Non-equilibrium ionic assemblies of oppositely charged nanoparticles† Rui Zhang, a Prateek K. Jha b and Monica Olvera de la Cruz * ab Structure and evolution kinetics of non-equilibrium clusters formed in a solution of oppositely charged nanoparticles are studied using a recently developed kinetic Monte Carlo simulation scheme (Jha et al., Soft Matter , 2012, 8, 227–234). A diverse range of dynamic cluster configurations are obtained by varying the interaction strength between nanoparticles, screening length, and packing density of nanoparticles. Structural details of the resulting clusters are obtained using the correlations of local bond orientational order parameters. At low-salt concentrations (weak screening), clusters with structures ranging from NaCl-type cubic aggregates to fibril-like chains are observed, while at high-salt concentrations (strong screening), disordered compact clusters are observed. A chain-folding barrier model is proposed to explain the kinetically trapped fibril-like assemblies. In higher-density solutions, large ionic clusters or percolated gel structures are observed. Our work demonstrates the structural richness of non-equilibrium ionic assemblies of oppositely charged nanoparticles and elucidates the effect of ion correlations on the determination of the structure of assemblies of oppositely charged nanoparticles. These “nanoionic composites” hold great promise in a variety of emerging applications such as templated polymerization of charged molecules and assembly of charged nano-objects. 1 Introduction Even the most sophisticated “top-down” design approaches have difficulties in creating highly ordered, perfect structures at the nanoscale. The common “bottom-up” strategy of self- assembly provides an alluding alternative. 1–3 An emerging theme in nanoscale self-assembly is the use of nanoparticles (NPs) as building blocks to create higher-order architectures. 4–7 In the past decade, chemists and materials scientists have achieved tremendous success in synthesizing NPs with precise control of the composition, size, shape, and functional mole- cules tethered to their surfaces. Many design rules have been established based on manipulating the strength and range of various physical interactions between NPs (e.g. van der Waals, steric, electrostatic, magnetic, etc.) by modifying the NP char- acteristics or environmental conditions. Thus, NP self-assembly is a promising route for fabrication of ordered, functional structures with representative applications in diagnostics, 8 catalytic systems, 9 nanoelectronics 10 and sensors. 11 Of partic- ular interest are studies on NP self-assembly in solutions of charged NPs (e.g. metal NPs 6 or NPs functionalized with charged ligands such as DNA 5,12 ). Recently, a diverse range of binary super-lattice structures have been reported by electro- static self-assembly of oppositely charged NPs 4,6,13 including unusual non-close-packed structures such as diamond-like lattices. 6 Moreover, computational researchers have applied the Madelung summation approach, 14 mean-eld theory 15 and computer simulations 14,16,17 to determine the stable super- lattice structures that agree with experimental ndings. However, super-lattices represent only equilibrium or static self- assembled structures (minimal free energy state) of oppositely charged NPs. The equilibration times for many relevant systems vary from hours to days, 4,6,13,14 and non-equilibrium structures/ patterns 18,19 are very different from those found at equilibrium. This has been demonstrated in recent theoretical studies of oppositely charged molecules on surfaces/interfaces, both under non-equilibrium 20,21 and equilibrium 22,23 conditions. To the best of our knowledge, a detailed study of non-equi- librium self-assembly of oppositely charged NPs in three- dimensions does not exist. One obstacle is the lack of compu- tational approaches that can track the evolution of ionic assemblies for a reasonably long time. Due to enormous length and time scale differences between NPs and solvent/salt mole- cules, an explicit solvent/salt simulation is essentially impos- sible. 24,25 In this work, we apply an implicit solvent and salt, coarse-grained model to study the aggregate structures and evolution of oppositely charged NPs at low to intermediate concentrations. We implement a recently developed kinetic a Department of Materials Science and Engineering, Northwestern University, Evanston, IL, 60208, USA. E-mail: [email protected]b Department of Chemical and Biological Engineering, Northwestern University, Evanston, IL, 60208, USA † Electronic supplementary information (ESI) available: (1) Four simulation movies and (2) a description of the movie details are included. See DOI: 10.1039/c3sm27529a Cite this: Soft Matter, 2013, 9, 5042 Received 2nd November 2012 Accepted 14th March 2013 DOI: 10.1039/c3sm27529a www.rsc.org/softmatter 5042 | Soft Matter , 2013, 9, 5042–5051 This journal is ª The Royal Society of Chemistry 2013 Soft Matter PAPER Published on 11 April 2013. Downloaded by University of Groningen on 25/03/2015 10:43:37. View Article Online View Journal | View Issue
Transcript
Soft Matter
PAPER
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aDepartment of Materials Science and Engine
IL, 60208, USA. E-mail: m-olvera@northwesbDepartment of Chemical and Biological
Evanston, IL, 60208, USA
† Electronic supplementary informationmovies and (2) a description of the m10.1039/c3sm27529a
Cite this: Soft Matter, 2013, 9, 5042
Received 2nd November 2012Accepted 14th March 2013
DOI: 10.1039/c3sm27529a
www.rsc.org/softmatter
5042 | Soft Matter, 2013, 9, 5042–50
Non-equilibrium ionic assemblies of oppositely chargednanoparticles†
Rui Zhang,a Prateek K. Jhab and Monica Olvera de la Cruz*ab
Structure and evolution kinetics of non-equilibrium clusters formed in a solution of oppositely charged
nanoparticles are studied using a recently developed kinetic Monte Carlo simulation scheme (Jha et al.,
Soft Matter, 2012, 8, 227–234). A diverse range of dynamic cluster configurations are obtained by
varying the interaction strength between nanoparticles, screening length, and packing density of
nanoparticles. Structural details of the resulting clusters are obtained using the correlations of local
bond orientational order parameters. At low-salt concentrations (weak screening), clusters with
structures ranging from NaCl-type cubic aggregates to fibril-like chains are observed, while at high-salt
concentrations (strong screening), disordered compact clusters are observed. A chain-folding barrier
model is proposed to explain the kinetically trapped fibril-like assemblies. In higher-density solutions,
large ionic clusters or percolated gel structures are observed. Our work demonstrates the structural
richness of non-equilibrium ionic assemblies of oppositely charged nanoparticles and elucidates the
effect of ion correlations on the determination of the structure of assemblies of oppositely charged
nanoparticles. These “nanoionic composites” hold great promise in a variety of emerging applications
such as templated polymerization of charged molecules and assembly of charged nano-objects.
