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Surface Tension of Ternary Solutions
Journal of Colloid and Interface Science 213, 360 –370 (1999) Article ID jcis.1999.6110, available online at http://www.idealibrary.com on
Surface Tension of Ternary Solutions Stanisław Lamperski Department of Physical Chemistry, Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznan´, Poland Received July 15, 1998; accepted January 28, 1999
A theory of the surface tension of ternary solutions is presented. It assumes different molecular sizes of individual components of the solution and takes into account intermolecular interactions both at the surface and in the bulk. The influence of these interactions on the surface tension behavior including the surface buffering effect and the surface phase transition is discussed. © 1999 Academic Press Key Words: surface tension; adsorption isotherm; ternary solution; intermolecular interactions; surface phase transition.
INTRODUCTION
In 1925 Seith (1) reported the results on surface tension of aqueous solutions of different organic solvents (surface active) with varying concentration of salts. He found that the surface tension curves plotted as a function of the surface active solvent concentration at different salt concentrations have a common point of intersection. This result was confirmed by Semenchenko (2) who referred to this phenomenon as the surface buffering effect. More recently Lebed and Eddin (3) observed a similar intersection of the surface tension curves for aqueous solutions of two organic liquids. The surface tension results for one of the systems they studied are shown in Fig. 1. This effect has not been explained theoretically. It motivated us to formulate a theory which would explain not only this effect, but would give a general description of the surface behavior of ternary solutions. Different equations and theories have been used to describe the surface tension of the ternary or multicomponent solutions. Li and Joos (4) extended the Szyszkowski (5) equation to the nonperfect systems. Shain and Prausnitz (6) derived a simple logarithmic dependence of the surface tension on the quotient of the interface and bulk activity of the component. Handa and Mukerjee (7) proposed a more extended form of this equation. Fu et al. (8) eliminated the surface activities from the Shain and Prausnitz equation assuming different sizes of individual components. Lamperski (9) derived equations for the surface composition and the surface tension of the irregular, athermal multicomponent solutions. Dynarowicz and Jawien´ (10) extended a regular solution theory to the ternary systems of different molecular sizes. Pandey and Pant (11) employed Flory’s (12, 13)
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statistical theory to obtain characteristic parameters of the ternary solutions, which allowed them to calculate the surface tension. Next Pandey et al. (14) extended that theory on quaternary systems. Cornellise et al. (15) and Hu et al. (16) used the Peng–Robinson (17) equation of state to calculate the surface tension of the binary and ternary systems. Zou and Stenby (18) developed a simplified gradient theory model, which predicts the surface tension for these systems with a reasonable accuracy. In a recent paper we developed a theory of the surface tension of two-component solutions (19) which takes into account different molecular sizes and intermolecular interactions. Generally it was an extension of the Bragg–Williams theory to the irregular solution. Using this theory we successfully described the surface tension of the formamide– butanol mixture. In this paper we extend this theory to the ternary solutions. THEORY
The definition of the surface tension suitable for our discussion was given by Eyring et al. (20). According to them, the surface tension, g, is
g5
O n ~G 2 G!, l
l
[1]
l
where G l and G are the mole Gibbs energies of molecules present in the l-th surface layer and in the bulk of the solution, respectively, and n l is the number of moles of the molecules present in the unit area of the l-th layer. Assuming that properties of the molecules present only in the first layer (l 5 1 [ s; the subscript “s” will indicate the variable which describes the surface phase properties) are different from the properties of the bulk molecules, and extending this definition on the N-component system we have
360
O n ~m N
g5
s,i
i51
s,i
2 m i !,
[2]
361
SURFACE TENSION OF TERNARY SOLUTIONS
different sizes, can be calculated from the theory of Flory (22),
O n ln v , N
Ds M 5 2R
i
i
[5]
i51
where v i is the volume fraction of the ith component,
O nV, N
v i 5 n iV i /
j
[6]
j
j51
V i is its mole volume and n i is the number of moles in the unit volume. Equation [5] can be adapted to the surface phase by replacing the volume fractions, n i , by the surface fractions, u i ,
O n ln u . N
FIG. 1. Surface tension of the water–acetone–propionic acid solution. Experimental data taken from Lebed and Eddin (3) and extrapolated to constant concentration of the propionic acid ( x HPr).
Ds M s 5 2R
s,i
i
[7]
i51
The surface fraction, u i , is defined where m s,i and m i are the chemical potentials of the component i in the surface layer and in the bulk. Equation [2] has been derived earlier by Schuchowitzky (21). According to him, m s,i is “the chemical potential of the component in the surface layer whose tension g would be relieved, the concentration and distribution of molecules remaining the same as initially.” To calculate the surface tension from Eq. [2] we need the chemical potentials m s,i and m i . The chemical potential of the i-th component of the multicomponent solution can be obtained from the Gibbs energy of mixing, Dg M
m i 5 m 0i 1
S D Dg M n i
,
[3]
T,p,n j Þn i
where m i0 is the standard chemical potential. As the Gibbs energy of mixing involves the enthalpy, Dh M, and entropy, Ds M, of mixing with temperature, T, Dg M 5 Dh M 2 TDs M,
[4]
we should know the exact formulae for these thermodynamical functions. The definition of the chemical potential given by Eq. [3] together with [4] is very general and concerns both bulk and surface solutions. The bulk entropy of mixing, when molecules of the individual components of the solution are of
On N
u i 5 n s,i S i /
s, j
S j,
[8]
j51
where S i is the area which one mole of the component i occupies at the interface. Derivation of the formula for the enthalpy of mixing is, in principle, the same for the surface and bulk solutions. So, we shall concentrate the discussion on the surface solution. Moreover, we shall neglect the order–disorder effects (due, e.g., to the H-bonds) supposing a random mixing of the solution components. Consideration of these effects in the general discussion of the surface behavior of the ternary solutions is of minor importance. In the analysis of the particular case these effects can be easily added to the theory by considering, e.g., associates. In order to determine Dh sM energies of the intermolecular interactions before (w 1 ) and after (w 2 ) random mixing is needed, Dh M s 5 w 2 2 w 1.
[9]
Before mixing a molecule interacts with molecules of the same type. Thus, the energy w 1 is given by
Oc N
w 1 5 N Av
j51
s, jj
w jj n s, j , 2
[10]
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STANISŁAW LAMPERSKI
where w jj is the energy of the intermolecular interactions between two molecules of the type j, c s,jj is the coordination number, and N Av is the Avogadro number. The whole expression is divided by 2 to avoid counting the same interactions twice. The energy of the interactions after the random mixing is
OO N
w 2 5 N Av
N
j51 k51
c s,soljk w jk n s, j , 2
[11]
where the mean coordination number, c s,soljk, indicates the average number of molecules k surrounding a molecule j in the perfect mixed surface solution. Earlier we introduced (19) the approximate dependence between the coordination numbers c s,soljk and c s, jk (the maximum number of molecules k that can surround the molecule j) c s,soljk < c s, jk u k .
OOc N
N
u w jkn s, j . 2
s, jk k
j51 k51
[17]
c s,ij 5 c s, ji r s, ji .
[18]
Now we can return to Eq. [3] for the chemical potential. Using Eqs. [4], [7], [9], [10], [13], [15], and [18], and after some simple arithmetic transformations, we obtain
Or N
u!
s, ji j
j51
H F O N
[13]
1 N Av
c s,ij u j w ij ~1 2 u i ! 2
j51, jÞi
O O
c s~1 2 u i ! 2 w ii 2 2 j52, jÞi N
2
j21
k51,kÞi
c su 2j w jj r s, ji 2
G
J
c s, jk n s, j u i u k w jk . [19] n s,i
[14]
In (19) we also derived two formulae which will be useful in further arithmetic transformations. The first gives the correlation between the surface fractions of two components, n s,i u 5 r s,iju i, n s, j j
di , dj
where d is the molecular diameter. Dividing Eq. [17] by the 0 corresponding equation for c s, ji , noting that n s,i } d i22 , and using Eq. [16], we obtain
0 m s,i 5 m s,i 1 RT~ln u i 2
Assuming that all molecules are spherical, the coordination number, c s, jj , for all kinds of molecules is the same: c s, jj 5 c s.
