and with a centered point charge of a positive divalent. Two configurations are considered: one called asymmetrical configuration with the hard sphere center reduced coordinates (0.25, 1.25) and (−0.25, −1.25) on the left and right planar wall, respectively, and the other symmetrical one with the relevant hard sphere center at the origin of coordinates. The periodic cell is a square with reduced size of 2 × 4, and other parameters are shown in the figure.

knowledge of interaction free energy of two infinite plates, which is normally much easier determined analytically or numerically than those involving colloid(s) with curved surface(s). Recent studies32 demonstrate that DA equally applies for colloids with surface roughness and/or charge heterogeneities, and this makes certain the validity of transferring theoretical arguments from one geometry to the other. To model adsorption of salt ions on the planar walls, we might as well assume that a fraction of the rod surface is tightly covered by divalent counterions, and the two planar walls are thus periodically adorned with hard spheres of a unit reduced diameter and a centered point charge of positive divalent. Two configurations, as shown in insets of Figure 1 (panels a and b), are considered to demonstrate the system energetics change induced by different arrangements of the counterions adsorbed on the rod surfaces, and the corresponding EPMF between the two planar walls and excess grand potential versus surface separation h* are plotted in Figure 1 (panels a and b). Figure 1 clearly illustrates that cross permutation of the counterions on the two surfaces is energetically favorable, and it causes an attractive EPMF at close separation, even if combination of the planar wall and the inlaid charged HS is electrically neutral; on 12492

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Figure 2. Effects of rod surface charge density on the EPMF induced by a 2:1 electrolyte solution. Bulk electrolyte concentration is c = 0.1 M, and other relevant parameters are described in the text.

and this understandably creates more chances for hydrogen bonding and causes a rising number of hydrogen bonds formed; second, the growth of |σ*| also means that the hydrogen acceptor will occur with more electrical quantity, and the hydrogen bonds formed tend to be strengthened. Moreover, Figure 2 shows that Hmin,i decreases with the increase of |σ*| and eventually reaches a constant value. The decrease of Hmin,i in the first stage can be interpreted as more close and firm bonding of the counterion as hydrogen to both hydrogen donor and hydrogen acceptor, and this undoubtedly originates from the rising of the electrostatic field induced by the hydrogen donor and hydrogen acceptor with higher |σ*|. As for the saturation phenomenon of Hmin,i at sufficiently high |σ*|, it should be due to the hard repulsion between the counterion and hydrogen donor and acceptor and that between the counterions adsorbed on two face-to-face surfaces of the two cylinder rods and dislocated some distance, as these hard repulsions prohibit infinite closeness between these surfaces. Our main discoveries include two scientific anomalies, as shown in Figures 3 and 4 and the effects of salt ion diameters on the EPMF, as exhibited in Figure 5. More specifically, one of

Figure 4. Bulk electrolyte concentration effects on the EPMF induced by 1:1 and 1:2 type electrolyte solutions, respectively. The |σ*| is assigned with a reasonably high value as 0.3 and other relevant parameters are described in the text.

the anomalies is about the electrolyte concentration (c) dependence of the LCA and illustrated in Figure 3. It is indicated that the LCA may weaken with rising of c, and this is completely different from general knowledge, which has been reached in colloid science and tells that the LCA intensifies with an increase of c. Figure 3 indicates that the common sense tendency does occur but only in the case of low |σ*|, and the abnormal concentration dependence appears in the high |σ*| extreme. By observing Figure 3 and Figures S1−S3 of the Supporting Information, one notes that different counterion valences correspond to different threshold values of |σ*|, above and below which the abnormal concentration dependence and common sense tendency apply, respectively; to be specific, for divalent and trivalent counterions, the threshold values are within |σ*| intervals of [0.15 0.2] and [0.1 0.125], respectively, and particularly, the threshold value interval seems not to be sensitive on the valence of the co-ion, as illustrated in Figure 3 and Figures S1 and S2 of the Supporting Information. We attribute the positive correlation for the low |σ*| zone to a fact that with an increase of c, more counterions will be adsorbed onto the rod surface, and this will produce more chance of formation of hydrogen bonds. As for the negative correlation for the high |σ*| zone, it seems to contradict the above explanation, but this is not true, as explained below. When |σ*| is high enough, the attractive electrostatic potential stemming from the rod surface charge and exerted on the counterion will be strong enough that the adsorbed counterions will always fill up the rod surfaces, and consequently, the counterion singlelayer adsorption capacity, which is responsible for the