1 Introduction
Even the most sophisticated “top-down” design approacheshave difficulties in creating highly ordered, perfect structures atthe nanoscale. The common “bottom-up” strategy of self-assembly provides an alluding alternative.1–3 An emergingtheme in nanoscale self-assembly is the use of nanoparticles(NPs) as building blocks to create higher-order architectures.4–7
In the past decade, chemists and materials scientists haveachieved tremendous success in synthesizing NPs with precisecontrol of the composition, size, shape, and functional mole-cules tethered to their surfaces. Many design rules have beenestablished based on manipulating the strength and range ofvarious physical interactions between NPs (e.g. van der Waals,steric, electrostatic, magnetic, etc.) by modifying the NP char-acteristics or environmental conditions. Thus, NP self-assemblyis a promising route for fabrication of ordered, functionalstructures with representative applications in diagnostics,8
catalytic systems,9 nanoelectronics10 and sensors.11 Of partic-ular interest are studies on NP self-assembly in solutions of
ering, Northwestern University, Evanston,
tern.edu
Engineering, Northwestern University,
(ESI) available: (1) Four simulationovie details are included. See DOI:
51
charged NPs (e.g. metal NPs6 or NPs functionalized withcharged ligands such as DNA5,12). Recently, a diverse range ofbinary super-lattice structures have been reported by electro-static self-assembly of oppositely charged NPs4,6,13 includingunusual non-close-packed structures such as diamond-likelattices.6 Moreover, computational researchers have applied theMadelung summation approach,14 mean-eld theory15 andcomputer simulations14,16,17 to determine the stable super-lattice structures that agree with experimental ndings.However, super-lattices represent only equilibrium or static self-assembled structures (minimal free energy state) of oppositelycharged NPs. The equilibration times for many relevant systemsvary from hours to days,4,6,13,14 and non-equilibrium structures/patterns18,19 are very different from those found at equilibrium.This has been demonstrated in recent theoretical studies ofoppositely charged molecules on surfaces/interfaces, bothunder non-equilibrium20,21 and equilibrium22,23 conditions.
To the best of our knowledge, a detailed study of non-equi-librium self-assembly of oppositely charged NPs in three-dimensions does not exist. One obstacle is the lack of compu-tational approaches that can track the evolution of ionicassemblies for a reasonably long time. Due to enormous lengthand time scale differences between NPs and solvent/salt mole-cules, an explicit solvent/salt simulation is essentially impos-sible.24,25 In this work, we apply an implicit solvent and salt,coarse-grained model to study the aggregate structures andevolution of oppositely charged NPs at low to intermediateconcentrations. We implement a recently developed kinetic
This journal is ª The Royal Society of Chemistry 2013
Monte Carlo (kMC) simulation scheme26 based on therenormalization of Smoluchowski diffusion equations to amaster equation with transition rates identical to the Glauberdenition of transition probabilities.27 For our model, thissimulation scheme is computationally more efficient thanBrownian dynamics (BD) simulation because it can employlonger time-steps while maintaining numerical stability.However, due to computational limitations associated withsimulation approaches aiming at realistic dynamics, our kMCscheme is not well-suited for the prediction of equilibriumstructures of NP assemblies. This is because the long-timebehavior of these systems is oen dictated by the very slowreorganization of crystalline domains into large, perfect crys-tals. Biased Monte Carlo approaches25 that employ articialcollective movements are thus better suited for equilibriumstudies, at the expense of their inability to predict “true”dynamic behavior far-from-equilibrium. Moreover, both kMCand BD methods neglect the long-range hydrodynamic inter-actions (HIs)25 between NPs due to solvent/salt ow. Sincecharged NPs involve long-range electrostatic interactions andwe are interested in short- to intermediate-time evolution of theionic cluster structures, it is prohibitively expensive to apply acoarse-grained approach that incorporates HIs, such as dissi-pative particle dynamics.25,28
Here, we report a rich range of non-equilibrium self-assem-bled structures (crystalline and disordered clusters, gels,chains, and networks) found in the kMC simulations. Thestructural signature can be controlled by tuning the Debyescreening length (salt concentration), interaction strengthbetween NPs (NP charge, solvent dielectric constant, andtemperature), and packing density of NPs. Chain-like assem-blies appear in magnetic NPs29–33 due to directional dipole–dipole interactions, and are observed at equilibrium and nearcriticality in simulations34–37 of size-asymmetric, oppositelycharged electrolytes using the primitive model. Interestingly, wend that kinetically trapped chain-like assemblies can alsoappear in size-symmetric, oppositely charged NPs as long-livednon-equilibrium structures in a broad range of parameterspace.
