4012
F. TSIENAND G.D. HALSEY,JR.
Simple Attractive-Disk Monolayer Isotherms with Phase Transitions
by F. Tsien and G. D. Halsey, Jr. Department of Chemistry, University of Washington, Seattle, Washington 98106
(Receiaed May 99, 1967)
Monolayer adsorption isotherms for the model of disks showing repulsion and inverse power attraction are discussed. The van der Waals (mobile) and Fowler (immobile) isotherms are put in compatible form. One adjustable parameter reflects the assignment of the reduced value of the van der Waals parameter a. Solid, liquid, and gaslike phases are identified and critical and triple points are calculated. The theory is applied to the data for krypton on exfoliated graphite, with reasonable success.
1. Introduction
Theoretical treatment for adsorption isotherms has been given in detail both for mobile and localized cases. Stebbins and Halseyl treated the phase transitions between the localized and mobile isotherms for hard disks on a structureless surface. We shall consider molecules here as disks with attractive force varying according to an inverse power law. The surface is assumed to be structureless to simplify the treatment.
scribed on a pairwise basis and in terms of the Sutherland potential
u(r) = -c*(r*/r)s U(T)
=
Q)
(r
> r*)
(r < r*)
We can then calculate a14where a = a‘/kT
2. Mobile Monolayers The two-dimensional analog of the van der Waals equation had been used to represent a mobile monol a ~ e r . ~It’ can ~ be expressed as (4
+ %)(A
- Nb)
=
NkT
(2.1)
where 4 is the spreading pressure, A is the area of the film, N is the number of molecules adsorbed, and a’ and b are parameters. b is sometimes referred to as the “co-area” per molecule, in analogy to the co-volume in the van der Waals equation. We shall choose b so that the area occupied by an adatom in the completed monolayer is b = 4/2(r*)2
(2.2)
where r* is the distance between the centers of atoms in closest array. If the adatom has no permanent dipole, the correction term a‘ is then the attraction which arises from the London dispersion forces. In calculating a’, we assume a uniform radial distribution function outside of r = r*, and zero inside. The molecular interaction will be deThe J W T of~Physical Chemistry
and
We shall, in the following calculations, assume the same bemperature dependence, but will express a as (2.7) where C is an adjustable parameter, and the reduced temperature
T* = kT/e* (2.8) With this relation for a/b, the two-dimensional equation of state eq 2.1 can be related to the adsorption isotherm (1) J. P. Stebbins and G . D. Halsey, Jr., J. Phys. Chem.,68, 3863 (1964). (2) T.L. Hill, Advun. Catalysis, 4, 211 (1952). (3) S. Ross and J. P. Olivier, “On Physical Adsorption,” Interscience Publishers, Inc., New York, N. Y.,1964. (4) T. L. Hill, J. Chem. Phys., 14, 441 (1946).
SIMPLE ATTRACTIVE-DISK MONOLAYER ISOTHERMS WITH PHASE TRANSITIONS
via the Gibbs adsorption equation which takes the form dt$ = lcTr d In p
which, after cancellation, becomes
(2.9)
K
where p is the gas pressure and l? is the surface concentration per unit area. Equations 2.1 and 2.9 give us the mobile isotherm In p = -In k,
e
e
+ In + -1-- 8 1-e
4013
2a -eb
=
($)/fzfv
=
A//[ A2
(2.10)
or
exp(-eE/2T) 1 - eXp(-&/T)
1’
where is the Einstein “characteristic temperature.” eE Will be approximated in the following fashion. The Lennard-Jones (6-12) potential is summed over the nearest neighbors in the hexagonal lattice and then differentiated twice with respect to r . This gives
(2.11)
(4.4)
where 0 is the fractional coverage. k, can be written as (2.12) where p0 is the standard chemical potential, x is the minimum energy required to evaporate an adsorbed atom from its lowest energy state in the monolayer, fi is the partition function for motion normal to the surface, and A =
h ( 2 ~ m T) k ‘Ie
(2.13)
3. Localized Monolayers We assume that the sites are fixed and that each adsorbed molecule interacts with its nearest neighbors only. Each laktice site has z nearest neighbor sites. The approximate adsorption isotherm for a random distribution as derived by Fowler and Guggenheims (referred to as the Fowler isotherm from now on), after rearranging and taking z = 6 for a triangular close packed two-dimensional lattice, has the form In p = - 1 n k ~
+ In
6
(4.3)
(4.5) where h
A* = T*
Ila
values for argon and krypton, calculated by this method, are 46 and 36”K, respectively.’ These values are close to the ones used by McAlpin and Pierotti,!j which are 45 and 34”K, respectively. K can now be expressed as BE
11 - exp(-
The adsorption isotherms can be expressed in a much simpler form if we define p* = p k ,
(4.8)
The mobile isotherm is then
where k F = fz.f,,fi
[exp(-x/kT)
(3.2)
and fi and f, are the partition function for motion parallel to the surface. Equation 3.1, combined with the Gibbs adsorpt,ion equation, yields
4. Corresponding States For convenience, we define K as6
( 5 ) R. H. Fowler and E. A. Guggenheim, “Statistical Thermodynamics,” Cambridge University Press, London, 1949. (6) We have used the same form of definition for K as by Stebbins and Halsey. However, K in their eq 31 and 32 as well as in Figure 4 should be replaced by 1 / K . (7) Values of e* and T* are taken from J. 0. Hirschfelder, C. F,; Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids, John Wiley and Sons, Inc., New York, N. Y., 1954. (8) J. J. McAlpin and R. A. Pierotti, J . C h m . Phys., 41, 68 (1984).
