A New Approach to Interfacial Energy

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

181, 259–274 (1996)

0377

A New Approach to Interfacial Energy I. Formulation of Interfacial Energy 1 TADAO SUGIMOTO Institute for Advanced Materials Processing, Tohoku University, Katahira 2-1-1, Aobaku, Sendai 980-77, Japan Received November 7, 1995; accepted February 26, 1996

This is a new fundamental theory of the interfacial energy including the explicit formulation of interfacial energy in a single formula, which involves the essential concept of the interfacial energy and all elements of the Gibbs, Langmuir, and Szyszkowski adsorption equations. The present thermodynamic theory is based on deductive conclusions that the intrinsic interfacial chemical potential of each component is generally higher than the bulk one at equilibrium and that the source of the interfacial energy is the excess chemical potential of each interfacial component. In addition, from an insight into the nature of interfaces, an interfacial double layer model has been proposed. The derived fundamental equation of the specific interfacial energy ( g ) at equilibrium is given as g Å N si ( msi A 0 mai ) / N sj ( msj B 0 mbj ), where N si and N sj are the two-dimensional maximum molar densities of arbitrary components i and j in the surface layers of a limited thickness, sA and sB , of two adjoining bulk phases a and b, respectively; mi and mj are the chemical potentials of components i and j in the phases indicated by the superscripts. Hence, g consists of the specific surface energies of the sA- and sB-layers, gA (ÅN si ( msi A 0 mai ) Å N sA ( msAA 0 maA )) and gB (ÅN sj ( msj B 0 mbj ) Å N sB ( msBB 0 mbB )), respectively, where A and B are the matrix components of the a- and b-phases, respectively. q 1996 Academic Press, Inc. Key Words: interfacial energy; surface energy; interfacial tension; surface tension; interfacial chemical potential; adsorption equation; Gibbs adsorption equation; Langmuir equation; Szyszkowski equation.

1. INTRODUCTION

As is well known, the Gibbs adsorption equation at constant temperature and pressure is given by d g Å 0∑ Gi dmi ,

[1.1]

where g is the specific interfacial energy or interfacial tension of an interface, Gi is the surface excess of component 1 Presented in part at the 47th Symposium on Colloid and Interface Chemistry, Okayama, October 8–10, 1994.

i, and mi is the chemical potential of component i at the interface but assumed to be common in the entire system at equilibrium (1). This is a kind of the Gibbs–Duhem equation in the interface at dT Å 0 and dP Å 0. It can be derived from the derivatives of the following fundamental equation for interface phase s by setting dT Å 0 and dP Å 0, U s Å TS s 0 PV s / gA s / ∑ mi n si ,

where U s , S s , V s , A s , and n si are the internal energy, overall entropy, volume, surface area, and mole number of component i in the interface phase (2, 3). In Eq. [1.2], an additional energy term gA s is located, as different from the corresponding equation for a bulk phase. It conforms well with our experience that there is undoubtedly some excess energy in the interface over the bulk phases, as explicitly observed as interfacial tension. As a result, gA s is added as the extra energy of the interface independently of the interfacial chemical potentials on the presumption that the interfacial chemical potential mi of each component is equal to the corresponding bulk one. Based on this treatment of the interfacial energy, modern thermodynamics of inhomogeneous systems has neatly been constructed since the past century and many interfacial phenomena have been elucidated quite successfully (1–10). But, in this conventional treatment, g is given only as an overall extra energy of the interface without further specification. However, if it were possible to define g as a function of more elementary variables, the benefit would be invaluable for understanding the backgrounds of diverse interfacial phenomena as well as the physical meaning of the interfacial energy itself. In particular, the explicit formulation of g is expected to shed light on the individual behavior of interfacial molecules which is thermodynamically unclear at present. Therefore, it seems worthwhile to discuss the background of the interfacial energy. Consider a binary inhomogeneous system consisting of two liquid phases. The interface must have been created in order to minimize the contact area between the two kinds

259

AID

JCIS 4284

/

6g12$$$221

06-03-96 07:39:38

[1.2]

0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

coidas

AP: Colloid

260

TADAO SUGIMOTO

of molecules lacking in affinity to each other, and thus the molecules remaining in contact with the different kinds of molecules at the interface must retain an excess free energy due to the lack of intermolecular affinity. Hence, according to the thermodynamic terminology, the excess free energy is basically a chemical energy of the individual molecules at the interface and it must correspond to the interfacial energy. In other words, the interfacial energy is ascribed to the total of the excess chemical potentials of the interfacial molecules over the bulk ones. It is no wonder if we consider that the chemical potential of each component is a partial molar free energy as a function of the partial molar entropy and partial molar enthalpy related to the molecular interactions consisting of chemical interactions characterized by the reactivity and physical interactions due to the van der Waals potential, electrostatic potential, etc. If we interpret the interfacial chemical potential and interfacial energy in this manner, the interfacial tension may be explained in terms of a counterforce to the external work to increase the number of the interfacial molecules of a high chemical potential at the expense of the bulk molecules of the lower chemical potential. According to this interpretation, gA s in a multicomponent system is directly written as gA s Å

∑ ( msi 0 mia )n si ,

This formulation of internal energy is the same as that for bulk phases. If the interfacial chemical potential mi in Eq. s [1.2] is represented by ms= i and if gA in Eq. [1.3] is inserted into Eq. [1.2], Eq. [1.2] is rewritten as a s U s Å TS s 0 PV s / ∑ msi n si / ∑ ( ms= i 0 mi )n i .

[1.5]

Since the last term in Eq. [1.5] is zero due to the presumption that mis= Å mia , Eq. [1.2] is apparently congruent with Eq. [1.4]. On the other hand, the last term in Eq. [1.4] corresponds to the Gibbs free energy of the interface phase given by G s Å ∑ msi n si .

[1.6]

The total differential of G s is given by dG s Å 0S sdT / V sdP / ∑ msi dn si

[1.7]

and msi is defined by

AID

JCIS 4284

/

6g12$$$222

06-03-96 07:39:38

,

[1.8]

s

T ,P ,n j

s G s= Å ∑ ms= i ni ,

[1.9]

*

the total differential of G s is given by dG s= Å 0 S sdT / V sdP 0 A sd g / ∑ msi dn si . *

[1.10]

Thus, ms= i is defined by ms= i Å

S D ÌG s= Ìn si

,

[1.11]

T ,P ,g,n js

where *

[1.4]

S D ÌG s Ìn si

where n sj represents the mole number of every interfacial component except component i. Hence the interfacial chemical potential msi is defined in exactly the same way as the bulk one. In this sense, msi is an ordinary intrinsic chemical potential. In the meantime, if the thermodynamic potential s s* (ms= as i n i in Eq. [1.2] is represented by G

G s Å G s 0 gA s .

[1.3]

where msi and mia are the newly defined interfacial chemical potential and bulk chemical potential of component i, respectively. Also, U s can immediately be written by U s Å TS s 0 PV s / ∑ msi n si .

msi Å

[1.12]

Note that, in the formulation of mis= , g is treated as an independent variable besides T, P, n si , and n sj , in contrast to the case of mis in Eq. [1.8]. Thus, mis= depends on g. In other words, mis= is not determined only by the state of the interface phase. On the other hand, since we have to specify a certain domain as an interface phase distinguished from the bulk phases to define G s , S s , V s , and n si , the state of the interfacial zone is determined only by T, P, and n si for all interfacial components, and thus the intrinsic interfacial chemical potential msi can be determined by these variables independently of mia and g. More* over, though G s ( ÅgA s / G s ) is an independent component * of U s from Eq. [1.4], gA s and G s are interdependent. * Namely, G s depends on the definition of gA s and vice versa. Therefore, since the conventional definition of gA s may be rephrased as an overall excess free energy of the interfacial molecules over the same numbers of the bulk molecules of * s each component, G s ( Å(ms* in Eq. [1.2]) must be i ni * a s s (mi n i , so that mi is always equal to mai , regardless of the state of the interfacial component i. Thus, it is obvious that mis= Å mai is not a deductive conclusion like the phase rule for the equilibrium between bulk phases, but a premise to define g. In other words, mis= is not a so-called intrinsic chemical potential, but a parameter identical to mai . In this context, Davies and Rideal (5) explained the equality of chemical potentials in the surface and bulk in a vapor– liquid system of a single component: the intermolecular distance of the surface molecules of a high chemical potential