2 Model and methods
We study a system of an equal number of positively and nega-tively charged NPs, with equal valence Z (Z+ ¼ �Z� ¼ Z) anddiameter s. They interact via a screened Coulomb (Yukawa)potential13,38 controlled by two dimensionless parameters – Z2lB/s and ks. Here, lB is the Bjerrum length and k�1 is the Debyescreening length; these parameters characterize the strengthand range of electrostatic interactions, respectively, and aregiven by lB ¼ e2/4p303rkBT and k�1 ¼ 1=
the salt concentration, 3r is the dielectric constant of thesolvent, 30 is the vacuum permittivity, and e is the electroniccharge.38 The use of a Yukawa potential in contrast to a moreaccurate scheme (e.g. Ewald summation) and the implicitsolvent and salt model limits the applicability of this model tolow to moderate salt concentration, and to cases where solvent-induced uctuations are not important. In our simulation, a
This journal is ª The Royal Society of Chemistry 2013
convenient time unit s0 ¼ s2/D is introduced, where D is thediffusion coefficient of free NPs in solution.26 For typical NPsystems, s0 � 10�5 to 10�1 s.
All simulations start from a random distribution of NPsinteracting via Yukawa potentials. In essence, this approximatesa process in which the NPs initially interact weakly (dis-assembled) and are later fast-“quenched” to interact via theYukawa potential used in simulations (i.e., a homogeneousphase is not stable). NPs spontaneously aggregate intosymmetry-breaking non-equilibrium structures. One possibleexperimental strategy to mimic the simulations reported in thispaper would be to add solvents with lower dielectric constant orto decrease temperature in order to enhance the interactionstrength. A general non-equilibrium theory of colloid/NPdynamics has been proposed by Medina-Noyola et al.39,40 toaddress the irreversible processes associated with dynamicarrest transitions. However, it is very challenging to theoreti-cally predict detailed cluster structures and computer simula-tions provide a favorable alternative.
2.1 NP interactions
Positively and negatively charged NPs interact via a screenedCoulomb (Yukawa) potential:13,41,42
UijðrÞkBT
¼
8><>:
ZiZjlB=s
ð1þ ks=2Þ2e�kðr�sÞ
r=s; r$ s
N; r\s
(1)
where r is the distance between NPs. To avoid expensivecomputation of pairwise interactions, we use a cut-off distancercut (¼2.5 s or a larger value such that |Uij(rcut)| # 0.1 kBT). Thatis, shied Yukawa potentials are applied in simulation,
Uij;shiftðrÞ ¼�Uij � 3ij r# rcut0 r. rcut
(2)
where 3ij ¼ Uij(rcut) is a small tail correction added to avoiddiscontinuity of potential at r ¼ rcut. Short-range attractionssuch as van der Waals (vdW) forces are oen present in NPsolutions and favor particle aggregation.26 However, since thegoal of this work is to study the dynamic self-assembly driven bycompeting electrostatic interactions, we exclude short-rangeattractions to isolate the effects of electrostatic interactions. Wenote that for all systems considered in this work, the strength ofvarious electrostatic interactions is very strong (�20 to 130 kBT),which is much higher than the typical vdW interaction strengthbetween two NPs.
2.2 Kinetic Monte Carlo simulation
Details of the kMC simulation approach have been discussedpreviously.26 In short, the kMC scheme is derived from arenormalization approach wherein the Smoluchowski diffusionequations are renormalized to amaster equation with transitionrates identical to the Glauber denition of transition probabil-ities.27 We perform three-dimensional kMC simulations for N ¼500 binary charged NPs (250 each of positively and negativelycharged particles) placed in a cubic box with periodic boundaryconditions. The NP density is dened as h ¼ ps3N/6V, where N
is the total number of NPs and V is the system volume. Weanalyze nite size effects and do not observe appreciabledifference between the results obtained by N ¼ 500 and N ¼1000 simulations. One kMC step consists of the following: (1)randomly picking one particle from the system; (2) attempting amove of the chosen particle of a xed step size a, where thedirection of movement is decided by the choice of the azimuthalangle q ¼ 2pu and the polar angle f ¼ arccos(2v � 1), u and vbeing random variables in (0, 1); (3) calculating the energychange DU caused by this move and accepting the move withthe Glauber denition of transition probability.
