J . Phys. Chem. 1987, 91, 353-356
353
Free Energy and Vapor Pressure of Sessile Drops. 1. Rapidly Established Contact Angle Equilibrium' Malcolm E. Schrader* David Taylor Naval Ship Research and Development Center, Bethesda, Maryland 20084
and George H. Weiss National Institutes of Health, Bethesda, Maryland 20014 (Received: March 27, 1986; In Final Form: July 7 , 1986) The free-energy change accompanying formation of a sessile drop on a given area of surface, by means of continuous vapor adsorption to the initially evacuated surface, is examined. Spontaneous adsorption with a continuous decrease of free energy occurs as the pressure is increased from zero topo to form an adlayer. As the sessile drop forms with an increase in pressure abovepO, the free energy increases from its minimum at po to that characteristic of the sessile drop. An expression for the latter as a function of the equilibrium contact angle and drop diameter is derived by considering the free energies of formation of the solid-liquid interface and of the liquid-vapor interface. By utilization of the free-energy expression, an equation for the vapor pressure of a rapidly equilibrating sessile drop at its equilibrium contact angle is obtained. It is found that the vapor pressure of this sessile drop is always greater than po and corresponds to that obtained when the Kelvin equation is applied to a point on the liquid-vapor interface. It is noted that the thermodynamics of this approach to sessile drop formation neither requires nor precludes formation of an autophobic adsorbed layer at po.
Adsorption isotherms of gases to solid adsorbents have various shapes2 characterized by differences in the low-, middle-, and high-pressure regions. When the adsorption of a vapor is studied with a view to understanding its ultimate condensation to liquid, the high-pressure region near po is critically important. Behavior of vapor adsorption isotherms in this pressure region can be classified into two overall categories-those that intersect the pressure line at p o and those that approach po asymptotically (Figure 1). The occurrence of an adsorption limit at po indicates that the vapor at p o is in equilibrium with an adsorbed layer of finite size and thickness on the solid surface that has no further tendency to spontaneously adsorb vapor. The asymptotic approach, on the other hand, indicates the ability to continuously adsorb vapor until the film acquires, in its outer region, the characteristics of bulk liquid. At this stage, the vapor can continue to condense without a change in free energy. Since the isotherm must either intersect po or approach it asymptotically, it is clear that conventional adsorption isotherms predict either liquid formation as a film (zero contact angle) or no liquid formation at all at po. There is no provision for sessile drop formation. An adsorption isotherm describing the "coexistence at po" of sessile drops and vapor in equilibrium with a solid surface was suggested by Derjaguin and Zorin3 in 1957. The isotherm (described as based on the ideas of Frumkin4) was proposed as a possible, although not preferred, explanation of data obtained by Derjaguin and Zorin on adsorption of polar vapors to various solid surfaces. The adsorption isotherm crosses the p o line into the supersaturation region, returns, crosses into the subsaturation region and then returns to approach the po line asymptotically from lower pressures. Adamson and Ling5 subsequently proposed that an isotherm of this general shape (Figure 2) provides the description of the formation all sessile drops by means of adsorption and condensation of vapor to solid surfaces. They point out that the isotherm is generated by a Polanyi potential approach to adsorption and suggest an empirical potential in which the attraction term varies with the inverse cube of the distance from the surface, and the repulsion decays exponentially from the surface. Subsequently, Adamsod substituted an attractive exponential term for the inverse cube at small distances, leading to the expression toe-ax - @e-d -I-g / X 3
where X is the distance from the surface and a, g, a , p, and
eo
*Present address: Department of Inorganic and Analytical Chemistry, Hebrew University, Givat Ram, Jerusalem, Israel.