coidas

AP: Colloid

A NEW APPROACH TO INTERFACIAL ENERGY

is enlarged to equate their chemical potential with that of the bulk molecules by increasing the partial molar entropy, and they attributed the surface tension to an attractive force between the surface molecules due to the intermolecular spacing. According to this argument, the interfacial chemical * potential ms defined by Eq. [1.11] appears to be treated like * an intrinsic chemical potential. However, if ms behaves like s* an intrinsic chemical potential, the m of the surface molecules must rather increase when they are more or less pulled apart from the original stable positions at the bottom of each potential well. If the displacement of the surface molecules is much more enlarged by pulling off the intermolecular bondings so as to lower the chemical potential to the level of the bulk molecules through increasing its partial molar entropy against the great increase of the partial molar enthalpy, the subsurface molecules underneath the great spaces created by the enlarged intermolecular spacing of the surface molecules must emerge as new surface molecules of a high chemical potential. Thus, the same procedure must be applied to the new surface layer again. However, the repetition of such a procedure may create an extremely porous or gaslike surface layer of a considerable thickness with a gradient of density. But there is no evidence of such a surface structure of a liquid phase. Hence, we must strictly distinguish * * ms from ms and note that ms Å ma is only a premise to define g. Let us discuss the nature of the intrinsic interfacial chemical potential msi . We hereafter refer to the intrinsic chemical potential simply as ‘‘chemical potential’’ unless it is required to be specified. First, we consider for simplicity a closed vapor–liquid inhomogeneous system of a single component in a given container at dT Å 0 and dP Å 0. For the equilibrium between the bulk phases of the liquid and its vapor, the chemical potential of vapor molecules must be, of course, equal to that of the liquid molecules; i.e., dG Å ( ma 0 mb)dn Å 0,

[1.13]

where dG denotes the change of the Gibbs free energy of the total system by the evaporation of the liquid molecules from bulk phase b into vapor phase a by dn moles; ma and mb are the chemical potentials of the molecules in the vapor and bulk phases, respectively. The equation suggests that if ma and mb are different, unilateral transfer of the molecules, either evaporation of the liquid or condensation of the vapor, may occur to change ma until both chemical potentials become equal to minimize the total free energy. The unilateral molecule transfer between the bulk phases is possible, because there is no restriction for the volume change of both bulk phases. Next, let us direct attention to the change of the total free energy by the change of the mole number of the molecules in the surface zone of the liquid phase. Since

AID

JCIS 4284

/

6g12$$$222

06-03-96 07:39:38

261

the increase of the mole number of the surface molecules, dn s , is always associated with the loss of the same amount of liquid molecules in the bulk (dn b Å 0dn s), dG Å msdn s / mbdn b Å ( ms 0 mb)dn s .

[1.14]

Since the surface chemical potentials ms and mb are both kept constant, G must linearly change with the change of n s . If ms ú mb , G takes its minimum value at the minimum of n s or A s determined by the geometric restriction from the shape of the container and the relative volume of the liquid. After G and A s have reached their minima, the freedom left for the surface molecules is only that of molecular exchange between the surface and bulk. However, such a molecular exchange has no effect on the total free energy G even though ms x mb; i.e., dG Å ( ms 0 mb)dn / ( mb 0 ms)dn Å 0.

[1.15]

Equation [1.15] shows that the equilibrium of the total system is established even in the presence of the surface molecules of a specially high chemical potential. It is no wonder, if we recall that, originally, thermodynamics requires only the minimum total free energy for the equilibrium of the entire system within the degree of freedom allowed for the system, which does not necessarily mean the equality of the chemical potential of each constituent in the whole system. Moreover, let us discuss whether or not the chemical potential of each component in the interfacial zone can be equal to the bulk one in a multicomponent inhomogeneous system. Consider a binary system of water ( a-phase) and oil ( bphase), in which water molecules (A) and oil molecules (B) are slightly dissolved in the opposite phases. Then, it is assumed that the mole fractions of water xA and oil xB continuously change in the interfacial zone, but always xA / xB Å 1. In general, the chemical potential of a constituent at a given position z is definitely determined by the composition at z and some long-range interactive potential field at z exerted from both bulk phases. Thus, each of the chemical potentials of water and oil molecules, mA and mB , is completely specified as a function of xA and z. For molecules A and B sufficiently away from the interfacial zone and thus free from the long-range interaction from the opposite bulk phases, their z-independent equilibrium mole fractions x Aa , x Ba , x Ab , and x Bb and chemical potentials mAa , mBa , mAb , and mBb are determined from a set of the relations mAa Å mAb , mBa Å mBb , x Aa / x Ba Å 1, and x Ab / x Bb Å 1. Here, it should be noted that the last four relations can be achieved by changing each volume of the two bulk phases. Since these chemical potentials in the bulk phases are independent of z, they are determined only by the compositions of the uniform bulk phases and thus they are fixed independently of the potential

coidas

AP: Colloid

262

TADAO SUGIMOTO

field near the interface. If the chemical potential of water at an arbitrary position z in the interfacial zone, mA (xA , z), is equal to the predetermined mAa ( ÅmAb ), i.e., mA (xA , z) Å mAa ,

[1.16]

xA is determined as a single function of z by this relation. However, since xB Å 1 0 xA , the chemical potential of oil molecules, mB (xB , z), is also determined as a function of z at the same time. Thus, it is generally impossible to equate mB (xB , z) additionally with mBb ( ÅmBa ) unless xA Å x Aa , xB Å x Ba , or xA Å x Ab , xB Å x Bb . In other words, if we assume that the chemical potentials of water and oil are both uniform throughout the system, the composition must be kept constant in each bulk phase up to the boundary of these bulk phases and discontinuously switched at the boundary. However, if the mole fraction of the matrix component of a bulk phase is uniform up to the discontinuous boundary, its surface chemical potential must be higher than the bulk one because of its higher standard chemical potential at the surface in direct contact with the opposite phase. Thus, at least, one of the components must have an interfacial chemical potential different from the bulk one, regardless of the mode of compositional change across the interface. It may readily be understood that the situation is the same for a general multicomponent system if we consider that ( xi Å 1 and msi Å mia Å mib for all components are incompatible with each other when the equilibrium between the bulk phases has been established by changing each volume. As a consequence, the phase rule of the uniform intrinsic chemical potential can generally be applied to the equilibrium between bulk phases but not to the equilibrium between bulk and interface phases. Next, let us proceed to discuss the equilibrium of molecular exchange between a section of a sufficiently small thickness Dz taken at an arbitrary position z parallel to the interface in the interfacial zone and an arbitrary position sufficiently away from the interfacial zone (z Å 0` ) in a bulk phase a in a multicomponent inhomogeneous system of two bulk phases a and b after the interfacial area is fixed at the minimum. First, we define the maximum molar density of component i at z, Ni , and its volume fraction yi as 1 Ni å Vi

[1.17]

mi , Ni

[1.18]

and yi Å

Ni value is close to the molar density of the pure substance of component i, it is basically a function of the composition and position z, since Vi is a function of these variables. If Az denotes the surface area of the section at z, mi the chemical potential of i at z in the section, and mia at z Å 0` in the bulk, the change of the Gibbs free energy G of the system by the molecular exchange at dT Å 0 and dP Å 0 is given by dG Å ∑ Az Ni ( mi 0 mia ) Dzdyi .

[1.19]

The change of volume fraction yA of the matrix component A of the a-phase is given by dyA Å 0∑ *dyi ,

[1.20]

where (* means the summation for all nonmatrix components, since ( yi Å 1. Hence, one obtains dG Å ∑ * Az[Ni ( mi 0 mia ) 0 NA ( mA 0 mAa )] Dzdyi .

[1.21]

If the system is in equilibrium (dG Å 0), the following relation holds for an arbitrary component i: Ni ( mi 0 mia ) Å NA ( mA 0 mAa ).

[1.22]

This equation shows that if one of the components has an interfacial chemical potential different from its bulk one, all the other components also have different interfacial chemical potentials from each bulk one. Since it has already been deduced that at least one component must have an interfacial chemical potential different from its bulk one, mi x mia holds for all components. Since the specific interfacial energy g corresponds to the excess free energy in the interfacial zone per unit area, g is given by

*0` ∑ Az Ni ( mi 0 mia )yi dz /`

gÅ

As

,

[1.23]

where A s denotes the surface area of the interface, and mia Å mib due to the equilibrium between the two bulk phases. Also, since the interfacial zone is sufficiently narrow in general, Az Å A s . Thus it follows from Eq. [1.23] with Eq. [1.22] that gÅ

*

/`

Ni ( mi 0 mia )dz,

[1.24]

0`

respectively, where Vi and mi are the partial molar volume of component i and its mole number per unit volume, or molar density, in the section at z, respectively. Though the

AID

JCIS 4284

/

6g12$$$223

06-03-96 07:39:38

where subscript i denotes an arbitrary component i. This is the general equation of g at equilibrium. As g is normally

coidas

AP: Colloid

A NEW APPROACH TO INTERFACIAL ENERGY

positive, there must be at least one position in the interfacial zone which commonly gives the maximum mi for all components from Eqs. [1.22] and [1.24]. Taking the maximum position as z Å 0, we may define the thickness of the interfacial zone, d, as

*0` Ni ( mi 0 mia )dz /`

då

Ni (z Å 0)( mi (z Å 0) 0 mia )

,

[1.25]

where Ni (z Å 0) and mi (z Å 0) are Ni and mi at z Å 0, respectively. And mi (z Å 0) is redefined as an overall interfacial chemical potential in the interfacial zone, msi . Namely, msi å mi (z Å 0).

