Quasicrystal formation in binary soft matter mixtures

Using a strategy that may be applied in theory or in experiments, we identify the regime in which a model binary soft matter mixture forms quasicrystals. The system is described using classical density functional theory combined with integral equation theory. Quasicrystal formation requires particle ordering with two characteristic length scales in certain particular ratios. How the length scales are related to the form of the pair interactions is reasonably well understood for one-component systems, but less is known for mixtures. In our model mixture of big and small colloids confined to an interface, the two length scales stem from the range of the interactions between pairs of big particles and from the cross big-small interactions, respectively. The small-small length scale is not significant. Our strategy for finding quasicrystals involves tuning locations of maxima in the dispersion relation, or equivalently in the liquid state partial static structure factors

Density functional theory for the crystallization of two-dimensional dipolar colloidal alloys

Two-dimensional mixtures of dipolar colloidal particles with different dipole moments exhibit extremely rich self-assembly behaviour and are relevant to a wide range of experimental systems, including charged and super-paramagnetic colloids at liquid interfaces. However, there is a gap in our understanding of the crystallization of these systems because existing theories such as integral equation theory and lattice sum methods can only be used to study the high temperature fluid phase and the zero-temperature crystal phase, respectively. In this paper we bridge this gap by developing a density functional theory (DFT), valid at intermediate temperatures, in order to study the crystallization of one and two-component dipolar colloidal monolayers. The theory employs a series expansion of the excess Helmholtz free energy functional, truncated at second order in the density, and taking as input highly accurate bulk fluid direct correlation functions from simulation. Although truncating the free energy at second order means that we cannot determine the freezing point accurately, our approach allows us to calculate \emph{ab initio} both the density profiles of the different species and the symmetry of the final crystal structures. Our DFT predicts hexagonal crystal structures for one-component systems, and a variety of superlattice structures for two-component systems, including those with hexagonal and square symmetry, in excellent agreement with known results for these systems. The theory also provides new insights into the structure of two-component systems in the intermediate temperature regime where the small particles remain molten but the large particles are frozen on a regular lattice

Power law polydispersity and fractal structure of hyperbranched polymers

Using the complementary approaches of Flory theory and the overlap function, we study the molecular weight distribution and conformation of hyperbranched polymers formed by the melt polycondensation of A-RN0-Bf - 1monomers in their reaction bath close to the mean field gel point pA= 1, where pAis the fraction of reacted A groups. Here f ≥ 3 , N0is the degree of polymerisation of the linear spacer linking the A group and the f-1 B groups and condensation occurs exclusively between the A and B groups. For ε ≡ (1-pA), ≪ we assume that the number density of hyperbranched polymers with degree of polymerisation N generally obeys the scaling form n (N) = N-τf (N/Nl) and we explicitly show that this scaling assumption is correct in the mean field regime (here Nlis the largest characteristic degree of polymerisation and the function f (N/Nl) cuts off the power law sharply for N>Nl). We find the upper critical dimension for this system is dc= 4, so that for d≥ dcthe mean field values for the polydispersity exponent and fractal dimension apply: τ = 3/2 , df= 4. For d = 3, mean field theory is still correct for ε > εGwhere εG≅ N0-1is the Ginzburg point; for ε < εG, mean field theory applies on small mass scales NNcwhere Nc≅ N03is a cross-over mass. Within the Ginzburg zone (i.e., dNcare non-Gaussian with fractal dimension given by df= d (for d = 2,3,4). Our results are qualitatively different from those of the percolation model and indicate that the polycondensation of ABf-1, unlike polymer gelation, is not described by percolation theory. Instead many of our results are similar to those for a monodisperse melt of randomly branched polymers, a consequence of the fact that

General Theory for Capillary Waves and Surface Light Scattering

We present a general theory for capillary waves and surface quasi-elastic light scattering for an isotropic liquid interface with adsorbed surfactant. We first examine the validity of three constitutive models for isotropic interfaces in the Newtonian regime, namely those of Scriven, Goodrich, and Kramer. Scriven's constitutive model contains three interfacial constants: the equilibrium surface tension γ, the interfacial dilational viscosity ζs, and the interfacial shear viscosity ηs. Goodrich's model and Kramer's model contain an additional interfacial constant: the transverse viscosity ηN, which is the dissipative counterpart of γ. We find that while Scriven's model satisfies frame invariance, the transverse viscosity term proposed by Goodrich and Kramer violates frame invariance. We therefore conclude that ηN is unphysical and that the Scriven model represents the most general constitutive model for isotropic interfaces in the Newtonian regime. Using Scriven's model as a starting point, we calculate the stress boundary conditions for capillary waves and generalize our results to include various interfacial relaxation processes, including diffusive interchange of surfactants (both in the absence and presence of adsorption barriers) and surfactant chain reorientation and relaxation. We then derive the dispersion relation and the power spectrum for capillary waves satisfying these boundary conditions. We find that, in all cases, the transverse viscoelasticity of the interface is controlled to leading order by the unperturbed equilibrium surface tension γ0 rather than a complex surface tension γ* = γ + iωηN, which is widely used in the literature for analyzing surface light scattering results. We reanalyze surface light scattering results for a wide range of interfacial systems where unphysical results (e.g., negative dilational viscosities) have been reported in the literature and find that these unphysical results are removed when we reparametrize the transverse viscoelasticity using γ0 rather than γ*

Monte Carlo simulation scheme for dendrimers satisfying detailed balance

A lattice Monte Carlo scheme for simulating dendrimers that is widely referenced in the literature is that of Mansfield and Klushin (Macromolecules 1993, 26, 4262). However, we show that this scheme does not obey a detailed balance and propose a modification to the original scheme that fixes this problem. To demonstrate the importance of detailed balance to the simulation results, we calculate the radius of gyration and structure factor for ideal dendrimers using our improved model and compare our results to Mansfield and Klushin's original scheme and exact analytical calculations. Excellent agreement is found between our model and the exact analytical calculations, while surprisingly large discrepancies are found between Manfield and Klushin's original scheme and the exact calculations. Our study highlights the importance of detailed balance generally to Monte Carlo simulations of dendrimers and the need to check previous results for nonideal dendrimers obtained from nondetailed balance schemes; we discuss the extension of our model to the nonideal dendrimer case

Rheology and molecular weight distribution of hyperbranched polymers

We study the melt rheology and molecular weight distribution of four short chain branched hyperbranched polyesters with different molecular weights and containing branched monomers of various alkyl chain lengths n (2 → 4; n is the number of CH2 groups in the alkyl chain). We find that the molecular weight distribution for all our samples obeys the static scaling form n(M) ∼ M-τexp(-M/Mchar) where n(M) is the number density of hyperbranched polymers with mass M, Mchar is the largest characteristic molecular weight, and τ is the polydispersity exponent. The values of τ for all our samples (either 1.35 or 1.55) are close to but not the same as the mean field value of τ = 1.5, a consequence of the fact that our polymers were synthesized under non-mean-field polycondensation conditions. For all our samples, we found that the rheology at low and intermediate frequencies could be modeled accurately using a dynamic scaling theory based on the Rouse model. This confirms that these hyperbranched polymers behave as polymeric fractals which are essentially unentangled. For these polymers, the fractal dimension in the melt was found to be consistent with the hyperscaling relation for hyperbranched polymers df = 3, although we found rheology to be rather insensitive to df for our system