Kinetic Oscillations on Pt(100): Theory

Paper Information

Citation: Imbihl, R., Cox, M. P., Ertl, G., Müller, H., & Brenig, W. (1985). Kinetic oscillations in the catalytic CO oxidation on Pt(100): Theory. The Journal of Chemical Physics, 83(4), 1578-1587. https://doi.org/10.1063/1.449834

Publication: The Journal of Chemical Physics 1985

Related Work: See also Oscillatory CO Oxidation on Pt(110) for the same catalytic system on a different crystal face, demonstrating that surface phase transitions drive oscillatory behavior across multiple platinum surfaces.

What kind of paper is this?

Theory ($\Psi_{\text{Theory}}$).

This paper derives a microscopic mechanism from first principles to explain experimentally observed kinetic oscillations. It relies heavily on formal analysis, including a Linear Stability Analysis of a simplified model to derive eigenvalues and identify Hopf bifurcations (transition from stable nodes to limit cycles). The primary contribution is the mathematical formulation of the surface phase transition, rather than a new general-purpose algorithm or experimental discovery.

What is the motivation?

Experimental studies had shown that the catalytic oxidation of Carbon Monoxide (CO) on Platinum (100) surfaces exhibits temporal oscillations and spatial wave patterns at low pressures ($10^{-4}$ Torr). While the individual elementary steps (adsorption, desorption, reaction) were known, the mechanism driving the periodicity was not understood. Prior models relied on indirect evidence; this work aimed to ground the theory in new LEED (Low-Energy Electron Diffraction) observations showing that the surface structure itself transforms periodically between a reconstructed hex phase and a bulk-like 1x1 phase.

What is the novelty here?

The core novelty is the Surface Phase Transition Model. The authors propose that the oscillations are driven by the reversible phase transition of the Pt surface atoms, which is triggered by critical adsorbate coverages:

1. State Dependent Kinetics: The hex and 1x1 phases have vastly different sticking coefficients for Oxygen (negligible on hex, high on 1x1).
2. Critical Coverage Triggers: The transition is not instantaneous but depends on whether local CO coverage exceeds a critical threshold ($U_{a,grow}$) or falls below another ($U_{a,crit}$).
3. Trapping-Desorption: The model introduces a “trapping” term where CO diffuses from the weakly-binding hex phase to the strongly-binding 1x1 patches, creating a feedback loop.

What experiments were performed?

As a theoretical paper, the “experiments” were computational simulations and mathematical derivations:

• Linear Stability Analysis: They simplified the 4-variable model to a 2-variable system to analytically determine the conditions for oscillations (existence of unstable stationary points).
• Hysteresis Simulation: They simulated temperature-programmed variations to match experimental CO adsorption hysteresis loops, fitting the critical coverage parameters ($U_{a,grow} \approx 0.5$).
• Reaction-Diffusion Simulation: They numerically integrated the full set of 4 coupled differential equations over a 1D spatial grid (40 compartments) to reproduce temporal oscillations and propagating wave fronts.

What were the outcomes and conclusions drawn?

• Mechanism Validation: The model successfully reproduced the “sawtooth” waveform of the oscillations observed in work function measurements.
• Phase Transition Role: Confirmed that the “slow” step driving the oscillation period is the phase transformation, specifically the requirement for CO to build up to a critical level to nucleate the reactive 1x1 phase.
• Spatial Self-Organization: The addition of diffusion terms allowed the model to reproduce wave propagation, showing that defects at crystal edges can act as “pacemakers” or triggers for the rest of the surface.

Reproducibility Details

To faithfully replicate this study, one must implement the system of four coupled differential equations. The hardware requirements are negligible by modern standards.

Models

The system tracks four state variables:

1. $u_a$: CO coverage on the 1x1 phase (normalized to local area $a$)
2. $u_b$: CO coverage on the hex phase (normalized to local area $b$)
3. $v_a$: Oxygen coverage on the 1x1 phase (normalized to local area $a$)
4. $a$: Fraction of surface in 1x1 phase ($b = 1 - a$)

The Governing Equations:

CO coverage on 1×1 phase: $$ \frac{\partial u_a}{\partial t} = k_1 a p_{CO} - k_2 u_a + k_3 a u_b - k_4 u_a v_a / a + k_5 \nabla^2(u_a/a) $$

CO coverage on hex phase: $$ \frac{\partial u_b}{\partial t} = k_1 b p_{CO} - k_6 u_b - k_3 a u_b $$

Oxygen coverage on 1×1 phase: $$ \frac{\partial v_a}{\partial t} = k_7 a p_{O_2} \left[ \left(1 - 2 \frac{u_a}{a} - \frac{5}{3} \frac{v_a}{a}\right)^2 + \alpha \left(1 - \frac{5}{3}\frac{v_a}{a}\right)^2 \right] - k_4 u_a v_a / a $$

The Phase Transition Logic ($da/dt$):

The growth of the 1x1 phase ($a$) is piecewise, defined by critical coverages:

• If $u_a/a > U_{a,grow}$: autocatalytic growth with $\partial a/\partial t = k_8 a (1-a)$
• If $(u_a/a)/U_{a,crit} + (v_a/a)/V_{a,crit} < 1$: decay to hex with $\partial a/\partial t = -k_8 a$
• Otherwise: $\partial a/\partial t = 0$

Algorithms

• Time Integration: Runge-Kutta-Merson routine.
• Spatial Integration: Crank-Nicholson algorithm for the diffusion term.
• Time Step: $\Delta t = 10^{-4}$ s.
• Spatial Grid: 1D array of 40 compartments, total length 0.4 cm (each compartment 0.01 cm).
• Boundary Conditions: Closed ends (no flux). Defects simulated by setting $\alpha$ higher in the first 3 “edge” compartments.

Data

Replication requires the specific rate constants. Note: $k_3$ and $\alpha$ are fitting parameters.

Citation

@article{imbihl1985kinetic,
  title={Kinetic oscillations in the catalytic CO oxidation on Pt(100): Theory},
  author={Imbihl, R and Cox, MP and Ertl, G and M{\"u}ller, H and Brenig, W},
  journal={The Journal of Chemical Physics},
  volume={83},
  number={4},
  pages={1578--1587},
  year={1985},
  publisher={American Institute of Physics}
}