The Becker–Döring equations with monomer input, competition and inhibition

Abstract

Abstract. We investigate the Becker-Döring model of nucleation with three generalisations; an input of monomer, an input of inhibitor and finally, we allow the monomers to form two morphologies of cluster. We assume size-independent aggregation and fragmentation rates. Initially we consider the problem of constant monomer input and determine the steady-state solution approached in the large-time limit, and the manner in which it is approached. Secondly, in addition to a constant input of monomer we allow a constant input of inhibitor, which prevents clusters growing any larger and this removes them from the kinetics of the process; the inhibitor is consumed in the action of poisoning a cluster. We determine a critical ratio of poison to monomer input below which the cluster concentrations tend to a non-zero steady-state solution and the poison concentration tends to a finite value. Above the critical input ratio, the concentrations of all cluster sizes tend to zero and the poison concentration grows without limit. In both cases the solution in the large-time limit is determined. Finally we consider a model where monomers form two morphologies, but the inhibitor only acts on one morphology. Four cases are identified, depending on the relative poison to monomer input rates and the relative thermodynamic stability. In each case we determine the final cluster distribution and poison concentration. We find that poisoning the less stable cluster type can have a significant impact on the structure of the more stable cluster distribution; a counter-intuitive result. All results are shown to agree with numerical simulation.

Symmetry-breaking in Chiral Polymerisation

“…This dual role leads to some subtle and interesting effects since an abundance of, say, right-handed polymers makes it unlikely for any left-handed polymeric sequences to form, and the majority of left-handed monomers produced will be consumed by inhibiting right-handed homochiral sequences -a 'double-whammy' effect. This form of competition is distinct to the models of nucleation involving competition we have analysed previously; for example in Wattis (1999) and in Bolton & Wattis (2004) there is only one monomer which assembles to form two morphologies of cluster. Competition is thus between the growth of one type of cluster and that of another.…”

Mathematical Models of the Homochiralisation of Crystals by Grinding

“…It may then be possible to use grinding to suppress the growth of one polymorph and allow a less stable form to be expressed. In this case, the growth and fragmentation rates of the two polymorphs will differ, we denote the two polymorphs by x and y following Bolton and Wattis ( 2004 ). In place of a , b , α , ξ , β we have a x , r , a y , r , b x , r , α x , r , etc.…”

Similarity solutions of a Becker–Döring system with time-dependent monomer input

“…We consider the initial data c r (0) = 0 for all r. The first moment of the distribution, M 1 , which is the total mass in the system (1.5) then satisfiesṀ 1 = αt w , thus 6) provided w > −1. It is also useful to define the quantity M 0 given by…”