JOURNAL OF CHEMICAL PHYSICS
VOLUME 109, NUMBER 9
1 SEPTEMBER 1998
Effect of gravity on contact angle: A theoretical investigation C. A. Warda) and M. R. Sasges Thermodynamics and Kinetics Laboratory, Department of Mechanical Engineering, University of Toronto, 5 King’s College Road, Toronto M5S 3G8, Canada
~Received 5 August 1997; accepted 29 May 1998! Using the Gibbs description of an interphase, the necessary conditions for equilibrium of a closed, two-phase fluid system in the presence of gravity are the Laplace and Young equations and a condition on the chemical potentials. The last condition has been neglected in all previous examinations of contact angles in a gravitational field. After introducing explicit expressions for the chemical potentials, we find that the condition on the chemical potentials can be used to determine the pressure profile within the system. In a ‘‘two-interface’’ system in which a liquid phase is both above and below a vapor phase and the vapor phase forms a solid–vapor interphase in one region, the pressure profile in the liquid phases is the same as it would have been if the vapor phase were not there; thus in a gravitational field, the pressure is smaller in the liquid phase above the vapor phase than it is in the liquid phase below the vapor phase. This results in the contact angle at the upper three-phase line necessarily being smaller than that at the lower three-phase line. This difference in contact angles is conventionally referred to as contact angle hysteresis; however, we show that it is simply an equilibrium property of a capillary system in a gravitational field. The contact angle difference predicted to exist in the presence of gravity does not violate the Young equation, but the Young equation does impose a restriction on the equilibrium adsorption isotherms at the solid–vapor and solid–liquid interfaces. © 1998 American Institute of Physics. @S0021-9606~98!51033-1#
I. INTRODUCTION
valid. Since Gibbs’ original derivation, there have been a number of challenges to the validity of the Young equation.3–8 These have usually been raised in reference to the effect of gravity on the contact angle, to contact angle hysteresis, or to the effect of drop size on contact angle. However, none of the previous investigations has taken into account the full coupling of the Young equation to the other conditions for equilibrium. For example, the experimental measurements of the contact angle have usually been carried out under nonequilibrium conditions where a sessile droplet is changing mass ~as a result of evaporation or condensation!. In some cases, an attempt has been made to approximate an equilibrium circumstance by having in the system both a sessile droplet of the liquid being studied and a pool of the liquid.9 Such a system is not necessarily in thermodynamic equilibrium, since Eq. ~1! is not necessarily satisfied. Without this equilibrium being satisfied, one would expect either evaporation or condensation of or on the droplet. In other cases, contact angles are reported with the droplet exposed to air.10 In this case, the disequilibrium is usually even stronger. Thus, strictly speaking, the Young equation does not apply to either of these nonequilibrium circumstances. Generally, this nonequilibrium effect is thought to be unimportant; however if it were, one would not expect to see a dependence of contact angle on droplet size. The experimental observations have been to the contrary. A number of investigations have indicated that for larger droplets, the ~advancing! contact angle is almost independent of the droplet radius, but as the droplet becomes smaller ~i.e., as the nonequilibrium effect becomes larger! the contact angle changes
Gibbs determined the necessary conditions for equilibrium1 in a system subjected to gravity that contains a line of contact between solid, liquid, and vapor phases. He assumed each interface could be replaced by a mathematical surface and that the interfacial energy did not depend explicitly on the surface curvature. For a smooth, homogeneous, rigid and nonvolatile ~i.e., ‘‘ideal’’! solid surface, he obtained three necessary conditions for equilibrium: ~1! the Young equation; ~2! the Laplace equation; and ~3! a relation between the chemical potentials of the fluid components and the gravitational intensity:
m kj 1W k gz5l †k ,
j5L,V,SV,SL,LV,
k51,2,...,r, ~1!
where m is the chemical potential and W is the molecular weight; a superscript SV, SL, or LV refers a property to the solid–vapor, solid–liquid, or liquid–vapor interface, and a superscript V or L refers a property to the vapor or liquid phases; a subscript on a property refers it to one of the r chemical species in the system; g is the gravitational intensity, z is the vertical position in the field, and l †k is a constant. It should be emphasized that ~as Gibbs pointed out2! these conditions for equilibrium form a coupled system of equations. Unless all three conditions are satisfied, there is no thermodynamic reason to think any of them would be a!
