Equilibrium properties of two-species reactive lattice gases on random catalytic chains

Dmytro Shapoval, Maxym Dudka, Olivier Bénichou, and Gleb Oshanin

Phys. Rev. E 102, 032121 – Published 14 September 2020

Abstract

We focus here on the thermodynamic properties of adsorbates formed by two-species $A + B \to ⊘$ reactions on a one-dimensional infinite lattice with heterogeneous “catalytic” properties. In our model hard-core $A$ and $B$ particles undergo continuous exchanges with their reservoirs and react when dissimilar species appear at neighboring lattice sites in presence of a “catalyst.” The latter is modeled by supposing either that randomly chosen bonds in the lattice promote reactions (Model I) or that reactions are activated by randomly chosen lattice sites (Model II). In the case of annealed disorder in spatial distribution of a catalyst we calculate the pressure of the adsorbate by solving three-site (Model I) or four-site (Model II) recursions obeyed by the corresponding averaged grand-canonical partition functions. In the case of quenched disorder, we use two complementary approaches to find exact expressions for the pressure. The first approach is based on direct combinatorial arguments. In the second approach, we frame the model in terms of random matrices; the pressure is then represented as an averaged logarithm of the trace of a product of random $3 \times 3$ matrices—either uncorrelated (Model I) or sequentially correlated (Model II).

Received 18 May 2020
Accepted 18 August 2020

DOI:https://doi.org/10.1103/PhysRevE.102.032121

Physics Subject Headings (PhySH)

Classical statistical mechanics

Disordered systems Lattice gases

Exact solutions for many-body systems Random matrix theory Transfer matrix calculations

Statistical Physics & Thermodynamics

Authors & Affiliations

Dmytro Shapoval ^1,2, Maxym Dudka ^1,2,3, Olivier Bénichou⁴, and Gleb Oshanin⁴

¹Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii Street, UA-79011 Lviv, Ukraine
²𝕃4 Collaboration & Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry, Europe
³Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
⁴Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), 4 Place Jussieu, 75252 Paris Cedex 05, France

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 102, Iss. 3 — September 2020

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
One-dimensional lattice containing $N$ adsorption sites in contact with vapor phases of $A$ and $B$ particles (blue and gray circles, respectively). Some fraction of bonds between the neighboring sites possesses special catalytic properties (thick black lines). $A$ and $B$ particles undergo continuous exchanges with their vapor phases, maintained at constant chemical potentials $μ_{A}$ and $μ_{B}$ , respectively, adsorb onto empty lattice sites, and desorb from the lattice. $A$ and $B$ particles appearing simultaneously at neighboring sites connected by a catalytic bond [case (a)] react instantaneously, and the reaction product leaves the system. $A$ and $B$ particles adsorbed on the neighboring sites connected by a noncatalytic bond coexist [case (b)].
Reuse & Permissions
Figure 2
One-dimensional lattice containing $N$ adsorption sites in contact with vapor phases. Sites with catalytic properties are marked by thick crosses. $A$ and $B$ particles are depicted by blue and gray circles, respectively. The particles' configuration, which corresponds to an immediate reaction, is realized in case (a), while in case (b) the neighboring $A$ and $B$ particles do not react.
Reuse & Permissions
Figure 3
Annealed disorder case. (a), (b) Disorder-averaged density $n_{A}$ as a function of activity $z_{A}$ for fixed $z_{B} = 15$ (a) and as a function of activity $z_{B}$ for fixed $z_{A} = 15$ (b), for three values of the mean concentration $p = 0.1, 0.5, 0.9$ of the catalytic bonds (Model I, red dashed curves) and catalytic sites (Model II, blue solid curves). (c, d) Disorder-averaged density $n_{A}$ as a function of the mean concentration $p$ of catalytic bonds or catalytic sites [the same color code as in panels (a) and (b)] for three values of $z_{A}$ and $z_{B} = 15$ (c) and for three values of $z_{B}$ and $z_{A} = 15$ (d). (e), (f) Logarithm of the compressibility $ϰ_{A}$ for mean concentration $p = 0.7$ of the catalytic bonds or sites as a function of the activity $z_{A}$ (e) or $z_{B}$ (f). From left to right, the curves correspond to $z_{B} = 1, 5, 10, 20$ (e) and to $z_{A} = 1, 5, 10, 20$ (f).
Reuse & Permissions
Figure 4
Annealed versus quenched disorder. (a) Disorder-averaged density $n$ for Model I as a function of the mean concentration $p$ of the catalytic bonds for three values of activity $z = z_{A} = z_{B}$ . (b) Disorder-averaged density $n$ for Model II as a function of the mean concentration $p$ of the catalytic sites for three values of the activity $z = z_{A} = z_{B}$ .
Reuse & Permissions
Figure 5
Model I. Annealed (red dashed curves) versus quenched (blue solid curves) disorder. (a), (b) Disorder-averaged density $n_{A}$ as a function of the mean concentration $p$ of catalytic bonds for three values of $z_{A}$ and $z_{B} = 15$ (a) and for three values of $z_{B}$ and $z_{A} = 15$ (b). (c), (d) Logarithm of the compressibility $ϰ_{A}$ as a function of the mean concentration $p$ of catalytic bonds for three values of $z_{A}$ and $z_{B} = 15$ (c) and for three values of $z_{B}$ and $z_{A} = 15$ (d).
Reuse & Permissions
Figure 6
Model II. Annealed (red dashed curves) versus quenched (blue solid curves) disorder. (a, b) Disorder-averaged density $n_{A}$ as a function of the mean concentration $p$ of catalytic sites for three values of $z_{A}$ and $z_{B} = 15$ (a) and for three values of $z_{B}$ and $z_{A} = 15$ (b). (c), (d) Logarithm of the compressibility $ϰ_{A}$ as a function of the mean concentration $p$ of catalytic sites for three values of $z_{A}$ and $z_{B} = 15$ (c) and for three values of $z_{B}$ and $z_{A} = 15$ (d).
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics