Langmuir 1993,9, 50-54
Equilibrium of Capillary Systems with an Elastic Liquid-Vapor Interface Dongqing Li and A. W.Neumann' Department of Mechanical Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4 Received March 5,1992.In Final Form: July 16,1992 Surface elasticity plays an important role in many surface phenomena. In this paper, the effects of an elasticliquid-vapor interfaceon capillaryand wetting phenomenaare studied. The mechanicalequilibrium conditionsare derived for did-liquid-vapor system with an elastic liquid-vapor interface. As examples, application of these general equations to spherical sessile drop systems and capillary rise systems shows the elastic effecte of the liquid-vapor interface on both the Laplace pressure and the contact angle.
n
Introduction Generallyspeaking, surfacetension is the work to create a unit area of new surface, and surface stress is related to the work of deforming a surface. These two quantities will be numerically equal when atomic mobilities are sufficientlyhigh to preserve the microscopicconfiguration of the surface followingthe deformation. This is true for many liquid-vapor interfaces whose surface tensions are constant, independent of surface deformation. However, there are many other liquid-vepor systems where the value of the surfacetension of the liquid-vapor interfacedepends upon the deformation of the surface. For example, some polymer or protein solution systems where molecules adsorbed at the solution-vapor interface may form an elastic 'skin". This is also the case of spread monolayers of insoluble substances (e.g., certain kinds of surfactants) on liquid-vapor interfaces. For instance, for dilute aqueous solutions of a methylcellulose polymer, the equilibrium monolayer is perfectly elastic with a surface elastic modulus of 9.5 dyn/cm for a 2.4 X 10-3 wt 7% solution.' For aqueoussolutionsof surfactants of the type RO[CH2CH20l,,H, it was found that the surface elasticity depends also on the surface age.2 For a solid-liquid-vapor system with a liquid-vapor interface which exhibits elasticity, an immediate concern is the effect of the elasticity of the liquid-vapor interface on the basic thermodynamic relations of such a system, such as the Laplace equation of capillarity and the Young equation. Sofar,the elasticityof the liquid-vapor interface has not been considered in the Laplace equation and the Young equation. Therefore, it is the purpose of this study to establish a thermodynamic model for an elastic liquidvapor interface-contact angle system and to evaluate the effecta of the elasticity on the fundamental aspects of capillary and wetting phenomena.
Vapur
Liquid Solid
Figure 1. A sessile drop resting on a planar solid surface in equilibrium with a vapor phase. and the vapor phase, and it has a homogeneousand smooth surface. Therefore, both the solid-liquid and the solidvapor interfaces can be treated as a simple, homogeneous surface phase defined in Gibbsian surface thermodynami c ~i.e., ; ~the fundamental equations in a differential form can be written as solid-liquid interface
r
solid-vapor interface
where the subscripts sl and sv denote the solid-liquid and the solid-vapor interfaces, respectively; U,S, A,and N are the internal energy, entropy, surface area, and mole number of the interface; T, y,and C( are the corresponding intensive parameters,the temperature, the surface tension, and the chemical potential. As can be seen clearly from these equations, the only mechanical work mode for these two interfaces is their surface tensions. In analogy to the grand canonical free energy of a bulk phase4
Thermodynamic Model Consider a sessile drop resting on a planar solid surface in equilibrium with a vapor phase (see Figure 1). The liquid-vapor interface of the sessile drop possesses elasticity, and hence ita surface energy depends on not only the area but also the deformation or the strain of the interface. The solid substrate, on the other hand, is an ideal solid in the sense that it is sufficientrigid, immiscible, and chemically inert to both the liquid forming the drop
the grand canonical free energies for these two interfaces can be written as
For an elastic liquid-vapor interface, since the degree of deformation and hence the surface strain generally
(1) Abraham, B. M.; Kettereon, J. B.; Behroozi, F. Langmuir 1986,2, 602. (2) Luc888Bn, J.; Gilee, D.J . Chem. Soc., Faraday Trans. 1 1975, 71, 217.
(3) Gibbs, J. W. The Scientific Papers of J. Willard Gibb; Dover:
New York, 1981; Vol. 1, p 219.
