Charge-regulation effects in nanoparticle self-assembly
CCharge-regulation effects in nanoparticle self-assembly
Tine Curk ∗ and Erik Luijten † Department of Materials Science & Engineering,Northwestern University, Evanston, Illinois 60208, USA Departments of Materials Science & Engineering,Engineering Sciences & Applied Mathematics, Chemistry,and Physics & Astronomy, Northwestern University, Evanston, Illinois 60208, USA (Dated: September 21, 2020)Nanoparticles in solution acquire charge through dissociation or association of surface groups.Thus, a proper description of their electrostatic interactions requires the use of charge-regulatingboundary conditions rather than the commonly employed constant-charge approximation. We im-plement a hybrid Monte Carlo/Molecular Dynamics scheme that dynamically adjusts the charges ofindividual surface groups of objects while evolving their trajectories. Charge-regulation effects areshown to qualitatively change self-assembled structures due to global charge redistribution, stabiliz-ing asymmetric constructs. We delineate under which conditions the conventional constant-chargeapproximation may be employed and clarify the interplay between charge regulation and dielectricpolarization.
Most solvated materials and biomolecules acquirenonzero charge due to dissociation or association ofcharged surface groups. For example, proteins, DNA,and silica nanoparticles attain their charge via protondissociation [1, 2]. The magnitude of the surface chargeis determined by the solution conditions, such as pH,but also by the presence of other charged entities in thevicinity, via charge regulation (CR). Although this phe-nomenon has been theoretically studied since the seminalwork by Kirkwood in the 1950s [3], the assumption thatall objects carry a constant charge is still widely employedand little is known about the many-body effects of CRon electrostatic aggregation.Traditionally, the Poisson equation is solved usingthe constant-charge (CC) or the constant-potential (CP)boundary condition. The CR boundary condition yieldsa solution that falls between these two limiting cases [4,5]. Charge-regulation effects have been shown to changepolyelectrolyte [6] and polymer brush [7, 8] phase be-havior and enhance protein–protein interactions [1, 9].Poisson–Boltzmann theory has elucidated the strong de-pendence of surface charge on pH, but is limited to theweak coupling regime and to static and simple geome-tries such as flat surfaces [2], a spherical particle [10], ora pair of particles [11, 12].Particle-based simulations avoid the approximationsinherent to a mean-field approach. However, whereasmolecular dynamics (MD) and Monte Carlo (MC) simu-lations of solvated systems with explicit charges are stan-dard, acid–base dissociation is rarely taken into account.Notable exceptions include hybrid techniques for atom-istic simulations in a constant-pH ensemble [13–15] andMC investigations of polyelectrolytes [16, 17] or planarsurfaces [18, 19]. Computational cost has limited thesestudies to relatively small systems, so that many-bodyeffects of CR have not been investigated and its con-sequences for the self-assembly of charged objects are largely unknown.Charge regulation is particularly relevant for aggrega-tion owing to the relation between the charge distribu-tion and the structure of the aggregate, which requiresself-consistent solution of the problem. Here, we assessthe effects of CR by investigating a fully dynamic sys-tem of up to 100 objects with more than 30,000 explicit,dissociable sites. By implementing an efficient and par-allelizable hybrid MD–MC scheme we examine how ag-gregation and self-assembly are affected by CR. In ad-dition, we combine our scheme with the Iterative Dielec-tric Solver (IDS) [20, 21], a boundary-element method, toexplore how dielectric polarization, another intrinsicallymany-body problem [22], affects CR.We consider spherical particles with a fixed density ofsurface-attached dissociable groups. Each group, e.g.,a weak acid, can be neutral or charged with a unitcharge q . The probability α i that a group i is chargeddepends on the equilibrium constant pK i and the chem-ical potential of the dissociated ion µ , but also on thelocal electrostatic potential ψ ( r i ) at the position r i ofthe group [4, 5, 16], α i − α i = 10 − pK i e − βµ ± βψ ( r i ) q , (1)where β ≡ / ( k B T ) is the inverse temperature and the ± applies to negatively (acid) and positively (base) chargedgroups, respectively. Note that ∆pK i = pK i + βµ log ( e )is independent of the choice of units. For acid dissocia-tion in an aqueous solution µ = − pH k B T ln(10). Equa-tion (1) applies to every dissociable group in the system,so that the full set { α i } determines the surface chargedensity ρ ( r ) = ∓ (cid:80) i α i q δ ( r i − r ). Thereby, these equa-tions provide a self-consistent boundary condition for thePoisson equation, ∇ · [ ε ( r ) ∇ ψ ( r )] = − ρ ( r ), with ε ( r ) thelocal permittivity.We apply this scheme to objects immersed in a mono-valent electrolyte, represented via the primitive model, a r X i v : . [ c ond - m a t . s o f t ] S e p − − po t en t i a l o f m ean f o r c e V ~ constant chargecharge regulating d / ( R+r ) − − c ha r ge q ~ L FIG. 1. Effect of charge regulation on pairwise interactions.Top: Potential of mean force V between a large particle withradius R