A. Milchev

Fluctuations and lack of self-averaging in the kinetics of domain growth

The fluctuations occurring when an initially disordered system is quenched at timet=0 to a state, where in equilibrium it is ordered, are studied with a scaling theory. Both the mean-sizel(t)d of thed-dimensional ordered domains and their fluctuations in size are found to increase with the same power of the time; their relative size fluctuations are independent of the total volumeLd of the system. This lack of self-averaging is tested for both the Ising model and the φ4 model on the square lattice. Both models exhibit the same lawl(t)=(Rt)x withx=1/2, although the φ4 model has “soft walls”. However, spurious results withx≷1/2 are obtained if “bad” pseudorandom numbers are used, and if the numbern of independent runs is too small (n itself should be of the order of 103). We also predict a critical singularity of the rateR∝(1−T/Tc)v(z−1/x),v being the correlation length exponent,z the dynamic exponent.Also quenches to the critical temperatureTc itself are considered, and a related lack of self-averaging in equilibrium computer simulations is pointed out for quantities sampled from thermodynamic fluctuation relations.

Monte-Carlo simulation of the Cahn-Hillard model of spinodal decomposition

Abstract A lattice version of the nonlinear Cahn-Hilliard equation describing spinodal decomposition of binary alloys is studied via Monte-Carlo simulation, considering a two-dimensional system at critical concentration. The linearization approximation is found to be rather inaccurate even if the final temperature of the quench is very low and thermal fluctuations are unimportant. The structure function is found to exhibit anisotropy consistent with the chosen square lattice geometry. The relation of this study to Ising model simulations and experimental work is briefly discussed.

Finite-size scaling analysis of the Φ4 field theory on the square lattice

AbstractMonte-Carlo calculations are performed for the model Hamiltonian ℋ = ∑i[(r/2)Φ2(i)+(u/4)/gF4(i)]+∑<ij> (C/2)[Φ (i)−Φ(j)]2 for various values of the parametersr, u, C in the crossover region from the Ising limit (r→-∞,u+∞) to the displacive limit (r=0). The variableφ(i) is a scalar continuous spin variable which can lie in the range-∞<φ(i)<+∞, for each lattice site (i).φ(i) is a priori selected proportional to the single-site probability in our Monte Carlo algorithm. The critical line is obtained in very good agreement with other previous approaches. A decrease of apparent critical exponents, deduced from a finite-size scaling analysis, is attributed to a crossover toward mean-field values at the displacive limit. The relation of this model to the coarse-grained Landau-Ginzburg-Wilson free-energy functional of Ising models is discussed in detail, and, by matching local moments 〈Φ2(i)〉, 〈Φ4(i)〉 to corresponding averages of subsystem blocks of Ising systems with linear dimensionsl=5 tol=15, an explicit construction of this coarse-grained free energy is attempted; self-consistency checks applied to this matching procedure show qualitatively reasonable behavior, but quantitative difficulties remain, indicating that higher-order terms are needed for a quantitatively satisfactory description.

Droplet spreading: A Monte Carlo test of Tanner’s law

The spreading of polymer droplets under conditions of complete wetting on an ideally flat and structureless solid substrate has been studied by computer simulation, using a coarse-grained bead–spring model of flexible macromolecules. Evidence is obtained that a power law close to Tanner’s law for the growth of the lateral droplet radius {r(t)∝t0.14} and contact angle {θ∝t−0.31} holds on nanoscopic scales. We observe the formation of a precursor film around the spreading droplet and find that the film attains diffusive dynamics at late times.

Semiflexible polymers grafted to a solid planar substrate: Changing the structure from polymer brush to “polymer bristle”

Monte Carlo simulations are presented for a coarse-grained model of polymer brushes with polymers having a varying degree of stiffness. Both linear chains and ring polymers grafted to a flat structureless non-adsorbing substrate surface are considered. Applying good solvent conditions, it is shown that with growing polymer stiffness the brush height increases significantly. The monomer density profiles for the case of ring polymers (chain length N(R) = 64) are very similar to the case of corresponding linear chains (N(L) = 32, grafting density larger by a factor of two) in the case of flexible polymers, while slight differences appear with increasing stiffness. Evidence is obtained that the chain dynamics in brushes is slowed down dramatically with increasing stiffness. Very short stiff rings (N(R) ≤ 16) behave like disks, grafted to the substrate such that the vector, perpendicular to the disk plane, is oriented parallel to the substrate surface. It is suggested that such systems can undergo phase transitions to states with liquid crystalline order.