(4) Callen, H. B. Thermodynamics and An Introduction to Thermostatistics, 2nd ed.;John Wiley 8 Sons: New York, 1985. (d
1993 American Chemical Society
Langmuir, Vol. 9, No. 1, 1993 51
Equilibrium of Capillary Systems change from place to place, the fundamental equation has to written in density form with respect to an element surface area dA1,. Under the assumption of moderate curvature and absence of chemicalreactions and electrical and magnetic interactions, the energy form of the fundamental equation for such an element can be written as
-4k-
Solid
8T
UlV = u l v ( ~ 1 v , c l l , ~ 1 2 , ~ 2 1 , ~ 2 2 , ~ ~ , l v , ~ 2 , 1 v , . . . , ~ ~(6) ~v) where the subscript lv denotes the liquid-vapor interface, ulvis the area density of the internal energy of the liquidvapor interface, $1, is the entropy density, nk,lv (k = 1,2, ...,r) is the mole density of the kth component, and t i j (ij = 1 , 2 ) is the surface strain tensor component. As ulvis defined as the energy density with respect to an element of surface area, dAlv, the total internal energy of this thin liquid f i i phase can be written as
Figure 2. An illustration of a virtual variation of the sessile drop configuration.
vapor interface is then given by n
(13)
Mechanical Equilibrium Conditions
(7)
For the system discussed above, the thermal and chemical equilibrium conditions are the same as those in The differential form of the fundamental equation can be the absence of elasticity; Le., the temperature T and the written as chemical potential pi (i 1,2, r) are constant through all phases in the system. However, the mechanical equilibrium conditions are different and are more interesting here. The mechanical equilibrium conditions for such a solid-liquid-vapor system can be derived by where the intensive parameters are defined as follows: minimizing the total grand canonical free energy of the system, with intensive parameters in the grand canonical temperature T = (aulv/aslv),ij,n,Jv free energy being ~onstant.~ The total grand canonical free energy of the system can chemical potential pk = ( a u l v / a n k J v ) ~ , , , , , n(9) ~ + h ~ be written as
...,
surface stress tensor
gi, = 2(aul,l~cij), ,v,,r r+,,,nhh
n = $2, + n, + n,
+ Q81 + 51,
(14)
It has been shown that5 gij = ysij + ay/acij i, j = 1 , 2
(10) The last term in eq 10 represents the surface elasticity effecte; i.e., the surface tension varies with various kinds of surfacedeformation: stretching, compreasing,twisting, andbending, etc. huation 10 givesa relationshipbetween surfaces t r w g i j and surface tension y. Generally, surface stress and surface tension will be numerically equal only if y is unaffected by the deformation, i.e., when surface atomic mobilities are high and there is no long-range correlation in atomic positions. In these cases, EEyldtij = 0, so that g = y. However, in the cases where surfactant or biologicalmoleculeaadsorbedat a liquid-vapor interface form a network, the long-range correlation in atomic or molecular positions and the low molecular mobilities will result in a surface deformation dependence of the local configuration around any particular molecule in the interfacial region. Therefore, EEy/atij # 0. For such an elastic liquid-vapor interface, the Euler form of the internal energy density can be written as 2
r
The density of the grand canonical free energy of the liquid-vapor interface can be obtained by the Legendre transformation: , 2 The total grand canonical free energy of the liquid(5) Blakely, J. M. Introduction to the Properties of Crystal Surfaces; Pergamon Prw: New York, 1973.
Jhd
- Ysv) Ual
where the subscripts 1, v, lv, sl, and sv denote the liquid phase, the fluid phase, the liquid-vapor interface phase, the solid-liquid interface phase, and the solid-vapor interface phase. The equilibrium condition is that the variation of the total grand canonical free energy is zero; i.e. 60 = 0 (16) In the above, the symbol 6 refers to a variation of the system in the sense of a virtualwork variation. The symbol d refers to an element or differential of a quantity. Combining eq 14 with eq 15 will give
6J(yd
-)7,
d A . 1 ~0 (16)
A virtual variation of the system is illustrated in Figure 2. The virtual variation terms in eq 16 canthen be obtained under the following considerations. The variation of the volume of the liquid drop is due to the variation of the liquid-vapor interface position. Therefore
where 6N1, is the normal component of motion of the element of the liquid-vapor interface into the fluid. The variation of the volume of the fluid phase is ale0 due to the variation of the liquid-vapor interface pition,
52 Langmuir, VoZ. 9, No. 1, 1993
Li and Neumann
but in an opposite direction to the variation of the liquid volume. Therefore
Z
4
The variation of the area and the strain of the liquidvapor interface includestwo p a r k one due to the variation of the position of the three-phase intersection, the other due to the variation of curvature,or the shape, of the liquidvapor Hence
Figure 3. A spherical sessile drop surface in the spherical R = constant for the drop surface. coordinate system (R,&@>.