c s,ij < c s
[12]
Using this dependence Eq. [11] takes the form
w 2 5 N Av
Thus, we derived in (19) the following approximate expression:
The expression in the first pair of parentheses gives the entropy contribution to the chemical potential when the individual components of the surface solution are of different sizes while the next expressions describe the contribution due to the intermolecular interactions. For a three-component mixture composed of molecules A, B, and C, Eq. [19] can be written in the well-known form
[15] 0 m s,i 5 m s,i 1 RT ln a s,i
[20]
with 0 r s,ij 5 n s,i /n s,0 j ,
with the surface activities, a s,i , given by [16]
0 where n s,i stands for the number of moles of the pure component i present in the unit area of the surface layer. The parameter r s,ij shows how many molecules i occupy the same area as one molecule j. The second formula concerns the correlation between the coordination number c s,ij and c s, ji . This correlation can be calculated exactly for the lattice model (23). However, the lattice model is appropriate for the chain polymer solutions, not for our system with the spherical molecules of different sizes.
ln a s,A 5 ln u A 1 u B~1 2 r s,BA! 1 u C~1 2 r s,CA! 2 u 2B A s,AB 2 u 2C A s,AC 2 u BuC B s,A
[21]
ln a s,B 5 ln u B 1 v A~1 2 r s,AB! 1 u C~1 2 r s,CB! 2 u 2Ar s,AB A s,AB 2 u 2C A s,BC 2 u AuC B s,B [22] ln a s,C 5 ln u C 1 u A~1 2 r s,AC! 1 u B~1 2 r s,BC! 2 u 2Ar s,AC A s,AC 2 u 2Br s,BC A s,BC 2 u AuB B s,C, [23]
363
SURFACE TENSION OF TERNARY SOLUTIONS
solution B 1 C into two solutions: A 1 B and A 1 C. The parameter B s,i can be expressed by a linear combination of the parameters A s,ij :
where A s,AB 5 A s,AC 5
1 ~c w 1 c sw BBr s,BA 2 2c s,ABw AB! 2kT s AA
[24]
1 ~c w 1 c sw CCr s,CA 2 2c s,ACw AC! 2kT s AA
[25]
1 A s,BC 5 ~c w 1 c sw CCr s,CB 2 2c s,BCw BC! 2kT s BB Bs,A 5
[30]
B s,B 5 r s,AB~ A s,AB 2 A s,AC! 1 A s,BC
[31]
B s,C 5 r s,AC~ A s,AC 2 A s,AB! 1 r s,BC A s,BC.
[32]
[26]
1 ~c w 1 r s,CAc s,CBw CB 2 c s,ABw AB 2 c s,ACw AC! [27] kT s AA
Bs,B 5
B s,A 5 A s,AB 1 A s,AC 2 A s,BC
1 ~c w 1 r s,CBc s,CAw CA 2 c s,BAw BA 2 c s,BCw BC! [28] kT s BB
1 Bs,C 5 ~c w 1 r s,BCc s,BAw BA 2 c s,CAw CA 2 c s,CBw CB!. [29] kT s CC The parameter A s,ij corresponds to the parameter cw/kT of the Bragg–Williams theory (24) for a two-component solution when the mole volumes of both components are equal. When the molecular sizes of the components are different, A s,ij describes a change in the intermolecular interactions energy for the process of mixing such amounts of the molecules i and j which occupy the same area at the surface layer. This energy is related to one molecule i. For A s,ij , 0 the separated components are thermodynamically more stable than their mixture (reduced solubility) while for A s,ij . 0 the mixture is more stable than the separated components. The negative A s,ij gives the positive deviations from the Raoult law and vice versa. For a three-component system we have a new parameter, B s,i , in the expression for the chemical potential. The parameter B s,A, for example, describes the negative value of the change in the energy for the following process:
In the strictly regular solution the molecular sizes of the individual components are the same. Thus, the surface fractions, u i , are equal to the corresponding mole fractions, x s,i , r s,ij 5 1, and Eqs. [21]–[23] take the form ln a s,A 5 ln x s,A 2 x 2s,B A s,AB 2 x 2s,C A s,AC 2 x s,Bx s,C B s,A
[33]
ln a s,B 5 ln x s,B 2 x 2s,A A s,AB 2 x 2s,C A s,BC 2 x s,Ax s,C B s,B
[34]
ln a s,C 5 ln x s,C 2 x 2s,A A s,AC 2 x 2s,B A s,BC 2 x s,Ax s,B B s,C
[35]
with
B s,A 5
A s,AB 5
cs ~w AA 1 w BB 2 2w AB! 2kT
[36]
A s,AC 5
cs ~w AA 1 w CC 2 2w AC! 2kT
[37]
A s,BC 5
cs ~w BB 1 w CC 2 2w BC! 2kT
[38]
cs ~w AA 1 w CB 2 w AB 2 w AC! kT 5 A s,AB 1 A s,AC 2 A s,BC
B s,B 5
cs ~w BB 1 w CA 2 w BA 2 w BC! kT 5 A s,AB 1 A s,BC 2 A s,AC
B s,C 5
[40]
cs ~w CC 1 w BA 2 w CA 2 w CB! kT 5 A s,AC 1 A s,BC 2 A s,AB.
Each circle in this diagram represents an amount of molecules which occupies the same area of the surface layer. The energy is related to the pair of the molecules i. For B s,A , 0 component A tends to be separated from the mixture B and C while for B s,A . 0 component A leads to the separation of the
[39]
[41]
These results are consistent with the corresponding equations given by Prigogine (25). From Eq. [19] we can derive expressions for the surface activities of the two-component (A 1 B) solution, ln a s,A 5 ln u A 1 u B~1 2 r s,BA! 2 u 2BA s,AB
[42]
ln a s,B 5 ln u B 1 u A~1 2 r s,AB! 2 u 2Ar s,AB A s,AB,
[43]
364
STANISŁAW LAMPERSKI
where A s,AB is given by [24]. We obtained the same result earlier (19). For the regular solutions the above equations reduce to the formulae given by Bragg and Williams ln a s,A 5 ln x s,A 2 x 2s,B A s,AB
[44]
ln a s,B 5 ln x s,B 2 x 2s,A A s,AB
[45]
with A s,AB defined by Eq. [36]. Thus, we have shown that the chemical potentials derived from Eq. [19] for some simple systems are consistent with the solutions already known. The chemical potential, m i , of the component i present in the bulk of the solution can be calculated from the formulae analogous to those derived above when dropping out the subscript “s” and replacing the surface fractions, u i , by the volume fractions, v i , when necessary. The bulk parameter r ij is defined r ij 5 n 0i /n 0j ,
[46]
where n i0 is the number of moles of the pure component i in the unit volume. When the surface and bulk chemical potentials are defined, we can return to the formula for the surface tension. If we express n s,i , by a function of u i , 0 n s,i 5 n s,i u i,
[47]
Eq. [2] takes the form
On
u ~ m s,i 2 m i!.
0 s,i i
[48]
i51
Using the definition for the surface tension of the pure component i, 0 0 0 g i 5 n s,i ~G s,i 2 G i ! 5 n s,i ~ m s,i 2 m 0i !,
[49]
we obtain
O u Sg 1 n N
g5
i
i51
S D S D g uA
T,a A,a B,a C, u B
g uB
T,a A,a B,a C, u A
50
[51]
5 0.
[52]
The solution of these equations, assuming that there are no vacancies in the surface layer,
u A 1 u B 1 u C 5 1,
[53]
leads to the adsorption isotherm described by a system of equations
F
uC gA 2 gC aC 1 r s,AC 2 1 r s,AC 5 r s,AC exp RTn 0s,C uA aA 1 ~ u 2A 2 u 2C!r s,AC A s,AC 1 u 2A~r s,BC A s,BC 2 r s,AC A s,AB! 1 u B~ u Cr s,AC B s,A 2 u AB s,C!#
[54]
F
uC aC gB 2 gC 1 r s,BC 2 1 r s,BC 5 r s,BC exp RTn 0s,C uB aB 1 ~ u 2B 2 u 2C!r s,BC A s,BC 1 u 2Ar s,AC~ A s,AC 2 A s,AB! 1 u A~ u Cr s,BC B s,B 2 u BB s,C!# .
