Surfactants in solution are known to arrange themsein a variety of structures, such as micelles, vesicles,bilayers.1–6 Design and fabrication of material compositcan benefit from proper manipulation of these seassembling properties of surfactants which may give risestructural order over many length scales.7 For example,semiconductor nanocrystals could be arranged in highlydered structures by coating them with surfactants.7 Synthesisof cobalt clusters by encapsulating them in inverted miceis another example where self-assembling propertiesused.8 More recently this shape directing property has beexploited in synthesis of mesoporous sieves9–13 where onecan achieve large pore diameters which are structurallydered over 100–200 Å with a similar level of perfectionwell-established microporous solids of pore diameter inrange of only 5–10 Å. The organic surfactants are usedtemplatesand the inorganic particles, usually silicates, ecase the templating structure to produce organic-inorgacomposite materials. Such self-assembling processes oin all form of biological systems where it has been realizthat organized organic arrays are an important part of inganic nucleation and phase formations in biosystems knasbiomineralization. Synthesis processes, such as ones mtioned above, quite naturally have relied on the central theof biomineralization and have tried to usebiomimeticapproaches using supramolecular preorganized orgaggregates.14
Recent discoveries of mesoporous molecular sieves hmade an important addition to this idea. Instead of usorganic materials as apassivestructure directing element,more general pathway ofcooperative self-assemblyhas been
10280021-9606/98/108(24)/10281/13/$15.00
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proposed10,11 where the structure directing organic elemenare allowed to dynamically evolve along with the inorganhost particles. This has lead to a wider tailoring capacitystructures which are not only ordered over larger lenscales but the patterns and periodicity so obtained hopened up a wide arena to explore. Naturally, this geneized route of cooperative self-assembly10,11has gained wide-spread attention among scientific community for its prosptive application in preparing nanoscale materials. Texperimental pathways leading to such self-assembstructures are very diverse and complex. Each surfactant-system in solution introduces competing interactions of dferent length scales arising out of different componenThey involve hydrophobic and hydrophilic interactions, vder Waals interactions, Coulombic interactions and hydrobonding at different stages of the assembly process. A pnomenological model which captures these characteristicsurfactant-host self assembly via different pathways is narally worthy of detail investigations.
The purpose of this article is twofold. First, to developphenomenological model for neutral and ionic surfactawhich is particularly well suited for multimicellar systemWe demonstrate that the proposed model captures the etials of surfactant self-assembly for both neutral and iosurfactants and is very efficient to study multimicelle sytems. We then address the issue of cooperative self-assein presence of the host particles via different pathways.fore we describe our model and present our results it willappropriate to summarize some of the relevant, previwork here. It is noteworthy in this context that a vast amouof literature has been addressed to the self-assembly offactants at the oil–water interface. For these three compo
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b
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10282 J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
oil-water-surfactant systems we refer to the review articleGompper and Schick15 and references cited there.16 In thisarticle we focus on self-assembly of neutral as well as iosurfactants immersed in one component solvent only.then investigate the role of host particles in cooperativecellization of surfactants in their presence. Cooperative sassembly of surfactants influenced by the presence ofhost particles is a new subject of research and there eonly a few theoretical studies in the literature. Analytreatments4,17 of the self-assembly process quite often avothe inherent complicated many body interactions and instrely on space filling packing arguments. These calculatihave been able to predict micellization and are very usguides for further detailed numerical investigations. Duetheir intrinsic complications, it is very hard to extend anlytic approaches to more realistic models. But numeriwork has been able to bridge the gap to some extent betwmany experiments and analytic theories. They can be broclassified into two types. The first category deals withreal-istic systems with fairly large number of interaction paraeters to mimic the actual structures and interactions ofsurfactant molecules.18,19This class of models is better suitewhen one is interested in properties of an isolated macromecule. But cooperative effects may appear at a late stagthe self-assembly process only after surfactants form milar or aggregates of other shapes. Studies of dynamicsuch processes may be severely restricted by computesources if detailed geometries and the interactions are inporated into the numerical scheme. The alternate routestart with a simplified model emphasizing more macroscoand universal properties, leaving aside the details but caping the essence of the physical phenomena. We will adthe second approach in this work. Surfactants interactingsimple Lennard–Jones~LJ! potential have been used asparadigm for most of the previous lattice and off-lattiMonte Carlo ~MC!, and molecular dynamics studies.20–26
