Topological structure prediction in binary nanoparticle superlattices.

Systems of spherical nanoparticles with capping ligands have been shown to self-assemble into beautiful superlattices of fascinating structure and complexity. In this paper, I show that the spherical geometry of the nanoparticle imposes constraints on the nature of the topological defects associated with the capping ligand and that such topological defects control the structure and stability of the superlattices that can be assembled. All these considerations form the basis for the orbifold topological model (OTM) described in this paper. The model quantitatively predicts the structure of super-lattices where capping ligands are hydrocarbon chains in excellent agreement with experimental results, explains the appearance of low packing fraction lattices as equilibrium, why certain similar structures are more stable (bccAB6vs. CaB6, AuCu vs. CsCl, etc.) and many other experimental observations.

Determination of anharmonic free energy contributions: Low temperature phases of the Lennard-Jones system.

We investigate a general method to calculate the free energy of crystalline solids by considering the harmonic approximation and quasistatically switching the anharmonic contribution. The advantage of this method is that the harmonic approximation provides an already very accurate estimate of the free energy, and therefore the anharmonic term is numerically very small and can be determined to high accuracy. We further show that the anharmonic contribution to the free energy satisfies a number of exact inequalities that place constraints on its magnitude and allows approximate but fast and accurate estimates. The method is implemented into a readily available general software by combining the code HOODLT (Highly Optimized Object Oriented Dynamic Lattice Theory) for the harmonic part and the molecular dynamics (MD) simulation package HOOMD-blue for the anharmonic part. We use the method to calculate the low temperature phase diagram for Lennard-Jones particles. We demonstrate that hcp is the equilibrium phase at low temperature and pressure and obtain the coexistence curve with the fcc phase, which exhibits reentrant behavior. Several implications of the method are discussed.

Separation of the Stern and diffuse layer in coarse-grained models: the cases of phosphatidyl serine, phosphatidic acid, and PIP2 monolayers.

We present here a method to separate the Stern and diffuse layer in general systems into two regions that can be analyzed separately. The Stern layer can be described in terms of Bjerrum pairing and the diffuse layer in terms of Poisson-Boltzmann theory (monovalent) or strong coupling theory plus a slowly decaying tail (divalent). We consider three anionic phospholipids: phosphatidyl serine, phosphatidic acid, and phosphatidylinositol(4,5)bisphosphate (PIP2), which we describe within a minimal coarse-grained model as a function of ionic concentration. The case of mixed lipid systems is also considered, which shows a high level of binding cooperativity as a function of PIP2 localization. Implications for existing experimental systems of lipid heterogeneities are also discussed.

General solution to the electric double layer with discrete interfacial charges.

We provide extensive molecular dynamics simulations of counterion and coion distributions near an impenetrable plane with fixed discrete charges. The numerical results are described by an explicit solution that distinguishes the plasma (√(A(c))/σ>3) and the binding regime (√(A(c))/σ<3) where σ is the ion diameter and A(c) = ∣e∕ν∣ (ν is the surface charge density). In the plasma regime, the solution consists of a product of two functions that can be computed from simpler models and reveals that the effects of the discreteness of the charge extends over large distances from the plane. The solution in the binding regime consists of a Stern layer of width σ and a diffuse layer, but contrary to standard approaches, the strong correlations between ions within the Stern layer and the diffuse layer require a description in terms of a "displaced" diffuse layer. The solution is found to describe electrolytes of any valence at all concentrations investigated (up to 0.4M) and includes the case of additional specific interactions such as van der Waals attraction and other generalizations. We discuss some open questions.

Dynamics and statics of DNA-programmable nanoparticle self-assembly and crystallization.

DNA linker mediated self-assembly is emerging as a very general strategy for designing new materials. In this Letter, we characterize both the dynamics and thermodynamics of nanoparticle-DNA self-assembly by molecular dynamics simulations from a new coarse-grained model. We establish the general phase diagram and discuss the stability of a previously overlooked crystalline phase (D-bcc). We also characterize universal properties about the dynamics of crystallization. We point out the connection to f-star polymer systems and discuss the implications for ongoing experiments as well as for the general field of DNA mediated self-assembly.

Design of polymer nanocomposites in solution by polymer functionalization.

Polymer nanocomposites, materials combining polymers and inorganic components such as nanosized crystallites or nanoparticles have attracted significant attention in recent years. A successful strategy for designing polymer nanocomposites is polymer functionalization via attaching functional groups with specific affinity for the inorganic component. In this paper, a systematic investigation by molecular dynamics of polymer functionalization for design of composites combining nanosize crystallites with multiblock polymers in solution is presented. It is shown that functionalization is an example of active self-assembly, where the resulting polymer nanocomposite exhibits a different type of order than the original pure polymer system (without inorganic components). Optimal polymer architectures and concentrations are identified appropriate for different applications, alongside an in-depth analysis on the origin and stability of the resulting phases as well as its experimental implications.

Electrostatic correlations at the Stern layer: physics or chemistry?