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/98/109(9)/3651/10/$15.00
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© 1998 American Institute of Physics
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with drop radius ~at some temperatures going up, and at others going down!. Rather than the dependence of contact angle on droplet size being interpreted as a nonequilibrium effect, it is currently being interpreted as a ~positive or negative! line tension effect.10,11 If the expression used by Gibbs for the interfacial energy is augmented to include an explicit dependence on surface curvature, then an additional term appears in the Young equation. This term is curvature dependent and would mean physically that there was a tension in the three-phase line of contact.10 There is, however, considerable disagreement about the magnitude of the line tension and even about its sign. Further, there is a correlation between the degree of disequilibrium in the system when the contact angle is measured and the sign reported for the line tension. For room temperature systems, those9 who attempted to maintain equilibrium in the system during contact angle measurements ~by maintaining a pool of liquid in the system! obtained a contact angle dependence on drop size that would mean the line tension was ‘‘negative,’’ and those10 who measured contact angles in more strongly nonequilibrium circumstances found values that correspond to positive values of the line tension. In view of the correlation between the degree of disequilibrium and the sign of the line tension, the experimental evidence for or against the existence of a line tension cannot be said to be conclusive. If line tension does exists, then it should be measured under equilibrium conditions, and from a theoretical viewpoint, if it does exist the interfacial energy must depend explicitly on curvature. It has been claimed12 that the phenomenon of contact angle hysteresis ‘‘cannot be explained’’ by the original formulation of the capillarity theory given by Gibbs. However, it seems too early to draw this conclusion, since none of the previous analyses have taken into account the full coupling between the three conditions for equilibrium. To investigate the effect of gravitational potential on the contact angle, we use the original description of a surface phase that was given by Gibbs, and consider a simple capillary system in a gravitational field. The system consists of a cylinder that has an ‘‘ideal’’ 13 solid surface, and is exposed to two liquid phases that are separated by a vapor phase ~see Figs. 1 and 2!. At the upper three-phase line, there is a contact angle, u u , and at the lower three-phase line, there is a second one, u l . In this configuration, the line tension would be acting perpendicular to the solid and would have no effect on either contact angle. We show that under equilibrium conditions, a difference between these contact angles does exist that depends on the intensity of the gravitational field. Thus, this difference in contact angles does not result from line tension. Normally, this difference in contact angles would be called ‘‘contact angle hysteresis,’’ but since the difference is simply an equilibrium property of the system that name seems inappropriate.
C. A. Ward and M. R. Sasges
aligned with the cylinder axis @see Fig. 1~A!#. The interface between the liquid and vapor phases will be assumed to be axisymmetric, and the internal energy of the liquid–vapor interface not to depend explicitly on the interface curvature. The dividing surface approximation is also adopted to describe the solid–liquid and solid–vapor interfaces. For these interfaces, the position of the dividing surface is taken to be such that there is no adsorption of the solid component. Hence, the intensive ~per unit area! internal energy of any of the three surface phases is assumed to depend upon the intensive entropy, s j , of the phase and on the number ~per unit area! of moles of each component adsorbed, n lj . Thus, for either a surface or a bulk phase, we may write u j 5u j ~ s j ,n 1j ,n 2j ,...n rj ! ,
~2!
j5LV,SV,SL,L,V.
For the surface phases the properties appearing in Eq. ~2! are understood to be expressed per unit interfacial area, and for the bulk phases are understood to be expressed per unit volume. From the definition of the intensive properties ds j 5
S D
1 du j 2 Tj
S D
r
m ij dn ij , Tj
(i
j5L,V,LV,LS, or SV. ~3!
From the Euler relation for a surface phase, one finds that r
g 5u 2Ts 2 ( n ij m ij , j
j
j
~4!
j5LV,SV, or SL
i51
and from the Euler relation for a bulk phase, one finds that the pressure may be written:
S
r
P 52 u 2Ts 2 j
j
j
( n ij m ij
i51
D
,
~5!
j5L or V.
Since the system is in a gravitational field, the system has an energy that arises from the field. The potential energy per unit mass would be gz, where g is the magnitude of the gravitational acceleration and z is the elevation of the mass in the field. The potential energy per unit volume for the bulk phase and per unit area for the surface phase is given by
S( D r
c 5zg j
n ij W i ,
i
j5L,V,LV,SV, or SL,
~6!
where W is the molecular weight of the substance. The total energy, E, for the system may be written E5
E
VL
1 1
~ u L 1 c L ! dV1
E E
A LV
A SL
E
VV
~ u V 1 c V ! dV
~ u LV 1 c LV ! dA1
E
A SV
~ u SV 1 c SV ! dA
~ u SL 1 c SL ! dA
~7!
and the total number of moles of one component II. EQUILIBRIUM IN A SYSTEM EXPOSED TO A GRAVITATIONAL FIELD
Suppose a liquid and a vapor phase are held in a cylindrical container that is closed to mass and energy transport, and that the container is present in a gravitational field
N i5
E
VL
1
n Li dV1
E
A SL
E
VV
n SL i dA.
n Vi dV1
E
A LV
n LV i dA1
E
A SV
n SV i dA ~8!
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J. Chem. Phys., Vol. 109, No. 9, 1 September 1998
C. A. Ward and M. R. Sasges
FIG. 1. Geometry of the system. ~A! General geometry of the container and liquid–vapor interface. ~B! Virtual displacement of the three-phase line. ~C! Virtual displacement of the interface. ~D! Differential geometry of the interface ~Ref. 14!.
