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Electrostatic Interactions between Janus ParticlesJoost de Graaf,1, a) Niels Boon,2 Marjolein Dijkstra,1 and Rene van Roij2, b)1)Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University,
Princetonplein 5, 3584 CC Utrecht, The Netherlands2)Institute for Theoretical Physics, Utrecht University,
Leuvenlaan 4, 3584 CE Utrecht, The Netherlands
(Dated: January 8, 2014)
In this paper we study the electrostatic properties of ‘Janus’ spheres with unequal charge densities on bothhemispheres. We introduce a method to compare primitive-model Monte Carlo simulations of the ionic doublelayer with predictions of (mean-field) nonlinear Poisson-Boltzmann theory. We also derive practical DLVO-like expressions that describe the Janus-particle pair interactions by mean-field theory. Using a large set ofparameters, we are able to probe the range of validity of the Poisson-Boltzmann approximation, and thusof DLVO-like theories, for such particles. For homogeneously charged spheres this range corresponds well tothe range that was predicted by field-theoretical studies of homogeneously charged flat surfaces. Moreover,we find similar ranges for colloids with a Janus-type charge distribution. The techniques and parameters weintroduce show promise for future studies of an even wider class of charged-patterned particles.
PACS numbers: 82.70.Dd, 87.16.D-, 61.20.Qg, 41.20.-q
I. INTRODUCTION
Electrostatic interactions in suspensions of charged col-loids are of paramount importance to the structure andphase behaviour of such systems1–12. The Derjaguin Lan-dau Verwey Overbeek (DLVO)13,14 theory is the mostwell-known theory by which the pair interactions betweenscreened charged particles can be described. However,DLVO theory only describes the interactions betweenhomogeneously charged spherical particles and is there-fore a monopole theory. The rapidly-growing zoo15,16 ofnew colloidal particles demands similar theories that areequipped to deal with anisotropy in shape and size.For classical, spherically charged, particles, DLVO the-
ory has proven to be a powerful means to describethe electrostatic properties of systems; mostly for high-polarity solvents, low surface charge, and high ionicstrength. The nonlinear Poisson-Boltzmann (PB) ap-proach17–19 extends this range, although analytic re-sults are only possible for a limited number of systems.PB theory is based on a mean-field approximation thatignores ion-ion correlations, which is justified only forsufficiently high temperatures, not too apolar media,and monovalent ions at reasonable concentrations. Inregimes where ion-ion correlations are important Strong-Coupling (SC) theory20–22 may be applied. Several mod-ifications to PB theory exist8,23–30, including modifica-tions of the traditional PB theory that account for finite-size ions.In the computational field, both Monte Carlo (MC)
and Molecular Dynamics (MD) techniques have been de-veloped to analyse charged particles suspended in an elec-trolyte. In primitive-model simulations, ions are taken
a)Electronic mail: [email protected])Electronic mail: [email protected]
into account explicitly, whilst the solvent is modelled asa dielectric continuum31–36. Ewald Sums37 usually pro-vide the basis for calculating the long-ranged Coulombcontribution to the total energy of a system with peri-odic boundary conditions. However, employing EwaldSums is computationally expensive and the systems thatcan be studied in the primitive model are therefore typ-ically small. When studying charged colloids suspendedin an electrolyte it is desirable to coarse-grain the sys-tem and use the much shorter ranged effective interac-tions between the particles instead of accounting for theions explicitly. Most simulation studies therefore con-sider systems where the interactions between the col-loids can be modelled by a DLVO pair potential12,38–41.We note that faster Ewald-Sums implementations exist,e.g., mesh-based approaches42,43, but these cannot out-perform a course-grained method that does not take theions into account at all in a system of charged colloidalparticles.
In this paper we investigate the range of applicabil-ity for nonlinear PB theory to accurately describe thebehaviour of the ion density around charged heteroge-
neous particles. This allows us to quantify the parameterregime for which a (multipole-expanded) DLVO approx-imation may be applied to describe pair interactions incoarse-grained simulations, since (the electrostatic partof) such DLVO theories can be derived using PB approx-imations. Charge-patterned particles have already beenstudied using the DLVO approximation by partitioningthe surface charge over a finite number of point-Yukawacharges with different prefactors44 to obtain effective pairinteractions. For charge-patterned particles this approx-imation still generally results in an expensive calculationof the pair interaction as a function of the separation andorientation. Moreover, the point-Yukawa description in-adequately accounts for the hard core of the particle,i.e., it implicitly assumes that ions can penetrate the col-loid45. In this paper a correct, simple, and therefore com-
2
putationally far more efficient DLVO-multipole approx-imation is derived for the effective interaction betweentwo charge-patterned particles. The multipole-based ef-fective interactions have an enormous potential for usein simulation studies to explore the phase behaviour ofpreviously inaccessible systems.For this study we focus on particles with a Janus-type
charge pattern. The term Janus refers to the two-facedRoman god of doors and was introduced to describe col-loid properties in 198846. A Janus particle46–48 consistsof two opposing parts (usually hemispheres) with differ-ent properties for the wetting, charge, chemical function-ality, etc. The past decade has seen a marked increasein the ability to synthesize such Janus colloids49–55 andtheir use in self-assembly experiments. Many interest-ing structures have been found44,56 and questions havebeen raised on how to approach simulations of such sys-tems. With our study we aim to address some of thesequestions for charged Janus particles in an electrolyte, inmuch the same way as the pioneering simulation studiesthat probed the applicability of the common DLVO/PBapproximation for homogeneously charged particles31,33.In Section II we introduce the methods by which we
compute the ion density around charged Janus particles:primitive-model MC simulations (II A) and nonlinear PBtheory (II B). We discuss the results of our investigationin Section III, which is divided into four parts. In Sec-tion III A we introduce the method, based on Fourier-Legendre (FL) mode decomposition, by which we com-pare the MC and PB results. This method is applied fora homogeneously charged particle in Section III B, wherewe also investigate the relation to the field-theoretical re-sults of Refs. 57 and 58 for homogeneously charged flatsurfaces. In Section III C we extend our results to a Janusdipole and show that there is remarkable correspondencewith the results for a homogeneously charged sphere. Weconsider a particle with a single charged hemisphere inSection IIID. Throughout these sections we give explicitrecipes for calculating the pair interactions between suchparticles within the PB approximation that we are test-ing. The interested reader is referred to the Appendixfor a derivation of these pair interactions. We discussour findings, comment on the potential synergy betweensimulation methods and theoretical results, and presentan outlook in Section IV.
II. SIMULATIONS AND THEORY
In the following we consider a system of sphericalcharge-patterned colloids with radius a suspended in anelectrolyte. The colloid volume fraction is denoted by η.We studied three types of charge distribution for the col-loids. (i) A homogeneous surface charge of Ze, with Z >0 the number of charges and e the elementary charge. (ii)A perfectly antisymmetric surface charge, with chargesZe/2 and −Ze/2 homogeneously distributed over theparticle’s upper and lower hemisphere, respectively. (iii)
A homogeneously charged upper hemisphere with chargeZe and a completely uncharged lower hemisphere. Unlessstated otherwise Z = 100 throughout this paper. We as-sume that there is a perfect dielectric match between thecolloid, the ions, and the medium to avoid any dielectricboundary effects.