N
g5
obtained by minimizing the surface tension against the surface layer composition (9)
i
0 s,i
RT ln
D
a s,i , ai
[50]
where a i is the bulk activity of the component i in the bulk. The equilibrium composition of the surface layer can be
[55]
Assuming that the bulk activities are known and constant, we can solve these equations for the surface fractions. Unfortunately, Eqs. [54] and [55] together with [53] cannot be solved analytically and the application of the numerical methods is needed. We have used the Newton– Raphson method for a system of nonlinear equations (26). The needed mole numbers, n s,i , were calculated from the densities and mole masses (19). When the surface composition is known the surface activities can be calculated from Eqs. [21]–[23] and the surface tension from Eq. [50]. For a two-component solution the adsorption isotherm takes the form
F
uB gA 2 gB aB 5 exp 1 r s,AB ~1 2 u B! r s,AB a rAs,AB RTn 0s,B
G
2 1 1 ~1 2 2 u B!r s,AB A s,AB .
[56]
365
SURFACE TENSION OF TERNARY SOLUTIONS
When the components are of the equal size, this isotherm reduces further to
F
G
aB x s,B gA 2 gB 5 exp 1 ~1 2 2x s,B! A s,AB . 1 2 x s,B a A RTn 0s,B
[57]
Next, substituting
S
b 5 exp
gA 2 gB 1 A s,AB RTn 0s,B
D
[59]
and assuming that at low concentrations of the solute (component B) the bulk activity of the solvent (A) is close to 1 (a A 5 1) we obtain the Frumkin isotherm (27) x s,B 5 ln a Bb 1 Ax s,B. 1 2 x s,B
[60]
On the other hand when we assume that A s,AB 5 0 and substitute DG 5 ~ g A 2 g B!/n 0s,B
[61]
into Eq. [56], we obtain
S
D
uB DG aB 5 exp 1 r s,AB 2 1 . ~1 2 u B! r s,AB a rAs,AB RT
[62]
It is similar to the adsorption isotherm derived by Levine and Fawcett (28) for the electrode– electrolyte interface. However, in our isotherm the activity of A is raised to the power r s,AB, which does not take place in the formula derived by Levine and Fawcett. For comparable diameters of the solution components (r s,AB ' 1) the expression exp(r s,AB 2 1) can be expanded into the power series. Using the first two terms of this series and assuming that at low concentrations a A ' 1 one obtains from Eq. [62] the Flory–Huggins isotherm:
S D
uB DG 5 a Bexp . r s,AB~1 2 u 2 ! r s,AB RT
Interaction pair
A s,ij
A ij
K
Acetone–water Acetone–propionic acid Water–propionic acid
3.2725 3.0645 1.5961
21.2946 3.2055 20.3070
22.53 0.96 25.20
[58]
A 5 22 A s,AB
ln
TABLE 1 The Values of the A s,ij and A ij Parameters Obtained from the Nonlinear Least-Square Fitting for the Water–Acetone–Propionic Acid System
[63]
We have shown above that the adsorption isotherm [56] is a generalization of the Flory–Huggins and Frumkin (27) isotherms. It includes both the different size effect and the intermolecular interactions. On the other hand, the adsorption isotherm given by a system of Eqs. [53]–[55] is the extension of isotherm [56] to the ternary solution.
RESULT AND DISCUSSION
The theory described in this paper can be used in different ways. When the bulk activities of the individual components of the ternary solution are known, one can calculate the values of the three parameters, A s,ij , which describe the molecular interactions at the surface. These values can be obtained by fitting the theoretical surface tension to the experimental data, as it has been described earlier (19) for the binary solutions. This method gives the most precise estimate of the molecular interactions at the surface. When the bulk activities are not known, we can calculate the A s,ij and A ij parameters by fitting the theoretical results to the experimental data. However, the number of the adjustable parameters increases to six now. Moreover, each parameter of the pair A s,ij and -A ij has a similar influence on the surface tension (see Fig. 4) which can lower the reliability of the results. We have applied this method to the ternary system investigated by Lebed and Eddin (3): water (w) - acetone (a) propionic acid (HPr) mixture. At T 5 293.15 K the pure liquids are characterized by the surface tensions g w 5 0.07275, g a 5 0.0237, and g HPr 5 0.0267 [N/m] and densities g w 5 0.9971, g a 5 0.7899, and g HPr 5 0.9942 [g/cm 3] (3, 29). The densities, together with the corresponding mole masses, are 0 needed to calculate the mole numbers n s,i and n i0 . We obtained 0 0 0 25 n s,w 5 1.576 3 10 , n s,a 5 6.184 3 10 26, n s,HPr 5 6.129 3 0 0 0 26 2 4 10 [mole/m ], and n w 5 5.535 3 10 , n a 5 1.360 3 10 4, n HPr 4 3 5 1.342 3 10 , [mole/m ]. The values of the A s,ij and A ij parameters obtained from the nonlinear least-square fitting are given in Table 1, while the theoretical and experimental surface tensions are shown in Fig. 2 by solid lines and graphic characters, respectively. We can expect some correlation between the bulk and surface parameters, A s,ij 5 KA ij ,
[64]
where K describes the difference between the surface and bulk mean coordination numbers. According to Patterson and Rastogi (30) a molecule moving from the bulk to the surface loses approximately 29% of its nearest neighbors which means that K ' 0.71. According to our calculations for the formamide–
366
STANISŁAW LAMPERSKI
FIG. 2. Surface tension of the water–acetone–propionic acid solution. Fit of the theoretical surface tension (solid lines) to the experimental data (3) (graphic characters).
n-butanol system, K is 0.41 which means that K comprises also other effects. It seems that the reasonable values for K should range from 0.4 to 0.7. As follows from Table 1 this criterion is only approximately kept for the acetone–propionic acid interactions. Finally, the theory can be used to analyze the general surface behavior of the ternary systems and this is what we shall concentrate on in this section. We will discuss the influence of intermolecular interactions on surface tension of the solution by varying values of the A s,ij and A ij parameters. To make the discussion closer to the real system we assume that the surface 0 tensions, g i , and the surface, n s,i , and bulk, n i0 , numbers of moles of the individual components of the hypothetical ternary solution, A 1 B 1 C, are the same as those of water (A), acetone (B), and propionic acid (C). At first we will consider a system with no intermolecular interactions, which will be our reference system. Dependence of the surface tension on the concentration is shown in Fig. 3. The volume fractions are recalculated into the bulk mole fractions. As the molecules B and C are larger than A, they are more easily adsorbed than A. Small bulk concentrations of B and C give a great increase in their surface concentration (dotted lines in Fig. 10). This results in a rapid fall in the surface tension observed mainly at low concentrations of B and C. Parameters A s,AB and A AB describe intermolecular interactions between molecules A and B in the surface layer and in the bulk of the solution, respectively. For a negative A s,AB, the miscibility of A and B on the surface is reduced. The molecules are pushed from the surface to the bulk of the solution. Thus,
FIG. 3. Surface tension of the ternary solution for A s,AB 5 0, A s,AC 5 0, A s,BC 5 0, A AB 5 0, A AC 5 0, A BC 5 0, T 5 293.15 K, g A 5 0.07275, g B 5 0 0 0 0.0237, g C 5 0.0267 [N/m], n s,A 5 1.576 3 10 25, n s,B 5 6.184 3 10 26, n s,C 5 26 2 4 4 0 0 6.129 3 10 [mole/m ], and n A 5 5.535 3 10 , n B 5 1.360 3 10 , n C0 5 1.342 3 10 4, [mole/m 3].
we need more energy to create the surface, which results in the increase in the surface tension (solid lines in Fig. 4). At low concentrations of B, the slope of the surface tension with
FIG. 4. Surface tension of the ternary solution for A s,AB 5 23 (solid lines) and A AB 5 3 (dotted lines). The other parameters are as in Fig. 3.