and a small particle (red sphere) with constantcharge q s and radius r , normalized by the coupling strength λ ( ˜ V = V /λ ). Charge regulation is realized through dissociablesites that can be either neutral or charged (white and blue,respectively, in the inset) and results in a significantly en-hanced attraction (blue line) compared to a constant charge q L = − q s on the large particle (black line). Bottom: Cor-responding (normalized) total charge on the larger particle,˜ q L = q L /q s (dashed blue line). The parameters employedhere correspond to a 6-nm silica particle in deionized waterat pH = 7 [12], ∆pK = − . , q s = 12 q , R = 4 l B , 2 r = l B ,pI = 6 . λ = 32 k B T . with chemical potential pI [23]. Our hybrid MD–MCmethod works as follows. The system configurationevolves via conventional MD using the velocity-Verletalgorithm, with parameters and potentials described inthe Supplemental Material. After every n MD time steps, n MC MC steps are performed, where each step samplesthe charging state of a dissociable group, Eq. (1), orinserts/deletes salt ions. Unlike the reaction-ensemblemethod or the constant-pH ensemble method, which arerestricted to a specific range of pH values and salt concen-trations [16], this scheme consistently implements bothsalt ion insertion and solvent dissociation via pK s andis thus valid for any salt concentration or pH. In ad-dition, within the primitive model we treat the disso-ciated charges and the monovalent salt ions as equiva-lent, which increases the performance of our MC schemecompared to existing implementations [24]. Since MCand MD steps both require an Ewald summation andthus have the same computational complexity, we set n MD = n MC = 1 /δt , with δt the MD time step, ensur-ing that the performance of the hybrid scheme is alwayswithin a factor two of the optimal performance.We begin by exploring the influence of CR on a spher-ical particle covered with n ss = 792 surface sites (Fig. 1,inset). To highlight effects of CR on the electrostatic in-teractions, we initially disregard polarization effects aswell as London dispersion forces and evaluate the aver- age charge (cid:104) q L (cid:105) on this particle. Evidently, this chargedepends on the dissociation constant and the solutionconditions; in the Supplemental Material we examine pHdependence and provide a comparison to Debye–H¨uckeltheory. Here, however, we are interested in a more subtleeffect: How does the charge, and thereby interactions, de-pend on the presence of other charged entities? We add asmall particle with constant charge equal in magnitude tothe charge on the isolated large particle, q s = −(cid:104) q L (cid:105) d →∞ ,and apply the metadynamics method [25] to calculate thepotential of mean force (PMF) between the two particles,normalized by the magnitude of the Coulomb energy atcontact λ = q / [4 πε ( R + r )] under CC conditions (Fig. 1).Charge regulation results in a nearly twofold increasein the interaction strength at contact compared to theCC approximation. This arises due to redistribution ofcharges on the sphere—an effect similar to polarization ofconducting objects—and due to the change in the totalcharge q L on the large particle, which depends on theproximity of the point charge q s . In the absence of CReffects, an equivalent conductive sphere would yield anincrease in the interaction strength by a factor 1.6 [22].At higher ionic strengths the electrostatic interactionsare screened and weakened. Crucially, however, in thepresence of CR the interaction at contact remains abouttwice stronger than the corresponding interaction underCC conditions, even at physiological salt concentrationconditions (see Supplementary Material).A central challenge in the rational design of materialsis the prediction of structure. Our findings for a particlepair indicate that CR may significantly affect aggrega-tion. Moreover, dielectric polarization has been shownto induce large-scale changes to self-assembled struc-tures through local redistribution of charge within parti-cles [22, 26]. Since charge regulation allows global redis-tribution of charge, we may expect it to be an even morepowerful factor. Thus, we turn to many-body effects andself-assembly of multiple particles. Binary mixtures ofsize-asymmetric particles give rise to a plethora of self-assembled structures [27, 28]. We focus on a prototypicalsystem of spherical particles with size ratio 1:7, motivatedby the observation that CR appreciably changes the pairinteraction when a small particle is positioned at a dis-tance d ≈ R/ ρ = 0 . R − and cou-pling strength λ = 64 k B T . In the conventional (CC)case, the particles form a compact, symmetric aggre-gate (Fig. 2a). However, when the charge is no longerkept fixed and identical on each large particle, the CRsimulation shows an extended conformation (Fig. 2b).Interestingly, this is accompanied by symmetry break-ing: The average net charge of the three large parti-cles in Fig. 2b is (cid:104) q L (cid:105) = { . q , . q , . q } for the CCCR (c) (e)(d) (f)(a)(b)
FIG. 2. Self-assembly of binary aggregates. Small particles(red) carry a constant charge q s = 16 q while large particleseither have a constant charge q L = − q s (CC, blue spheres inpanels a,c,e) or are charge-regulating (CR, spheres with neu-tral (white) or charged (blue) surface sites, panels b,d,f). Inthe CR case, ∆pK = − .