The variations of the solid-liquid interface area are due to the variation of the position of the three-phase intersection, i.e. 6sA,(Td - YSV)
6T
(20)
where JL denotes the integration over the entire length of the three-phase line, dL is the element of length of the three-phase (solid-liquid-vapor) intersection line, 6T is the virtual motion of this line normal to dL along the solid and in a direction which increases with Asl,8 is the contact angle, and R1 and R2 are the principal radii of curvature of dA1,. Substituting eqs 17-20 into eq 16 and collecting similar terms yields
Since the variations are arbitrary, it is necessary and sufficient that each integrand in the two terms in eq 21 be zero. This gives the following two equations:
Equation 22 is the equilibrium condition governing the liquid-vapor meniscus, valid for any point of the liquidvapor interface. It has a similar form to the classical Laplace equation of capillarity: PI- Pv = Ylv(l/R1+
(24)
However, ylv in the Laplace equation is replaced by C(1/2)gijin eq 22. Recalling eq 10, eq 22 predicts that the shape of the liquid-vapor interface is controlled by both the surface tension and surface elasticity. This implies that, in the presence of elasticity, the “surface tension” of the liquid-vapor interfacemeasured by drop shape analysis (suchas ADSA8) is actually a summation of the true surface tension and the elasticity effects. Equation 23 is the mechanical equilibrium condition for any point at the line of the solid-liquid-vapor threephase intersection. It has a similar form to the classical Young equation: Ylv cos 8 = Y,
- Ysl
(25)
Again ylv in the Young equation is now replaced by (6) Johnson, R. E., Jr. J. Phys. Chem. 1959,63,1655. (7) Rice, J. In Commentary on the Scientific Writings of J. Willard Cibbs. Volume I , Thermodynamics;Donnan,F. G., Haas, A., Us.; Yale University Press: New Haven, CT, 1936; p 13. (8) Cheng, P.; Li, D.; Boruvka, L.; Rottenberg, Y.; Neumann, A. W. Colloids Surf. 1990,43, 151.
C(1/2)gij in eq 23 which directly accounts for the effects of surface elasticity of the liquid-vapor interface on the equilibrium contact angle. It should be noted that the value of C(I/B)gijis generally a function of the position on the liquid-vapor interface; therefore, C(1/2)gij in eq 23 should take its value at the three-phase line. Generally, dyl,/dcij in stress tensor, gij, characterizes the elasticity of an interface, i.e., the dependenceof surface tension on the deformation of the interface. As seen from eq 10, for an ideal liquid-vapor system where dylv/dcij = 0, C(1/2)gij = ylv, and then eqs 22 and 23 will revert to the Laplace equation, eq 24, and the Young equation, eq 25, respectively. For the cases of dyly/dcij # 0, one has to obtain the expressions for cij in order to show what eqs 22 and 23 imply explicitly. As there is no general expression for cij, we will further illustrate eqs 22 and 23 through the following simple examples.
Case 1: Spherical Surface For simplicity, let us consider as an example a spherical sessile drop of radius R. The principle of this example applies to other cases where the general geometry requires more complicated analysisfor the expressions of the strain tensor. In this example, the elasticity may result from a monolayer of surfactant on the liquid-vapor interface. When the drop volume undergoes a smallchange, the state of strain of the liquid-vapor interface will change correspondingly. For a spherical sessile drop, it is convenient to use a spherical coordinate system (see Figure 3). Furthermore, we will assume that the strain tensor does not contain any shear components (i.e., cij = 0 when i # j ) and denote the normal strain components by €11 = eM (strain in 4 direction) and €22 = egg (strain in @ direction). First, we will derive the expressions of cM and egg for a point (4, @) at such a spherical surface when the drop volume changes. The symmetry of the spherical surface makes uniform for all values of 4. Therefore, by definition €-
= As(&
(26)
where S4 is the arc length of the profile of the spherical sessile drop and As4 is the change of this arc length (see Figure 4a). More explicitly
s4= me
(27)
where 8 is the equilibrium contact angle. Now, consider that the drop volume undergoes a small change. This will cause a change in radius from R to R + AR; as a result of the liquid-vapor surfacetension dependenceon the surface deformation, ylv changes, and hence the contact angle also changes, say, from 8 to 8 + A@. Therefore
Langmuir, Vol. 9, No.1,1933 53
Equilibrium of Capillary Systems S, = 2 R e
S
b
Figure 5. An illustration of capillary rise of a liquid at a vertical wall.