Figure 3. Bulk electrolyte concentration effect on the EPMF induced by a 2:1 electrolyte solution. The calculations are for values of |σ*| belonging to two different regions, and other relevant parameters are described in the text. 12493

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not possible, as the two models used in the two studies are different. More specifically, the RLPE is modeled in ref 33 as N spheres (monomers), each with a charge, connected through springs with a large spring constant. The rigidity of the polymer backbone is ensured by a three-body interaction with a large force constant, whereas the polyelectrolyte is modeled as an infinitely long charged cylinder in the present work. Obviously, the two models clearly differ from each other in length, shape, and rigidity, etc. It also should be noted that the counterion condensation theory also predicts the LCA for univalent counterions.34 It is experimentally confirmed15,35 that the LCA occurs in the presence of a univalent counterion; in particular, the literature15,35 document shows clear examples of attraction at distances much farther than the essential contact distance found in the present work and such attraction in more dilute concentrations than the present extreme concentrations, such as 2 M. The author believes that the difference between calculated and experimentally measured attractions arises from some uncontrollable factors, which makes the experimentally measured attractions inevitably infiltrated by other types of force, such as solvation force, hydrophobical force, electric dipole force, depletion force, van der Waals force, etc., and the fact that only very few divalent counterions can in practice precipitate DNA and no univalent counterions do so even at very high concentrations just goes to show that these uncontrollable factors invalidate a naı̈ve comparison between theory and experiment. On the other hand, theoretical approaches play an indispensable role in exploring the individual electrostatic effect by adjusting relevant parameters and, thus, can supply clear demonstration of the electrostatic effects uncontaminated by other factors. Influence of the counterion diameter dcation on the LCA is depicted in Figure 5, in which one changes dcation from below the reference length to above the reference length and keeps the co-ion’s diameter dcation unchanged and vice versa. It is noted that whether high or low |σ*| situations (Hmin,i), the nearest surface separation at which the EPMF is the most attractive always increases with rising of dcation. This is in sharp contrast to the case of the co-ion diameter and is strong evidence of the counterion serving in the role of hydrogen atom as increase of the hydrogen atom diameter must mean rising of the length of the hydrogen bond, equivalent to Hmin,i, as it is. As for the correspondence between the LCA well depth and dcation, depending on whether low or high |σ*| zone is under consideration, it can be monotonic and nonmonotonic. It is gradually becoming more accepted that the LCA originates from electrostatic correlation effects; in the classical DFT language, the electrostatic correlation effects originate from the term Fcoupling in the free energy expression (see Methods), and indeed, if one gets rid of Fcoupling in the calculation, the LCA disappears immediately. Obviously, strength of Fcoupling depends and Δc(1)coupling (r), on the size of both coefficients, Δc(1)coupling α αβ and according to the language of liquid state theory, the latter actually measures the cross correlations between the electrostatic and steric interactions. Therefore, the following corresponding relation is clear: the coefficients will disappear as the ion size reduces to zero and will gradually become more noteworthy as the size increases. One may argue that the larger counterion should contribute to the repulsive interaction and, consequently, will weaken the LCA; however, this is not the actual situation. Although an increase of the ion size induces a rising of repulsion interaction, the rising occurs in bulk and near the rod simultaneously; as a result, the excess grand potential