The total entropy may be expressed similarly. This integral will be written in more detail for the system corresponding to Fig. 1~A!: S5
E
z0
0
1 1
s L p L 2 dz1
E E
H
zb
z0
s V L 2 p dz1
zb
0
E
zb
E f E
s L ~ L 2 2x 2I ! p dz1
E
fb
0
s LV 2 p x I R 1 d 1
zb
z0
s V p x 2I dz
H
zb
necessarily simultaneously valid. As will be seen, all three conditions must necessarily be satisfied before one can say that the system is in equilibrium. Boruvka and Neumann12 presented a derivation of the Young and Laplace equations; however, they did not obtain Eq. ~1! because they implicitly assumed the chemical potential function to be of the same functional form for all phases. This assumption would prevent them from obtaining a closed system of equations to define the equilibrium state for a three-phase system. We discuss this point further below. One can only speculate as to why confusion has arisen regarding the conditions for equilibrium in a three-phase system. In his classic work,1 Gibbs began his determination of the condition for equilibrium on page 314. He then obtained in that section the Young and Laplace equations, but he did not obtain Eq. ~1! in that section. For its derivation, he referred to an earlier section ~p. 276! in which only fluids were present, but he discussed the importance of Eq. ~1! in the system of equations for a three-phase system ~on p. 319!; thus, he was clearly aware of the necessity of using Eq. ~1! to define the equilibrium configuration of a three-phase system. However, he did not present an explicit expression for the chemical potentials, and the expressions for these functions will be seen to play a critical role in our considerations. To show that Young, Laplace, and Eq. ~1! define the necessary conditions for equilibrium, we apply the entropy postulate and require that the total entropy be an extremum subject to constraints of constant total energy, total volume, and total number of moles of each component. To satisfy the constraints, we introduce the Lagrange multipliers, l 0 ,l i ,i 51,2,...,r, and define the function r
S * [S2l 0 E2
( l iN i .
~10!
i51
To determine under what conditions S * has an extremum, we follow the standard calculus of variations procedure15 and introduce the set of comparison functions:
s SV 2 p Ldz
s SL 2 p Ldz1 p L 2 ~ s SL ! u z50 1 p L 2 ~ s SV ! u z5H ,
3653
~9!
where the subscript I or b on a quantity indicates that it is to be evaluated at the liquid–vapor interface or the three-phase boundary. In view of Eqs. ~2! and ~3!, the total entropy, S, may be seen to depend on u j ,n 1j ,n 2j ,...,n rj , as well as the volume of each phase but in view of Eqs. ~7! and ~8!, not all of these variables are independent. III. NECESSARY CONDITIONS FOR EQUILIBRIUM
The thermodynamic description of the surface phase given in Eq. ~2! is the same as that used by Gibbs.1 Using this description, Johnson5 presented a derivation of the Young equation, but did not obtain explicitly either the Laplace equation or Eq. ~1!. Thus, Johnson’s derivation could give the impression that the Young equation could be valid, but that the Laplace equation and Eq. ~1! were not
u j 5u ej 1 e 0 h 0 ~ z ! ,
j5L,V,LV,SV, or SL,
~11!
j 1 e ih i~ z ! , n ij 5n ie
j5L,V,LV,SV, or SL,
i51,2,...,r, ~12!
where the subscript e on a quantity indicates the equilibrium value of the variable and h 0 , h i are arbitrary functions. If the function t j is defined by r
t [s 2l 0 ~ u 1 c ! 2 ( l i n ij , j
j
j
~13!
j
i51
then S * may be written @see Fig. 1~A!#: S *5
E
z0
0
1 1
t L p L 2 dz1
Et Et H
zb
zb
0
E
z0
L p dz1
V 2
SL
zb
E tp f E t
t L ~ L 2 2x 2I ! p dz1
E
fb
0
t LV 2 p x I R 1 d 1
zb
z0
H
zb
V
x 2I dz
SV
2 p Ldz
2 p Ldz1 p L 2 ~ t SL ! u z50 1 p L 2 ~ t SV ! u z5H . ~14!
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C. A. Ward and M. R. Sasges
e 0 h 0 and e i h i . After inserting Eqs. ~11! and ~12! into Eq. ~14!, taking the partial derivative of the result with respect to e i , one finds
We note that z 0 , z b , and f b each depend on u j ,n ij . Since the number of moles in the respective phases are allowed to vary, the limits of the integrals listed in Eq. ~14! depend on
S D E S EF S EF EF S EF S ]S* ]ei
z0
5
0
e i 50
1
1
1 1
D
m Li h i 2 L 2l 0 W i gz2l i p L 2 dz1 T zb
z0
fb
0
H
zb
zb
0
hi 2
EF S zb
z0
D
S DG D S D G S D G S
m Li ] x 2I 2 V 2l W gz2l p x 1 t p dz1 0 i i I TL ]ei
S
2 p R 1x Ih i 2
S D
E St S E zb
50
~16!
e i 50
1
] x 2I ] x 2I 1tV ]ei i ]ei
2
0
j5L,V,LV,SV,SL
~17!
R 1x I
]fb ]ei
] x 2I ]e0
1tV
0
2 t LV
]fb ]e0
] R 1x I ]e0
2 t SV b
0
dz
0
d f 50,
~22!
0
]zb ]e0
1 t SL b
0
]zb ]e0
50.
When Eq. ~21! is used in Eq. ~17! and the result is used to simplify the expression for t j , one finds from Eq. ~13! that
t j 5s j 2
t L5
~18!
S D
r
n i m ij 1 j u 1 , Tj Tj i51
PL , TL
(
t V5
2 g SV SV t 5 SV , T ~19!
Eq. ~15! reduces to
S D S D S D
]fb ]zb ]zb 2 t SV 1 t SL 50. t LV b R 1~ f b ! b b ]ei ]ei ]ei
j5L,V,LV,SL,SV ~24!
PV , TV
t LV 5
2 g LV , T LV
j5L,V,LV,SV,SL,
~21!
2 g SL SL t 5 SL . T
~25!
If Eq. ~22! is multiplied by d e 0 and Eq. ~18! by d e i , the latter summed over i and added to the former, one finds
E
zb
z0
~20!