are constants. The first term is the attraction energy, the second is the repulsion term described as the "distortion potential", and the third is the attraction at larger distances. There are a number of physical postulates which are, explicitly or implicitly, incorporated into this approach. First, the assumption is made that the vapor pressure of all sessile drops is equal topo. Second, the potential approach to adsorption generally assumes that the influence of surface free energy, arising from creation or obliteration of surfaces, is negligible. Proceeding along these lines, with the point of view that at saturation the substance could just as well be liquid as vapor, the attractive potential which always exists between a solid surface and a gas molecule will inevitably tip the balance and cause liquefaction of the saturated vapor. On the basis of this consideration, the liquid would, of course, be in film (rather than drop) form since the attraction is exerted uniformly by the solid surface. Consequently, all vapors should condense as liquid films, i.e., maintain a zero contact angle on all solid surfaces. To account for nonformation of liquid film at po in many systems, it is concluded that all contact angle situations involve a structural rearrangement of vapor molecule^^-^ at the solid interface resulting, in effect, in an autophobic adsorbed layer. This results in what is termed a distortion potential, which opposes the attractive force. Adamson points out that if suitable constants are chosen, the general shape of the Figure 2 isotherm will prevail. The requirement that in all contact angle situations liquid-vapor molecules must adsorb to the solid surface to form an autophobic monolayer or multilayer is, for many but not all liquid-solid combinations, at variance with expectations based on intermolecular f o r c e ~ . ~ J ~ Results In this paper, a completely thermodynamic approach to sessile drop formation is proposed. In contrast to treatments based on (1) Presented in part at the 57th Colloid and Surface Science Symposium of the Colloid and Surface Chemistry Division of the American Chemical Society, Toronto, Ontario, Canada, June 12-15, 1983. (2) Brunauer, S . The Adsorption of Gases and Vapors; Princeton University Press: Princeton, NJ, 1945; Vol. 1. (3) Derjaguin, B. V.; Zorin, Z. M. Proc. Int. Congr. Surf. Act., 2nd 1957, 2, 145. (4) Frumkin, A. Zh. Fiz. Khim. 1938, 12, 337. (5) Adamson, A. W.; Ling, I. Adu. Chem. Ser. 1964, 43, 57. (6) Adamson, A. W. J. Colloid Interface Sci. 1968, 27, 180. (7) Tadros, M. E.; Hu, P.; Adamson, A. W. J. Colloid Interface Sci. 1974, 49, 184. (8) Hu, P.; Adamson, A. W. J. Colloid Interface Sci. 1977, 59, 605. (9) Girifalco, L. A,; Good, R. J. J . Phys. Chem. 1957, 61, 904. (10) Fowkes, F. M. Ind. Eng. Chem. 1964, 56(12), 40.
This article not subject to U S . Copyright. Published 1987 by the American Chemical Society
354
Schrader and Weiss
The Journal of Physical Chemistry, Vol. 91, No. 2, 1987
-------I 0
P
PO
Figure 1. Vapor adsortion isotherms: (a, left) liquid film forms at PO; (b, right) layer of limited thickness forms at po. r is the amount of vapor adsorbed per unit area, p the vapor pressure, and po the saturation vapor
pressure. Figure 3. Hypothetical pathway for adsorption of vapor to form sessile drop: (a) adsorbed layer; (b) liquid film; (c) sessile drop.
isotherm will run parallel to the p line at pa (Figure la). If, on the other hand, bulk liquid does not condense onto the surface, the isotherm intersects thepO line (Figure lb). The requirements for each of these two situations are clear-cut. If the sum of the interfacial tensions (defined here as interfacial free energy per unit area) of the solid-liquid and liquid-vapor interfaces is less than the tension of the clean solid surface YSL
+ Y L - Ys < 0
(1)
then there is negative free energy associated with the process and a liquid film will condense spontaneously on the surface (Figure la). If the reverse is true 0
1
Figure 2. Isotherm for coexistence of sessile drops with vapor at po. h is the 'film thickness".
