Finite-N validity of the large-N structured-phase coarsening exponents in the N-component Ginzburg–Landau model of self-assembled fluids, and the asymptotic growth law of the single-component Swift–Hohenberg equation

The N-component time-dependent Ginzburg–Landau model of self-assembled fluids, solved exactly by Marini Bettolo Marconi and Corberi (1995) in the large-N limit $N→∞$, predicts that coarsening in the structured (spatially modulated) phase obeys ordinary single-length scaling with dynamic exponent $z=2$ (characteristic length $L(t)∼t^{1/2}$), a structure-factor peak height growing as $t^{1/2}$ independent of dimension, irrelevance of the order-parameter conservation law, and a relaxation time $τ∼T_f^{-2}$. The letter closes by asking how generic this $t^{1/2}$ behavior is, whether the $N→∞$ results have any relevance at finite $N$ — notably $N=1$, where the topological defects suppressed at $N=∞$ are present — and it conjectures that the conflicting exponents found in single-component Swift–Hohenberg simulations arise from defect dynamics not captured by the large-N closure.

Background

Phase-ordering kinetics, or coarsening, is the study of how a system equilibrates after being rapidly cooled (quenched) into a region of its phase diagram with more than one stable phase: domains of the ordered phase form and grow, and the late-time state is described statistically by the structure factor $C(k,t)$ — the Fourier transform of the two-point correlation function of the order parameter $φ$ (the coarse-grained local density difference between the phases). For simple phase separation one expects a single characteristic length $L(t)∼t^{1/z}$, with dynamic exponent $z$, and dynamical scaling $C(k,t)=L(t)^{α}f(kL(t))$ with a time-independent scaling function $f$ and $α=d$ in $d$ dimensions.

Self-assembled fluids — microemulsions, ternary oil–water–surfactant mixtures in which amphiphilic surfactant molecules coat the interfaces and stabilize a spatially modulated (lamellar or striped) phase of fixed preferred wavelength — order at a finite wave number rather than at $k=0$, and so require a different coarse-grained description. Gompper and Schick (1990) formulated a Landau–Ginzburg theory of amphiphilic systems in which a term quadratic in $φ$ multiplies $(∇φ)^2$, penalizing sharp interfaces and favoring modulation. Its dissipative dynamics is a time-dependent Ginzburg–Landau equation with a $∇^4$ term, whose deterministic part coincides with the Swift–Hohenberg equation — the generic model for systems with a finite-wavelength instability, introduced in 1977 for Rayleigh–Bénard convection (the instability of a fluid layer heated from below). The order parameter may be non-conserved (NCOP: $φ$ is not conserved, as in convection) or conserved (COP: $φ$ obeys local conservation, as in a binary mixture or diblock copolymer); in simple phase separation these two cases give different exponents, e.g. $z=2$ for NCOP versus $z=3$ for COP (scalar order parameter, $d>1$).

Marini Bettolo Marconi and Corberi (1995) introduced an $N$-component version of this model, with a surfactant term $wφ^2(∇φ)^2/(2N)$, and solved its coarsening exactly in the large-$N$ (spherical-model) limit $N→∞$, where the interacting problem closes into a self-consistent, effectively linear equation for $C(k,t)$ with time-dependent coefficients $B(t)$ and $Q(t)$. The full definitions are given in Eqs. (1)–(7) of the source letter (arXiv:cond-mat/9505041): the Hamiltonian $H[φ]=\int d^dx\,[\tfrac12(\nabla^2φ)^2+\tfrac b2(\nabla φ)^2+\tfrac w{2N}(φ)^2(\nabla φ)^2+\tfrac r2(φ)^2+\tfrac g{4N}((φ)^2)^2]$; the Langevin equation $\partial φ_α/\partial t=-\Gamma\,\delta H/\delta φ_α+η_α$ with Gaussian white noise $η$, where the kinetic coefficient is a constant $\Gamma$ for NCOP and $-\Gamma\nabla^2$ for COP; and the closed large-$N$ equation $\frac{d}{dt}C(k,t)=-2\Gamma k^p[k^4+\dot B(t)k^2+\dot Q(t)]C(k,t)+2k^p\Gamma T_f$ with $p=0$ (NCOP) or $p=2$ (COP), where the self-consistency conditions are $\dot B=b+wS_0$ and $\dot Q=r+gS_0+wS_2$, with $S_0(t)$ and $S_2(t)$ the momentum-cutoff integrals of $C(k,t)$ and $k^2C(k,t)$. This extends the large-$N$ growth-kinetics method of Coniglio and Zannetti, who found (1989) that the conserved scalar model exhibits multiscaling: instead of ordinary scaling, the exponent of $t$ in $C(k,t)$ is itself a function of the scaling variable $x=k/k_m(t)$ (with $k_m$ the peak position), $C(k,t)∼t^{αφ(x)}$ with $φ(x)=1-(1-x^2)^2$; Coniglio, Ruggiero and Zannetti (1994) analyzed the corresponding crossover structure. Marconi and Corberi mapped the model's phase diagram into unstructured (ordering at $k=0$), structured (ordering at finite $k$) and tricritical Lifshitz-point regions (a Lifshitz point being the meeting point of the disordered, uniformly ordered and modulated lines). In the unstructured region they recover the standard results (NCOP: ordinary scaling with $z=2$, $α=d$; COP: multiscaling with $k_m^{-1}∼(t/log t)^{1/4}$); at the tricritical Lifshitz point the exponents change (NCOP: $z=4$ with scaling function $f(x)=e^{-x^4}$; COP: $L∼t^{1/6}$ with $φ(x)=(3x^2-x^6)/2$). The headline results concern the structured region: ordinary single-length scaling with $α=1$ and $z=2$ (the characteristic length, measured for instance by the inverse width of the structure-factor peak, grows as $t^{1/2}$), a peak height growing as $t^{1/2}$ — independent of dimension, unlike the $α=d$ of plain phase separation — irrelevance of the conservation law (COP and NCOP give the same exponents), and a time scale $τ∼T_f^{-2}$ to reach the scaling regime, where $T_f$ is the final temperature after the quench.