We introduce a minimal free energy describing the interaction of charged groups and counterions including both classical electrostatic and specific interactions. The predictions of the model are compared against the standard model for describing ions next to charged interfaces, consisting of Poisson-Boltzmann theory with additional constants describing ion binding, which are specific to the counterion and the interfacial charge ("chemical binding"). It is shown that the "chemical" model can be appropriately described by an underlying "physical" model over several decades in concentration, but the extracted binding constants are not uniquely defined, as they differ depending on the particular observable quantity being studied. It is also shown that electrostatic correlations for divalent (or higher valence) ions enhance the surface charge by increasing deprotonation, an effect not properly accounted within chemical models. The charged phospholipid phosphatidylserine is analyzed as a concrete example with good agreement with experimental results. We conclude with a detailed discussion on the limitations of chemical or physical models for describing the rich phenomenology of charged interfaces in aqueous media and its relevance to different systems with a particular emphasis on phospholipids.

Micellar crystals in solution from molecular dynamics simulations.

Polymers with both soluble and insoluble blocks typically self-assemble into micelles, which are aggregates of a finite number of polymers where the soluble blocks shield the insoluble ones from contact with the solvent. Upon increasing concentration, these micelles often form gels that exhibit crystalline order in many systems. In this paper, we present a study of both the dynamics and the equilibrium properties of micellar crystals of triblock polymers using molecular dynamics simulations. Our results show that equilibration of single micelle degrees of freedom and crystal formation occur by polymer transfer between micelles, a process that is described by transition state theory. Near the disordered (or melting) transition, bcc lattices are favored for all triblocks studied. Lattices with fcc ordering are also found but only at lower kinetic temperatures and for triblocks with short hydrophilic blocks. Our results lead to a number of theoretical considerations and suggest a range of implications to experimental systems with a particular emphasis on Pluronic polymers.

Self-assembled ordered polymer nanocomposites directed by attractive particles.

We theoretically investigate general conditions under which an inorganic phase can direct the self-assembly of an ordered polymer nanocomposite. For this purpose, we consider a solution of triblock copolymers forming a hexagonal phase of micelles and investigate the effect of adding attractive particles. We show that if the triblock is functionalized at its ends by attaching groups with specific affinity for the particles, thus effectively becoming a pentablock, the particles direct the self-assembly of the system into phases where both the polymers and the particles exhibit mesoscopic order. Different lamellar and gyroid phases (both with Ia3d and I4(1)32 space symmetries) are presented in detail. Our results show that functionalization is a very powerful route for directing self-assembly of polymer nanocomposites. We briefly discuss the connections with recent theoretical and experimental results in diblock melts with nanoparticles as well as for problems where polymers are used to template the growth of an inorganic phase in solution.

Effect of dipolar moments in domain sizes of lipid bilayers and monolayers.

Lipid domains are found in systems such as multicomponent bilayer membranes and single component monolayers at the air-water interface. It was shown by Keller et al. [J. Phys. Chem. 91, 6417 (1987)] that in monolayers, the size of the domains results from balancing the line tension, which favors the formation of a large single circular domain, against the electrostatic cost of assembling the dipolar moments of the lipids. In this paper, we present an exact analytical expression for the electric potential, ion distribution, and electrostatic free energy for different problems consisting of three different slabs with different dielectric constants and Debye lengths, with a circular homogeneous dipolar density in the middle slab. From these solutions, we extend the calculation of domain sizes for monolayers to include the effects of finite ionic strength, dielectric discontinuities (or image charges), and the polarizability of the dipoles and further generalize the calculations to account for domains in lipid bilayers. In monolayers, the size of the domains is dependent on the different dielectric constants but independent of ionic strength. In asymmetric bilayers, where the inner and outer leaflets have different dipolar densities, domains show a strong size dependence with ionic strength, with molecular-sized domains that grow to macroscopic phase separation with increasing ionic strength. We discuss the implications of the results for experiments and briefly consider their relation to other two dimensional systems such as Wigner crystals or heteroepitaxial growth.

Charge inversion at minute electrolyte concentrations.

Anionic dimyristoylphosphatidic acid monolayers spread on LaCl3 solutions reveal strong cation adsorption and a sharp transition to surface overcharging at unexpectedly low bulk salt concentrations. We determine the surface accumulation of La3+ with anomalous x-ray reflectivity and find that La3+ compensates the lipid surface charge by forming a Stern layer with approximately 1 La3+ ion per 3 lipids below a critical bulk concentration, ct approximately 500 nM. Above ct, the surface concentration of La3+ increases to a saturation level with approximately 1 La3+ per lipid, thus implying that the total electric charge of the La3+ exceeds the surface charge. This overcharge is observed at approximately 4 orders of magnitude lower concentration than predicted in ion-ion correlation theories. We suggest that transverse electrostatic correlations between mobile ions and surface charges (interfacial Bjerrum pairing) may contribute to the charge inversion.

Ground state of a large number of particles on a frozen topography.