SURFACE TENSION OF TERNARY SOLUTIONS
FIG. 5. Surface tension of the ternary solution for A s,AB 5 3. The other parameters are as in Fig. 3.
respect to x B is positive which indicates a negative adsorption of B. A similar surface tension behavior can be observed for the positive A AB (dotted lines in Fig. 4) although the mechanism of this surface tension increase is different: the positive A AB causes an increase in the mutual solubility of A and B in the bulk and thus stabilizes thermodynamically the A 1 B mixture. Again the molecules prefer the bulk of the solution to its surface. Addition of the component C reduces the effect of the A-B interactions, mainly in the bulk. The above examples show that the opposite effects occurring on the surface and in the bulk of the solution result in a similar surface tension behavior. Thus, we will restrict further discussion to the surface solutions. We will consider the influence of the bulk solution only in the cases when it is essentially different than that due to the surface solution. The positive values of A s,AB thermodynamically stabilize the surface solution of A and B. Molecules of the surface solution attract molecules from the bulk which produces a drop in the surface tension seen in Fig. 5. Small bulk concentrations of the component C have a weak influence on the surface tension, which means that the adsorbability of C is low. In fact, the thermodynamically stable surface solution of A and B does not let molecule C into the surface layer. The influence of the A s,AC and A AC parameters on the surface tension is well marked for x B 5 0. The negative A s,AC and positive A AC raise the surface tension with the increasing but low concentrations of C, and the adsorption of C is negative (the positive slope of the surface tension with respect to x C). For a negative A s,AC, the g -x B curves in Fig. 6 are nearly
367
FIG. 6. Surface tension of the ternary solution for A s,AC 5 22. The other parameters are as in Fig. 3.
independent of the concentration of C. They intersect at different points and finally at high concentrations of B we obtain a weak positive adsorption of C. A different picture is observed for a positive A AC (Fig. 7). The surface tension curves intersect nearly at one point at a very low concentration of B (at x B ' 0.006). Starting from this point the initially negative adsorption
FIG. 7. Surface tension of the ternary solution for A AC 5 2. The other parameters are as in Fig. 3.
368
STANISŁAW LAMPERSKI
FIG. 8. Surface tension of the ternary solution for A s,AC 5 2. The other parameters are as in Fig. 3.
FIG. 9. Surface tension of the ternary solution for A s,BC 5 22.67. The other parameters are as in Fig. 3.
of C changes into the strongly positive. The intersection of the surface tension curves does not correspond to that observed by Lebed and Eddin (3) because of two reasons:
at the phase separation of the surface solution. In that case we had a positive slope before the discontinuity point and a negative after it. In the present case the slope is negative all the time. To provide a better illustration of the surface behavior in this particular case, in Fig. 10 we present the dependence of the
(i) we have an increase in g at x B 5 0 with the increasing concentration of C while Lebed and Eddin (3) obtained here a surface tension drop; (ii) our intersection point occurs at a very low concentration of B ( x B ' 0.006) while they found it at x B ' 0.2. By increasing the value of A AC one can slightly shift the intersection point toward higher B concentrations, but not as much as necessary. For the positive values of A s,AC and low B concentrations we observe a rapid decrease in the surface tension with increasing concentration of C (Fig. 8). This shows the large adsorbability of C at low concentrations of B. The high adsorbability of C at low B concentrations due to the thermodynamical stability of the surface solution of A and C makes the component B exhibit negative adsorption in this region. At higher concentrations of B, the influence of component C is gradually reduced and the adsorption of C falls down while the adsorption of B becomes positive. The negative values of the A s,BC parameter increase the surface tension with increasing concentration of C at high x B. This effect is shown in Fig. 9. However, we also observe in this figure some unexpected surface tension behavior at low x B. At higher concentrations of C (curves 5 and 6) the slope of the surface tension with respect to x B is not continuous. We observed earlier (19) a discontinuity of the surface tension slope
FIG. 10. Surface fractions of the ternary solution at x C 5 0.25 for the parameters are as in Fig. 9 (solid lines) and are as in Fig. 3 (dotted lines).
369
SURFACE TENSION OF TERNARY SOLUTIONS
surface fractions u A, u B, and u C against x B at x C 5 0.25 (solid lines) and the corresponding results for the reference system with no intermolecular interactions (dotted lines). At a low x B, the decrease in the surface concentration of C and the increase in B is much lower than in the absence of B–C surface interactions. Molecules C, which are in the majority on the surface, hinder the increase in the surface concentration of B because of the repulsion between these molecules. The situation changes fundamentally when x B approaches some critical value (;0.2 in Fig. 10). Now, the increasing bulk concentration of B forces an increase in the concentration of B on the surface while this time molecules C are pushed out from the surface because of the repulsion between B and C. For A s,BC , 2 2.67, the curves u B and u C exhibit S loops characteristic of the phase transition (19). Interactions between B and C have a very small effect on the surface concentration of A. However, although this influence is small, we observe some discontinuity in the slope of u A with respect to x B at x B ' 0.2. The positive values of A s,BC lower the surface tension at high concentrations of B. This effect is shown in Fig. 11. It results from the formation of the thermodynamically stable solution of B and C on the surface. On the other hand a thermodynamically stable bulk solution of B and C ( A BC . 0) increases the surface tension. Let us note that by increasing the right-hand side of the surface tension curves when their extreme left points (at x B 5 0) are fixed, we can get a situation when all the curves intersect at one point (Fig. 12). This is the case observed by Lebed and Eddin (3). Changing the value of A BC, we can shift the intersection point within a certain range on the x B axis.
FIG. 12. Surface tension of the ternary solution for A BC 5 2. The other parameters are as in Fig. 3.
The intersection of the curves does not necessary take place at one point. For example for A BC 5 4 and the same concentrations of C as in Fig. 12, the curves intersect approximately in the range from x B 5 0.1 to 0.3. In this case it is difficult to say that the solution exhibits the surface buffering effect. In this section we have discussed the basic surface tension behavior exhibited by the ternary solutions. We did not analyze a superposition of two or more A s,ij and/or A ij parameters, but these effects can be deduced easily on the basis of the above discussion. At this stage, we did not also consider the hydrogen bond formation or the higher terms in the expression for the enthalpy of mixing (19, 31, 32) which may improve a theoretical description of the experimental results but they make the final form of the adsorption isotherm much more complicated. Finally, we hope that the present theory, alternatively extended by the order– disorder effects, will also be useful in the explanation of the surface tension of the systems which seem more difficult to describe theoretically like, e.g., along the binodal ˇ echova and Bartovska´ (33). curve measured recently by C CONCLUSIONS
Although the ternary systems exhibit a wide spectrum of the surface tension behaviors, with what has been shown in the previous section, we can formulate some general conclusions.
FIG. 11. Surface tension of the ternary solution for A s,BC 5 2.67. The other parameters are as in Fig. 3.
(i) Intermolecular attraction in the bulk of the solution increases the surface tension while the same interactions at the surface layer reduce it. (ii) Intermolecular repulsion in the bulk of the solution
370
STANISŁAW LAMPERSKI
reduces the surface tension while the same interactions at the surface layer increase it. (iii) Strong intermolecular repulsion at the surface layer leads to discontinuity in the surface tension due to the phase separation at the surface. (iv) The larger molecules are adsorbed easier. These conclusions are similar to those derived earlier (19) for binary systems and perhaps can also be extended on multicomponent solutions. The ternary systems may exhibit one particular behavior: a so-called buffering effect which can occur in two different ranges of concentration. ACKNOWLEDGMENTS Financial support from Adam Mickiewicz University, Faculty of Chemistry is greatly appreciated.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Seith, W., Z. Phys. Chem. 117, 257 (1925). Semenchenko, V. K., and Davidovskaya, E. A., Koll. Zeits. 73, 24 (1935). Lebed, V. I., and Eddin, F. S., Zh. Khim. Niev. Rast. 1, 141, 153 (1992). Li, B., and Joos, G. P., Colloids Surf. A 88, 251 (1994). von Szyszkowski, B., Z. Phys. Chem. 64, 385 (1908). Shain, S. A., and Prausnitz, J. M., AIChE J. 10, 766 (1964). Handa, T., and Mukerjee, P., J. Phys. Chem. 85, 3916 (1981). Fu, J., Li, B., and Zihao, W., Chem. Eng. Sci. 41, 2673 (1986). Lamperski, S., J. Colloid Interface Sci. 144, 153 (1991). Dynarowicz, P., and Jawien´, W., Colloids Surf. A 70, 171 (1993). Pandey, J. D., and Pant, N., J. Am. Chem. Soc. 104, 3299 (1982). Flory, P. J., J. Am. Chem. Soc. 87, 1833 (1965).
