Article pubs.acs.org/JPCA
Explicit Ions Condensation around Strongly Charged Polyelectrolytes and Spherical Macroions: The Influence of Salt Concentration and Chain Linear Charge Density. Monte Carlo Simulations Fabrice Carnal and Serge Stoll* Group of Environmental Physical Chemistry, F.-A. Forel Institute, University of Geneva, 10 Route de Suisse, 1290 Versoix, Switzerland ABSTRACT: The condensation of monovalent counterions and trivalent salt particles around strong rigid and flexible polyelectrolyte chains as well as spherical macroions is investigated by Monte Carlo simulations. The results are compared with the condensation theory proposed by Manning. Considering flexible polyelectrolyte chains, the presence of trivalent salt is found to play an important role by promoting chain collapse. The attraction of counterions and salt particles near the polyelectrolyte chains is found to be strongly dependent on the chain linear charge density with a more important condensation at high values. When trivalent salt is added in a solution containing monovalent salt, the trivalent cations progressively replace the monovalent counterions. Ion condensation around flexible chains is also found to be more efficient compared with rigid rods due to monomer rearrangement around counterions and salt cations. In the case of spherical macroions, it is found that a fraction of their bare charge is neutralized by counterions and salt cations. The decrease of the Debye length, and thus the increase of salt concentration, promotes the attraction of counterions and salt particles at the macroion surface. Excluded volume effects are also found to significantly influence the condensation process, which is found to be more important by decreasing the ion size.
1. INTRODUCTION Physicochemical properties of polyelectrolyte chains and, more generally, macroion properties arise from the long-range nature of Coulombic interactions leading to a large panel of compound associations. In addition to the interactions between colloidal systems, small charged mobile counterions and charged particles strongly interact in the macroion vicinity and form ionic clouds, hence modifying their reactivity and conformational properties. Counterions can also be completely dissociated into the bulk, which is entropically favorable. The ionic distribution and condensation of mobile ions around rigid polymeric systems was described by the Poisson− Boltzmann theory.1,2 In general, the nonlinear Poisson− Boltzmann equation is not easy to solve in closed form, and a linearization is commonly used as proposed by Debye and Hückel.3 For a symmetric system, the nonlinear Debye−Hückel equation provides good approximations to numerical Poisson− Boltzmann calculations.4 A new approach of counterion condensation around charged rods was developed by Manning, where the counterion distribution is treated in terms of the linear charged density parameter ξ. 5 For the author, condensation is observed if ξ exceeds a limiting value leading to a decrease of the chain effective charge. In later developments, the bound counterions were confined in a volume Vp around the backbone.5 This model is similar to the Oosawa model.6,7 The constraint implicating that counterions © 2012 American Chemical Society
fall into condensed and uncondensed categories represents the nonelectrostatic component of like-charge attraction theories. These nonelectrostatic elements result in attractive interactions between identically charged polyelectrolytes in monovalent salt as reviewed recently.8 To account for correlations between charged particles, Barbosa et al. modified the usual Poisson− Boltzmann theory and introduced a counterion correlation contribution to the local density functional.9 Other parameters were also introduced to distinguish specific counterion affinities with linear chains.10 The link between the number of condensed counterions and the adsorption excess per monomer was described by Mohanty et al., confirming the counterion condensation when ξ > 1, even if the system is infinitely diluted.11 The chain flexibility case was theoretically examinated by other groups, and a chain shrinking was predicted when counterion condensation is significant.12,13 Results provided by Poisson−Boltzmann theory and molecular dynamics simulations have shown a good agreement concerning rigid rods.14 Due to the lack of ionic correlations in Poisson−Boltzmann theory, a stronger condensation occurs by Special Issue: Herman P. van Leeuwen Festschrift Received: January 31, 2012 Revised: May 22, 2012 Published: May 22, 2012 6600
