Simulations of self-assembling systems

Current Opinion in Colloid & Interface Science 6 Ž2001. 357᎐365

Simulations of self-assembling systems Raj RajagopalanU Department of Chemical Engineering, Uni¨ ersity of Florida, Gaines¨ ille, FL 32611-6005, USA

Abstract The previous focus on the thermodynamics of self-assembly of surfactants in solution through simulations is now being expanded to include phenomena in the fluid dynamic regime. This expansion implies that a formal edifice must be built to link molecular dynamics smoothly to mesoscopic and macroscopic length and time scales. We summarize and comment on recent trends in this area along with new results based on classical approaches. The latter include molecular dynamics as well as off-lattice Monte Carlo simulations and lattice-based Guggenheim-type models. 䊚 2001 Elsevier Science Ltd. All rights reserved. Keywords: Self-assembly; Monte Carlo; Molecular dynamics; Mesoscopic simulations; Dissipative particle dynamics; Lattice gas; Lattice Boltzmann

1. Scope The self-assembly of surfactants in solution and the resulting equilibrium phases and non-equilibrium phenomena have been the subject of numerous contributions in the previous issues of this journal Že.g. w1,2x. and elsewhere Žsee, e.g. w3 ,4,5 x and references therein .. The structural complexity of the selfassembled aggregates and phases, their relation to the underlying energetic and entropic interactions, and the competing relaxation mechanisms are well known. Therefore, the need for theoretical and computational methods that make the interpretation of experimental observations easier and serve as predictive tools is readily apparent. Excellent treatments of the current status of theoretical advances on these topics are available in recent monographs and compendia w3 ,4,5 x. The experimental aspects are discussed in some of these references and in others w5 ,6x. The present review, however, is restricted exclusively to 䢇䢇

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Tel.: q1-352-392-0868; fax: q1-352-392-9513. E-mail address: [email protected] ŽR. Rajagopalan..

simulations, particularly to the most recent trends and developments in the simulation of self-assembling systems. The term self-assembly is rather broad and applies to spontaneous aggregation and order formation of constituents when they are mixed in correct proportions under appropriate conditions. The process is reversible and represents thermodynamic equilibrium. Additional structural changes may also occur when the constituents are subjected to non-equilibrium forces Že.g. shear.. The self-assembly process plays an important role in a number of widely varying contexts and the components involved could be synthetic surfactants Že.g. block co-polymers or short-chain surfactants. or biological species Že.g. self-assembly of the DNA double helix from two complementary oligonucleotide chains, the assembly of lipid molecules to form bilayers and membranes, protein folding and the stereoselectivity present in numerous receptor molecules.. The biologically interesting self-assembly processes are numerous and are important in their own right to deserve separate attention. Here we shall focus primarily on short-chain surfactants, although

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the methods and some of the results reviewed are clearly applicable to self-assembly of polymers as well. In addition to self-assembly mediated by other components such as electrolytes, oil or co-surfactants, there are also reasons to explore the influence of inorganic ‘host’ particles, which are known to affect the phase diagrams and the microstructure of the different phases w7x. In such cases, one typically starts with a mixture of surfactants, solvent and host particles, rather than with host particles and pre-organized surfactant aggregates. These mixtures are interesting not only technologically Žas they can be used, e.g. in the preparation of mesoporous sieves through surfactant-templating. but also theoretically. They are analogous to ternary mixtures of oil, water and surfactants, and form a variety of intricate structures Že.g. liquid-crystalline structures of different symmetry, bicontinuous structures, etc... The self-assembly process in this case is occasionally known as co-operative self-assembly, and we shall consider some recent simulations of such processes as well. An issue that has been beyond the reach of traditional simulation techniques is the fluid dynamics of self-assembling phases or phases with ordered domains. This is an extremely complex problem, and any meaningful examination of self-assembling phases under realistic processing conditions must consider the coarsening or structural rearrangements that would be induced by fluid dynamics. Therefore, an exciting aspect of simulations of self-assembling systems is the emerging trend toward mesoscopic methods wdissipative particle dynamics ŽDPD. and lattice-gas and lattice-Boltzmann methodsx, which seek to link microscopic phenomena to macroscopic Žparticularly fluid dynamic. aspects. The primary emphasis of this review will, therefore, be on this class of simulations. However, we shall also comment briefly on some of the reports based on the more traditional methods such as molecular dynamics ŽMD. and off-lattice and lattice Monte Carlo techniques. Of these, detailed simulations based on MD are particularly important since they supply the information needed for the coarse-graining incorporated into the mesoscopic techniques. The scope of the present review is restricted largely to reports that have appeared in print in the last 2 years, although older reports that are relevant to recent trends are also included. The term self-assembling systems is too broad for a single review, as noted above. Therefore, we restrict the focus further to self-assembly in the bulk Žand to short-chain surfactants.. A recent review in this journal w8x has covered self-assembly at surfaces and gasrliquid interfaces Žin addition to some aspects of mesoscopic simulations..