A. Ewald Sums and Monte Carlo Simulations
To study the systems described above by MC simu-lations we turn to the primitive model, for which theions are represented by charged spheres and the solventis treated as a dielectric continuum. To simplify the cal-culations we study only one of these particles, which welocate at the centre of a volume Vcell = 4πa3/(3η). Weapply periodic boundary conditions to this volume to ac-count for the fact that we are in principle interested in asystem which contains many colloids. The particle’s (het-erogeneous) surface charge is specified by 100 charge sitesdistributed over this surface, which can be positively ornegatively charged, or which do not have charge. Thesecharge sites on the colloid are chosen according to theoptimal packing of 100 points on a sphere59 to ensurethat they are spaced as homogeneously as possible.The number of free monovalent ions N in the volume
Vcell is fixed, i.e., we are interested in an average ionconcentrationN/Vcell for the system that we approximateby our one-colloid calculation. We only consider systemsfor which a monovalent salt has been added to enhancethe screening effected by the counter ions to the particle’scharge. The balance between the number of positive N+
and negative N− ions (N = N+ + N−) is such that thevolume, and thereby the entire system, is charge neutral.For the monopole and charged hemisphere we require Z+N+ −N− = 0 and for the Janus dipole N+ = N−.To sample phase space Monte Carlo (MC) simula-
tions are performed in the isothermal-isochoric (canon-ical, NV T ) ensemble. We consider a cubic simulationbox of length L = 50d (η = (4π/3)(a/L)3, Vcell = L3),for which we employ periodic boundary conditions. Hered is the ion diameter and we assume all ions to be thesame size. The spherical colloidal particle is located atthe centre of the box (the origin). The particle’s rota-tional symmetry axis is chosen parallel to one of the box’ribs for the Janus-type charge distributions.The ion-ion pair potential is a combination of a
Coulomb and a hard-core interaction part:
UII(ri, rj) =qiqje
2
4πǫ0ǫ
1
|ri − rj |+
{
∞, |ri − rj | ≤ d;0, |ri − rj | > d,
(1)
with ri and rj the position of ions i and j with respect tothe colloid’s centre, respectively. The function | · | givesthe Euclidean norm of a vector, qi = ±1 the sign of thei-th ion’s charge, ǫ0 the permittivity of vacuum, and ǫthe relative dielectric constant of the medium, ions, andparticle. The interaction between a charge qie on the
3
particle located at ri, with |ri| = a − d/2, and an ionwith charge qje located at rj is given by
USI(ri, rj) =qiqje
2
4πǫ0ǫ
1
|ri − rj |+
{
∞, |rj | ≤ a+ d/2;0, |rj | > a+ d/2.
(2)The coupling between periodicity and the (long-range)Coulomb interactions, Eqs. (1) and (2), is taken into ac-count using Ewald Sums with conductive boundary con-ditions37,60. The total electrostatic energy UC of a par-ticular configuration may be written as
4πǫ0ǫ
e2UC =
1
2L3
∑
k 6=0
4π
|k|2
∣
∣
∣
∣
∣
∣
N∑
j=1
qj exp (ik · rj)
∣
∣
∣
∣
∣
∣
2
exp
(
−|k|24γ
)
−√
γ
π
N∑
i=1
q2i +1
2
N∑
i6=j
qiqjerfc(√
γ|ri − rj |)
|ri − rj |, (3)
where summation is over both the ions and the chargesites, i.e., N is the total number of charges in the sys-tem, both free and fixed; k ≡ (2π/L)l is a Fourier spacevector, with l ∈ Z
3; γ is the Ewald convergence param-eter60; and erfc(·) is the complementary error function.One can safely ignore the site-site interactions in Eq. (3),because this gives a constant contribution to the electro-static energy UC. The self-energy term also drops out ofthe energy difference, on which the acceptance criterionfor the MC trial moves is based60.For our simulations we employ the following parame-
ters. (i) Each run consists of 100,000 MC equilibrationcycles, where 1 MC cycle is understood to be one trial(translation) move per free ion. (ii) This equilibration isfollowed by a production run of 250,000 MC cycles to de-termine the ensemble-averaged ion density profiles ρ±(r),with r the position with respect to the centre of the col-loid. (iii) The step size for the ion translational movesis in the range [0, 5d] and it is adjusted to yield an ac-ceptance ratio of 0.25. (iv) For the Ewald Sums the realspace cut-off radius for the third term in Eq. (3) is setto L/2.5, and we use γ = 0.03 and |l| < 6. This choiceof Ewald parameters gives a reasonable approximationto the value of the electrostatic interaction energy. Dou-bling and halving the number of cycles for several runsshowed that the MC parameters give sufficiently equi-librated results for most systems. A possible exceptionto the perceived equilibration is deep inside the strong-coupling regime, where ion-ion correlations play an im-portant role, as we will explain in Section III A.
B. The Poisson-Boltzmann Approach
The spherical particle of radius a in a cubic box(L×L×L) models a system with colloid volume fraction
η = (4π/3)(a/L)3. The equivalent system in PB the-ory is described using a spherical Wigner-Seitz (WS) cellmodel19,61–63, where the radius of the WS cell is given by
R =
(
3L3
4π
)1/3
= aη−1/3, (4)
and the colloid is located at the centre of the cell. Thechoice of R ensures that the volumes, and therefore theaverage density of colloids/ions is the same as in the cubicbox of the MC simulations. PB theory is applied, inaccordance with the procedure outlined in Refs. 64–66, todetermine the dimensionless electrostatic potential φ(r)and the associated ion density profiles ρ±(r) around thecolloid.
In our MC simulations the hard-core interaction be-tween the ions and the colloid prevent the ions fromapproaching the colloid’s centre closer than a distanceof a + d/2. We therefore assume the same sphericalhard-core exclusion volume for the point ions in PB the-ory. The colloid’s surface charge density is given by q(r),which is only nonzero when |r| = a; the spatial integralover q(r) gives the total colloid charge. The PB equationfor this system may now be written as
∇2φ(r) = 4πλBq(r) +
{
0 |r| ≤ a+ d/2κ2 sinh(φ(r)) |r| > a+ d/2
,
(5)where κ2 = 8πλBρs (such that κ−1 is the Debye screeninglength), with ρs the (yet unknown) bulk ion density andλB the Bjerrum length
λB =e2
4πǫ0ǫkBT, (6)
with kB Boltzmann’s constant and T the temperature.We impose the following boundary condition
∇φ(r) · r|r=R = 0, (7)
with r ≡ r/|r|, on the edge of the spherical cell to ensurethat the normal component of the electric field vanishesat the boundary, i.e., the WS cell is charge neutral.
To solve Eq. (5) with the above boundary conditionsthe charge density q(r) and the electrostatic potentialφ(r) are expanded into a complete set of Legendre poly-nomials as
q(r) =∑
ℓ
σℓδ(r − a)Pℓ(x); (8)
φ(r) =∑
ℓ
φℓ(r)Pℓ(x), (9)
with σℓ and φℓ(r) the surface-charge and potentialmodes, respectively. Here r ≡ |r| and x ≡ r · z, with z
the orientation of the colloid’s rotational symmetry axis.