5, which results in neutral structures, (cid:104) q L (cid:105) ≈ − q s . Whereas CC conditions give rise to compactstructures, CR leads to anisotropic and open assemblies. Thisobservation persists with increasing particle number: N p = 3,4, and 100 large and small particles in panels (a,b), (c,d), and(e,f), respectively, at ion concentration pI = 5 .
7. The CRimages are instantaneous realizations; the charge distributionon the large particles is continuously fluctuating. top, middle and bottom particles, respectively, indicatingthat CR stabilizes asymmetric, heterogeneously chargedstructures through a global redistribution of charge. Asimilar symmetry-breaking transition has recently beenreported for CR of membrane stacks [29]. A set of fourlarge and four small particles shows the same trend, form-ing a symmetric (square-like) structure forms under CCconditions (Fig. 2c), but an asymmetric structure un-der CR (Fig. 2d). This charge redistribution due to CRpersists for larger systems and gives rise to much moreextended structures than found in CC self-assembly. Asnoted, the CR-induced enhancement of pairwise inter-actions continues to hold at higher ionic strength. Thesame is true for the asymmetry imparted by CR (seeSupplemental Material).We illustrate this in a system of 100 large and 100small particles at a concentration ρ = 0 . R − (lateralsystem size L = 28 . R ). The structures are characterizedby the local coordination number z , which measures thenumber of small particles within a distance d n = δ + 2 σ (the first minimum of the radialdistribution function) from each large particle. Un-der CR conditions, one-dimensional string-like structuresappear ( (cid:104) z (cid:105) = 2 .
04, Fig. 2f), compared to folded two-dimensional hexagonal packed monolayers with (cid:104) z (cid:105) =2 .
80 that form in the simulations employing CC condi-tions (Fig. 2e).Arguably, open structures similar to Fig. 2f have beenobserved for conducting particles in a low-permittivity medium [26]. However, we emphasize that the under-lying mechanism is different. In the case of dielectricmismatch, the total charge on each individual particleis conserved and the polarization charge is redistributedacross the surface of the particle; the conductivity of theparticles then merely guarantees a constant potential oneach surface. This differ from the CR process, where theCP limit would be realized by globally grounding all par-ticles to a common potential, allowing free redistributionof charge among different objects and the solution. Sucha system has, to our knowledge, not been investigated.Of particular interest, then, is the question of the com-bined effect of CR and dielectric polarization. Like CR,polarization leads to charge redistribution and accom-panying strong many-body effects [22]. Moreover, theprerequisite condition, namely a strong permittivity con-trast between particles and the surrounding medium, oc-curs in numerous aqueous systems, including suspensionsof silica or polystyrene colloids and protein solutions. Toanswer whether CR or polarization dominates, we aug-ment our particle model with an additional boundary-element layer of 1472 patches uniformly distributed oneach sphere, positioned just below the CR layer in theinset of Fig. 3. Dielectric polarization charges are con-trolled by the mismatch ˜ ε , which denotes the ratio of thedielectric constants of the particle and the surroundingsolvent. After each MC and MD step, we employ theIDS to compute the induced charge on each surface el-ement. Conversely, these polarization charges are takeninto account when computing the dissociation probabil-ity of each surface group. We evaluate the role of di-electric effects by reexamining the system of Fig. 1 fortwo extreme cases: A small particle of fixed charge in-teracting with a large particle of either high permittiv-ity (i.e., nearly conducting; ˜ ε = 100) or low permittiv-ity (˜ ε = 0 . ε (cf. overlapping curves in Fig. 3).This observation is consistent with Kirkwood’s explana-tion of the dielectric increment of protein solutions [3].Charge regulation screens dielectric polarization, there-fore, in the far field proteins can behave as high-dielectricobjects even though the protein core has a dielectric con-stant significantly lower than water.In view of the potentially far-reaching consequencesof CR, it is an important question under which circum-stances its effects can be ignored. Figure 3 illustrates thatfor a small particle with relatively high density of surfacesites (a 6-nm silica particle in water), CR strongly af-fects the interactions. Conversely, we can estimate whenthe CC or CP approximation results in an error in theelectrostatic interaction that is smaller than the thermal d / ( R+r ) − − po t en t i a l o f m ean f o r c e V ~ CC, =0.01CC, =100CR, =0.01CR, =100
FIG. 3. Dielectric effects on the (normalized) potential ofmean force ˜ V for the particle pair described in Fig. 1, withthe additional condition that the large particle is polariz-able. Under constant-charge (CC) conditions, a particle withlow dielectric constant (˜ ε = 0 .