13. Abe, A., and Flory, P. J., J. Am. Chem. Soc. 87, 1838 (1965). 14. Pandey, J. D., Rai, R. D., and Shukla, R. K., J. Chem. Soc. Faraday Trans 1 85, 331 (1989). 15. Cornelisse, P. M. W., Peters, C. J., and Arons, J. D., Fluid Phase Equilib. 82, 119 (1993). 16. Hu, Y. Q., Li, Z. B., Lu, J. F., Li, Y. G., and Jin, Y., Chinese J. Chem. Eng. 5, 193 (1997). 17. Peng, D. Y., and Robinson, D. B., Ind. Eng. Chem. Fundam. 15, 59 (1976). 18. Zuo, Y. X., and Stenby, E. H., J. Chem. Eng. Jpn. 29, 159 (1996). 19. Lamperski, S., J. Colloid Interface Sci. 157, 32 (1993). 20. Jhon, M. S., van Artsdalen, E. R., Grosh, J., and Eyring, E., J. Chem. Phys. 47, 2231 (1967). 21. A. Schuchowitzky, Acta Physicochim. U.R.S.S. 19, 176 (1944). 22. Flory, P. J., in “Principles of Polymer Chemistry,” Chap. XII, Cornell Univ. Press, Ithaca, NY, 1953. 23. Prigogine, I., Bellemans, A., and Mathot, V., in “The Molecular Theory of the Solutions,” Sect. 3.5, North Holland, Amsterdam, 1957. 24. Hill, T. L., in “An Introduction to Statistical Thermodynamics,” Chap. XX, Dover, New York, 1986. 25. Prigogine, I., and Defay, R., in “Chemical Thermodynamics,” p. 257. Longmans, London, 1962. 26. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T., in “Numerical Recipes in Pascal. The Art of Scientific Computing,” Chap. 9.6, Cambridge Univ. Press, Cambridge, UK, 1989. 27. Frumkin, A. N., Z. Phys. Chem. 116, 446 (1925). 28. Levine, S., and Fawcett, W. R., J. Electroanal. Chem. 99, 265 (1979). 29. Weast, R. C., Ed., “CRC Handbook of Chemistry and Physics,” 60th ed., CRC Press, Boca Raton, FL, 1980. 30. Patterson, D., and Rastogi, A. K., J. Phys. Chem. 74, 1067 (1970). 31. Redlich, D., and Kister, A. T., Ind. Eng. Chem. 40, 345 (1948). 32. Randles, J. E. B., and Behr, B., J. Electroanal. Chem. 35, 389 (1972). 33. Cˇechova, M., and Bartovska´, L., Collect. Czech. Chem. Commun. 61, 489 (1996).
Surface Tension of Ternary Solutions Stanisław Lamperski Department of Physical Chemistry, Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznan´, Poland Received July 15, 1998; accepted January 28, 1999
A theory of the surface tension of ternary solutions is presented. It assumes different molecular sizes of individual components of the solution and takes into account intermolecular interactions both at the surface and in the bulk. The influence of these interactions on the surface tension behavior including the surface buffering effect and the surface phase transition is discussed. © 1999 Academic Press Key Words: surface tension; adsorption isotherm; ternary solution; intermolecular interactions; surface phase transition.
INTRODUCTION
In 1925 Seith (1) reported the results on surface tension of aqueous solutions of different organic solvents (surface active) with varying concentration of salts. He found that the surface tension curves plotted as a function of the surface active solvent concentration at different salt concentrations have a common point of intersection. This result was confirmed by Semenchenko (2) who referred to this phenomenon as the surface buffering effect. More recently Lebed and Eddin (3) observed a similar intersection of the surface tension curves for aqueous solutions of two organic liquids. The surface tension results for one of the systems they studied are shown in Fig. 1. This effect has not been explained theoretically. It motivated us to formulate a theory which would explain not only this effect, but would give a general description of the surface behavior of ternary solutions. Different equations and theories have been used to describe the surface tension of the ternary or multicomponent solutions. Li and Joos (4) extended the Szyszkowski (5) equation to the nonperfect systems. Shain and Prausnitz (6) derived a simple logarithmic dependence of the surface tension on the quotient of the interface and bulk activity of the component. Handa and Mukerjee (7) proposed a more extended form of this equation. Fu et al. (8) eliminated the surface activities from the Shain and Prausnitz equation assuming different sizes of individual components. Lamperski (9) derived equations for the surface composition and the surface tension of the irregular, athermal multicomponent solutions. Dynarowicz and Jawien´ (10) extended a regular solution theory to the ternary systems of different molecular sizes. Pandey and Pant (11) employed Flory’s (12, 13)
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statistical theory to obtain characteristic parameters of the ternary solutions, which allowed them to calculate the surface tension. Next Pandey et al. (14) extended that theory on quaternary systems. Cornellise et al. (15) and Hu et al. (16) used the Peng–Robinson (17) equation of state to calculate the surface tension of the binary and ternary systems. Zou and Stenby (18) developed a simplified gradient theory model, which predicts the surface tension for these systems with a reasonable accuracy. In a recent paper we developed a theory of the surface tension of two-component solutions (19) which takes into account different molecular sizes and intermolecular interactions. Generally it was an extension of the Bragg–Williams theory to the irregular solution. Using this theory we successfully described the surface tension of the formamide– butanol mixture. In this paper we extend this theory to the ternary solutions. THEORY
The definition of the surface tension suitable for our discussion was given by Eyring et al. (20). According to them, the surface tension, g, is
g5
O n ~G 2 G!, l
l
[1]
l
where G l and G are the mole Gibbs energies of molecules present in the l-th surface layer and in the bulk of the solution, respectively, and n l is the number of moles of the molecules present in the unit area of the l-th layer. Assuming that properties of the molecules present only in the first layer (l 5 1 [ s; the subscript “s” will indicate the variable which describes the surface phase properties) are different from the properties of the bulk molecules, and extending this definition on the N-component system we have
360
O n ~m N
g5
s,i
i51
s,i
2 m i !,
[2]
361
SURFACE TENSION OF TERNARY SOLUTIONS
different sizes, can be calculated from the theory of Flory (22),
O n ln v , N
Ds M 5 2R
i
i
[5]
i51
where v i is the volume fraction of the ith component,
O nV, N
v i 5 n iV i /
j
[6]
j
j51
V i is its mole volume and n i is the number of moles in the unit volume. Equation [5] can be adapted to the surface phase by replacing the volume fractions, n i , by the surface fractions, u i ,
O n ln u . N
FIG. 1. Surface tension of the water–acetone–propionic acid solution. Experimental data taken from Lebed and Eddin (3) and extrapolated to constant concentration of the propionic acid ( x HPr).
Ds M s 5 2R
s,i
i
[7]
i51
The surface fraction, u i , is defined where m s,i and m i are the chemical potentials of the component i in the surface layer and in the bulk. Equation [2] has been derived earlier by Schuchowitzky (21). According to him, m s,i is “the chemical potential of the component in the surface layer whose tension g would be relieved, the concentration and distribution of molecules remaining the same as initially.” To calculate the surface tension from Eq. [2] we need the chemical potentials m s,i and m i . The chemical potential of the i-th component of the multicomponent solution can be obtained from the Gibbs energy of mixing, Dg M
m i 5 m 0i 1
S D Dg M n i
,
[3]
T,p,n j Þn i
where m i0 is the standard chemical potential. As the Gibbs energy of mixing involves the enthalpy, Dh M, and entropy, Ds M, of mixing with temperature, T, Dg M 5 Dh M 2 TDs M,
[4]
we should know the exact formulae for these thermodynamical functions. The definition of the chemical potential given by Eq. [3] together with [4] is very general and concerns both bulk and surface solutions. The bulk entropy of mixing, when molecules of the individual components of the solution are of
On N
u i 5 n s,i S i /
s, j
S j,
[8]
j51
where S i is the area which one mole of the component i occupies at the interface. Derivation of the formula for the enthalpy of mixing is, in principle, the same for the surface and bulk solutions. So, we shall concentrate the discussion on the surface solution. Moreover, we shall neglect the order–disorder effects (due, e.g., to the H-bonds) supposing a random mixing of the solution components. Consideration of these effects in the general discussion of the surface behavior of the ternary solutions is of minor importance. In the analysis of the particular case these effects can be easily added to the theory by considering, e.g., associates. In order to determine Dh sM energies of the intermolecular interactions before (w 1 ) and after (w 2 ) random mixing is needed, Dh M s 5 w 2 2 w 1.
[9]
Before mixing a molecule interacts with molecules of the same type. Thus, the energy w 1 is given by
Oc N
w 1 5 N Av
j51
s, jj
w jj n s, j , 2
[10]
362
STANISŁAW LAMPERSKI
where w jj is the energy of the intermolecular interactions between two molecules of the type j, c s,jj is the coordination number, and N Av is the Avogadro number. The whole expression is divided by 2 to avoid counting the same interactions twice. The energy of the interactions after the random mixing is
OO N
w 2 5 N Av
N
j51 k51
c s,soljk w jk n s, j , 2
[11]
where the mean coordination number, c s,soljk, indicates the average number of molecules k surrounding a molecule j in the perfect mixed surface solution. Earlier we introduced (19) the approximate dependence between the coordination numbers c s,soljk and c s, jk (the maximum number of molecules k that can surround the molecule j) c s,soljk < c s, jk u k .