maximum number of hydrogen bonds likely to be formed, has reached its saturation value and will not be influenced by the electrolyte concentration. Instead, an increase of c will strengthen the screening effect following the classical formulas κ = (∑i=1Nqi2ρbi /εκT)1/2 and thus weaken the LCA as observed; however, for the low |σ*| situation, the rising of the hydrogen bond number dominates over the screening effect increase, and consequently, the positive correlation phenomenon results. Action of electrostatic screening in the LCA is also convincingly demonstrated in Figure S4 of the Supporting Information, which shows clearly that the multivalent co-ion (2:2 type electrolyte) tends to more productively suppress the LCA than the univalent co-ion (2:1 type electrolyte), even if the total concentration of the ions is lower in the former case than in the latter case. One notes that the double influence falls in with the above classical formulas for the screening constant (κ). The other anomaly is a discovery, as depicted in Figure 4, that the LCA occurs in the presence of only a univalent counterion on condition that both c and |σ*| rise to a certain degree, and this discovery is completely at variance with the conventional wisdom that the presence of high valence counterions is necessary for occurrence of the LCA. One notes that the hydrogen-bonding style mechanism can fully explain the anomaly from two aspects, and relevant particulars are given below. First, going up in c will increase the number of counterions adsorbed onto the rod surfaces, and therefore, more hydrogen bonds will be formed; second, rising of |σ*| simultaneously generates more chances of hydrogen bond formation by adsorbing more counterions onto the rod surfaces and increases the energy per hydrogen bond by raising the charge of the hydrogen acceptor and donor. As a result, the LCA finally appears when the electrolyte concentration mounts up to 2.0 M, given that |σ*| is assigned with a reasonably high value of 0.3. Figure 4 clearly shows that 1:1- and 1:2-type electrolytes of the same concentration elicit the LCA of different strengths; the former corresponds to a weaker LCA, while the latter is related to a more salient LCA by comparison. With consideration that the 1:2 type electrolyte is of higher ionic strength than the 1:1-type electrolyte of the same concentration, if |σ*| = 0.3 belongs to a high |σ*| interval for the univalent counterion, the LCA strength induced by the 1:2-type electrolyte should be smaller than that due to the 1:1-type electrolyte; however, inexistence of the above expectation indicates that |σ*| = 0.3 still belongs to a low |σ*| interval for the univalent counterion, and the counterion single-layer adsorption capacity does not yet reach its saturation value due to weaker direct Coulombic interactions between the rod surface and univalent counterions compared to that resulting from high-valence counterions. As a result, further increase of the counterion concentration will help in raising the counterion single-layer adsorption capacity and the net result is a rising of the number of hydrogen bonds formed and strengthening of the LCA. One can note that in a recent work,33 the EPMF between two similarly charged rodlike polyelectrolytes (RLPE) along the distance between their centers of mass is calculated by using a molecular dynamics simulation package LAMMPS, and existence of an attractive well in the presence of monovalent counterions is demonstrated for large linear charge densities of the polymer backbone. This discovery and our results are not inconsistent, in that two demand high surface charge densities as a basic prerequisite of appearance of the attraction well. However, a quantitative comparison between the two results is 12494

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the dropping of the saturation adsorption capacity actually has no effect on the counterion single-layer adsorption capacity, proportional to the number of hydrogen bonds likely to be formed, as it is. As a result, the electrostatic correlation influence has always taken the dominating role and the monotonic correspondence relation results. To provide further proof of the present hydrogen-bonding style mechanism, an investigation is performed regarding the influences of dielectric permittivity and temperature on the EPMF and the results are displayed in Figure 6. One clearly observes that the influences do not seem to be in accord with the classical screening formulas, which predicts that with the dropping of dielectric permittivity and temperature, the screening effects will become more noticeable, and depth of the attraction well will reduce; however, Figure 6 shows an opposite tendency. Thus, a careful analysis is worthwhile. One notes that in the system Hamiltonian, the dielectric permittivity appears in the denominator of the potential functions responsible for the interion interactions, dropping of its value certainly reinforces the direct Coulombic interactions among small charged ions and also the hydrogen bond strength; in fact, increase of the hydrogen bond strength is also clearly manifested by the decrease of Hmin,i, and this is particularly obvious for the low |σ*| zone and also detectable for the high |σ*| zone. Now that one insists on the leading role of the hydrogen-bonding formation in explaining the LCA, its direct increase in strength will understandably hide an increase of the indirect screening effect due to dropping of the dielectric permittivity and acting in the opposite direction and causes the “anomaly” demonstrated in Figure 6 (panel a and b). As for the anomaly shown in Figure 6 (panels c and d), there are other reasons. It is well-known that the rising of the temperature will intensify the random thermal motion of ions and tends to destroy the strict asymmetrical permutation of the counterion layer nearest the rod surface; the resulting outcome is coexistence of asymmetrical, symmetrical, and various indermediate configurations, and this of course will lower the number of hydrogen bonds formed and also weaken the hydrogen bond strength as marked by the rising of Hmin,i with the increase of temperature. Then, considering the leading role of the hydrogen-bonding formation as discussed, the weakened screening effect due to temperature rising should not be the leading role, and the net result is the weakening of the LCA with the increasing temperature, as shown in Figure 6 (panels c and d). To argue further for the hydrogen-bonding style mechanism from a molecular level, a long-range nonelectrostatic interaction denoted by utail_αβ(r) was added to the interion potential functions and expect to observe probable modification of the LCA strength; the resulting EPMFs for cases of both high and low |σ*| zones are displayed in Figure 7. The employed utail_αβ(r) is of a Yukawa form and can be repulsive or attractive by adjusting the energy parameter, ε. Apparently, the adsorbed counterions nearest the rod surface repel one another by virtue of their direct like charge interion electrostatic interactions, and this makes it harder to have complete coverage of the rod surface by the counterions. Intuitively, one may take it that the saturation coverage rate increases with dropping of ε, as the dropping of ε means that the interactions between counterions are prone to be more attractive or less repulsive; consequently, the counterions can be more effectively massed together by reducing ε, and an obvious and direct outcome is the rising number of hydrogen bonds likely to be formed, as well as