2. Why simulations? Simulations are not panacea, and they do not replace theories, and certainly not experiments. Simulations are best seen as computer Žor, ¨ irtual . experiments that serve as adjuncts to theory and real experiments and provide otherwise inaccessible Žor, not easily accessible . microscopic or macroscopic information that theorists and experimentalists can use. Experience has shown that even short simulations Že.g. spanning only nanoseconds. with highly simplified models of the constituents Že.g. bead-spring models of surfactants . or pre-organized units Že.g. constrained aggregates. provide useful guidelines concerning the size, shape, surface roughness and internal structure of the aggregates and can even help in clarifying experimental observations Žsee, e.g. w9x.. Simulations also can be useful in situations in which experiments are impractical Že.g. systems at high temperatures or pressures.. Simulations of self-assembly of surfactants can also provide important insights into the physics of fundamental processes such as geometry of random surfaces, structure and dynamics of crumpled surfaces and phase transitions in membranes. Moreover, as already noted, surfactant-mediated templating techniques have much to offer for the synthesis of novel nano-structured materials Žsee, e.g. w10᎐12x.. Finally, simulations allow one to go beyond the inevitable simplifications characteristic of theoretical formulations Že.g. mean-field approximations., so that the accuracy or acceptability of such approximations can be scrutinized systematically. These possibilities and promise, however, must be tempered with the realities of computational demands and resources. Therein lies the current thinking on computational approaches to self-assembling systems Žas well as to other problems.. In the present review, we begin with the current drive to wed microscopic Žatomistic. simulations to mesoscopic methods that seek to predict structure and properties of the systems at useful, macroscopic length and time scales. This will then be followed by summaries of some recent reports on atomistic simulations and coarsegrained versions.

3. Linking the microscale dynamics to macroscopic behavior: mesoscopic simulations Extensive experience with well-established molecular theories of simple fluids and with corresponding molecular simulation procedures does allow one to approach self-assembling systems at the molecular level with a detailed structural representation of the

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various components along with their energetic interactions. Indeed this is ideally the method of choice if one wishes to avoid Žthe often ill-defined or insufficiently defined. phenomenological parameters necessitated by coarse-graining Žsee, e.g. the discussions of lattice Monte Carlo simulations.. However, the computational task the atomistic approach requires is so demanding that such an ambitious program is beyond the reach of currently available computational resources. This implies that a certain level of spatial and temporal coarse-graining is inevitable. In a recent review in this journal, Shelley and Shelley w8x have already presented a schematic representation of the length scales and time scales involved in modeling surfactant solutions and describe the types of theoretical orientation Že.g. quantum mechanics, statistical mechanics or continuum mechanics. and coarsegraining Že.g. lattice models or dissipative particle dynamics simulations. needed for the various hierarchy of processes or phenomena. Therefore, we shall dispense with such details here, unless they are relevant for an understanding of the results or in the context of the types of questions explored in the simulations. ŽReaders unfamiliar with details of simulation methods may consult standard references such as w13᎐16x for background.. The current limitations of molecular dynamics ŽMD. simulations typically impose drastic simplifications that prevent the examination of certain aspects of structural or temporal features of the system. One extreme example is the a priori selection of the shape of the micelle itself in the simulations, which clearly precludes the potential use of the simulations to examine the self-assembly process itself. Relatively small-scale MD simulations Žtypically with ; 20 surfactants and ; 10 3 solvent molecules over a period of approx. 0.1 ns. have been successful in modeling the structure of micelles and their interactions with the solvent w17x, but the dynamical properties of interest require run times that are well beyond the typically accessible values. For example, the time scales of interest range from 0.1 ps for shape fluctuations to 1 ␮s for mesoscale events such as surfactant exchange with the bulk and hundreds or more seconds for macroscale phenomena such as shear-induced changes. The extreme demands of the high end of this scale is the driving force for the development of mesoscopic procedures such as lattice-gas w18,19 ,20,21x, latticeBoltzmann w22 ,23x and dissipative particle dynamics simulations w24᎐26 ,27x. The central question of interest here, which poses a worthwhile challenge for the coming years, is: can one develop a well-coordinated sequence of gradually increasing Žand overlapping. coarse-grained procedures that would allow one to move from the atomistic world to a macroscopic description? Some recent progress in this di䢇䢇

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rection has already suggested ways to move from MD to mesoscopic simulations such as dissipative particle dynamics ŽDPD. w19 ,22 ,26 ,27x. We shall return to this shortly. First, the mesoscopic models are so called because they retain Žor, start from. some of the elements of the microscopic approaches Že.g. some form of suitably defined effective pair-wise additive interaction forces among the elements, a Langevin-type governing equation of motion, etc.. while seeking macroscale behavior Že.g. cluster growth dynamics or Navier᎐ Stokes-level continuum dynamics.. At the outset, it should be emphasized that the scope of the current attempts to devise mesoscopic models exceeds what is of immediate interest to us here. However, such a broad scope is needed Žand is, in fact, essential . in order to place the mesoscopic simulation procedures on a solid theoretical footing, i.e. the formulation of the underlying equations must be consistent with global thermodynamic and dynamic conservation laws. Recent attempts along these lines have included formulations of lattice-gas models for amphiphilic fluid dynamics w18,19 x, in which methods to link the Hamiltonian of the hydrodynamic lattice-gas automata to hydrodynamic flow with conserved momentum are derived so that a self-consistent treatment of the hydrodynamics of the amphiphilic fluids is assured. The above lattice-gas model appears capable of describing phenomena such as the dynamics of phase separation and shear-induced sponge-tolamellar phase transition. A similar effort on lattice-Boltzmann models for amphiphilic solutions is also being made w22 ,23x. This model, an improvement over lattice-gas automata, is designed to assure independent mass conservation for each species Žin contrast to earlier models. and to accommodate self-consistent forces between the species. This lattice-Boltzmann model fills an important gap in the hierarchy of mesoscopic models, as it allows one to examine lamellar phases not realized in lattice-gas models. These recent developments in lattice-gas and lattice-Boltzmann models also shed light on the differences in the phenomenological behavior accessible through various mesoscopic formulations and identify issues for further investigation. Dissipative particle dynamics, introduced in the early 1990s w28x and reformulated subsequently in its current form w29x, is perhaps the most promising mesoscopic method for surfactant systems. ŽAn earlier review in this journal w30x is dedicated solely to DPD, but with a focus on the dynamics and rheology of polymers in solution and the dynamics of mesophase separation in block co-polymer melts.. DPD differs from MD in a number of respects, but perhaps the two most important differences are in the nature of the effective Žconservative. interaction force acting 䢇䢇

