- Email: [email protected]
Specific volume and entropy of the formation of the interface between two immiscible liquids
Specific Volume and Entropy of the Formation of the Interface between Two Immiscible Liquids V L A D I S L A V S. M A R K I N l AND A L E X A N D E R G. V O L K O V A. N. Frumkin Institute of Electrochemistry of the USSR Academy of Sciences, 31 Leninsky Prospekt, Moscow 117071, USSR
Received August 8, 1988; accepted April 5, 1989 A rigorous thermodynamic definition is given for the specific characteristics of interface formation: volume, entropy, and energy. The characteristics are calculated by the Hansen method. It is shown that the specificcharacteristics of interface formation coincide with the coefficientsof the Gibbs adsorption equation in the Hansen representation, only for the simplestcase of a two-componenttwo-phasesystem. In general, a system of equations is derived for determining the specificcharacteristics, and its solution is found for some of the most important particular cases, say, for a three-component two-phase system. These results are compared with other approaches which are not general enough and have strong limitations. © 1990 Academic Press, Inc. In the recent decade there has been a great surge of interest in defining the specific volume and entropy of the formation of the interface between two immiscible liquids. The problem was first considered by Gibbs ( 1 ) who defined the quantity (O~/Op) for a two-component system as a change in the volume when the interface area increases by unity. Later, Bridgm a n (2) derived the relation (OV/OA)T,p = (03"/Op)r~ for two immiscible liquids, and it was first noted in (3) that O~//Op has a dimension of length. The term "the volume of surface formation" was introduced by Motom u r a ( 4 - 6 ) who studied this quantity both theoretically and experimentally. He also introduced the concepts of "the entropy and energy of surface formation." An attempt to calculate the entropy of formation of the interface between two immiscible electrolyte solutions was made by Silva (7) who started his analysis with a simple particular case of binary electrolytes selectively soluble in one of the phases only. According to M o t o m u r a ' s definition, the specific volume (entropy, energy) of surface To whom all correspondence should be addressed.
formation is a change in the volume (entropy, energy) of the system when the two phases form an interface of unit area. While calculating these quantities, Motom u r a used the modern and very efficient Hansen method (8). This approach was criticized by G o o d (9-1 1 ), because M o t o m u r a introduced the new reference system. Motom u r a ' s approach is similar to the method of two dividing surfaces frequently employed in thermodynamics of insoluble films (12). In addition M o t o m u r a employed in calculations the quantities r ~ , I'~, Aya, and Ay b which do not have an unambiguous thermodynamic definition and cannot be found from experiment. For the sake of fairness, one should note that M o t o m u r a did not conceal the last circumstance (4). Furthermore, and most important, his mathematical definition of the specific volume and entropy of surface formation does not agree, for a general case, with the original concept. M o t o m u r a was right to admit that a change in the volume (and the entropy) of a system is related to the adsorption of the components at the interface, which results in a change of the molar volume of the components. But if the same c o m p o n e n t has
305 0021-9797/90 $3.00 Journal of Colloid andlmerface Science, VoL 135, No. 2, March 15, 1990
Copyright © 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
306
MARKIN
AND
different molar volumes in different phases, one must know which volume should be taken as a reference. In other words, one should know from which phase the substance comprising the interface excesses was taken or in what proportion this substance was distributed between the phases. By using relation (39) of Ref. (4), Motomura arbitrarily admitted that the substance was completely contained in one phase only. Clearly, this is not true for a general case and, therefore, this arbitrary assumption restricts the applicability of Motomura's relations to particular, though perhaps important, cases. Thus, calculation of the specific volume and entropy of surface formation is generally an open problem, and below we perform such a calculation. H a n s e n "s method. While calculating the specific characteristics describing an interface, we use, along with the well known Gibbs method, a new and very efficient Hansen method (8, 13-16). If the surface excesses are determined by the Gibbs method, the volume of a system acts as a preferable variable: just like the pressure differential, the volume does not enter the adsorption equation because it is assumed from the very beginning that the sum of volumes of homogeneous phases in the reference system is equal to the volume of the real system. In Hansen's method (8) the volume becomes a c o m m o n extensive variable, just like the masses of the components and the system entropy. In geometric language this means that from the very beginning the volume of the reference system is not assumed equal to the volume of the real system, and, therefore, one may speak not only about surface excesses of the components and entropy, but also about surface excess of the volume. Hansen's adsorption equation can be obtained by different methods (14, 16). To save space we start with Gibbs' adsorption equation with absolute adsorptions 1"~: m
dT = - s S d T -
~ Fid~i.
[11
i~l Journal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990
VOLKOV
Using the Gibbs-Duhem equations, m
- s ~ d T + dp - ~ c~d~i = 0 i=1 m
- s ¢ d T + dp - ~ c~id~i = 0,
[2]
i=1
one can eliminate any pair of differentials in Eq. [I], say d~j and d~k. After this transformation Eq. [I] takes the form d v = -s~j,k)dT + r~,k)@ --
['i(j,k)d#i, i4=j,k
[31 where s = s" q- S ~T ~(j,k) @ S~r~j,k~ S(j,k)
[4]
T s(j,k) = T ~(j,k) + ~ j , k ~
[51
Pi(j,k) = P i q- C~7(aj, k) ~- CfliT~(J,k)"
[61
The terms • (5,k~ =
(cyrk
-
drj)/(cfc{
- c~cy),
-r~(j.k) = (C~rj -- Cyrk)/(CyC~k- C~C~)
[7]
are the coefficients of transition from an arbitrary Gibbs initial reference system to the particular Hansen reference system with rj--- r k = 0 .
[8]
This is a conventional definition of a particular Hansen system; it is denoted by the symbol (j, k). By any of the indexesj and k, one can mean either entropy or volume. Then the criterion defining Hansen's reference system may take the form ( s , j ) , (V, k ) , or even (s, V). In the Gibbs method the reference system was defined by only one index (j), but the volume surface excess was assumed equal to 0. Hence, the Gibbs criterion (j) is equivalent to the Hansen criterion ( V , j ) or (j, V); that is why Gibbs' reference system is only a particular case of Hansen's reference system. Hansen's relative excesses, as well as Gibbs', are derivatives of the surface tension with respect to the relevant variables, with the only
307
IMMISCIBLE LIQUIDS INTERFACE
difference being that the derivatives are calculated when the other variables are fixed: S~j,k ) ~"
- ( OT / OT)p,m÷uj,u~
[9]
r~j,~) = (a'r/Op)r,.~j,.~
[10]
I',(,k) = -(O~/Om)r,p**~j,,k.