The off-lattice simulations are to be compared with nearneighbor lattice models both of which mimicneutralsurfac-tants only. Desplat and Care20 have performed lattice MCsimulation where the surfactants consist ofs particles withone of them serving as a head and the rests21 forming thetail of the surfactant. Their calculation qualitatively capturthe micellar size distribution as a function of temperatuHowever since surfactants are embedded on a thdimensional lattice it is not possible to extract informatiabout the detailed shapes of the micelles. Equilibrium prerties of nonionic micelles were also studied by Linse aco-workers and Mackie and co-workers by self-consistfield calculations and lattice Monte Carlo methods.21 Rectoret al. in their molecular dynamics~MD! simulations haveadopted the simplest two particle model representing anphiphile. In their treatment the surfactant consists of onedrophilic and another hydrophobic particle only joined byharmonic spring.22 They also use aNpT ensemble and Wi-dom test particle approach to determine the critical miceconcentration.4 Smit et al. also established the presencemicelles in a surfactant-solvent mixture in their Msimulation.23,24 Recent MD simulations by Palmer anLiu25,26 are more detailed compared to those mention
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above. They have incorporated additional bending eneterm and have considered surfactants of length 4 and 8spectively, immersed in a solvent.25 For a fixed set of parameters they observe cylindrical and spherical micellar assbly as a function of the surfactant concentration. They haalso extended the calculation to take into account the inganic precursors and concluded that beyond a critsurfactant-precursor interaction, the precursors have a dring effect on preserving the shapes of the micelles. Howesince the solvent molecules are kept explicitly in their modnumerical results are restricted to only a few micelles amicelle–micelle interactions are almost negligible.26
In all of the numerical studies mentioned above, the svent molecules are present explicitly in the calculatioHowever it is well known that for micellar self-assembly thratio of surfactants to solvents is rather small and hence mof the computing efforts are spent towards monitoringmotion of the solvent particles. Therefore even for themodels with simpler interactions it is hard to explore tphase diagram exhaustively. It will not be inappropriatemake some comments regarding the treatment of solventgrees of freedom in earlier numerical studies of surfactself-assembly. There have been many studies of thehydro-phobic effectand hydrophobic hydration4 which deal withthe concomitant changes occurring at the microscopic stture of water in the vicinity of surfactants.27 Experimentallyhowever micellar aggregation have also been seen in osolvents which indicate that self-assembly may not requthe detailed nature of the interaction of surfactants with wter. Previous numerical simulations which capture the esstial features of micellar aggregation also model the solvwith simple LJ potential. Hence one may think that tlength scales associated with detailed nature at the moleclevel contribute relatively weakly and those have been ingrated out in simple models of surfactant-solvent systecited above. Unlike many previous numerical studies of sfactant self-assembly, an important ingredient of our aproach is that we have gone one step further and have etively eliminated the solvent degrees of freedom completeThe effect of the solvent has been indirectly incorporatedappropriate set of phenomenological parameters amongsurfactants. Evidently the model as we will describe in tnext section contains less number of phenomenologicalrameters and the method has considerable computationavantages. For a range of parameters we observe micellagregation with similar cluster size distribution as foundearlier studies for neutral micelles where the solvent degrof freedoms were incorporated explicitly. Next we hashown thatinverted micellesand their distribution functionscome naturally into this scheme. We then extend these sies for ionic surfactantsby incorporating additional screeneCoulomb~SC! interaction among the head groups of the sfactants. Apart from micellization,ionic micelles exhibit ad-ditional structural ordering. Finally, we present results fohost encased micellization as a demonstration ofcooperativeself-assemblyalluded to above.
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pauouebT
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10283J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
II. MODEL
Our model consists of the surfactants and the hostticles only, but the interaction potentials describing the sfactants in our model are similar to those used in previstudies. In this article we consider a two-dimensional modThis captures the basic features of self-assembly and enaus to search the parameter space more exhaustively.model surfactant consists of chains of lengthNm (Nm57) ofwhich the first one or two are chosen to be head (h) sites~Nh51 or 2! and the rest of the monomers are treated as t(t) sites. We consider surfactant chains of fixed bonds of ulengths (l 051) but introduce bond bending energy. The ptential function for the model is then given by:
U5(i , j
N
f i j ~r i j !1( Rb~u i jk2u0!2, ~1!
where r i j is the distance between sitesi and j , u i jk is theangle subtended by three successive monomers in a gsurfactant, andu0 is the equilibrium value ofu i jk . In ourcalculations we have chosenu05p but kept Rb as aparameter.28 Note f i j (r i j ) is a pairwise potential acting between any two monomers and is of LJ form:
f i j ~r i j !5H 4e i j F S r i j
s i jD 12
2S r i j
s i jD 6G2f0~Ri j
c !, r i j ,Ri jc
0, r i j >Ri jc .
~2!