Polanyi theory which ignore the energy of surface formation, it is assumed that the reversible condensation process is controlled solely by surface free-energy changes. This is expected since, at equilibrium, there is no change in free energy on condensation of vapor to bulk liquid or evaporation of bulk liquid to vapor. Consequently, if there are any changes in the surface free energy during these bulk phase changes, they will constitute the totality of free-energy change and will control the process. Accordingly, condensation or evaporation of a vapor in equilibrium with its liquid can occur without change in free energy only if the phase change takes place in a manner that does not cause any change in surface free energy. This can occur, for example, if the condensation or evaporation takes place on a preexisting flat surface of its own liquid, so that there is no change in the nature or area of any surface. If, on the other hand, a new body of liquid is formed, free energy must be supplied to create its surfaces. For example, small free drops will distill onto large ones, since this continuously decreases the total surface free energy associated with the given amount of bulk liquid by decreasing the total surface area. Conversely, small free drops will not form spotaneously from large ones. Likewise, if liquid is to condense spontaneously onto a solid surface, the total free energy of the new interfaces formed must be less than that of the interfaces that have disappeared. To examine the surface free-energy changes taking place on adsorption and condensation, imagine a chamber containing a perfectly flat, clean, nondeformable solid surface in perfect vacuum. Connected to it through a valve is another chamber containing a liquid reservoir consisting of an infinitely large spherical drop suspended in space. Due to the size of the drop, the surface is essentially flat (or the radius is infinite) so that the reservoir is in equilibrium with its vapor at po,the standard saturation vapor pressure. By means of the valve we admit vapor, reversibly and isothermally, to the sample chamber in small increments, using a suitable analytical method to determine adsorption to the surface. If liquid has condensed onto the surface at po, the adsorption
YSL
+ Y L - Ys
'0
(2)
a film containing bulk liquid cannot form spontaneouly on the surface, and the vapor at po remains in equilibrium with an adsorbed layer of limited thickness (Figure lb). Formation of a film in this case would require an input of free energy (Le., would involve a positive free-energy change) equal to ysL + yL- ysV per unit area. Formation of a sessile drop with contact angle greater than 0' on this same unit area of surface will clearly require an even greater input of free energy, since the area of the curved liquid-vapor interface is greater than that of the straight liquid-vapor interface of the film. Consequently, all sessile drops in mechanical equilibrium (with no contact angle hysteresis) on ideal surfaces have a positive free energy of formation. The actual process of condensation of vapor to form a sessile drop covering an arbitrary area, with a given contact angle 8, will of course take place through initial formation of a very small drop, at contact angle 8 but covering a small portion of the area, followed by growth of the drop through condensation while maintaining the contact angle 8 until the area of surface is covered. For purposes of calculating the change in free energy (which depends only on the initial and final states), however, we follow a path of reversible condensation on a fixed area of solid surface. Thus, for an area of solid surface with radius r (Figure 3a), which contains an adsorbed (but not liquid) layer in equilibrium with vapor at po, we write for the surface free energy Fsv = +sv
(3)
The surface free energy of a liquid film (Figure 3b) condensed on this area is (4) + YL) The increase in surface free energy required to form the liquid film from the vapor-equilibrated surface is then given by
FF= *?(YSL
U S , ,
=
Tr2(ySL
+ Y L - YSV)
(5)
We now condense vapor onto the film until a sessile drop with contact angle 8 is formed (Figure 3c). The resulting surface free energy is equal to that of the solid-liquid interface plus that of
The Journal of Physical Chemistry, Vol. 91, No. 2, 1987 355
Vapor Pressure of Sessile Drops
various types of growth when dn moles of its vapor are added to (or subtracted from) a drop standing at its equilibrium contact angle. If there is no contact angle hysteresis and mechanical equilibrium of the drop is rapidly established (Figure 4a), we may differentiate with respect to the radius of the solid-liquid interface at constant contact angle. We then have
Now
Figure 4. Drop growth on condensation of dn moles of vapor: (a) constant 0; (b) constant r.
