Sticking Coefficients - Advances in Chemistry (ACS Publications)

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Sticking Coefficients FRANK M. WANLASS and HENRY EYRING University of Utah, Salt Lake City, Utah

Absolute reaction rate theory starts with the equilibrium concept for the transition state and then corrects the calculated equilibrium rate of change to nonequilibrium conditions by multiply­ Downloaded by UNIV QUEENSLAND on September 18, 2013 | http://pubs.acs.org Publication Date: June 1, 1961 | doi: 10.1021/ba-1961-0033.ch016

ing by a transmission coefficient, Ξ. For the stick­ ing of molecules to a surface, the sticking coeffi­ cient is just this transmission coefficient.

For the

reverse process, of sublimation or of evaporation, the transmission coefficient is likewise required to correct the calculated equilibrium rate which is in general equal to the sticking coefficient.

If

one regards a condensed phase as a giant mole­ cule, then evaporation or sublimation is a special kind of unimolecular reaction so that understand­ ing such processes illuminates the entire field of unimolecular reactions.

By taking account of the

fact that the evaporation of physically adsorbed nitrogen molecules on tungsten is competitive with passage to the chemisorbed atomic state, the sticking coefficient can be explained.

With in­

creased surface coverage, the nitrogen molecules are more loosely held with a corresponding ex­ ponential increase in the rate of evaporation. Thus an interesting beginning has been made in understanding of rates of surface adsorption.

por reactions such as H

2

+ I -^2HI 2

(1)

the specific rate at which product is formed at equilibrium is very closely the specific reaction rate in the absence of products. Thus in the usual expression for the specific reaction rate constant k

' = v JL k

140

In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.

(2)

WANLASS AND ΕΥ RING

141

Sticking Coefficients

the transmission coefficient, H, which corrects the equilibrium specific rate to nonequilibrium conditions is almost unity. On the other hand two hydrogen atoms, colliding in the absence of a third body, are stabilized to form a molecule only in the extremely rare circumstance that quadrupole radiation is emitted. Since about every eighth collision between three hydrogen atoms (3) at atmospheric pressure yields a stable molecule, a three-body collision is the important method for homogeneous recombination. The only factors difficult to estimate fairly accurately in Equation 2 are the activation energy, E , which occurs in K 9 , and the transmission coefficient, E. Thus in the recombination of atoms or radicals where E ^ 0 the only serious uncertainty is in Ξ. Similarly, the adsorption of molecules from a surface usually involves little or no activation energy, so that again the only uncertainty is in the sticking coefficient, S. S is just the transmission coefficient, E, for the adsorption process. e

0

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0

Understanding the sticking coefficient is not only important for dealing with adsorption; it is also fundamental to a proper understanding of the general theory of all association reactions. Pooh at Equilibrium When Equation 2 is written in the equivalent form (3) the quantity in parenthesis refers to the activated state and is the partition function for the initial state. When the activated complex and the initial state are separated in space, they may be subjected to different conditions. Thus in homogeneous diffusion or passage across a phase boundary as in sublimation or vaporization, the activated complex and reactants may be environments with dif­ ferent temperatures, different concentrations, different dielectric constants, etc. Thus the question arises as to whether the reactant continually equilibrates with its surroundings or whether the activated complex still retains the properties it had at the last potential minimum even as it reaches and passes through the transi­ tion state. The vaporization of a pure liquid or the reverse process, the condensation of the liquid, provides an interesting test of this delayed equilibration hypothesis. Thus as a hydrogen-bonded molecule, which vibrates in the liquid, separates from the surface it frees itself from the potential energy restrictions which prevented rotation. However, in so far as the evaporating molecule has insufficient collisions with neighbors to equilibrate to the free rotational partition function, f , of the gas, it will retain substantially the partition function, f of the condensed phase even in the activated complex. Consequently for the condensation process the usual g

h

formula for molecules colliding,

. ^ :, per second per square centimeter must V 2-7Γ mkT be multiplied by the factor fi/f to give the rate of sticking to the surface. Thus fi/fg is the sticking coefficient, S, for liquids (5) in agreement with the available experimental evidence. Atoms, which have only translational degrees of freedom, do not necessarily stick on each collision with a surface and only every eighth triple collision between three hydrogen atoms redistributes energy sufficiently to yield a molecule (3). Delayed nucleation on a solid surface or in a solution is another example where sticking of atoms may be incomplete. Delayed equilibrium of energy in the coordinate joining a nitrogen molecule to a surface may thus decrease sticking by another factor in addition to that for rotation. g

In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.