Problems consisting in finding the ground state of particles interacting with a given potential constrained to move on a particular geometry are surprisingly difficult. Explicit solutions have been found for small numbers of particles by the use of numerical methods in some particular cases such as particles on a sphere and to a much lesser extent on a torus. In this paper we propose a general solution to the problem in the opposite limit of a very large number of particles M by expressing the energy as an expansion in M whose coefficients can be minimized by a geometrical ansatz. The solution is remarkably universal with respect to the geometry and the interaction potential. Explicit solutions for the sphere and the torus are provided. The paper concludes with several predictions that could be verified by further theoretical or numerical work.

Salty solutions near a charged modulated interface.

Charged monolayers at a liquid-vapor interface may be found in a crystalline state, resulting in a surface density of charge that displays periodic modulations. In this paper we discuss how these modulations affect different thermodynamical and mechanical properties (compared with the equivalent uniform charge density) of a system consisting of the charged monolayer and a bulk solution including a finite concentration of counter-ions and co-ions. It is shown that very accurate results for low and moderate salt concentrations are possible within an expansion in the Fourier modes of the modulations, the Weak Amplitude Perturbation (WAP), if the finite size of the ions are included as a Stern layer. We conclude by discussing the implications and the relevance of these results for both theoretical studies and experiments.

Coarse-grained molecular-dynamics simulations of the self-assembly of pentablock copolymers into micelles.

Multiblock polymers in aqueous solution, where one or several blocks are hydrophobic, exhibit a rich variety of phases and states of aggregation. In this paper, we investigate a pentablock system ABCBA, where the B block is always hydrophilic and the A and C blocks have varying degrees of hydrophobicity depending on external conditions. We report coarse-grained molecular-dynamics simulations where the solvent is included explicitly and monomers interact via a 6-9 Lennard Jones potential function. The hydrophobic interaction is modeled by tuning the parameter controlling the strength of the interaction between the hydrophobic monomers and the solvent. We investigate the structure and morphology of the micelles for two concrete situations representing changes in temperature and the pH level. The simulated system is directly relevant to a recently synthesized pentablock system consisting of a triblock Pluronic with an added pH-sensitive end group [B. C. Anderson et al., Macromolecules 36, 1670 (2003)].

Induced crystallization of polyelectrolyte-surfactant complexes at the gas-water interface.

Synchrotron x-ray and surface-tension studies of a strong polyelectrolyte (PE) in the semidilute regime (approximately 0.1 M monomer charges) with varying surfactant concentrations show that minute surfactant concentrations induce the formation of a PE-surfactant complex at the gas-solution interface. X-ray reflectivity and grazing angle x-ray diffraction show the complex PE-surfactant resides at the interface and the alkyl chains of the surfactant form a two-dimensional liquidlike monolayer. With the addition of salt (NaCl), columnar crystals with distorted-hexagonal symmetry are formed.

Grain boundary scars and spherical crystallography.

We describe experimental investigations of the structure of two-dimensional spherical crystals. The crystals, formed by beads self-assembled on water droplets in oil, serve as model systems for exploring very general theories about the minimum-energy configurations of particles with arbitrary repulsive interactions on curved surfaces. Above a critical system size we find that crystals develop distinctive high-angle grain boundaries, or scars, not found in planar crystals. The number of excess defects in a scar is shown to grow linearly with the dimensionless system size. The observed slope is expected to be universal, independent of the microscopic potential.

Crystalline order on a sphere and the generalized Thomson problem.

We attack the generalized Thomson problem, i.e., determining the ground state energy and configuration of many particles interacting via an arbitrary repulsive pairwise potential on a sphere via a continuum mapping onto a universal long range interaction between angular disclination defects parametrized by the elastic (Young) modulus Y of the underlying lattice and the core energy E(core) of an isolated disclination. Predictions from the continuum theory for the ground state energy agree with numerical simulations of long range power law interactions of the form 1/r(gamma) (0

Universal negative poisson ratio of self-avoiding fixed-connectivity membranes.

We determine the Poisson ratio of self-avoiding fixed-connectivity membranes, modeled as impenetrable plaquettes, to be sigma = -0.37(6), in statistical agreement with the Poisson ratio of phantom fixed-connectivity membranes sigma = -0.32(4). Together with the equality of critical exponents, this result implies a unique universality class for fixed-connectivity membranes. Our findings thus establish that physical fixed-connectivity membranes provide a wide class of auxetic (negative Poisson ratio) materials with significant potential applications in materials science.

Tubular phase of self-avoiding anisotropic crystalline membranes.

We analyze the tubular phase of self-avoiding anisotropic crystalline membranes. A careful analysis using renormalization group arguments together with symmetry requirements motivates the simplest form of the large-distance free energy describing fluctuations of tubular configurations. The non-self-avoiding limit of the model is shown to be exactly solvable. For the full self-avoiding model we compute the critical exponents using an epsilon expansion about the upper critical embedding dimension for general internal dimension D and embedding dimension d. We then exhibit various methods for reliably extrapolating to the physical point (D=2,d=3). Our most accurate estimates are nu=0.62 for the Flory exponent and zeta=0.80 for the roughness exponent.