The Journal of
Physical Chemistry
0 Copyright, 2982, by the American Chemical Society
VOLUME 86, NUMBER 21
OCTOBER 14, 1982
LETTERS Nucleation by Supersaturated Partially Wetting Films T. J. Sluckln Faculty of Mathematical Studies, University of Southampton, Southampton SO9 5NH, United Kingdom (Received: June 17, 1982; I n Final Form: August 13, 1982)
Dash has classified film growth into three types. Types I and I1 correspond to total and partial wetting, respectively. In the type I1 regime, it is shown that there is a thermodynamiclimit on the thickness of an adsorbed film under supersaturation conditions. It is believed that this limit corresponds to wall-induced nucleation. where AT = T I ,- T , TI, is the type 11-type I transition The limit occurs at a critical overpressure Ap 0: temperature, and T is the ambient temperature.
(An2,
In this Letter I shall be concerned with the process of nucleation of a liquid in a slightly supersaturated gas. It is well-known that this nucleation process is radically affected by whether or not the vapor wets the walls of the container in which it is placed. If it does, a liquid film is formed on the walls of the container at pressures slightly less than the saturation vapor pressure, and, as the saturation vapor pressure is approached, the liquid film grows to macroscopic dimensions. In this way the nucleation problem is avoided and superheating cannot take place. If the vapor does not wet the wall, however, nucleation of the liquid is governed by the possibility of forming critical droplets; this may take place spontaneously or with the aid of impurities which act as nucleation centers. In either case the nucleation is governed by the size of the liquidvapor surface tension, and a finite amount of superheating may take place. There has recently been renewed interest in microscopic theories of adsorption and wetting. Dash2 has pointed out (1)See, e.g., J. J. Burton in ’Modern Theoretical Chemistry”, B. J. Berne, Ed., Plenum Press, New York, 1977. (2) J. G.Dash, Phys. Rev. B , 15, 3136 (1977).
that there can be three types of film growth near a surface near saturated vapor pressure. Type I film growth corresponds to complete wetting of the surface, and continuous film growth to macroscopic dimensions as the saturation vapor pressure is approached, as described above. Type I1 film growth involves the growth of a finitely thick film as saturation is approached, and corresponds to partial wetting of the surface by the vapor; the liquid-vapor interface will intersect the wall at some finite contact angle in this case. Type I11 film growth corresponds to the wall not being wet by the vapor; in this case there is no adsorption and the vapor does not wet the wall. Cahn3 and Sullivan4 have examined film growth from a microscopic viewpoint. In both works a transition between type I1 and type I film growth is predicted; the high-temperature phase is a total wetting phase. However, Cahn predicts that the transition will be first order, whereas Sullivan predicts a second-order transition. It is not clear to what extent these conclusions are model dependent; Cahn does not take account of interactions be(3)J. W.Cahn, J. Chem. Phys., 66,3667 (1977). (4)D.E.Sullivan, Phys. Reu. B , 20, 3991 (1979).
0022-365418212088-4089$01.2510 0 1982 American Chemical Society
4090
The Journal of Physical Chemistty, Vol. 86, No. 21, 1982
tween the adsorbate and the wall, whereas Sullivan, in order to consider a soluble model, dictates that the range of the wall-particle and particle-particle interactions be the same. Further work on this problem is needed; however, it seems likely that the nature of the transition is affected by the relative range of interaction of the walladsorbate and adsorbate-adsorbate interaction^.^ In this Letter I consider a situation in which the type 11-type I transition is second order. I shall consider film growth in the type I1 region, rather near the type I-type I1 transition. In this regime, as the saturation vapor pressure is approached from below, the adsorbed film grows to reach large but not macroscopic dimensions. As the transition is approached, the film thickness at saturation grows, finally reaching macroscopic dimensions at the transition. I now consider film growth in the supersaturated regime, still staying in the type I1 region. The adsorbed film grows further, but eventually is no longer stable at some finite overpressure. At this overpressure we expect wall-induced nucleation to set in. The further one is from the type I-type I1 transition, the greater that overpressure will be,at low temperature it seems likely that conventional nucleation mechanisms will once again be dominant. I now quantify these ideas, using Sullivan’s model of adsorption. I believe that the physical content of the results is not model dependent: they depend only on the wetting transition being second order. In this model the total grand thermodynamic potential fl is given by Q = Jdr
[fh[p(r)l
+ Y2Jw(r - r’) p(r) dr’) dr’ +
Figure 1. Function $(cLh*) for a slightly undersaturated vapor. The
dashed line is the function I(&*). The curves cross at ph*(0)< ph*, corresponding to class I1 film formation.
ph
[ ~ r- p) ~ r ) (1) ]
where is the local free energy density of a hard-sphere fluid of density p, w(r - r’) is the attractive part of the interparticle potential, U(r) is an external potential, and p is the ambient chemical potential. The interparticle potential is given by w(r) = -(a/4?rro3)(r0/r) exp[-(r/ro)l
Letters
(2)
Figure 2. Function $‘(p,,’) at critical overpressure. The curves I@,,*) and $(p,,*) just touch: at greater pressures they will cease to do so. However, ph(0) < &,+; the film remains in class I I and has reached its maximum thickness.
