The LiHoxY1-xF4 magnetic material in a transverse magnetic field Bx x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in a random disordered system. We show that the Bx-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline symmetries, generates, via the predominant dipolar interactions between Ho3+ ions, random fields along the Ising z direction. This identifies LiHoxY1-xF4 in Bx as a new random field Ising system. The random fields explain the rapid decrease of the critical temperature in the diluted ferromagnetic regime and the smearing of the nonlinear susceptibility at the spin-glass transition with increasing Bx and render the Bx-induced quantum criticality in LiHoxY1-xF4 likely inaccessible.

This study examines the numerical accuracy, computational cost, and memory requirements of self-consistent field theory (SCFT) calculations when the diffusion equations are solved with various pseudo-spectral methods and the mean-field equations are iterated with Anderson mixing. The different methods are tested on the triply periodic gyroid and spherical phases of a diblock-copolymer melt over a range of intermediate segregations. Anderson mixing is found to be somewhat less effective than when combined with the full-spectral method, but it nevertheless functions admirably well provided that a large number of histories is used. Of the different pseudo-spectral algorithms, the 4th-order one of Ranjan, Qin and Morse performs best, although not quite as efficiently as the full-spectral method.

Monte Carlo field-theoretic simulations (MC-FTS) are performed on melts of symmetric diblock copolymer for invariant polymerization indexes extending down to experimentally relevant values of N̅ ∼ 10 4 . The simulations are performed with a fluctuating composition field, W − (r), and a pressure field, W + (r), that follows the saddle-point approximation. Our study focuses on the disordered-state structure function, S(k), and the order−disorder transition (ODT). Although shortwavelength fluctuations cause an ultraviolet (UV) divergence in three dimensions, this is readily compensated for with the use of an effective Flory−Huggins interaction parameter, χ e . The resulting S(k) matches the predictions of renormalized one-loop (ROL) calculations over the full range of χ e N and N̅ examined in our study, and agrees well with Fredrickson−Helfand (F−H) theory near the ODT. Consistent with the F−H theory, the ODT is discontinuous for finite N̅ and the shift in (χ e N) ODT follows the predicted N̅ −1/3 scaling over our range of N̅ .

We investigate the properties of the XY pyrochlore antiferromagnet with infinite local 111 planar anisotropy. We identify the ground states and show that the configurational ground state entropy is subextensive. By computing the free energy due to harmonic fluctuations and by carrying out Monte Carlo simulations, we find that the model exhibits thermal order-by-disorder leading to low-temperature long-range order consisting of discrete magnetic domains. In doing so, we set aside doubts that order-by-disorder survives in the thermodynamic limit in this model. We compute the spin wave spectrum and show that thermal and quantum fluctuations select the same magnetic structure. With a previously unreported finite-size scaling analysis of Monte Carlo data, we confirm that the transition is first order for the XY model. Using Monte Carlo simulations, we find that the state selected by thermal fluctuations in this XY pyrochlore antiferromagnet can survive the addition of sufficiently weak nearest-neighbor pseudo-dipolar interactions or long-range dipolar interactions to the spin Hamiltonian. Quite interestingly, the resulting state selected by thermal order-by-disorder is metastable below some temperature. We discuss our results in relation to the Er 2 Ti 2 O 7 and Er 2 Sn 2 O 7 pyrochlore antiferromagnets.

Field-theoretic simulation (FTS) offers an efficient means of predicting the equilibrium behavior of high-molecular-weight structured polymers, provided one is able to deal with the strong ultraviolet (UV) divergence that occurs at realistic molecular weights. Here melts of lamellar-forming diblock copolymer are studied using a Monte Carlo version (MC-FTS), where the composition field fluctuates while the pressure field follows the mean-field approximation. We are able to control the UV divergence by introducing a new effective Flory− Huggins interaction parameter, χ e , thereby permitting MC-FTS for molecular weights extending down to values characteristic of experiment. Results for the disordered-state structure function, the layer spacing and compressibility of the ordered lamellar phase, and the position of the order−disorder transition (ODT) show excellent agreement with recent particle-based simulation. Given the immense versatility of FTS, this opens up the opportunity for quantitative studies on a wide range of more complicated block copolymer systems.