Figure 5. Influences of counterion and co-ion diameters on the EPMF induced by a 2:1 electrolyte solution. Other relevant parameters are shown in the figure and described in the text.

will not alter visibly as the inter-rod separation changes, and the final result is that the effect on the EPMF induced by the rising of the ion size is masked by the augmented electrostatic correlation effect due to rising of the ion size as expressed mathematically in eq M13 below. As a result, the sizes of Δc(1)coupling and Δc(1)coupling (r) depend α αβ heavily on existence of the repulsive interactions at close separation and more specifically, it grows with the rising of the ion HS diameter dαβ. The above analysis is clearly illustrated in Figure 5b by the discovery of a correspondence relation between the deep LCA well and the large value of dαβ in the case of low |σ*| zone. However, the monotonic correspondence relation becomes nonmonotonic in the case of the high |σ*| zone, as illustrated in the inset of Figure 5b, where it is shown that only for small size counterion, the correspondence relation is valid; when the counterion diameter arrives at some high degree, the LCA well depth decreases with a further increase of dcation. Existence of the extreme point indicates that another factor, working in an opposite direction, is playing more and more nonignorable role with an increase of the dcation. The factor should be the counterion single-layer saturation adsorption capacity which obviously drops with the increase of the counterion size, and whose reduction, in the case of high |σ*| zone where the counterion single-layer adsorption has reached the saturation adsorption capacity, directly diminishes the number of hydrogen bonds likely to be formed and eventually overwhelms the enhanced electrostatic correlation effects and induces the appearance of extreme point. However, the extreme phenomenon does not show in the case of low |σ*| zone, and this is because the counterion single-layer adsorption has not obtained its saturation value in the low |σ*| zone. Thus, 12495

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Figure 6. Influences of relative permittivity of medium and temperature on the EPMF induced by a 2:1 electrolyte solution. The calculations are for values of |σ*| belonging to two different regions, and other relevant parameters are shown in the figure and described in the text.

strengthening of the LCA. The above analysis is clearly illustrated in Figure 7a by the displayed correspondence between the deep LCA potential well and low value of ε. For the low |σ*| zone, although adsorption capacity of the counterion does not reach saturation, decrease of repulsion between the counterions will raise the single layer adsorption capacity, and a similar corresponding relation also exists and is displayed in Figure 7b. The only difference between the two |σ*| zones is the more evident decrease of Hmin,i with a reduction of ε in the low |σ*| zone than in the high |σ*| zone, and the origin of such difference is the more loose adsorption and weaker hydrogen bonding occurring in the low |σ*| situation that leaves space for the reduction of Hmin,i with a lowering of ε. In summary, the perfect explanation for the influences of interions energy parameter provides strong support to the validity of the hydrogen-bonding style mechanism. To further illustrate the qualitative foretelling and interpretation functions of the hydrogen-bonding style mechanism, the effects of rod diameter d*cylinder on the EPMF were calculated, and the results are displayed in Figure 8. It is graphically shown that the effects will enable the LCA strength decrease with an increase of dcylinder * for the case of the high |σ*| zone and nonmonotonically vary with dcylinder * for the low |σ*| zone. We think that the change of the rod diameter d*cylinder

leads to modifications of the LCA by two ways. First, under the same conditions, an increase of d*cylinder will raise the adsorption capacity per unit rod area of the counterions, and this is because, compared with the repulsion interactions between the counterions in bulk, the surface with the larger radius of curvature provides an adsorption environment in which the adsorbed counterions can avoid the repulsion interactions among the ions to a larger extent due to the fact that the counterions adsorbed on another facet of the cylinder will have to interact with those counterions adsorbed on the opposing facet of the cylinder across larger space occupied by the cylinder. The repulsion interactions are therefore weakened largely, and this is an energetically favorable process. Second, with the increase of dcylinder * , average distance between “hydrogen atom” and “hydrogen acceptor” tends to rise, and this certainly weakens the hydrogen bond. In summary, the first way obviously strengthens the LCA by increasing the number of hydrogen bonds likely to be formed, and the second one weakens the LCA by reducing the average energy per hydrogen bond formed; it is the relative weight of the two effects that determines the way the LCA changes with the d*cylinder. More specifically, in the case of the high |σ*| zone, the counterion adsorption capacity has reached its saturation value, and the first mechanism will not play the role; instead, the second way will dominate the variation of the LCA with the dcylinder * . The 12496