4
The Pℓ(·) are ℓ-th order Legendre polynomials, i.e.,
P0(x) = 1; (10)
P1(x) = x; (11)
P2(x) =1
2
(
3x2 − 1)
; (12)
P3(x) =1
2
(
5x3 − 3x)
; (13)
...
Pℓ(x) = 2ℓℓ∑
k=0
xk
(
ℓ
k
)( ℓ+k−12
ℓ
)
, (14)
where the expressions between brackets in Eq. (14) de-note binomial coefficients. The nonlinear PB equation[Eq. (5)] is likewise expanded using Fourier-Legendre(FL) mode decomposition and Taylor expansion of thesinh(·) term around the monopole potential φ0(r). Thehigher-order expansion coefficients contain products ofLegendre polynomials which effects couplings betweenthe various modes. These products must be rewritten asa sum of single Legendre polynomials67 to solve for theseparate modes using an iterative procedure. The modecoupling this induces necessitates the analysis of a signif-icant number of multipoles even if, for example, only thedipole mode is of interest. References 45 and 68 can beconsulted for more information on the procedure of modeexpansion to solve the PB equation for heterogeneouslycharged colloids.
It is important to note that the PB theory treatsthe screening ions in the grand-canonical (µV T ) ensem-ble. The MC simulations were however performed in thecanonical ensemble, where the number of ions is fixed,to allow for faster exploration of phase space. We fit thebulk ion concentration ρs in PB theory to ensure thatthe number of positive and negative ions in the WS cellcorresponds to the number of ions in the MC simulationbox. We consider this condition, coupled with the factthat we study the same colloid volume fraction η in bothapproaches, sufficient to justify comparison of the resultsin the two ensembles. The bulk ion concentration is fittedaccording to the criterion
N± = N±,PB ≡∫
drρ±,PB(r), (15)
where the integration is over the region |r| ∈ [a+d/2, R].One of the two equations is redundant, since solving forN+ is equivalent to solving for N−. The appropriatebulk ion concentration ρs, which comes into the right-hand side of Eq. (15) via the dependence of ρ±,PB(r) =ρs exp(∓φ(r)) on this concentration, is established usingan iterative procedure. All PB results presented in thispaper were obtained on an equidistant radial grid of 2,000points for |r| ∈ [a+d/2, R] by 5-th order Taylor expansionof sinh(·) using 6 multipole modes.
III. IONIC SCREENING OF JANUS PARTICLES
In this section we describe our results for the compar-ison of ion density profiles obtained by MC simulationsand by PB theory. A total of 99 systems are consid-ered for each of the three charge-patterned colloids. Weuse three particle radii a = 5d, 10d, and 15d. For everyparticle radius a, three salt concentrations are studied:125, 250, and 375 monovalent cations and anions, re-spectively, are added to the counter ions already presentin the system. This gives N± = 175, 300, and 425 for theJanus dipole. For the homogeneously and hemispheri-cally charged particle N+ = 125, 250, and 375, whenN− = 225, 350, and 475, respectively. We consider 11Bjerrum lengths λB/d = 0.01, 0.05, 0.1, 0.25, 0.5, 1.0,2.0, 4.0, 6.0, 8.0, and 10.0 for each a and N± combina-tion.We subdivided the results sections on the basis of the
surface-charge pattern studied. For each type of chargepattern, we consider the range where PB theory accu-rately describes the system. We also give a multipole-expanded DLVO description for the effective pair poten-tial between two such objects (for the Janus dipole andcharged hemisphere), which may be used in this range tomodel the pair interactions in coarse-grained simulations.
A. Method of Comparison
Figure 1 shows an example of our results for a typicalset of parameters: a = 10d, λB = d, and 250 added an-ions and cations, respectively. The azimuthal average ofthe net ionic charge [ρ+(r)−ρ−(r)] is shown for the Janusdipole (Fig. 1a) to give insight into the shape of the den-sity profiles. The multipole-expanded cation-density pro-files (Fig. 1b; ρ+,ℓ(r) for 0 ≤ ℓ ≤ 3) further illustrate thelevel of correspondence between the MC and PB resultfor this system. Due to the antisymmetry of the prob-lem combined with the fact that Pℓ(x) = (−1)ℓPℓ(−x)the anion densities follow as ρ−,ℓ(r) = ρ+,ℓ(r)(−1)ℓ forthe Janus dipole. The system is in the sufficiently di-lute and weak-coupling regime for the ion-ion interactions(α ≈ 0.06), which explains the good agreement betweenboth methods. Here we use the association-parameterα ∈ [0, 1] introduced in Ref. 36, which gives the equilib-rium fraction of the available ions in the electrolyte thathave formed pairs, to quantify the extent to which strong-coupling effects occur. The definition of α in terms of ourvariables reads
α = 1− 1
2Kρs
(
√
1 + 4Kρs − 1)
; (16)
K =π
2
∫ λB
d
dr r2 exp
(
2λB
r
)
, (17)
where the fitted value for ρs is used. Here K is theequilibrium constant for the formation of Bjerrum pairs:dipole-like clusters of two oppositely charged ions that
5
Figure 1. A comparison between MC and PB results showingthe ion densities around a Janus dipole for N± = 300, λB = d,and a = 10λB. In (a) the contour plot shows the net chargedensity ρ+(r)− ρ−(r): the MC result (left) and the nonlinearPB solution (right). In (b) the same data is represented us-ing a Fourier-Legendre mode expansion of the charge densityρ±,ℓ(r), for ℓ = 0, 1, 2, and 3. Note that the negative modesin (b) can in this case be mapped onto the positive modes bymultiplication with (−1)ℓ.
