^{1,a)}and Moshe Sheintuch

^{1}

### Abstract

We analyze the behavior of a microkinetic model of a catalyticreaction coupled with weak enthalpy effects to show that under fixed gas-phase concentrations it can produce moving waves with an intrinsic length scale, when the underlying kinetics is oscillatory. The kinetic model incorporates dissociative oxygen adsorption, reactant adsorption and desorption, and surface reaction. Three typical patterns may emerge in a one-dimensional system (a long wire or a ring): homogeneous oscillations, a family of moving waves propagating with constant velocities, and patterns with multiple source∕sink points. Pattern selection depends on the ratio of the system length to the intrinsic wave length and the governing parameters. We complement these analysis with simulations that revealed a plethora of patterned states on one- and two-dimensional systems (a disk or a cylinder). This work shows that weak long-range coupling due to high feed rates maintains such patterns, while low feed rates or strong long-range interaction can gradually suppress the emerging patterns.

This work was supported by the Technion’s Fund for the Promotion of Research. M.S. is a member of the Minerva Center of Nonlinear Dynamics. O.N. was partially supported by the Center for Absorption in Science, Ministry of Immigrant Absorption State of Israel.

I. INTRODUCTION

II. PROBLEM STATEMENT

A. Mathematical model

B. Analysis of lumped systems

III. INTRINSIC LENGTH SCALE AND LINEAR ANALYSIS

A. Qualitative analysis

B. CO-oxidation model (1D system)

IV. NUMERICAL SIMULATIONS

A. Behavior on a Ring

B. Behavior on a disk or a cylinder

V. PATTERN SUPPRESSION BY LIMITED FEED RATE

VI. CONCLUDING REMARKS

### Key Topics

- Bifurcations
- 21.0
- Catalysis
- 18.0
- Surface patterning
- 18.0
- Boundary value problems
- 11.0
- High pressure
- 8.0

## Figures

(Color) Typical breathing pattern [(a), experiments (Ref. 15); (b), simulations (Ref. 11)] in the case of strong gas-phase coupling and multiwave patterns [(c, d), present work] in the case of weak coupling. Each row represents equally intervaled snapshots of the temperature on a disk surface during one quasiperiod. Simulations in row (b) were conducted with fixed-temperature boundary conditions and , , . Simulations in rows (c, d) were conducted with no-flux boundary conditions and , , (c), and (d); ( is a convective velocity). (a, b), 6.4 cm (c, d); reactor length (a, b), 0.3 cm (c, d).

(Color) Typical breathing pattern [(a), experiments (Ref. 15); (b), simulations (Ref. 11)] in the case of strong gas-phase coupling and multiwave patterns [(c, d), present work] in the case of weak coupling. Each row represents equally intervaled snapshots of the temperature on a disk surface during one quasiperiod. Simulations in row (b) were conducted with fixed-temperature boundary conditions and , , . Simulations in rows (c, d) were conducted with no-flux boundary conditions and , , (c), and (d); ( is a convective velocity). (a, b), 6.4 cm (c, d); reactor length (a, b), 0.3 cm (c, d).

Typical bifurcation diagram of the lumped oscillatory model [Eqs. (3) and (4) with ] showing the homogeneous steady state solutions of (a) and temperature (b) as functions of with various gas temperatures [ (1); 483 K (2), 486 K (3), and 493 K (4)]. Solid and dashed lines denote stable and unstable steady states, respectively. Points mark the Hopf bifurcation points .

Typical bifurcation diagram of the lumped oscillatory model [Eqs. (3) and (4) with ] showing the homogeneous steady state solutions of (a) and temperature (b) as functions of with various gas temperatures [ (1); 483 K (2), 486 K (3), and 493 K (4)]. Solid and dashed lines denote stable and unstable steady states, respectively. Points mark the Hopf bifurcation points .

Typical bifurcation diagrams showing (a) the homogeneous steady state and oscillatory branches, and (b) the periods of oscillations of the lumped system [Eqs. (3) and (4)] as functions of . Dashed-dotted lines correspond to a constant-velocity rotating pulse solution of the ODE system [Eqs. (6) and (7)] on a ring with . . Other notations as in Fig. 2.

Typical bifurcation diagrams showing (a) the homogeneous steady state and oscillatory branches, and (b) the periods of oscillations of the lumped system [Eqs. (3) and (4)] as functions of . Dashed-dotted lines correspond to a constant-velocity rotating pulse solution of the ODE system [Eqs. (6) and (7)] on a ring with . . Other notations as in Fig. 2.

Bifurcation map of the lumped system bounded by Hopf bifurcation lines showing the subdomains (i–iv) of the distributed system [Eqs. (3) and (4)] behavior.

Bifurcation map of the lumped system bounded by Hopf bifurcation lines showing the subdomains (i–iv) of the distributed system [Eqs. (3) and (4)] behavior.

Typical spatial profiles in a moving pulse solution (a) and schematic phase plane (b) showing null curves (solid and dashed lines) and (dash-dotted line) of learning model (5). In (b) thick solid and dashed lines are calculated with a fixed value that correspond to the ignited and the extinguished fronts (denotes by arrows), respectively.