(
+ cos 8)
(6)
+ 2cos 8 - C O S
6’)
(17)
Therefore
the new liquid-vapor interface. Since, for minimum liquid-vapor surface area for given solid-liquid area, the drop must be a spherical segment, the liquid vapor surface area is equal to 2ar2/(1
(
”(ArY,D))o = 2nryL 1
an
J8
-
2-3
r
COS
8
+ cos3 8
Simplifying, we have
The free energy of the drop therefore equals
(!E) 9
r
(7) The increase in surface free energy on transforming the film to the drop is then given by
The total surface free-energy change in forming a drop from the solid surface equilibrated with vapor is then ~ S V , D=
AFSV.F + AFF,D
(9)
Now, from Young’s YL
cos 8 = Ysv - YSL
(11)
By substitution AFSV,D
= “r2YL(
+
2 cos 0 - cos 8
Now, the relation between the change in free energy and the vapor pressure is d(AF)SV,D = R T In p / p o dn where po is the initial pressure, p the final pressure, T the temperature, R the molar gas constant, and n the number of moles converted from po top. Letting n = V/Vmwhere Vis the volume and V , the molar volume of the liquid, and ignoring the volume of the admolecules adsorbed at PO, while noting that the volume of the sessile drop is, for a spherical segment Tr3
V=
3
(2 - 3 COS 8
+ cos3 8)
sin3 8
(14)
we have ar3
n = 3 Vm
(2 - 3
COS
e + cos3 8)
sin3 8
(15)
Formulas may now be obtained for a sessile drop undergoing (11) Young, T.Philos. Trans. R . SOC.London 1805, 95,65. (12) Bangham, D. H.; Razouk, R. I. Trans. Faraday SOC.1937,33,1459.
Therefore
We note at this point that if there is a very large hysteresis effect, the drop edge will be unable to advance when dn moles are added, and the drop will stick to the same solid surface area. Since the drop volume increases while the solid-liquid interfacial area remains constant, the contact angle must increase (Figure 4b). The differentiation therefore takes place with respect to the contact angle at constant radius of the solid-liquid interface. This case will be treated in a forthcoming publication.
Discussion Effect of the Solid-Liquid Interface. Noting, on examination of (21), that r, the radius of the solid-liquid interface, is equal to sin 8 times the radius of curvature of the liquid-vapor interface, it can be seen that (21) has the form of the Kelvin equation applied to the liquid-vapor interface alone. One must be careful, however, not to conclude that in this case the vapor pressure is established by the liquid-vapor interface alone. In the numerator of,(12), the term containing 2/( 1 cos 8) represents the free energy of the liquid-vapor interface, while -(cos 8) arises from the freeenergy contribution of the solid-liquid interface as given by (1 l ) , the Young equation. Without the -(cos e) term, the trigonometric expression in (19) does not reduce to sin 8. Molecular Nature of the Interface. The formation of the flat solid-liquid interface and the formation of the curved liquid-vapor interface are both an integral part of the process of sessile drop formation which always requires a net input of surface free energy and yields a vapor pressure greater than po. The situation may be viewed as follows. A bulk film comes into existence on a solid surface when sufficient energy is given off by solid-liquid interaction to create a new liquid-vapor interface. When that energy is inadequate, liquid can deposit only in the form of sessile drops. It makes no difference whether the lack of interactive energy is due to the intrinsic lack of attraction between molecules of the original solid surface and those of the liquid or to formation at po of an autophobic layer of the liquid molecules. In either case, free energy must be supplied to form the sessile drop and its vapor pressure not only “crosses the p o line” but remains higher than po after it is completely formed. Free-Energy Minimum and Stability of the Sessile Drop. The free energy of sessile drop formation by means of vapor adsorption
+
J . Phys. Chem. 1987, 91, 356-362
356
means of spontaneous mechanical adjustment of the shape of an existing drop. We have a situation then in which a sessile drop existing in a metastable high free-energy state at pressure above po adjusts itself mechanically to a configuration with the minimum amount of free energy possible for a drop of that volume (of that substance) in contact with that type of solid surface. The freeenergy minimum of the system at po is of course the deeper one, while that arising from the shape and contact angle of the drop may be regarded as a secondary minimum. The reason this situation can exist, of course, is because evaporation of the sessile drop from its pressure of p'to po is generally a very much slower process than the mechanical adjustment of the shape of a drop on contacting a solid surface.
r
P'
P
Figure 5. Isotherm for formation of a single sessile drop. p'is the vapor pressure of the sessile drop. The dotted line does not necessarily represent the actual pathway from po to p'.