By taking the partial derivative of S * with respect to e 0 and making similar arguments as those applied to Eq. ~15!, one arrives at the following requirements: 1 , Tj
~23!
0
i
x I ~ f b ! 5L,
~15!
and when this result is compared with Eqs. ~4! and ~5!, one finds
dz
d 50.
0
] x 2I ]e0
t LV b R 1~ f b !
In addition, since
l 05
fb
1
LV
S S
S D D D
D S DD F S DG S D S D S D
2
L
z0
tL 2
d f 1 ~ 2 p t LV b ! R 1~ f b ! x I~ f b !
m SV m SV ]zb i i SL 2 2 2l W gz2l p L dz12 p L t 1 p L h 2 2l 0 W i gz2l i . 0 i i i b T SV ]ei T SV
m ij 2l 0 W i gz2l i 50, Tj
fb
D D
DG
D
m Li 2l 0 W i gz2l i p L 2 dz TL
hi 2
E S S D S DD E F t S ] ]e D G f z0
zb
hi 2
m SV m SV ]zb i i SV 2 2 2l W gz2l p L dz22 p L t 1 p L h 2 2l 0 W i gz2l i 0 i i i b T SV ]ei T SV
and since the limits of the integrals also depend on these arbitrary functions, we must have zb
S DG
hi 2
and since h i is an arbitrary function, the only way that Eq. ~16! can be satisfied for all values of this function is if 2
E S H
m LV ] R 1x I i LV LV 2l 0 W i gz2l i 12 p t T ]ei
Since S * is an extremum when e i vanishes, we require
]S* ]ei
D
m Li ] x 2I 2 2 L h i 2 L 2l 0 W i gz2l i ~ L 2x I ! p 1 t 2 p dz T ]ei
~ 2 t L 1 t V ! 2x I dx I dz1
E
fb
0
2 t LV d ~ R 1 x I ! d f 50. ~26!
Since we have considered variations about the equilibrium configuration, there is a resulting displacement of the interface. From Fig. 1~C!, the following relations are found for this displacement: R 1 d f 5sin f dz
~27!
and
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J. Chem. Phys., Vol. 109, No. 9, 1 September 1998
dx I 5sin f dR 1 .
C. A. Ward and M. R. Sasges
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~28!
Also, from Fig. 1~A! it may be seen that x I 5R 2 sin f .
~29!
When Eqs. ~27!, ~28!, and ~29! are used in Eq. ~26!, one finds
ES zb
z0
~ 2 t L 1 t V ! 1 t LV
S
1 1 1 R1 R2
DD
dz50.
~30!
Since this integral must vanish for all values of the limits, the argument of the integral must vanish. In view of the relations given in Eq. ~25!, one finds from Eq. ~30! that P VI 2 P LI 5
S
D
1 1 1 g LV , R1 R2
~31!
which is the Laplace equation. If Eq. ~23! is multiplied by d e 0 and Eq. ~20! by d e i , the latter summed over i and added to the former, one finds SV SL t LV b R 1 ~ f b ! d f b 2 t b d f b 1 t b dz b 50.
~32!
It should be noted that the quantities appearing in Eq. ~32! are all evaluated at the three-phase line. Since Eq. ~27! also applies at the three-phase line R 1b
dfb 5sin f b dz b
~33!
and
p f b5 2 u . 2
~34!
FIG. 2. Pressure variation with elevation in a two-interface configuration.
than approximately 36°, then the thermodynamically more stable configuration is the two-interface configuration shown in Fig. 2. This prediction has been found to be supported by experimental studies.16,17 If the gravitational intensity is no longer zero, we investigate the relation that would have to exist between the contact angles at the upper and lower interfaces in order for equilibrium to exist.
After using Eqs. ~33! and ~34! in Eq. ~32! and the relations given in Eq. ~25!, one finds SL LV g SV b 2 g b 5 g b cos u ,
~35!
which is the Young equation. From Eqs. ~21! and ~17! one finds a result equivalent to Eq. ~1!:
S D
li m ij 1W i gz5 2 , l0
j5L,V,LV,SV,SL.
~36!
For the isolated system, Eqs. ~21!, ~31!, ~35!, and ~36! constitute the necessary conditions for equilibrium. It should be noted that unless all these conditions are satisfied, the entropy would not necessarily be an extremum. IV. CONSEQUENCES OF THE NECESSARY CONDITIONS
We now relax one of the constraints and suppose the system is exchanging energy with a surrounding reservoir of known temperature. Then the system and reservoir constitute an isolated system, and when equilibrium is reached they would have the same temperature. The necessary conditions for equilibrium for the isothermal system then would be Eqs. ~31!, ~35! and ~36!. Using the necessary conditions for equilibrium, a stability analysis16,17 has been reported of the configuration shown in Fig. 1, and it has been found that if: ~1! the gravitational intensity is reduced to zero, and ~2! the contact angle is less
A. Pressure profile in a capillary system
If the differential of Eq. ~5! is taken and the result combined with Eq. ~3!, then one obtains the Gibbs–Duhem relation for either bulk phase, n j d m j 52s j dT1d P j ,
j5L or V.
~37!
