Figure 7. Circular islands of condensed phase, one monolayer thick, are seen in a sea of the uncondensed monolaver phase. Election micrograph of a monolayer of stearic acid at 70 dynes per cm. spread on water ( 9 )
I
”A specialist in nucleation and condensationtheory and a specialist in physical adsorption theory combine their separate approaches to two-dimensional film formation on the surface of an adsorbent.
THE INTERFACE SYMPOSIUM-4
ADSORPTION AND CONDENSATION PROCESSES JOHN B. HUDSON n the present paper, we consider the relation of proce m of physical adsorption to the occurrence of firstorder phase changes in systems where surface considerations play a significant role. We begin with a review of the phenomena involved, some of which have long been recognized and treated as first-order phase changes, though others have not been treated in those terms. The thermodynamictreatment of phase changes, applied to these systems, yields a generalized phase diagram that takes into account the presence of a chemically inert adsorbent surface. This phase diagram enables us to see clearly the relation between bulk and monolayer condensation, and the conditions wherein to expect one or the other. We consider next the kinetics of the phenomena and the added effectsthereby introduced into our predictions. We consider finally the application of this treatment to systems of practical interest and show its relevance to processes of technological importance.
I
Systems of lnlemsl
What systems can be considered as undergoing firstorder phase changes? The criterion for such a change is that matUial be transferred fmm one thermodynamically well-defined phase to another by a p r o w involving a discontinuous change in the state functions of the material 80 transferred. In other words, for such a process to occur at constant T and p, there must be a AH and a AS of transformation. This contrasts with the concept of a second-order phase change, in which oceurs a discontinuous change in the partial derivatives of the state functions, but not in the state functions themselves. Obvious and well defined examples of first-order
SYDNEY ROSS changes are the bulk-phase transformations, such as ice e water; water e steam; or dry ice e carbon dioxide gas, wherein one macroscopic bulk-phase is transformed to another with the attendant change in the bulk thermodynamic properties. The process of physical adsorption, involving the transfer of sorbate molecules from a bulk vapor to a two-dimensional adsorbed phase, with attendant discrete changes in the enthalpy and entropy of the sorbed material, is another and, at least in the context of this symposium, a fairly common example of the first-order phase change:
* .a.M In addition to theae well-known examples, we may similarly consider several other processes occurring in systems where a one-component, sorbable material is in contact with a chemically inert surface. For example, the vertical discontinuities obaerved in adsorption isotherms of gases on solids, and the horizontal d h n t i n u i ties observed during the compression of monolayers spread on liquids, both display the thermodynamic characteristica of a &&order phase change-the change in these cases being from a two-dimensional gas to some sort of two-dimensional condensed phase. For monolayers spread on water, the correctness of this interpretation has been shown by direct observation with an electron microscope. Figure 1 (9)is an electron micrograph of a film of stearic acid spread on water, observed at a degree of compression in the middle of the two-phase region. Circular “islands” of condensed phase, one monolayer thick, are Seen in a “sea” of the uncondensed monolayer phase. Similar photographs show that the VOL 5 6
NO. 1 1
NOVEMBER 1 9 6 4
31
!
islands grow reversibly from the uncondensed phase as the area of the film is decreased at constant temperature. The same kind of analogy to bulk systems, for gases adsorbed on solid surfaces, can be shown by comparison of t h e p V behavior of examplesof the bulk system and the sorption-system, which shows a twodimensional transition of adsorbed gas to monolayer condensed-phase. This comparison is shown in Figure 2 (h, 70). The bulkphase data are presented in terms of density rather than volume to facilitate comparison with the conventional p-9 curve for the adsorbed p h m . Note that both isotherms contain a region of low slope at low pressure corresponding to compression of the uncondensed phase; a vertical discontinuity in density at the equilibrium pressure for formation of the condensed phase, during which the condensed phase grows at the expense of the gas; and a further slow rise following the phase change, attributable to compression of the condensed phase (plus, in the two-dimensional case, possible beginning of second-layer formation). We may range still further from the usually recognized phase transformations to consider such processes as the spreading of a bulk liquid or solid to form a monolayer (gaseous or condensed, depending on the pressure) on a foreign surface, solid or liquid; or the reverse of these processeethe growth of bulk phases from monolayers as the pressure is increased. In all cases involving twodimensional phases, the surface energy of the substrate involved, and more particularly the way in which thissurface energy changes as gaseous or condensed layers form upon it, is critical to the production of these phases and the transformations that may occur among them, as dealt with late
in Gibbs free energy, AG,involved in transferring a given amount of material from one phase to another by an isothermal, isobaric proms. This free-energy change A@v), characteristic of consists of a term AH = A E the difference in the enthalpy of material in the present and newly formed phases; a term - T U , characteristic of the difference in entropy between the two phases at the temperature of transformation; and, where a surface phase is involved, the term A(Ay), the change in the total surface free energy when a monolayer phase is formed on a previously dean substrate. The bulk variables, H a n d S,are expressed in terms of a chemical potential, #@ = ( W / & I ) ~ ,per ~ , mole ~ ~ ,thus: #p = H - TS. This g~ is related to the standard state of the gas phase at one atmosphere by the relation y = #p, R T In p. The surface area is expressed in terms of adsorbate molecular density on the surface, r mole/q. cm., that is, A = n/r. The condition for equilibrium between any two phases is thus