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between the DPD ‘particles’ Žwhich are collections of molecular units, larger than the number typically used in united-atom models in off-lattice, coarse-grained models discussed later in this review. and in the way dissipative effects are implemented. In contrast to MD, the conservative interaction force in DPD is soft and repulsive and represents an effective, pair-wise interaction between the DPD particles. The softer form of the force allows longer time steps, and in combination with the Langevin-type Ži.e. Brownian dynamics. equation of motion, permits exploration of larger time scales than possible with MD. In addition, the random and dissipative forces, which collectively form a ‘thermostat’ and satisfy the conditions necessary to produce a canonical ensemble, are related to each other in such a way that the momentum is locally conserved so that the hydrodynamic flow effects on macroscopic length scales can be reproduced. A critical and very illuminating review of DPD was presented a few years back by Groot and Warren w31 x. The sample application they examined was for a homopolymer melt, but many of the issues discussed Že.g. the formation of micelles and mesophases or the potential problems with time scale separation between particle diffusion and momentum diffusion. are also relevant to surfactant systems. The implementation of DPD to surfactant systems has just begun. For example, a recent report w32 x examines the dynamics of coarsening of a polydomain smectic and the formation of a microdomain smectic under shear for a minimal model of an amphiphile Ža rigid dimer with a hydrophilic and a hydrophobic group. at fairly high concentrations. Building DPD from molecular dynamics is a challenging problem, which is beginning to receive attention. In an excellent, carefully laid-out exposition, Flekkøy et al. w26 x lay the groundwork for linking MD to DPD through a systematic coarse-graining procedure. A precise definition of ‘mesoscale’ is formulated, and the phenomenological parameters in DPD are related to the average microscopic fluxes of conserved quantities. For instance, the interparticle forces are constructed from the viscosity obtained from MD, and the interparticle energy transfer is related to the heat conductivity, also derived from MD. The DPD derived in the above paper is a representation of the underlying MD given by the hydrodynamic values of the fluxes. The development of DPD, lattice-gas and latticeBoltzmann models is far from complete. For example, much work remains to be done even on the numerical procedures, since even some ‘routine’ steps such as the integration procedures applied to the DPD dynamical equations may introduce artifacts in the physical properties derived from the simulations w27 x. Implementation schemes that are free of inconsisten䢇䢇

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cies Že.g. lack of microscopic reversibility . also need attention w33 x. However, the stage is set for rapid, practical developments in going from molecular simulations to the macroscopic, hydrodynamic regime. A caveat, however, is in order. Even surfactant insertion or deletion and micelle disintegration fall within the domain of mesoscopic methods in terms of length and time scales, but MD simulations currently available typically cover only approximately 10 2 ps ᎏ times that are too short for providing the type of structural and dynamical information needed for formulating mesoscopic equations. A systematic effort to bridge the gap remains open. 䢇

4. Probing dynamics at the microscale: atomistic simulations Although the previous MD simulations have been successful in predicting size distributions, structural details of the aggregates, micellization conditions, etc., they have often ignored molecular-level details such as ion binding and chemical specificity. A recent work w34 x seeks a fully atomistic and relatively large-scale Ž15 000 atoms. description of the structure and dynamics of Ž n-nonyltrimethylammonium chloride and eruyl bisw2-hydroxyethylxmethylammonium chloride. micelles using molecular trajectories over a time scale of approximately 3 ns. Although still very short, this time scale does allow fast dynamical processes such as monomer insertion or removal and local fragmentation and growth of the aggregates to be monitored and provides insights into the dominant kinetic mechanisms far from equilibrium Že.g. Becker᎐Doring ¨ kinetics dealing with monomer exchange between aggregates as opposed to Smoluchowski dynamics describing coagulation-fragmentation between clusters .. Such simulations form the first step toward the testing of the Aniansson᎐Wall kinetic model w35᎐38x and its extensions w39x. The simulations, in this case, reveal a Smoluchowski dynamics far from equilibrium and a Becker ᎐ Doring mechanism Žonly step-by-step ¨ monomer exchange. near equilibrium. Moreover, for over-sized aggregates, the Becker᎐Doring kinetics is ¨ associated with a slow expansion᎐contraction process, with a characteristic time scale of approximately 500 ps, whereas a characteristic time of only approximately 50 ps is needed for shape fluctuations for aggregates close to the mean size. The results also suggest that hydrogen bonding of co-surfactants with the head groups of the Žprimary. surfactants could play a key role in the stability of the aggregates. Another largely overlooked topic that has received some attention recently is the formation of reverse micelles in non-polar media. There have been some previous reports on this topic dating back to Žat least. 䢇