OOc N
N
u w jkn s, j . 2
s, jk k
j51 k51
[17]
c s,ij 5 c s, ji r s, ji .
[18]
Now we can return to Eq. [3] for the chemical potential. Using Eqs. [4], [7], [9], [10], [13], [15], and [18], and after some simple arithmetic transformations, we obtain
Or N
u!
s, ji j
j51
H F O N
[13]
1 N Av
c s,ij u j w ij ~1 2 u i ! 2
j51, jÞi
O O
c s~1 2 u i ! 2 w ii 2 2 j52, jÞi N
2
j21
k51,kÞi
c su 2j w jj r s, ji 2
G
J
c s, jk n s, j u i u k w jk . [19] n s,i
[14]
In (19) we also derived two formulae which will be useful in further arithmetic transformations. The first gives the correlation between the surface fractions of two components, n s,i u 5 r s,iju i, n s, j j
di , dj
where d is the molecular diameter. Dividing Eq. [17] by the 0 corresponding equation for c s, ji , noting that n s,i } d i22 , and using Eq. [16], we obtain
0 m s,i 5 m s,i 1 RT~ln u i 2
Assuming that all molecules are spherical, the coordination number, c s, jj , for all kinds of molecules is the same: c s, jj 5 c s.
c s,ij < c s
[12]
Using this dependence Eq. [11] takes the form
w 2 5 N Av
Thus, we derived in (19) the following approximate expression:
The expression in the first pair of parentheses gives the entropy contribution to the chemical potential when the individual components of the surface solution are of different sizes while the next expressions describe the contribution due to the intermolecular interactions. For a three-component mixture composed of molecules A, B, and C, Eq. [19] can be written in the well-known form
[15] 0 m s,i 5 m s,i 1 RT ln a s,i
[20]
with 0 r s,ij 5 n s,i /n s,0 j ,
with the surface activities, a s,i , given by [16]
0 where n s,i stands for the number of moles of the pure component i present in the unit area of the surface layer. The parameter r s,ij shows how many molecules i occupy the same area as one molecule j. The second formula concerns the correlation between the coordination number c s,ij and c s, ji . This correlation can be calculated exactly for the lattice model (23). However, the lattice model is appropriate for the chain polymer solutions, not for our system with the spherical molecules of different sizes.
ln a s,A 5 ln u A 1 u B~1 2 r s,BA! 1 u C~1 2 r s,CA! 2 u 2B A s,AB 2 u 2C A s,AC 2 u BuC B s,A
[21]
ln a s,B 5 ln u B 1 v A~1 2 r s,AB! 1 u C~1 2 r s,CB! 2 u 2Ar s,AB A s,AB 2 u 2C A s,BC 2 u AuC B s,B [22] ln a s,C 5 ln u C 1 u A~1 2 r s,AC! 1 u B~1 2 r s,BC! 2 u 2Ar s,AC A s,AC 2 u 2Br s,BC A s,BC 2 u AuB B s,C, [23]
363
SURFACE TENSION OF TERNARY SOLUTIONS
solution B 1 C into two solutions: A 1 B and A 1 C. The parameter B s,i can be expressed by a linear combination of the parameters A s,ij :
where A s,AB 5 A s,AC 5
1 ~c w 1 c sw BBr s,BA 2 2c s,ABw AB! 2kT s AA
[24]
1 ~c w 1 c sw CCr s,CA 2 2c s,ACw AC! 2kT s AA
[25]
1 A s,BC 5 ~c w 1 c sw CCr s,CB 2 2c s,BCw BC! 2kT s BB Bs,A 5
[30]
B s,B 5 r s,AB~ A s,AB 2 A s,AC! 1 A s,BC
[31]
B s,C 5 r s,AC~ A s,AC 2 A s,AB! 1 r s,BC A s,BC.
[32]
[26]
1 ~c w 1 r s,CAc s,CBw CB 2 c s,ABw AB 2 c s,ACw AC! [27] kT s AA
Bs,B 5
B s,A 5 A s,AB 1 A s,AC 2 A s,BC
1 ~c w 1 r s,CBc s,CAw CA 2 c s,BAw BA 2 c s,BCw BC! [28] kT s BB
1 Bs,C 5 ~c w 1 r s,BCc s,BAw BA 2 c s,CAw CA 2 c s,CBw CB!. [29] kT s CC The parameter A s,ij corresponds to the parameter cw/kT of the Bragg–Williams theory (24) for a two-component solution when the mole volumes of both components are equal. When the molecular sizes of the components are different, A s,ij describes a change in the intermolecular interactions energy for the process of mixing such amounts of the molecules i and j which occupy the same area at the surface layer. This energy is related to one molecule i. For A s,ij , 0 the separated components are thermodynamically more stable than their mixture (reduced solubility) while for A s,ij . 0 the mixture is more stable than the separated components. The negative A s,ij gives the positive deviations from the Raoult law and vice versa. For a three-component system we have a new parameter, B s,i , in the expression for the chemical potential. The parameter B s,A, for example, describes the negative value of the change in the energy for the following process:
In the strictly regular solution the molecular sizes of the individual components are the same. Thus, the surface fractions, u i , are equal to the corresponding mole fractions, x s,i , r s,ij 5 1, and Eqs. [21]–[23] take the form ln a s,A 5 ln x s,A 2 x 2s,B A s,AB 2 x 2s,C A s,AC 2 x s,Bx s,C B s,A
[33]
ln a s,B 5 ln x s,B 2 x 2s,A A s,AB 2 x 2s,C A s,BC 2 x s,Ax s,C B s,B
[34]
ln a s,C 5 ln x s,C 2 x 2s,A A s,AC 2 x 2s,B A s,BC 2 x s,Ax s,B B s,C
[35]
with
B s,A 5
A s,AB 5
cs ~w AA 1 w BB 2 2w AB! 2kT
[36]
A s,AC 5
cs ~w AA 1 w CC 2 2w AC! 2kT
[37]
A s,BC 5
cs ~w BB 1 w CC 2 2w BC! 2kT
[38]
cs ~w AA 1 w CB 2 w AB 2 w AC! kT 5 A s,AB 1 A s,AC 2 A s,BC
B s,B 5
cs ~w BB 1 w CA 2 w BA 2 w BC! kT 5 A s,AB 1 A s,BC 2 A s,AC
B s,C 5
[40]
cs ~w CC 1 w BA 2 w CA 2 w CB! kT 5 A s,AC 1 A s,BC 2 A s,AB.
Each circle in this diagram represents an amount of molecules which occupies the same area of the surface layer. The energy is related to the pair of the molecules i. For B s,A , 0 component A tends to be separated from the mixture B and C while for B s,A . 0 component A leads to the separation of the
[39]
[41]
These results are consistent with the corresponding equations given by Prigogine (25). From Eq. [19] we can derive expressions for the surface activities of the two-component (A 1 B) solution, ln a s,A 5 ln u A 1 u B~1 2 r s,BA! 2 u 2BA s,AB
[42]
ln a s,B 5 ln u B 1 u A~1 2 r s,AB! 2 u 2Ar s,AB A s,AB,
[43]
364
STANISŁAW LAMPERSKI
where A s,AB is given by [24]. We obtained the same result earlier (19). For the regular solutions the above equations reduce to the formulae given by Bragg and Williams ln a s,A 5 ln x s,A 2 x 2s,B A s,AB
[44]
ln a s,B 5 ln x s,B 2 x 2s,A A s,AB
[45]
with A s,AB defined by Eq. [36]. Thus, we have shown that the chemical potentials derived from Eq. [19] for some simple systems are consistent with the solutions already known. The chemical potential, m i , of the component i present in the bulk of the solution can be calculated from the formulae analogous to those derived above when dropping out the subscript “s” and replacing the surface fractions, u i , by the volume fractions, v i , when necessary. The bulk parameter r ij is defined r ij 5 n 0i /n 0j ,
[46]
where n i0 is the number of moles of the pure component i in the unit volume. When the surface and bulk chemical potentials are defined, we can return to the formula for the surface tension. If we express n s,i , by a function of u i , 0 n s,i 5 n s,i u i,
[47]
Eq. [2] takes the form
On
u ~ m s,i 2 m i!.
