Langmuir 2005, 21, 10871-10873
10871
How Many Langmuirs Are Required for Monolayer Formation? A. E. Bea, I. M. Irurzun, and E. E. Mola* Instituto de Investigaciones Fisicoquı´micas Teo´ ricas y Aplicadas (INIFTA), CONICET, Facultad de Ciencias Exactas Universidad Nacional de La Plata, Repu´ blica Argentina Received May 23, 2005. In Final Form: August 22, 2005 In the present work, we provide the exact answer to the title question employing a probabilistic approach. n (1/i), The average number of Langmuirs 〈L〉 required for monolayer formation was found to be equal to ∑i)1 i.e., the armonic series up to the nth term, where n is the number of adsorption sites. This result is particularly useful when a reduced number of adsorption sites is considered, such as adsorption on small terraces of nanoscopic dimensions where the value of n could be in the range of a few thousands sites. In this case, the use of integrated equations derived from the mean-field approach would provide completely misleading results.
1. Introduction The mean-field approach is usually employed in kinetic chemistry to derive differential equations governing the temporal evolution of adsorbed chemical species. This theoretical approach is based on the assumption that the number of molecules or adsorption sites is large; that is, this is valid on the mesoscopic scale. However, the development of experimental techniques capable of exploring nanometric scales, together with the increasing interest in the development of catalysts at the same scale, make it necessary to evaluate fractional coverages at the nanometer scale. Kinetic models developed in the meanfield approach have usually been used to evaluate variations in small surfaces of about 5 nm1 or even less. Though the mean-field approach is clearly unapplicable to such small surfaces, it is still possible to derive exact results employing a probabilistic approach.2 In the present paper, we determine the average number of Langmuirs and the time t to saturate a metal surface by a gas of molecules that require a single adsorption site. We assume irreversible adsorption on a two-dimensional lattice of an arbitrary symmetry (square, hexagonal, honeycomb, etc.) with n adsorption sites. From the kinetic theory of gases, it is known that the average number of collisions z per unit area and unit time is
z ) cN/4
(1)
where c ) (8kBT/πm)1/2 is the molecular average velocity and N is the number of gas molecules per unit volume. Therefore, cN/4n ) cp/4kBTn is the average number of Langmuirs per unit time, where p is the gas pressure and kB is Boltzmann’s constant. In ultrahigh vacuum experiments, it is very well accepted that in order to work under clean surface conditions the time required for the experiment should be much less than the time t required for monolayer completion. This time t is assumed to be of the order of * To whom correspondence should be addressed. e-mail address:
[email protected]; Fax: +54 221 4254642 (1) Sachs, C.; Hildebrand, M.; Volkening, S.; Wintterlin, J.; Ertl, G. Science 2001, 293, 1635. (2) Irurzun, I. M.; Ranea, A. V.; Mola, E. E. Chem. Phys. Lett. In press, 2005.
t∼
4kBTn pc
(2)
On the other hand, in the mean field approach, the rate of adsorption coverage is written as
dθ ) k(1 - θ)p dt
(3)
where k ) c/4kBTgn ) [(2πmkBTg)1/2 n]-1 is the flux of gas molecules in units of monolayers (ML) per second, m is the gas molecule mass, and Tg is the gas-phase temperature. If we assume that n ) 1015 sites/cm2, m ) 5 × 10-23 g and Tg ) 300 K, then k ) 2.78 × 105 mbar-1 s-1 ML-1, and by integrating the above equation, we found that
ln(1 - θ) ln(1 - θ) ) -[(2πmkBTg)1/2 n] (4) t)kp p and therefore a complete monolayer is obtained at t f ∞ and requieres an average number of Langmuirs 〈L〉 f ∞. This is of course a consequence of the mean field requirement to consider the coverage as a continuous variable. 2. Exact Mumber of Langmuirs Required for Monolayer Formation The purpose of the present work is to calculate the exact number of Langmuirs required for monolayer formation. In the sequential process of filling a surface, there are n equivalent possibilities to place the first particle (molecule) on the lattice and we shall call them configuration 1, C(1). For the present purposes there is an equivalence between occupied and unoccupied lattice sites. There are n - 1 possibilities to place the second one and to arrive at configuration 2, C(2), and one possibility to remain in C(1). In general, there will be n - 2, n - 3, n - (i - 1), ..., 1 possibilities to place the third, the fourth, ..., the ith, ..., the nth molecule, respectively and to arrive at C(3), C(4), ..., C(i), ..., C(n)*. And consequently, there will be 2, 3, ..., i - 1, ..., n - 1 possibilities to remain in C(2), C(3), ..., C(i - 1), ..., C(n - 1). The asterisk identifies the final configuration, that is, when the surface is completely saturated.