The letter closes by asking how generic this $t^{1/2}$ behavior is, and whether the $N→∞$ results have any relevance for finite $N$ — in particular $N=1$, the physically relevant single-component case, in which topological defects of the stripe pattern (dislocations, grain boundaries and disclinations — the point-like and line-like singularities of the orientational order of the stripe field) are present but are suppressed in the $N→∞$ limit. The question is sharp because numerical studies of the single-component stochastic Swift–Hohenberg equation report different growth exponents: Elder, Viñals and Grant (1992) found the structure-factor length exponent $x_s=1/4$ at small noise amplitude and $1/5$ at low temperature, with defects that are hard to anneal; Cross and Meiron (1995) found $1/5$ for both potential and nonpotential dynamics (nonpotential meaning the dynamics derives from no free-energy function); Hou, Sasa and Goldenfeld (1997) confirmed these values, found logarithmic corrections at deep quench, and confirmed that the peak height scales like the domain size (consistent with $α=1$) while the length grows with $z=4$–$5$ rather than $2$; Christensen and Bray (1998) argued from a phase-equation calculation that the growth passes through a $t^{1/4}$ transient whose crossover to the asymptotic $t^{1/2}$ is estimated to occur only around $t∼10^7$ for $ε=0.25$ (the distance from onset), beyond numerical reach; Boyer and Viñals (2001, 2002) found $z≈3$ close to onset ($ε=0.04$), interpreted via grain-boundary motion, and, away from onset, slower noise-dependent growth attributed to grain-boundary pinning and glassy dynamics; and Qian and Mazenko (2003) found defect-density, energy and orientational-correlation exponents near $1/3$ with a structure-factor exponent near $1/5$, dominated by grain boundaries and dislocations. Experiments on diblock-copolymer stripe ordering (Harrison et al. 2000) observe $x=1/4$. The finite-$N$ question and the reconciliation of these exponents with the large-$N$ $z=2$ remain unresolved.

Problem Statement

The problem originates in the closing paragraph of Marconi and Corberi's 1995 letter, which asks how generic the $t^{1/2}$ behavior is and whether the $N→∞$ results have any relevance at finite $N$ — notably $N=1$, where the topological defects absent from the $N→∞$ solution are present — and how they square with the exponents observed in single-component Swift–Hohenberg simulations: determine the phase-ordering kinetics of the $N$-component Ginzburg–Landau model of self-assembled fluids at finite $N$, and in particular:

(1) Whether the large-$N$ structured-phase results — ordinary single-length scaling with $α=1$ and $z=2$ (characteristic length $L(t)∼t^{1/2}$), a structure-factor peak height growing as $t^{1/2}$ independent of dimension, irrelevance of the order-parameter conservation law, and $τ∼T_f^{-2}$ — persist at finite $N$, notably at $N=1$, where the topological defects absent from the $N→∞$ solution are present.

(2) Whether and how the large-$N$ results extend to finite $N$ in the rest of the phase diagram — in particular whether the multiscaling-versus-ordinary-scaling crossovers found at $N=∞$ in the unstructured and tricritical Lifshitz-point regions survive when $N$ is finite.

(3) The true asymptotic growth law of the single-component ($N=1$) Swift–Hohenberg dynamics in the modulated phase — reconciling the large-$N$ prediction $z=2$ with the exponents reported in simulations and experiments ($z=3$, $4$, $5$, i.e. length exponents $1/3$–$1/5$) — including whether, as the source letter conjectures, the discrepancy is due to topological-defect dynamics not captured by the large-$N$ closure.

These three items are one scientific objective: to establish whether, and in what regime, the exactly solvable large-$N$ description is a faithful guide to the physical single-component modulated phase.

References

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