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observed with divalent counterions, as well as the formation of aggregates with trivalent counterions.33 Large macroion clusters were observed at the macroion isoelectric concentration followed by a redissolution when multivalent cations are in excess due to the charge inversion phenomenon.34 In the case of two highly charged macroions in their neutralizing divalent counterions solution which are in excess, Messina et al. demonstrated by molecular dynamics simulations the existence of one overcharged and one undercharged macroion resulting in a natural long-range attraction between them.35 If oppositely charged macroions are mixed, the counterions attenuate the interactions between them, but do not avoid aggregates formation. In this case, larger clusters appear when positively and negatively charged macroions are in equivalent quantity.36 In nature, the macroion charge distribution usually shows a discrete behavior. This phenomenon also plays a role in the counterion condensation process as shown by Qamhieh et al. by Monte Carlo simulations.37 Indeed, a stronger monovalent and multivalent counterion accumulation was found when discrete macroion charges were considered with a more pronounced difference comparing with conventional central charge distribution for higher counterion valence. Moreover, the structural confinement of binding sites to the body of nanospheres impacts the complexation with metal ions as shown by van Leeuwen et al.38 In this case, a multivalent metal ion will accumulate in the monovalent ionic cloud situated around the nanosphere as well as within the body of the soft sphere. In this paper, the condensation of explicit monovalent counterions and trivalent salt particles is investigated using Monte Carlo simulations. The cases of rigid rods, flexible polyelectrolyte chains, as well as spherical macroions are studied. For polyelectrolyte chains, we focus on the influence of trivalent salt, flexibility, and Manning parameter on the counterion condensation and chain conformation. In the macroion case, the influence of the Debye length, macroion charge, and explicit particle size on the effective macroion charge is specifically investigated. Our simulations are compared to the Manning theory25,39 and to Monte Carlo simulations performed by other groups.16,37
simulations when couterion valence increases. Salt valency and concentration clearly influence the fraction of ions that is under the chain influence and thus chain conformational behavior as shown by Monte Carlo simulations.15−18 Trivalent cations allow one to observe chain collapse. The translational entropy per counterion decreases with the increase of counterion valence. The free volume for ions in solution also plays a role in condensation processes as studied by Hsiao using molecular dynamics simulations.19 On one hand, the author found the replacement of the all the monovalent ions by large tetravalent counterions. On the other hand, the solution entropy increases with the decrease of tetravalent ion size leading to a reduction of interparticle correlations, making the condensation process more difficult. Other important factors such as the contour length are also implicated. Monte Carlo simulations showed an increase of the neutralized charge fraction by approximately 3% for a 3-fold increase in chain length due to the diminished importance of the chain end effect.20 Indeed, more counterions are attracted at the chain center due to stronger electrostatic potentials.21 Specific chain conformations such as toroidal and hairpin structures are the result of strong adsorption of trivalent and tetravalent counterions around semiflexible chains as reported by various research groups by molecular dynamics simulations.22−24 Counterion condensation around macroions has also been an important field of research for the past few years. As for polyelectrolyte chains, Manning theoretically explored the condensation on spherical macroions from the point of view of standard counterion condensation theory.25,26 The distinction between small and large spheres is made. The radius of small spheres is small compared to the Debye length, whereas the size of large spheres is comparable to the thickness of the diffuse ion cloud surrounding it. The effective macroion charge after condensation and the critical surface charge density needed to observe counterion condensation were given. Belloni deduced a criterion for spheres analogous to Manning condensation theory for linear polymeric systems.27 Counterion condensation is in this case dependent on the macroion charge and radius as well as the counterion valence. The effect of multivalent salt addition on the effective macroion charge was investigated later on the basis of ion chemical potential.28 Indeed, the chemical potential is uniform in solution, which is not the case considering the ion concentration. The macroion effective charge can thus be obtained from the system electroneutrality condition. Another Poisson−Boltzmann approach was proposed by Borukhov on the basis of variational free energy.29 Excluded volume interactions reduce the condensation degree, which is attributed to an effective increase of the macroion size. A significant deviation from the original Poisson−Boltzmann equation is found for strong electrostatic potentials. Considering an energetic definition of the effective colloidal charge, Diehl and Levin30 simulated aqueous colloidal suspensions containing monovalent and multivalent counterions using Monte Carlo techniques. The authors found a larger charge renormalization with the increase of valence confirming the stronger colloid-counterion coupling. Considering systems with several spherical macroions, the long-range nature of electrostatic interactions are responsible of repulsions between the colloids even if they are neutralized by monovalent counterions. This behavior was confirmed in different studies using Monte Carlo simulations.31,32 In presence of multivalent counterions, an attraction at short macroion separation was