+
+
.G
* ,G
(1)
Equilibrium relations among the phases that may be found in systems of interest to the study of surface processes can now be written. These equilibrium relations can then be used to construct phase diagrams indicating the phase equilibria involved in systems where surface phases are present. Consider first a system of a onecomponent vapor phase in contact with an energetically uniform, chemically inert surface. In this system, for the transformation bulk vapor % bulk crystal, we have, at equilibrium
AG = (a8@- n y ) = 0
Thermodynmmic'Basis
Let us review briefly the thermodynamic basis for fistorder phase changes, considering in all cases the change
where n is the number of moles transferred, or : p
= PI
+ R T l n ($Po)
. ci Y
6 d
P
PRESSURE, YY.HG 32
(PI
INDUSTRIAL AND ENGINEERING CHEMISTRY
(3)
PRESSURE, MM.HG
X 10' (p)
(4)
trated by a phase diagram generalid to take account of the presence of the surface and the phases that form upon it. The phase rule, generalized to account for interfacial phases, states that (2)
F=C-P+2+i
'I
IEMPERATUII
Figure 3. Gcwnliudphmc dingrmn for the case %Po< sPo
where 3pois the equilibrium vapor pressure of the bulk crystal phase. Again, similarly, for the transformation bulk vapor e adsorbed gas
AG = (n.4
+ n/f/r)
- (ncp
+ n.Of/r)
= 0
(5)
or : Y = /H
+ -r1 (/f- ."f)vw8 + RT ln p
(6)
=
This may readily be rearranged to give: =
-
(2f - a0fl 0 aP
(7)
- #P
which is a statement of the Gibbs adsorption theorem. These first two transformations were fairly obvious examples; a slightly less obvious situation is that of forming a monolayer condensed phase on the adsorbent surface-a twodimensional liquid or crystal if you willfrom the bulk vapor phase. For this case:
AG =
hL1d+ nd,'f/..lr)l - [dew) + n(."f/='r)l
= *'H
1
+wlr -
)fO..
=P,
=
+ R T l n (&
o
(8)
(9)
where .,,'r is the molecular density on the surface at saturated monolayer concentration and is the pressure of the gas in equilibrium with condemd monolayer. We now have equilibrium relations among four phases: bulk vapor, bulk crystal, adsorbed gas, and monolayer condensate. Similar relations involving bulk liquid phases and/or phase transitions from one monolayer condensed phase to another could likewise be formulated but need not be further elaborated. These equilibrium relations can be graphically illus-
where i is the number of interfaces of different .y or dy/dp. For the system we have been considering, in which C = 1, i = 1, making only the one further assumption that Z ~ is O always less than a l a , the resulting generalized phase diagram is shown in Figure 3. In addition to the familiar stability curves for the bulk phases we have an additional line representing the equilibrium between monolayer gaseous and condensed phases. Furthermore, the areas in which the surface is exposed to the bulk vapor phase are now two-phase regions, a condition allowed by the extra degree of freedom introduced by the term i = 1 in the generalized phase rule. This phase diagram can be used to trace the course of any p-T process in thin system. For example, an adsorption isotherm at TAis represented by the vertical line so labeled. In such a process at low pressures an adsorbed gas is present, increasing in density a9 the pressure is increased. When the pressure exceeds the equilibrium value for the two-dimensional condensed phase, ,pa, this phase will nucleate and grow until a complete condensed monolayer is formed. As the pressure is increased still further, additional adsorption takes place on this monolayer condensate. When the pressure exceeds the equilibrium value for the bulk condensed phase, Sf* the appropriate phase (bulk solid or bulk liquid, depending on the temperature) will nucleate and gmw. An experimental example of such an isotherm, taken from the published literature, (7) is given in Figure 4 for krypton at 77' K. on P-33 (2700°), a graphite of high surface homogeneity. Again, an isotherm taken at Tc would show the genera1 rise in coverage with pressure observed at lower temperatures, but no discontinuity associated with the formation of a monolayer condensate would be observed, as Tc > zT..it, the critical temperature for the twodimensional phase transition. I t has been shown (3) that ST,,* is approximately half of sT..it. Figure 5 is an experimental isotherm corresponding to this example taken from the published literature (7) and represents nitrogen on graphite at 77' K. The condensation of a fatty-acid monolayer spread on water, as shown in Figure 1, is an additional example of the same adsorbed gas F? monolayer condensate transition, occurring as the film pressure, in this case the p w u r e against the barrier confining the film, is increased beyond Z ~ O . We now consider a phase diagram for a second case : that in which ,pa is greater than apo. The diagram is shown in Figure 6. Isotherms for this case, at TAfor example, would show limited adsorption up to the equilibrium pressure of the bulk condensed phase, with much less than monolayer coverage even at that point. As the equilibrium pressure is exceeded, the appropriate three-dimensional phase will nucleate and grow. The VOL 56
NO. 1 1
NOVEMBER 1 9 6 4
33
illustrative experimental isotherm for this case is shown in Figure 7 for water on graphite (77), which is a thiid example taken from the published literature. Thus it is seen that not only is the course of wellrecognized phase transformations such as melting and boiling described by a path on the generalized phase diagram; but also transitions among surface phases, such as are encountered in adsorption from the vapor or the spreading of monolayers on liquid substrates, are equally well specified. By extending the above treatment to multicomponent systems one could also cover such processes as wetting, detergency, and the stabilization of emulsions.