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Figure 8. Influences of the cylinder-reduced diameter dcylinder * on the EPMF induced by a 2:1 electrolyte solution. Calculations are for values of |σ*| belonging to two different regions, and other relevant parameters are shown in the figure and described in the text.

Figure 7. Influences of energy parameter ε of the long-range nonelectrostatic interaction on the EPMF induced by a 2:1 electrolyte solution. The calculations are for values of |σ*| belonging to two different regions, and other relevant parameters are shown in the figure and described in the text.

equivalent to a decrease of T or εr in the case of the single influence. In accordance with Figure 6, in the case of the combined influence the LCA should strengthen as T is raised, and this is exactly exemplified in Figure 9. Two of the commonly used occurrence mechanisms for the LCA are a Oosawa model49 and a Wigner crystal model,37,38 and it may be interesting to compare the present mechanism with them. In the Oosawa model, thermal fluctuations create transient regions of high and low counterion densities on the colloids, and it is the transient complementary counterion density profiles which stirs up the LCA when two colloids approach each other; consequently, the Oosawa model predicts a stronger LCA under a higher T49,39,40 due to augmented fluctuations in the counterion density. This clearly is in contradiction with the experimental finding41 that a decrease in T causes a decrease in the critical divalent counterion concentration for aggregation of fd viruses and the present prediction, as discussed in Figure 6, that a decrease of T will strengthen the LCA. In the Wigner crystal model, counterions condensed on colloid surface are considered to form Wigner crystals; when the distance between two colloids decreases to a lattice constant of the Wigner crystals, cross correlation of the condensed counterions occurs, and cohesive energy of the Wigner crystals causes the LCA. Agreeably, the Wigner crystal model predicts a weaker LCA under a higher T because of decomposition38,39 of the crystal lattice structure due to augmented thermal motion of the counterions and explains the temperature effects on the LCA experimentally reported in ref 41 and theoretically predicted in the present work. However, the experimentally relevant LCA occurs for values

results are as exhibited in Figure 8a. On the other hand, in the case of the low |σ*| zone, the actual counterion adsorption capacity has not reached its saturation value, and there is room for further rising of the adsorption by the first way; as a result, the two ways simultaneously contribute but work in the opposite direction, and the nonmonotonic dcylinder * dependence of the LCA finally results. In the real electrolyte solution, relative permittivity of medium and solution temperature do not vary independently; one such example is given in the reference, in which the medium relative permittivity εr is empirically assumed to change with the temperature T as follows: εr = 78.5(298/T)1.4. After considering the function relation, a combined influence of εr and T on the EPMF can be calculated, and the results in the case of a 2:1 electrolyte solution are illustrated in Figure 9. By a comparison between Figures 9 and 6, one concludes that the combined and single influence due to εr or T are similar, except that the LCA strengthens as T rises in the case of the combined influence. From a statistical mechanics point of view, intermolecular potential [e.g., the interion potential function uαβ(r) under the present consideration) plays the role in the form of βuαβ(r); if uαβ(r) does not include long-range nonelectrostatic interaction [i.e., utail_αβ(r) in M10 disappears], both εr and T play their roles by the form of the product εrT. The above function relation gives εrT = 78.5 × 2981.4/T0.4, and thereby, the product εrT decreases with T; in other words, when considering the combined influence, an increase of T is 12497

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cannot justify the alteration of Hmin,i with the electrolyte and surface charge conditions, which actually indicates that specification of Hmin,i is a result of many-body correlation effects, rather than a single-body neutralization effect. Consequently, the oversimplified and qualitative concept of the surface charge neutralization is not consistent with the present hydrogen-bonding style mechanism, which is based on first-principle statistical mechanics calculations, and it is not instructive to calculate the fraction of the surface charge neutralization in the present explanation framework, which needs not be such a concept.