Volume 71, Number 12
November 1967
4014
F. TSIENAND G. D. HALSEY,JR.
and the Fowler isotherm becomes In p* = In K
In pB* = In p G *
",>
+ In (1-
6
-
-0
T*
(4.10)
and
(7.2) The expression (PB = (PG is equivalent to saying that the area BDEB is equal to the area EFGE. The spreading pressure can be calculated via the Cibbs adsorption equation (PB
5. Classical Case The classical law of corresponding states is correct when the mass of the molecule is sufficiently large, or when A* is negligibly small. In such a case, we can expand the exponentiation term e' = 1 x so that 1 e-' = x. The ratio K can then be approximated as
+
=
=
kTr
A*2 2
1
3
(5.1)
(PO
d(P = k T r d In p
4TT*
K=
(7.1)
(7.3)
b In p -de be
(7.4)
Integration of eq 7.4 gives
4 A* (7.5)
after expansion
when applied to the mobile isot.herm
*TT* K =
A*2
(T) 1
2
(I--Bs1
= 30/T*
(5.3)
so that our classical Fowler isotherm then becomes In p* = In 30
- In T* + In
"8)
-
(1
6 -0 T*
(5.4)
6. Critical Conditions Much work has been done on the two-dimensional condensation for the mobile isothermsa and the Fowler isotherm.s Designating the reduced critical temperature by T,* 4 T,*(mobile) = -C 27
(6.1)
T,*(Fowler) = 1.5
(6.2)
so that for T* lower than 4/27 C and 1.5 first-order phase changes occur for both the mobile and Fowler isotherms. Under these conditions, we will show that first-order phase changes occur between the mobile and Fowler is0therms.
7. Phase Transitions We will refer this section to Figure 1, and the subscripts A, B, C , etc., to the points A, B, C, etc. For the mobile isotherm ABCDEFGH, the criteria for phase transition are satisfied when the two phases have equal pressure, and dso when they have equal spreading pressure. Expressed in mathematical form, they are The J O U Tof ~P h y a h l Chmistry
$eB2)]
=
o
(7.6)
where ,'I is the maximum surface concentration as the pressure approaches infinity. The criteria for phase transitions are also satisfied between the mobile and Fowler isotherms. Taking the two hypothetical isotherms ABGH (mobile) and IJKO (Fowler), we vi7ill find that at points H and N In pH* = In px*
(7.7)
and (PH
=
(7.8)
4N
where, after integration
and (PN = r,kT[-ln
(1 -
e,> - -eN'] 3 T*
(7.10)
At sufficiently low coverage, the stable isotherm is the mobile isotherm, and at extremely high coverage, the stable isotherm is the Fowler isotherm. Therefore, the stable isotherm path must then be ABEGHNO. It can also be shown that, at any given p * , the stable isotherm path has a lower chemical potential than the other isotherm paths. We shall designate the expanded phase AB, the condensed phase GH, and the ordered phase NO as the two-dimensional gas phase, liquid phase, and solid phase, respectively.
SIMPLEATTRACTIVE-DISK MONOLAYER ISOTHERMS WITH PWSETRANSITIONS
Under certain conditions, when the mobile transition pressure is close to the Fowler transition pressure, there is also a phase transition between the unstable mobile isotherm and the stable Fowler isotherm. Thus the unstable two-dimensional gas-solid transition is represented by the path CM. When T* gets sufficiently low, another type of phase transition takes place. The equilibrium between the two isotherms takes place before the mobile transition. The stable isotherm path now follows PQLO which has the lower chemical potential. OL is then the twodimensional gas-solid transition.