- 2rrRsin B-
I
Figure 4. Geometry of a spherical sessile drop: (a) arc length S;, (b) arc length Sg.
AS, = 2(R + AR)(e + A8) - 2R8 = 28 AR + 2R A0 (28) where the higher order small term (AR A8) is neglected, and t, = ASJS, = m/R + Ae/e (29) Similarly, the symmetryof the spherical surface makes e55 uniform for (or independent of) all values of @. Therefore, for a point (t$, @),we have
= AS,/s, (30) where So is the length of a circle of a constant value of t$ on the surface of the spherical sessile drop (see Figure 4b) and AS, isthe length change of this circle due to the change in drop radius. From elementary calculus S, = 2rR sin t$ AS, = 2r sin t$ AR (31) Therefore '5,
e, = AS,/S, = AR/R (32) Through some mathematical manipulations and neglecting higher order small terms, it can be readily shown that
g, = Ylv
= Ylv + R(aYlv/aR) + wylv/ae) (34)
+ a~lv~ae5, = rlV + R(arlv/aR)
(35) = e54 = 0 and J = 1/R1+ 1/Rz = 2/R, so that eqs 22 and 23 can be further written as
Recallthat in this case
p1- P, = ( ~ I R ) ( Y+~ R(arlv/aR) ~ + ~ i / 2 ) 8 ( a ~ ~ ~ 1 a(36) m (VI,,
= M/R
gM
ylV+R(hIvIaR)
(38)
Therefore (39) Pi- Pv= (2/R)(ylV+ R(hlv/aR)) Comparing eq 39 with the classical Laplace equation, - Pv) eq 24, it is readily seen that the Laplace pressure (PI across an elastic liquid-vapor interface depends not only on the surfacetension, ylv,but a h o n the surface elasticity, a.ylvlaR. It also should be noted that eqs 36 and 39 are derived only for spherical liquid-vapor interfaces. Therefore, the parameters on the right-hand side of these equations and hence PI- Pv are constant for the entire surface. For more general, nonspherical drops, the radiue of curvature of the liquid-vapor interface changes from place to place; hence, the surface tension, ylv, and the surface elastic effects can be expected to be different at different points of the surface.
Case 2: Capillary Rise on a Vertical Plate
Therefore g# = Ylv + aYlv/a'#
of the interface. With respect to the term of aTlv/a8in eq 36, it should be pointed out that, unlike a system with an ideal liquid-vapor interface, the Laplace preseure (PI P,) across an elastic liquid-vapor interface also depends on the boundary condition of this interface, Le., the equilibrium contact angle, 8, at the three-phase contact line. This is one of the special features of the surface elasticity and may be understood as the long-range dependenceof the local confiiation aroundany molecule on the deformation of the entire interface. For a given solid-liquid system, rsv, rd,and hence the right-hand side of eq 37 should be constant. Therefore, the left-hand side of eq 37 shows a dependence of contact angle on the drop size due to the presence of surface elasticity. Of course, %lv/a8 in eq 36 will disappear in the caw of a spherical droplet or bubble. For such a case, e55 and g50 are the same as given in eqs 32 and 35. However, S, = 2rR, AS, = 2 s AR, and
+ R(arlv/aR) + ~i/2)e(aylV/ae)) COS 8 = rSv - ySl
(37)
As seen from the classical Laplace equation, eq 24, and the Young equation, eq 25, the curvature of the liquidvapor interface affects only the Laplace pressure (PI-Pv), but not the contact angle. This, however, is not the case for the system discussed in this paper. Apparently, drd dR in eqs 36 and 37 reflects the curvature (drop size) or the bending effects of an elastic liquid-vapor interface on the Laplace pressure and on the contact angle; Le., the contact angle is no longer independent of the curvature
This type of system is of general interest and, fortunately, does not require complicated geometrical analysis. Analysis for capillary rise systems with an ideal liquidvapor interface can be found elsewhere.9 Figure 5 illuetrates a capillary rise system consisting of a vertical plate, a liquid phase, and a fluid phase;the liquid-vapor interface possesses elasticity. In Figure 5, h is the height of capillary rise, 8 is the contact angle, and the parameter t$ is the angle between the normal at a point P and the z d e . When P is on the z axis, z = h and t$ = 90° - 8. Apparently, for such an interface, one of the radii of curvature is infinite, i.e., 1/Rz = 0; therefore, the mean curvature J = 1/R1 + 1/Rz = UR1, or for simplicity, J = 1/R. In this case, it is easy to understand that the surface strain exists only along the curved profile of the liquidvapor interface, as shown in the r-z plane of Figure 5. As the curvature changes, the strain is no longer a uniform, (9) Neumann, A.