■

CONCLUSIONS The present paper is summarized as follows: (1) The EPMF is investigated theoretically between two similarly charged rods (modeling DNA) immersed in a PM electrolyte solution, and two scientific anomalies are discovered. First, contrary to popular notion that the LCA strengthens with increase of the bulk electrolyte concentration, it is disclosed that opposite effect unexpectedly occurs if the two rod surfaces involved are sufficiently charged and the popular view only applies for the case of low rod surface charge quantity. Second, it is found that the univalent counterion induces the LCA solely only if the bulk electrolyte concentration rises sufficiently and the rod surface charge quantities are high, and this challenges the popular view that the presence of polyvalent counterions is a precondition for occurrence of the LCA. (2) A hydrogenbonding style mechanism is advanced for the aim of revealing the origin of the LCA. There is little guesswork involved in the present mechanism as it has an energetic argument as first calculated in the present paper by 3D classical DFT. By appealing to fairly common-sense concepts such as bond energy, bond length, number of hydrogen bonds formed, and counterion single-layer saturation adsorption capacity, the present mechanism self-consistently explains the two scientific anomalies. (3) In addition, effects of counter- and co-ion diameters in eliciting the LCA are first investigated in this work. It is found that the LCA strengthens without exception with an increase of the co-ion diameter; however, for the case of counterions, the monotonic corresponding relation only applies for the low |σ*| zone, and it becomes nonmonotonic when the rod surface charge quantity is sufficiently high. Furthermore, one observes that although the nearest surface separation Hmin,i, at which the EPMF is the most attractive, actually does not change with alteration of the co-ion’s diameter (danion), but it indeed increases with the counterion’s diameter. Encouragingly, the hydrogen-bonding style mechanism does the best job of explaining these new observations self-consistently. (4) Effects of the rod surface charge density, co-ion valence, relative permittivity of medium, temperature, nonelectrostatic interion interaction, and rod diameter in modifying the LCA are systematically investigated, and several novel phenomena are first confirmed. All of these observations are self-consistently explained by the hydrogen-bonding style mechanism, and this supplies strong support to the hydrogen-bonding style mechanism for the LCA.

Figure 9. Combined influence of temperature and relative permittivity of water medium on the EPMF induced by a 2:1 electrolyte solution. The calculations are for values of |σ*| belonging to two different regions, and other relevant parameters are shown in the figure and described in the text. In the process of the calculations, the relative permittivity of the water medium is not fixed and changes with the temperature by a formula, as shown in the figure.36

of Ξ < 100 (the coupling parameter Ξ is defined in Methods), and even concerns the regime of Ξ ∼ 10−2 to 10−1.42−46 Simulations indicate47 that the snapshots for Ξ = 0.5 and 100 show liquid behavior, and only when Ξ runs up to 105 does crystalline order eventually occur. Thus, it seems that the Wigner crystallization is not connected or responsible for the LCA48 for cases experimentally realized. Using both the Oosawa model and Wigner crystal to formulate logical explanations for other LCA effects as disclosed presently is beyond the present scope. It seems the two models may not be the most direct solutions to these effects; however, it is gratifying that by appealing to fairly common-sense concepts such as bond energy, bond length, number of hydrogen bonds formed, and counterion single-layer saturation adsorption capacity, the present mechanism self-consistently and clearly explains all LCA phenomena. Some readers may feel that the present hydrogen-bonding style mechanism makes it seem that a fraction of the counterions are “bound” on the surface and neutralizes a fraction of the surface charges. In fact, essence of the present hydrogen-bonding style mechanism is to lower the system energy level by an appropriate space arrangement of the electrolyte ions, but the most stable configuration does not demand that the salt ions have to be in contact with the surface to neutralize a fraction of the surface charges. Otherwise, one

■

METHODS

In the classical DFT framework, one works in a grand canonic ensemble, and a corresponding grand potential Ω[{ρα}] is supposed to be a functional of a single-particle density distribution ρα of component α. Given an expression for 12498

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Ω[{ρα}], one ascertains the equilibrium microscopic configuration of the system by minimizing Ω[{ρα}], and then determines the grand potential itself by substituting the equilibrium ρα(r) into the expression of Ω[{ρα}]. From the calculated grand potential and density distributions, one can easily acquire all of the other relevant thermodynamic quantities as of surface tension, surface adsorption, solvation free energy, and the EPMF as a function of surface separation, a central theme of this study. The grand potential Ω[{ρα}] is related to an intrinsic Helmholtz free energy F[{ρα}] via a Legendre transform: Ω[{ρα }] = F[{ρα }] +