2. MODEL Monte Carlo simulations are performed using three-dimensional off-lattice coarse-grained models at 298 K. The Metropolis algorithm40 is specifically used in the canonical ensemble. All of the objects, such as monomers, spherical macroions, counterions, and salt particles, are represented by impenetrable hard spheres and evolve in a periodic cubic box. The solvent, here water, is treated implicitly as a dielectric medium with a relative dielectric permittivity constant εr = 78.54 taken as that of water. Within the box, all objects are charged and interact with each other via a pairwise full Coulomb electrostatic potential and excluded volume potential defined as ⎧ ∞, rij < R i + R j ⎪ ⎪ Uij(rij) = ⎨ zizje 2 ⎪ , r ≥ R i + Rj ⎪ 4πε0εr rij ij ⎩
(1) −19
where e is the elementary charge (1.6× 10 C), ε0 is the permittivity of the vacuum (8.85× 10−12 C V−1 m−1), zi,j is the charge carried by monomers, macroions, counterions, or salt 6601
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particles, rij is the distance between them (center−center), and Ri,j is their radii. Uij is thus positive or negative when repulsive or attractive interactions are considered. The total energy Etot for a given conformation is given by the sum of the pairwise potentials Uij. Simulations of isolated rigid rods, flexible chains, and spherical macroions surrounded by explicit counterions and salt particles are considered here. Rigid rods as well as spherical macroions do not move during the whole Monte Carlo runs, while the position of flexible polyelectrolytes is modified by specific movements such as kink-jump, end-bond, reptation, and partially clothed pivot.41−43 All counterions and salt particles are moved through the box by translation movements. Polyelectrolyte chains are represented as a succession of 50 freely jointed monomers carrying one negative charge. The monomer radius may vary within the range of 2−7 Å. Counterions and salt particles always carry a charge on their center of +1 for the counterions, +3 for the salt cations and −1 for the salt anions. Their radii are fixed and equal to 2, 2.5, and 2 Å for counterions, salt cations, and salt anions, respectively. As simulations at various trivalent salt concentrations are carried out, the parameter β already used elsewhere is introduced according to16 β = 3Ntri /Nbead
The calculation of average values (observables), such as running coordination numbers, radial distribution functions, or radii of gyration, requires a period of energy stabilization. For the various simulation runs, the equilibration period is 2.5 × 105 Monte Carlo steps followed by a production period of 7.5 × 105 steps.
3. RESULTS AND DISCUSSION 3.1. Rigid Rod Simulation. Strong rigid rods of 50 monomers are considered here. Equilibrated conformations are represented in Table 1 considering various trivalent salt Table 1. Monte Carlo Simulations of Rigid Rods Surrounded by Explicit Monovalent Counterions and Trivalent Salt Particlesa
(2)
where Ntri and Nbead are the number of trivalent salt cations and monomers. The charge equivalence is reached when the number of charges resulting from the salt cations and chain backbone are equivalent, i.e., when β = 1. The spherical macroions have radii of 5, 20, and 73 Å, and their negative charges are homogeneously distributed since they are situated at the macroion center. The charge range considered here is [−11, −216], allowing one to adjust the surface charge density σ [mC/m2]. The added salt is strictly monovalent, and counterions may be mono-, di-, or trivalent. All of the counterions and salt particles have radii of 0.5 or 2 Å. When strong electrostatic interactions occur between charged polymers or spherical macroions with surrounded particles, the chain linear charge density or macroion surface charge density are expected to decrease by the condensation of ions from the solution. This phenomenon has been extensively and theoretically studied by Manning.25,44 Ions are then electrostatically bound to the chain/macroion or free in bulk. The determination of the fraction of bound ions is of main importance. However, in the simulation area, the definition of condensed and free ions is not well established, leading to different condensation criteria. For the simulation of polyelectrolyte chains, Belloni’s criterion is used in this paper.27 It consists of localizing the inflection point of the counterion running coordination number as a function of logarithmical radial distances. This criterion reproduces the salt-free Poisson−Boltzmann limit. Within the limit of high salt concentration, the inflection points disappear. Considering the counterion condensation around spherical macroions, Belloni’s criterion is not used here due to the poor representation of the inflection points. Thus, a geometrical criterion is used instead. Counterions are then considered condensed if they are situated within a distance of 3 Å from the macroion surface.37 This criterion is arbitrary here. Within this limit, the entire peak of counterion radial distribution functions is included. For the sake of clarity, the same limit value is used for the various simulations considering spherical macroions.
a
Counterions are represented in purple, salt cations in cyan, and salt anions in orange.