Heree i j , s i j , andRi jc are the LJ parameters and cutoff di
tances for the pair of monomersi and j , respectively. Theaddition of the termf0(Ri j
c ) causes the interaction to vanissmoothly at a distanceRi j
c s i j and beyond. In order to modesurfactants in a solvent one needs two values of the cuparameterRi j
c . A cutoff Ri jc 521/6s i j introduces a purely re
pulsive interaction, whereas a choice ofRi jc 52.5s i j intro-
duces an attractive LJ tail. In this article we consider surftants with hydrophilic heads and hydrophobic taiTherefore when these surfactants are in water there iseffective attraction among the hydrophobic tails, a head-and a head-head repulsion. In our method these couldmodeled with appropriate cutoff parametersRhh
c 521/6shh ,Rht
c 521/6sht , and Rttc 52.5s tt , respectively. To model the
host particles (p) we choose simple monomers interactiamong themselves and with the surfactants with LJ potenThe interaction of the host particles among themselveswith the heads are taken to have both repulsive and attracparts. However the strengths of the interactionepp and ehp
are in general different. The interactions of the host particwith the tails are always repulsive. Units of length and ttemperature (T) have been chosen ass tt ande tt /kB , respec-tively.
For ionic surfactants there is an additional SC interactamong the heads. We use the standard Debye-Huckel~D-H!form4 given by
Ui jsc~r i j !5u0
exp~2kr i j !
r i j, ~3!
where the interaction strengthk21 is the D-H screeninglength andu0 is a function of the total charge of a surfactan
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We will useu0 andk as phenomenological parameters. Wshould also state that there are counterions in the solvencharge neutrality which contribute to the screening butnot appear in our calculations explicitly.
The Monte Carlo moves consist of off-lattice counteparts of forward and backwardslithering-snakereptationmoves29 of the surfactants, andkink jumps30 of the individualmonomers. In one forward reptation move a given surfactis chosen at random and is translated by an amount; l 0 inany direction. A kink jump is the off-lattice counterpart othe Verdier-Stockmeyer model30 which consists of puttingthe inneri th particle to its mirror image position along thbond joining its adjacent monomers satisfying the followiequation:
Ri85Ri 111Ri 212Ri , ~4!
and the end particles are then rotated according to
R185R21c1
RNm8 5RNm
1cm , ~5!
wherec1 andcm are two randomly chosen vectors of lengl 051.
A single reptationcauses the whole surfactant to mowhereas a single kink jump makes one monomer to flip onTherefore a single MC step in our model consists of oreptation andNm21 kink jumps chosen at random. Periodboundary conditions are applied and a link list31 is used forMC updates. However the link list used here is different ththe one often used in MD. When a given move is accepthe neighbor list for that and the neighboring cells is updaimmediately. In order to calculate the energy change forreptation it is not necessary to calculate the energy chafor all theNm monomers in that chain. A careful observatioshows that the change in energy is given by
DE5@e181eNh118 #2@eNh1eNm
#, ~6!
whereei andei8 correspond to the energy of thei th monomerfor the old and the new configurations, respectively. The nconfigurations are accepted by the standard Metropolis rNow we will discuss our simulation results.
III. RESULTS
A. Thermodynamic considerations
Before we describe our results let us first briefly reviesome of the earlier analytic results due to Israelachvili aco-workers.4,17 For other detailed work we refer the readerthe work of Blankschtein and co-workers,32 and Ben-shauland co-workers.33 An important quantity in the theory ocluster aggregation is the dependence of chemical poteon aggregation numberN. Equilibria among different aggregates imply
m5m101kT log~X1!
5m201
1
2kT logS X2
2 D5¯5mN0 1
kT
NlogS XN
N D ~7!
or
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oe
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10284 J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
m5mN5mN0 1kT logS XN
N D5const, N51,2,3,... ~8!
wheremN is the mean chemical potential of the aggregateaggregation numberN and mN
0 is the standard part of thchemical potential. NoteXi / i is the mole fraction of thei thspecies so that the total concentrationC of surfactants isgiven by
(i 51
`
Xi5C. ~9!
From Eq.~7! it is easy to get the following equation:
XN
N5FX1 expS m1
02mN0
kT D GN
. ~10!
The above equation gives the necessary conditions foraggregates to form. SinceXi,1 ; i , it follows that if m i
0 is aconstant
XN
N5X1
N!1. ~11!
