Discussion notes on “Spurious character of singularities associated with phase transitions in cylindrical pores”, by K. Binder et al.

Abstract

Comments are provided for [6]. It would be of some interest if the quasi-one-dimensional nature of phase coexistence in a cylindrical pore could be mapped directly onto an exactly solvable one-dimensional model, to thereby compare the observed cluster statistics with rigorous results for isodesmic chemical equilibria (linear self-assembly). For example, a simple coarsegraining of simulation density profile data into coin-shaped discs should allow for a direct mapping onto the one-dimensional lattice-gas (LG) model, [1]. Early simulation observations of the multi-domain structure of phase-coexistence in cylindrical pores showed a qualitative resemblance to the polydisperse domains of one-dimensional selfassembled cluster distributions [2]. With modern computer resources and a suitable mapping it should be possible to collect entire equilibrated cluster distributions from simulations of standard models used by physical chemists. Of most relevance here are cases where the self-assembled phase wants to wet the pore wall, since then there can be no pore-induced nucleation barrier to interfere with one-dimensional self-assembly. The physical chemistry of one-dimensional self-assembly has previously been elucidated from exact solutions of the cluster distributions of one-dimensional lattice-gas models. These results include the effect of solvent-induced self-assembly on isodesmic chemical equilibria [3], the effect of altering the binary equilibrium constant to depart from isodesmic behaviour [4] and the effects of confinement and/or a finite amount of solute [5]. The last situation is directly relevant to the ‘transition’ between normal isodesmic behaviour and ‘polymerisation.’ Namely, if there is sufficient space and amount of solute then the well-known polydisperse exponential cluster distribution is found. Here, the maximum in the isodesmic cluster distribution will always be isolated monomers. However, if the system runs out of space or sufficient solute to be able to generate the largest cluster lengths then the exponential distribution of cluster sizes will rapidly switch over to the opposite case of polymerisation, where the most probable cluster is the longest one possible. Due to the one-dimensional geometry, this ‘transition’ cannot be a true phase-transition, but it is still a clearly identifiable qualitative change in behaviour. In particular, a cross-over line in phase space can be a e-mail: jrh.leeds@gmail.com 244 The European Physical Journal Special Topics uniquely defined by solving the one-dimensional LG model for the ‘transition’ states at which isolated monomers are equally probable as the largest possible cluster, [5]. Note that this cross-over is controlled by the ratio of the free-energy cost of a coarsegrained disc free-surface to the temperature energy scale kBT . This appears to be the only reason why linear polymerisation is far more dominant in chemical self-assembly than observed in physical self-assembly!