[1.26]

Hence, Eq. [1.24] can be rewritten as g Å Ni (z Å 0) d ( msi 0 mia ).

[1.27]

In the existing thermodynamic theory of interfaces, the composition of the interfacial component is usually assumed to be changed continuously across the dividing surface. If the interface is actually continuous in terms of composition and chemical potential, the overall interfacial chemical potential of component i, msi , must be defined as the maximum chemical potential of component i at a certain point in the interfacial zone, and the thickness of the interfacial zone, d, defined by Eq. [1.25] is uniform and independent of the size of each component molecule from Eq. [1.22]. However, if the interface is discontinuous, at least in terms of thermodynamics, the treatment of these parameters will be different as to be shown in the following paragraphs. Finally, let us consider the state of the interface. Though we have not specified hitherto the state of the interface in order to maintain the generality of the argument, it is worthwhile for understanding the entire backgrounds of interfacial phenomena to discuss the continuity of the interface. The continuous interface convention is convenient for a general treatment of interfaces, but, it is also true that it has evoked the continual controversies and excessive sophistication on the treatments of the boundary of adjoining phases. If the interface is basically discontinuous, the repeated arguments on the Gibbs dividing surface of some inherent ambiguity (1, 7, 11, 12) can be terminated. Originally, an interface is created when the affinity between different kinds of matrix molecules is much smaller than that between the same matrix molecules and thus the contact area of the matrix molecules of different kinds must be minimized in order to minimize the total free energy of the system at equilibrium. In other words, the number of the matrix molecules in touch with a different kind of matrix molecule must be as small as possible. This means that the interface must be discontinuous

AID

JCIS 4284

/

6g12$$$223

06-03-96 07:39:38

263

with the minimum surface area. Moreover, if we consider a pure water system, the chemical potential of water molecules in the surface monolayer must be especially high due to the asymmetric environment as compared to that in the bulk phase including the molecular layer just underneath the top layer. On the other hand, some of the vapor molecules may be adsorbed on top of the surface monolayer of the water, but the adsorbed amount seems to be extremely limited because they are stabilized only at the bottom of each molecule attached to the surface. And, the molecules in the vapor phase are still more diluted because of being completely free from stabilization by matrix molecules. Hence, the environments for a molecule in the adjoining four regions including the bulk liquid, surface monolayer, adsorption layer of the vapor molecules (gas-side surface monolayer), and bulk vapor phase are distinctively different. Similarly, in a general multicomponent inhomogeneous system, the compositions of the components in the adjoining four regions across the interface are also expected to be discrete from one another. Therefore, at least from a chemical viewpoint, the interface seems basically discontinuous with two adjoining surface monolayers discrete in composition on its each side. However, if long-range interactions such as the van der Waals potential across the interface, polarization, perturbation in molecular orientation due to the self-rearrangement of interfacial molecules, etc., have some effect on the environment of the bulk molecules near the interface, there may be more or less tailings of the compositional and chemical potential distributions on both outer sides of the adjoining surface monolayers. However, this effect may be rather minor as compared to the great chemical potential difference between the surface molecules in direct contact with different matrix molecules of the opposite phases and the molecules in each second molecular layer. For instance, the thick˚ from the X-ray ness of free surface of water is ca. 3.2 A reflectivity measurement (13), which is almost equal to the ˚ . We may call the calculated thickness of a monolayer 3.1 A pair of the adjoining two surface layers including the surface monolayers with some tailings of compositional and chemical potential distributions on each outer side as the ‘‘interfacial double layer.’’ Thus the term ‘‘surface layer’’ used in this paper means an effective surface layer and it is not necessarily limited to the surface monolayer. For precise measurement of the thickness of an interfacial zone, special care must be taken in establishing the complete equilibrium. For instance, in a vapor–liquid system, the thickness of the vapor-side surface layer in equilibrium is often confused with that in the steady state of evaporation. Though we must compile a good deal of reliable data using, e.g., X-ray reflectivity measurements (13, 14), ellipsometry (15–18), etc., prior to the final conclusion on the issue of whether or not the interface is actually continuous, the interfacial double model which can be applied even to the

coidas

AP: Colloid

264

TADAO SUGIMOTO

component molecule in the sA- and sB-layers is mostly fixed on the flat interface so that mi is kept constant at least over the molecular thickness of component i. Thus, d si A and d si B strongly depend on the molecular thickness of each component in the interfacial double layer, in contrast to the uniform thickness of the interfacial zone, d, in the continuous interface model. Then we further define the two-dimensional maximum molar densities of component i, N si A and N si B , in the sA- and sB-layers, respectively, as N si A å Ni (z Å 0 e ) d si A ;

FIG. 1. Compositional profiles of the interfacial double layer model (I. D. L. Model) and continuous interface model (C. I. Model) in a binary inhomogeneous system: xA and xB are the mole fractions of components A and B as the matrices of phases a and b, respectively.

interfaces subjected to some interfacial perturbation will prove its validity in elucidation of the backgrounds of many interfacial phenomena, including adsorption in general, characteristic behavior of interfacial molecules at different kinds of interfaces, pressure effects on interfacial energy, etc. In the interfacial double layer model illustrated in Fig. 1, the origin of the z-coordinate is taken at the definite boundary, and the overall chemical potentials of component i in surface layers sA and sB , msi A and msi B , correspond to their chemical potentials at z Å 0 e and /e, respectively, where e is an infinitesimal distance from the boundary: msi A å mi (z Å 0 e );

msi B å mi (z Å /e ).

[1.28]

Likewise, the thicknesses of component i in the sA- and sBlayers, d si A and d si B , are defined as

*0` Ni ( mi 0 mia )dz

N si B Å Ni (z Å /e ) d si B .

[1.30]

Though N si A and N si B are the inverses of partial molar area at the interface as functions of the compositions in the sAand sB-layers, respectively, they are close to the two-dimensional molar density of the pure substance of component i. And, d si A and d si B are both equal to the molecular thickness of component i if the effect of the long-range interaction can be neglected. In addition, since there seems no fear of confusion, N si A and N si B will commonly be denoted by N si for simplicity. The N si is mainly determined by the molecular area of component i though partly subject to the long-range effects. And it may change more or less with the progress of adsorption, but virtually be kept constant during the early stage of adsorption and the final stage close to the saturated adsorption, in both of which the activity coefficient of each component is also kept constant. On the other hand, the volume fractions of component i within the infinitesimal intervals [ 0 e, 0] and [0, /e], yi (z Å 0 e ) and yi (z Å /e ), are equal to the respective area fractions in the interfacial sublayers, y si A and y si B , as y si A Å yi (z Å 0 e );

y si B Å yi (z Å /e ).

[1.31]

Hence, the area fractions, y si A and y si B , are used for yi (z Å 0 e ) and yi (z Å /e ) in the interfacial double layer model, while y si in the continuous interface model is the overall volume fraction of component i in the interfacial zone of thickness d. From Eqs. [1.28] – [1.31], the two-dimensional surface concentrations of component i in the sA- and sBlayers, G si A and G si B , are generally given by

0e

d

sA i

å

sA i

Ni (z Å 0 e )( m

a i

0m )

*0` Ni ( mi (z) 0 mia )yi dz 0e

;

G

*/e Ni ( mi 0 mib )dz

sA i

s i

sA i

s i

sB i

Å N y

å

mi (z Å 0 e ) 0 mia

/`

d si B å

Ni (z Å /e )( m

sB i

b i

0m )

,

[1.29]

where Ni (z Å 0 e ) and Ni (z Å /e ) are the maximum molar densities of component i at z Å 0 e and /e, respectively. In the interfacial double layer model, the ‘‘head’’ of each

AID

JCIS 4284

/

6g12$$$223

06-03-96 07:39:38

*/e Ni ( mi (z) 0 mib )yi dz

,

[1.32a]

.