[11]
Definition of the specific characteristics of surface formation. Let the specific volume of surface formation be denoted by r~orm and the specific entropy of surface formation by S~or,~. In accordance with the definition of the specific volume of surface formation, the calculation of this quantity is carried out with temperature, pressure, and the total number of moles of all the components in the entire system kept constant: T sform = ( O V / O A ) T , p , [ m ] .
[12]
The symbol [nil is defined as IF/el "= h i , n 2 . . . . .
Gibbs' description. For the beginning we shall take an arbitrary Gibbs reference system having the total volume 17, phase volumes V" and V ~, and surface excesses Fi. If the numbers of moles of the Component i in the Phases o~ and/3 are n7 and n~ and their molar volumes are v~ and v~, then the volume of the system is V = ~ (vTn7 + v~n~i).
The derivative of volume with respect to interface area is equal to (OV/OA)T,p,[ni]
= Z [l')~(On~/OM)T,p,[ni] k
+ v~(On~/OA)T,p,~.i~].
[17]
The derivatives (On~/OA)T,p,r,,l and (One~ OA )T,p,t,~l remain to be found. To this end, we use the condition that chemical potentials of the same uncharged component in different phases vary equally:
F/r,
where r is the number of components. The term r~o~m can be expressed through the change in surface tension as a function of pressure if we use the thermodynamical potential of the system G which has the differential
( O # ; / O A ) T , p , [ n i ] = ( O U ~ / O A ) T , p , [ n i ].
By using cross-differentiation we get
[18]
By expressing the chemical potentials ~ in terms of the molar fractions xy of the components, and taking into account that
dxT=(1/n")
dG = - S d T + Vdp + ~dA + ~ ttidn~. [13]
~ ( r j ~ - x;)dnm,
[19]
m
where n" = ~mnm and 6jm is the Kronecker symbol which is 1 f o r j = m and 0 f o r j :~ m, we obtain
~-~orrn = ( O V / O A ) r , p , [ n ~ ] = (O"y/Op)T,A,[n~] ,
[16]
i
[14]
( 1/n~)(Onm/OA)r,p,[n~] and
m
S~o~m = ( OS/ OA )T,p,I,,~ J = -(OT/OT)p,A,[nd.
[15]
The conditions indicated by subscripts are most convenient for experimental investigation of the dependence of surface tension on temperature and pressure. Thus, volume and entropy of surface formation are directly determined in experiment. Let us now analyze how these quantities are related to Gibbs' and Nansen's surface excesses.
= ~ ( 1/n~)(On~/OA)T,p,t< m
J
Now we can consider the equation of material balance for the Component rn: nm=
nm+
n~ + rmA.
[21]
Journal of ColloM and Interface Science, Vol. 135, No. 2, March 15, 1990
308
MARKIN
AND
VOLKOV
Differentiating it with respect to A, one obtains:
are partially soluble in each other. Under these conditions the solution is of the form
( On~n/ OA )T,p, tn~] + ( O n ~ / OA )r,p,~,l
a - f o . . -- - [ r ~ ( c ~
"q- ~m -{- A(OPm/Om)T,p,[n~l
= 0.
(Onm/A)T,p,[n~] + (OnBrn/A)T,p,[ni] + P m
= O.
[231 In this section we restrict our consideration to the case of soluble components. Using the systems of linear Eqs. [20 ] and [23] we can readily determine the derivatives in question, and by substituting them into Eq. [17] we can solve our problem. As follows from Eq. [23], here the terms --( Onm/ OA )T,p,[ni] and - ( O n ~ / OA )T,p,i~,] show the amount of Component m adsorbed per unit area of the dividing surface from the Phases a and/3, respectively. This question was not considered in Motomura's theory (7). Note that these quantities may have different signs. So if they are positive, they are really adsorptions from corresponding phases. If they are negative, they are rather desorptions from interface. These two terms can even compensate for each other; it would mean that in this case there is no resulting adsorption of a given component, but rather a transition of it from one phase into another due to formation of a new interface. Unfortunately, the solution of Eqs. [ 20 ] and [23] in a general form, though not difficult in principle, yields rather cumbersome expressions. In order to grasp the general idea of the final results, we consider a few simple examples. B i n a r y system. This case is of special interest. Let us assume that Component 1 is the main component in the Phase a, and Component 2 in the Phase ~, and that the liquids Journalof Colloidand InterfaceScience,Vol. 135, No. 2, March
15, 1990
c~) + r2(c~
( c ~ { c ~ - c~c{)
[22]
This is a general balance equation. But very often it can be simplified if Component m is soluble at least in one phase and the total amount of this component in the system is large compared with surface e x c e s s PmA. Then the last term in Eq. [22 ] can be omitted and one obtains
-
-
e~l)l/
T (s1 , 2 ) .
[241
Thus, the specific volume of surface formation in a binary system equals Hansen's volume excess with respect to Components 1 and 2. This is an obvious result which could have been obtained directly from Eq. [16 ]. When considering the pressure derivative of surface tension we can note that according to the phase rule the state of a two-phase binary system is determined by two intensive parameters. If temperature and pressure are taken as such parameters, then: (O'y/Op)T,A,[ni] = (O'y/Op) = a-~1,2). [25]
It is this result that was obtained by a different method in Eq. [25]. Next, we derive equations for the masses of Components 1 and 2 adsorbed from Phases a and t3 per unit area on the dividing surface:
(On ? / OA ) r.p,.,,n2 = x~f(x~P2 - X~FI)/(X~{X~2 -- X~lX~), [261
= X~I(X~F1 -- x T r 2 ) / ( x T x ~ 2 - x ~ x ~ ) ,
[27]
( On~/ OA ) r,p,.1,.~ = x~(x~Pz - x~Pl)/(xTx~
- x~x'~), [28]
= x~(x~P1 - x ° { P z ) / ( x ~ x ~ - x ~ x ~ ) .
[29]
Once these quantities have been determined we can find not only the volume of surface formation, a-Sfo~m,but also the contribution to this quantity of each of the phases, r ~o~ and a- fBorm: "/'form = 1) 1
nl
p,nl,n 2
-}- l)~( Ort'~/ OA )T,p, nl,n2 -- ( c ~ l r 2 -
c~V~)/(c~c~ - c ~ d ) .
[301
IMMISCIBLE
LIQUIDS I N T E R F A C E
And in an analogous way
T~orm : (C~F1 - c~F2)/(c~c~ - c~c$). [31] It can be readily shown that Y~orm :
T~orm + Tform.
[32]
Furthermore T~orm = T~1,2 )
and
7--ff~orrn= 7-~1,2)-
[33]
It should be noted that the expression for the volume of surface formation contains no derivatives of the Otz~/Oxk type, which are present in Eqs. [20] and [23]. This is a remarkable feature of the binary system for which Eqs. [20] and [23] are singular. This means that the molar fractions of the components and, consequently, the chemical potentials of the binary system do not undergo the slightest change during interface formation. They remain strictly constant: d~7 = d ~ = 0.
