Unified Theoretical Framework for Polycrystalline Pattern Evolution

Abstract

The rate of curvature-driven grain growth in polycrystalline materials is well-known to be limited by interface dissipation. We show analytically and by simulations that, for systems forming modulated phases or non-equilibrium patterns with crystal ordering, growth is limited by bulk dissipation associated with lattice translation, which dramatically slows down grain coarsening. We also show that bulk dissipation is reduced by thermal noise so that those systems exhibit faster coarsening behavior dominated by interface dissipation for high Peierls barrier and high noise. Those results provide a unified theoretical framework for understanding and modeling polycrystalline pattern evolution in diverse systems over a broad range of length and time scales.PACS numbers: 61.72. Mm, 05.40.Ca, 61.72.Hh, 62.20.Hg Polycrystalline patterns are observed in very diverse systems including crystalline solids [1], colloidal systems [2,3], various spatially modulated phases of macromolecular systems such as diblock copolymers [4,5], and nonequilibrium (NE) dissipative structures [6]. When grain boundaries (GBs) between domains of different crystal orientation are mobile, those patterns generally coarsen in time to reduce GB length or area by elimination of smaller grains. This coarsening behavior has been extensively studied because of its practical importance for engineering polycrystalline materials [7] and its fundamental relevance for our general understanding of nonequilibrium ordering phenomena.The ordering dynamics of modulated phases and NE patterns has been extensively studied theoretically [8][9][10][11][12][13][14][15][16] in the framework of model equations of the formwhere ψ is an order parameter appropriate to each system that can be globally conserved (n = 1) or non-conserved (n = 0), η is a noise uncorrelated in space and time with a variance determined by the fluctuation-dissipation relation η( r, t)η( r , t) = 2αT (−∇ 2 ) n δ( r − r )δ(t − t ), and F is a Lyapounov functional with a minimum in a lattice ordered state. Eq. (1) has also been proposed as a theoretical framework− the phase-field-crystal (PFC) model− to study polycrystalline materials on diffusive time scales with ψ representing the crystal density field [17,18]. While Eq. (1) has been traditionally studied for purely relaxational (p = 0) dynamics [8][9][10][11][12][13][14][15][16][17], propagative (p = 1) wave-like dynamics has also been introduced in the PFC framework to mimic phonon-mediated relaxation of the strain field [18].Extensive computational studies of Eq. (1) have shown that the characteristic domain or grain size for both roll patterns [9][10][11][12] and hexagonal lattices [13][14][15][16] grows ∼ t q . The exponent q is typically much smaller than the q = 1/2 value expected for "normal grain growth" in polycrystalline materials [19], and depends on parameters and noise strength [15,16]. While there have been theoretical attempts to explain those exponents for roll patterns [9][10][11][12], the origin of this sluggish (low q) coarse...

Pattern formation mechanisms in sphere-forming diblock copolymer thin films

“…Here φ A and φ B are the local densities for the A and B blocks, respectively, and f is the volume fraction of one block in the copolymer. The dynamics can be described by the following time-dependent equation for a conserved order parameter [27] ∂ψ ∂t…”

Consistent Hydrodynamics for Phase Field Crystals

“…Grain rotation.-To test the theory numerically, we study the dynamics of a rotated circular crystalline grain embedded in a crystalline matrix. Although experimental studies of polycrystalline patterns suggest that smaller grains usually disappear at the boundary of two larger grains rather than in the middle of a single matrix [21], the rotated grain remains important for understanding grain boundary motion and has been studied theoretically [22] using molecular dynamics simulations [23,24] and PFC models [25,26].…”

Morphological Instability of Grain Boundaries in Two-Phase Coherent Solids

“…However, remarkably, a wide range of high-angle GBs have been shown both computationally and experimentally to exhibit coupling [4][5][6][7][8][9][10][11], making it a general phenomenon. Coupling has also been shown to influence the mechanical and coarsening behaviors of both small and large assemblies of fine grains [9,12,13,16,17]. Those studies have focused primarily on single-phase polycrystalline materials.…”