are closely bound due to the strong interaction en-ergy35,36. In Ref. 36 it was shown that α . 0.5 im-plies that strong-coupling effects are not relevant, i.e.,the lower the value of α the lower the concentration ofBjerrum pairs (for α = 1 all ions have formed pairs andhigher order clusters).The local dimensionless charge density for an equiva-
lent, homogeneously charged colloid is y ≡ ZλB/(κa2) =
4πσλB/κ ≈ 5.2 for the parameters of Fig. 1. The param-eter y can be used to estimate the level of nonlinearityin the system and since y ≈ 5.2 exceeds unity, mode cou-pling occurs45,68. We will come back to this parameter inthe context of heterogeneous surface charges later. Forthe Janus dipole of Fig. 1 we can clearly observe a non-linear effect, namely limr↓a ρ±,0(r) 6= ρs, despite the factthat the charge on the colloid has no intrinsic monopolecomponent. Nonvanishing quadrupole (ℓ = 2) modes arealso induced by mode coupling (nonlinearity)45.In order to quantify the difference between results ob-
tained by MC simulations and by PB theory we comparethe difference in the distribution of ions in the doublelayer directly for each mode. To that end we introducethe so-called difference functions fℓ, which can be ap-plied to a general Janus particle with QU unit charges onthe upper hemisphere and QL unit charges on the lower
hemisphere, respectively as
fℓ =4π
|QU|+ |QL|
∫ L/2
a+d/2
dr r2 |ρℓ,MC(r) − ρℓ,PB(r)| ,
(18)where the ionic charge modes are defined by ρℓ(r) =ρ+,ℓ(r) − ρ−,ℓ(r) and where we use the labels MC andPB to indicate the origin of the respective profiles. Equa-tion (18) has the property that all fℓ are 0 when the twoprofiles are exactly the same and that at least one fℓ > 0
when they are not. Because we compare results for thecubic geometry of the simulation box to the spherical ge-ometry of the WS cell in PB theory, the upper integrationboundary is set to L/2 < R. In principle the difference inshape and associated boundary conditions imply that wecompare a simple-cubic crystal of colloids with a liquid ofcolloids at the same volume fraction. However, due to theseparation of the particles and the level of ionic screeningthe results are virtually independent of the shape of thevolume when we compare up to r = L/2. This is thereason why we only consider systems with added salt.In Eq. (18) the functions fℓ are ‘normalized’ by |QU|+
|QL| such that for a homogeneously charged particlef0 = 2, for the worst-case scenario of full discrepancybetween the MC and PB results. Because the countercharge in the double layer should compensate for the netcharge on a colloid, each |ρ0(r)| separately contributes atmost |QU|+ |QL|, which explains the normalization. Anexample of a significant mismatch between the MC andPB results is found deep in the strong-coupling regimewhere the MC method predicts a total condensation ofcounter ions in a very small region close to the surface,whilst PB theory predicts that the counter charge is lo-cated in a diffuse layer around the colloid of significantwidth. For higher order modes the value of fℓ is bounded,but the range is not necessarily [0, 2]. To get a feelingfor the order of magnitude of fℓ in the case of a goodagreement between PB and MC results, we mention thatf0 ≈ 0.003, f1 ≈ 0.073, f2 ≈ 0.006, and f3 ≈ 0.057for the parameter set shown in Fig. 1, whilst Fig. 2dshows profiles with f0 ≈ 0.99, f1 ≈ 0.027, f2 ≈ 0.034,and f3 ≈ 0.031. Note that although f0 in Fig. 2d con-firms the huge mismatch between PB and MC resultsfor the ℓ = 0 mode, the fℓ with ℓ > 0 are still small.This suggests that the spherical symmetry of the problemis only slightly broken within MC simulations and thusthat strong coupling does not induce significant multi-polar charge distributions inside the screening cloud of ahomogeneously charged particle.
B. Homogeneously Charged Spherical Particles
To prove that the difference functions introduced inEq. (18) give a useful description of the deviation betweenthe MC and PB results, we investigate the monopole de-viation f0 for homogeneously charged spherical particles,in Fig. 2a. We compare the behaviour of f0 to field-theoretical predictions57,58, which are also aimed at es-tablishing a range of validity for PB theory, and showthat there is a good correspondence between the tworanges.In Refs. 57 and 58 the parameter regimes are investi-
gated, for which various theoretical approximations givetrustworthy results for the effective ion distribution of ahomogeneously charged flat surface. For these flat sur-faces parameter space is partitioned into three pieces, seeFig. 2a. (I) A region where the Debye-Huckel (DH) ap-
6
Figure 2. A comparison of the data obtained by PB theoryand by MC simulations of homogeneously charged spheresaccording to the difference function f0 of Eq. (18), whichquantifies the deviation in the distribution of charge in theionic double layer. We show f0 as a function of κµ, the ra-tio of the Gouy-Chapmann and the Debye length, and Ξ,the strong-coupling parameter, for several of the systems westudied. The field-theoretical prediction of Refs. 57 and 58for homogeneously charged flat surfaces partitions parameterspace into three regimes, as is indicated by the continuous andthe dashed line. The Debye-Huckel (DH), Poisson-Boltzmann(PB), and Strong-Coupling (SC) approximations should beused to obtain acceptable results in the respective domains.The value of the PB-SC divide is denoted by Ξ∗
≈ 10. (b-d)Three samples of the ion profiles that are obtained by PB the-ory (full curves) and MC simulations (dashed curves), showingcation and anion densities. Here f0 = 0.03, 0.39, and 0.99.The discontinuity in the first derivative of the MC ρ+,0 profilein (d) is caused by the positive ions condensing on the surfaceof the colloid in combination with the binning procedure weapplied to average the ion densities. A similar discontinuityis present in the ρ−,0 results, although this is not visible onthe scale of this plot.
proximation69 can be used, the screening is linear. (II) Aregion where the charge of the surfaces becomes higher,the nonlinear PB equation18,19 has to be solved in orderto determine the effective electrostatic interactions. (III)A region in which the ion-ion correlations close to the sur-face require the use of Strong-Coupling (SC) theory20–22.
In Fig. 2 the parameters κµ = 2/y and Ξ = (y/2)κλB rep-resent parameter space in a ‘field-theoretical language’(see Ref. 58), with y ‘our’ local dimensionless charge den-sity [as defined just below Eq. (17)] and µ the Gouy-Chapmann length. The Gouy-Chapmann length mustnot to be confused with the ionic chemical potential,which is also usually denoted by µ. Ξ is a measure for theparticle’s surface charge, expressed in terms of the Bjer-rum length. PB theory produces satisfactory results forΞ < Ξ∗, with Ξ∗ a cut-off value: Ref. 58 sets the valueof Ξ∗ ≈ 10 for the transition between the PB and SCregimes. The value of κµ is of minor importance whenΞ < Ξ∗ as PB theory can be straightforwardly applied tothe DH region in the low-charge limit.
By comparing the MC and PB ion profiles for the 99systems containing a homogeneously charged sphere thatwe have investigated, we found f∗
0 = 0.1 to be a goodboundary value for the regime (f0 < f∗
0 ) in which PB-theory accurately describes the ion density profiles. Fig-ures 2(b-d) show three example ion profiles to illustratethe possible level of deviation corresponding to a particu-lar f0 value: f0 = 0.05, 0.39, and 0.99, respectively. Notethat our choice of f∗
0 = 0.1 is meant to imply that belowthis value PB theory gives a good description, however,for f0 > f∗
0 PB theory may give a reasonably accuratedescription (see Fig. 2c), but this is not a given and SCtheory may be required to give a good description. Usingour f∗
0 values as a function of y and Ξ, see Fig. 2a, welocate the PB-SC divide at Ξ∗ ≈ 1. Minor changes inthe value of f∗
0 do not significantly change the locationof the PB-SC transition in parameter space. However,since what is considered an unacceptable level of the dis-crepancy between PB and MC results is dependent onthe quantities/behaviour we are interested in, there is adegree of arbitrariness to our result. Nevertheless, ourapproach to this problem and our choice for f∗
0 appearsjustified since we obtain a similar partitioning of parame-ter space as was found in Ref. 58. This was to be expectedfor a homogeneously charged sphere, since there is onlya geometrical difference with respect to a homogeneouslycharged plate, which for sufficiently large spheres can beconsidered small close to the sphere’s surface. Our resultsshow that even for relatively small spheres (compared tothe size of the ions) there is qualitative agreement.
