TIME
7. Comparison with Data and Limits to the Analysis
We have in Figs. 3.9 and 3.10 a plausible explanation for the puzzling experi- mental observations cited at the beginning of this chapter. To a first approxi- mation, the logarithm of the global time constant, lnTexp• appears linear against 1/ Ts over a substantial range and this suggests that an activated law like (3.1) is responsible. When the observations are analyzed in this way the energy of activation is found to be correlated with, but less than, -µ0 , the equilibrium chemical potential. We can now understand this result in terms of the inherent nonlinearity of the equation of state for an interacting system, and in particular, the sign of its curvature in the vapor pressure-coverage plane. Because the iso- therm is nonlinear there is no single simple time constant but rather a succes- sion of local time constants as a function of time, each related to the slope of
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state. Such a time may be defined, however, as the total change in coverage necessary to reach steady state divided by the initial isothermal desorption rate - a quantity which is independent of n0-n55 only for independent particles obey- ing first-order kinetics. Changes in this time with desorption temperature, (3.73), then reflect not only the dependence of the local time scale, T1c 1(n0 ) , on temperature but also the variations brought about in the global steady state condition. For nonlinear isotherms of positive curvature these two factors com- pete against one another, (3.76), in determining the apparent activation energy in an Arrhenius parameterization, (3.1), of Texp· For the FHH model, E / jµ0
I
lies between 0.7 and 1 in the same temperature range as the experimentally quoted value of two-thirds, and T0 is about 10-11 sec versus the observed range of 10-9 - 10-10 sec, in qualitative but not necessarily quantitative agreement with the data.A more detailed comparison is presented in Fig. 3.14 which shows one set of time constant data from Sinvani et al. contrasted with the predictions of the FHH model under similar conditions. The agreement in absolute magnitude is good at the lowest substrate temperatures and remains within a factor of five or so out to the highest temperature point. The slope of the data and model calculations clearly differ, however, being about 0.6 in units of jµ0
I
for the former and about 0.8 for the latter (see the middle solid line of Fig. 3.10 ). This discrepancy should not be surprising in view of our assumptions: after all, the true equation of state of the film is unknown, the sticking coefficient may not be constant and equal to one as we have supposed, and our definition of the time constant, though appealingly simple, is still somewhat crude.That the data lie above the calculated curve, that is, that the observed time constants appear to be longer than the model's predictions, could be due to our
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token, the difference in slope might be attributable to a sticking coefficient which is a decreasing function of temperature. One must also bear in mind that, as mentioned at the conclusion of section 3, the isothermal approximation at t=O provides an upper limit to the maximum desorption rate of the exact solu- lions. The predictions for Terp based on this approximation may then be too small on account of both a tendency of the boundary layer method to overesti- mate the maximum slope of the exact desorption curve and the increasing failure of the film to desorb isothermally at higher substrate temperatures, as decoupling becomes more difficult. The likelihood of error in this direction is further reinforced by the capacity of the binding energy to grow without bound in the FHH model which, at high temperature, leads to an underestimate of the change in coverage between equilibrium and steady state.
Alternatively, the substrate temperature assignment in the experiments may be suspect at high heater powers. If it were necessary to shift the experi- mental points to lower substrate temperatures, they would then be in better accord with the calculated curve. The data also lie in a regime where the number of collisions among desorbing particles as they leave the surface is no longer small, and the significance of this fact with respect to the measurements is not easy to assess. These and other points of experimental detail will be discussed at greater length in the following chapter.
References
Bender, C. M. and Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers (MacGraw Hill, New York, 1978).
Bienfait, M. and Venables, J. A., Surf. Sci. 64, 425 ( 1977).
Dash, J. G., Films on Solid Surfaces (Academic Press, New York, 1975).
Libraries Inc., 1980).
Kruyer, S., K. ned. Akad. Wet. B58, 73 (1955).
Lin, C. C. and Segal, L. A., Mathematics Applied to Deterministic Problems in the Natural Sciences (MacMillan, New York, 1974).
Nagai, K., Shibanuma, T. and Hashimoto, M., Surf. Sci. 145, L459 (1984).
Opila, R. and Gomer, R., Surf. Sci. 112, 1 (1981).
Sinvani, M., Taborek, P. and Goodstein, D., Phys. Rev. Lett. 48, 1259 (1982).
Venables, J. A. and Bienfait, M., Surf. Sci. 61, 667 (1976).
1. Introduction
At various places in our exposition we have alluded, as necessary, to one or another aspect of the way in which the desorption time constant experiments are conducted. In the present chapter we elaborate considerably on these issues, describing the important points of experimental technique in detail, and draw attention to those features which, in part, suggested that a new series of meas- urements be undertaken. The major results of those new experiments are then presented along with a discussion and analysis within the framework of the ther- mal model.