In other wordsXN /N!X1 , so that the large aggregates arare. It generally follows that in order to get aggregatesappreciable size,mN
0 has to decrease as a function ofN. Animportant aspect of this functional dependence is micellition versus complete phase separation. Israelachvili andworkers have shown that ifmN
0 5m`0 1A/Np then for p,1
there is a phase separation. It is noteworthy that in this cmN
0 goes tom`0 at largeN. Micellization on the contrary will
occur if mN0 either exhibits a minimum, or becomes consta
for finite aggregation numberN. For surfactants formingspherical micelles, a reasonable assumption is that the attive energy among the hydrophobic tails is proportionalthe surface areaga and the repulsive contribution arises dto the hydrophilic head groups is proportional toK/a so that
mN0 5ga1
K
a, ~12!
wherea is the effective surface area that a surfactant ocpies at the surfactant–solvent interface.34 These two oppos-ing terms immediately give a minimum inmN
0 at the optimalsurface areaa05A(K/g). The validity of this simple ther-modynamic and geometric packing argument can be cpared with the cluster statistics obtained from our simulatresults. In our simulation we have calculated average eneper surfactant chainEN for a cluster of aggregation numbeN. We will see thatEN is a very useful quantity to understand the underlying physics of micellization. The depedence ofEN on the aggregation numberN, bending coeffi-cient Rb , andT is extremely useful in tailoring shapes ansizes in the aggregation process.
B. Neutral surfactants
We first present results for the neutral surfactants. Omodel of neutral surfactant consist of chains withNm57where the first two sites are considered as head sites anrest of the monomers consist of the tail particles. The
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parameters are summarized in Table I. The simulation is pformed for 200 surfactant chains (Nc5200) confined in asquare box of length 100 so that the surfactaconcentration35 ~number of surfactants per unit area! is 0.02.Initial configurations are generated by first choosing the sfor the surfactant heads randomly and generating the resthe chain particles with an off-lattice two-dimensional seavoiding walk. We then equilibrate the system with purerepulsive interaction for all the particles. This makes the sfactants to be uniformly distributed. We explore the phadiagram at this fixed concentration as a function of tempeture (T) and bending energy parameterRb .
First we show some typical snapshots that we get frthe off-lattice simulation of surfactants without explicit incorporation of the solvent particles. We have investigasurfactant self-assembly for a wider range ofT andRb . Herewe show only some of the most relevant results. Figurshows typical snapshots of micellar aggregation a500 000 MC steps forT50.5 and for different values of thebending energy parameter (Rb). We notice with increasingthe valueRb from 0.1 to 1.0, the shapes of the clustechange from circular to more rectangular structures. We wcome back to this issue later. In order to understandeffect of each parameter on micellization in detail we harelied on two quantities. The first one is the average eneper surfactant chainEN for aggregates of given sizeN. Thesecond quantity is the corresponding cluster distributfunction for which we have used a simple distance criterto determine whether a surfactant belongs to a given clusFor hydrophobic tails, two surfactant chains are considein the same cluster if any two pair of monomers from thetwo surfactant tails lie within the attractive cutoff distan~2.5!. In order to determineEN and the cluster distributionfunction an ensemble average is taken over three to sixof independent runs. For higher temperature we hchecked that statistics for two independent runs are almidentical. A time averaging on the cluster distribution is peformed over different time windows. Invariance of thecluster distributions ensures that a steady-state distribuhas been achieved.
Figure 2 shows typical snapshots of micellar aggregatfor the temperatures 0.8, 0.6, 0.45, and 0.4, respectivelyRb50.2 and in Fig. 3 we show the corresponding time avaged cluster distribution functions for the entire run~circles!,for the last 100 000~squares!, and the last 50 000~dia-monds!, respectively, by calculating the distribution functioin every 500 MC steps. The squares and the circles aremost in identical position and therefore can be interpretedthe steady-state cluster distribution. The circles may beevant for those experiments where data is taken for the enspan of the experiments. At higher temperature the clu
TABLE I. Interaction parameters for the neutral surfactants.
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10285J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
FIG. 1. Snapshots of micellar aggregation for a 2% surfactant solution with chain lengthNc57 at a temperature 0.5~in reduced unit! at the end of MC time500 000. The MC simulations were done for 1400 monomers confined in a two-dimensional box of length 100~in units of LJ parameters! with periodicboundary conditions along bothx and y directions. The figure shows snapshots for different values ofRb : ~a! Rb50.1, ~b! Rb50.2, ~c! Rb50.6, and~d!Rb51.0.
e
uisa
deserr
.thim
oft ist
nits--rmaltivelyera-
fect
nd-leey
distribution decreases monotonically. As the temperaturlowered the cluster distribution exhibits a peak. Figure 3~c!is very similar to the cluster distribution observed in previosimulations with explicit solvent degrees of freedom. Itcharacterized by a large number of free surfactant chainsa characteristic peak at a larger value ofN. We should men-tion that while the cluster distributions remain in a steastate, breaking of clusters into pieces and coalescencsmaller clusters proceed simultaneously with aggregatedifferent sizes being in chemical equilibrium with each othA time development of such events is shown in Fig. 4 foparticular set.
