Renormalization-theoretic analysis of non-equilibrium phase transitions: I. The Becker-Döring equations with power law rate coefficients

Abstract

We study in detail the application of renormalisation theory to models of cluster aggregation and fragmentation of relevance to nucleation and growth processes. We investigate the Becker-Döring equations, originally formulated to describe and analyse non-equilibrium phase transitions, and more recently generalised to describe a wide range of physicochemical problems. In the present paper we analyse how the systematic coarse-graining renormalisation of the Becker-Döring system of equations affects the aggregation and fragmentation rate coefficients. We consider the case of power-law size-dependent cluster rate coefficients which we show lead to only three classes of system that require analysis: coagulationdominated systems, fragmentation-dominated systems and those where coagulation and fragmentation are exactly balanced. We analyse the late-time asymptotics associated with each class. PACS

Cluster growth poised on the edge of break-up: Size distributions with small exponent power-laws

“…A similar result has also been obtained by using generating functions and scaling ansatzs [15]. In summary, the previous results have pointed out three growth regimes for α ≤ 1 (for α > 1 normalizable solutions are not found), depending on the exponents of the reaction rates [15][16][17]. In the asymmetric case α > µ detachment dominates, and very slow or stagnated growth is expected.…”

“…Equations similar to REs Eqs. (1)-(2) have been studied previously by using asymptotic matched expansion and large-time asymptotic expansions [16], as well as by using renormalization methods [17]. In both cases, three classes of growth are found: aggregation-dominated and detachment-dominated cases, and the case where aggregation and detachment are exactly balanced.…”

“…In the opposite case, α < µ, no steady state solutions exist. In the symmetric case α = µ with 0 ≤ α ≤ 1, however, a steady state is obtained [15][16][17] and one may expect size distributions of a scaling form. A convenient definition of scaling form is n s (θ ) = θ s −2 f (s/ s ), where f (x) is the scaling function and x = s/ s is defined in terms of average size s = s 2 n s / sn s (for details, see Ref.…”

The Becker–Döring equations with exponentially size-dependent rate coefficients

“…Case VI: η < γ < 0 In this case the system evolves to the steady-state solution (3.6), which for large aggregation numbers r has the asymptotic behaviour c sss r ∼ (J/ac 1 )e −γr and is thus divergent (since γ < 0). At large times the kinetics can be determined by considering ψ(r, t) = c r (t)/c sss r which satisfies ∂ψ ∂t = and the transformations χ = y/t and ψ = g(χ) yield 25) and hence…”

“…Results for the case of rates which depend on size in a power-law fashion (a case of direct relevance to classical nucleation theory) have been derived by King & Wattis [13,26]. Coveney, Wattis and Bolton have shown that by eliminating a proportion of the concentration variables, a coarse-grained mesoscopic description can be obtained [10,24,25,6]. The resulting model also has the form of a Becker-Döring system, with modified aggregation and fragmentation rate coefficients; the coarse-graining procedure can be reapplied and thus has the form of a renormalisation transformation, with each application filtering out more of the detailed dynamics of the cluster size distribution.…”