p ¼�1þ exp
�DU
kBT
���1
(3)
A kMC sweep contains N such kMC steps and corresponds toa step in time given by26
Dt ¼ a2
12D¼ s0
12
�as
2(4)
Note that the kMC simulation26 does not allow unphysicalviolations of the hard-core constraint (zero acceptance proba-bility); one does not need to substitute the hard-core contribu-tion to the Yukawa potential by a steep repulsive function as isoen done for Brownian dynamics.43
Systems of a wide numerical range of ks (from 1 to 5) aresimulated. For each system, ten independent simulations areperformed to obtain time-dependent cluster statistics. Eqn (4)shows that the time step Dt is proportional to the step sizesquared (a2). However, we note that eqn (4) is derived26 basedon the assumption that the energy change associated with aMonte Carlo move is reasonably small. This assumption isalso used while renormalizing the diffusion equation to derivethe transition probability (eqn (3)). In the original paperdescribing the scheme,26 a step size of a ¼ 0.1 s was found tobe a good choice for simulations of particles interacting byshort-ranged Lennard-Jones potentials. However, a lower stepsize (a ¼ 0.02 s) was used in a later publication,44 whereadditional bonded-interactions and ow-induced interactionswere involved. Since a longer particle displacement (larger a)leads to a larger energy change in general, both eqn (3) and (4)are less accurate for larger step size. In all simulations, wehave chosen a step size small enough such that the systematicerror generated by a nite step size is comparable to thestatistical error, and large enough such that the method iscomputationally efficient. We call it the “proper” step size. Wehave performed a series of test simulations using differentstep sizes and found that the magnitude of the “proper” stepsize a* depends on both the contact energy (dened as u* ¼(Z2lB/s)/(1 + ks/2)2 in units of kBT) and the range (1/ks in unitsof s) of the Yukawa potential. An empirical relationship forthe “proper” step size, a*/s ¼ 0.36/(u*ks), is obtained basedon the results of test simulations. We have conrmed thatthere is no appreciable difference between all the results (e.g.,average cluster size, fraction of crystalline particles, etc.)produced by simulations that employ step sizes of a ¼ a* or a¼ a*/2, thus giving condence in the above relationship forthe “proper” step size. However, the difference between the
5044 | Soft Matter, 2013, 9, 5042–5051
results of a ¼ a* and a ¼ 2a* simulations is signicant.Moreover, we nd that the MC acceptance ratio p for a longaggregation time is always close to 0.35 in simulations usingstep size ¼ a*, and the average distance between centers oftwo neighboring NPs (hlbondi) is smaller (tighter packing) forlarger u* and ks following a universal relationship for ourYukawa system: a* z (hlbondi � s)/4. It would be interesting tocheck whether p(a ¼ a*) z 0.35 and a* z (hlbondi � s)/4 aregood criterions of “proper” step size for other NP systems(particles contain a hard core with diameter s) where particlesaggregate to form clusters.
An alternative computational approach to study our systemis Brownian dynamics simulations. Sanz et al. have studiedthe gelation of oppositely charged colloids that interact via theYukawa potential (sharp so-repulsion) by Brownian dynamics(BD) simulation43 with a time step dened as Dt/s0 ¼ 7 �10�6/u*. Taking this as the “proper” time step for BD, one canobtain the relationship DtkMC/DtBD z 1400/u*(ks)2. Thismeans for all Yukawa conditions that we discuss in this paper,DtkMC is at least 1.5 times of DtBD. For low-ks systems, wherewe observe interesting cubic and bril-like clusters, DtkMC isan order of magnitude larger than DtBD. However, it appearsthat in the case of very short-range and strong interactions,kMC is numerically less efficient than BD. Since hydrody-namic interactions are not included in kMC and BD simula-tions, both of them are unable to reproduce the correctscaling of cluster diffusion coefficient Dc with cluster size nc(e.g., Dc � 1/nc
3/5 for chains (Zimm theory45) and Dc � 1/nc1/3
for compact clusters). Instead, effective diffusivity in bothkMC and BD scales as Dc � 1/nc.46,47 More work is needed toexplore into these issues.