10.1021/la051368i CCC: $30.25 © 2005 American Chemical Society Published on Web 09/30/2005
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Langmuir, Vol. 21, No. 23, 2005
Bea et al. n
1
≈ ∫1 ∑ i)1 i Figure 1. Chain of configurations from a clean surface, C(0), up to a saturated surface, C(n)*. The probability to remain in a given configuration and to arrive at the next one are also indicated.
Figure 1 shows the sequential adsorption process as a chain of configurations where the rings are linked by the probability p(i f i + 1) to arrive at C(i + 1) from C(i). The figure also shows the probability p(i f i) to remain in the same configuration. p(i f i + 1) and p(i f i) are written as
p(i f i + 1) )
n-i n
i p(i f i) ) n
(6)
mn-2 ) 1 +
n-2 2 mn-2 + mn-1 n n
(8)
mn-3 ) 1 +
n-3 3 mn-3 + mn-2 n n
(9)
n-i i mn-i + mn-i+1 n n
(10)
and in general
n + mn-i+1 i
1
∑ i)1 i
(11)
If the average number of adsorption attempts 〈m〉 from C(0) to C(n)* is normalized by the number of adsorption lattice sites n, we obtain the average number of Langmuirs 〈L〉 ) 〈m〉/n for lattice saturation n
〈L〉 )
1
∑ i)1 i
(14)
4kBTg n ln n pc
(15)
(2πmkBTg)1/2 n ln n tML ≈ p
(16)
From this equation, we conclude that the time required for monolayer formation does not depend only on the pressure p, the temperature Tg, mass molecule m, and (linearly) the number of adsorption sites as is usually believed, but it depends in a much more complex form on the number of adsorption sites; that is, it is proportional to
tML ∼ n ln n
(17)
The unexpected factor ln n plays the role of some kind of magnification factor. We can now ask ourselves what the coverage θ(tML) (from the mean-field point of view) at the exact time tML for monolayer formation is. We can answer this question by equating the right-hand side of eqs 4 and 16. The following result is obtained:
(18)
3. Conclusions
The boundary condition of this recursive equation is mn ) 0. Therefore mn-1 ) n, mn-2 ) n/2 + n, mn-3 ) n/3 + n/2 + n ..., m0 ) n/n + n/(n - 1) + n/(n - 2) + n/(n - 3) + ... + n/3 + n/2 + n. The average number of adsorption attempts 〈m〉 ) m0 from C(0) (the clean lattice) until lattice saturation C(n)* will then be n
(13)
In Table 1 the goodness of the approximation given by eq 14 can be compared with the exact result, eq 12. The average time tML for monolayer formation is, from eq 2 and eq 14
θ(tML) ) 1 - n-1
eq 10 can be rewritten as follows
〈m〉 ) m0 ) n
) ln n
or
(7)
mn-i )
i
〈L〉 ≈ ln n
(5)
n-1 mn-1 n
mn-i ) 1 +
di
Therefore, the average number of Langmuirs to saturate a lattice of any symmetry is an extensive quantity given by the following equation:
tML ≈
The average number of adsorption attempts, mi, from the configuration C(i) until the saturated configuration, C(n)*, can be written as
mn-1 ) 1 +
n
(12)
If the number of adsorption sites is very large, say of the order of 1015 per cm2, then the summation can be approached as follows:
For practical purposes the mean field equation can be used to calculate tML assuming in advance a θ value (