Typical spatial profiles in a moving pulse solution (a) and schematic phase plane (b) showing null curves (solid and dashed lines) and (dash-dotted line) of learning model (5). In (b) thick solid and dashed lines are calculated with a fixed value that correspond to the ignited and the extinguished fronts (denotes by arrows), respectively.

Typical neutral curves corresponding to the Hopf points (lower boundary of the oscillatory domain, solid line) and (upper boundary, dashed line). , ; .

Typical neutral curves corresponding to the Hopf points (lower boundary of the oscillatory domain, solid line) and (upper boundary, dashed line). , ; .

Effect of the rotation velocity on the period-one rotating pulse solution [Eqs. (9) and (10)] showing (a) the maximal and (b) the minimal concentration , (c) the rotation period [], and (d) the ratio. Points in (c) denote the results of numerical simulations of the system [Eqs. (3) and (4)] on a ring. Dotted line in (d) denotes the asymptotic value of the temporal period of the corresponding lumped model [Eqs. (3) and (4)]. , .

Effect of the rotation velocity on the period-one rotating pulse solution [Eqs. (9) and (10)] showing (a) the maximal and (b) the minimal concentration , (c) the rotation period [], and (d) the ratio. Points in (c) denote the results of numerical simulations of the system [Eqs. (3) and (4)] on a ring. Dotted line in (d) denotes the asymptotic value of the temporal period of the corresponding lumped model [Eqs. (3) and (4)]. , .

Effect of the ring perimeter on pattern selection within the oscillatory domain: homogeneous oscillations [(a) ], pseudohomogeneous oscillations [(b) ], period-one rotating waves [(c) ], multiperiodic rotating waves [(d) ], patterns with various numbers of sink∕source points [(f) , ; (g) , ; (h) , ]. Patterns (a)–(c) and (e)–(g) are simulated with gradually varying IC; IC for pattern (d) are obtained by reproducing a period-one solution (c); pattern (h) is simulated with a stepwise IC. Time . , .

Effect of the ring perimeter on pattern selection within the oscillatory domain: homogeneous oscillations [(a) ], pseudohomogeneous oscillations [(b) ], period-one rotating waves [(c) ], multiperiodic rotating waves [(d) ], patterns with various numbers of sink∕source points [(f) , ; (g) , ; (h) , ]. Patterns (a)–(c) and (e)–(g) are simulated with gradually varying IC; IC for pattern (d) are obtained by reproducing a period-one solution (c); pattern (h) is simulated with a stepwise IC. Time . , .

A typical sequence of pattern transformation on a ring of fixed perimeter with increasing : (a) moving waves , (b) irregular moving waves (60 Pa), (c) a multiperiodic pattern with two pairs of sinks∕sources (200 Pa), (d) a condensed sinks∕sources pattern (440 Pa), (e) a multispot irregular pattern (450 Pa), (f) homogeneous oscillations (500 Pa). . In plates (a) and (b) only a half of the ring is shown. Time (a), 25 s (b, c), 50 s (d), 100 s (e), and 150 s (f).

A typical sequence of pattern transformation on a ring of fixed perimeter with increasing : (a) moving waves , (b) irregular moving waves (60 Pa), (c) a multiperiodic pattern with two pairs of sinks∕sources (200 Pa), (d) a condensed sinks∕sources pattern (440 Pa), (e) a multispot irregular pattern (450 Pa), (f) homogeneous oscillations (500 Pa). . In plates (a) and (b) only a half of the ring is shown. Time (a), 25 s (b, c), 50 s (d), 100 s (e), and 150 s (f).

A typical sequence of 2D spatiotemporal pattern transformation on a wide cylinder with increasing : (a) an irregular cellularlike pattern , (b) an irregular targetlike pattern (100 Pa), (c) a regular two-target pattern (300 Pa), (d) a clusterlike pattern (480 Pa). , , . Row (a) shows a fragment of the surface.

A typical sequence of 2D spatiotemporal pattern transformation on a wide cylinder with increasing : (a) an irregular cellularlike pattern , (b) an irregular targetlike pattern (100 Pa), (c) a regular two-target pattern (300 Pa), (d) a clusterlike pattern (480 Pa). , , . Row (a) shows a fragment of the surface.

Effect of initial conditions [(a) symmetric and (b) asymmetric] on spatiotemporal patterns on a disk. , , .

Effect of initial conditions [(a) symmetric and (b) asymmetric] on spatiotemporal patterns on a disk. , , .

(Color) Effect of the long-range interaction strength on pattern selection showing the pattern suppression with decreasing convective velocity . Each row represents pattern transformation during one period of oscillations for conditions when the relevant four-variable system exhibits an imperfect (a) and regular (b)–(d) target patterns. , (a), 300 Pa (b)–(d); (a), (b), (c), (d). , , (annular reactor width). Flow in horizontal direction from left.

(Color) Effect of the long-range interaction strength on pattern selection showing the pattern suppression with decreasing convective velocity . Each row represents pattern transformation during one period of oscillations for conditions when the relevant four-variable system exhibits an imperfect (a) and regular (b)–(d) target patterns. , (a), 300 Pa (b)–(d); (a), (b), (c), (d). , , (annular reactor width). Flow in horizontal direction from left.

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