R. Rajagopalan r Current Opinion in Colloid & Interface Science 6 (2001) 357᎐365

the late 1980s Žw40᎐42x and references therein ., but this area has some current relevance in view of the drive toward environmentally benign processing technologies. For instance, there is an interest in exploring the use of carbon dioxide as a solvent medium because it is non-toxic and inflammable. Since most commercially available surfactant molecules do not form stable reverse micelles, synthesis of special surfactants that find CO 2 thermodynamically hospitable is being investigated by a number of research groups Žsee, e.g. w43,44x., but very little is known about such surfactants and their aggregation behavior. Atomistic simulations can provide useful information in these cases. Following some preliminary tests for CO 2-philic surfactants w45x, Salaniwal et al. w46 ,47 x report some more detailed investigations of equilibrium structure and aggregation dynamics of a hybrid surfactant molecule wŽ C 7 F 1 5 .Ž C 7 H 1 5 . CHSO 4y Na q x in a waterrcarbon dioxide mixture. The surfactant chosen is structurally relatively small, but the small size makes the computations manageable. As an additional benefit, a direct comparison of the results with experiments is possible since this system has been studied using small-angle neutron scattering recently w48x. The results illustrate the need for sufficiently large systems Ži.e. the number of surfactant chains and water and CO 2 molecules. in the simulations to avoid finite-size dependence of the shape and size of the reverse micelles, and the larger simulations do provide structural information consistent with experimental observations. The simulations, in addition, provide information not accessible to experiments, such as the existence of fairly large interfacial region Ž; 1᎐1.5 nm. in the aqueous region and the strong orientation of the water molecules in response to electric fields of the anionic head groups and Naq ions Žin contrast to the ‘bulk’ region inside the core, which exhibits a hydrogen-bonded network.. The results also show that the perfluoralkane tails display a more extended conformation than the alkane tails Žin contrast to what is observed in vacuo., thus indicating that the former is more CO 2-philic. The results on the dynamics reveal a Smoluchowski-type diffusion-limited aggregation, but on a faster time scale relative to that of aqueous surfactant systems. 䢇

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5. Equilibrium systems: off-lattice Monte Carlo simulations The fact that mesoscopic simulations are still in developmental stage provides a strong incentive to continue to explore coarse-grained, equilibrium techniques. An example of current interest in this respect is co-operative self-assembly, which was already mentioned in this review. There exist only a few theoreti-

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cal studies on this topic w49x and even fewer computational studies for guiding theory and experiments. Since co-operative effects manifest themselves at a late stage in the self-assembly process, an examination of such effects must resort to Monte Carlo ŽMC. simulations. For example, even the mere inclusion of solvent molecules Ževen in spatially coarse-grained lattice models discussed later in this review. increases the computational effort significantly. However, micellar self-assembly occurs at very low surfactant concentrations Ži.e. at very low surfactant-to-solvent ratios.. Therefore, attempts have been made to eliminate the solvent degrees of freedom completely by using effective phenomenological parameters for solventrsurfactant interactions w50x. The surfactants in this approach are treated as a connected string of beads, with coarse-grained beadrbead interactions, bond-bending energies, etc. The presence of solvent, for example, is accounted for through an effective attraction among hydrophobic tail beads and through headrhead and tailrtail repulsions. ŽOther components, such as the host particles in the case of cooperative self-assembly, are represented as additional monomers or multi-bead units with appropriate phenomenological interactions. . Ionic systems include interactions of the screened-Coulomb type. Clearly, encapsulation of contaminants can also be studied using such an approach. The above model has also been used w51 x for studying aggregation and bilayer formation in ionic surfactant solutions and to examine the morphology of surfactant-host composites. Although the computations are restricted to two-dimensional systems, characteristic structural features of ionic micelles are produced. By appropriately adjusting the screenedCoulombic interaction parameter one can also go smoothly from neutral to ionic systems. Nevertheless, how useful these are in practical contexts and in the interpretation of real experimental observations remains to be tested. The results of Bhattacharya and Mahanti w51 x for co-operative self-assembly suggest that the surfactant aggregates act as sources of quenched disorder for the motion of the host particles. Extensions of the work would allow an examination of what sizes and concentrations of the host particles are needed for producing the desired microstructures. Much remains to be explored concerning the success of models that retain some level of surfactant structure Žas in the above off-lattice simulations and the lattice simulations described below.. For example, these models have been successful in predicting the critical micelle concentration ŽCMC. and its temperature dependence. However, these models have not been examined systematically for the variations of the CMC with the molecular architecture of the surfac䢇䢇

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tant molecules and the experimentally observed closed-loop co-existence curves. ŽIsing-type site-bond models of surfactants, which lack structural features, have been shown to display closed-loop co-existence curves w3 ,52x.. 䢇䢇