[34]
This holds true for a binary system only. The addition of one more component essentially alters the situation. Nevertheless, the results obtained for the binary system can be used as a basis for describing more complicated systems. For this purpose Hansen's method is useful.
Use of the Hansen method for describing the volume of surface formation. If the phase volumes in Hansen's reference system, defined by the criterion ([1], [2]), equal V(~,2) and V ~,2), the volume of the real system is V = V(al,2) -~- V~I,2 ) ~- 7"~1,2)A
= Z (vTn7 + v~n~i) + rs(,,2)A.
[35]
309
It should be noted that the summation is carried over all the components, including 1 and 2, although they are not adsorbed in Hansen's system under consideration. However, as the area of the interface changes, all the components, including 1 and 2, are redistributed between the phases. This leads to a change in the volume of the system. The main system of Eqs. [20] and [23], for determining the derivatives On~/OA and On~/OA varies only slightly: Eq. [20] remains unchanged, while instead of Eq. [23] we have (Onm/OA)T,p,[nil + (On~/OA)T,p,[ni] -[- I~m(l,2) = 0.
[38]
Here we also neglect the term A(O~m(1,2)/ O.4)r,p,[,,] for the reason that all the components are soluble and the system volume is large. The possible role of this term is considered later. By solving the system of Eqs. [ 20 ] and [ 38 ], and substituting the results into Eq. [17], we obtain the volume of surface formation. We now illustrate this scheme by a concrete example. Ternary two-phase system. Let us assume that the system consists of Components 1, 2, and 3, each of which may, in principle, be present in any of the two phases. Components 1 and 2 are solvents which mainly determine Phases a and ft. The problem can be solved without approximation; however, the final equations are again very cumbersome. To simplify them, we consider dilute solutions: x2 ~ 1, x~ ~ 1, x{ ~ 1, x3~ ~ 1. Furthermore, we assume that the solutions are ideal, i.e., tzi = #/0 + R T In xi.
/zal,2
The total amount of Component i is
ni = n7 + n~i + I'i(1,2)A.
[36]
If i equals 1 or 2, the last term in this sum disappears. In this case the volume of surface formation is s o~ 7"(orrn = 7-(1,2) -1- ~ [ rotk(Onk/OA)T,p,[ni] k
+ V~k(On~/OAr,p,[,,1].
From these assumptions we obtain the volume of surface formation in the form of a series s s "/-form = 7 " ( 1 , 2 ) + [(v~ -
o~ c~ {[l)3c3V(1,2)-~-
'/)3~3c3flV~1,2)]
v~;)dv~1,2) + (v~ - v~)c~Z5,2)]
× (x~3 - x~) } r3(~,~)/(c~zS,2) + c~3v~1,~)). [391
[37]
In deriving this equation we assumed that the volumes of Phases o~ and/3 are comparable. JournalofColloidandInterfaceScience,Vol. 135, No. 2, March 15, 1990
310
MARKIN
AND
It is important to note that the volume of surface formation depends not only on the intensive variables, but also on the extensive quantities, V ~1,2)and V f~,2). This dependence can disappear. If, however, we neglect the mutual solubility of Components 1 and 2 and assume that Component 3 is present only in the phase o~, the volume r~o~mwill no longer depend on V~1,2) and V~1,2). In this case, T ~OITn = "/'form S - - U3c~F 3 ( 1 , 2 ) .
[40]
For the particular case in question, this equation was obtained by Motomura (7). Let us now assume that the volume of one phase considerably exceeds that of the other phase: V" ~ V e. Then, returning to the starting system, we have Z~o~n =
7/'(1,2) '
-
V~ 3 1n3(1,2).
[41]
Here the volume ~-~o~mdoes not depend on the extensive variables since during the formation of a new surface area the adsorption takes place only from the large phase/3. In the experiment, the volume of surface formation is determined by the pressure dependence of surface tension. Experiments are usually carried out in such a way that the quantity r(o~m is determined directly. It should be noted that the volume of surface formation is very small. For some binary systems, studied in (4-6, 17-22), this quantity is positive and does not exceed 0.04 nm. For decanol-water and dodecanol-water binary systems, r~o~mis negative and amounts to -0.0087 and -0.0680 nm, respectively (5, 6). The volumes of surface formation have been thoroughly measured in numerous experiments carried out in Motomura's (4-6, 17-20) and Lin's (21, 22) laboratories. For ternary and multicomponent systems containing very active surfactants, the adsorption of these surfactants may be quite significant, and the term v3~F3(1,2)in Eq. [41] may be very high. According to Motomura's estimate (11), for the reasonable values of the parameters the term can be as high as 1 rim. In this case, however, Hansen's volume excess Journal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990
VOLKOV
7~1,2 ) will increase considerably, so that the experimentally measured quantity • ~o~mneed not change noticeably. Indeed, for some surfactants at the hexane-water interface the quantity r~o~mis rather low and does not exceed 0.03 n m ( 1 7 ) .
Other characteristics of surface formation. Up to now we have derived relations only for the specific volume of surface formation. However, using these relations and substituting entropy for the volumes, we can easily derive analogous expressions of the specific entropy for surface formation:
Sfor~ = S'(1,2) + ~ [s~(On~/OA)r,p, tn,l k
+ s~(On~/OA)r,p,tm].
[42]
System [20], [41], and [42] makes it possible to calculate the specific entropy of interface formation. Since the energy of the system can be represented as Urorm = 3' +
TsSform-P~'}orm,
[431
the method can be extended to the specific energy of surface formation. It is assumed, of course, that temperature and pressure in the system remain unchanged, and that the total masses of the components are kept constant. For the specific Helmholtz free energy we obtain J~orm = " / - P~orm.
[44]
The Gibbs free energy can be defined differently (23). If the energy G is defined in terms of the variables T, p, A, and ni we have for this characteristic gfo~m = %
[451
Finally, if the free energy G is defined in terms og T, p, and ni, the corresponding energy of surface formation vanishes: g f o r l n ~--- 0 .