[1.32b]

/`

G

sB i

Å N y

å

mi (z Å /e ) 0 mib

If we use Eqs. [1.32a] and [1.32b], Eq. [1.23] is rewritten as

coidas

AP: Colloid

265

A NEW APPROACH TO INTERFACIAL ENERGY

gÅ

∑ ( msi 0 mia )n si A / ∑ ( msi 0 mib )n si B As

, [1.33]

where n si A and n si B are the mole numbers of component i in the sA- and sB-layers, respectively; msi is the overall interfacial chemical potential in the interfacial double layer model defined as msi å

sA i

m n n

sA i sA i

sB i sB i

/m n /n

sB i

.

[1.34]

Equation [1.33] can generally be rewritten with arbitrary components i and j in the a- and b-phases as gÅ

∑ N si ( msi A 0 mia )y si A / ∑ N sj ( msj B 0 mjb )y sj B . [1.35]

This is a general expression of g, which holds even in nonequilibrium states. If the system is in equilibrium (d g Å 0), it follows from ( dy si A Å 0 and ( dy si B Å 0 that N si ( msi A 0 mia ) Å N sA ( msAA 0 mAa ); N sj ( msj B 0 mjb ) Å N sB ( msBB 0 mBb )

[1.36]

and g Å N si ( msi A 0 mia ) / N sj ( msj B 0 mjb ),

[1.37]

where mia Å mib and the subscripts A and B denote the matrix components of the a- and b-phases, respectively. In this context, Butler (19) already referred to the surface or interfacial chemical potential as early as 1932. He derived the following equation from a hypothesis of surface monolayers in which each component is presupposed to have an exceptionally high chemical potential and the surface areas occupied by the individual components are interdependent, gÅ

msi 0 mia , Ai

[1.38]

where msi is the chemical potential of component i in the interfacial monolayers, mia the chemical potential of component i in the bulk phases, and Ai the partial molar surface area of component i. Although his argument was quite suggestive, as was discussed by Verschaffelt (20), Kofoed and Villadsen (21), Rusanov (22), and Eriksson (23–25), it does not appear to be accepted nowadays, probably because it basically lacks in generality in contrast to current thermodynamics which imposes no special condition such as the specific surface monolayer on the state of the interface, and because it apparently conflicts with the conventional theorem of the uniform chemical potential since no analysis has ever been made on the nature of the intrinsic interfacial chemical

AID

JCIS 4284

/

6g12$$$224

06-03-96 07:39:38

potential in comparison with the conventional interfacial chemical potential defined by Eq. [1.11]. As a result, the interfacial chemical potential msi is now reinterpreted as a convenient parameter defined by msi å mi / gAi ,

[1.39]

where mi is the conventional chemical potential of component i common in the entire system (24). Namely, g is regarded as an elementary variable and msi is treated as a subordinate function of g, in contrast to the treatments of Butler’s original paper and the present theory. This reversed interpretation, which is though in conformity with the conventional treatment, may not lead us to the understanding of the essential backgrounds of g itself and the related interfacial phenomena. However, there is one more basic problem in his treatment: surface monolayers on each side of the interface are regarded as a single uniform interfacial layer and he applied Eq. [1.38], as it is, to his experiments. Obviously, this treatment is inadequate, since the chemical potentials of the components in the two monolayers, sA and sB , must be different due to the incompatibility of msi A Å msi B for all components with ( x si A Å 1 and ( x si B Å 1, and thus the average mole fraction (x si A / x si B )/2 ( Å x si ) cannot be used as the activity of msi . The rigorous distinction between the chemical potentials in the sA- and sB-layers is of vital importance for the precise analysis on the behavior of interfacial molecules. On the other hand, the Langmuir equation as another important adsorption equation is given by uÅ

KC , 1 / KC

[1.40]

where u is the coverage of an adsorbed species, C is the concentration of the species in the bulk phase, and K is a constant (26–28). This equation was derived by assuming the chemical equilibrium of adsorption and desorption of the adsorbate. Since the starting point for the derivation of this equation is entirely different from that of the Gibbs equation, they have been regarded as two independent equations. Before the advent of the Langmuir equation, Szyszkowski (29) had proposed the following empirical relation between g and C from his capillary measurements on aqueous solutions of fatty acids mainly in the ranges of saturated adsorption,

S

g C Å 1 0 K1 log10 1 / g0 K2

D

,

[1.41]

where g 0 is the initial g of the pure solvent, and K1 and K2 are constants. Probably, this equation seems to have been

coidas

AP: Colloid

266

TADAO SUGIMOTO

induced from the combination of the initial condition, g Å g 0 at C Å 0, and the linearity of g against log10C for C/K2 @ 1. If we combine Eqs. [1.1] and [1.40] for a two-component system, we find the following equation corresponding to Eq. [1.41], g Å g 0 0 N sRT ln(1 / KC),

[1.42]

where N s is the two-dimensional maximum molar density of the solute in the interface ( u Å G /N s). This is also referred to as the Szyszkowski equation. Since one of the Gibbs, Langmuir, and Szyszkowski equations can be obtained if the other two equations are given, two of them are independent equations and they are at least mathematically of equal weight. But from a physical viewpoint, the most fundamental independent adsorption equations hitherto known may be the Gibbs and Langmuir equations. However, since g and u change at the same time with adsorption, there must be some correlation between the apparently independent equations and even these most fundamental equations may describe only specific aspects of a more essential background. Besides, it is rather surprising that the Langmuir equation generally describing the equilibrium of chemical species between an interface and bulk phase has never been substantiated by thermodynamics. This may imply that there is some unknown barrier between the Langmuir equation and conventional thermodynamic treatment for interfaces. All of these physical analyses on the concepts of the interface and interfacial energy are based on a presumption that some interfacial energy is present at the interface of an inhomogeneous system. However, it seems more important to formulate the fundamental theory of the interfacial energy from a chemical viewpoint how the interfacial energy is formed. Thus, the main objectives of this theory are to derive the fundamental equation of the interfacial energy through a direct chemical approach to g and make clear the underlying principle of adsorption through the insights into the chemical behavior of individual interfacial components. We will find that the derived equation of g which agrees with Eq. [1.37] involves all elements of the Gibbs, Langmuir, and Szyszkowski adsorption equations as well as the essential concept of the interfacial energy in its single formula. Finally, the significance of N si in the interfacial double layer model and the conventional continuous interface model will be discussed and some comments on the known adsorption equations, such as the Gibbs, Langmuir, and Szyszkowski equations will be given in light of the present theory. 2. FUNDAMENTAL THEORY

ergy through a kind of chemical process of activated surface molecules. Consider the work necessary for the formation of a surface by breaking the molecular bonds of a pure solid or liquid consisting of matrix molecules A. If the chemical potential of A for 1 mole in the bulk is represented by m0A and that of the activated surface molecules A* by m0A * , the work to form 1 mole of surface molecules A*, wA * , is given by wA * Å m0A * 0 m0A .

Similarly, the work for the formation of 1 mole of active surface molecules B* from bulk molecules B, wB * , is given by wB * Å m0B * 0 m0B .

AID

JCIS 4284

/

6g12$$$225

06-03-96 07:39:38

[2.2]

Here we assume that A and B are immiscible with each other and then bring the two surfaces into contact. The work for the formation of 1 mole of molecular pairs AB of the interface from active molecules A* and B*, DwAB , is given by DwAB Å m0AB 0 m0A * 0 m0B * ,

[2.3]

where m0AB is the chemical potential of molecular pairs AB and the two-dimensional sizes of molecules A and B are provisionally assumed to be the same for simplicity. Thus, the total work for the formation of 1 mole of AB pairs from A and B in each bulk phase, wAB , is wAB Å wA * / wB * / DwAB Å m0AB 0 m0A 0 m0B .