2.3 Analysis of ionic clusters
We study the cluster size distribution by sampling the clustersformed at different times in the simulation run.26 Two particlesare considered to be in the same cluster if the distance betweenthem is less than 1.5 s. At any given time instant, the numberaverage cluster size is dened asMn ¼
Pjnc, j/
Pj1, where nc, j is
the number of particles in the j-th cluster.In order to characterize the order (crystallinity) of a compact
cluster, we compute the correlation of local bond orderparameters.48–50 A bond orientational order parameter qlm(i) forparticle i is dened as
qlmðiÞ ¼ 1
NbðiÞXNbðiÞ
j¼1
Ylm
rij�; (5)
where Ylm(rij) are the spherical harmonics of order l, rij is theunit vector connecting the particle i with its neighbor particle j,and Nb(i) is the total number of nearest neighbors of particle i.Two particles are taken as neighbors with each other if thedistance between them is smaller than 1.1 s and if so these twoparticles form a bond. We choose this small value to enforcethat neighbors belong to the rst nearest neighbor shell (all areoppositely charged) regardless of the cluster structure details.The two particles i and j are connected if the correlationbetween the bond orientational order given by
This journal is ª The Royal Society of Chemistry 2013
is larger than 0.5. A particle is considered as ordered if it isconnected to at least half of its neighbors. Otherwise, it isconsidered as disordered. The 4th-order bond order parameterq4(l ¼ 4) best distinguishes the different cluster structuresobserved in simulation and is used to analyze a cluster's order(crystallinity). The radial distribution function is averaged overall particles in the simulation box at a given instant of time (not
Fig. 1 Typical ionic cluster simulation snapshots (h ¼ 0.004) for three ks ¼ 1 systemindicated (in unit of s0). Snapshots are zoomed in to show individual clusters in detailis used to indicate the depth of particles from the top plane (closer or farther). RedRadial distribution function between two positively charged particles g++ (by symmeZ2lB/s ¼ 81. Dotted lines represent the first three g+� peaks (r=s ¼ 1;
ffiffiffi3
p;ffiffiffi5
p) and t
This journal is ª The Royal Society of Chemistry 2013
time-averaged) and obtained by the standard binningprocedure.
3 Results and discussion
All systems that we report in this work have well-known equi-librium phase behavior:16 a gas phase coexists with a CsCl-typesuper-lattice phase. Under non-equilibrium conditions, weinstead have observed a variety of dynamic self-assemblystructures. In what follows, we summarize in detail the struc-tural signatures of the time-dependent NP assemblies underdifferent ks, Z2lB/s and h conditions, which delineates impor-tant non-equilibrium aspects of system properties,
s: (a) Z2lB/s ¼ 81, (b) Z2lB/s ¼ 225, and (c) Z2lB/s ¼ 300 at the aggregation timeand do not show the entire simulation box. The contrast (bold or faded) of particlesand blue colors represent positively and negatively charged NPs, respectively. (d)try, g++ ¼ g��) and two oppositely charged particles g+� at t ¼ 1000 s0 for ks ¼ 1,he first three g++ peaks (r=s ¼ ffiffiffi
Fig. 2 (a) Fraction of ordered particles forder and (b) fraction of particles in chainfchain as a function of aggregation time for ks ¼ 1 systems (h ¼ 0.004) withdifferent Z2lB/s values. Inset: average chain size as a function of Z2lB/s at the timewhen fchain reaches its maximum.
Fig. 3 Typical ionic cluster simulation snapshots (h¼ 0.004) at t¼ 200 s0 for twoks ¼ 5 systems: Z2lB/s ¼ 160 and Z2lB/s ¼ 440.
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supplementary to the existing knowledge of self-assembly ofoppositely charged NPs under equilibrium conditions.
We start the discussion at one specic dilute NP density h
¼ 0.004. For each ks, we rst determine the approximatevalue of Z2lB/s beyond which NPs start aggregating to formionic clusters. This critical value of Z2lB/s increases with ks
(from z50 for ks ¼ 1 to z120 for ks ¼ 5). We then simulatea series of Z2lB/s conditions in the aggregation regime foreach ks. The cluster size/structure and its evolution kineticsshow a rich ks- and Z2lB/s-dependence. Two distinct kineticprocesses are generally present – fusion occurs when two(diffusive) clusters merge into one larger cluster if they getsufficiently close, isomerization takes place when particleswithin a cluster change their relative positions and recong-ure the cluster.
Fig. 1 presents simulation snapshots with increasing timefor three Z2lB/s values at ks¼ 1. For this weak screening system,we nd two distinct long-lived cluster structures. In the case ofZ2lB/s¼ 81 (Fig. 1a), NPs transiently aggregate into short chains(see snapshot at t ¼ 5 s0) in the early aggregation stage. Theseelongated congurations are short-lived and evolve into morecompact congurations. At the end of the simulation run,multiple ordered clusters emerge (see snapshot at t ¼ 1000 s0,and Movie S1 in the ESI†), which are NaCl-type cubic crystallitesas conrmed by the radial distribution function calculation(Fig. 1d). Fig. 1b and c show snapshots under two higher Z2lB/sconditions (225 and 300). It is evident that chains sustain for alonger time for larger magnitudes of Z2lB/s. The snapshot forZ2lB/s ¼ 300 at long time is in sharp contrast to the Z2lB/s ¼ 81case. At the end of the simulation run, bril-like clusters, ratherthan NaCl-type crystalline clusters, are widespread in thesimulation box (see snapshot at t ¼ 200 s0, and Movie S2 in theESI†).