Thus, we present a strict definition of the specific characteristics of fiat interface formation. It is more consistent than the defini-
311
IMMISCIBLE LIQUIDS INTERFACE
tions proposed both by Motomura and Silva (4, 7). One of the main distinctions is that in present calculations we have taken into account the "origins" of the surface excesses, in the sense that their fractions were found which belonged to different phases prior to surface formation. Another essential distinction is that component redistribution occurring during the formation of a new surface was taken into account. That is why in Eqs. [40] and [42] summation is performed over all the components, even over those whose adsorption is 0. The elasticity of surface excesses. The previous calculations were mainly carried out under an assumption that the amount of substance adsorbed at the interface is small in comparison with the total amount of substance in the overall system. That is why we assume that the surface excesses remain unchanged under surface extension, i.e.,
( OPk(1,2)/ OA )r,p,[~] = O. The assumption is quite reasonable, provided the components have finite solubility in at least one of the phases. However, sometimes the system can contain an impurity of such a high surface activity that virtually all the impurity is located at the interface. In this case the extension of the surface leads to a considerable change in at least one of the surface excesses which precisely describes this surfactant. Hence, substance balance is described in Hansen's representation by the equation
A(OPm(l,2)/OA)T,p,[;i] =
0.
[46]
The last term in this equation, A(OPm(1,2)/ 0A)r,p,l~,], following the Gibbs terminology, may be called the elasticity of surface excess for Component m. Consequently, the analysis performed above referred to 0 elasticity of surface excesses. The elasticity is maximal if all Component rn is located at the interface. Then, l~m{l,2)A = const and
A(OPrn(1,2)/OA)T,p,[ni]=
- l[~rn(l,2).
A(OT"S(l,2)/OA)T,p,[ni] q- ~ [Vm(On~n/OA)r,p,[ni]
T ~OiT~ ~ 9"(1,2) S q-
m
+ v~(On~/aA)r,p,[,~].
[481
Thus, this expression also reveals the elasticity effect: the t e r m A(OT"s(1,z)/OA)T,p,[ni] is the elasticity of the excess volume with respect to Components 1 and 2, Comparing these equations with relations of the preceding section, we see that here the s u m s Fm(1,2) + A(OI~m(1,2)/OA)T,p,[ni]and r{ 1,2)+ A (Or*(t,2)/OA) T,p,tnil act in place of the quantities I~m(1,2) and r{l,2), respectively. Hence, the solutions obtained above will be valid if we make the corresponding substitution. Let us consider, as an example, a ternary system, and let Component 3 be entirely confined at the interface. This means that its elasticity is maximal and that F ~ , z ) + A(OPmO,2)/OA)r,p,[nil = 0.
[49]
It follows then from the relations of the preceding section
(On m/ OA ) r,p,[~] + ( On~,/ OA ) T,p,[n~] ~- F m ( l , 2 ) ~-
Hence, the elasticity of any surface excess ranges from 0 to -Fm(l,2). Equation [20], for determining the derivatives Onm/OA and On~m/OA, also remains unchanged. The solution of this system should be substituted into the expression for the specific volume of surface formation which now takes the form (instead of [ 30 ] )
[47]
r~o~m = rS(~,2) + A(ar~{~,2)/aA)r,p,[,~].
[50]
This is an important result because ifa monolayer of poorly soluble substance is located at the interface, the excess volume r{l,2), with respect to the two solvents, can be rather large about the monolayer thickness. Nevertheless, the volume of surface formation, r(orm, is small because the elasticity of the excess volume r{~,2) strongly reduces this sum. Note in conclusion that the concept of the elasticity of surface excesses, which was deJournal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990
312
MARKIN AND VOLKOV
scribed above, is a direct consequence o f the Gibbs elasticity o f films, which is k n o w n ( 1 ) to be defined by A(O3~/OA). It can be demonstrated that these quantities are closely interrelated: films are mechanically elastic only if surface excesses possess elasticity. As an illustration, we consider a ternary two-phase system, one o f the c o m p o n e n t s o f which possesses high surface activity. The H a n s e n adsorption equation is o f the form
dy
= -sS{1,2)dT + rs(1,2)dp
-
-
F3(l,2)d/2 3.
[511 The chemical potential o f the third c o m p o nent, #3, depends on temperature, pressure, and its surface excesses: #3 = tz3( T , p, F30,2)).
[52]
F r o m the preceding equation, one easily finds the Gibbs film elasticity A ( Oy / OA ) T,v, lnil =
-AI'3(
1,2)( OIA3/OA ) T,p,[ni]
= - A (0Y3(1,2)/014) T,p,I,,I × (0#3/0 In I~3(l,2))T,p.
[53]
Thus, the Gibbs elasticity o f a film is proportional to the elasticity o f the surface excess of the poorly soluble c o m p o n e n t . This relation is also o f great help for various model calculations. REFERENCES 1. Gibbs, J. W., "Collected Works." Dover, New york, 1961.
Journal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990
2. Bridgman, P. W., "The Physics of High Pressure." p, 382. Bell, London, 1952. 3. Defay, R., Prigogine, I., Bellemans, A., and Everett, D. H., "Surface Tension and Adsorption." Longmans, London, 1966. 4. Motomura, K., J. Colloid Interface Sci. 64, 348 (1962). 5. Motomura, K., Aratono, M., Matubayasi, N., and Matuura, R., J, Colloid Interface Sci. 67, 247 (1978). 6. Motomura, K., Matubayasi, N., Aratono, M., and Matuura, R., J. Colloid Interface Sci. 64, 356 (1978). 7. Silva, F., Rev. Port. Quim. 26, 25 (1984). 8. Hansen, R. S., J. Phys. Chem. 66, 410 (1962). 9. Good, R. J., PureAppl. Chem. 48, 427 (1976). 10. Good, R. J., J. Colloid Interface Sci. 85, 128 (1982). 11. Good, R. J., J. ColloidlnterfaceSci. 110, 298 (1986). 12. Rusanov, A. I., "Phasengleigewichteund Grenztl~ichemer Scheinungen." Academie Verlag, Berlin, 1978. 13. Markin, V. S., and Volkov, A. G., Elektrokhimiya 24, 318 (1988). 14. Markin, V. S., and Volkov, A. G., Elektrokhimiya 24, 325 (1988). 15. Markin, V. S., and Volkov, A. G., Elektrokhimiya 24, 478 (1988). 16. Markin, V. S., and Volkov, A. G., in "The Interface Structure and Electrochemical Processes at the Boundary between Two Immiscible Liquids" (V. E. Kazarinov, Ed.), VINITI, Moscow, 1988. 17. Motomura, K., J. Colloid Interface Sci. 110, 294 (1986). 18. Motomura, K., Adv. Colloid Interface Sci. 12, 1 (1980). 19. Matubayasi, N., Motomura, K, Aratono, M., and Matuura, R., Bull. Chem. Soc. Japan 51, 2800 (1978). 20. Matubayasi, N., Motomura, K., Kaneshina, S., Nakamura, M., and Matuura, R., Bull. Chem. Soc. Japan 50, 523 (1977). 21. Lin, M., J. Chim. Phys. 76, 61 (1979). 22. Lin, M., Filpo, J. L., Monsoura, P., and Baret, J. F., J. Chem. Phys. 71, 2202 (1979). 23. Jaycock, M. J., and Parfitt, G. D. "Chemistry of Interfaces." Wiley, New York, 1981.