142

ADVANCES I N CHEMISTRY SERIES

Sticking Probability for N on Bare Tungsten as a Function of Temperature 2

Figure 1 shows the results of two experiments (2, 4) for the temperaturedependent sticking probability, S, of N on a clean tungsten surface. Curve A is drawn through the experimental points obtained by Ehrlich; curve Β is drawn through Kisliuk's data. Ehrlich claims his data, in addition to giving temperature dependence, is correct to within 5% in magnitude. Kisliuk's data are proportional only to S, thereby giving only the temperature dependence. It is important to note in Figure 1 that both curves show a decrease with tem­ perature, and it should be possible to fît Β smoothly onto A by multiplying by a suitable scale factor, possibly as shown by the dashed line. To explain the data shown in Figure 1 the temperature dependence of fs/f is needed. The rotational partition function for a diatomic molecule that is free is 2

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ff

2IkT y

~ σ*«

The rotational partition function for an N molecule bound to a tungsten surface is more in doubt, but it would probably have very little temperature dependence if the molecule could turn or vibrate on the surface only with difficulty. 2

S

.5 4 .3 .2 .1 0

J

I

200

Figure 1.

I

I

400

I

I

600

L

800

1000

1200

Comparison of data on sticking probability, S, of nitrogen on tungsten

It, therefore, might be reasonable for S = const 1 / Γ . At least S now quali­ tatively decreases with temperature, as indicated in Figure 1. Further investiga­ tion is proceeding to see how well the data shown in Figure 1 can be fitted quanti­ tatively and what account should be taken of slowness of translational degrees of freedom of the nitrogen to equilibrate.

Temperature Coefficient and Surface Coverage Dependence of Sticking Coefficient of N on Tungsten 2

When a nitrogen molecule hits a tungsten surface, there will be some probability that it will be physically adsorbed. We denote this by S and expect it will be essentially equal to the ratio of the adsorbed to the gas rotational parti­ tion functions of nitrogen. After the molecule is physically adsorbed, it will have to diffuse thermally to a strong binding site where it can stay permanently and thus be considered chemically bound. The rate of finding a particular type of 0

In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.

WAN LASS AND EYRING

143

Sticking Coefficients

permanent site should be k±

^1 — -^, where σο is the number of permanent σ

sites per square centimeter of surface and σ is the number already filled. There is the competing probability, k , of a molecule's leaving the surface before it is chemically bound. The sticking probability will, therefore, be 2

ki 5

=

_

Λ

( l

Δ

-

_££/_ +

Rates ^ and k will both be of the form e- ^ and ve- . We are going to assume that the desorption energy depends on surface coverage in the manner E = E — ota, as Becker (J) found for cesium atoms on tungsten. Rewriting Equation 4 so that the dependence on surface coverage is plain, we get ο (5) be* * 1 + — — 2

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(4)

*2

2

v

E

kT

E2/kT

20

1

S and h will both have temperature dependence, but we will not worry about checking their temperature dependence here. If b/σο is small enough, S should be the ordinate intercept of the curves in Figure 2; therefore S should be easy to estimate for each curve. 0

0

0

^T«243°K

12

20 6 0 100 140 180 220 240 *I0"

Figure 2. Temperature coefficient and surface coverage dependence of sticking coefficient of nitrogen on tungsten In the following, Equation 5 will be fitted to the 2 9 8 ° Κ curve in Figure 2 For this curve we take S = 0.286. Equation 5 can be rewritten as 0

be σο

a

So - S

a

—

σ

0.286 - S

S

(6)

S

There are three parameters, b, a, and σ , to determine in this equation; therefore three points on the experimental curve will be needed. When the coordinates of three experimental points are put into Equation 6 three times, one obtains 0

26 be^* 260 σ - 117

b e ^

σ - 72 0

0

133.

b e

u

* «

153 σ - 145 0

243 43

(7)

Eliminating b and a between the previous equations, one obtains *ba

e

=

8.69

\ ση — 12 J

p73« = 56.5

- 145\ \ σο —72 )

ίσο

im 12)

In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.

(8) (9)

144

ADVANCES IN CHEMISTRY SERIES

It is found that σο = 262 satisfies Equation 9, and using this value from Equation 8 we get a — 0.042, and from Equation 7 we get h — 0.926. Therefore, Equation 5 becomes S

1

=

So ~

0.926* ·

.

Λ

1

0

0420

W ^ T

+

From Equation 10 it is justifiable that we assumed b/a

0°) 0

found to have the value

to be small, since it was

· Now we can write Equation 5 as

In (σο — σ) + In ^°

^ — In b + ασ

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ο or 0 oo/: In (262 -

σ) + In —

ο

(11)

= \n b +