∑ ∫ drρα (r)[uα(r) − μα ] α

Flong_tail[{ρα }] =

αβ

(M4)

where the long-range nonelectrostatic interactions, utail_αβ(r), among ions at a larger separation is of a Yukawa type in this investigation and given as ⎧ r < dαβ 0 ⎪ utail_αβ(r ) = ⎨ ⎪ * (r /dαβ − 1)]/r r > dαβ ⎩ εαβ dαβ exp[−καβ (M5)

Similarly, FCoul is given in the mean field approximation as25−27

(M1)

FCoul[{ρα }] =

1 2

∑ ∫ d r ∫ d r′ρα (r)ρβ (r′)uCoul_αβ(|r − r′|) αβ

(M6)

where uCoul_αβ(r) denotes the potential due to direct Coulombic interactions among salt ions and is given by ⎧ 0 r < dαβ ⎪ uCoul_αβ(r ) = ⎨ ⎪ q q / rε r > d αβ ⎩ α β

(M7)

where qα is the electric charge of α component in units of Coulomb, and ε = ε0εr is permittivity of the solution with vacuum permittivity (ε0) and relative permittivity (εr) of the solution medium. We apply a second-order functional perturbation expansion approximation to deal with Fcoupling:24−27

∫ drρα (r)[ln(ρα (r)λα2) − 1] + Fex[{ρα }]

α

(M2)

where λα is the thermal de Broglie wavelength of the αth component, kB is the Boltzmann constant, and T is absolute temperature. The excess contribution Fex originates from internal interactions within the system and is in general unknown. For the charged system under consideration, it comprises of Fshort_hr (due to highly steep repulsive interactions occurring when ions approach each other closely), Flong_tail (due to long-range nonelectrostatic interactions appearing when ions are separated far), FCoul (due to direct Coulombic interactions among ions), and Fcoupling (due to coupling of Coulombic and short-range highly steep repulsive interactions):

Fcoupling[{ρα }] = Fcoupling[{ραb }] +

∫ dr

∑ −kBT Δcα(1)coupling[ρα (r) − ραb ] α

+ 1/2

∬ dr dr′ ∑ ∑ − kBT α

(2)coupling Δcαβ (| r

β

− r′|)[ρα (r) − ραb ][ρβ (r′)

− ρβb ]

Fex[{ρα }] = Fshort_hr[{ρα }] + Flong_tail[{ρα }] + FCoul[{ρα }] + Fcoupling[{ρα }]

∑ ∫ dr ∫ dr′ρα (r)ρβ (r′)

utail_αβ(|r − r′|)

where μα is the chemical potential for the αth component and the summation is over all components; uα(r), an external potential acting on the αth component consists of an electrostatic part qαv(r) as will be expatiated upon below, due to the presence of external charges such as a charged surface, a macro-ion, and a point charge, etc. or their arbitrary combinations and a nonelectrostatic part ϕα(r), due to a confining geometry or combination of several confining geometries. F[{ρα}] consists of an ideal gas contribution and an excess contribution Fex, viz. F[{ρα }] = kBT ∑

1 2

(M8)

where Δc(1)coupling and Δc(2)coupling (r) are coupling parts of a bulk α αβ first and second-order direct correlation functions (DCF), respectively, and Δc(2)coupling (r) is obtainable in the following αβ way:

(M3)

The four terms are not known exactly except for FCoul, and making appropriate approximations for these unknowns is thus one of themes in the classical DFT field. Below, we will discuss our approaches for dealing with them one by one. In the present paper, each component particle is with an explicit hard sphere (HS) core with an HS diameter dαα = dα for the αth component. What Fshort_hr represents is actually an excess Helmholtz free energy of the HS fluid mixture with HS diameter dα and can be accurately estimated by the best available Rosenfeld type functional;30 we will use a recently proposed version of the Rosenfeld type functional to deal with Fshort_hr,27 and the numerical details can be consulted from relevant literature and not be repeated here.24,27,30 We utilize one mean field approximation as often used in the classical DFT literature to deal with Flong_tail:24

(2)coupling (2) (2) Δcαβ (r ) = cαβ (r ) + βuCoul_αβ(r ) − cHS αβ(r )

(M9)

c(2) αβ (r)

where the bulk second-order DCF is for an overall neutral mixture of charged hard spheres (i.e., the bulk electrolyte solution in the PM) and is analytically available in a form of mean spherical approximation (MSA).28 It should be pointed out that to avoid double counting of Fshort_hr [{ρα}] and FCoul[{ρα}] in the excess free energy expression eq M3, both −βuCoul_αβ(r) and c(2) HS_αβ(r) are subtracted in eq M9, and the two subtracted quantities exactly correspond to the two terms of c(2) αβ (r) relevant to FCoul[{ρα}] and Fshort_hr [{ρα}], respectively. The Δcα(1)coupling is obtained by functional integration of Δc(2)coupling (r), and Fcoupling [{ρbα}] is obtainable αβ by the fluctuation route and the expression for Δc(2)coupling (r). αβ 12499