Received August 8, 1988; accepted April 5, 1989 A rigorous thermodynamic definition is given for the specific characteristics of interface formation: volume, entropy, and energy. The characteristics are calculated by the Hansen method. It is shown that the specificcharacteristics of interface formation coincide with the coefficientsof the Gibbs adsorption equation in the Hansen representation, only for the simplestcase of a two-componenttwo-phasesystem. In general, a system of equations is derived for determining the specificcharacteristics, and its solution is found for some of the most important particular cases, say, for a three-component two-phase system. These results are compared with other approaches which are not general enough and have strong limitations. © 1990 Academic Press, Inc. In the recent decade there has been a great surge of interest in defining the specific volume and entropy of the formation of the interface between two immiscible liquids. The problem was first considered by Gibbs ( 1 ) who defined the quantity (O~/Op) for a two-component system as a change in the volume when the interface area increases by unity. Later, Bridgm a n (2) derived the relation (OV/OA)T,p = (03"/Op)r~ for two immiscible liquids, and it was first noted in (3) that O~//Op has a dimension of length. The term "the volume of surface formation" was introduced by Motom u r a ( 4 - 6 ) who studied this quantity both theoretically and experimentally. He also introduced the concepts of "the entropy and energy of surface formation." An attempt to calculate the entropy of formation of the interface between two immiscible electrolyte solutions was made by Silva (7) who started his analysis with a simple particular case of binary electrolytes selectively soluble in one of the phases only. According to M o t o m u r a ' s definition, the specific volume (entropy, energy) of surface To whom all correspondence should be addressed.
formation is a change in the volume (entropy, energy) of the system when the two phases form an interface of unit area. While calculating these quantities, Motom u r a used the modern and very efficient Hansen method (8). This approach was criticized by G o o d (9-1 1 ), because M o t o m u r a introduced the new reference system. Motom u r a ' s approach is similar to the method of two dividing surfaces frequently employed in thermodynamics of insoluble films (12). In addition M o t o m u r a employed in calculations the quantities r ~ , I'~, Aya, and Ay b which do not have an unambiguous thermodynamic definition and cannot be found from experiment. For the sake of fairness, one should note that M o t o m u r a did not conceal the last circumstance (4). Furthermore, and most important, his mathematical definition of the specific volume and entropy of surface formation does not agree, for a general case, with the original concept. M o t o m u r a was right to admit that a change in the volume (and the entropy) of a system is related to the adsorption of the components at the interface, which results in a change of the molar volume of the components. But if the same c o m p o n e n t has
305 0021-9797/90 $3.00 Journal of Colloid andlmerface Science, VoL 135, No. 2, March 15, 1990
Copyright © 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
306
MARKIN
AND
different molar volumes in different phases, one must know which volume should be taken as a reference. In other words, one should know from which phase the substance comprising the interface excesses was taken or in what proportion this substance was distributed between the phases. By using relation (39) of Ref. (4), Motomura arbitrarily admitted that the substance was completely contained in one phase only. Clearly, this is not true for a general case and, therefore, this arbitrary assumption restricts the applicability of Motomura's relations to particular, though perhaps important, cases. Thus, calculation of the specific volume and entropy of surface formation is generally an open problem, and below we perform such a calculation. H a n s e n "s method. While calculating the specific characteristics describing an interface, we use, along with the well known Gibbs method, a new and very efficient Hansen method (8, 13-16). If the surface excesses are determined by the Gibbs method, the volume of a system acts as a preferable variable: just like the pressure differential, the volume does not enter the adsorption equation because it is assumed from the very beginning that the sum of volumes of homogeneous phases in the reference system is equal to the volume of the real system. In Hansen's method (8) the volume becomes a c o m m o n extensive variable, just like the masses of the components and the system entropy. In geometric language this means that from the very beginning the volume of the reference system is not assumed equal to the volume of the real system, and, therefore, one may speak not only about surface excesses of the components and entropy, but also about surface excess of the volume. Hansen's adsorption equation can be obtained by different methods (14, 16). To save space we start with Gibbs' adsorption equation with absolute adsorptions 1"~: m
dT = - s S d T -
~ Fid~i.
[11
i~l Journal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990
VOLKOV
Using the Gibbs-Duhem equations, m
- s ~ d T + dp - ~ c~d~i = 0 i=1 m
- s ¢ d T + dp - ~ c~id~i = 0,
[2]
i=1
one can eliminate any pair of differentials in Eq. [I], say d~j and d~k. After this transformation Eq. [I] takes the form d v = -s~j,k)dT + r~,k)@ --
['i(j,k)d#i, i4=j,k
[31 where s = s" q- S ~T ~(j,k) @ S~r~j,k~ S(j,k)
[4]
T s(j,k) = T ~(j,k) + ~ j , k ~
[51
Pi(j,k) = P i q- C~7(aj, k) ~- CfliT~(J,k)"
[61
The terms • (5,k~ =
(cyrk
-
drj)/(cfc{
- c~cy),
-r~(j.k) = (C~rj -- Cyrk)/(CyC~k- C~C~)
[7]
are the coefficients of transition from an arbitrary Gibbs initial reference system to the particular Hansen reference system with rj--- r k = 0 .
[8]
This is a conventional definition of a particular Hansen system; it is denoted by the symbol (j, k). By any of the indexesj and k, one can mean either entropy or volume. Then the criterion defining Hansen's reference system may take the form ( s , j ) , (V, k ) , or even (s, V). In the Gibbs method the reference system was defined by only one index (j), but the volume surface excess was assumed equal to 0. Hence, the Gibbs criterion (j) is equivalent to the Hansen criterion ( V , j ) or (j, V); that is why Gibbs' reference system is only a particular case of Hansen's reference system. Hansen's relative excesses, as well as Gibbs', are derivatives of the surface tension with respect to the relevant variables, with the only
307
IMMISCIBLE LIQUIDS INTERFACE
difference being that the derivatives are calculated when the other variables are fixed: S~j,k ) ~"
- ( OT / OT)p,m÷uj,u~
[9]
r~j,~) = (a'r/Op)r,.~j,.~
[10]
I',(,k) = -(O~/Om)r,p**~j,,k.
[11]
Definition of the specific characteristics of surface formation. Let the specific volume of surface formation be denoted by r~orm and the specific entropy of surface formation by S~or,~. In accordance with the definition of the specific volume of surface formation, the calculation of this quantity is carried out with temperature, pressure, and the total number of moles of all the components in the entire system kept constant: T sform = ( O V / O A ) T , p , [ m ] .