[2.4]

Here, DwAB may be negative or positive. But wAB must be positive due to the very small affinity between A and B which is a necessary condition for the formation of an inhomogeneous system. Since the specific surface (or interfacial) energy g is the work for the formation of a surface (or interface) for unit area from the bulk phase(s), or the total excess energy of the surface molecules (or interfacial molecular pairs) over the total energy of the constituent molecules in the bulk phase(s) per unit area, the specific surface energies of the a-phase, g 0A * , and b-phase, g 0B * , and the specific interfacial energy, g 0AB , are given by g 0A * Å N s0 ( m0A * 0 m0A ),

[2.5]

g 0B * Å N s0 ( m0B * 0 m0B ),

[2.6]

g 0AB Å N s0 ( m0AB 0 m0A 0 m0B ),

[2.7]

and

2.1. Interfacial Energy of One-Component Bulk Phases

Let us formulate the interfacial energy from a chemical standpoint, focusing on the formation of the interfacial en-

[2.1]

respectively, where N s0 is the two-dimensional molar density

coidas

AP: Colloid

267

A NEW APPROACH TO INTERFACIAL ENERGY

or mole number of the surface molecules (or interfacial molecular pairs) per unit area. However, as discussed in the Introduction, if we take into account the physical long-range effects, the surface layers may not be limited to the surface monolayers. Thus, N s0 is assumed to be equal to or somewhat larger than the corresponding value of the surface monolayer. And, the specific surface (or interfacial) energy will be referred to simply as surface (or interfacial) energy in this paper, unless it is necessary. Also, g 0AB will be written as g 0 . If we define the interface phase as a double layer consisting of two adjoining surface layers sA of the a-phase of molecules A and sB of the b-phase of molecules B, m0AB may A for molecules A in be divided into two components: m0,s A B for molecules B in the s -layer as the sA-layer and m0,s B B A / m0,sB . m0AB Å m0,s A B

[2.8]

g 0 Å g 0A / g 0B ,

[2.9]

Hence,

where A 0 m0 ); g 0A Å N s0 ( m0,s A A

B 0 m0 ). g 0B Å N s0 ( m0,s B B

[2.10]

preceding Section 2.1, g in the course of this process is given by g Å ∑ N si ( msi A 0 mia )y si A / ∑ N sj ( msj B 0 mjb )y sj B .

Thus the change of g, d g, is given by d g Å ∑ N si ( msi A 0 mia )dy si A /

A 0 m0 ); g 0A Å N sA ( m0,s A A

s A

B 0 m0 ), g 0B Å N sB ( m0,s B B

d g Å ∑ * [N si ( msi A 0 mia ) 0 N sA ( msAA 0 mAa )]dy si A

∑ * [N sj ( msj B 0 mjb ) 0 N sB ( msBB 0 mBb )]dy sj B . [2.14]

When the system reaches the equilibrium, d g Å 0 for the changes of any y si A and y sj B , so that

[2.11]

s B

where N and N are the two-dimensional molar densities of pure molecules A and B at the given temperature and pressure in the sA- and sB-layers, respectively. From an experimental point of view, the overall interfacial energy g 0 may be directly measured, but the individual components, g 0A and g 0B , are interdependent and they cannot be determined separately. Nevertheless, the physical meaning of each surface energy as a component of the interfacial energy is definite and the merit of the expression of g 0 as the sum of g 0A and g 0B in Eq. [2.9] is that g 0A and g 0B can formally be treated individually. In this treatment, for the molecules in a surface layer of the interfacial double layer, the opposite surface layer is regarded to be a uniform plane as a part of their environment determining their chemical potentials.

AID

JCIS 4284

/

6g12$$$225

06-03-96 07:39:38

N si ( msi A 0 mia ) Å N sA ( msAA 0 mAa ),

[2.15a]

N sj ( msj B 0 mjb ) Å N sB ( msBB 0 mBb ).

[2.15b]

Inserting Eqs. [2.15a] and [2.15b] into Eq. [2.12], one finally obtains g Å N si ( msi A 0 mia ) / N sj ( msj B 0 mjb ),

[2.16]

where i and j are arbitrary components including matrix components A and B. This equation in agreement with Eq. [1.37] derived from the physical analyses on the presumption of the presence of the interfacial energy is the most fundamental formula in this theory. Equation [2.16] shows that g consists of the two surface energies of the sA- and sB-layers, gA and gB , as

2.2. Interfacial Energy of Multicomponent Bulk Phases

If we introduce some foreign substances into a system consisting of a-, b-, and s-phases of the matrix molecules, they will diffuse to the interface, exclude a corresponding amount of the interfacial matrix molecules, and finally reach the equilibrium among these phases. On the analogy of the

∑ N sj ( msj B 0 mjb )dy sj B , [2.13]

which corresponds to the change of the total free energy (the Gibbs free energy when dT Å 0 and dP Å 0 or Helmholtz free energy when dT Å 0 and dV Å 0) of the system per unit area of the interface by the exchange of the components between the s-phase and bulk phases a and b. Using (* dy si A / dy sAA Å 0 and (* dy sj B / dy sBB Å 0 due to ( y si A Å 1 and ( y sj B Å 1, where (* means the summation for all nonmatrix components,

/

In an actual system, however, the two-dimensional sizes of A and B are generally different so that g 0A and g 0B may be written as

[2.12]

g Å gA / gB ,

[2.17]

gA Å N si ( msi A 0 mia ) Å N Aa ( msAA 0 mAa ),

[2.18a]

gB Å N sj ( msj B 0 mjb ) Å N sB ( msBB 0 mBb ).

[2.18b]

where

coidas

AP: Colloid

268

TADAO SUGIMOTO

Equation [2.16] involves all elements of the Gibbs, Langmuir, and Szyszkowski equations in its single formula (see section 3.3 under Discussion). If we write g as a function of the chemical potentials of the matrix molecules in particular, g Å N sA ( msAA 0 mAa ) / N sB ( msBB 0 mBb ),

[2.19]

or more concretely g Å g 0 / N sA RT ln

a sAA a sBB s / N RT ln , B a Aa a Bb

[2.20]

where a sAA , a Aa , a sBB , and a Bb are the activities of the matrix molecules A and B in the respective phases shown by the superscripts. If the mole fractions of A in the a-phase and B in the b-phase are very close to unity, as rather usual in the study of adsorption, a Aa and a Bb can be approximated as unity. In this case, Eq. [2.20] reduces to g Å g 0 / N sA RT ln a sAA / N sB RT ln a sBB .

[2.21]

This equation shows that g 0 g 0 is determined solely by the maximum molar densities of the matrix molecules and their activities in the interface, irrespective of the compositions of the other components both in the interface and in the bulk phases, though the activities of the matrix molecules themselves in the interface are governed by the compositions of the bulk phases. In addition, if (* x si A ! 1 and (* x si B ! 1 for the mole fractions of the nonmatrix interfacial components, a sAA á 1 0 (* x si A á 1 0 (* G si A /N sA and a sBB á 1 0 (* x si B á 1 0 (* G si B /N sB so that Eq. [2.21] may be approximated as

S

g Å g 0 / N sA RT ln 1 0

S

/ N sB RT ln 1 0

∑ * G si A N sA

∑ * G si B N sB

D

S DS D a si A a ia

∑ * Gi RT, [2.22]

where (* Gi Å (* G si A / (* G si B . As long as an inhomogeneous system remains entirely in equilibrium, the interfacial area must be kept constant at the minimum value with the progress of adsorption until the interfacial energy g reaches zero, which means that the chemical potential of each component becomes uniform throughout the entire system. When g is going to be negative, the bulk phases may start mixing together in a form of emulsion in order to reduce the total free energy by increasing the interfacial area. But it may cause regaining of g due to the reduced concentration of the adsorbed molecules, and

AID

JCIS 4284

/

6g12$$$225

2.3. Interfacial Energy as a Function of the Bulk-Phase Compositions

From Eq. [2.18a], it follows that

D

á g0 0

thus the interfacial area may be balanced at a certain level to keep g Å 0. In this stage, the interfacial area is a function of the total amount of the adsorptive in the system and the chemical potential of each component is uniform everywhere in the whole system. Although this paper does not deal with multilayer adsorption in detail, Eqs. [2.16] and [2.20] can be applied to such a case as well, but multilayer adsorption itself has no direct influence on g unless a sAA /a Aa and a sBB /a Bb are changed. In general, adsorptive molecules which tend to be adsorbed in multilayer have originally rather strong affinity with each other and thus at least some of them are already associated or linearly linked together in the bulk phase. The chemical potential of these linked species must be different from that of their monomers. Thus, we may treat these associated species as a different component and the multilayer adsorption can be regarded as the adsorption of these linked species. As a consequence, the multilayer adsorption can be treated in the same way as usual monomeric adsorption in a multicomponent system, though we have to take into account the equilibrium of their association in the bulk phase. Since the concentration of monomeric species is usually much higher than that of the linked molecules when the absolute concentration of the monomeric species is sufficiently low, multilayer adsorption often appears to start after the saturation of the single layer adsorption since the concentration of the linked species becomes high enough. But of course there is no distinction in treatment between direct adsorption of the linked species and stepwise adsorption of the monomeric species at equilibrium.

