We will see that both the special boundary conditions for the experiments. i.e. the presence of the surrounding gas) and the non-linear properties of. In Chapter 4, we review the experimental techniques with particular attention to the circumstances necessary for a proper evaluation of the data.
Detailed Balance
Collisions Between Atoms
This difference in scattering probabilities stems from the fact that there is an exponentially greater number of states available to the wall if it absorbs energy instead of losing it, i.e. the wall acts as a heating bath. As a consequence of the detailed balance relationship, a gas in equilibrium with a surface maintains an equilibrium distribution of energies.
Evaporation from a Liquid
The average energy removed by a liquid per evaporating atom is given in terms of the energies of the free particles, dp), by. As before, the average energies introduced above can be expressed as averages of the holding probability with respect to equilibrium and incident fluxes.
Adsorption and Desorption
Comsa, David, and Rendulic (1977)] then studied the angular dependence of the average kinetic energy for the Dz molecule desorbing from Ni. The discussion is focused only on the stationary properties of the distribution and the attachment probabilities.
Thermodynamic Systems in External Fields
P is the vapor pressure of the film and P 0 the saturated vapor pressure of the bulk phase. We will always be concerned about deviations from equilibrium caused by a sudden heating of the substrate on which the film rests.
Introduction
The model predictions compare remarkably well with the results of both types of experiment. The model we present in this paper provides a qualitative and semi-quantitative understanding of the results of these experiments.
The macroscopic continuum model 1. Basic equations
The film temperature approaches that of the heater with very little energy escaping into the gas. Energy can be stored either in raising the temperature of the film (represented by CN) or in heat of desorption (CL).
Results
So if we evaluate F(t) at t = tP, we should get a result proportional to the maximum height of the bolometer signal for each mode. Most of the heat returns to the crystal, creating a signal more similar to the vacuum result.
Discussion
Nevertheless, the gross dependence expressed by (49) roughly corresponds to the time constant T 1 of the model. For all the numerical results cited, we took as the number density in the film (28] (cf.
P 9 (Torr)
Boundary Layer Theory of the Nonlinear Equations
We further note that our choice of time scale, t*, also differs from that used in the linear case and can be written using (3.35a) as. Neglecting the terms responsible for the coupling of temperature and film thickness changes, as we did in the linear case, the lowest order equations become for the outer region. Changes in film thickness are coupled to changes in temperature through the latent heat term in the energy equation.
Assuming that quasi-equilibrium is maintained, the basic equation for isothermal desorption in the presence of ambient gas is Our notations for the surface coverage in the above formula differ from our previous emphasis on the film thickness, o(t), and our previous use of the symbol n to denote the density of the bulk fluid to emphasize the complete generality of (3.46) and, in particular, its independence from the bulk continuum point of view or any other assumptions about the film's equation of state.).
SURFACE COVERAGE
Local Linearization of the Rate Equation
In terms of the parameterization (3.62), it is the slope of the adsorption isotherm that determines the local time scale that governs the approach to the steady state. From this construction it is evident that (n;s-ni) will be less than (n55-ni) for isotherms of positive curvature, the linearized estimate of the total amount to be volatilized is less than the true value. Obviously, the simple geometric interpretation in terms of the isotherm (Fig. 3.3) is no longer adequate.
The FHH isotherm must asymptote at high coverage to the total vapor pressure P 0(T), so the curvature of the isotherm must be negative at high coverage. Furthermore, there is no simple connection between (3.74b) and the equilibrium value of the chemical potential, -µJ(T 0,n0.
TIME
Comparison with Data and Limits to the Analysis
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 produced in the global steady state. However, the slope of the data and model calculations clearly differs, being approx. 0.6 in units of jµ0I for the former and approx. 0.8 for the latter (see the middle solid line in Fig. 3.10. The likelihood of error in this direction is further enhanced by the capacity of the binding energy to grow unbound in the FHH model, which at high temperature leads to an underestimation of the change in coverage between equilibrium and steady state.
At various places in our presentation we have, where necessary, alluded to some aspect of the manner in which desorption time constant experiments are carried out. The main results of these new experiments are then presented together with a discussion and analysis within the thermal model framework.