NaCl-type cubic clusters are observed at t ¼ 1000 s0 for thethree lowest Z2lB/s-conditions (60, 81 and 115) that we havesimulated. To quantify the cluster order evolution, we apply thelocal bond orientational order parameter method49,50 (Section2.3) to calculate the fraction of ordered NPs ( forder) at differenttimes. This method determines a particle as “ordered” if itsposition is correlated with its neighbors' positions and has beenwidely used to study particle order in compact clusters/densephases.43,49,50 As Fig. 2a shows, forder nearly attains a plateauvalue in the time window 10–200 s0 and then steadily increasesuntil the end of simulation. Among the three cases, Z2lB/s ¼ 81represents an optimal condition where the crystallizationoccurs most rapidly. Another interesting cluster congurationfound in the ks ¼ 1 system is the ionic chain (total number ofbonds¼ nc� 1, where nc is the number of NPs in the cluster). InFig. 2b, the fractions of NPs in chain clusters ( fchain) as afunction of the aggregation time for four Z2lB/s values (81, 160,225 and 300) are shown. In the calculation, we require that achain contains at least four particles. We nd that themaximum value of fchain, the time at which this maximum isattained, and the average chain size (inset) all increase withincreasing Z2lB/s values. Note that even for the case of thelargest Z2lB/s ¼ 300, fchain goes to zero at t � 100 s0. In general,simulations predict that chains disappear due to one of the two
5046 | Soft Matter, 2013, 9, 5042–5051
possibilities. First, it can isomerize (by local folding) to morecollapsed conguration where this process occurs lessfrequently for larger Z2lB/s. Second, it can fuse into anothercluster with the new (larger) cluster not being a chain (see MovieS2 in the ESI† for bril-like cluster evolution).
Aggregate structures formed under strong screening condi-tions exhibit many different features than those formed in theweak screening case discussed above. For example, for all Z2lB/svalues (up to 440) that we simulate in the ks¼ 5 system, neithercrystalline nor bril-like clusters are observed. Instead, compactdisordered ( forder < 0.1 at long time) clusters emerge. Fig. 3shows snapshots for two different Z2lB/s conditions at t¼ 200 s0(also see Movie S3 in the ESI† for a simulation run under Z2lB/s¼ 160 condition). In general, we nd that ionic chains do notform when the screening is strong, even at the early aggregationstage.
Nowwe discuss the reasons why NPs form ionic chains ratherthan random Bjerrum pairs,51,52 where one positive particle is
This journal is ª The Royal Society of Chemistry 2013
connected to only one negative particle, and why chains are onlyobserved for low ks and sustain longer for higher Z2lB/s.Consider twopositive and twonegativeNPs in the k¼ 0 limit. Thetotal ionic energy is �2kBTZ
2lB/s if they form two random Bjer-rumpairs. But if they forma “+�+�” chain, the energy is lower by
an amount13kBTZ2lB=s. Note that although the whole system is
electroneutral, the energy is attractive for both cases due to ioniccorrelations. This implies that when the value of Z2lB/s is largeenough, Coulomb energy would win over entropy, which leads tochains being more favored than separate Bjerrum pairs (thesame conclusion holds for nite k). To examine kinetic evolutionof the tetramer-chain under different ks and Z2lB/s conditions,we compare ionic energies of two pentamer congurations(chain growth) in Fig. 4a. In Fig. 4b, we compare ionic energies ofthe tetramer-chain with its two isomerized states. Fig. 4a clearlyshows that, under low ks conditions, attaching aNP to the end ofa chain is energetically favorable than attaching it to the middle
Fig. 4 (a) Ionic energy analysis of two pentamer clusters formed by adding onepositively charged NP to either the middle (left cartoon) or the end (right cartoon)of a tetramer-chain (top cartoon). Both pentamers have lower ionic energy thanthe tetramer. DE1 (solid curve) represents the ionic energy difference between thelinear pentamer and the tetramer. DE2 (dashed curve) represents the ionic energydifference between the branched pentamer and the tetramer. (b) Ionic energyanalysis of three isomerized tetramer clusters. DE (dash curve) represents the ionicenergy difference between the lowest-energy square structure (right cartoon)and the kinetically favored linear structure (left cartoon). Eb (solid curve) repre-sents the transition-state barrier height. DE1, DE2, DE and Eb are all scaled bykBTZ
2lB/s and the scaled values only depend on ks.
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of the chain. For the latter case, the attraction between the freeNP and its oppositely charged neighbor on the chain is largelycompensated by repulsions from the two close like-charged NPson the chain. Hence chains favor linear growth for weakscreening systems, which is consistent with our simulationresults. However, a chain is not in theminimum energy state. AsFig. 4b shows, the lowest-energy tetramer conguration is acompact square-like structure. We denote the ionic energydifference between the chain structure and compact structure asDE. Interestingly, a chain needs to overcome an energy barrier totransit to the compact structure. The middle cartoon in Fig. 4brepresents one plausible transition-state conguration. Its ionicenergy is higher than that for the linear structure byEb.DE andEbas a function of ks are plotted in Fig. 4b. Both quantities scalewith Z2lB/s, and the magnitude of Eb sharply decreases with ks.Without daunting calculation of ionic energies for larger sizedclusters, we argue that this chain-folding barrier model quali-tatively captures the trend for all scale clusters, i.e., the energybarrierwhich one chainorbril-like cluster needs toovercome toisomerize to more compact conguration scales with Z2lB/s andstrongly decays with ks where ksT1. This particular energeticfeature of ionic NP clusters explains our simulation results.