If the liquid is assumed to have a constant isothermal compressibility, k, and a pressure range is considered for which u k ~ P2 P ` ! u !1,
~38!
then the Gibbs–Duhem relation may be integrated to obtain
m L 5 m ~ P ` ,T ! 1 n L` ~ P L 2 P ` ! ,
~39!
n L`
is the specific volume of the liquid phase at the where saturation pressure, P ` , corresponding to the temperature. If the vapor phase is approximated as an ideal gas, then integrating the Gibbs–Duhem relation gives
F G
V ¯ T ln P , m V 5 m ~ P ` ,T ! 1R ~40! P` where ¯R is the ideal gas constant. It should be noted that the functional forms of the chemical potentials of the liquid and vapor phases are different. This difference plays a critical role in what follows. For the liquid phases, Eq. ~1! gives
m ~ P ` ,T ! 1 n L` ~ P L 2 P ` ! 1Wgz5l † .
~41!
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Since l † is a constant, this equation may be evaluated at two different vertical positions, z 1 and z 2 , and then subtracted to give P L~ z 2 ! 2 P L~ z 1 ! 5
2Wg
n L`
~ z 2 2z 1 ! .
~42!
P Lu 5 P ` 1
FS D
G
2Wg ~ z 2 2z 1 ! . ¯R T
~43!
If Eq. ~1! is applied at any position on either side of the liquid–vapor interface, then one finds
m L ~ T, P LI ! 5 m V ~ T, P VI ! ,
~44!
where the subscript I on a property indicates that the property is to be evaluated at the interface but in the phase indicated by the superscript. Since the chemical potential functions have different dependencies on the pressure ~i.e., functional forms! one finds from Eqs. ~39! and ~40! that a certain relation must exist between the pressures in the respective phases at the interface:
F
G
n L` ~ P LI 2 P ` ! . ¯R T
P VI 5 P ` exp
~45!
From the Laplace equation the pressure difference across the interface may be expressed in terms of the mean radius of curvature, R LV , and if Eq. ~45! is then used to eliminate, P VI , one finds P LI 5 P `
F
G
n L` 2 g LV exp ~ P LI 2 P ` ! 2 LV . ¯R T R
~46!
Note that if the value of the mean radius of curvature is known, Eq. ~46! may be solved ~iteratively! for P LI . With this value of P LI , one may use Eq. ~45! to obtain the corresponding value of P VI . Thus, for a given temperature P LI and P VI may be viewed as functions of R LV . Let a property associated with the upper liquid–vapor interface be indicated by a subscript u, and one with the lower interface by a subscript l. On the upper interface, the relation between the liquid and vapor phase pressures can be determined by rewriting Eq. ~45!, P Lu 5 P ` 1
¯R T
n L`
F G
ln
P Vu
P`
,
~47!
and similarly on the lower interface P Ll 5 P ` 1
¯R T
n L`
F G
ln
P Vl
P`
.
~48!
In the vapor phase, the relation between the pressure at a point on the upper interface and that at a point on the lower interface, a distance h below, is seen from Eq. ~43! to be given by
F G
P Vu 5 P Vl exp
2Wgh . ¯R T
After Eq. ~49! is substituted into Eq. ~47! one finds
~49!
ln
n L`
P Vl
P`
2
Wgh
n L`
~50!
.
After making use of Eq. ~48!, Eq. ~50! may be written P Lu 5 P Ll 2
Following the same procedure in the vapor phase yields P V @ z 2 # 5 P V @ z 1 # exp
F G
¯R T
Wgh
n L`
~51!
.
For a given value of the pressure in the liquid phase at the upper interface, eq. ~51! indicates that the pressure in the liquid phase at the lower interface would be the same as that which would have existed at the lower interface if the intervening region had been filled with liquid.
B. Relation between mean radii of curvature and separation distance
We may now develop a method to determine the curvature at one interface in terms of that at the other and the difference in interface elevation. If Eq. ~31! is applied at the lower interface and then, using Eqs. ~49! and ~50!, the pressures in the liquid and vapor phases at the lower interface are written in terms of those at the upper interface, one finds
F GS
P Vu exp
D
Wgh Wgh 2 g LV 2 P Lu 1 L 5 LV . ¯R T n` Rl
~52!
After applying Eq. ~31! at the upper interface, solving for P Lu , and substituting the result into Eq. ~52!, one can solve for R LV l : 2 g LV R LV u
R LV l 5 2g
LV
2
Wgh
n L`
LV V R LV u 1R u P u
S
Wgh exp 21 ¯R T
D
. ~53!
Recall that for given values of R LV u and T, the value of can be determined from Eqs. ~45! and ~46!. Thus, Eq. in terms of four vari~53! represents an expression for R LV l ables, the upper interface radius, R LV , the vertical separation u of the interfaces, h, the temperature, and g. We may now calculate the equilibrium, axial pressure profile for an isothermal capillary system that is in the twointerface configuration. Suppose the axial mean radius of curvature at the upper interface is known and the lower interface is at a known distance h away. Knowledge of R LV u allows P Lu and P Vu to be determined from Eqs. ~45! and ~46!. Then, from knowledge of h, the pressure in the liquid phase at the lower interface may be determined from Eq. ~51! and the mean radius of curvature, R LV from Eq. ~53!. Since the l pressures in the liquid phase at the interface would then be known, Eqs. ~42! and ~43! could then be used to calculate the pressure in each phase away from the interface. Using this method, a pressure profile has been calculated for a particular case and the result is shown in Fig. 2. Although the vapor phase pressure appears constant in Fig. 2, it does follow the relation given by Eq. ~43!. P Vu
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J. Chem. Phys., Vol. 109, No. 9, 1 September 1998
C. A. Ward and M. R. Sasges
C. Necessary difference in upper and lower contact angles
As may be seen from Eq. ~53!, the mean radii of curvature of the upper and lower interfaces are not in general equal. If the values of T,L,H,N,g, u u are given, we may now determine the effect of the difference in the mean radii of curvature on the contact angles at the two three-phase lines. In order to determine the magnitude of the contact angle difference, it is necessary to find the shape of the liquid– vapor interface under the influence of gravity. The radial coordinate of the interface is a function of f, and of the radius of the curvature that originates at the cylinder axis ~see Fig. 1!: x I 5sin f R 2 ~ f ! .