[12]
The symbol [nil is defined as IF/el "= h i , n 2 . . . . .
Gibbs' description. For the beginning we shall take an arbitrary Gibbs reference system having the total volume 17, phase volumes V" and V ~, and surface excesses Fi. If the numbers of moles of the Component i in the Phases o~ and/3 are n7 and n~ and their molar volumes are v~ and v~, then the volume of the system is V = ~ (vTn7 + v~n~i).
The derivative of volume with respect to interface area is equal to (OV/OA)T,p,[ni]
= Z [l')~(On~/OM)T,p,[ni] k
+ v~(On~/OA)T,p,~.i~].
[17]
The derivatives (On~/OA)T,p,r,,l and (One~ OA )T,p,t,~l remain to be found. To this end, we use the condition that chemical potentials of the same uncharged component in different phases vary equally:
F/r,
where r is the number of components. The term r~o~m can be expressed through the change in surface tension as a function of pressure if we use the thermodynamical potential of the system G which has the differential
( O # ; / O A ) T , p , [ n i ] = ( O U ~ / O A ) T , p , [ n i ].
By using cross-differentiation we get
[18]
By expressing the chemical potentials ~ in terms of the molar fractions xy of the components, and taking into account that
dxT=(1/n")
dG = - S d T + Vdp + ~dA + ~ ttidn~. [13]
~ ( r j ~ - x;)dnm,
[19]
m
where n" = ~mnm and 6jm is the Kronecker symbol which is 1 f o r j = m and 0 f o r j :~ m, we obtain
~-~orrn = ( O V / O A ) r , p , [ n ~ ] = (O"y/Op)T,A,[n~] ,
[16]
i
[14]
( 1/n~)(Onm/OA)r,p,[n~] and
m
S~o~m = ( OS/ OA )T,p,I,,~ J = -(OT/OT)p,A,[nd.
[15]
The conditions indicated by subscripts are most convenient for experimental investigation of the dependence of surface tension on temperature and pressure. Thus, volume and entropy of surface formation are directly determined in experiment. Let us now analyze how these quantities are related to Gibbs' and Nansen's surface excesses.
= ~ ( 1/n~)(On~/OA)T,p,t< m
J
Now we can consider the equation of material balance for the Component rn: nm=
nm+
n~ + rmA.
[21]
Journal of ColloM and Interface Science, Vol. 135, No. 2, March 15, 1990
308
MARKIN
AND
VOLKOV
Differentiating it with respect to A, one obtains:
are partially soluble in each other. Under these conditions the solution is of the form
( On~n/ OA )T,p, tn~] + ( O n ~ / OA )r,p,~,l
a - f o . . -- - [ r ~ ( c ~
"q- ~m -{- A(OPm/Om)T,p,[n~l
= 0.
(Onm/A)T,p,[n~] + (OnBrn/A)T,p,[ni] + P m
= O.
[231 In this section we restrict our consideration to the case of soluble components. Using the systems of linear Eqs. [20 ] and [23] we can readily determine the derivatives in question, and by substituting them into Eq. [17] we can solve our problem. As follows from Eq. [23], here the terms --( Onm/ OA )T,p,[ni] and - ( O n ~ / OA )T,p,i~,] show the amount of Component m adsorbed per unit area of the dividing surface from the Phases a and/3, respectively. This question was not considered in Motomura's theory (7). Note that these quantities may have different signs. So if they are positive, they are really adsorptions from corresponding phases. If they are negative, they are rather desorptions from interface. These two terms can even compensate for each other; it would mean that in this case there is no resulting adsorption of a given component, but rather a transition of it from one phase into another due to formation of a new interface. Unfortunately, the solution of Eqs. [ 20 ] and [23] in a general form, though not difficult in principle, yields rather cumbersome expressions. In order to grasp the general idea of the final results, we consider a few simple examples. B i n a r y system. This case is of special interest. Let us assume that Component 1 is the main component in the Phase a, and Component 2 in the Phase ~, and that the liquids Journalof Colloidand InterfaceScience,Vol. 135, No. 2, March
15, 1990
c~) + r2(c~
( c ~ { c ~ - c~c{)
[22]
This is a general balance equation. But very often it can be simplified if Component m is soluble at least in one phase and the total amount of this component in the system is large compared with surface e x c e s s PmA. Then the last term in Eq. [22 ] can be omitted and one obtains
-
-
e~l)l/
T (s1 , 2 ) .
[241
Thus, the specific volume of surface formation in a binary system equals Hansen's volume excess with respect to Components 1 and 2. This is an obvious result which could have been obtained directly from Eq. [16 ]. When considering the pressure derivative of surface tension we can note that according to the phase rule the state of a two-phase binary system is determined by two intensive parameters. If temperature and pressure are taken as such parameters, then: (O'y/Op)T,A,[ni] = (O'y/Op) = a-~1,2). [25]
It is this result that was obtained by a different method in Eq. [25]. Next, we derive equations for the masses of Components 1 and 2 adsorbed from Phases a and t3 per unit area on the dividing surface:
(On ? / OA ) r.p,.,,n2 = x~f(x~P2 - X~FI)/(X~{X~2 -- X~lX~), [261
= X~I(X~F1 -- x T r 2 ) / ( x T x ~ 2 - x ~ x ~ ) ,
[27]
( On~/ OA ) r,p,.1,.~ = x~(x~Pz - x~Pl)/(xTx~
- x~x'~), [28]
= x~(x~P1 - x ° { P z ) / ( x ~ x ~ - x ~ x ~ ) .
[29]
Once these quantities have been determined we can find not only the volume of surface formation, a-Sfo~m,but also the contribution to this quantity of each of the phases, r ~o~ and a- fBorm: "/'form = 1) 1
nl
p,nl,n 2
-}- l)~( Ort'~/ OA )T,p, nl,n2 -- ( c ~ l r 2 -
c~V~)/(c~c~ - c ~ d ) .
[301
IMMISCIBLE
LIQUIDS I N T E R F A C E
And in an analogous way
T~orm : (C~F1 - c~F2)/(c~c~ - c~c$). [31] It can be readily shown that Y~orm :
T~orm + Tform.
[32]
Furthermore T~orm = T~1,2 )
and
7--ff~orrn= 7-~1,2)-
[33]
It should be noted that the expression for the volume of surface formation contains no derivatives of the Otz~/Oxk type, which are present in Eqs. [20] and [23]. This is a remarkable feature of the binary system for which Eqs. [20] and [23] are singular. This means that the molar fractions of the components and, consequently, the chemical potentials of the binary system do not undergo the slightest change during interface formation. They remain strictly constant: d~7 = d ~ = 0.
