Langmuir 1992,8, 2448-2454
2448
Chemical Potential and Diffusion Properties of Dispersed Colloidal Matter Y. Zimmels Department of Civil Engineering and Department of Chemical Engineering (Secondary Affiliation), Technion IIT, Haifa 32000, Israel Received April 7, 1992. I n Final Form: July 24, 1992 The chemical potential of colloidal matter, dispersed as monomers or as aggregates in bulk and in contact with solid surfaces, is considered. It is shown that for colloidal particles in a dispersion, the chemical potential is a function of the energy densities associated with the colloidal particles inclusive of their interfacial energy density. For colloidal fluid particles that are in contact with a solid surface, the chemicalpotential of the colloidalmatter is shownto depend on the rate of change of all interfacesinvolved with the volume enclosed by them. This impliesthat the solid-gas,solid-liquid, and liquid-gasinterfaces affect collectively the chemical potential of the matter in contact with the solid surface, and hence they also change its tendency to condensate,aggregate,evaporate,or dissolve. The extended chemical potential is used to derive diffusion coefficientaof colloidal matter which is shown to consist of four specific diffusion coefficients. The net force available from a field to drive colloidal particles in the direction of the concentration gradient is shown to be diminished by the effect of counterdiffusion,which is inherently induced by this very field. The effect of aggregationin bulk and at the interfacesis considered, regarding the chemical potential as well as diffusion and hydrodynamic properties of the colloidal particles. In conclusion, the interpretation of some field effecta is outlined.
Introduct ion The behavior of colloidal systems depends on their chemical potential which affects their tendency to diffuse and aggregate. Therefore, it is of interest to find a general formulation of the chemical potential of colloidal particles which can then be used to analyze their diffusion and aggregation properties. To this end, a careful thermodynamic formulation of the colloidal system is required which accounts for the system variables and makes use of appropriate thermodynamic potentials. In this work a different approach for the formulation and use of thermodynamic potentials is described and implemented for the derivation of generalized chemical potentials of colloidal systems. These chemical potentials are then applied in conjunction with hydrodynamic hindrance factors for the derivation of diffusion coefficients of colloidal particles in dispersion as well as for the analyses of diffusion properties, aggregation, and effect of force fields in related systems.
Theory 1. Thermodynamics of Monodisperse Colloidal Dispersions. The thermodynamics of colloidal dispersions is considered using two alternative approaches. In the first approach, the Legendre transformation of the internal energy which is associated with the colloidal particles is used to formulate a thermodynamic potential which is then used to derive the chemical potential. In the second approach, the chemical potential is derived by evaluation of the energy changes occurring due to mass transfer. The chemicalpotential is consideredwith respect to the matter of a single colloidal particle and then for the whole colloidal assembly. 1.1. Legendre Transformation of the Internal Energy. The first step in the analysis is to formulate the thermodynamicpotential which best describesthe colloidal system. There is no restriction on the type of interfaces involved. For example, the colloidal particle can either
be dimersed in a continuous Dhase or be in contact with a solid surface. In the former case, a single interface is formed between the colloidal particle and the continuous phase, whereas in the latter case, three interfaces exist between the three phases involved. The formulation of the thermodynamic potential requires the identification of the unconstrained variables that characterize the colloidal system. These variables control the thermodynamic behavior of the system in contrast to those that are fixed by internal or external constraints. The internal energy of a thermodynamic system is a function of its extensive variables1
U = U(Xo,XI,...,X,,A1,...,A,) where Xi,i = 0, 1,..., t, are extensive variables pertaining to the bulk properties whereas AI, ...,A,,, denote interfacial areas and m is an integer. The intensive conjugates of Xi are defiied by ti = aUIdXi where all Xj j # i are held constant i j = 0, 1,..., t . The Legendre transformation of U wherein XO, ..., Xn are replaced by their intensive conjugates 50, ..., &, is given' by eq 1
H(€o,...,€n,Xn+lt...,Xt,Al,...,A,) = U(Xo,...,X,,Al,..,Am)n
t
m
Equations 1and 2 provide the basis for the formulation of thermodynamic potentials. (1) Callen, H.B. Thermodynamics; John Wiley: New York, 1960.
0743-7463/92/2408-2448$03.00/00 1992 American Chemical Society
Langmuir, Vol. 8, No. 10,1992 2449
Chemical Potential of Colloidal Matter 6V,6 N
6V, 6N
I
6 V=ydx=const.
b
a
Figure 1. A spherical colloidal particle divided into volume element 6 Vcontainingeach 6N moles of matter: (a) 6 Vis defined between two-planarcross sections of the sphere; (b) GVis defined by an arbitrary shape. In both cases GV is fixed.