We now discuss theN dependence ofEN . Figure 5shows theEN as a function ofN at different temperaturesFigure 5 should be compared with Fig. 3 which showscorresponding cluster distributions. Here we notice anportant result. For micellization, the dependence ofEN on Nhas to become flat at a finiteN, or as a function ofN exhibit
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s
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a minimum.36 We notice in Fig. 3 that at a temperature0.8, the distribution comprises mostly of monomers as iapparent from the scale of Fig. 3~a!. The corresponding plofor EN also monotonically decreases withN. According tothe simple theorya la Tanford and Israelachviliet al.sketched above,3,4 a minimum ~at N5M , say! occurs be-cause of the competition of the attractive and repulsive uin the surfactants. ForN,M the hydrophobic energy is increased where as forN.M , the geometric constraint increases the free energy. For higher temperature the theenergy is of the same order of magnitude of the attracinteraction energy (e tt51) and hence the system effectivebehaves as a purely repulsive one. With decrease of tempture the attractive interaction overcomes the thermal efand a minimum appears inEN as shown in Figs. 5~b! and5~c!. With further decrease of temperature the energy lascape for EN becomes more rugged exhibiting multipminima. Our simulations very clearly demonstrate the k
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of length
10286 J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
FIG. 2. Snapshots of micellar aggregation for a 2% surfactant solution with chain lengthNc57 for Rb50.2 and for temperatures 0.8, 0.48, 0.45, and 0.4~inreduced unit!, respectively, at the end of MC time 500 000. The MC simulations were done for 1400 monomers confined in a two-dimensional box100 ~in units of LJ parameters! with periodic boundary conditions along bothx andy directions.
onheinkeme
f t
l t
th
-T
i
-at
In anof
n-d-r-
natethe
mether
eald
features of the physical processes controlling micellizatiOne needs an attractive interaction which can overcome tmal fluctuations and a repulsive steric repulsion for modelmicellization of surfactants. We should mention that unliearlier simulations with explicit solvent degrees of freedoour effective LJ parameters should be considered as tempture dependent when one takes into account the effect oimplicit solvent condition as the temperature changes.
Next, we have studied the the effect ofRb on micelliza-tion. From Fig. 1 it is clear that by increasingRb , the shapeof the micelles gradually changes from being sphericarectangular structures. For comparison with Fig. 2~c! (T50.45, Rb50.2) we have shown another snapshot forsame temperature bur forRb50.6 in Fig. 6~a!. A comparisonof Fig. 2~c! and Fig. 6~a! very clearly shows how the bondbending energy influences the shapes of the micelles.cluster distribution and theEN;N is also qualitatively dif-ferent. A larger bending energy introduces polydispersitythe cluster distribution as shown in Fig. 6~b!. Again one can
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see how it occurs from theEN;N plot shown in Fig. 6~b!.Comparing Fig. 6~c! with the corresponding Figs. 3~c! and5~c! ~for T50.45) we notice that Fig. 6~a! lacks a sharpminimum, it contains multiple shallow energy valleys instead. This could be qualitatively understood by lookingthe configurations of Fig. 6~c!. With a larger value ofRb thesurfactants arrange themselves in a rectangular shape.ideal situation for a very large bending energy the shapethe individual surfactants will be dominated by bending eergy and therefore individual surfactants will become rolike. Therefore the cluster energy will be minimized by aranging the surfactants such that heads of the altersurfactants lie on one side. It is then easy to check thatcost of putting one more surfactant will be roughly the saas the average surfactant energy in that cluster, or in owords EN;EN11 , which implies that theEN vs N will ex-hibit plateaus. What we have just stated is true in the idlimit for very large aggregates and for infinitely rigid ro
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ent timeions.
10287J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
FIG. 3. Cluster size distributions for Figs. 2~a!–2~d!. Circles represent the time averaged distribution for the entire run. Squares and diamonds represaveraged distribution for the last 100 000 and 50 000 time steps. The ensemble average is performed over three to six different initial configurat
llo
i-th
rom
willandus,
ce
re-
urar-
l
shaped surfactants. In our case the above argument wimodified by the surface effects and also by the finite valuethe bending energy which is reflected in Fig. 6~c!, where onesees a relatively flatEN;N compared to that in Fig. 5~c!.These studies show how the rigidity of the surfactants~con-trolled throughRb) can introduce polydispersity on micellzation. An important aspect of our numerical results isplot EN;N as a function ofT and Rb which explains whycluster distributions in various cases are different.
C. Inverted micelles
Since our model captures the regular micellization pcess, it is almost self-evident how one obtains inverted
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e
-i-
celles in this scheme. For the sake of completeness webriefly describe it here. Surfactants with attractive headsrepulsive tails produce inverted micelles as expected. Thinverted micelles can be obtained with the following choifor the cutoff parametersRhh
c 52.5shh , Rttc 521/6s tt , Rht
c
521/6sht . Figure 7 shows a typical snapshot and the corsponding distribution function.