We have observed crystalline cubic clusters for the ks ¼ 1system (Fig. 1a) and disordered compact clusters for the ks ¼ 5system (Fig. 3). This suggests that a longer range of electrostaticinteractions (low ks) favors ordered cluster structure. In Fig. 5,we compare the forder vs. t (time) curves for different ks systemsat xed Yukawa contact energy u* ¼ 36 (a moderate value atwhich NPs aggregate to form compact clusters in all systems; forthe ks ¼ 1 system, this is the case for Z2lB/s ¼ 81). We nd thatforder increases with time where ks(2. At ksT2, clusters do not
Fig. 5 Fraction of ordered particles forder as a function of aggregation time underdifferent ks conditions. For all systems (h ¼ 0.004), the contact energy is fixed at36 kBT. Statistics is made for all clusters (upper panel) and only neutral clusters(lower panel), respectively.
Fig. 6 Typical simulation snapshots for systems with density h ¼ 0.019 and 0.065 at the aggregation time indicated (in unit of s0): (a) ks ¼ 1, Z2lB/s ¼ 81; (b) ks ¼ 1,Z2lB/s ¼ 300; and (c) ks ¼ 5, Z2lB/s ¼ 220. The whole simulation box is shown for all systems.
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evolve into more ordered structures. Poorer order is developedfor higher ks. The basic trend does not change even if onlyneutral clusters (lower panel of Fig. 5) are counted. (We havechecked the inuence of cluster valence on the cluster orderand found that neutral clusters, i.e., clusters containing anequal number of positively and negatively charged NPs, yieldthe highest forder value for all systems. In this case, forderrepresents the fraction of ordered NPs in all neutral clusters.) Sotwo qualitatively different cluster evolution pictures are gener-ated. For systems where the Debye screening length is longerthan the particle radius (ks(2), compact clusters evolve intomore ordered/crystalline structures with time (e.g. see MovieS1†). While for systems where the Debye screening length isshorter than the particle radius, clusters do not evolve intoordered cubic assemblies (e.g. see Movie S3†). Therefore, forexperimentalists who hope to create crystalline ionic clusters,weak screening is desired.
Finally, we address how packing fraction of NPs affects theproperties of ionic assemblies. Based on kMC simulations atmultiple densities, we have found that Mn universally increaseswith h at the same aggregation time, which is a trend consistentwith the attractive sphere system.26Whilehhas a strong inuence
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on the cluster size, the structural signatures (crystallinity andanisotropy) of the assemblies are essentially controlled by theYukawa condition (Z2lB/s and ks). To illustrate this point, typicalsnapshots at two densities higher than 0.004 are presented inFig. 6. Three specic Yukawa conditions are selected, underwhich we have observed that, for h ¼ 0.004, NPs aggregate intolong-lived cubic, bril-like, and disordered ionic clusters,respectively. For ks ¼ 1 and Z2lB/s ¼ 81 (Fig. 6a), NaCl-type crys-talline order is obtained at all densities. For the disorderedsystem characterized by ks¼ 5 and Z2lB/s¼ 220 (Fig. 6c), forder isnot improved by increasing the density of NPs. Finally, for ks¼ 1and Z2lB/s¼ 300, an interesting network structure is formed at h¼ 0.019 (Fig. 6b), which is due to the interconnectionof elongatedbril-like clusters in dense solutions. At h ¼ 0.065, NPs formpercolated ionic gels for all systems. A rich range of crystallinityand porosity of the gels can be realized (bottom row of Fig. 6).
4 Conclusions
We have performed kinetic Monte Carlo simulations to studythe non-equilibrium self-assembly process of oppositelycharged NPs with equal size and valence. By systematically
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investigating the inuence of the three control parameters –
Z2lB/s (Yukawa potential strength), ks (inverse Debye screeninglength) and h (NP density), we have observed a rich variety ofdynamically assembled structures. In dilute solutions, ionicclusters formed by the two types of NPs exhibit varied order andanisotropy. Under weak screening conditions (low salt concen-tration), both ordered NaCl-type crystalline clusters and aniso-tropic bril-like chains are observed. The bril-like clusters arekinetically trapped structures and sustain for a long time atlarge Z2lB/s (e.g. Fig. 1c) because the energy barrier betweenelongated clusters and the minimum energy compact congu-ration scales with Z2lB/s. However, this energy barrier sharplydecreases with increase in ks (Fig. 4); no ionic chains areobserved for the strong screening case. Comparing compactclusters under different ks conditions, we found an order–disorder transition occurring at ks � 2. For smaller ks, thefraction of ordered particles forder gradually increases with time,and a clear tendency toward NaCl-type crystalline arrangementis observed. On the other hand, for ksT2, forder does notincrease with time and the clusters tend to be disordered.Simulations at higher NP densities show that the average clustersize increases with NP density, eventually leading to theformation of a percolated structure at very high density. Bycontrolling the Yukawa conditions (Z2lB/s and ks), we haveobtained a variety of percolated structures such as crystallineionic gels and porous ionic networks.