concentrations (β = 0, 0.3, 3) and Manning linear charge density parameters ξ = lB/b, where lB is the Bjerrum length (7.14 Å here) and b is the distance between successive charges on the rod. The case of ξ = 1 represents the limit of counterion condensation defined by Manning.39 In Table 1, the variation of Manning parameter is controlled by adjusting the monomer size, thus explaining the visual size differences between the polyelectrolyte monomers and sizes of the counterions, salt cations, and salt anions. Globally, electrostatic interactions between counterions and salt cations with the chain backbone are increased with the Manning parameter, leading to the adsorption of a higher particle fraction. For a given ξ, monovalent counterions (purple) are progressively replaced by trivalent salt cations (cyan) with the increase of the salt concentration due to stronger electrostatic interactions of the trivalent cations with the charged monomers. Manning Parameter Influence on the Counterion Distribution. Figure 1 represents the number of monovalent counterions that are within bins of 0.1 Å around the rod versus the distance from the rod surface for systems without salt (β = 0) and for different Manning parameters (ξ = [0.51,1.78]). Manning theory predicts a counterion condensation if ξ > 1. The limit of counterion condensation ξ = 1 is here represented by gray symbols. It has to be noted that the number of counterions begins to increase only at 2 Å due to the 6602
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the whole simulation box. This phenomenon is more visible for ξ = 0.71 due to the lower attraction with the chain backbone. When simulations are carried out within the Manning condensation domain (ξ > 1), the points are concentrated around the chain backbone, confirming the strong attraction of counterions within this range. Salt Concentration Influence on the Number of Condensed Counterions and Salt Fraction. To better understand the trivalent salt influence on the monovalent counterion condensation, the fraction of condensed monovalent counterions plus trivalent salt cations as a function of salt concentration (β) is represented in Figure 3a for various ξ values. Complementarily, the fractions of separated counterions and salt cations are presented in Figure 3b,c. Figure 1. Number of monovalent couterions that are within bins of 0.1 Å around the chain backbones as a function of the distance from the chain surface in the salt-free case.
counterion excluded volume. In an ideal system where ξ ≤ 1, the counterions are homogeneously distributed in the system, and the corresponding curve in Figure 1 is flat. This behavior is observed at the lower Manning parameters, i.e., when ξ =[0.51,0.71]. When ξ increases at the limit of Manning condensation domain (ξ = 1), the curve is no more fully flat but remains quite regular. Thus, when ξ = 1, the counterions are significantly attracted to the chain monomers. When the Manning condensation domain is reached, the slope of the curves rapidly increases when the distance to the chain decreases due to strong counterion condensation. Manning Parameter Influence on the Counterion Interaction Potential Energy. The interaction potential energy of one monovalent counterion with a rod of 50 monomers is now quantified in Figure 2 for various Manning linear charge
Figure 2. Interaction potential energy of one monovalent counterion and one rod of 50 monomers as a function of the distance between the counterion and the nearest monomer (center−center).
Figure 3. Fraction of condensed (a) monovalent counterions plus trivalent salt cations, (b) monovalent counterions, and (c) trivalent salt cations as a function of trivalent salt concentration.
density parameters ξ. The potential energy of one monovalent counterion is represented as a function of the distance between the counterion and the nearest monomer (center−center). Attractive electrostatic interactions between monomers and counterions is negative. Thus, the interaction potential energy decreases with the increase of ξ (and consequently counterion condensation). The limits of interaction potential energy are then −14.34, −8.67, and −6.41 kBT for ξ = 1.78, 1, and 0.71. For ξ ≤ 1, the counterions do not stay near the rod and move through the box leading to a point homogeneous distribution in
Considering the condensation of counterions plus salt cations (Figure 3a), the charge equivalence β = 1 plays the role of a limit between two domains. Within the range β = [0,1], and when ξ > 1, the monovalent counterions are progressively replaced by the trivalent salt cations due to stronger electrostatic interactions. Consequently, the fraction of condensed counterions plus salt globally decreases within the range β = [0,1] since one trivalent salt cation replaces several monovalent counterions, which are released in bulk. When salt excess occurs (β > 1), such a decrease is limited since the 6603
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3.2. Flexible Polyelectrolyte Simulation. Simulations are carried out for flexible polyelectrolyte chains of 50 monomers moving with specific movements through the box. The amount of trivalent salt change is expressed with the parameter β. Table 2 provides an illustration of equilibrated conformations for