For completeness we comment on the accuracy of ourMC result deep inside the strong-coupling regime. TheMC results show that a layer of oppositely charged ionscan form on the surface of the charged particle. Theinteraction between the charges (sites and ions) is suchthat the free ions effectively condense on the particle,see Refs. 8, 26, 70–73 for a more comprehensive accountof this phenomenology. The ions in the electrolyte ex-perience similarly strong interactions and form Bjerrumpairs. Since we only consider single particle MC trialmoves, the formation of Bjerrum pairs interferes with theexploration of phase space in the strong-coupling limit.The clusters hardly move, because most single particlemoves that would break up a cluster are rejected based on
7
the energy difference. This results in an ill-converged en-semble average, when the Bjerrum-pair concentration ishigh (α & 0.5)36. The problem can be overcome by intro-ducing cluster and association-dissociation moves for theBjerrum pairs to obtain a more efficient sampling35,36.However, we do not believe that ion condensation andBjerrum-pair formation will influence our result with re-gard to the location of Ξ∗, since these effects only startto play a role for Ξ ≫ Ξ∗, since then α > 0.5.
C. Janus-Dipole Charge Distributions
1. Comparison of the Monte Carlo and
Poisson-Boltzmann Results
In order to investigate the range of applicability of the(nonlinear) Poisson-Boltzmann approximation, we applyour method of comparison from Section (III A) to higherorder Fourier-Legendre (FL) modes of the ion density forthe case of a Janus dipole. Figure 3 shows the deviationparameter fℓ for ℓ = 1, 3 and the large set of param-eters we studied. Note that the Janus dipole bears noeven multipole moments [ρ−,ℓ(r) = ρ+,ℓ(r)(−1)ℓ] due toits antisymmetric charge distribution. To apply a rep-resentation similar to the one used in Ref. 57 for Janusparticles, we introduce the following modified dimension-less parameters: yΣ = 2/(κµΣ) ≡ (|QU|+ |QL|)λB/(κa
2)and ΞΣ ≡ (yΣ/2)κλB. The sum of the absolute valueof the charge on each hemisphere is used, rather thanthe total charge (which would be zero in the case of aJanus dipole). For pure monopoles yΣ and ΞΣ reduce tothe original parameters y and Ξ. We prefer to expressour results in terms of the dimensionless (absolute) localcharge density yΣ rather than in terms of κµ = 2/yΣ,since the former is a more natural quantity for PB the-ory of colloid systems. The use of κµ in Fig. 2 was toillustrate the correspondence between our results and thepartitioning given in Ref. 58.For the dipole term (ℓ = 1) we observe trends in the
value of f1 (Fig. 3) as a function of y and Ξ similar tothose observed for the value of f0 of a homogeneouslycharged sphere (Fig. 2a). The onset of a strong differencein the correspondence between the two results for thedipole mode occurs at f∗
1 ≈ 0.1. For the octupole (ℓ = 3)term, the crossover value f∗
3 for an appreciable level ofdeviation appears to be slightly larger than 0.1, but onthe strength of our results it is difficult to state this withcertainty.Based on Fig. 3, a regime can be distinguished for the
leading dipole term where PB theory yields accurate re-sults for the charge profiles in the electric double layer(ΞΣ < Ξ∗
Σ ≈ 1). For the ℓ = 1 through ℓ = 5 modes(ℓ = 5 not shown here) the correspondence between MCand PB results is also sufficient when ΞΣ < Ξ∗
Σ. Themodified parameter ΞΣ therefore appears useful to de-scribe parameter space for dipolar Janus particles withregards to quantifying the region where PB theory can
Figure 3. The deviation fℓ in the double layer, determined us-ing the MC simulations and PB theory, for a Janus dipole asa function of the modified charge density yΣ and the modifiedstrong-coupling parameter ΞΣ. The subgraphs show the re-sults for the first two odd FL modes (ℓ = 1, 3), correspondingto the dipole and octupole term, respectively.
be used to describe the system.
2. Multipole-Expanded DLVO Approximation for the
Janus Dipole
Equations describing the electrostatic pair-interactionof (spherical) colloidal particles with a substantial dipolarcontribution to the surface charge are derived in the Ap-pendix. This derivation is performed within the Poisson-Boltzmann approximation, and the resulting equationscan therefore be considered as an extension of DLVOtheory towards inhomogeneously charged particles. Themonopole-monopole, monopole-dipole, and dipole-dipole
8
interactions are given respectively by
βVMMij (Rij) = λB
exp(−κRij)
Rij
ZYi ZY
j , (19a)
βVMDij (Rij) =
λBexp(−κRij)
R2ij
(
(pYi · Rij)Z
Yj − (pY
j · Rij)ZYi
)
, (19b)
βVDDij (Rij) =
λBexp(−κRij)
R3ij
(
(1 + κRij)(pYi · p
Yj )
− (3 + 3κRij + (κRij)2)(pY
i · Rij)(pYj · Rij)
)
, (19c)
with Rij the distance vector between particle i and j,
Rij ≡ |Rij |, and Rij = Rij/Rij . We also introducethe ‘Yukawa-monopoles’ ZY
i and ‘Yukawa-dipoles’ pYi ,
which, for Janus spheres of radius a, are given by
ZYi = (QU +QL)
exp(κa)
1 + κa, (20a)
pYi = (QU −QL)
3a exp(κa)ni
4(2 + ǫc/ǫ)(1 + κa) + (κa)2,
(20b)
with ni is the particle’s symmetry axis, which points tothe northern hemisphere, QU the total charge on the up-per hemisphere, QL the total charge on the lower hemi-sphere, and ǫc/ǫ the ratio between the relative dielectricconstant of the colloidal particle and that of the sur-rounding medium, which we choose 1 unless stated dif-ferently. However, Eqs. (19) and (20) are, as is typicalfor DLVO theory, only valid for sufficiently small charges,since linearised PB theory (Debye-Huckel approxima-tion) was employed in the derivation of these equations.Fortunately, charge renormalisation has proven to be auseful tool in broadening the range of applicability to-wards particles with a higher charge19,24. This renor-malisation procedure was extended towards dipoles andhigher multipoles in Ref. 45.
D. Hemispherical Charge Distributions
1. Comparison of the Monte Carlo and
Poisson-Boltzmann Results
For the hemispherical charge distribution, we also com-pared PB results with those of MC simulations, in orderto determine the regime of validity of the PB-multipoleexpansion. We again considered 99 systems and foundthat for the even modes the level of deviation f∗
ℓ ≈ 0.1sets a rough upper bound to the applicability of PB the-ory. For the odd modes the correspondence between theMC and PB results seems to hold for slightly higher val-ues of the deviation parameter: f∗
ℓ ≈ 0.25. Using thesetwo values of f∗ we can roughly locate the range of va-lidity of the PB result in the region ΞΣ . 1. The effectsof strong coupling are however far more apparent for our
hemispherical charge distribution than for the other twodistributions we considered. This is because in the MCsimulations we used 50 divalent charge sites on the upperhemisphere instead of monovalent sites. Our results forthe hemisphere are therefore less convincing than for theother two charge distributions, but our preliminary indi-cation is that the range in which the PB approximationis valid is roughly the same as that for the monopole andJanus dipole.
2. Multipole-Expanded DLVO Approximation for the
Hemispherical Charge Distribution
As mentioned in the previous section, one cannotalways resort to standard DLVO theory for particleswith an anisotropic surface charge, because dipole,quadrupole, or higher order multipole charge distribu-tions are relevant. Consequently, two-body interactionsof the form monopole-dipole, dipole-dipole, monopole-quadrupole, etc., should be considered as well.