[34]
This holds true for a binary system only. The addition of one more component essentially alters the situation. Nevertheless, the results obtained for the binary system can be used as a basis for describing more complicated systems. For this purpose Hansen's method is useful.
Use of the Hansen method for describing the volume of surface formation. If the phase volumes in Hansen's reference system, defined by the criterion ([1], [2]), equal V(~,2) and V ~,2), the volume of the real system is V = V(al,2) -~- V~I,2 ) ~- 7"~1,2)A
= Z (vTn7 + v~n~i) + rs(,,2)A.
[35]
309
It should be noted that the summation is carried over all the components, including 1 and 2, although they are not adsorbed in Hansen's system under consideration. However, as the area of the interface changes, all the components, including 1 and 2, are redistributed between the phases. This leads to a change in the volume of the system. The main system of Eqs. [20] and [23], for determining the derivatives On~/OA and On~/OA varies only slightly: Eq. [20] remains unchanged, while instead of Eq. [23] we have (Onm/OA)T,p,[nil + (On~/OA)T,p,[ni] -[- I~m(l,2) = 0.
[38]
Here we also neglect the term A(O~m(1,2)/ O.4)r,p,[,,] for the reason that all the components are soluble and the system volume is large. The possible role of this term is considered later. By solving the system of Eqs. [ 20 ] and [ 38 ], and substituting the results into Eq. [17], we obtain the volume of surface formation. We now illustrate this scheme by a concrete example. Ternary two-phase system. Let us assume that the system consists of Components 1, 2, and 3, each of which may, in principle, be present in any of the two phases. Components 1 and 2 are solvents which mainly determine Phases a and ft. The problem can be solved without approximation; however, the final equations are again very cumbersome. To simplify them, we consider dilute solutions: x2 ~ 1, x~ ~ 1, x{ ~ 1, x3~ ~ 1. Furthermore, we assume that the solutions are ideal, i.e., tzi = #/0 + R T In xi.
/zal,2
The total amount of Component i is
ni = n7 + n~i + I'i(1,2)A.
[36]
If i equals 1 or 2, the last term in this sum disappears. In this case the volume of surface formation is s o~ 7"(orrn = 7-(1,2) -1- ~ [ rotk(Onk/OA)T,p,[ni] k
+ V~k(On~/OAr,p,[,,1].
From these assumptions we obtain the volume of surface formation in the form of a series s s "/-form = 7 " ( 1 , 2 ) + [(v~ -
o~ c~ {[l)3c3V(1,2)-~-
'/)3~3c3flV~1,2)]
v~;)dv~1,2) + (v~ - v~)c~Z5,2)]
× (x~3 - x~) } r3(~,~)/(c~zS,2) + c~3v~1,~)). [391
[37]
In deriving this equation we assumed that the volumes of Phases o~ and/3 are comparable. JournalofColloidandInterfaceScience,Vol. 135, No. 2, March 15, 1990
310
MARKIN
AND
It is important to note that the volume of surface formation depends not only on the intensive variables, but also on the extensive quantities, V ~1,2)and V f~,2). This dependence can disappear. If, however, we neglect the mutual solubility of Components 1 and 2 and assume that Component 3 is present only in the phase o~, the volume r~o~mwill no longer depend on V~1,2) and V~1,2). In this case, T ~OITn = "/'form S - - U3c~F 3 ( 1 , 2 ) .
[40]
For the particular case in question, this equation was obtained by Motomura (7). Let us now assume that the volume of one phase considerably exceeds that of the other phase: V" ~ V e. Then, returning to the starting system, we have Z~o~n =
7/'(1,2) '
-
V~ 3 1n3(1,2).
[41]
Here the volume ~-~o~mdoes not depend on the extensive variables since during the formation of a new surface area the adsorption takes place only from the large phase/3. In the experiment, the volume of surface formation is determined by the pressure dependence of surface tension. Experiments are usually carried out in such a way that the quantity r(o~m is determined directly. It should be noted that the volume of surface formation is very small. For some binary systems, studied in (4-6, 17-22), this quantity is positive and does not exceed 0.04 nm. For decanol-water and dodecanol-water binary systems, r~o~mis negative and amounts to -0.0087 and -0.0680 nm, respectively (5, 6). The volumes of surface formation have been thoroughly measured in numerous experiments carried out in Motomura's (4-6, 17-20) and Lin's (21, 22) laboratories. For ternary and multicomponent systems containing very active surfactants, the adsorption of these surfactants may be quite significant, and the term v3~F3(1,2)in Eq. [41] may be very high. According to Motomura's estimate (11), for the reasonable values of the parameters the term can be as high as 1 rim. In this case, however, Hansen's volume excess Journal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990
VOLKOV
7~1,2 ) will increase considerably, so that the experimentally measured quantity • ~o~mneed not change noticeably. Indeed, for some surfactants at the hexane-water interface the quantity r~o~mis rather low and does not exceed 0.03 n m ( 1 7 ) .
Other characteristics of surface formation. Up to now we have derived relations only for the specific volume of surface formation. However, using these relations and substituting entropy for the volumes, we can easily derive analogous expressions of the specific entropy for surface formation:
Sfor~ = S'(1,2) + ~ [s~(On~/OA)r,p, tn,l k
+ s~(On~/OA)r,p,tm].
[42]
System [20], [41], and [42] makes it possible to calculate the specific entropy of interface formation. Since the energy of the system can be represented as Urorm = 3' +
TsSform-P~'}orm,
[431
the method can be extended to the specific energy of surface formation. It is assumed, of course, that temperature and pressure in the system remain unchanged, and that the total masses of the components are kept constant. For the specific Helmholtz free energy we obtain J~orm = " / - P~orm.
[44]
The Gibbs free energy can be defined differently (23). If the energy G is defined in terms of the variables T, p, A, and ni we have for this characteristic gfo~m = %
[451
Finally, if the free energy G is defined in terms og T, p, and ni, the corresponding energy of surface formation vanishes: g f o r l n ~--- 0 .
Thus, we present a strict definition of the specific characteristics of fiat interface formation. It is more consistent than the defini-
311
IMMISCIBLE LIQUIDS INTERFACE
tions proposed both by Motomura and Silva (4, 7). One of the main distinctions is that in present calculations we have taken into account the "origins" of the surface excesses, in the sense that their fractions were found which belonged to different phases prior to surface formation. Another essential distinction is that component redistribution occurring during the formation of a new surface was taken into account. That is why in Eqs. [40] and [42] summation is performed over all the components, even over those whose adsorption is 0. The elasticity of surface excesses. The previous calculations were mainly carried out under an assumption that the amount of substance adsorbed at the interface is small in comparison with the total amount of substance in the overall system. That is why we assume that the surface excesses remain unchanged under surface extension, i.e.,
( OPk(1,2)/ OA )r,p,[~] = O. The assumption is quite reasonable, provided the components have finite solubility in at least one of the phases. However, sometimes the system can contain an impurity of such a high surface activity that virtually all the impurity is located at the interface. In this case the extension of the surface leads to a considerable change in at least one of the surface excesses which precisely describes this surfactant. Hence, substance balance is described in Hansen's representation by the equation
A(OPm(l,2)/OA)T,p,[;i] =
0.