06-03-96 07:39:38

where K ia å exp

F

a Aa a sAA

vi

Å K ia ,

A 0 m0,a ) 0 ( m—oi ,sA 0 m—oi ,a ) / mi ( m0,s A A RT

[2.23]

G

, [2.24]

in which mi å

N sA . N si

[2.25]

Here, m—oi denotes the standard chemical potential of component i at xi Å 1 extrapolated from xi ! 1 in contrast to m0i which is also the standard chemical potential of component i at xi Å 1 but that of its pure substance. We use m0i for

coidas

AP: Colloid

269

A NEW APPROACH TO INTERFACIAL ENERGY

S

D

matrix components and m—oi for nonmatrix components, since, initially, xi Å 1 for the former and xi Å 0 for the latter. Also, it should be noted that K ia and the corresponding K ib are not the equilibrium constants of component i peculiar to each matrix but proper to the interface consisting of matrices A and B and their interface. In the present theory, it is assumed that K ia @ 1 and K ib @ 1 for nonmatrix components, as is usual in the study of adsorption.

∑ * K ia x ia g Å g / N RT ln 1 0 1 / ∑ * K ia mi x ia

Multicomponent System of Low Adsorption Level

Let us consider the interfacial energy in a practically important two-component system, in which only one nonmatrix component i is involved and the solubility of component i in the b-phase is negligible as compared to that in the aphase. In this case, the interfacial energy is changed by the adsorption of component i solely from the a-phase. If we consider the slight dissolution of each matrix component into the opposite phase, this is a ternary system. But, usually, we can neglect the contribution of the intermixing of matrix components (see Section 3.1 under Discussion). (a) Low adsorption range (x si A ! 1). In this range, sA x i is given by

In this case ( (* x si A ! 1), a si A á x si A , a sAA á x sAA , a ia á x , and a Aa á 1. Thus it follows from Eq. [2.23] that a i

∑ x si A Å ∑ K ia x ia (1 0 ∑ * x si A ) mi Å 1.

[2.26]

Since (* x si A ! 1, this equation can be approximated as

∑ K ia x ia (1 0 mi ∑ * x si A ) Å 1.

[2.27]

Thus one finds the following relationship between the total mole fraction of the nonmatrix components in the sA-layer and their mole fractions in the a-phase as

∑ * x si A Å

∑ * K ia x ia , 1 / ∑ * K ia mi x ia

[2.28]

0

S

/ N sB RT ln 1 0

x

a i

ÅK x

a i

S

D

∑ * K ia x ia 10 . 1 / ∑ * K ia mi x ia mi

S

∑ * K ia x ia gA Å g / N RT ln 1 0 1 / ∑ * K ia £i x ia

x si A Å

S

∑ * K ib x ib 1 / ∑ * K ib m *i x ib

D

[2.30]

D

,

[2.31]

where K ib is the equilibrium constant of component i for the equilibrium between the sB-layer and b-phase, corresponding to K ia for the equilibrium between the sA-layer and aphase, and m *i å N sB /N si . Consequently,

AID

JCIS 4284

/

6g12$$$226

.

[2.32]

K ia x ia 1 / K ia mi x ia

[2.33]

from Eq. [2.28] or [2.29]. On the other hand, x si A in terms of Gi is written as x si A Å

Gi . N 0 ( mi 0 1) Gi

[2.34]

s A

1 1 2mi 0 1 1 Å a s / , Gi K i N AVA Ci N sA

sB i

and gB Å g 0B / N sB RT ln 1 0

D

Thus, Eq. [2.33] is transformed as

Similarly, the corresponding equations for (* x and x are also obtained. Thus, gA and gB as functions of the compositions of the bulk phases are given by s A

1 / ∑ * K ib m *i x ib

[2.29] sB i

0 A

∑ * K ib x ib

Two-Component System

since K Aa Å 1, mA Å 1, and x Aa á 1. From Eqs. [2.23] and [2.28], the individual x iaA is given by sA i

s A

06-03-96 07:39:38

[2.35]

where VA is the molar volume of matrix molecules A and Ci is the molar concentration of component i in the a-phase (Ci á x ia /VA ). If Gi is measurable, the reciprocal plot of 1/ Gi vs 1/Ci gives K ia N sA from the slope of the linear part of the low Ci range and (2mi 0 1)/N sA from the intercept at C 01 Å 0 extrapolated from the linear part. The curve of i G 01 vs C 01 must become flat as C 01 approaches zero due i i i to the saturation of the adsorption, and its extrapolation to C 01 Å 0 gives 1/N si . Hence, N si , N sA , mi , and K ia are all i determined. By use of these parameters, g at a low adsorption level can be calculated by the equation

S

g Å g 0 / N sA RT ln 1 0

K ia x ia 1 / K ia mi x ia

D

[2.36]

or g Å g 0 0 N si RT ln(1 / K ia mi x ia ),

coidas

AP: Colloid

[2.37]

270

TADAO SUGIMOTO

the latter of which is obtained when we use a sAA á (1 / K ia mi x ia ) 0 ( 1 / mi) a Aa

[2.38]

combining Eqs. [2.23] and [2.33]. When x ia becomes so small as K ia mi x ia ! 1, Eqs. [2.36] and [2.37] can be approximated as g Å g 0 0 N sA RTK ia x ia .

[2.39]

(b) Saturated adsorption range (x si A á 1). range, msAA and msi A are given by A / RT ln f ( 0 ) x sA msAA Å m0,s A A A

In this

[2.40]

an arbitrary range of x si A can be calculated from the measurement of g and a ia . Also, the parameters such as K ia , N si , N sA , and N sB determined by the measurement of either Gi or g in each two-component system can be used for the calculation of g of a multicomponent system in the range of (* x si A ! 1 and (* x si B ! 1 with Eq. [2.32]. 2.4. Activity Coefficients of Interfacial Chemical Potentials

The activity coefficients of interfacial components are important not only as a criterion of their molecular interactions in interfaces but also for the calculation of g at an arbitrary adsorption level on the basis of Eq. [2.16]. From Eq. [2.18a], an elementary interfacial energy gA is generally written as

and sA i

m

u,sA i

Åm

/ RT ln f

(1) i

,

[2.41]

respectively, where f A( 0 ) is the activity coefficient of x sAA at x sAA Å 0 and f i( 1 ) is that of x si A at x si A Å 1. Also, it follows from Eq. [2.23] that (1) i a i

f a

S

a f

(0) A

D

mi

a A

x sAA

Å K ia .

[2.42]

g Å g 0 0 N si RT ln

K ia 0 N si RT ln a ia . f i( 1 )

a g 0A Å N si ( m—oi ,sA 0 m—oi ,a ) / N *s i RT ln K i .

AID

JCIS 4284

/

6g12$$$227

06-03-96 07:39:38

[2.45]

Hence,

or f isA Å

[2.43]

This equation shows that g as a function of ln a ia is lowered with the slope of N si RT in the saturated adsorption range in contrast to the very low adsorption range in which N sA instead of N si is responsible for the change of g as suggested by Eq. [2.39]. By plotting g 0 0 g against x ia with Eq. [2.39] in the low adsorption range, we can obtain N sA K ia . And, by plotting g 0 0 g against ln a ia with Eq. [2.43] in the saturated adsorption range, we obtain N si and K ia / f i( 1 ) . If it can be assumed that the matrix molecules in a liquid–liquid or gas–liquid interface are randomly oriented, N sA may be given by N sA Å / 3 01 y 02 N , where yA is the molecular volume of a matrix A molecule A, and N is the Avogadro number. In any case, if N sA is known, K ia and f i( 1 ) can be determined besides N si from the measurement of g as a function of a ia without the data of Gi . As it is generally difficult to directly measure Gi in liquid–liquid and gas–liquid interfaces, this procedure seems convenient for the determination of the parameters of these interfaces. Once K ia and N si have been obtained, Gi in

[2.44]

Initially, a si A /a ia Å K ia from Eq. [2.23] and thus

gA Å g 0A / N si RT ln

As a consequence, g ( Åg 0 / N sA RT ln ( f A( 0 ) x sAA /a Aa )) as a function of a ia is given by

a si A . a ia

gA Å N si ( m—oi ,sA 0 m—oi ,a ) / N si RT ln

f isA x si A , K ia a ia

S

K ia a ia gA 0 g 0A exp sA xi N si RT

D

.

