scholarly journals Quasi-steady state reduction for the Michaelis–Menten reaction–diffusion system

2018 ◽  
Vol 56 (6) ◽  
pp. 1759-1781
Author(s):  
Martin Frank ◽  
Christian Lax ◽  
Sebastian Walcher ◽  
Olaf Wittich
2009 ◽  
Vol 19 (08) ◽  
pp. 2637-2675 ◽  
Author(s):  
STEFAN WEHNER

This review summarizes recent studies on the catalytic CO oxidation on Iridium(111) surfaces. This was investigated experimentally under ultrahigh vacuum (UHV) conditions using mass spectroscopy to detect gaseous products and photoelectron emission microscopy (PEEM) to visualize surface species. The underlying reaction–diffusion system based on the Langmuir–Hinshelwood mechanism was analyzed numerically.The existence of bistability for this surface reaction was shown in experiment. For the first time the effect of noise on a bistable surface reaction was examined. In a surface science experiment the effects on product formation and the development of spatio-temporal patterns on the surface were explored.Steady state CO2rates were measured under constant flux of the CO + O mixture as a function of sample temperature (360 K < T < 700 K) and gas composition, characterized by the molar fraction of CO in the feed gas (0 ≤ Y ≤ 1). The reaction reveals bistability in a limited region of Y and T. A rate hysteresis with two steady state rates was observed for cycling Y up and down, one of high reactivity (upper rate, oxygen covered surface) and one of low reactivity (lower rate, CO covered surface). The position of the hysteresis loop shifts to higher Y values and decreases in width with increasing temperature. For small CO content in the feed gas the CO2rate is proportional to Y3/2. At 500 K extremely slow Y cycling measurements (about 100 hours per direction) were done and showed that bistability still exists and no slowly changing transients were observed. The requirements for the speed with which experiments can be executed without producing experimental artifacts were explored. Since over-sampling alters the measured hysteresis loop, a maximum rate for changing the gas composition in Y cycling experiments was determined.The influence of noise on the reaction rates and the formation of spatio-temporal patterns on the surface was explored by superposing noise of Gaussian white type on Y and on T. Noisy Y (deviation Δ Y) represents multiplicative and additive noise, noisy T (deviation Δ T) multiplicative noise only. Noisy T enters the reaction via the rate-determining step, the observed CO2rates become noisy for low temperatures (around 420 K) when the surface is dominantly oxygen covered ( CO + O reaction step is rate-limiting) and for higher temperatures (around 500 K) when the surface is dominantly CO covered ( CO desorption step is rate-limiting).The effect of noisy Y was examined for a sample temperature of 500 K and is dependent on the selected average gas composition. In the regions with one steady state CO2rate (outside the hysteresis) the recorded rates were noisy. The probability distribution of the rates is Gaussian shaped for the upper rate (below hysteresis) and asymmetric for the lower rate (above hysteresis). For large noise strength bursts, short-time excursions to and above the upper rate, were observed.Inside the hysteresis small noise made the steady state rates noisy, but above a Y-dependent Δ Y transients from the locally stable to the globally stable rate branch were observed. These transients take up to several ten thousand seconds and become faster with increasing noise. For larger Y noise strength bursts and switching between both steady state rates were detected.Photoelectron emission microscopy (PEEM) was done to visualize spatio-temporal adsorbate patterns on the surface as expected from the observations in the CO2rate measurements. CO - and oxygen-covered regions on the Ir (111) surface are visible in PEEM images as gray and black areas as a consequence of their work function contrast. Islands of the adsorbate, corresponding to the globally stable branch, are formed in a background of the other one. The long transient times are the result of the extremely slow domain wall motion of these islands (around 0.05 μm s-1). In the hysteresis region CO oxidation on Iridium(111) surfaces is dominated by domain formation and wall motion for small to moderate noise strength. The island density increases with noise, but the wall velocity is independent of applied Δ Y. For larger noise amplitudes, fast switching between oxygen- and CO -dominated surfaces is observed as well as nucleation and growth of the minority phase in the majority phase.In the numerically analyzed reaction–diffusion system all parameters were taken from the experiment. Modeling the reaction–diffusion system shows qualitative up to quantitative agreement with the experimental observations. The length scale for the modeling grid is determined from wall velocity seen in the experiments.


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