majority of monovalent counterions have been replaced by trivalent cations below β = 1. Considering the value ξ = 1, the fraction of condensed counterions plus salt is similar when β = 0 or 1, and a decrease is first observed followed by an increase. This behavior suggests that a limited amount of monovalent counterions evolves near the rod when ξ = 1. As soon as trivalent salt is introduced, the few monovalent counterions are directly replaced by trivalent cations, leading to a decrease of the fraction of condensed counterions plus salt. The following curve increase corresponds to the condensation of trivalent cations, which have more affinities with the rod than monovalent counterions. If the chain linear charge density is set to ξ = 0.51 or 0.71, the curve strictly increase within β = [0,1]. Indeed, the condensation domain is not reached in these cases, and the affinity of monovalent counterions with the rod is very low. The fraction of condensed counterions plus salt increases within β = [0,1], which is the result of the increase of trivalent cations in the system and thus the condensation process. The replacement of monovalent counterions by trivalent cations can also be observed considering separated condensed fractions. In Figure 3b, a decrease occurs within β = [0,1] for ξ = [0.71,1.78] since monovalent counterions near the backbone are rapidly replaced by trivalent cations due to stronger affinity. In this figure, we observe a stronger monovalent counterion condensation for high ξ values. The simulation carried out for ξ = 0.71, i.e., below the Manning condensation range, also allowed a slight amount of condensed monovalent counterions to appear in the case without salt (β = 0). It has to be noted that the inflection point defined by Belloni’s criterion for this situation is very weak and thus not accurate. Considering a saltfree system, an inflection point is normally found only within the Manning condensation range. As it corresponds to the Poisson−Boltzmann limit, the behavior of counterion condensation in our case is expected to slightly differ due to the finite size of the system. When ξ = 0.51 and β = 0, no counterion condensation is observed. Our results show a very weak counterion condensation, even when ξ < 1. Comparing with Manning theory, the two domains corresponding to condensed and noncondensed can be distinguished, but the results are not strictly identical. Indeed, Manning theory describes the condensation around an infinite rod, while our rod is finite. Furthermore, the excluded volume is present in our simulations and varies with ξ. For small ξ values, the excluded volume of the monomers is larger, which can slightly modify the counterion distribution in the system and around the rod. Belloni’s criterion used here is a criterion among others, and the comparisons between different simulations with the same criterion would be more adequate. Curves for trivalent salt cations (Figure 3c) show a decrease of the condensed fraction with the increase of β for the whole ξ values. Indeed, the number of trivalent cations that can be condensed is limited by the global chain charge and conformation. Two different regimes with positive and negative curvatures may be observed within β = [0,2]. Indeed, the curve curvature is negative when strong condensation is observed, and positive within the low condensation domain. When a few trivalent cations are added (β = 0.3), they are immediately condensed around the rod within the strong condensation domain, resulting in a condensed salt fraction of 1. If the low condensation domain is considered, the condensed salt fraction does not reach 1 with smaller values for lower chain linear charge densities.
Table 2. Equilibrated Conformations of Flexible Polyelectrolyte Chains Surrounded by Explicit Monovalent Counterions (Purple), Trivalent Salt Cations (Cyan), and Monovalent Salt Anions (Orange)
several trivalent salt concentrations and Manning linear charge density parameters ξ. The Manning parameter is controlled by adjusting the monomer size. Globally, a higher counterion and salt adsorption is observed with the increase of the Manning parameter, and monovalent counterions are progressively replaced by trivalent salt cations with the increase of β. Equilibrated chain conformations are the result of the subtle intrachain repulsive interactions and monomer−counterions or salt cations attractive interactions. It is well-known that multivalent salt is necessary to induce folded chain conformations.16,18,45 Thus, when β = 0, the monovalent counterion condensation does not allow one to counterbalance the intrachain energy, resulting in stretched chains even within ξ > 1 domain. When trivalent salt is introduced (β = 0.3), the charge equivalence is not reached, but locally stretched segments can be observed (for instance when ξ = 0.71). Folded structures are shown here only when trivalent salt concentration is higher than the charge equivalence within the Manning condensation domain (ξ = 1.78, β = 3). Manning Parameter and Salt Concentration Influence on the Chain Conformations. To gain insight into the effects of trivalent salt cations and Manning parameter on the chain conformational changes, the renormalized mean square radius of gyration ⟨R2g⟩ as a function of trivalent salt concentration is calculated. Curves are presented in Figure 4 for various Manning parameters ξ. Considering the situation when β = 0, we observe a significant decrease of chain dimension with the increase of ξ due to the attraction of monovalent counterions near the chain. When β ≠ 0, the presence of trivalent salt cations induces important conformational changes within β = ]0,1], i.e., before the charge equivalence. Indeed, chain compactions are observed 6604
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Figure 4. Renormalized chain mean square radius of gyration as a function of trivalent salt concentration for various Manning parameters ξ.