I
Kinetic Basis
1
I I
The argument to this point has been purely thermodynamic; it has developed the relations between various adsorption and condensation processes occurring at surfaces. The argument is now extended to cover the added complication of the kinetic effects during monolayer and bulk-phase changes, and how they influence the behavior in a given real system. The equations that were written earlier for phase equilibria between various bulk and surface phases in the system chosen all relate to phases that are inlinite in extent; no account was taken of the presence of phase boundaries, e.g., between adsorbed gas and monolayercondensate in the monolayer case. Nevertheless, any new condensed phase must form by the nucleation of a minute particle of that phase and by its subsequent growth to macroscopic size. When this is taken into account our picture of the system in the vicinity of monolayer-phase or of bulk-phase transformations is quite different. Consider the free energy change involved in forming a small particle of the bulk-solid phase at a pressure a
little greater than 8p0 on the isotherm previously drawn (TAon Figure 3). For simplicity we will assume the particle to be a cylinder of height h, and radius r. For this case
AG =
+
~ G I J f. - .f)
+ Za-rhUJ + dhd.
-d
(10)
In this equation the second term is inherently positive (unfavorable), the third term is inherently negative, and in most cases, the fist term is also inherently positive. Note that the surface energy terms, which are always unfavorable to growth of the particle are in the second power of particle size, while the volume free energy term, which is favorable to growth when $0
I I I I I Ik-I I PI I L-U-UII I .---
0
0.2
0.4
0.6
REMTNE PRESSURE
0.8
Figure 7 . Illusfrotive isothnm-watn on graphite (77)
I .o
We have developed the concept of applying the thermodynamics and, in terms of a theoretical model, the kinetics of first-order phase changes to processes occurring in systems where surface phases are involved, with a few illustrative examples from published experimental data. The results obtained are in effect predictions of the behavior of general types of systems, and they can be used to answer certain questions. Consider first whether monolayer condensation will occur in a given system. No unequivocal criterion yet exists for this process. The scanty data that are available, notably that for chloroform vapor (8) on graphite and for water vapor on graphite (77), indicate that, should the heat of adsorption exceed the corresponding change of enthalpy for bulk condensation, monolayer condensation will occur at a pressure less than that at which bulk condensation cccurs. Should the reverse relation for the heats pertain, no monolayer condensation is to be expected. That is, if the adsorbateadsorbent interaction is strong, considerable adsorption and probably monolayer condensation are favored at VOL 5 6
NO
11
NOVEMBER 1964
35
pressures belorn7 3;bO. If the interaction is weak, an adsorbate concentration sufficient for monolayer condensation is never reached below $0. Considering further the question of the extent to which adsorption takes place as the pressure approaches the bulk-phase equilibrium value, $0, here again a relation between the heats of adsorption and of bulk condensation appears to hold: a heat of adsorption greater than the heat of bulk condensation implies extensive adsorption (0 > 1) at pressures below $0. A second generalization is that if the surface energy of the bulk adsorbate phase be small compared to that of the adsorbent surface, high coverage at low pressures, and monolayer condensation at temperatures below are favored. The latter rule is rather more speculative than the former, as few pertinent experimental data are available at present. Turning to predictions based on our kinetic treatment, we see that if the surface energy of the adsorbent be large compared to that of the bulk phase nucleating on it, then rapid nucleation at small supersaturations will be favored ; and vice-versa. These two opposite situations can each be used to advantage: the first in cloud seeding, where the object is to maximize the nucleation probability; the second in the coating of windshield surfaces with a low energy film, where the object is to minimize or completely inhibit condensation, which is effected by the extremely slow rate of nucleation on the coated surface. In addition to the general rules deduced so far, continued theoretical and experimental development of this subject will doubtless lead to other significant correlations : for example, the relation between monolayer condensation and spreading or wetting; the relative magnitudes of the supersaturation required for nucleation of bulk and of monolayers in a given system; and a theory of physical adsorption in terms of the more pertinent parameter zpo rather than &, which has been used in many previous treatments. Furthermore, extension of what we have developed here to multicomponent systems, such as those involving a surfaceactive agent at the interface between two bulk or monolayer condensed phases, will serve to clarify the processes occurring in such phase changes as the monolayer condensed + bulk condensed phase change observed in detergency and the reverse bulk condensed phase + monolayer phase transformation involved in the spreading of a material by means of a wetting agent. Thus it appears that results to date, while they do indicate that relations such as those we seek do indeed exist, are not adequate for complete specification of the systems involved. These results must be greatly amplified, both through additional experimental study of systems in which phase changes involving monolayer phases occur and through continued development of the theoretical treatment. Only then will more concrete generalizations regarding many systems be justified. NOMENCLATURE A = Area of surface in sq. cm. C = Number of components in a system E = Internalenergy 36
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
f
= Surface free energy, ergs/sq. cm.