Equation 1can be applied to any increment or element 6 of the extensive variables as follows: dx
U(6Xo,...,6X,,6Al, ...,6Am)-
Cfk6 x k (3) k=O
Hence the differential change pertaining to this element is given by
The symbol6 can denote an increment or an element which is part of the existing colloidal particle and in general of a thermodynamic system. For example, 6N denotes part of the colloidal particle that consists of 6N mole. Summation of all such parts gives the complete colloidal particle; see Figure 1. Equations 3 and 4 provide a fundamental description of a thermodynamic system through the reference to each of its parts. The next step is to identify the variables that should be transformed. For the case of a single incompressible colloidal particle, the unconstrained variables are T, P, p, and AI, ...,Am, i.e. temperature, pressure, chemicalpotential, and interfacial areas, respectively. Here the pressure is of the medium surrounding the colloidal particle. Using in eq 4 the notation XO= S, f o = T, X1 = V, 51 = -p, X2 = N, f2 = p, where S is entropy, gives
6 V=ydx=const.
Figure 2. A two-dimensionalsystem comprisinga dispersion of colloidal particles in an external field. The system is discretized into fixed volume elements 6V by planar cross sections set perpendicular to the field. The insert shows a volume element 6V which is dx wide and y high.
with solid surfaces depends on all interfaces involved, inclusive of the flat ones. This rather unexpected result is also corroborated by the second approach for the derivation of the chemical potential, given in the sequel. Consider a systemcomprising of a dispersion of colloidal particles which is enclosed by a diathermal wall. The system2is in an external force field f~~which has a value f~ inside the dispersion. The system (see Figure 2) is discretized into fixed volume elements 6V. The extensive variable correspondingto f~ is X F . Here the unconstrained , AI, ...,A m but they have a variables are T, P, p, f ~and different meaning. The chemical potential is now dependent on concentration-dependent energy densities. Application of eq 5 for the above set of variables gives d6H(T,p,p,fF,Al,...,A,,,) = -6s d T + 6V dp - 6N dp m
At equilibrium and for constant T and p m
4 N d p - 6XF d& + Cyi d6Ai = 0
d6H(T,p,p,Al,...,A,,,) = -6s d T + 6V dp - 6N dp +
(9)
i=l
m i=l
where Ti values are the interfacial energy per unit area of the ith interface. For a system at equilibrium and at constant temperature and pressure, eq 5 reduces to eq 6 which involves only interfacial effects
Equation 9 must be satisfied for any arbitrary choice of the pairs pt, f~ and pa, AI, ...,Am, where p = pt + pa. The subscripts f and a in pt and pa denote that these parts of the chemical potential originate from the field and from interfacial effects, respectively. Hence, eqs 10and 11must be satisfied simultaneously -6N dpt - 6XF dfF = O
m
dpa = uNzyid6Ai/6v
-6N dpa +
i=l
where subscript a denotes interfacial effects, U N is molar volume of the colloidalparticle material, and use was made of vN6N = 6V. Taking a differential increment or element 6justifies using 6Ai/6V= dAi/dVand setting 6Vfixed gives m
(7)
Equation 7 shows that the chemicalpotential is a function of the rate of change of all interfacial areas whether curved or flat with their volume element. Equation 7 predicts that the chemicalpotential of colloidalparticles in contact
(10)
m
Cyid6Ai = 0
(11)
i=l
The ith interfacial area of 6N moles of colloidal particles which are dispersed in 6V is given by 6Ai = 6N&Aci
(12)
where & is Avogadro's number and Aci is the ith interfacial area associated with a single colloidal particle. Note that here a colloidalparticle is considerede o n e unit and hence 1mol of colloidal particles consists of N such units. If Aci, i = 1, t are fixed then differentiatingeq 12and combining
...,
(2) Zimmels, Y.J. Colloid Interface Sci. 1989, 130, 386-404.
Zimmels
2450 Longmuir, Vol. 8,No. 10,1992
(dN < 0) that subsequently move into the volume dVN which is occupied by the displaced mass. The internal work dWp done by the N dN moles in pushing the -dN moles at pressure P outside the system is given by
number m
+
number 1 interface number 2 interface
Figure 3. An open system of a liquid phase which is enclosed by a boundary comprised of m interfaces. the result with eq 11 yields m
yiAcid In C
dp, = 1=1
where in deriving eq 13,use was made of d In 6N = d In C, since 6V is fixed and C = 6N/6V. Integration of eq 13 gives
dW,=-PdVN=-PvNdN,dN