Figure 4. (a) A sketch of two interacting particles with radiusa, both having charge on only one hemisphere. To eliminatethe dipole moment we relocate the centre of the charge dis-tribution at a distance b (along the particle’s rotational sym-metry axis) from the geometrical (hard-core) centre of theparticle. This choice results in a distance Rij between thecharge distributions. Graphs (b) and (c) show the suggestedvalue of b and the Yukawa weight factor C(κa), respectively,as a function of the colloid radius a in terms of the Debyescreening length κ−1, for a wide range in ǫc/ǫ, the ratio be-tween the relative dielectric constant of the particle and thatof the surrounding medium. In (c), we indicate C = 1, whichis the weight in case of the regular DLVO equation, using adashed line.
Figure 4a shows two spherical particles with a positivehemispherical charge distribution, i.e., one side is chargedthe other is uncharged. Instead of choosing the centre of
9
the charge distribution to coincide with the geometricalcentre of a spherical particle one is free to place this pointanywhere inside the particle. The most natural locationis the point for which the (Yukawa-)dipole moment van-ishes and thereby all dipole interactions. With this choicethe electrostatic two-body interaction in terms of only amonopole-monopole term, is expected to be maximallyaccurate. This point is located on the rotational sym-metry axis of the hemispherically charged colloid. Forsufficiently large interparticle distances we obtain the fol-lowing interaction potential for the shifted-monopole ap-proximation:
βVij(Rij) ≈(
Zeκa
1 + κaC(κa)
)2
λBexp(−κRij)
Rij, (21)
with Rij the shifted centre-to-centre distance, Z the par-ticle charge, and C(κa) a renormalisation factor. Thisrenormalisation factor depends on the ratio of the parti-cle radius a and the Debye screening length κ−1 only andC = 1 in case of regular DLVO theory. We can numeri-cally determine the distance b from the particle’s geomet-rical centre, where we should place the shifted monopolesuch that the dipole term vanishes. Figure 4b shows theratio b/a as a function κa for several values of ǫc/ǫ. Notethat b/a goes to 1/2 for small κa, which is the (Coulomb)result that is obtained for unscreened particles. Weshow the corresponding renormalisation factors C(κa) inFig. 4(c). We find C(κa) . 1, because the charge islocated closer to the (new) centre of the charge distri-bution than for homogeneously charged particles. Notethat the interaction potential is independent of the orien-tation of the particles, when rotated around the shifted(monopole-charge) centre. The monopole-monopole ap-proximation therefore does not capture the full interac-tion between the two hemispherically charged colloids.To capture the orientational dependence higher ordermodes (octopole and higher) are required. However, for amonopole-monopole only approximation Eq. (21) is max-imally accurate by design. In the Appendix we explainhow to calculate and, thereby, also how to minimize theYukawa dipole moment for any type of charge distribu-tion and hard-core shape.
IV. CONCLUSION AND OUTLOOK
In this paper we studied the range in parameter spacefor which the nonlinear Poisson-Boltzmann (PB) theoryaccurately describes the behaviour of the ions around aJanus charge-patterned spherical colloid in a 1:1 elec-trolyte. We used primitive-model Monte Carlo (MC)simulations to establish the ion density around such acharged particle for a huge set of parameters. By alsocomputing the ion density for the same parameters andcomparing the two results, we were able to establish aregime in which this PB theory gives a good approxima-tion for the ion distribution. This comparison is based
on Fourier-Legendre (FL) decomposition of the MC iondensity to determine the contribution of the monopole,dipole, quadrupole, . . . charge terms. The theoretical ap-proach also relies on FL decomposition and this enablesus to quantify the differences on a mode-by-mode basis.
For a homogeneously charged sphere we compared ourrange of validity for PB theory to the range found inRefs. 57 and 58 for a system of homogeneously chargedflat plates in an electrolyte. There is a remarkable corre-spondence between the two ranges, especially consideringthe small size of the colloids that we studied in relationto the size of the ions. For such small spheres a greaterdeviation with respect to the results of a flat-plate calcu-lation could reasonably be expected. We were also ableto show that the range in which the PB results accu-rately describe the ion density around a spherical Janus-dipole is similar to that found for the homogeneouslycharged sphere. For particles with only one (homoge-neously) charged hemisphere, there is an indication thatthe regime in which PB theory can be applied matchesthe regime found for the two other particles.
In the PB-regime that we obtained, we can use sim-ple (multipole-expanded) DLVO-like equations, which wederived in this paper, to describe the interactions be-tween two particles with a Janus-type charge. We gaveexplicit expressions for the monopole and dipole inter-actions, since these terms are typically dominant forJanus particles. These electrostatic interactions resem-ble well-known Yukawa interactions, and reduce to thesein the homogeneous charge limit. For the Janus-dipoleobtaining the (multipole-expanded) DLVO expression isrelatively simple. To accurately model a hemisphericalcharge distribution using only a monopole-term is a lit-tle more complicated. The key step proved to be shiftingthe centre of the charge distribution from the centre ofthe particle towards the charged hemisphere in order toeliminate the Yukawa-dipole contribution.
Our analysis forms a basis of a good understandingof the range in parameter space for which the PB ap-proximation can be applied to describe the behaviour ofheterogeneously charged colloids. This is, for instance,relevant to the study of such particles using simulations,where PB-theory-based effective interactions can be usedto study the phase behaviour of such particles in theright regime. Note that we only considered equilibriumion density profiles of stationary colloids. The rotationalmovement of mobile charge-patterned colloids can occuron time scales that would lead to an out-of-equilibriumdouble layer. What effect the out-of-equilibrium ion den-sity would have on the screening of the particle and howsuch effects should be incorporated into effective inter-action potentials used in simulations, is left for futureinvestigation.
10
V. ACKNOWLEDGEMENTS
M.D. acknowledges financial support by a “Neder-landse Organisatie voor Wetenschappelijk Onderzoek”(NWO) Vici Grant and R.v.R. by the Utrecht Univer-sity High Potential Programme and by a NWO ECHOgrant.
APPENDIX: JANUS PARTICLES AND THE
DEBYE-HUCKEL APPROXIMATION - DLVO THEORY
FOR PATCHY COLLOIDS
In this appendix we derive the (multipole-expanded)DLVO-like equations for general charge distributions andwe show how this leads to Eqs. (19) - (20). Before that weset the stage by first examining the quality of the dipole-only approximation for a Janus-type charge distribution.
Figure 5. A contour plot of the net charge density ρ+(r) −ρ−(r) around an antisymmetric Janus particle for the param-eters Z = 10, N± = 425, λB = d, and a = 10λB, showing theprofile that follows from a mode expansion up to ℓ = 6 on theleft, on the right only the dipole mode is plotted.