W.Ado. Colloid Interface Sci. 1974,4,
106.
Li and Neumann
64 Langmuir, Vol. 9, No.1, 1993
one, and it changes from place to place along the surface. Considering the horizontal plane as a reference state of the liquid-vapor interface, then by definition, the local strain of the liquid-vapor interface is given by
From elementary calculus
dzldx = -tan 4 tan2 4 + 1 = l/cos24 Therefore, we can show that c = l/cos 4 - 1
(41)
At the three-phase contact line, using the conditionthat 4 = 90° - 0 and denoting liquid-vapor interfacial tension
by ylv,e,eq 42 can be written as
Now, the Laplace equation, eq 22, and the Young equation, eq 23, can be written as eqs 44 and 45, respectively.
Generally, in the presence of surface elasticity, many aspects of surface phenomena have to be reconsidered. For example, even measuring surface tension yiv and contact angle B by using the conventional methods now becomes more complicated. At least, one cannot be sure that what is measured is pure surface tension rather than a mix of surface tension and surface elastic effects. For the systems of capillary rise at a vertical plate with an ideal liquid-vapor surface,as the liquid surfacetension is a constant for the entire surface, integrating the classical Laplace equation yields9 sin B = 1- p,gh2/2ylV (46) Equation 46 allows determination of 8 from measuring the height of the capillary rise, h. However, this not the case of an elastic liquid-vapor interface. First of all, as the form of eq 44 is different from that of eq 24, eq 46 is no longer applicable; second, (ylv + (l/a(sin +/cos2 4)-
(dyl,/a+)) in eq 44 is not a constant; therefore, one cannot directly integrate eq 44 to obtain an analytical solution similar to eq 46. By comparing eq 45 and eq 37, one can also realize the follwing implication. For the same pair of liquid and solid, yavand ysl will be the same; thus, the right-hand sides of eqs 45 and 37 are identical. However, because of the geometricdifference of the liquid-vapor interfacesbetween a capillaryrise system and a sessiledrop system,the surface deformationsand hence the terms representingthe elastic effects in eqs 45 and 37 are different. Therefore, it can be easily seen that the contact angles in these two cases generally will have to be different in order to equalize the left-hand sides of these two equations. This implies that, in the presence of an elasticliquid-vapor interface, contact angles measured by using the capillary rise method will be different from those measured from a sessile drop for the same solid-liquid system. Alternatively,this is to say that the contact angles will depend on the geometric configurations of the elastic liquid-vapor interface. To verify this prediction experimentally would require development of an appropriate method to measure the contact angles of the capillary rise system in the presence of surface elasticity. To distinguish the effects of surfacetension and surface elasticity experimentally,one may consider the following approach A spherical shape can be approximately assumed for a sufficiently small pendent drop of water so that eq 39 is applicable. An elastic skin may be formed on this drop by depositing certain insoluble surfactant onto the watel-vapor interface. Using drop shape analysis (such as ADSAS), one can measure the value for the expression on the right-hand side of eq 39 as a function of the drop radius. This can be done by adjusting the drop volume slightly while the amount of surfactant on the surface is constant. As the expression on the righthand side of eq 39 involves only R and ylV(R),one can determine ylv = f ( R )numerically from the experimental data. In this way the true surface tension and the surface elasticityR (hlv/4R) (bothare functionsof the drop radius) can be distinguished. A similar treatment can be made for sessile drop systems as well. Through the foregoing discussions,it is clear that surface elasticity will have significant effects on both the Laplace pressure and the contact angle. This fact introduces more complications into the surface tension and contact angle measurements. As the existing techniques may provide informationonly about the simultaneouseffecta of surface tension and surface elasticity, one has to find other means to distinguish the elasticity contribution from the results of the conventional experiment methods.
Acknowledgment. This study was supported by the Medical Research Councilof Canada (Grant No."2).