D. Ionic surfactants
We now show results for the ionic surfactants. Omodel ionic surfactant consists of one head and six tail pticles. Ionic heads are taken to be bigger37 than the tail par-ticles with a choiceshh52s tt which leads to a natura
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10288 J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
FIG. 4. Snapshots of surfactant aggregation at different time for parameters corresponding to Fig. 2~c!.
eea
tro
fo
t ram
enth
itatuthwain
inis-alhemer
ois are-inget the
ione of
tomeize
choice ofsht51.5s tt to satisfy the usual average rule for thLJ particles. We have adjusted the bond length for the hto the first tail particle to be 1.5l 0 as well.38 The SC param-eters that we have used are listed in Table II. We also induce a second cutoff distanceRi j
sc for the SC interaction fornumerical expediency. We have performed simulationsvarious values of the screening parameterk. For a detailcomparison with the neutral surfactant results we presensults for the same concentration and choice of the LJ pareters corresponding to the Fig. 1~a! (T50.5 andRb50.2).Figures 8~a! and 8~b! show the snapshots fork50.5 and 0.1,respectively, foru0510.0. With an increasing value ofk21,the screening length, an order begins to appear in the sthat the distribution for the average separation betweenionic micelles becomes narrower. To give a more quanttive answer we have calculated the time averaged strucfactor S(k) for these two cases and compared them withstructure factor for the neutral surfactants. Figure 9 shothese structure factors. Compared to the neutral surfactthe ionic surfactants exhibit sharper peaks with increas
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d
-
r
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ntsg
value ofk21. We have checked that thek value for the peakcorresponds to the average separation. This order is remcent of the two-dimensional ordering of a two-dimensionscreened Coulomb gas right above its melting point. Talgorithms that we use here either update a single monoor one surfactant chain only. If we introduceadditionalmoves for the center of massesfor the micelles after they areformed we believe this local ordering will finally go over ta hexagonally ordered structure. To our knowledge thisnew result compared to the earlier work on micellar agggation. Figure 10 shows cluster distributions correspondto Fig. 8. We notice from Fig. 8 that the number of fresurfactant chains is less compared to the neutral case asame temperature@Fig. 1~a!#. It is likely that in presence ofthe screened Coulomb interaction the entropic contributto the free energy is less for the surfactants. In presencthe SC interaction it becomes harder for the surfactantsmove from one configuration to another one with the saenergy. Therefore they take the alternate route to minimthe energy from the attractive tail-tail interactions.
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n for the
10289J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
FIG. 5. EN;N for Figs. 2~a!–2~d!. Circles represent the full time averaged distribution. Squares and diamonds represent time averaged distributiolast 100 000 and 50 000 time steps. The ensemble average is also performed over different initial configuration.
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E. Neutral surfactants and host particles
We now discuss the role of external host particlesaggregation of neutral surfactants. This brings us to theportant issue of cooperative versus biomimetic pathwayobtain mesoporous sieves mentioned in the introduction.11 Inorder to address this issue we have considered two possimulation pathways. The first pathway is to start withpreformed micellar arrangementof surfactants@as shown inFig. 1~a!# and allow inorganic host particles to interact withese micellar aggregates. In the second case, we start wrandom configuration of the surfactant-host system. This sond pathway corresponds to the cooperative self-assemcase. If one adds host particles at a stage when the sutants have already self-assembled into micelles, then theparticles will either form aggregates among themselvesthey may encase these micelles to minimize the energy.will depend on the interaction strengths,39 ehp and epp . Atypical snapshot of this pathway is shown in Fig. 11. Herchoice ofehp.epp makes the host particles mostly decora
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the micelles rather than forming clusters of their own.choice ofehp5epp51 will allow many host-particle aggregates along with isolated host particles in the systemshown in Fig. 11~b!. We have checked that for a very largvalue of ehp host particles have more drastic effect on dforming the micelles.26
Now we discuss the configuration obtained for the cooerative self-assembly process. It is nota priori obvious thatin the presence of the host particles from the very beginna random initial distribution of surfactants will evolve intmicelles. Figure 12 demonstrates the existence of these anate generalized biomimetic and less obvious route. Halso micellization proceeds unhindered and we obtain hencased micelles for the same set of parameters as in FigIt should be noted here that the experimental pathways leing to such self-assembling structures are very diversecomplex. Depending on experimental conditions one maymay not obtain the same final structures by using two diffent pathways, even for the same surfactant-host stoichi
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10290 J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
etry. It is exciting to note that this essential complexitypresent in our simple model. As an example, we havecluded another snapshot of the cooperative self assemprocess in Fig. 12~b! for the same set of parameters as in F12~a!, but for a lower temperature (T50.25). Instead of mi-cellization we now get wormlike structures seen in soexperiments.40 It seems clear that this simple model h
FIG. 6. ~a! Snapshot of micellar aggregation forRb50.6 atT50.45.~b! Thecorresponding cluster distribution, and~c! EN as a function of aggregationnumberN.