6. Equilibrium systems: lattice Monte Carlo simulations Further simplifications can be accomplished by replacing the continuous space with a discretized lattice of suitable geometry. As well known, lattice simulations have been used successfully in polymer science for many years for investigating universal properties of single chains, polymer layers and solutions and melts. Their applications in studies of the self-assembly of co-polymers have also been extensive w53x. The use of lattice Monte Carlo simulations for surfactant solutions have also increased in recent years, as has already been noted in this journal w8,54x. Here, we shall restrict ourselves to a few recent additions to the literature. Short reviews of progress prior to 1999 and a more comprehensive introductory treatment may be found elsewhere w8,54,55 x. A fundamental unresolved question with the use of lattice models is their utility. As well-known from cellular automaton models, even a highly simplified set of simple, local rules applied to lattice systems can produce global features and patterns that mimic real systems remarkably closely. However, the true test of the usefulness of such models is how good they are in predicting the critical details Žsuch as phase diagrams, equilibrium properties and the dynamics ᎏ with dynamic Monte Carlo simulations.. In contrast to atomistic simulations, lattice simulations target global, universal behavior and properties rather than microscopic specificity and dynamics. They could be quite useful in testing critical assumptions in analytical Žusually mean-field. theories. For example, Rodriguez-Guadarrama et al. w56x have explored the possibility of using data generated from three-dimensional lattice Monte Carlo ŽLMC. simulations to test and extend some of the commonly used empirical criteria for micellization. The size distributions of aggregates obtained from the simulations can be combined with the mass action model to obtain the coefficients of an empirical equation due to Goldstein w57x for the Gibbs energy of micellization of non-ionic surfactants. Goldstein’s model takes into account the existence of three principal forces involved in the process of aggregation, namely, the solvophobic effect, the reduction in configurational entropy, and finally, the creation of a solvophobic 䢇

interface. Using the simulation results and Goldstein’s model, one can formulate generalized correlations for the principal driving forces for micellization as functions of the lengths of the head group and the tail of the amphiphiles, so that one can predict the properties of the aggregates for any amphiphile for a given dimensionless interaction energy Žthe so-called interchange energy in the Guggenheim lattice model.. One can also use the information obtained from the simulations for the CMC of different amphiphiles to determine the approximate Gibbs free energy changes involved in transferring the head and the tail of the amphiphile to the aggregates. A comparison with the experimental results reported in the literature for alkyl ethoxylate surfactants shows that the free energy changes are very similar to those obtained from the simulations. The above lattice formulation has also been applied to the phase behavior of surfactant ᎐solute᎐solvent systems and to the solubilization of solutes in the micelles w58,59x. Recently, lattice MC simulations have also been used to monitor energy fluctuations Žand hence the specific heat. and size fluctuations of surfactant aggregates w60 x. The results show a peak in the specific heat Žarising from the energy fluctuations . at the onset of micellization, corresponding to the simultaneous occurrence of a knee in the cluster distribution function ŽCDF.. This observation confirms the Nagarajan-Ruckenstein proposal of early 1980s that the knee in the CDF signals the onset of micellization w61x ᎏ a proposal that had been criticized as non-universal by Ben-Naim and Stillinger w62x. The moments of the CDF obtained from the simulations also confirm the mean-field predictions of Blankschtein et al. w63x that the various moments can be expressed solely in terms of the second moment. Collectively, the above LMC simulations illustrate how coarse-grained lattice models can be used effectively to formulate guidelines for theories and experiments. In addition to the ones reviewed above, there is one other recent report that deserves mention here. Floriano et al. w64 x have suggested a method based on grand-canonical Žlattice . Monte Carlo simulations that combines free-energy information from a series of simulations using small systems by histogram reweighting. The procedure allows rapid determination of the CMC and is less susceptible to metastability and hysteresis effects. In addition, micellization and macroscopic phase separations can be clearly distinguished. Preliminary results, for non-ionic systems, indicate that simple packing considerations can be misleading in describing the self-assembly process. In particular, a complex interplay between enthalpic and entropic factors determines the mean aggregation number. For example, the tails of individual Žfree. 䢇

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R. Rajagopalan r Current Opinion in Colloid & Interface Science 6 (2001) 357᎐365

surfactants assume more compact conformations at low temperatures, as expected. Although this might suggest, based on packing arguments, that the aggregation numbers should be correspondingly lower, the observed mean aggregation numbers actually show a significant increase with decreasing temperatures.

7. Closing remarks and future directions The above discussion illustrates that off-lattice and lattice simulations of coarse-grained surfactants have much to offer in mapping molecular-level features to phenomenological observations. Much remains to be done on mapping the properties of coarse-grained models to real systems Žand vice versa., and attempts to bridge this gap rigorously and formally are not quite on the same level as in the case of mesoscopic simulations. In fact, only very recently, has there been some systematic activity in scrutinizing the quality of Žmean-field. theories of lattice chains Že.g. Guggenheim’s random mixing model, quasichemical theory, Flory᎐Huggins theory, lattice cluster theory, etc.. using rigorous simulations w65 ,66 x, although these attempts have been largely restricted to polymer chains, with some minor exceptions. Much remains to be done on this topic and on lattice and off-lattice models for surfactant solutions. However, currently the momentum and excitement reside in the area of mesoscopic simulations, with a focus on formally linking molecular dynamics to hydrodynamics through structural and dynamic coarsegraining. One of the appealing aspects of mesoscopic methods such as DPD is the access they provide to studies of non-linear phenomena Že.g. coupling between flow and mesostructural phases. for which theoretical treatments are hard. Systematic examinations of the individual strengths and weaknesses of various mesoscopic schemes such as lattice-gas and lattice-Boltzmann algorithms and DPD remain open. On the theoretical side, additional work is needed for establishing the theoretical basis for coarse-graining procedures Že.g. how MD simulations can be used to develop expressions for the conservative force in DPD w67x.. On the numerical side, the influence of numerical artifacts requires further attention. Additional attention is also needed on the inclusion of electrostatic interactions in mesoscale simulations. Mesoscale models that preserve fluid dynamics are indispensable for studying collective properties of complex fluids, particularly fluids with self-assembling species. This area is expected to grow rapidly in the coming years. Papers of particular interest within the annual period of review have been highlighted as: 䢇

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References and recommended reading 䢇 䢇䢇

of special interest of outstanding interest

w1x Soten I, Ozin GA. New directions in self-assembly: materials synthesis over ‘all’ length scales. Curr Opin Coll Interface Sci 1999;4:325᎐337. w2x Corti M, Zemb T. Self-assembly under the influence of weak or long-range forces. Curr Opin Coll Interface Sci 2000;5:1᎐4. w3x Gompper G, Schick M. Self-assembling amphiphilic systems. 䢇䢇 London: Academic Press, 1994. Although this volume predates the 2-year period covered by the present review, it is included here because of its comprehensive focus on the theoretical side of self-assembly. It describes three principal theories of self-assembly in amphiphilic systems: Ži. microscopic ŽIsing-type and Guggenheim-type. models; Žii. the intermediate length scale Ginzburg᎐Landau models; and Žiii. the larger length scale interfacial Žmembrane. models. The underlying concepts behind these models, the relationship among them and their advantages and disadvantages are discussed. w4x Gompper G, Schick M. Lattice theories of microemulsions. In: Gelbart WM, Ben-Shaul A, Roux D, editors. Micelles, membranes, microemulsions and monolayers. New York: Springer-Verlag, 1994:395᎐426. w5x Kumar P, Mittal K, editors. Handbook of microemulsion 䢇䢇 science and technology. New York: Marcel Dekker, 1999. A fairly comprehensive volume, focusing primarily on microemulsions. Contains theoretical as well as experimental contributions. w6x Laughlin RG. The aqueous phase behavior of surfactants. San Diego: Academic Press, 1994. w7x Palmer BJ, Liu J. Effects of solute᎐surfactant interactions on micelle formation in surfactant solutions. Langmuir 1997;12:6015᎐6021. w8x Shelley JC, Shelley MY. Computer simulation of surfactant solutions. Curr Opin Coll Interface Sci 2000;5:101᎐110. w9x Karaborni S, Esselink K, Hilbers PAJ et al. Simulating the self-assembly of gemini Ždimeric. surfactants. Science 1994;266:254᎐256. w10x Hue QS, Margolese DI, Ciesla U et al. Generalized synthesis of periodic surfactant inorganic composite-materials. Nature 1994;368:317᎐321. w11x Tanev PT, Pinnavaia TJ. A neutral templating route to mesoporous molecular-sieves. Science 1995;267:865᎐867. w12x McGrath KM, Dabbs DM, Yao N, Aksay IA, Gruner SM. Formation of a silicate L-3 phase with continuously adjustable pore sizes. Science 1997;277:552᎐556. w13x Heermann DW. Computer simulation methods in theoretical physics. 2nd edition Berlin: Springer-Verlag, 1990. w14x Allen MP, Tildesley DJ. Computer simulation of liquids. Oxford: Oxford University Press, 1987. w15x Frenkel D, Smit B. Understanding molecular dynamics simulations: from algorithms to applications. San Diego: Academic Press, 1996. w16x Haile JM. Molecular dynamics simulation: elementary methods. New York: Wiley, 1992. w17x Kuhn H, Breitzke B, Rehage H. The phenomenon of water penetration into sodium octanoate micelles studied by molecular dynamics computer simulation. Colloid Polym Sci 1998;276:824᎐832. w18x Boghosian BM, Coverney PV, Love P, Maillet JB. Mesoscale modeling of amphiphilic fluid dynamics. Mol Simulat 2001;26:85. w19x Boghosian BM, Coveney PV, Love PJ. A three-dimensional 䢇䢇 lattice-gas model for amphiphilic fluid dynamics. Proc R Soc London 2000;456A:1431᎐1454.