[2.46]

[2.47]

Specifically, the activity coefficient of matrix component A in the sA-layer, f isA , is given by f AsA Å

S

a Aa gA 0 g 0A exp sA xA N sA RT

D

,

[2.48]

since K Aa Å 1. Since the parameters such as K ia and N si are determined by use of Eqs. [2.39] and [2.43], the interfacial activity coefficients f isA and f AsA can be calculated with Eqs. [2.47] and [2.48] if the data of G si A , a ia , a Aa , and gA 0 g 0A are all available. And it is possible when the adsorption occurs only from the a-phase, viz., gA 0 g 0A Å g 0 g 0 and G si A Å Gi . Let us consider the determination of the activity coefficients of interfacial components in a practically important two-component system at dT Å 0 and dP Å 0 for cases in which the data of either g or Gi are unavailable. Case 1. g is unavailable.

coidas

AP: Colloid

271

A NEW APPROACH TO INTERFACIAL ENERGY

It follows from the Gibbs–Duhem equation in the sAlayer that N si y si A d ln a si A / N sA y sAA d ln a sAA Å 0.

[2.49]

This is none other than the Gibbs adsorption equation (Eq. [1.1]) for a two-component system. Since mi x si A , 1 / ( mi 0 1)x si A

[2.57]

mi dg / N sA RTd ln a ia . ( mi 0 1)dg / N sA RT(d ln a si 0 d ln a Aa )

[2.58]

y sAA Å

Differentiating Eq. [2.18a], [2.50] x sAA Å

Combination of Eqs. [2.49] and [2.50] yields f

F*S

a Aa Å s exp 0 x AA

xa i

y

Combination of Eqs. [2.48] and [2.58] yields

d ln a Aa dx ia

sA A

0

/

y si A Å

one finds x si A from Eq. [2.56] as

a si A a sAA N si d ln a Å N sA d ln a . ai aA

sA A

1 0 x si A ; 1 / ( mi 0 1)x si A

D G

y si A d ln a ia dx ia mi dx ia

f AsA Å a Aa [2.51]

F

( mi 0 1)dg / N sA RT(d ln a ia 0 d ln a Aa ) mi dg / N sA RTd ln a ia 1 exp

and f isA Å

F*S xa i

K ia a ia exp 0 x si A

d ln a Aa dx ia

mi y sAA

0

D G

d ln a ia dx ia dx ia

/ y si A

.

[2.52]

a i

f

sA A

a i

a A

K ia Å

and f isA

a A

F*

1 Å s exp 0 x AA

xa i

0

F*

K ia x ia Å s exp 0 xi A

y si A dx ia mi x ia

xa i

0

y si A a dx i x ia

G G

[2.53]

,

[2.54] f isA a ia , Ci . f BsB

Case 2. Gi is unavailable. It follows from Eq. [2.20] that a sAA . a Aa

[2.55]

Eliminating a sAA and a si A from Eqs. [2.49], [2.50], and [2.55], one obtains dg Å 0RT(N sA y sAA d ln a Aa / N si y si A d ln a ia ).

AID

JCIS 4284

/

6g12$$$228

.

[2.59]

S D x si A x ia

Å10 a

x i r0

1 N RT s A

S D dg dx ia

[2.60] a

x i Å0

from Eqs. [2.23] and [2.55] or by the plotting of Eq. [2.39]. Hence, even if Gi or x si A is unavailable, it can be calculated as a function of g, a ia , and a Aa by the aid of Eq. [2.58]. 3. DISCUSSION

respectively. Hence, the activity coefficients f AsA and can be obtained by Eqs. [2.51] and [2.52] with G si A , and a Aa , or by Eqs. [2.53] and [2.54] with G si A and Similarly, f BsB and f isB can be obtained. Once f AsA and are determined, g can be calculated by Eq. [2.20].

dg Å N sA RTd ln

D

Similarly, f isA is given as a function of a ia , a Aa , and g from Eqs. [2.47] and [2.58], where x si A is given by Eq. [2.58] with x si A Å 1 0 x sAA and K ia is given by

If x ! 1 so that a á x , a á 1, and É dln a É ! É dln a ia É, Eqs. [2.51] and [2.52] reduce to a i

S

g 0 g0 N sA RT

G

[2.56]

06-03-96 07:39:38

3.1. N si in the Interfacial Double Layer Model

We have proposed a new model of the interface, the interfacial double layer model, consisting of two surface layers of different bulk phases. In fact, from a chemical viewpoint, the origin of the interfacial energy is the specifically high chemical potentials of the interfacial matrix molecules due to the small affinity between the matrix molecules of the opposite phases in contact. Thus, the thickness of each surface layer may be limited close to a single molecular layer thickness. But, if there are considerable tailings of chemical potential distribution of each component on each side of the interface due to the long-range interactions, N si values must be increased. Thus, N si values should basically be determined by experiment. On the other hand, the thermal fluctuation of the interfacial matrix molecules or the roughness of solid surfaces may increase the N si values of all interfacial components, while the solvation of nonmatrix components must decrease their N si values. And the orientation or order-

coidas

AP: Colloid

272

TADAO SUGIMOTO

ing of an interfacial component must change its N si value. Furthermore, the matrix component of a bulk phase may at least theoretically be dissolved to some extent in the opposite phase and some part of it may be adsorbed back to the surface of the original bulk phase. Although this effect can usually be neglected, if necessary, the matrix component slightly dissolved in the opposite phase must be treated as a nonmatrix component in the opposite phase in equilibrium with the matrix component of the mother phase. In any case, it is worthwhile to compare experimental / 3 01 N si values with e 02 N which corresponds to the N si value i of a randomly oriented component i in a single molecular layer, since it may provide us with information on the states of the interfacial molecules. 3.2. Surface Excess and Dividing Surface of Gibbs’ Theory

Gibbs (1) assumed an interfacial zone in which the composition of each matrix component continuously changes and represented the surface excess of component i by Gi . On the other hand, Gi in the present theory denotes the surface concentration of component i in the interface. If the surface excess and surface concentration of component i in an interfacial zone are designated by G *i and Gi , respectively, there is the following relationship between them,

/

V

s a

As

Å 0∑ Gi dm , a i

JCIS 4284

/

6g12$$$229

gA Å N si ( msi A 0 mia ) Å

∑ y si A N si ( msi A 0 mia ) Å

∑ G si A ( msi A 0 mia ) [3.3]

so that

Å 0∑ G si A dm ia ,

[3.4]

since (G si A dm si A Å 0 from the Gibbs–Duhem equation at dT Å 0 and dP Å 0 in the sA-layer, and (N si ( msi A 0 mia )dy si A Å N si ( msi A 0 mia ) ( dy si A Å 0 due to ( dy si A Å 0. Similarly, dgB Å 0(G si B dm ib . Thus, [3.5]

since mia Å mib ( Åmi ) and Gi Å G si A / G si B . [3.2]

since (C ia dm ia Å 0 and (C ib dm ib Å 0 at dP Å 0 and dT Å 0 (Gibbs–Duhem equations). Thus, the surface excess can be replaced by the surface concentration in the Gibbs adsorption equation. And it is obvious that Eq. [3.2] holds regardless of the thickness of the interfacial zone. Although Eq. [3.2] holds irrespective of the exact position of the boundary of two bulk phases, each G *i depends on the choice of the boundary position. Gibbs (1) introduced the notion of the dividing surface as a definition of the boundary of two bulk phases in contact, which is usually chosen for the surface excess of a matrix component to be zero. However, even in a binary system, the location of the dividing

AID

If we reinterpret the Gibbs adsorption equation given by Eq. [1.1] in terms of the intrinsic chemical potential, the mi in Eq. [1.1] must be the chemical potential of component i in the bulk phases. And, the Gibbs equation with mi in the bulk phases can readily be derived from Eq. [2.16] or [2.18] as

d g Å 0∑ G si A dm ia 0 ∑ G si B dm ib Å 0∑ Gi dmi ,

a i

∑ C ia dm ia / V bs ∑ C ib dm ib

Gibbs Equation

[3.1]

where V as and V bs are the volumes of the a-side and b-side interfacial zones, respectively, A s is the interfacial area, and C ia and C ib are concentrations of component i in the bulk phases a and b, respectively. As far as the Gibbs adsorption equation is concerned, d g Å 0∑ G *i dm Å 0∑ Gi dm

3.3. Notes on the Known Adsorption Equations

dgA Å ∑ G si A (dm si A 0 dm ia ) / ∑ N si ( msi A 0 mia )dy si A

V as V bs G *i Å Gi 0 s C ia 0 s C ib , A A

a i

surface for a matrix component is generally different from that for the other matrix component (12). In a multicomponent system, each dividing surface of the matrices may move according to the composition of each bulk phase. Therefore, as long as we assume a continuous interfacial zone, the boundary of the bulk phases cannot exactly be defined. In the interfacial double layer model, on the other hand, the boundary of two bulk phases is definite and its location is independent of the compositions.