~54!
From the differential geometry for a given interface, as shown in Fig. 1~D!, the following relations can be found: dx I 5R 1 ~ f ! cos f d f ,
~55!
dz I 5R 1 ~ f ! sin f d f .
~56!
The Laplace equation relates the liquid–vapor interface curvatures to the pressures of the liquid and vapor regions, while Eqs. ~42! and ~43! yield the variation of these pressures with elevation. Combining these three equations, and using the radius of the cylindrical container, L, as the length scale for the problem, Eqs. ~55! and ~56! can be written as dx 8 5
cos f d f sin f q~ f !2 x8
~57!
value of R LV a has been assumed, the maximum radial dimension of the interface will be equal to unity when the turning angle reaches a value of p /22 u : x 8 ~ f max! 51.
p L z z 0l n L`
and ~58!
where x 8 and z 8 are the nondimensionalized radial and vertical coordinates of the interface and the following definitions have been introduced: q ~ f ! [2 k ~ f !
B[
WgL 2
n L` g LV
k ~ f ! [exp
L
~ k ~ f ! 21 ! L P La
Ra
g LV
LV 1
~59! ~60!
,
S
1Bz 8 ~ f ! ,
D
2WgL ~ z 8 ~ f ! 2z 8a ! , ¯R T
~61!
where the subscript a indicates that a quantity is to be evaluated on the interface at the cylinder axis. and the shape of the liquid–vapor The value of R LV a interface may be found as follows. An initial value for R LV a is assumed, and the corresponding value of liquid pressure on the axis then calculated from Eq. ~46!. Using this calculated value of the axial pressure, Eqs. ~57! and ~58! can be integrated numerically to obtain x 8 ( f ) and z 8 ( f ). If the correct
~62!
If the equality indicated in Eq. ~62! is not met, a new value of R LV a is chosen and the process repeated until equality exists. An interface calculated in this way satisfies all the necessary conditions for equilibrium that are applicable at that interface. The other interface may be determined from an iterative procedure. Let the elevation of the first interface be z 0 , and make the hypothesis that a second interface exists at a separation distance h away. By using Eq. ~53!, the mean curvature at the second interface may be determined and Eqs. ~57! and ~58! integrated numerically. The pressures in the bulk phases throughout the system may be determined from Eqs. ~42! and ~43!. The number of moles in this hypothesized configuration may be determined. The volumes of the adsorbed phases are neglected according to the Gibbs dividing surface approximation, and the number of moles in the adsorbed phases may be neglected compared to those in the liquid and vapor phases. The density of the vapor varies with pressure according to the ideal gas law, and it is assumed that over the range of pressures considered the liquid specific volume is equal to n L` . Referring to Fig. 2 and proceeding analogously to Eq. ~9!, the number of moles in this hypothetical configuration may be expressed as N5
sin f d f , dz 8 5 sin f q~ f !2 x8
3657
1
E
1
E
1
pLz
~ z 0l 1h !
z bu ~ z 0l 1h !
z bu
n L` 1
n L`
~ H2z 0l 2h ! 1
E
z 0l
p ~ L 2 2x Iu ! dz1
PV p x 2Iu dz1 ¯R T
E
z bu
z bl
1
z bl
E
n L` z bl
z 0l
p ~ L 2 2x Il ! dz
PV p x 2Il dz ¯R T
PV p L 2 dz. ¯R T
~63!
After carrying out the integrals of Eq. ~63! numerically, one can compare the calculated number of moles with the number known to exist in the system. If these quantities are not the same, then a new separation distance is hypothesized and the process repeated until the equilibrium separation that satisfies the constraint is found. Since the liquid density has been assumed constant, the choice of z 0 does not effect the separation distance. The cylindrical volume being considered is characterized by its height and radius, H and L. Using the calculation technique outlined and given values of T, L, H, N, g, and u u , contact angles at the lower interface have been calculated. The position of the lower interface, z 0 , has little effect on the contact angle difference. The important factor is the separation distance between the two interfaces. For a given volume, the separation distance depends strongly on the cylinder diameter and the number of moles in the system, but is almost independent of how much of the liquid phase is above or below the vapor phase. Thus, in Fig. 3 the difference in contact angles of the upper and lower interfaces has been
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J. Chem. Phys., Vol. 109, No. 9, 1 September 1998
FIG. 3. Contact angle difference vs interface separation: ~A! for a 1.2-mmdiam cylinder and ~B! for a 2.0-mm-diam cylinder, each partially filled with water at 25 °C. The dashed portions of the curves indicate the conditions under which the pressure in the vapor phase becomes greater than the saturation vapor pressure. Curves have been plotted for upper contact angles of 0°, 10°, 20°, 30°, 40°, and 60°.