[46]
The last term in this equation, A(OPm(1,2)/ 0A)r,p,l~,], following the Gibbs terminology, may be called the elasticity of surface excess for Component m. Consequently, the analysis performed above referred to 0 elasticity of surface excesses. The elasticity is maximal if all Component rn is located at the interface. Then, l~m{l,2)A = const and
A(OPrn(1,2)/OA)T,p,[ni]=
- l[~rn(l,2).
A(OT"S(l,2)/OA)T,p,[ni] q- ~ [Vm(On~n/OA)r,p,[ni]
T ~OiT~ ~ 9"(1,2) S q-
m
+ v~(On~/aA)r,p,[,~].
[481
Thus, this expression also reveals the elasticity effect: the t e r m A(OT"s(1,z)/OA)T,p,[ni] is the elasticity of the excess volume with respect to Components 1 and 2, Comparing these equations with relations of the preceding section, we see that here the s u m s Fm(1,2) + A(OI~m(1,2)/OA)T,p,[ni]and r{ 1,2)+ A (Or*(t,2)/OA) T,p,tnil act in place of the quantities I~m(1,2) and r{l,2), respectively. Hence, the solutions obtained above will be valid if we make the corresponding substitution. Let us consider, as an example, a ternary system, and let Component 3 be entirely confined at the interface. This means that its elasticity is maximal and that F ~ , z ) + A(OPmO,2)/OA)r,p,[nil = 0.
[49]
It follows then from the relations of the preceding section
(On m/ OA ) r,p,[~] + ( On~,/ OA ) T,p,[n~] ~- F m ( l , 2 ) ~-
Hence, the elasticity of any surface excess ranges from 0 to -Fm(l,2). Equation [20], for determining the derivatives Onm/OA and On~m/OA, also remains unchanged. The solution of this system should be substituted into the expression for the specific volume of surface formation which now takes the form (instead of [ 30 ] )
[47]
r~o~m = rS(~,2) + A(ar~{~,2)/aA)r,p,[,~].
[50]
This is an important result because ifa monolayer of poorly soluble substance is located at the interface, the excess volume r{l,2), with respect to the two solvents, can be rather large about the monolayer thickness. Nevertheless, the volume of surface formation, r(orm, is small because the elasticity of the excess volume r{~,2) strongly reduces this sum. Note in conclusion that the concept of the elasticity of surface excesses, which was deJournal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990
312
MARKIN AND VOLKOV
scribed above, is a direct consequence o f the Gibbs elasticity o f films, which is k n o w n ( 1 ) to be defined by A(O3~/OA). It can be demonstrated that these quantities are closely interrelated: films are mechanically elastic only if surface excesses possess elasticity. As an illustration, we consider a ternary two-phase system, one o f the c o m p o n e n t s o f which possesses high surface activity. The H a n s e n adsorption equation is o f the form
dy
= -sS{1,2)dT + rs(1,2)dp
-
-
F3(l,2)d/2 3.
[511 The chemical potential o f the third c o m p o nent, #3, depends on temperature, pressure, and its surface excesses: #3 = tz3( T , p, F30,2)).
[52]
F r o m the preceding equation, one easily finds the Gibbs film elasticity A ( Oy / OA ) T,v, lnil =
-AI'3(
1,2)( OIA3/OA ) T,p,[ni]
= - A (0Y3(1,2)/014) T,p,I,,I × (0#3/0 In I~3(l,2))T,p.
[53]
Thus, the Gibbs elasticity o f a film is proportional to the elasticity o f the surface excess of the poorly soluble c o m p o n e n t . This relation is also o f great help for various model calculations. REFERENCES 1. Gibbs, J. W., "Collected Works." Dover, New york, 1961.
Journal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990
2. Bridgman, P. W., "The Physics of High Pressure." p, 382. Bell, London, 1952. 3. Defay, R., Prigogine, I., Bellemans, A., and Everett, D. H., "Surface Tension and Adsorption." Longmans, London, 1966. 4. Motomura, K., J. Colloid Interface Sci. 64, 348 (1962). 5. Motomura, K., Aratono, M., Matubayasi, N., and Matuura, R., J, Colloid Interface Sci. 67, 247 (1978). 6. Motomura, K., Matubayasi, N., Aratono, M., and Matuura, R., J. Colloid Interface Sci. 64, 356 (1978). 7. Silva, F., Rev. Port. Quim. 26, 25 (1984). 8. Hansen, R. S., J. Phys. Chem. 66, 410 (1962). 9. Good, R. J., PureAppl. Chem. 48, 427 (1976). 10. Good, R. J., J. Colloid Interface Sci. 85, 128 (1982). 11. Good, R. J., J. ColloidlnterfaceSci. 110, 298 (1986). 12. Rusanov, A. I., "Phasengleigewichteund Grenztl~ichemer Scheinungen." Academie Verlag, Berlin, 1978. 13. Markin, V. S., and Volkov, A. G., Elektrokhimiya 24, 318 (1988). 14. Markin, V. S., and Volkov, A. G., Elektrokhimiya 24, 325 (1988). 15. Markin, V. S., and Volkov, A. G., Elektrokhimiya 24, 478 (1988). 16. Markin, V. S., and Volkov, A. G., in "The Interface Structure and Electrochemical Processes at the Boundary between Two Immiscible Liquids" (V. E. Kazarinov, Ed.), VINITI, Moscow, 1988. 17. Motomura, K., J. Colloid Interface Sci. 110, 294 (1986). 18. Motomura, K., Adv. Colloid Interface Sci. 12, 1 (1980). 19. Matubayasi, N., Motomura, K, Aratono, M., and Matuura, R., Bull. Chem. Soc. Japan 51, 2800 (1978). 20. Matubayasi, N., Motomura, K., Kaneshina, S., Nakamura, M., and Matuura, R., Bull. Chem. Soc. Japan 50, 523 (1977). 21. Lin, M., J. Chim. Phys. 76, 61 (1979). 22. Lin, M., Filpo, J. L., Monsoura, P., and Baret, J. F., J. Chem. Phys. 71, 2202 (1979). 23. Jaycock, M. J., and Parfitt, G. D. "Chemistry of Interfaces." Wiley, New York, 1981.