Apparatus and Techniques
A schematic representation of the experimental arrangement inside the cell has already been shown in the inset of the figure. This result is one third of the corresponding value for a quiescent gas of the same density. The scale factor in the argument of the exponent has changed from (3m I r-r' I 2 to (3mz2, which differ unless the detector happens to be directly above the point in question and the algebraic drop at long times has shifted) to the gentler (t - t')-3 behavior.
The actual voltage generated is obtained by multiplying S (t) by Ad, the area of the detector in cm2, and by K, the unknown sensitivity of the bolometer. The reason this is useful is that the propagators are just functions of the difference, t-t'.
TIME (µ.sec)
They continue to grow and make up a significant portion of the total surface under each waveform. If the rate of this change is long (but not too long) compared to the width of the propagator, then by the time tp becomes comparable to this width, all fast and easily desorbed particles have already contributed to the signal. If the desorption time rate were to increase further, then the ratio of peak to tail amplitude would shift in favor of the latter until, in the limit of a truly constant desorption rate, the signal would go.
This explains the jump in the signal at t=tp as well as the initial shift of all the spectra at t=O despite their return to the prepulse baseline at long times. The logarithm of the rate of change of surface coverage with time (the net desorption rate) is illustrated in Fig.
It would be attractive to be able to quote these results as functions of surface coverage rather than as functions of time, and to be able to reconstruct the adsorption isotherm (with assumptions about the adhesion coefficient) from them. The asymmetric error bars reflect conservative estimates of the uncertainty introduced by the difficulties we have alluded to regarding the saturation zone, numerical differentiation, and non-isothermal behavior at very short times. The solid line gives the corresponding isothermal prediction of the detail-balance model, with a=l and the FHH equation of state, for the same ambient conditions.
A closer examination of the data at very low power shows that the agreement with theory is remarkably good, perhaps better than we might expect given our choice of state equation. We now turn our attention to the properties of the time-of-flight spectrum as a function of the pulse repetition period.
R E PETITION PERIOD ( msec)
Given the import of our explanation for the shape of the replenishment curve versus tr at low power, our -17 db data now seems puzzling. For example, starting at the equilibrium coverage n0, a pulse of width tp will take us to the point n (tp) = n1 on the Ts isotherm before dropping back to the To isotherm where film reaccumulation begins. During periods of desorption this quantity is proportional to the height of the Ts isotherm above the horizontal line at P/.Ts.nss) (as illustrated for the specific point n2) and, conversely, during periods of readsorption it is the distance from the To isotherm with respect to the horizontal line at P J(T 0,n0 ) that is relevant (as shown for the point ni).
Due to the positive curvature of the isotherm, the actual vapor pressure at n0, P J(Ts,n0) exceeds the scaled value [(n0-nss)/(n2-nss)]PJ(Ts.n2). As a consequence, the initial desorption rate (ie, the distance to the line at PJ(Ts,nss)) in the trc experiment exceeds that of the scaled tr experiment.
PULSE WIDTH (µsec)
The original motivation here was to study the systematics of the time constant under different initial conditions of the film as well as different desorption temperatures. While the local time constant certainly changes according to variations of the chemical potential of the film with coverage as indicated in 3.68 ), there is no corresponding simple prediction for the global time constant. The situations we have previously discussed correspond to one of two simple limits: the time constant experiment), or tp/Ts»l, so that.
The whole situation, summarized by , is analogous to what happens when we periodically switch a capacitor between two parallel branches of an electric circuit, one containing a battery emf V 0 = n0-nss with a series resistance R 0 , and the other containing a resistance value Rs< Of the various time constants we have discussed, each has its own domain of applicability and relates to specific experimental circumstances. Is the behavior of the sticking coefficient different, by orders of magnitude, from what we assumed. In retrospect, it seems that we would have obtained a more penetrating test of our assumptions if we had simultaneously studied the angular dependence of the desorption rate. Both determinations would depend on the quasi-equilibrium assumption, but they would be independent of the equation of state if the sticking coefficient varied only insignificantly with coverage. To the extent that accurate values for these quantities can be combined with comparable knowledge of the sticking coefficient, we can expect to make realistic predictions.REPETITION PERIOD ( msec)
![Fig. 9. (a) Comparison of predicted reflection coefficient with data of Dietsche and Kinder [ 15] at T = 1.04 K](https://thumb-ap.123doks.com/thumbv2/123dok/10412701.0/61.922.245.554.294.786/fig-comparison-predicted-reflection-coefficient-data-dietsche-kinder.webp)