aO,i
=
.ef
.,lf
Surface free energy of substrate entirely free of adsorbed phase (equivalent to J ) = Surface free energy of substrate covered with adsorbed monolayer a t coverage = 8 = Surface free energy of adsorbed-condensed monolayer a t coverage = unity Surface free energy of the edge of the adsorbed-condensed monolayer Surface free energy of x / interface ~ Surface free energy ofx/g interface Surface free energy ofbare substrate Surface free energy of the edge of the x-phase Number of degrees of freedom in a system Gibbs free energy Free energy difference between vapor phase and nucleus of critical dimensions Height of nucleus of condensed phase Height ofnucleus of critical dimensions Number of interfaces to be considered in applying phaserule to a system Nucleation rate, Equation 13 Arbitrary constant in Equation 13 Boltzmann’s constant ( = 1.38 X ergs/deg.) A generalized component: any molecular species that undergoes a phase change Number of moles of material transferred from one phase to another Number of molecules transferred from one phase to another Xumber of phases, both bulk and interfacial, in a system Pressure of gas Pressure of gas in equilibrium with two-dimensional phase Pressure of gas in equilibrium with three-dimensional phase Pressure of gas at critical supersaturation of two-dimensional condensed phase Pressure of gas a t critical supersaturation of three-dimensional condensed phase (nucleus) Radius ofnucleus of a condensed phase Entropy Temperature in degrees Kelvin Critical temperature of a phase change between twodimensional states Critical temperature of a phase change between threedimensional states Three-dimensional phase, in process of nucleating Interface between x and substrate Interface between x and its own vapor Molecular density in two-dimensional phase (moles/ sq. cm.) Molecular density in adsorbed-condensed monolayer at coverage = unity Fractional coverage in adsorbed-monolayer phase Chemical potential Chemical potential of substrate entirely free of adsorbed phase (equivalent to a ~ ) Chemical potential of substrate covered with-adsorbed monolayer at coverage = 0 Chemical potential of bare solid substrate Chemical potential of adsorbed-condensed monolayer a t coverage = unity Chemical potential of gas phase Chemical potential of gas phase at standard state Chemical potential of x-phase
REFERENCES (1) Amberg, C. H., Spencer, W. B., Beebe, R. .4., Can. J . Chem. 93, 305 (1955). (2) Crisp, D. J., pp. 17-35, “Surface Chemistry,” Butterworth’s Scientific Publn., London, 1949. (3) DeBoer, J. H., p . 147, “The Dynamical Character of Adsorption,” Clarendon Press, Oxford, 1953. ( 4 ) Hirth, J. P., Pound, G. M., Progress in ‘Materials Science 11, Macmillan, New York (1963).
(5) Hudson, J. B., J.Phys. Chem. 67,1884 (1963). (6) “International Critical Tables,” McGraw-Hill, New York, 1928. (7) Joyner, L. G . , Emmett, P. H., J. Am. Chem. Soc. 70,2353 ( 1 9 4 8 ) . (8) Machin, W. D., R05.9,S., Proc. Roy‘.SOC. (London) 265A,455 (1962). (9) Ries, H. E., Jr., Kimball, W. A., Nature 181, 901 (1958). (10) Ross, S., Winkler, W., J . Am. Chem. Soc. 76, 2837 (1954). (11) Youn G. J., Chessick, J,:J., Healy, F. H., Zettlemoyer, A. C., J . Phys. Chenz. 58, 313 8954).
