A numerical implementation of self-consistent mean-field theory for the structural phase behavior of block copolymers is proposed. Our scheme does not require a priori assumptions of the underlying mesoscopic symmetries. The method potentially enables us to characterize, with high accuracy, the structural phase diagram of block copolymers with significant architectural complexity. We illustrate the method by applying it to a triblock copolymer system.

We present an improved algorithm of the self‐consistent mean‐field implementation that has been recently proposed for the calculation of block copolymer self‐assembly. Without requiring prior knowledge of the symmetry of the mesophase segregation, the algorithm is numerically stable and significantly faster than previously proposed methods. These advantages provide a valuable tool for combinatorial screening of novel stable and metastable structural phases of block copolymers. We apply the method and demonstrate complex mesophases in linear, asymmetric triblock copolymer melts. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1777–1783, 2002

Statistical mechanics of the discrete nonlinear Schrödinger equation is studied by means of analytical and numerical techniques. The lower bound of the Hamiltonian permits the construction of standard Gibbsian equilibrium measures for positive temperatures. Beyond the line of $T=\infty$, we identify a phase transition through a discontinuity in the partition function. The phase transition is demonstrated to manifest itself in the creation of breatherlike localized excitations. Interrelation between the statistical mechanics and the nonlinear dynamics of the system is explored numerically in both regimes.

We derive underdamped evolution equations for the order-parameter (OP) strains of a proper ferroelastic material undergoing a structural transition, using Lagrangian variations with Rayleigh dissipation, and a free energy as a polynomial expansion in the ${N=n+N}_{\mathrm{op}}$ symmetry-adapted strains. The $N_{\mathrm{op}}$ strain equations are structurally similar in form to the Lagrange-Rayleigh one-dimensional strain dynamics of Bales and Gooding (BG), with “strain accelerations” proportional to a Laplacian acting on a sum of the free-energy strain derivative and frictional strain force assuming geometric linearity. The tensorial St. Venant’s elastic compatibility constraints that forbid defects, are used to determine the n non-order-parameter strains in terms of the OP strains, generating anisotropic and long-range OP contributions to the free energy, friction, and noise. The same OP equations are obtained by either varying the displacement vector components, or by varying the N strains subject to the $N_c$ compatibility constraints. A Fokker-Planck equation, based on the BG dynamics in more than one dimension with noise terms, is set up. The BG dynamics corresponds to a set of nonidentical nonlinear (strain) oscillators labeled by wave vector $\overrightarrow{k},$ with competing short- and long-range couplings. The oscillators have different “strain-mass” densities $\rho \mathrm{}\left(k\right)\mathrm{}\sim {1/k}^2$ and dampings $\sim 1/\rho \mathrm{}\left(k\right)\mathrm{}\sim k^2,$ so the lighter large-$k$ oscillators equilibrate first, corresponding to earlier formation of smaller-scale oriented textures. This produces a sequential-scale scenario for post-quench nucleation, elastic patterning, and hierarchical growth. Neglecting inertial effects yields a late-time dynamics for identifying extremal free-energy states, that is, of the time-dependent Ginzburg-Landau form, with nonlocal, anisotropic Onsager coefficients that become constants for special parameter values. We consider in detail the two-dimensional (2D) unit-cell transitions from a triangular to a centered rectangular lattice ${(N}_{\mathrm{op}}{=2,n=1,N}_c=1)$ and from a square to a rectangular lattice ${(N}_{\mathrm{op}}{=1,n=2,N}_c=1)$ for which the OP compatibility kernel is retarded in time, or frequency dependent in Fourier space (in fact, acoustically resonant in $\omega /k).\mathrm{}$ We present structural dynamics for all other 2D symmetry-allowed proper ferroelastic transitions: the procedure is also applicable to the 3D case. Simulations of the BG evolution equations confirm the inherent richness of the static and dynamic texturings, including strain oscillations, domain-wall propagation at near-sound speeds, grain-boundary motion, and nonlocal “elastic photocopying” of imposed local stress patterns.

It has long been known that double-stranded DNA is subject to temporary, localized openings of its two strands. Particular regions along a DNA polymer are destabilized structurally by available thermal energy in the system. The localized sequence of DNA determines the physical properties of a stretch of DNA, and that in turn determines the opening profile of that DNA fragment. We show that the Peyrard-Bishop nonlinear dynamical model of DNA, which has been used to simulate denaturation of short DNA fragments, gives an accurate representation of the instability profile of a defined sequence of DNA, as verified using S1 nuclease cleavage assays. By comparing results for a non-promoter DNA fragment, the adenovirus major late promoter, the adeno-associated viral P5 promoter and a known P5 mutant promoter that is inactive for transcription, we show that the predicted openings correlate almost exactly with the promoter transcriptional start sites and major regulatory sites. Physicists have speculated that localized melting of DNA might play a role in gene transcription and other processes. Our data link sequence-dependent opening behavior in DNA to transcriptional activity for the first time.