plotted as a function of the ratio of the separation distance to the cylinder radius. For the case considered, the value of the contact angle difference is seen to be always positive, indicating that at equilibrium the lower contact angle is predicted to be larger than the upper. The dashed portions of these figures indicate conditions where the vapor phase pressure is greater than the saturation pressure corresponding to the temperature. The iterative procedure used to calculate the liquid– vapor interface corresponding to a given contact angle also enables the dependence of pressure on contact angle to be determined. Recall that once the interface that corresponds to a given contact angle has been calculated, the pressures on the interface at the axis of the cylinder are also known. These pressures have been tabulated for water in a 1.2-mm-diam cylinder at 25 °C for both upper and lower interfaces, and the results may be seen in Fig. 4. For contact angles less than 90°, the pressure in the vapor is less than the saturation pressure, while for contact angles greater than this value, the pressure in the vapor phase is greater than the saturation value. To determine the relation between the contact angles of the upper and lower interface in general, one may again consider Eq. ~53!. Since for all systems of physical interest,
C. A. Ward and M. R. Sasges
FIG. 4. ~A! Equilibrium pressure in the vapor phase vs upper contact angle. ~B! Equilibrium pressure in the liquid phase vs upper contact angle. These calculations have been carried out for water in a 1.2-mm-diam cylinder at 25 °C. For contact angles less than 90°, the pressure is less than the saturation value, while for larger contact angles, the pressure is greater than the saturation value.
Wgh !1, ¯R T
~64!
the exponential appearing in Eq. ~64! may be expanded and only the first term retained. Then, if the vapor is approximated as an ideal gas, Eq. ~53! may be written R LV l
5 R LV u
S
1 12
R LV u Wgh LV 2g
S
1
1 L2 V n` n
DD
~65!
and since 1 1 @ n L` n V
~66!
one may conclude from Eq. ~65! that LV R LV l .R u .
~67!
Since the mean radius of curvature at the lower interface is larger at each point of the interface than the mean radius of curvature directly above on the upper interface, it follows that the contact angle on the lower interface will be larger than that of the upper interface. The effect of gravity on the predicted difference in contact angles may be determined from Eq. ~53!. As may be seen there, when g vanishes the mean radius of curvature at the lower interface become equal to mean radius of curvature
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J. Chem. Phys., Vol. 109, No. 9, 1 September 1998
C. A. Ward and M. R. Sasges
directly above on the upper interface. Since the mean radii are equal at all corresponding points on the two interfaces, the contact angles would also be the same.
In Sec. IV, it was shown that if the contact angle at one of the interfaces was given along with the experimentally controllable variables T,L,H,N,g, then the contact angle at the other interface could be predicted. And it has been seen that Eq. ~1! and the Laplace equation lead to the prediction of the contact angle at the lower interface being larger than that at the upper. To determine if this predicted contact angle difference is consistent with the Young equation, we first consider the Gibbs adsorption equation. After differentiating Eq. ~4! and combining with Eq. ~3!, one finds ~68!
j5SL,SV,SL.
When Eq. ~1! is applied at the three-phase line
m SV 5 m SL 5 m LV 5 m L 5 m V .
~69!
After making use of Eqs. ~69! and ~37!, Eq. ~68! may be written n SV d P. nV
d g SV 5 ~ 2s SV 2n SV s SV ! dT2
~70!
For an isothermal change of state, Eq. ~70! may be applied at the upper three-phase line and then integrated to obtain S0 g SV l 2 g 52
E
n SV d P2 nV
V Pu
0
E
V Pl V Pu
n SV dP nV
~71!
and following the same procedure at the upper interface gives S0 g SV u 2 g 52
E
n SV d P. nV
V
Pu
0
~72!
After subtracting Eq. ~72! from Eq. ~71!, one finds that the difference in solid–vapor surface tension at the upper and lower interfaces may be written in terms of the isotherm relation for the solid–vapor interface, n SV (T, P V ): SV g SV l 2 g u 52
E
V
Pl V
Pu
n SV d P. nV
~73!
If the same procedure is followed at the solid–liquid interface, then the result may be written in terms of the isotherm relation at the solid–liquid interface, n SL (T, P L ): SL g SL l 2 g u 52
E
L
Pl L
Pu
n SL n L` d P
~74!
and after subtracting Eq. ~74! from Eq. ~73! one finds SL SV SL ~ g SV l 2 g l ! 2 ~ g u 2 g u ! 52
E
V
Pl V
Pu
n SV d P1 nV
SE
1 2 g LV
V
Pl V
Pu
n SV d P1 nV
E
L
Pl L
Pu
D
n SL n L` d P . ~76!
Since it has been predicted that the contact angle at the lower interface is greater than that at the upper, the left-hand side of Eq. ~76! must be less than zero. Thus,
V. YOUNG EQUATION CONSIDERATIONS
d g j 52s j dT2n j d m j ,
cos u l 2cos u u 5
3659
E
L
Pl L
Pu
n SL n L` d P. ~75!
By combining the Young equation with Eq. ~75!, one obtains
E
L
Pl L
Pu
n SL n L` d P,
E
V
Pl V
Pu
n SV d P. nV
~77!