06-03-96 07:39:38

Langmuir Equation If we regard adsorption as a substitutional reaction of surface matrix molecules A by adsorbates Xi , the chemical equilibrium may be written as X

a i

/ mi A sA _ X

sA i

/ mi A a .

[3.6]

Thus the equilibrium condition is written in terms of intrinsic chemical potentials as mia / mi msAA Å msi A / mi mAa .

[3.7]

This is identical to Eq. [2.15a]. And, if we regard A sA and

coidas

AP: Colloid

273

A NEW APPROACH TO INTERFACIAL ENERGY

X isA as different chemical species from A a and X ia of different chemical potentials, respectively, Eq. [3.7] is a rather familiar formula of an ordinary chemical equilibrium. If N si happens to be equal to N sA or mi Å 1 in the low adsorption range in a two-component system, it follows from Eq. [2.33] that x si A Å

K ia x ia . 1 / K ia x ia

[3.8]

This is none other than the Langmuir equation (Eq. [1.40]), and Eq. [3.7] is the thermodynamic expression thereof. Thus, at least, the concept of the Langmuir equation is as general as the Gibbs equation, as firmly supported by thermodynamics. However, in the case of adsorption from a liquid phase, the original Langmuir equation can be applied only when mi happens to be close to unity and, in addition, only in a relatively low adsorption range. But, in the case of adsorption of one kind of molecules from a gas phase, £i Å 1 is always satisfied so that the Langmuir equation can be applied as long as the adsorption level is sufficiently low or the interaction between the adsorbed molecules is negligible. Detailed discussion in this respect will be given in Part II of this series. In this context, it seems noteworthy that, originally, Langmuir (26) derived his equation (Eq. [1.40]) on an assumption of the equilibrium of adsorption and desorption of gaseous molecules of one kind, i.e., in the absence of matrix molecules to be excluded by the adsorption of the gas molecules: viz., ka y ia (1 0 y si ) Å kd y si , where ka and kd are the rate constants of adsorption and desorption, respectively. This relation is in accord with mi Å 1 which will be derived from the extension of the present theory in Part II. Szyszkowski Equation The general forms of the Szyszkowski equation are represented by Eqs. [2.37] and [2.43], as derived directly from Eq. [2.16], while the application of the conventional Szyszkowski equation, [1.41] or [1.42], is mostly limited to the cases in which the conventional Langmuir equation [1.4] can be applied. 3.4. The Significance of the Present Theory

Equation [2.16] which contains all elements of the Gibbs, Langmuir, and Szyszkowski equations directly leads us to these equations, while it is impossible to reach Eq. [2.16] conversely from them. This fact may imply that Eq. [2.16] is the most fundamental equation of adsorption as well as an explicit formula of the interfacial energy at equilibrium. Moreover, the derivation of Eq. [2.16] and the thermodynamic explanation of the well-established Langmuir and Szyszkowski equations, both of which basically conflict with

AID

JCIS 4284

/

6g12$$$229

06-03-96 07:39:38

conventional thermodynamics due to its presumption of uniform chemical potentials of the components, are possible only by the treatment of g as a function of the excess chemical potentials of the interfacial constituents. In addition, the theoretical limitations of the original Langmuir and Szyszkowski equations have been manifested in light of the present theory. As a consequence, the introduction of the concept of the intrinsic interfacial chemical potential and the direct formulation of g as a function of the intrinsic chemical potentials of the interfacial and bulk components are essential for understanding the individual behavior of the interfacial molecules and its effects on the interfacial energy. Another benefit of the present theory may be its inherent simplicity based on a single definition of the chemical potentials of components applied commonly to the bulk and interface phases. In addition, since G s can be defined without g which involves mia as well as msi and thus G s is a function of only msi independent of mia (see Eq. [1.6]), it is not needed to introduce any additional chemical potentials such as, e.g., the cross chemical potential which correlates the bulk and interface, the complete chemical potential of a bulk component consisting of the ordinary bulk chemical potential and cross chemical potential (30), etc. By use of the intrinsic interfacial chemical potentials, many excessively sophisticated or erroneous interpretations of interfacial phenomena will be simplified or corrected, and more straightforward understanding will be possible. Furthermore, the essential understanding of the nature of the interfacial energy is expected to lead us to novel development of interface science in theory and experiment, as will be shown in the later parts of this series. 4. CONCLUSIONS

(1) The specific interfacial energy has been formulated as a function of the intrinsic chemical potentials of the interfacial and bulk components by a single formula, Eq. [2.16], on the basis of the interfacial double layer model. (2) It is possible to describe the individual behavior of the interfacial molecules and correlate it with the change of the interfacial energy by Eq. [2.16] which involves all elements of the Gibbs, Langmuir, and Szyszkowski adsorption equations in itself. (3) If we account for the interfacial tension from a thermodynamic viewpoint, it may be interpreted as a counterforce to the external work to increase the number of the interfacial molecules of a high chemical potential at the expense of the bulk molecules of the lower chemical potential. REFERENCES 1. Gibbs, J. W., ‘‘The Scientific Papers of J. Willard Gibbs,’’ Vol. 1, p. 219. Dover, New York, 1961.

coidas

AP: Colloid

274

TADAO SUGIMOTO

2. Guggenheim, E. A., Trans. Faraday Soc. 39, 397 (1940). 3. Guggenheim, E. A., ‘‘Thermodynamics,’’ 7th ed., p. 45. Elsevier, Amsterdam, 1985. 4. Lewis, G. N., and Randall, M., ‘‘Thermodynamics,’’ 2nd ed., p. 470. McGraw–Hill, New York, 1961. 5. Davies, J. T., and Rideal, E. K., ‘‘Interfacial Phenomena,’’ p. 4. Academic Press, New York, 1961. 6. Kruyt, H. R., ‘‘Colloid Science,’’ p. 116. Elsevier, Amsterdam, 1952. 7. Adamson, W. A., ‘‘Physical Chemistry of Surfaces,’’ p. 68. Wiley, New York, 1976. 8. Atkins, P. W., ‘‘Physical Chemistry,’’ p. 237. Freeman, San Francisco, 1978. 9. Hiemenz, P. C., ‘‘Principle of Colloid and Surface Chemistry,’’ p. 274. Marcel Dekker, New York, 1977. 10. Everett, D. H., ‘‘Basic Principles of Colloid Science,’’ p. 63. Royal Chem. Soc., London, 1988. 11. Hansen, R. S., J. Phys. Chem. 66, 410 (1968). 12. Turkevich, L. A., and Mann, J. A., Langmuir 6, 445 (1990). 13. Braslau, A., Deutsch, M., Pershan, P. S., Weiss, A. H., Als-Nielsen, J., and Bohr, J., Phys. Rev. Lett. 54, 114 (1985).

AID

JCIS 4284

/

6g12$$$230

06-03-96 07:39:38

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Thomas, B. N., and Rice, S. A., J. Chem. Phys. 86, 3655 (1987). Beaglehole, D., Physica B 100, 163 (1980). Beaglehole, D., Physica B 112, 320 (1982). Meunier, J., J. Phys. (Paris) Lett. 46, 1005 (1985). Meunier, J., in ‘‘Physics of Amphiphilic Layers’’ (J. Meunier, D. Langevin, and N. Boccara, Eds.), p. 118. Springer-Verlag, Berlin, 1987. Butler, J. A. V., Proc. R. Soc. A 135, 348 (1932). Verschaffelt, J. E., Bull. Cl. Sci. Acad. R. Belg. 22, 373, 390, 402 (1936). Kofoed, J., and Villadsen, J., Acta Chem. Scand. 12, 1124 (1958). Rusanov, A. I., Kolloidn. Zh. 24, 309 (1962). Eriksson, J. C., Ark. Kemi 25, 331 (1966). Eriksson, J. C., Ark. Kemi 25, 343 (1966). Eriksson, J. C., Ark. Kemi 26, 49 (1966). Langmuir, I., J. Am. Chem. Soc. 38, 2221 (1916). Langmuir, I., J. Am. Chem. Soc. 39, 1848 (1917). Langmuir, I., J. Am. Chem. Soc. 40, 1361 (1918). Von Szyszkowski, B., Z. Phys. Chem. 64, 385 (1908). Defay, R., and Prigogine, I., ‘‘Surface Tension and Adsorption,’’ pp. 53, 60. Longmans, London, 1966.

coidas

AP: Colloid