We argue that the kMC scheme is more efficient and numer-ically stable than the conventionalBrowniandynamics technique(Section 2.2). However, a direct comparison of accuracy andperformance of the two approaches is not attempted. It isimportant to make such comparisons in systems where thecomputational costs for bothapproaches arenot intense to gainadetailed understanding of the differences between the twomethods. One limitation of the kMC scheme is that it is not well-suited in describing the behavior at very long times, since themethod only employs simple single-particle displacements. Forinstance, as discussed in Section 2.2, kMC fails to predict correctcluster diffusivity scaling for large clusters. We note that theintroduction of collective movesmight enable kMC to reproducecorrect cluster diffusivity scaling.However, incorporation of suchcluster movements is not straightforward as we need to preserverealistic dynamics. The latter requirement is important if far-from-equilibrium structures are of interest. For example, non-physical collective moves may bypass the kinetically trappedcongurations, such as the chains reported in this paper. Usingthe cluster evolution mechanism identied in our simulations,we argue that the structural features of the ionic assembliesreported here are insensitive to enhanced cluster diffusivity.First, single particles (“monomeric cluster”) diffuse much fasterthan big clusters and “monomer addition” is the dominantprocess at the early stage, which is well-described by kMC.Second, in all fusion processes observed in our simulationswhere two big clusters (A and B) fuse together to form a newcluster (C), the structural signature (crystalline, bril-like ordisordered compact) of cluster C is the same as clusters A and B.We do not expect pronounced variations in the structural signa-tures if the frequency of fusion is higher due to enhanced cluster
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diffusivity. Third, we like to mention the recent experimentalwork of Velev et al., where they report long-lived chain-like clus-ters composed of oppositely charged, size-symmetric colloidalnanoparticles under low ionic strength conditions,18 thusproviding an experimental support to our simulation ndings.
Although we have studied a considerably wider parameterspace than previously done, more variations remain possible forfurther exploration. For example, NPs can be designed to havemacroscopic net charge (the charge neutrality of the wholesystem is maintained by counterions and salts), i.e., N+Z+ +N�Z� s 0. In this situation, each ionic cluster may have netcharge of the same sign and generate strong enough repulsionbetween each other to prevent fusion and stabilize the clustersize. Movie S4 (see the ESI†) shows an illustrative case (N+ ¼ N�,2Z+ ¼ �Z�). In this 2 : 1-NP mixture, Mn stops increasing withtime and nally reaches a steady value before the end of thesimulation. However, cluster isomerization still occurs, and thetotal potential energy keeps driing, which indicates that thesystem is out of equilibrium. Chains have considerably longerlife due to their large amount of net charge, which are nowenergetically favored when compared with compact congura-tions. At this point, it would be instructive to explore the effectsof altering the NP size ratio and/or shape. There are few studiesthat explore such effects even for equilibrium super-latticeproperties. We hope that the non-equilibrium self-assemblystudy reported here would motivate further future explorationsof the dynamic (ionic cluster,18 gel19) structures formed byoppositely charged NPs for an even broader range of conditions.
Finally, we note thatWhitesides et al.53 have recently reporteda systematic experimental study on 2D electrostatic-self-assembly (ESA) of oppositely charged macroscopic objects,where both crystalline lattices and ionic chains were observed.ESA has so far been primarily used to create a large variety ofordered structures with building blocks in awide range of lengthscales (molecular,54 nanoscale,4,6 microscale13 and macro-scale53,55). It is conceivable that analogous ESA phenomenawould emerge in systems with distinctive scales. In fact, webelieve some general principles of ESA canbe established in nearfuture. In the nanoworld, our simulation ndings reveal thepromise of non-equilibrium self-assembly of oppositely chargedNPs in several emerging applications. For example, the self-assembled ionic chains and crystals can be used as “nanoionictemplates” to control the polymerization process of chargedmolecules and the assembly of charged nano-objects.56 Analo-gous to the DNA-templated synthesis,57,58which takes advantageof the complementarity of the four types of nucleotides, it ispossible here that the complementarity of positively and nega-tively charged units of the ionic template can be utilized.54
Positively (negatively) charged molecules/nano-objects areattracted to the neighborhood of negatively (positively) chargedNPs constituting the template. Applying a catalyst to inducereactions between molecules/nano-objects in the vicinity of thetemplate can eventually transfer the pattern of a template to theformed polymers (polyelectrolytes) and “nano-molecules”. Bycontrolling such systems' details (e.g., size ratio of the chargedmolecule and NP), the properties of the resulting products (e.g.,the block length of polymers) can thus be nely controlled.
We would like to thank Vladimir Kuzovkov, Jos Zwanikken,Subas Dhakal, Guillermo Ivan Guerrero and Creighton Thomasfor stimulating discussions, William Kung and Arjan Quist forcritical reading of the manuscript, and the anonymous reviewerfor insightful comments and valuable suggestions to improvethe quality of the paper. The simulations were performed in theQuest cluster at Northwestern University and in TARDIS cluster,funded by the Office of the Director of Defense Research andEngineering (DDR&E) and the Air Force Office of ScienticResearch (AFOSR) under Award number FA9550-10-1-0167. Wethank the nancial support of the Non-Equilibrium EnergyResearch Center (NERC), which is an Energy Frontier ResearchCenter funded by the U.S. Department of Energy, Office ofScience, Office of Basic Energy Sciences under Award numberDE-SC0000989.
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