and ph*(o) is given by the crossing point Of $&,) where I(ph)
=
(ph
-p -2%~)~
and I(&,) (8)
The wall is at z = 0 in the x-y plane, and the wall-particle potential is given by
For type I1 and type I films the lower crossing point must be taken. The type 11-type I transition corresponds to
W )= -ew exp[-(z/r0)l
2 e w / a = Pl(T12)
(3)
Sullivan defines the hard-sphere chemical potential phb)
= afh(p)/ap
(4)
and the hard-sphere pressure Ph(p) defined implicitly by aph(P)/aphb)
of
=p
(5)
It is important to note that &(p) is a monotonic function p. The density profile p ( z ) is implicitly defined by
(9)
where p1 is the liquid density. Now $(&*) has a minimum at ph* = &+,and at TI2,&+ = &,*(o). A graphical construction for ph*(o) in the case of type I1 films is shown in Figure 1. In general, if p < ps, the saturation vapor pressure chemical potential, $(&,+) > 0, whereas, if 1.1 > ps, $’(&+) < 0. Let us now grow the film with p > pB,p > p 8 ,but still with boundary conditions at z = appropriate to the vapor phase. The boundary condition for ph*(O) can still and #(ph*) still cross. For a be satisfied so long as I(&*) high enough overpressure this will no longer happen (see Figure 2). The critical condition will be that the curves I(&,*) and $’(&,*) just touch. The condition that this be the case is I(ph*)
= $‘(bh*)
(10)
(5) P. Taramna and R. Evans, Mol. Phys., submitted for publication. (6) For simple fluids the hard-sphere equation of state is well described by the Carnahan-Starling equation P h b ) = pkBT(1
+ v + 7’ - V? /(I - v ) ~
where r ) = 77p03/6, and u is the hard-sphere diameter. Within this model, Sullivan has calculated that a/keT$ = 11.102.
(11)
After a little simplification these conditions may be written as 2cw[ew - p h *
+ p1 =
- ph*l
(12)
The Journal of Physical Chemistty, Vol. 86, No. 21, 1982 4091
Letters
2tw = ap*
(13)
At temperature T , we are in the type I1 regime and E,
< tc(T) = b/2)Pl(T)
(14)
where e c ( T ) is that value of t, required a t temperature T to cause the type I-type I1 transition to take place at temperature T. Then let e, = d T ) + Ae,(T) (15) Now at temperature T12= T - AT, the value tW gives rise to the type I-type I1 transition. Then e, = (a/2)PI(T12) (16) Expanding to first order in A T
which must be supplemented by condition 13, which may be rewritten by using eq 22c as 2At = aAp* = a(ap*/aPh*)APh* (29) Equations 28 and 29 may now be combined in order to express Ap as a function of At. It is more transparent to present the results as a result for the critical overpressure Ap as a function of AT, the temperature drop from the type 11-type I transition: Ap and A T are then both experimental quantities. If we note that the hard-sphere compressibility Xh is defined by
it is relatively straightforward to show from eq 28 and 29 that - 1
and combining eq 14-17 yields (18)
= -(a/2)[dpi/aTIAT
where [ap,/aTI is taken along the coexistence curve. Now define the critical overpressure A p by AP=P-Ps
(19)
Acl = P - PS
(20)
where ps is the saturation vapor pressure, and for small Ap, Ap, the Gibbs-Duhem relation yields AP/& = P&T) (21) where p,(T) is the gas density at temperature T. Similarly we may define A h * = Ph* - Ph+
(224
APh* = Ph* - Ph+
(2%)
&h* = Ph* - PI
(22c)
= Pi
(23)
AP*/&*
where &,+ is defined for t = tc. Now if we define F(c,ph*,P) = '/Zz[I= 2e[c - ph*
+ P] + a(&,*
-p)
(24)
the osculation conditions 10 and 11 reduce to
F=0
1
(25)
dF/aph* = 0
Now F(ec,ph+,ps) = (aF/allh*)(tc,Ph+,Ps) = 0
(26a)
= (aF/ae) (tc,ph+&Ls)
(26b)
because at TI, the osculation conditions are also fulfilled. Similarly from eq 25 above F(b#h*#) = 0 (27) Expanding eq 27 in lowest-order powers of At, A&*, Ap around eq 26a yields
This is the principal result in this Letter. There are a number of points which should be made in connection with this result. It is not clear to what extent the result is dependent on the particular model of adsorption used. It may also be that the instability of the thermodynamic equations for the film profile, which takes place in the metastable gas regime, is not in fact physical and therefore does not have the nucleation-induced properties that we ascribe to it. However, as we have pointed out, there are physically intuitive reasons for supposing that nucleation near a wall is more likely near the wetting transition. An order of magnitude estimate of the size of the predicted effect from eq 31 gives Ap/(An2
-
1-10 atm (deg K)2
(32)
This is rather a large effect; however, observation is aided by the quadratic dependence of Ap on AT. The result, of course, is not unique to liquid nucleation from a gas and indeed generalizes to any type of new phase nucleation near a wetting transition. It seems possible that an analogous phenomenon has already been observed. It is known that liquid 4He is class I1 against a number of substrates with respect to the growth of solid 4He films. Balibar et al.' have observed that solid 4He nucleation takes place at a finite, small, and reproducible overpressure. While the analysis of this letter cannot directly be applied to their experiments-at the relevant temperatures helium exhibits strong quantum effects and the solidliquid coexistence curve is more or less independent of temperature-we may expect that similar qualitative considerations apply. The most important of these is the conclusion that a fiiite overpressure is enough to cause the new phase to nucleate, that this nucleation is somehow wall induced, and that the amount of overpressure required is proportional to (A#, where t is a wetting parameter and At is the difference between the ambient and critical wetting parameters. Acknowledgment. I thank S. Balibar and R. Evans for useful remarks.
aF -Ap acl
= 0 (28)
(7) S.Balibar, B.Castaing, and C. Laroche, J.Phys. Lett. (Orsay, fi.), 41, L-283 (1980).