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ample potential to predict new phases as a function offerent parameters and concentrations. Such studies maytain information to promote further experiments.
IV. DISCUSSION AND SUMMARY
We have presented detailed numerical results for miclization of surfactants using off-lattice Monte Carlo methoAn important ingredient of our method is that unlike maprevious numerical studies we have eliminated the solvdegrees of freedom. The interaction of the surfactants wthe solvent particles is implicit in the phenomenological prameters of the model. Since micellization occurs at low sfactant concentration, this method has a distinct compu
FIG. 7. ~a! Snapshot of micellar aggregation and~b! the correspondingcluster distribution atT50.25. The rest of the parameters are the same aFig. 1.
TABLE II. Interaction parameters for the ionic surfactants.
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10291J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
tional advantage since no time is spent to monitor the solvdegrees of freedom. This helps us not only to study multicelle systems for both ionic and nonionic surfactants but athe cooperative self-assembly of the surfactants in the pence of host particles. For statistical properties, e.g., cludistribution andEN , an ensemble averaging for different intial conditions was also feasible. First we consider neusurfactants modeled via simple LJ interactions. For neumicelles, the cluster-size distribution obtained from thsimulation is qualitatively similar to previous studies whesurfactant-solvent interactions are treated explicitly. Tgives us confidence in our simplified yet computationavery efficient model. We demonstrated that the self-assemof surfactants into inverted micelles is naturally embeddedthis scheme. Next we extend these calculations for ionicfactants. Here micellization occurs with additional ordericoming out of the longer range screened Coulomb intetion. We then study the role of additional host particlesmimic recent experiments on surfactant-silicate coopera
FIG. 8. ~a! Snapshots of micellar aggregation for ionic surfactants at theof MC time 500 000. The SC parameters are~a! u0510.0, k50.5, andRhh
sc530.0, ~b! u0510.0, k50.1, andRhhsc530.0, respectively. The rest o
the parameters are the same as in Fig. 1.
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self-assembly. For certain parameter values we still obunhindered self-assembly of surfactants into host-encamicelles. We also note that the thickness of the walls andshape of the encased micelles can be tailored by pro
d
FIG. 9. Time averaged structure factors for neutral and ionic surfactaThe dashed line~squares! is for neutral surfactants whose typical snapshoshown in Fig. 1~a!. The solid line~circles! is for ionic surfactants whosetypical configuration is shown in Fig. 8~b!. The peak corresponds to thaverage separation among the micelles which is sharp and narrow duepresence of screened Coulomb interaction. The time average is takevery 500th step over the last 50 000 MC steps.
FIG. 10. Cluster distribution of ionic surfactants for Figs. 8~a! and 8~b!.
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10292 J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
choice of the parameters. These findings may have relevto guide new experiments. Finally we demonstrated thattroducing host particles on the preformed micellar phasealso obtains host encased micellar structures. This is content with recent experiments that the final configuratiowhich are similar can be obtained via different pathwaysdetailed study of inverted micelles along with the investigtions of cooperative self-assembly of ionic micelles with hparticles is currently under preparation which we will repin a separate publication.
FIG. 11. Effect of host particles~shaded squares! on preformed micelles forthe first pathway discussed in the text. Figure shows snapshot for 410particles for ~a! epp51, ehp52 and ~b! epp51, ehp51. The rest of theparameters are the same as in Fig. 1.
TABLE III. Interaction parameters for the host particles.
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ACKNOWLEDGMENTS
We thank Dr. Eric Prouzet, Dr. Peter Tanev, and DZhen Wang for numerous discussions and bringing toattention other experimental work. We especially thank Pfessor T. J. Pinnavaia for giving details of his work and fcomments on our manuscript. This work has been suppoby National Science Foundation Grant numbers CH9633798~A.B. and S.D.M.!, DMR-9413513~A.C.!. Partialcomputer support through Michigan State Universitygratefully acknowledged. A.C. thanks Michigan State Uversity for the hospitality shown during the present work.
1For reviews, see,Surfactants in Solution, edited by~a! K. L. Mittal and P.Bothorel~Plenum, New York, 1985! and~b! K. L. Mittal and B. Lindman~Plenum, New York, 1984!, and other volumes in this series;Physics ofAmphiphiles: Micells, Vesicles and Microemulsions, edited by V. DeGior-gio and M. Corti~North-Holland, Amsterdam, 1985!.