We have shown that the mode expansion up to ℓ = 6gives good agreement between MC simulations and PBtheory regarding the ion profiles in a well-defined regimeof parameters. For Janus particles in general the mostdominant multipoles are the monopole and/or the dipole;note, however, that for ‘pure’ Janus dipoles the monopoleterm vanishes. To show this Fig. 5 plots the PB resultfor the local ionic charge densities around such a purelydipolar Janus particle, using a smaller charge (Z = 10)than elsewhere in this paper. The number of ions addedto the system, N± = 425, corresponds to κa ≈ 2.8. Wefind that yΣ = 0.36 and ΞΣ = 0.05 for this system, whichimplies that we are within the regime where PB-theory isapplicable according to our analysis. We are aware thata multipole expansion of Yukawa-like interaction poten-tials does not necessarily converge in general74,75. In thisparticular case, however, we catch most of the physics ofthe interacting Janus particles, by treating the particleas a ‘pure’ dipole (without higher order modes). To il-lustrate this, the left side of Fig. 5 shows the ion chargedensity up to ℓ = 6, which we proved to be sufficient
for good correspondence with MC results in the regimewhere PB-theory is applicable. The right side shows thedipole (ℓ = 1) mode only. The differences are thereforedue to the missing ℓ = 3 and ℓ = 5 terms. The dipoleapproximation overestimates the electrostatic potentialin the axial direction, whilst it underestimates the elec-trostatic potential in the perpendicular direction.With this in mind we explain the way to extend the
applicability of DLVO theory to anisotropically chargedparticles. The result of setting up this theory allows usus to find explicit equations for the monopole-dipole anddipole-dipole interaction potential between Janus parti-cles as a function of their orientation. These expres-sion are similar to the well-known expressions for theinteraction of unscreened dipoles. We begin by consid-ering the effective electrostatic energy of an extendedcharge configuration eq(r) in a 1:1 electrolyte with bulkconcentration 2ρs - we do not incorporate hard-core ef-fects at present. By using the electrostatic energy fromCoulomb’s law combined with the ideal-gas entropy forthe monovalent ions, we find that the grand potential ofthe charge configuration inside a two-component mono-valent ion mixture inside a solvent is given by
βH =
∫
dr∑
α=±
ρ±(r)
(
ρ±(r)
ρs− 2
)
+1
2
∫
dr (ρ+(r)− ρ−(r) + q(r))φ(r), (22)
where ρ±(r) are the ion densities and where the dimen-sionless electrostatic potential is
φ(r) = λB
∫
dr′ρ+(r
′)− ρ−(r′) + q(r′)
|r− r′| . (23)
Note that we applied the Debye-Huckel approximation;the ‘usual’ entropic term of the ions is linearised viaρ±(r)(log(ρ±(r)/ρs) − 1) ≈ ρ±(r)(ρ±(r)/ρs − 2), imply-ing that we assume the ion densities not to vary muchfrom the bulk value ρs. For a more detailed derivation ofEq. (22) see for example Ref. 66.We consider a system that consists of a collection of
M (fixed) charges located at rci, with i = 1, . . . , M anindex. These charges should not be considered as pointcharges, but rather as localized charge distributions qi(r)inside associated volumes Vi, which are non overlappingand centred at rci, as is sketched in Fig. 6. Each qi(r) isonly nonzero inside Vi, and the total (non-ionic) charge
density is hence written as q(r) =∑M
i=1 qi(r).Since we have not included any hard-cores yet, the ion
densities follow immediately from setting the functionalderivative of Eq. (22) w.r.t. the ion densities to zero,
δH/δρ±(r) = 0, yielding ρ±(r) = ρs(1 ∓∑M
i=1 φi(r)),with
φi(r) = λB
∫
Vi
dr′ qi(r′)exp(−κ|r′ − r|)
|r′ − r| , (24)
where κ2 = 8πλBρs. By assuming that the charges qi(r)within the volumes Vi have a fixed position, Eq. (22) can
11
Figure 6. A sketch of two interacting charge distributions(grey) qi(r) and qj(r), inside the enclosing volumes Vi andVj , respectively. These volumes are rod-like in this particularexample to resemble rods with one charged ‘head’ and haveboundaries that are indicated by dashed instead of solid linessince we do not consider hard cores while calculating the iondensities connected to these charge distributions. The centresof the charge distributions can be chosen arbitrarily and areindicated by ri and rj .
be written, up to a self-energy constant, as
βH = λB
∑
i<j
∫∫
Vi,Vj
dr dr′ qi(r)qj(r′)exp (−κ|r′ − r|)
|r′ − r| .
(25)Equations (24) and (25) were derived using the Taylor-expanded (quadratic) Hamiltonian (22) and thereforegive appropriate results only in case the dimension-less electrostatic potential φ(r) remains sufficiently smallw.r.t. unity.
We will now use the result of Eq. (25) to obtain ananalogous theory for charge distributions with associatedhard-core volumes. This is done by ‘freezing’ the ionic
charge profiles inside the volumes Vi and adding these tothe charge configurations qi(r), such that the new chargedistributions qi(r) are found. Effectively we compensatefor the fact that ions in the Yukawa approximation canpenetrate the hard particle. We thus consider a new com-bined charge distribution qi(r) that is exactly the distri-bution we are interested in, by compensating the fixedcharge for the ion profiles it induces.
Starting with the obtained ion densities for the systemwithout hard cores, we split the entire system volume Vinto Vout, consisting of all points r outside the volumesVi for all i, and the volume Vin of points that are insideone of these volumes. Note that Vin is the complementof Vout. The ion densities are also split into ρout± (r) and
ρin±(r) such that ρout± (r) + ρin±(r) = ρ±(r) and ρout± (r) = 0
for all r inside Vin, and ρin±(r) = 0 for all r inside Vout.
Eq (22) may therefore be rewritten as
βH =
∫
Vout
dr∑
α=±
ρout± (r)
(
ρout± (r)
ρs− 2
)
+
∫
Vin
dr∑
α=±
ρin±(r)
(
ρin±(r)
ρs− 2
)
+
∫
dr1
2(ρout+ (r)− ρout− (r) + q(r))φ(r), (26)
with
φ(r) = λB
∫
dr′ρout+ (r′)− ρout− (r′) + q(r′)
|r− r′| , (27)
and the ‘new’ colloidal charge distributions
q(r) = q(r) + ρin+(r)− ρin−(r)
=
M∑
i=1
qi(r), (28)
for which qi(r) ≡ qi(r) + ρin+(r)− ρin−(r) when r inside Vi.Within the DLVO-approximation, the net ionic charge
density ρin+(r) − ρin−(r) in volume Vi is only induced bythe fixed charges qi(r) in that volume itself (large inter-particle distances). One therefore finds for r in Vi that
qi(r) ≈ qi(r)− 2ρsφi(r)
= qi(r)− 2λBρs
∫
Vi
dr′ qi(r′)exp(−κ|r′ − r|)
|r′ − r| .