0 s,i i
[48]
i51
Using the definition for the surface tension of the pure component i, 0 0 0 g i 5 n s,i ~G s,i 2 G i ! 5 n s,i ~ m s,i 2 m 0i !,
[49]
we obtain
O u Sg 1 n N
g5
i
i51
S D S D g uA
T,a A,a B,a C, u B
g uB
T,a A,a B,a C, u A
50
[51]
5 0.
[52]
The solution of these equations, assuming that there are no vacancies in the surface layer,
u A 1 u B 1 u C 5 1,
[53]
leads to the adsorption isotherm described by a system of equations
F
uC gA 2 gC aC 1 r s,AC 2 1 r s,AC 5 r s,AC exp RTn 0s,C uA aA 1 ~ u 2A 2 u 2C!r s,AC A s,AC 1 u 2A~r s,BC A s,BC 2 r s,AC A s,AB! 1 u B~ u Cr s,AC B s,A 2 u AB s,C!#
[54]
F
uC aC gB 2 gC 1 r s,BC 2 1 r s,BC 5 r s,BC exp RTn 0s,C uB aB 1 ~ u 2B 2 u 2C!r s,BC A s,BC 1 u 2Ar s,AC~ A s,AC 2 A s,AB! 1 u A~ u Cr s,BC B s,B 2 u BB s,C!# .
N
g5
obtained by minimizing the surface tension against the surface layer composition (9)
i
0 s,i
RT ln
D
a s,i , ai
[50]
where a i is the bulk activity of the component i in the bulk. The equilibrium composition of the surface layer can be
[55]
Assuming that the bulk activities are known and constant, we can solve these equations for the surface fractions. Unfortunately, Eqs. [54] and [55] together with [53] cannot be solved analytically and the application of the numerical methods is needed. We have used the Newton– Raphson method for a system of nonlinear equations (26). The needed mole numbers, n s,i , were calculated from the densities and mole masses (19). When the surface composition is known the surface activities can be calculated from Eqs. [21]–[23] and the surface tension from Eq. [50]. For a two-component solution the adsorption isotherm takes the form
F
uB gA 2 gB aB 5 exp 1 r s,AB ~1 2 u B! r s,AB a rAs,AB RTn 0s,B
G
2 1 1 ~1 2 2 u B!r s,AB A s,AB .
[56]
365
SURFACE TENSION OF TERNARY SOLUTIONS
When the components are of the equal size, this isotherm reduces further to
F
G
aB x s,B gA 2 gB 5 exp 1 ~1 2 2x s,B! A s,AB . 1 2 x s,B a A RTn 0s,B
[57]
Next, substituting
S
b 5 exp
gA 2 gB 1 A s,AB RTn 0s,B
D
[59]
and assuming that at low concentrations of the solute (component B) the bulk activity of the solvent (A) is close to 1 (a A 5 1) we obtain the Frumkin isotherm (27) x s,B 5 ln a Bb 1 Ax s,B. 1 2 x s,B
[60]
On the other hand when we assume that A s,AB 5 0 and substitute DG 5 ~ g A 2 g B!/n 0s,B
[61]
into Eq. [56], we obtain
S
D
uB DG aB 5 exp 1 r s,AB 2 1 . ~1 2 u B! r s,AB a rAs,AB RT
[62]
It is similar to the adsorption isotherm derived by Levine and Fawcett (28) for the electrode– electrolyte interface. However, in our isotherm the activity of A is raised to the power r s,AB, which does not take place in the formula derived by Levine and Fawcett. For comparable diameters of the solution components (r s,AB ' 1) the expression exp(r s,AB 2 1) can be expanded into the power series. Using the first two terms of this series and assuming that at low concentrations a A ' 1 one obtains from Eq. [62] the Flory–Huggins isotherm:
S D
uB DG 5 a Bexp . r s,AB~1 2 u 2 ! r s,AB RT
Interaction pair
A s,ij
A ij
K
Acetone–water Acetone–propionic acid Water–propionic acid
3.2725 3.0645 1.5961
21.2946 3.2055 20.3070
22.53 0.96 25.20
[58]
A 5 22 A s,AB
ln
TABLE 1 The Values of the A s,ij and A ij Parameters Obtained from the Nonlinear Least-Square Fitting for the Water–Acetone–Propionic Acid System
[63]
We have shown above that the adsorption isotherm [56] is a generalization of the Flory–Huggins and Frumkin (27) isotherms. It includes both the different size effect and the intermolecular interactions. On the other hand, the adsorption isotherm given by a system of Eqs. [53]–[55] is the extension of isotherm [56] to the ternary solution.
RESULT AND DISCUSSION
The theory described in this paper can be used in different ways. When the bulk activities of the individual components of the ternary solution are known, one can calculate the values of the three parameters, A s,ij , which describe the molecular interactions at the surface. These values can be obtained by fitting the theoretical surface tension to the experimental data, as it has been described earlier (19) for the binary solutions. This method gives the most precise estimate of the molecular interactions at the surface. When the bulk activities are not known, we can calculate the A s,ij and A ij parameters by fitting the theoretical results to the experimental data. However, the number of the adjustable parameters increases to six now. Moreover, each parameter of the pair A s,ij and -A ij has a similar influence on the surface tension (see Fig. 4) which can lower the reliability of the results. We have applied this method to the ternary system investigated by Lebed and Eddin (3): water (w) - acetone (a) propionic acid (HPr) mixture. At T 5 293.15 K the pure liquids are characterized by the surface tensions g w 5 0.07275, g a 5 0.0237, and g HPr 5 0.0267 [N/m] and densities g w 5 0.9971, g a 5 0.7899, and g HPr 5 0.9942 [g/cm 3] (3, 29). The densities, together with the corresponding mole masses, are 0 needed to calculate the mole numbers n s,i and n i0 . We obtained 0 0 0 25 n s,w 5 1.576 3 10 , n s,a 5 6.184 3 10 26, n s,HPr 5 6.129 3 0 0 0 26 2 4 10 [mole/m ], and n w 5 5.535 3 10 , n a 5 1.360 3 10 4, n HPr 4 3 5 1.342 3 10 , [mole/m ]. The values of the A s,ij and A ij parameters obtained from the nonlinear least-square fitting are given in Table 1, while the theoretical and experimental surface tensions are shown in Fig. 2 by solid lines and graphic characters, respectively. We can expect some correlation between the bulk and surface parameters, A s,ij 5 KA ij ,
[64]
where K describes the difference between the surface and bulk mean coordination numbers. According to Patterson and Rastogi (30) a molecule moving from the bulk to the surface loses approximately 29% of its nearest neighbors which means that K ' 0.71. According to our calculations for the formamide–
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FIG. 2. Surface tension of the water–acetone–propionic acid solution. Fit of the theoretical surface tension (solid lines) to the experimental data (3) (graphic characters).
n-butanol system, K is 0.41 which means that K comprises also other effects. It seems that the reasonable values for K should range from 0.4 to 0.7. As follows from Table 1 this criterion is only approximately kept for the acetone–propionic acid interactions. Finally, the theory can be used to analyze the general surface behavior of the ternary systems and this is what we shall concentrate on in this section. We will discuss the influence of intermolecular interactions on surface tension of the solution by varying values of the A s,ij and A ij parameters. To make the discussion closer to the real system we assume that the surface 0 tensions, g i , and the surface, n s,i , and bulk, n i0 , numbers of moles of the individual components of the hypothetical ternary solution, A 1 B 1 C, are the same as those of water (A), acetone (B), and propionic acid (C). At first we will consider a system with no intermolecular interactions, which will be our reference system. Dependence of the surface tension on the concentration is shown in Fig. 3. The volume fractions are recalculated into the bulk mole fractions. As the molecules B and C are larger than A, they are more easily adsorbed than A. Small bulk concentrations of B and C give a great increase in their surface concentration (dotted lines in Fig. 10). This results in a rapid fall in the surface tension observed mainly at low concentrations of B and C. Parameters A s,AB and A AB describe intermolecular interactions between molecules A and B in the surface layer and in the bulk of the solution, respectively. For a negative A s,AB, the miscibility of A and B on the surface is reduced. The molecules are pushed from the surface to the bulk of the solution. Thus,
FIG. 3. Surface tension of the ternary solution for A s,AB 5 0, A s,AC 5 0, A s,BC 5 0, A AB 5 0, A AC 5 0, A BC 5 0, T 5 293.15 K, g A 5 0.07275, g B 5 0 0 0 0.0237, g C 5 0.0267 [N/m], n s,A 5 1.576 3 10 25, n s,B 5 6.184 3 10 26, n s,C 5 26 2 4 4 0 0 6.129 3 10 [mole/m ], and n A 5 5.535 3 10 , n B 5 1.360 3 10 , n C0 5 1.342 3 10 4, [mole/m 3].
we need more energy to create the surface, which results in the increase in the surface tension (solid lines in Fig. 4). At low concentrations of B, the slope of the surface tension with
FIG. 4. Surface tension of the ternary solution for A s,AB 5 23 (solid lines) and A AB 5 3 (dotted lines). The other parameters are as in Fig. 3.