2G. J. T. Tiddy, Phys. Rep.57, 1 ~1980!.3C. Tanford,The Hydrophobic Effect~Wiley, New York, 1980!.
ost
FIG. 12. Cooperative self-assembly for surfactant-host system for theond pathway mentioned in the text. Figure shows snapshots for~a! epp
51, ehp52, T50.5,~b! epp51, ehp52, T50.25. Here the host particles arpresent from the very beginning of the self-assembly.
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nj
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,
da
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10293J. Chem. Phys., Vol. 108, No. 24, 22 June 1998 Bhattacharya, Mahanti, and Chakrabarti
4J. Israelachvili,Intermolecular and Surface forces~Academic, New York,1985!.
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6R. G. Laughlin,The Aqueous Phase Behavior of Surfactants~Academic,New York, 1994!.
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10Q. Huo, D. I. Margolese, U. Ciesla, D. G. Demuth, P. Feng, T. E. GierSeger, A. Firouzi, B. F. Chmelka, F. Schuth, and G. D. Stucky, ChMater.6, 1176~1994!; Q. Huo, D. I. Margolese, U. Ciesla, P. Feng, T.Gier, D. G. Demuth, T. E. Gier, P. Seger, R. Leon, P. M. Petroff,Firouzi, F. Schuth, and G. D. Stucky, Nature~London! 368, 317 ~1994!;A. Monnier, F. Schuth, Q. Huo, D. Kumar, D. Margolese, R. S. MaxweG. D. Stucky, M. Krishnamurty, P. Petroff, A. Firouzi, M. Janicke, B. MChmelka, Science261, 1299~1993!.
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Vol. 16 ~Academic, New York, 1994!. An extensive lattice Monte Carlowork for the oil-water-surfactant system have been done by Larson~seeRef. 16!.
16R. G. Larson, L. E. Scriven, and H. T. Davis, J. Chem. Phys.83, 2411~1985!; R. G. Larson, J. Chem. Phys.89, 1642~1988!; 91, 2479~1989!;96, 7904~1992!; J. Phys. II France6, 1441~1996!.
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ences therein.28Following Palmeret al. ~Ref. 25!, we have chosen a bending potential th
favors 180° angles in the tail rather than 120° angles. This increaseseffective length of the surfactants by preventing sharp bends in theAlso, as mentioned by Palmeret al. ~Ref. 25!, simulations indicate that70%–80% of dihedral angles in the hydrocarbon tail are in thetransconfiguration, which supports the use of a straight tail.
29F. T. Wall and F. Mandel, J. Chem. Phys.63, 4592~1975!.30P. H. Verdier and W. H. Stockmayer, J. Chem. Phys.36, 227 ~1962!.31G. S. Grest, B. Dunweg, and K. Kremer, Comput. Phys. Commun.55, 269
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34Actually for other type of repulsive of interactions, e.g.,K/ap, wherep.1 the argument still holds good~Ref. 4!.
35The value of the surfactant concentration 0.02 is certainly above critmicellar concentration. Our choice was guided partially by the earwork of Rectoret al. ~Ref. 22! and from preliminary runs at several concentrations.
36We have tried to understand the physics from the quantityEN . However,it is possible to calculatemN
0 2m10;N @Eq. ~10!# from the normalized
cluster distribution. We have calculatedmN0 2m1
0 from the cluster distribu-tion data (Xi) obtained from our simulation and checked that the depdence ofmN
0 2m10 is a monotonically decreasing function ofN, consistent
with previous work~Ref. 21!.37Usually the ionic head surfactants have bigger heads compared to hy
carbon tail particles.38When the bond length becomes unequal on either side of a monome
Eq. ~4! for kink jump does not hold good. For thei th particle in a givenchain and choosing the origin at thei 21 particle the generalized kinkjump algorithm becomes u→2u. Here u5cos21(di 11,i 21
•di , j 21)/(udi 11,i 21uudi ,i 21u), anddi , j is the vector along the bond from thi th to the j th monomer. For fixed bond lengths the above equation splifies to Eq.~4! without going through the more expensive computatiof the angle. We have incorporated these generalized kink jump movethe tail particle next to the ionic head.
39Since one of our objectives is to look for host encased micellar aggretion a natural choice is to take the head-particle and particle-particleteractions attractive and choose tail-particle interaction to be repul~Rhp
c 52.5shp , Rppc 52.5spp , Rtp
c 521/6s tp).40If one incorporates additional Monte Carlo moves for the clusters the
selves, it is possible that the surfactant-host system may attain a diffestate. Here we have shown only the preliminary results for cooperaself-assembly. The details of the surfactant-host self-assembly arerently under preparation.
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