Equation ~77! represents a restriction on the isotherm relations at the solid–liquid and solid–vapor interface that must be satisfied if the Young equation is to be consistent with the predicted contact angle difference. Further analysis would require the availability of theoretical isotherm relations at the solid–liquid and solid–vapor interface. However, at present such relations do not exist. One might consider applying either the Brunauer–Emmett–Teller ~BET!18 or the Frenkel–Halsey–Hill ~FHH!19–21 relation at the solid–vapor interface; however, as seen in Fig. 4, the pressure in the vapor phase deviates from the saturation vapor pressure by only approximately 1024 %. Since the BET and the FHH equations each predict a nonphysical ~i.e., infinite! value of n SV when the vapor pressure is equal to the saturation pressure, these equations are of doubtful validity when the vapor pressure is so close to the saturation vapor pressure. VI. DISCUSSION AND CONCLUSION
If the two-interface configuration indicated in Fig. 2 is in equilibrium while in a gravitational field, it must satisfy ~1! the Young equation, ~2! the Laplace equation, and ~3! Eq. ~1!. From the last of these conditions, the difference in the pressure in the liquid phase at the upper and at the lower interface is the same as that which would have existed at these positions if the system had been filled with liquid @see Eq. ~51!#. By contrast, the pressure in the vapor phase is almost independent of position. Thus, the difference in pressure across the upper interface is larger than that across the lower interface. When these differences in pressure are used in the Laplace equation, one finds that the contact angle at the upper interface is necessarily smaller than that at the lower interface. As may be seen in Fig. 3, the magnitude of this contact angle difference may be tens of degrees even in simple systems. The magnitude of the predicted contact angle difference is such that it should be possible to examine these predictions experimentally.22 This difference in contact angles would normally be referred to as contact angle ‘‘hysteresis.’’ The analysis presented herein shows that the difference in contact angles simply results from the upper three-phase line being at a higher position in the gravitational field than the lower three-phase line, and that the difference in contact angles at these lines is a thermodynamic property of the system. Thus, using the word hysteresis in naming the phenomena seems to be inappropriate. When the Young equation is used to examine the mechanism by which this contact angle difference can exist in an equilibrium system, it is found that the adsorption isotherms
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J. Chem. Phys., Vol. 109, No. 9, 1 September 1998
at the solid–liquid and solid–vapor interfaces must satisfy Eq. ~77!. Thus, the contact angle difference does not necessarily violate the Young equation, but it does establish certain conditions that the adsorption isotherms must satisfy. ACKNOWLEDGMENTS
The authors wish to thank the Canadian Space Agency and the Natural Science and Engineering Research Council for their financial support of this work. J. W. Gibbs, Trans. Conn. Acad. III, 108 ~1876!; recently republished in The Scientific Papers of J. Willard Gibbs ~Ox Bow, Woodbridge, CT, 1993!, Vol. 1, p. 219. 2 In Ref. 1, p. 319. 3 B. A. Pethica and Y. J. P. Pethica, Proceedings of the Second International Congress on Surface Activity, London ~Butterworths Scientific, London, 1957!, Vol. 3, p. 131. 4 J. J. Bikerman, in Ref. 3, p. 125. 5 R. E. Johnson, J. Phys. Chem. 63, 1655 ~1959!. 6 W. J. Herzberg and J. E. Marian, J. Colloid Interface Sci. 33, 161 ~1970!. 1
C. A. Ward and M. R. Sasges H. Fujii and H. Nakae, Philos. Mag. A 72, 1505 ~1995!. Y. Liu and R. M. German, Acta Mater. 44, 1657 ~1996!. 9 M. Yekta-Fard and A. B. Ponter, J. Colloid Interface Sci. 126, 134 ~1988!. 10 J. Gaydos and A. W. Neumann, J. Colloid Interface Sci. 120, 76 ~1987!. 11 R. J. Good and M. N. Koo, J. Colloid Interface Sci. 71, 283 ~1979!. 12 L. Boruvka and A. W. Neumann, J. Chem. Phys. 66, 5464 ~1977!. 13 C. A. Ward and A. W. Neumann, J. Colloid Interface Sci. 49, 286 ~1974!. 14 F. Bashforth and J. C. Adams, An Attempt to Test the Theories of Capillary Action by Comparing the Theoretical and Measured Forms of Drops of Fluid ~Cambridge University Press, Cambridge, 1883!. 15 R. Weinstock, Calculus of Variations ~Dover, New York, 1974!, p. 48. 16 C. A. Ward, D. Yee, M. R. Sasges, L. Pataki, and D. Stanga, ASME, AMD-Vol. 154/FED-Vol. 142 ~1992!, p. 111. 17 M. R. Sasges, C. A. Ward, H. Azuma, and S. Yoshihara, J. Appl. Phys. 79, 8770 ~1996!. 18 S. Brunauer, P. H. Emmett, and E. Teller, J. Am. Chem. Soc. 60, 309 ~1938!. 19 J. Frenkel, Kinetic Theory of Liquids ~Clarendon, Oxford, 1946!, p. 308. 20 G. D. Halsey, Jr., J. Chem. Phys. 16, 931 ~1948!. 21 T. L. Hill, J. Chem. Phys. 17, 590 ~1949!. 22 M. R. Sasges and C. A. Ward, J. Chem. Phys. 109, 3661 ~1998!, following paper. 7 8
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