The structure, stability, and reorganization of lamella-forming block copolymer thin film surface topography ("islands" and "holes") were studied under boundary conditions driving the formation of 0.5 L thick structures at short thermal annealing times. Self-consistent field theory predicts that the presence of one perfectly neutral surface renders 0.5 L topography thermodynamically stable relative to 1 L thick features, in agreement with previous experimental observations. The calculated through-film structures match cross-sectional scanning electron micrographs, collectively demonstrating the pinning of edge dislocations at the neutral surface. Remarkably, near-neutral surface compositions exhibit 0.5 L topography metastability upon extended thermal treatment, slowly transitioning to 1 L islands or holes as evidenced by optical and atomic force microscopy. Surface restructuring is rationalized by invoking commensurability effects imposed by slightly preferential surfaces. The results described herein clarify the impact of interfacial interactions on block copolymer self-assembly and solidify an understanding of 0.5 L topography, which is frequently used to determine neutral surface compositions of considerable importance to contemporary technological applications.

Self-consistent field theory (SCFT) is used to study the step edges that occur in thin films of lamellar-forming diblock copolymer, when the surfaces each have an affinity for one of the polymer components. We examine film morphologies consisting of a stack of ν continuous monolayers and one semi-infinite bilayer, the edge of which creates the step. The line tension of each step morphology is evaluated and phase diagrams are constructed showing the conditions under which the various morphologies are stable. The predicted behavior is then compared to experiment. Interestingly, our atomic force microscopy (AFM) images of terraced films reveal a distinct change in the character of the steps with increasing ν, which is qualitatively consistent with our SCFT phase diagrams. Direct quantitative comparisons are not possible because the SCFT is not yet able to probe the large polymer/air surface tensions characteristic of experiment. ■ INTRODUCTIONThe behavior of block copolymer melts on solid substrates has received considerable attention over the last two decades motivated by a wide range of applications. 1−3 Not surprisingly, the earliest work 4−8 involved the simplest conceivable system, thin films of symmetric AB diblock copolymer on flat substrates. In the bulk, symmetric diblock copolymer forms a lamellar phase of alternating A-and B-rich domains, where each period, L 0 , consists of two monolayers. The main effect of confining the diblock copolymer to a thin film is to orient the domains. When the substrate and air surfaces of the film each have an affinity for one of the polymer components, the lamellae orient in the plane of the film, thereby forming a parallel lamellar (L ν ∥ ) phase consisting of ν diblock monolayers. Because of the strong tendency to maintain the preferred domain size of the bulk, the film height is quantized, H ≈ L 0 ν/ 2. Even values of ν are selected when the two surfaces have an affinity for the same component, and odd values occur when the two surfaces have opposite affinities. Usually the amount of material cast on the substrate is incommensurate with the preferred domain size, which leads to macrophase separation into thin and thick regions (i.e., terraces) separated by steps of height L 0 . The topography of the air surface changes from islands to a bicontinuous network to holes as the area of the thicker region increases. 4 The behavior becomes particularly interesting with the formation of elaborate terraced structures, when droplets 9−11 or rings 12 of diblock copolymer are deposited on the substrate.The presence of step edges separating different film thicknesses is ubiquitous in block copolymer films, and yet they have received relatively little theoretical attention. A step edge between the L ν ∥ and L ν+2 ∥ phases will typically involve ν continuous monolayers plus one semi-infinite bilayer as depicted schematically in Figure 1. Ausserréet al. 5 initially argued that the bilayer should exist one monolayer below the air surface, rather than on top as in Figure 1. Fu...

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