0
0.2
4015
0.4
8
016
I 0
0:6
Figure 3. Plot of coverage a t which transition occurs as a function of reduced temperature for C = 4.2.
IT I"
Tr.39 T9.44
0.2
nl
I
$
-5
-4
-3
-2
-I*
0
I
2
3
Ln P Figure 4. Adsorption isotherm for reduced temperature of 0.35, 0.39, and 0.44. C = 4.2.
0 Figure 1. Phase transitions between mobile and Fowler isotherms, indicated by long dotted lines, as explained in text.
4 :O
C
6:O
Figure 5. Plot of reduced critical temperature and reduced triple point as a function of C. The ratio of triple point and critical temperature is also shown.
I
0.25
0.30
0.3,5
T
0.40
0.45
Figure 2. Two-dimensional reduced phase diagram for C = 4.2.
A typical phase diagram is shown for C = 4.2 (Figure 2). The corresponding coverage for which the phase transitions occur as a function of reduced temperature is shown in Figure 3. Some typical isotherms Volume 71, Number 1.8 November 1967
4016
are shown in Figure 4. For analysis of data, however, C is available as an adjustable parameter. It determines the values of the reduced triple point and critical temperatures, and thus the ratio of the unreduced temperatures. To show the dependence on C, T* of the critical point and of the triple point are plotted as a function of C, in Figure 5. The ratio of the critical temperature to the triple point is also shown.
8. Application to Data for Krypton on Graphite The recent data of Thomy and Duvalg resemble the isotherms of Figure 4 and appear to be suitable for analysis. Although these data do not quite reach a two-dimensional triple point at their lowest temperature of 77.4”K,the short vertical riser that appears to correspond to a “solid-liquid” equilibrium is almost on top of the longer “gas-liquid” riser at that temperature. A short extrapolation suggests a triple-point temperature of about 77°K. The observed critical temperature is about 86°K. The ratio of 0.9 (Figure 5 ) yields a value of C = 3.2 These results may be contrasted with the values for bulk krypton where the ratio of triple-point to critical temperature is 0.55 for a value of C = 4.1. From the reduced value of the critical temperature at C = 3.2,e*/k is calculated to be 180”. This result is close to the free gas pair interaction energy of 171”,’but rather far from the pair interaction calculated for the case of krypton on a graphite surface (144”).*
9. Discussion Both the mobile and immobile isotherms that are used in this analysis are of the lowest degree of approximation to the exact isotherms, and at least are consistent. The application of a Sutherland potential to the mobile case and a harmonic oscillator model to the solid is not consistent. At high temperature (T* = 30), eq 5.3 predicts a reversal in the stability of the low den-
The Journal of Physical Chemistry
F. TSIENAND G. D. HALSEY, JR.
sity mobile film for this reason. This temperature would be of the order of 5000°K for krypton and is thus not physically important. Reference to Figure 5 indicates that if C < 3 no liquid ever forms, and the transition is always from mobile gas to solid, at any value of T*. There are thus no critical or triple-point temperatures below this value. For C > 6, the liquid phase is stable down to T* = 0, and only the critical temperature remains. I t should be noted that if we use eq 2.3 to calculate C, a value of 1.8 is obtained. This value is below the range of interest, and this is the reason we have left C as an empirical parameter. A similar difficulty is encountered if the equation is used in three-dimensionaI form to estimate van der Waals constants. The existence of an unstable transition from an overcompressed gas to the solid (Figure 1) may have a bearing on the difficulty of observing the two-step gasliquid-solid condensation. If the equilibrium solidliquid region is typically as short as it is for krypton on graphite, the triple point could easily be lost if nucleation for the liquid is difficult. Although a true critical point would then be missing, a crude sampling of data would indicate that one existed. All that actually would happen is that a wide step would change into a narrow one just below 0 = 1. This phenomenon would occur near the lattice critical temperature, and the narrow step might be missed in the onset of secondlayer formation, and so one could be misled into seeing ‘(T,.” Note that aside from this possibility, a direct manifestation of a lattice critical temperature is not present in our results. In order to find such a temperature, one must explore the region where K in eq 5.3 is less than unity, and thus the underlying surface favors a lattice gas. (9) A. Thorny and X. Duval, Colloques Internationaux Du Center National de La Recherche Scientifique, No. 152, 1965.