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Describes the theoretical basis and computer implementation of a three-dimensional hydrodynamic lattice-gas model for amphiphilic solutions. Examples of binary systems Ž oil ᎐ water and surfactant ᎐water. and ternary systems are given, and the formation of spherical and worm-like micelles and sponge phases is illustrated. Boghosian’s web site, Žhttp:rrphysics.bu.edur; brucebrMolSimr. Žas of April 2001., presents additional information and numerous graphical illustrations of the phases observed. w20x Emerton AN, Weig FWJ, Coveney PV, Boghosian BM. The shear-induced isotropic-to-lamellar transition in a lattice-gas model of ternary amphiphilic fluids. J Phys Condens Matter 1997;9:8893᎐8905. w21x Maillet JB, Coveney PV. Two-dimensional hydrodynamic lattice-gas simulations of binary immiscible and ternary amphiphilic fluid flow through porous media. Phys Rev E 2000;62:2898᎐2913. w22x Chen HD, Boghosian BM, Coveney PV, Nekovee M. A 䢇䢇 ternary lattice Boltzmann model for amphiphilic fluids. Proc R Soc Lond 2000;456A:2043᎐2057. The formulation of a lattice-Boltzmann model for a ternary amphiphilic system is presented, and the use of the model is illustrated. The results are compared with existing predictions of lattice-gas models. w23x Nekovee M, Coveney PV, Chen HD, Boghosian BM. Lattice Boltzmann model for interacting amphiphilic fluids. Phys Rev E 2000;62:8282᎐8294. w24x Boek ES, Coveney PV, Lekkerkerker HNW, van der Schoot P. Simulating the rheology of dense colloidal suspensions using dissipative particle dynamics. Phys Rev E 1997;55: 3124᎐3133. w25x Flekkøy EG, Coveney PV. From molecular dynamics to dissipative particle dynamics. Phys Rev Lett 1999;83:1775᎐1778. w26x Flekkøy EG, Coveney PV, De Fabritiis G. Foundations of 䢇䢇 dissipative particle dynamics. Phys Rev E 2000;62:2140᎐2157. An excellent contribution which seeks to provide an atomistic foundation for dissipative particle dynamics, so that a precise definition ‘mesoscale’ can be formulated. It also establishes clear links between the phenomenological parameters used in the DPD algorithm and the averages of microscopic fluxes of the conserved quantities. The authors describe how the above averages could be related to either the local hydrodynamics or to the specific microscopic interactions relevant to the system. The differences between the proposed DPD method and the previous versions are also emphasized. w27x Besold G, Vattulainen I, Karttunen M, Polson JM. Towards 䢇 better integrators for dissipative particle dynamics simulations. Phys Rev E 2000;62:R7611᎐R7614. Numerical artifacts Žand hence incorrect physical properties. that could result from the integrators used in DPD are discussed and an alternative is proposed. w28x Hoogerbrugge PJ, Koelman JMVA. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhys Lett 1992;19:155᎐160. w29x Espanol ˜ P, Warren PB. Statistical mechanics of dissipative particle dynamics. Europhys Lett 1995;30:191᎐196. w30x Warren PB. Dissipative particle dynamics. Curr Opin Coll Interface Sci 1998;3:620᎐624. w31x Groot RD, Warren PB. Dissipative particle dynamics: bridg䢇䢇 ing the gap between atomistic and mesoscopic simulation. J Chem Phys 1997;107:4423᎐4435. Presents a critical review of DPD. The dissipative force and the random force in DPD have to satisfy certain conditions to guarantee that the system satisfies statistical mechanics corresponding to canonical ensembles. This paper presents the physical basis of the needed conditions and formulates algorithms for arbitrary time steps. The negative consequences of failure to satisfy the conditions are also explained.

w32x Jury S, Bladon P, Cates M et al. Simulation of amphiphilic 䢇 mesophases using dissipative particle dynamics algorithm. Phys Chem Chem Phys 1999;1:2051᎐2056. Illustrates the application of DPD for amphiphilic solutions using a minimal model for the surfactant structure. Results for formation of smectic phases under flow and for orientation and reorganization of a large domain under shear are presented. Simulated phase diagrams are also shown. w33x Pagonabarraga I, Hagen MHJ, Frenkel D. Self-consistent dissipative particle algorithms. Europhys Lett 1998;42: 䢇 377᎐382. An implementation scheme for DPD that preserves microscopic reversibility is suggested and is compared with previous versions. w34x Maillet J-B, Lachet V, Coveney PV. Large scale molecular 䢇 dynamics simulations of self-assembly processes in short and long chain cationic surfactants. Phys Chem Chem Phys 1999;1:5227᎐5290. A report on large-scale simulations of structural and dynamical properties of a short-chain and a long-chain cationic surfactant. The self-assembly processes such as surfactant insertion or removal and growth and fragmentation of micelles are examined and the kinetics of the processes near and far from equilibrium are also studied. w35x Aniansson EAG, Wall SN. Kinetics of step-wise micelle association. J Phys Chem 1974;78:1024᎐1030. w36x Aniansson EAG, Wall SN. Correction and improvement of kinetics of step-wise micelle association. J Phys Chem 1975;79:857᎐858. w37x Aniansson EAG, Wall SN. Kinetics of formation of ionic micelles. 2. Analysis of time constants: reply. Ber Bunsen Phys Chem 1977;81:1293᎐1294. w38x Aniansson EAG. Theory of micelle formation kinetics. Ber Bunsen Phys Chem 1978;82:981᎐988. w39x Wattis JAD, Coverney PV. General nucleation theory with inhibition for chemically reacting systems. J Chem Phys 1997;106:9122᎐9140. w40x Linse P. Molecular dynamics study of the aqueous core of a reversed micelle. J Chem Phys 1989;90:4992᎐5004. w41x Brown D, Clarke JHR. Molecular dynamics simulation of a model reverse micelle. J Phys Chem 1991;92:2881᎐2888. w42x Tobias DJ, Klein ML. Molecular dynamics simulations of a calcium carbonatercalcium sulfonate reverse micelle. J Phys Chem 1996;100:6637᎐6648. w43x Cooper AI, Londono JD, Wignall G et al. Extraction of a hydrophilic compound from water into liquid CO 2 using dendritic surfactants. Nature 1997;389:368᎐371. w44x DeSimone JM, Guan Z, Elsbernd CS. Synthesis of fluoropolymers in supercritical carbon dioxide. Science 1992;257: 945᎐947. w45x Salaniwal S, Cui ST, Cummings PT, Cochran HD. Self-assembly of reverse micelles in waterrsurfactantrcarbon dioxide systems by molecular simulation. Langmuir 1999;15: 5188᎐5192. w46x Salaniwal S, Cui ST, Cummings PT, Cochran HD. Molecular 䢇 simulation of a dichain surfactantrwaterrcarbon dioxide system. 1. Structural properties of aggregates. Langmuir 2001;17:1773᎐1783 see ref. 47. w47x Salaniwal S, Cui ST, Cummings PT, Cochran HD. Molecular 䢇 simulation of a dichain surfactantrwaterrcarbon dioxide system. 2. Self-assembly and aggregation dynamics. Langmuir 2001;17:1784᎐1792. The above two papers report reverse-micellar self-assembly of a surfactant in carbon dioxide. The alkane tails are modeled using the united-atom concept, and the sulfate head group is a fully atomistic representation. The exchange kinetics is relatively fast in this system, which makes molecular dynamics simulations easier.