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To summarize, the interion potential function uαβ(r) under consideration is uαβ(r ) = ∝

In accordance with the definition, the reduced EPMF per unit area and in unit of kBT is calculated as u*elec (H ) = Ω*ex (H ) − Ω*ex (Hmax )

r < dαβ

uCoul_αβ(r ) + utail_αβ(r ) r > dαβ

where Hmax is large enough that uelec * (H) will no longer change when Hmax increases progressively. It should be noted that as the reported EPMF u*elec is measured per unit surface area and in units of kBT, the real EPMF in units of kBT should be equal to uelec * A, where the surface area A for a cylinder of length lcylinder and diameter dcylinder is defined as A = πdcylinderlcylinder; as a result, the EPMF per unit cylinder length and in units of kBT is u*elecπdcylinder and at least around 1.0, so they are physically meaningful.

(M10)

Minimizing the grand potential of eq M1 with respect to the density distributions and evaluating the chemical potential for bulk densities, one obtains the equations determining the equilibrium density distributions of all components: ρα (r) = ραb exp[− ϕα(r)/kBT − qα[ψ (r) − ψbulk ]/kBT

■

+ cα(1)short_hr[r; {ρα }] − cα(1)short_hr({ραb }) +∑

∫ dr′[Δcαβ(2)coupling(|r − r′|)

Figures S1−S4 analyzed in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

(M11)

■

where c(1)short_hr is calculated from the HS density functional α approximation used,27,30 and throughout the paper, superscript b denotes a quantity of bulk state,and its absence a quantity in the inhomogeneous state. ψ(r), the mean electrostatic potential, due to both the external charges as well as the internal ionic distributions, is given as ψ (r) = v(r) +

α

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS It is the author’s great pleasure to thank the two anonymous referees for thoughtful, constructive, and professional comments. This project is supported by the National Natural Science Foundation of China (Grants 21173271 and 21373274).

q ρ (r′)

∫ dr′ ∑ ε|αr α− r′|

ASSOCIATED CONTENT

S Supporting Information *

β

− utail_αβ(|r − r′|)/kBT ](ρβ (r − ) − ρβb )]

(M13)

(M12)

■

One notes that ψ(r) can also be calculated by a Poisson’s equation subjected to boundary conditions specified by the external charges responsible for v(r), and the two approaches are actually equal since they all can be deduced from the Coulomb law and principle of superposition. Throughout the paper, the confining geometries are all hard, and thus, ϕα(r) is infinite when the ions come into contact with the rods and zero otherwise. Throughout most of the present calculations, the long-range nonelectrostatic interactions utail_αβ(r) is absent; for very limited purposes, utail_αβ(r) is present, then it is truncated and shifted at rcut_αβ and for simplicity ε11 = ε22 = ε (Arabic numerals 1 and 2 denote cation and anion species, respectively). The potential parameters for unlike particle pairs, such as dαβ, εαβ, and rcut_αβ, are determined by a Lorentz−Berthlot rule (i.e., arithmetic mean is employed for dαβ and rcut_αβ) and geometric mean for εαβ. In the text, the superscript * marks a reduced quantity; more specifically, one uses d = 4.2 Å as the unit of length, and correspondingly, reduced density ρ* are defined as ρd3. The nearest surface separation h and cylinder diameter (dcylinder) are reduced as h* = h/d and d*cylinder = dcylinder/d, respectively. Surface charge density σ of the rod is reduced as σ* = σd2/e (e is electron charge in unit of Coulomb). Excess grand potential per unit surface area Ωex is reduced as Ωex * = Ωexd2/kBT, and Ωex = Ω − Ωb is easily calculated by importing the equilibrium density distributions and bulk densities into the grand potential expression, respectively. Some parameters are fixed throughout the calculations, unless otherwise stated, and they are rcut−αβ * = 6, T = 298 K, εr = 78.5, and d*cylinder = 5. In addition, the aforementioned coupling parameter Ξ is defined as Ξ ≡ 2πq3l2Bσ with q as the counterion valence and lB = e2/4πεkBT as the Bjerrum length.

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