due to stronger monomer−trivalent cations interactions. When simulations are made out of the Manning condensation domain, the increase of the number of trivalent cations influences less importantly the chain conformation. Within this domain, monovalent counterions are poorly attracted by the backbone, and chain conformations are more stretched than in the Manning condensation domain. Nevertheless, the trivalent cations cause the decrease of chain dimension from fully stretched to locally folded structures. In the Manning condensation domain, the chains are already locally folded when β = 0 due to the presence of monovalent counterions, and, thus, fully folded structures are observed when β = 1 is reached. Trivalent cations then allow a better chain compaction below the Manning condensation domain and when β = [0,1]. Above the charge equivalence, trivalent salt cations are in excess and remain in bulk, leading to a very limited chain dimension variation with β. Manning Parameter Influence on the Condensed Counterion Fraction. The fraction of condensed monovalent counterions as a function of the Manning parameter ξ is presented in Figure 5a for systems without salt (β = 0). For comparison, the results for rigid rods and the Manning theoretical fraction of condensed counterions fξ =1 − 1/ξ are also included. The number of monovalent counterions that are within bins of 0.1 Å around the flexible chain as a function of the distance from the backbone surface are calculated in Figure 5b. Manning theoretical prediction leads to a counterion condensation when ξ>1. Thus, no condensation occurs within the range ξ = [0,1] (Figure 5a). The fraction of monovalent condensed counterions decreases with the Manning parameter for our simulations, but a small counterion fraction remains attracted near the chain even below the Manning condensation domain (for instance, ξ = 0.71). In this case, the inflection point was not well-defined. When ξ = 0.51, no counterion condensation is observed for rigid and flexible chains. Within the ξ range investigated here, no significant slope change can be seen for our simulations, but the main tendency is achieved. It has to be noted that the conditions are not the same due to the finite polyelectrolyte length and box size in the simulation case, thus favoring the counterion condensation. Chain flexibility brings an extra degree of freedom. As a result, the monomers can adopt favorable positions around explicit particles leading to a slight increase of counterion fraction, which falls within Belloni’s condensation criterion.
Figure 5. (a) Fraction of condensed monovalent couterions as a function of the Manning parameter for flexible, rigid, and theoretical Manning cases. (b) Monovalent couterion number within bins of 0.1 Å around the backbone as a function of the distance from chain surface.
The monovalent counterion number, which is within bins of 0.1 Å around the chain (Figure 5b), indicates the same behavior compared with rigid rods (Figure 1). Indeed, curves are flat when ξ = 0.51 and 0.71 (i.e., below the Manning condensation domain). However, the curve slopes are more important within the Manning condensation domain. Comparing with Figure 1, the counterion number near the backbone is higher in the flexible case for all the Manning parameters, confirming the results of Figure 5a, i.e., the stronger monovalent counterion adsorption. The addition of trivalent salt cations in the flexible case has the same global influence compared with rigid rods (Figure 3). Thus, these results are almost identical and are not presented here. 3.3. Spherical Macroion Simulations. The condensation phenomenon of counterions and salt particles around one spherical macroion is investigated here. The amount of monovalent salt (and, consequently, the Debye screening length) as well as macroion radius, macroion charge, and macroion monovalent counterions are input parameters and vary for the various simulations. To quantify the condensation of monovalent counterions and salt cations, these species are considered bound to the macroion surface when they evolve within a distance of 3 Å from the macroion surface.37 Comparisons with Manning theory are made.25 Thus, two different cases are observed. A sphere is considered small when the radius is small compared to the Debye screening length (κ·a ≪ 1) and large when the radius is comparable to the Debye screening length (κ·a ≥ 1 and κ·lB ≪ 1). Here κ represents the inverse of the Debye screening length, a is the macroion radius, and lB is the Bjerrum length. Theoretical expressions are found 6605
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to express the effective macroion charge after condensation as well as the critical surface charge density needed to achieve counterion condensation. The effective macroion charge Neff is given by25 N eff = −(2a /zlB) ln(κa) for small spheres (κ ·a ≪ 1)