Structural properties of concave cylindrical brushes interacting with free chains

We present a self-consistent field theoretical study of the microstructure of concave cylindrical brushes as a function of the cylinder radius, grafting density, grafted chain length, and the solvent quality. We show that the results for the radial monomer density profile and the distribution of the free ends are in good agreement with the corresponding molecular dynamics results. Part of the investigation is focused on the conformational behavior of a free macromolecule in a cylindrical brush. A central result is the observed non-monotonous variation of the size of a free chain in a brush-coated tube when the tube radius is systematically changed. An interpretation of this behavior which differs qualitatively from that of a polymer in cylindric confinement is suggested in terms of scaling theory and rationalized by considering the overlap between the free polymer and the grafted chains as a function of the tube radius.

Escape transition of a polymer chain: Phenomenological theory and Monte Carlo simulations

The escape transition of a polymer mushroom (i.e., a flexible polymer chain of length N end-grafted onto a flat repulsive surface), occurring when a piston of radius R which is much larger than the size of the mushroom (R0≈aNν, here a is the segment length and ν≈3/5) but much smaller than the linearly stretched chain (Rmax=aN), compresses the polymer to height H, is investigated for good solvent conditions. We argue that in the limit of N→∞ a sharp first-order type transition emerges, characterized in the isotherm force fvs. height H by a flat region from Hesc,t=Ĥ1[N/(R/a)]ν/(1-ν) to Himp,t=Ĥ2[N/(R/a)]ν/(1-ν), with (Ĥ2-Ĥ1)/Ĥ1≈0.26.Monte Carlo methods are developed (combining configurational bias methods with pivot- and random-hopping moves) which allow the study of this transition for chain lengths up to N=1024. It is found that even for such long chains the transition is still slightly rounded. The expected scaling of the transition heights with N and R is nevertheless verified. We show that the transition shows up via a double-peak structure of the radial distribution function of the monomers underneath the piston.

Hydrokinetic simulations of nanoscopic precursor films in rough channels

We report on simulations of capillary filling of highly wetting fluids in nanochannels with and without obstacles. We use atomistic (molecular dynamics) and hydrokinetic (lattice Boltzmann; LB) approaches which indicate clear evidence of the formation of thin precursor films, moving ahead of the main capillary front. The dynamics of the precursor films is found to obey a square-root law like that obeyed by the main capillary front, , although with a larger prefactor, which we find to take the same value for the different geometries (2D–3D) under inspection. The two methods show a quantitative agreement which indicates that the formation and propagation of thin precursors can be handled at a mesoscopic/hydrokinetic level. This can be considered as a validation of the LB method and opens the possibility of using hydrokinetic methods to explore space–time scales and complex geometries of direct experimental relevance. Then, the LB approach is used to study the fluid behaviour in a nanochannel when the precursor film encounters a square obstacle. A complete parametric analysis is performed which suggests that thin-film precursors may have an important influence on the efficiency of nanochannel-coating strategies.

Forced imbibition—a tool for separate determination of Laplace pressure and drag force in capillary filling experiments

When a very thin capillary is inserted into a liquid, the liquid is sucked into it: this imbibition process is controlled by a balance of capillary and drag forces which are hard to quantify experimentally, particularly considering flow on the nanoscale. By computer experiments using a generic coarse-grained model, it is shown that an analysis of imbibition forced by a controllable external pressure independently quantifies the Laplace pressure and Darcys permeability as relevant physical parameters governing the imbibition process. From the latter one may then compute the effective pore radius, effective viscosity, dynamic contact angle and slip length of the fluid flowing into the pore. In determining all these parameters independently, the consistency of our analysis of such forced imbibition processes is demonstrated.

The escape transition of a polymer: A unique case of non-equivalence between statistical ensembles

A flexible polymer chain under good solvent conditions, end-grafted on a flat repulsive substrate surface and compressed by a piston of circular cross-section with radius L may undergo the so-called “escape transition” when the height of the piston D above the substrate and the chain length N are in a suitable range. In this transition, the chain conformation changes from a quasi-two-dimensional self-avoiding walk of “blobs” of diameter D to an inhomogeneous “flower” state, consisting of a “stem” (stretched string of blobs extending from the grafting site to the piston border) and a “crown” outside of the confining piston. The theory of this transition is developed using a Landau free-energy approach, based on a suitably defined (global) order parameter and taking also effects due to the finite chain length N into account. The parameters of the theory are determined in terms of known properties of limiting cases (unconfined mushroom, chain confined between infinite parallel walls). Due to the non-existence of a local order parameter density, the transition has very unconventional properties (negative compressibility in equilibrium, non-equivalence between statistical ensembles in the thermodynamic limit, etc.). The reasons for this very unusual behavior are discussed in detail. Using Molecular Dynamics (MD) simulation for a simple bead-spring model, with N in the range 50