R. Rajagopalan r Current Opinion in Colloid & Interface Science 6 (2001) 357᎐365 w48x Eastoe J, Bayazit Z, Martel S, Steytler DC, Heenan RK. Droplet structure of water-in-CO 2 microemulsion. Langmuir 1996;12:1423᎐1424. w49x Nagarajan R, Ruckenstein E. Theory of surfactant self-assembly: a predictive molecular thermodynamic approach. Langmuir 1991;7:2934᎐2969. w50x Bhattacharya A, Mahanti SD. Self-assembly of neutral and ionic surfactants: an off-lattice Monte Carlo approach. J Chem Phys 1998;108:10281᎐10293. w51x Bhattacharya A, Mahanti SD. Self-assembly of ionic surfac䢇䢇 tants and formation of mesostructures. J Phys Condens Matter 2001;13:1413᎐1428. A two-dimensional, off-lattice Monte Carlo study of co-operative self-assembly. The results show additional local ordering among the host particles as the concentration is increased. It is suggested that the ordering of the host particles can be predicted from known phase diagrams by treating the micelles as sources of quenched disorder. w52x Wenzel W, Ebner C, Jayaprakash C, Pandit R. Critical micelle concentration from a lattice gas model. J Phys Condens Matter 1989;1:5245᎐5250. w53x Binder K, Muller M. Monte Carlo simulation of block copo¨ lymers. Curr Opin Coll Interface Sci 2000;5:314᎐322. w54x Larson RG. Simulations of self-assembly. Curr Opin Coll Interface Sci 1997;2:361᎐364. w55x Rajagopalan R, Rodriguez-Guadarrama LA, Talsania SK. 䢇 Lattice Monte Carlo simulations of micellar and microemulsion systems. In: Kumar P, Mittal KL, editors. Handbook of microemulsion science and technology. New York: Marcel Dekker, 1999:105᎐137. A review of lattice-based Monte Carlo models focusing primarily on micellar solutions. w56x Rodriguez-Guadarrama LA, Talsania S, Mohanty KK, Rajagopalan R. Thermodynamics of aggregation of amphiphiles in solution from lattice Monte Carlo simulations. Langmuir 1999;15:437᎐446. w57x Goldstein RE. Model for phase-equilibria in micellar solutions of nonionic surfactants. J Phys Chem 1986;84: 3367᎐3378. w58x Talsania S, Rodriguez-Guadarrama LA, Mohanty KK, Rajagopalan R. Phase behavior and solubilization in surfactantsolute-solvent systems by Monte Carlo simulations. Langmuir 1998;14:2684᎐2693.

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w59x Talsania SK, Wang Y, Rajagopalan R, Mohanty KK. Monte Carlo simulations for micellar encapsulation. J Coll Interface Sci 1997;190:92᎐103. w60x Bhattacharya A, Mahanti SD. Energy and size fluctuations of 䢇 amphiphilic aggregates in a lattice model. J Phys Condens Matter 2000;12:6141᎐6160. Lattice Monte Carlo simulations are used to examine energy fluctuations as a function of temperature. The peak in the energy fluctuation is related to the onset of micellization. The results are used to test previous claims concerning CMC and to examine predictions based on mean-field concepts. w61x Ruckenstein E, Nagarajan R. Critical micelle concentration: transition point for micellar size distribution. J Phys Chem 1975;79:2622᎐2626. w62x Ben-Naim A, Stillinger F. Critical micelle concentration and the size distribution of surfactant aggregates. J Phys Chem 1980;84:2872᎐2876. w63x Blankschtein D, Thurston GM, Benedek GB. Theory of phase-separation in micellar solutions. Phys Rev Lett 1985;54:955᎐958. w64x Floriano MA, Caponetti E, Panagiotopolous AZ. Micelliza䢇 tion in model surfactant systems. Langmuir 1999;15: 3143᎐3151. A combination of the grand-canonical Monte Carlo method and histogram reweighting is introduced to predict self-assembly of lattice surfactants. The method allows rapid estimation of CMC’s from relatively small simulations. w65x Buta D, Freed KF, Szleifer I. Thermodynamic properties of lattice polymers: Monte Carlo simulations and mean-field 䢇 theories. J Chem Phys 2000;112:6040᎐6048 see ref. 66. w66x Buta D, Freed KF, Szleifer I. Monte Carlo test of lattice cluster theory: thermodynamic properties of binary polymer 䢇 blends. J Chem Phys 2001;114:1424᎐1431. The above two papers initiate a systematic examination of lattice theories of chain-like molecules using Monte Carlo simulations and Widom’s insertion method for determining chemical potentials. The focus, however, is restricted to relatively longer chains Žwith approx. 50 or 100 beads per chain. than one would need for surfactants of the type relevant to the present review. w67x Forrest BM, Suter UW. Accelerated equilibration of polymer melts by time-coarse-graining. J Chem Phys 1995;102: 7256᎐7266.