(3)
N eff = −(2a /zlB)(1 + κa) ln(κlB) for large spheres (κ ·a ≥ 1 and κ ·lB ≪ 1)
(4)
and the critical surface charge density σcrit according to σcrit = −
e ln(κa) 2πzlBa
σcrit = −
e(1 + κa) ln(κlB) 2πzlBa
for small spheres
for large spheres
(5)
(6)
Equilibrated conformations are presented in Table 3 for the small and large sphere cases with various surface charge Table 3. Equilibrated Conformations Corresponding to Manning Small (Radius = 5 Å) and Large (Radius = 73 Å) Sphere Casesa Figure 6. Effective macroion charge in function of the Debye length for the small sphere case with two ion radii (0.5 and 2 Å). σ = (a) −560.93 and (b) −1427.81 mC/m2. Results calculated by Manning theory are also presented.25
omitted. Thus these species in solution can move closer to the macroion surface, leading to a more efficient charge screening. Similarly, simulations with smaller ion radii (0.5 Å) allow observing a more efficient condensation around the macroions compared with those of 2 Å, in good agreement with the fact that small ions can come closer to the macroion surface. Comparing the cases with σ = −560.93 and σ = −1427.81mC/m2, attractive electrostatic interactions between the ions and macroions are stronger in the second case, leading to a more efficient ion condensation. Indeed, considering that the bare macroion charges are −11 and −28 in Figure 6a,b, the fraction of charge reduction due to counterions and salt cations is more important in panel b. Moreover, the difference between ion radii of 0.5 and 2 Å is more important with higher bare macroion charges. Although it is found that there is more condensation when the bare charge is larger, the Manning prediction for the two cases is that the effective charge is the same, i.e., independent of bare charge. Visually, it looks like the simulation curves for the two values of bare charge (for a given ion size) are not very different. The fact that the effective charge may not be strongly dependent on bare charge represents an interesting point. Large Sphere Case. Figure 7 represents the effective macroion charge after counterion and salt cation condensation as a function of the Debye length for a macroion radius of 73 Å. Surface charge densities of (a) −36.84 and (b) −51.67 mC/m2 corresponding to bare charges of −154 and −216 are considered. Counterion and salt radii are 0.5 and 2 Å. For comparison, the effective macroion charges calculated by Manning theory are also shown. For both surface charge densities, the variation of the effective macroion charge with κ−1 is weak. Due to the low σ values considered here, the differences between ion radii of 0.5
a
The spherical particles are surrounded by monovalent couterions (purple) and salt (cyan).
densities σ. When a = 5 Å, the macroion surface charge density variation is important between the two cases considered here, leading to a significant increase of counterion condensation. For the large sphere case, both surface charge densities are low, and the counterion condensation difference is not easily visible. Small Sphere Case. The effective macroion charge after counterion and salt cation condensation as a function of the Debye length is presented in Figure 6 for macroion radii of 5 Å. The bare macroion charges fixed at the center are (a) −11 and (b) −28, corresponding to surface charge densities of −560.93 and −1427.81 mC/m2. Counterion and salt radii of 0.5 and 2 Å are considered. For comparison, the effective macroion charges calculated by Manning theory are also represented. For both macroion charge densities, the same global behavior between simulations and Manning theory is observed. The Debye length decreases with the increase of salt concentration. Thus, the screening of macroion charge is more effective when κ−1 is small, resulting in a stronger counterion and salt cation condensation. Consequently, the absolute value of the effective macroion charge decreases with the decrease of Debye length. The condensation predicted by Manning theory is found to be more important compared with simulations. Manning theory represents the ions in solution as point charges without excluded volume, and the correlations between them are 6606
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Figure 7. Large sphere case. Effective macroion charge as a function of the Debye length with ion radii of 0.5 and 2 Å. σ = (a) −36.84 and (b) −51.67 mC/m2. Curves calculated by Manning theory are also presented.25
Figure 8. (a) Fraction of counterions within a distance r from the macroion center for monovalent, divalent, and trivalent counterions, and (b) the corresponding radial distribution functions.