01) has a diminished attrac-tion (open circles) and a high-dielectric particle (˜ ε = 100)has an enhanced attraction (asterisks). Both cases are su-perseded by the attraction strength under charge-regulating(CR) conditions. Here, the high-permittivity case still isslightly stronger, but the effect is barely visible on the scaleof the graph. The inset illustrates the CR-BEM model. TheBEM layer of polarizable surface patches (˜ ε = 100) is placedat a distance R/ energy k B T . We place a point charge q at a distance d from a charge-regulating surface with a mean surfacecharge density σ and a maximum (fully ionized) chargedensity σ . In a mean-field approximation, α i = σ/σ and the surface capacitance, determining the linear re-sponse of the surface charge, then follows from Eq. (1)as C = ∂σ∂ψ = σ (1 − σ/σ )( − βq ). In the absence ofionic screening, the change in the potential at the sur-face due to charge q is ∆ ψ = q/ (4 πεd ). The chargeproduced by the surface capacitance will be containedwithin an area of size ∼ d since d is the relevant lengthscale. The additional charge density due to CR is thus σ CR ∼ − q ˜ C/ [ d (1 + ˜ C )] [30], with the dimensionless ca-pacitance ˜ C ≡ − Cdl B / ( q β ). We observe that the CCapproximation is valid if the CR charge is sufficientlysmall, − σ CR qd (cid:28) q /l B or ˜ C (cid:28) / (˜ q − q ≡ q/q (cid:112) l B /d . Conversely,the CP limit implies an image charge q im ∼ − q , becausefor a single flat surface the global CP limit is equal tothe local CP condition and can therefore be captured bya single image charge. The CP limit is justified when( σ CR d − q im ) q/d (cid:28) q /l B or ˜ C (cid:29) ˜ q −
1. The CP con-dition screens any possible dielectric polarization chargesand thus dielectric effects can be neglected. These twoconditions can be parametrized by just two dimensionlessvariables, ˜ C and ˜ q , allowing us to delineate the differentregimes in Fig. 4.This schematic also allows us to estimate the impor- ��� � �� ����������������� ������ � ˜ � � � � � ����� � �� � � � � �� � ˜ CCorCP CPCC & dielectric CR & dielectric
FIG. 4. Schematic diagram showing the applicability of theconstant-charge (CC) and constant-potential (CP) approxi-mations of CR. For sufficiently strong charges and interme-diate values of the surface capacitance, the use of a full CRsolver—as proposed in the main text—is required. In ad-dition, in the two regions denoted by “dielectric” dielectricpolarization effects can be important. tance of CR for different particle sizes. Notably, the PMFin Fig. 3 was obtained for a particle of only a few Bjer-rum lengths in diameter ( R = 4 l B ), corresponding to˜ C ∼ . q ∼ k B T . Rescaling the (linear) systemsize d → γd while keeping the Coulomb energy constantimplies q → γ / q and, therefore, σ → γ − / σ . Thisrescaling keeps ˜ q constant, but the surface capacitance˜ C ∼ − σdl B /q changes as ˜ C → γ − / ˜ C [31]. Thus,the CC approximation becomes increasingly more accu-rate as the particle size increases, which helps explainwhy the CC approximation works rather well for pre-dicting experimentally observed crystal structures andclusters of micron-sized colloidal particles [27, 28]. Onthe other hand, nanoscale particles will generally exhibitvery strong CR effects; e.g., our results (Fig. 2b) providea possible explanation for the chain formation observedin nanoparticle assembly [32, 33].In summary, we have implemented a hybrid MD–MCtechnique for resolving CR effects in arbitrary dynami-cal systems. Utilizing this method, we have shown thatCR-induced many-body effects can qualitatively alter thepredicted self-assembled structures via stabilization ofasymmetrically charged aggregates. 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