Sticking coefficient

Sticking coefficient is the term used in surface physics to describe the ratio of the number of adsorbate atoms (or molecules) that adsorb, or "stick", to a surface to the total number of atoms that impinge upon that surface during the same period of time.^[1] Sometimes the symbol S_c is used to denote this coefficient, and its value is between 1 (all impinging atoms stick) and 0 (no atoms stick). The coefficient is a function of surface temperature, surface coverage (θ) and structural details as well as the kinetic energy of the impinging particles.

Derivation

When arriving at a site of a surface, an adatom has three options. There is a probability that it will adsorb to the surface ( $P_{a}$ ), a probability that it will migrate to another site on the surface ( $P_m$ ), and a probability that it will desorb from the surface and return to the bulk gas ( $P_d$ ). For an empty site (θ=0) the sum of these three options is unity.

P_{a}+P_{m}+P_{d}=1

For a site already occupied by an adatom (θ>0), there is no probability of adsorbing, and so the probabilities sum as:

P_{d}'+P_{m}'=1

For the first site visited, the P of migrating overall is the P of migrating if the site is filled plus the P of migrating if the site is empty. The same is true for the P of desorption. The P of adsorption, however, does not exist for an already filled site.

P_{m1}=P_{m}(1-\theta )+P_{m}'(\theta )

P_{d1}=P_{d}(1-\theta )+P_{d}'(\theta )

P_{a1}=P_{a}(1-\theta )

The P of migrating from the second site is the P of migrating from the first site and then migrating from the second site, and so we multiply the two values.

P_{m2}=P_{m1}\times P_{m1}=P_{m1}^{2}

Thus the sticking probability ( $s_{c}$ ) is the P of sticking of the first site, plus the P of migrating from the first site and then sticking to the second site, plus the P of migrating from the second site and then sticking at the third site etc.

s=P_{a}(1-\theta )+P_{m1}P_{a}(1-\theta )+P_{m1}^{2}P_{a}(1-\theta )...

s=P_{a}(1-\theta )\sum _{n=0}^{\infty }P_{m1}^{n}

There is an identity we can make use of.

\sum _{n=0}^{\infty }x^{n}={\frac {1}{1-x}}\forall x<1

\therefore s=P_{a}(1-\theta ){\frac {1}{1-P_{m1}}}

The sticking coefficient when the coverage is zero $s_{0}$ can be obtained by simply setting $\theta =0$ . We also remember that

1-P_{m1}=P_{a}+P_{d}

s_{0}={\frac {P_{a}}{P_{a}+P_{d}}}

{\frac {s}{s_{0}}}={\frac {P_{a}(1-\theta )}{1-P_{m1}}}{\frac {P_{a}+P_{d}}{P_{a}}}

If we just look at the P of migration at the first site, we see that it is certainty minus all other possibilities.

P_{m}1=1-P_{d}(1-\theta )-P_{d}'(\theta )-P_{a}(1-\theta )

Using this result, and rearranging, we find:

{\frac {s}{s_{0}}}=[1+{\frac {P_{d}'\theta }{(P_{a}+P_{d})(1-\theta )}}]^{-1}

{\frac {s}{s_{0}}}=[1+{\frac {K\theta }{1-\theta }}]^{-1}

K{\overset {\underset {\mathrm {def} }{}}{=}}{\frac {P_{d}'}{P_{a}+P_{d}}}

References

↑ sticking coefficient IUPAC Compendium of Chemical Terminology 2nd Edition (1997), Accessed 30 September 2008

King-Ning Tu, James W. Mayer, and Leonard C. Feldman, in Electronic Thin Film Science for Electrical Engineers and Materials Scientists, Macmillan, New York, 1992, pp. 101–102.

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