Using the same criterion for counterion condensation as Qamhieh et al.,37 39% of monovalent, 75% of divalent, and 94% of trivalent counterions are situated within a shell of 3 Å around the macroion. A good agreement is found with the results of the authors (41%, 76%, and 94%). Simulations with the same parameters were also carried out in a spherical box, and identical results were found, even if the volume was 2 times smaller.
and 2 Å are also very weak. Considering the bare macroion charges, the fraction of charge reduction by counterion and salt cation condensation is barely more important when the charge is −216. On one hand, the absolute value of the effective macroion charge slightly increases with the Debye length increase in our simulations. On the other hand, the curve calculated by Manning theory seems to adopt a different behavior since the absolute value of the effective charge decrease is observed with κ−1. Figure 7 for the large sphere case seems to exhibit a stronger dependence of the effective charge on bare charge, contrary to the prediction. Counterion Charge Influence on Condensation. Figure 8a represents the fraction of counterions within a distance r from the macroion center, and Figure 8b represents the corresponding radial distribution functions. Macroion radii of 20 Å and counterion radii of 2 Å are considered here. No salt is added. The charge of counterions varies between +1 and +3, and the bare macroion charge is −60. Simulations with these parameters have already been carried out by Qamhieh et al.37 The only difference is that a cubic box is used here instead of a spherical cell. Consequently, the box volume is more important in our case, and the fraction of counterions is not strictly 1 when the box limit is reached due to spherical shells. The condensation difference between monovalent and multivalent counterions is quite important. As expected, the electrostatic interactions between the macroion and the multivalent counterions are stronger, resulting in a more efficient condensation. Indeed, the fraction of condensed trivalent cations reaches 95% very quickly in Figure 8a. When the radial distribution functions are calculated (Figure 8b), the peak difference between monovalent and trivalent counterions is considerable, confirming the strong adsorption behavior of trivalent particles.
4. CONCLUSION Using Monte Carlo simulations, the condensation of monovalent counterions and trivalent salt particles around strong isolated rigid and flexible polyelectrolyte chains as well as spherical macroions was studied. In the chain case, the influence of parameters such as trivalent salt concentration, Manning parameter, and chain flexibility was investigated. Considering macroion simulations, the Debye length variation, macroion charge, as well as counterion size and charge were of main interest. Results were compared with theoretical models proposed by Manning and also with Monte Carlo simulations carried out by other research groups.16,25,37,39 The condensation around isolated chains reveals interesting points. Chain screening effect by trivalent salt cations, which is more efficient due to stronger electrostatic interactions, allows one to observe the replacement of monovalent counterions by salt cations within the range β = [0,1], i.e., below the charge equivalence. This counterion substitution is strongly dependent on the Manning parameter ξ. Thus, a quasi linear regime is observed considering the fraction of condensed trivalent salt cations within the range β = [0,2] around the condensation limit defined by Manning theory (Figures 3c). The change of the slope curvature below and above this limit shows two different adsorption domains. Indeed, trivalent salt cations substitute several monovalent counterions in the Manning 6607
dx.doi.org/10.1021/jp3010019 | J. Phys. Chem. A 2012, 116, 6600−6608
The Journal of Physical Chemistry A
Article
condensation domain and less than one below ξ = 1. The results presented here show that long-range electrostatic interactions between chains and charged ions influence the ion distribution in the system. Limited counterion condensation is observed for ξ values that are below the condensation domain (for instance, ξ = 0.71) due to the finite size of the chains. The influence of chain flexibility and possible monomer rearrangement around charged ions allows observing a stronger condensation. In the case of spherical macroions, the fraction of macroion charge neutralized by monovalent counterions and salt particles determined by Monte Carlo simulations and Manning theory25 is found to be important. Indeed, Manning describes charged ions in solution as point charges without excluded volume and correlations between them. Thus, condensation phenomenon occurs more efficiently. For small and large spherical macroions described by Manning, simulations reported here show a stronger ion condensation with the salt concentration and consequently the Debye length. Thus, the fraction of neutralized macroion charge is higher. The bare macroion charge also plays a role since a higher condensed ion fraction is achieved when charge is increased. The effect of ion excluded volume is not to be omitted in condensation processes since results presented here show a significant increase of the macroion charge neutralization with the decrease of the size of counterions and salt particles. This effect is particularly important considering the small sphere case. The effective macroion charge defined by Manning theory shows the same global behavior compared with our simulations in the small sphere case, i.e., a reduction with the increase of salt concentration. Within the large sphere domain, the opposite behavior is observed, leading to an increase of charge macroion with the decrease of the Debye length.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors are grateful to G. S. Manning, in particular, D. Palomino, and F. Loosli for their encouragement and stimulating discussions. We also gratefully acknowledge the financial support received from the following source: Département de l’Instruction Publique de l’Etat de Genève and the Swiss National Science Foundation (Project 200021_135240).
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