(29)
The second line in Eq. (26) becomes a constant that mayregarded as a self-energy term and therefore can be ig-nored to yield
βH =
∫
Vout
dr∑
α=±
ρ±(r)
(
ρ±(r)
ρs− 2
)
+1
2
∫
dr (ρ+(r)− ρ−(r) + q(r))φ(r), (30)
with the restriction that ρ±(r) = 0 if r is inside Vin. Com-paring Eq. (22) with Eq. (30), the effective interactionHamiltonian between hard particles with charge densi-ties qi(r) can be recognized in the latter. The effectiveinteraction energy for such a system can thus be obtainedfrom Eq. (25) with qi(r) the solution of Eq. (29), withqi(r) the actual charge density of interest. For particleswith hard-core volume Vi and corresponding charge den-sities qi(r) it is necessary to first solve qi(r) from Eq. (29).By finding qi(r) we obtain the charge distribution thatcompensates for the self-induced ion densities, since theseare required in to determine the effective interaction inEq. (25).At this point we have found a procedure to calculate
the electrostatic interactions between inhomogeneouslycharged colloidal particles with hard cores, within the
12
Figure 7. A sketch of two interacting charge distributions qiand qj with hard-core volumes Vi and Vj , respectively, sepa-rated by a distance Rij . We also show the vectors si and sj ,which have their origins at rci and rcj , respectively.
DLVO approximation that we discussed earlier. In orderto isolate specific multipole interactions between colloids,Eq. (25) is rewritten as
βH = λB
∑
i<j
∫∫
Vi,V,j
dsidsj
{
qi(rci + si)qj(rcj + sj)
·exp(−κ|Rij + sj − si|)|Rij + sj − si|
}
, (31)
with Rij = rcj − rci the colloid-colloid distance andsi = r−rci the coordinate relative to rci. Both are shownin Fig. 7. The Yukawa interaction can then be expandedinto spherical harmonics around these centres76, whichare rci and rcj . For this we must assume that the dis-tance from any point in Vi to its centre rci is less than thedistance to any other centre rcj . This however automati-cally holds for equi-sized spherical volumes if one choosesthe centres rci exactly in the middle of the spheres.Here, the monopole and dipole terms are of main inter-
est. Eq. (31) can therefore be written, up to a monopole-monopole, monopole-dipole, and dipole-dipole interac-tion term as
H =∑
i<j
VMMij (Rij) + VMD
ij (Rij) + VDDij (Rij). (32)
The resulting interaction terms are given by Eq. (19), inwhich the ‘Yukawa monopole’ and ‘Yukawa dipole’ arenow (for the general charge distribution)
ZYi =
∫
Vi
dsi qi(rci + si)sinh κsiκsi
; (33a)
pYi = 3
∫
Vi
dsi qi(rci + si)
(
coshκsi − 1κsi
sinhκsi
(κsi)2
)
si,
(33b)
with si = |si|. Note that Eqs. (33a) and (33b) reduce tothe well-known expressions for Coulomb monopole and
dipole in the limit κa ↓ 0, see Ref. 77. Also the in-teraction terms (19) give the well-known result for pureCoulomb systems in this limit. Finally, the electrostaticpotential around charge distribution i can be expandedas
φi(rci + si) = ZYi λB
exp(−κsi)
si
+ (pYi · si)λB
exp(−κsi)
s2i(1 + κsi)
+ O(ℓ ≥ 2), (34)
in which quadrupole and higher order multipoles are notincluded.As an example we show that the interaction between
particles with a homogeneous charge distribution is inagreement with DLVO theory. By considering a spher-ical colloid with a hard core radius a and a charge Zdistributed homogeneously over its surface and choosingrci = 0 for convenience, the charge distribution is givenby qi = Z/(4πa2)δ(si−a). It can be shown that Eq. (29)is solved by
qi(si) =
{
Z4πa2 δ(si − a) + ZλB2ρs
a(1+κa) if si ≤ a,
0 if si > a.(35)
Note that the ‘added’ charge density inside the colloidshas the same sign as the surface charge and exactly can-cels the ionic charge due screening. From Eq. (33a) oneobtains ZY
i = Z exp(κa)/(1 + κa) and this gives theDLVO result as Eq. (19a) shows.The Yukawa monopole and the Yukawa dipole can also
be extracted from the solution for the electrostatic poten-tial outside a spherical particle, using Eq. (34), withoutcalculating the charge density qi(r) explicitly. Namely,we can often solve the (linearized) Poisson-Boltzmannequation around a single particle mode-by-mode, andthen read the multipole moments from the mode am-plitudes in the final solution. As an example we con-sider spherical particles with a fixed surface charge dis-tribution that is rotationally symmetric around the unitvector ni, e.g., a Janus particle. The electrostatic po-tential in the vicinity of the single particle can be ex-panded as φ(si, xi) =
∑∞ℓ=0 φℓ(si)Pℓ(xi), with xi = si·ni,
and Pℓ(xi) the ℓ’th order Legendre polynomial66. Nowℓ = 0 and ℓ = 1 correspond to the monopole and thedipole contribution to the electrostatic potential, respec-tively. The multipole mode functions, which solve thelinearised PB equation, behave as φℓ(si) ∼ sℓi for r < aand φℓ(si) ∼ ki(κsi) for r > a, with ki the i’th modi-fied spherical Bessel function. By applying the boundarycondition
limsi↓a
φ′(si, xi) = limsi↑a
φ′(si, xi)− 4πλBσ(xi), (36)
with the prime (′) denoting the radial derivative (w.r.t.si), and σ(xi) the surface-charge density, a solution to theelectrostatic potential in terms of a multipole expansion
13
is obtained. Since the modified spherical Bessel functionscan also be recognized in Eq. (34), one is able to derivethat the monopole and the dipole moment should relateto the expansion of the surface-charge density, σ(xi) =∑∞
ℓ=0 σℓPℓ(xi), by
ZYi = 4πa2σ0
exp(κa)
1 + κa; (37)
pYi = 4πa3σ1
exp(κa)
3 + 3κa+ (κa)2ni. (38)
In the case of Janus particles with charge densities σU ≡QU/(4πa
2) and σL ≡ QL/(4πa2) on the upper and lower
hemisphere respectively, one finds σ0 = 12 (σU + σL) and
σ1 = 34 (σU − σL).
Throughout this paper we assumed that the dielec-tric constant of the particles matched the dielectric con-stant of the solvent. However, Eq. (38) can easily be ex-tended to particles that have a dielectric mismatch withthe solvent. We modify Eq. (38) by including a dielectricjump in the boundary condition [Eq. (36)], such that theYukawa multipoles for these particles can be obtained aswell. As is known from the DLVO equation, we find thatthe Yukawa monopole is unaffected by the value of thedielectric constant inside the particle, whilst the Yukawadipole changes. Equation (38) becomes
pYi = 4πa3σ1
exp(κa)
(2 + ǫc/ǫ)(1 + κa) + (κa)2ni, (39)
with ǫc/ǫ the ratio of the relative dielectric constant in-side and outside the particles. Note that the dipolemoment becomes very small if the interior of the col-loid is very polar w.r.t. the surrounding medium, e.g.,(Pickering) emulsions of water in oil. Consequently themonopole-dipole and dipole-dipole interactions will benegligible for these systems, even in case of an asymmet-ric charge distribution. However, if the dielectric con-stant of the colloid is comparable or smaller than thedielectric constant of the solvent a significant dipolar in-teraction may arise.
CONTRIBUTIONS
J.d.G performed the MC simulation studies, N.B. per-formed the PB-theory calculations, and they collaboratedfully on the analysis of their findings and on the prepara-tion of the manuscript; both contributing equally. M.D.and R.v.R. primarily supervised J.d.G. and N.B., respec-tively. All authors discussed the results and commentedon the manuscript.
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