SURFACE TENSION OF TERNARY SOLUTIONS
FIG. 5. Surface tension of the ternary solution for A s,AB 5 3. The other parameters are as in Fig. 3.
respect to x B is positive which indicates a negative adsorption of B. A similar surface tension behavior can be observed for the positive A AB (dotted lines in Fig. 4) although the mechanism of this surface tension increase is different: the positive A AB causes an increase in the mutual solubility of A and B in the bulk and thus stabilizes thermodynamically the A 1 B mixture. Again the molecules prefer the bulk of the solution to its surface. Addition of the component C reduces the effect of the A-B interactions, mainly in the bulk. The above examples show that the opposite effects occurring on the surface and in the bulk of the solution result in a similar surface tension behavior. Thus, we will restrict further discussion to the surface solutions. We will consider the influence of the bulk solution only in the cases when it is essentially different than that due to the surface solution. The positive values of A s,AB thermodynamically stabilize the surface solution of A and B. Molecules of the surface solution attract molecules from the bulk which produces a drop in the surface tension seen in Fig. 5. Small bulk concentrations of the component C have a weak influence on the surface tension, which means that the adsorbability of C is low. In fact, the thermodynamically stable surface solution of A and B does not let molecule C into the surface layer. The influence of the A s,AC and A AC parameters on the surface tension is well marked for x B 5 0. The negative A s,AC and positive A AC raise the surface tension with the increasing but low concentrations of C, and the adsorption of C is negative (the positive slope of the surface tension with respect to x C). For a negative A s,AC, the g -x B curves in Fig. 6 are nearly
367
FIG. 6. Surface tension of the ternary solution for A s,AC 5 22. The other parameters are as in Fig. 3.
independent of the concentration of C. They intersect at different points and finally at high concentrations of B we obtain a weak positive adsorption of C. A different picture is observed for a positive A AC (Fig. 7). The surface tension curves intersect nearly at one point at a very low concentration of B (at x B ' 0.006). Starting from this point the initially negative adsorption
FIG. 7. Surface tension of the ternary solution for A AC 5 2. The other parameters are as in Fig. 3.
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FIG. 8. Surface tension of the ternary solution for A s,AC 5 2. The other parameters are as in Fig. 3.
FIG. 9. Surface tension of the ternary solution for A s,BC 5 22.67. The other parameters are as in Fig. 3.
of C changes into the strongly positive. The intersection of the surface tension curves does not correspond to that observed by Lebed and Eddin (3) because of two reasons:
at the phase separation of the surface solution. In that case we had a positive slope before the discontinuity point and a negative after it. In the present case the slope is negative all the time. To provide a better illustration of the surface behavior in this particular case, in Fig. 10 we present the dependence of the
(i) we have an increase in g at x B 5 0 with the increasing concentration of C while Lebed and Eddin (3) obtained here a surface tension drop; (ii) our intersection point occurs at a very low concentration of B ( x B ' 0.006) while they found it at x B ' 0.2. By increasing the value of A AC one can slightly shift the intersection point toward higher B concentrations, but not as much as necessary. For the positive values of A s,AC and low B concentrations we observe a rapid decrease in the surface tension with increasing concentration of C (Fig. 8). This shows the large adsorbability of C at low concentrations of B. The high adsorbability of C at low B concentrations due to the thermodynamical stability of the surface solution of A and C makes the component B exhibit negative adsorption in this region. At higher concentrations of B, the influence of component C is gradually reduced and the adsorption of C falls down while the adsorption of B becomes positive. The negative values of the A s,BC parameter increase the surface tension with increasing concentration of C at high x B. This effect is shown in Fig. 9. However, we also observe in this figure some unexpected surface tension behavior at low x B. At higher concentrations of C (curves 5 and 6) the slope of the surface tension with respect to x B is not continuous. We observed earlier (19) a discontinuity of the surface tension slope
FIG. 10. Surface fractions of the ternary solution at x C 5 0.25 for the parameters are as in Fig. 9 (solid lines) and are as in Fig. 3 (dotted lines).
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SURFACE TENSION OF TERNARY SOLUTIONS
surface fractions u A, u B, and u C against x B at x C 5 0.25 (solid lines) and the corresponding results for the reference system with no intermolecular interactions (dotted lines). At a low x B, the decrease in the surface concentration of C and the increase in B is much lower than in the absence of B–C surface interactions. Molecules C, which are in the majority on the surface, hinder the increase in the surface concentration of B because of the repulsion between these molecules. The situation changes fundamentally when x B approaches some critical value (;0.2 in Fig. 10). Now, the increasing bulk concentration of B forces an increase in the concentration of B on the surface while this time molecules C are pushed out from the surface because of the repulsion between B and C. For A s,BC , 2 2.67, the curves u B and u C exhibit S loops characteristic of the phase transition (19). Interactions between B and C have a very small effect on the surface concentration of A. However, although this influence is small, we observe some discontinuity in the slope of u A with respect to x B at x B ' 0.2. The positive values of A s,BC lower the surface tension at high concentrations of B. This effect is shown in Fig. 11. It results from the formation of the thermodynamically stable solution of B and C on the surface. On the other hand a thermodynamically stable bulk solution of B and C ( A BC . 0) increases the surface tension. Let us note that by increasing the right-hand side of the surface tension curves when their extreme left points (at x B 5 0) are fixed, we can get a situation when all the curves intersect at one point (Fig. 12). This is the case observed by Lebed and Eddin (3). Changing the value of A BC, we can shift the intersection point within a certain range on the x B axis.
FIG. 12. Surface tension of the ternary solution for A BC 5 2. The other parameters are as in Fig. 3.
The intersection of the curves does not necessary take place at one point. For example for A BC 5 4 and the same concentrations of C as in Fig. 12, the curves intersect approximately in the range from x B 5 0.1 to 0.3. In this case it is difficult to say that the solution exhibits the surface buffering effect. In this section we have discussed the basic surface tension behavior exhibited by the ternary solutions. We did not analyze a superposition of two or more A s,ij and/or A ij parameters, but these effects can be deduced easily on the basis of the above discussion. At this stage, we did not also consider the hydrogen bond formation or the higher terms in the expression for the enthalpy of mixing (19, 31, 32) which may improve a theoretical description of the experimental results but they make the final form of the adsorption isotherm much more complicated. Finally, we hope that the present theory, alternatively extended by the order– disorder effects, will also be useful in the explanation of the surface tension of the systems which seem more difficult to describe theoretically like, e.g., along the binodal ˇ echova and Bartovska´ (33). curve measured recently by C CONCLUSIONS
Although the ternary systems exhibit a wide spectrum of the surface tension behaviors, with what has been shown in the previous section, we can formulate some general conclusions.
FIG. 11. Surface tension of the ternary solution for A s,BC 5 2.67. The other parameters are as in Fig. 3.
(i) Intermolecular attraction in the bulk of the solution increases the surface tension while the same interactions at the surface layer reduce it. (ii) Intermolecular repulsion in the bulk of the solution
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reduces the surface tension while the same interactions at the surface layer increase it. (iii) Strong intermolecular repulsion at the surface layer leads to discontinuity in the surface tension due to the phase separation at the surface. (iv) The larger molecules are adsorbed easier. These conclusions are similar to those derived earlier (19) for binary systems and perhaps can also be extended on multicomponent solutions. The ternary systems may exhibit one particular behavior: a so-called buffering effect which can occur in two different ranges of concentration. ACKNOWLEDGMENTS Financial support from Adam Mickiewicz University, Faculty of Chemistry is greatly appreciated.
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