to an initially evacuated surface consists, essentially, of two steps. The firsts involves spontaneous adsorption of vapor, with accompanying decrease of surface free energy, until po is reached. The decrease in free energy, -AF,is also called re,the spreading pressure, which is always positive (and never quite equal to zero, since there is always some spontaneous adsorption). The second step involves increasing the pressure past P O , with accompanying increase in free energy, to form the sessile drop at p' (Figure 5). A minimum in total surface free energy consequently exists at PO. The sessile drop is therefore metastable and will evaporate into an infinite volume of vapor at po or distill onto the surface of an infinitely large sphere of liquid. In is important, however, to distinguish this minimum in surface free energy from that which is involved in the Young equation. The Young equation can be derived by considering the value of cos B required to minimize the total free energy of the three interfaces (Le., solid-vapor, solid-liquid, and liquid-vapor). This minimum is arrived at by
Conclusions 1. All sessile drops on homogeneous, smooth, nondeformable solid surfaces have a positive free energy of formation, which can be expressed as a function of the equilibrium contact angle and drop size. The expression contains two terms, one representing the free energy of the solid-liquid interface and the other that of the liquid-vapor interface. 2. The vapor pressure of these sessile drops is greater than po. They are consequently metastable and will distill back to a reservoir of liquid of vapor pressure po. Their existence in reproducible configurations results from the fact that mechanical equilibrium governing the contact angle and shape of the drop can be rapidly established, while distillation at these small pressure differences is usually relatively slow. 3. The expression for the vapor pressure of a sessile drop in rapidly adjustable equilibrium configuration is the same as that for the Kelvin equation applied to the liquid-vapor interface alone. This result, however, is derived only when the free-energy change at the solid-liquid interface is included along with that at the liquid-vapor interface. 4. The thermodynamics of sessile drop formation neither require nor preclude formation of an autophobic adsorbed layer at po.
Size and Shape of Sodium Deoxycholate Mlcellar Aggregates G. Esposito,+E. Giglio,*N. V. Pavel,**and A. Zanobit Eniricerche S.p.A., 00015 Monterotondo, Roma, Italy, and Dipartimento di Chimica, Uniuersitd di Roma "La Sapienza", 00185 Roma, Italy (Received: April 15, 1986: In Final Form: August 13, 1986)
From SAXS and ESR measurements carried out on sodium deoxycholate (NaDC) aqueous solutions as a function of NaDC and NaCl concentrations and temperature we have deduced the mean hydrodynamic radius, shape, and aggregation number of the micellar aggregates. The validity of a helical model previously proposed for NaDC micellar aggregates was satisfactorily checked. Some limits of the experimental techniques are discussed. A qualitative agreement between the data obtained from SAXS and ESR measurements was observed. The extent of micellar growth increases with the concentration of added NaCl and as the temperature is lowered. Much smaller changes than for the ionic strength were detected by increasing the NaDC concentration within the range 0.05-0.15 M.
Introduction In a recent article a helical model was proposed for describing the structure of sodium deoxycholate (NaDC) micellar aggregates in aqueous solutions.' This approximate model was assumed mainly on the basis of X-ray studies carried out on sodium deoxycholate and rubidium deoxycholate* (RbDC) crystals. The helix has the lateral surface covered by deoxycholate anions (DC-) with the hydroxyl and the angular methyl groups protruding toward the inside and the outside of the helix, respectively (Figure
'Eniricerche S.p.A.
* Universitii di Roma "La Sapienza".
1). Its interior is filled with cations surrounded by water molecules. Ion-ion and ion-dipole interactions together with a close net of hydrogen bonds stabilize the helix which, astonishingly, presents the nonpolar face of DC- oriented toward the aqueous medium. However, hydrophilic groups of the NaDC helical aggregate can be approached by the water molecules toward both the bases of the helix and the relatively large empty regions ~
~~
~~
~
~~
(1) Conte, G.; Di Blasi, R.; Giglio, E.; Panetta, A,; Pavel, N. V . J . Phys. Chem. 1984.88, 5720. (2) Campanelli, A. R.;Candeloro De Sanctis, S.;Giglio, E.; Petriconi, S. Acta Crystallogr., Sect. C: Cryst. Struct. Commun. 1984, C40, 631.
0022-3654/87/2091-0356$01.50/0 0 1987 American Chemical Society
