КОЛЛОИДНЫЙ ЖУРНАЛ, 2007, том 69, № 5, с. 581-585
УДК 541.182
THERMODYNAMICS OF MICELLES IN SURFACTANT SOLUTION:
LATTICE MEAN-FIELD THEORY © 2007 Zaid A. Al-Anber
Chemical Engineering Department, Faculty of Engineering Technology, Al-Balqa Applied University P. Box: 15008 Marka-Amman 11134, Jordan Поступила в редакцию 15.11.2006 г.
We apply the lattice self-consistent mean-field theory to study thermodynamics of micelles in surfactant solution. The model surfactants used are H4T4 and H2T4. The formation of spherical micelles is considered. The effect of the head length on the thermodynamics stability of the micellar solution is examined. The critical micelle concentration is studied at different length of head, fractional charge, solvent quality and univalent salt concentration. A smaller critical micelle concentration is associated with a larger aggregation number and the smallest micelles are found at lowest univalent salt concentration.
1. INTRODUCTION
The behavior of surfactant in a solution plays an important role in different practical applications such as washing, cleaning, dispersing, emulsifying, foaming, drying, and pharmaceutical technology. The surfactant in aqueous solution forms micelles at concentration up to and just above the so-called critical micelle concentration.
The behavior of non-ionic surfactant is widely studied using lattice mean field theories [1-4] and molecular simulations [5-9]. Since the surfactant chains in the most of these applications are ionic, the scientific workers start to pay an attention to consider charges in their simulations. Marko and Rabin [10] describe the microphase-segrega-tion properties of diblock copolymers composed of a neutral polymer joined to a polyelectrolyte. They also construct a theory for weak concentration fluctuations in order to determine the limit of stability of the mixed phase. Dan and Tirrell [11] investigate the self-assembly of block copolymers with a strongly charged and a hydrophobic block in a selective polar Solvent. They investigated the aggregation of diblock copolymers consisting of a neutral block and a polyelectrolyte block in aqueous salt solution by means of a scaling model. The conformational equilibria in the charged region of spherical ionomeric colloids are studied by Ronis [12] using a self-consistent mean-field Monte Carlo method. Bohmer et al. [13] extended the lattice mean-field theory to study the charged chains and applied it to adsorption of weakly charged polyelec-trolyte at planar and oppositely charged surfaces. Recently, Linse and coworkers [14-16] are used the lattice self-consistent mean-field (SCF) theory to study the behavior of micelle in aqueous solution composed of diblock co-polymers with charged hydrophilic blocks.
The aim of our paper is to gain insight in how far the self-consistent mean-field technique can be used to study the micellization of short surfactants. Therefore, we follow the development of lattice self-consistent mean-field
theory by Linse et al. [16] to study the micellization of model surfactants such as H4T4 and H2T4, and then to study the effects of univalent salt concentration on the micelle formation of these model surfactants; in particular, to study the salt concentration effects on the stability and critical micelle concentration of micellar systems.
We briefly present the lattice self-consistent mean-field theory and define the important parameters of the study. In the subsequent section, the SCF theory is provided, followed by results and discussions.
2. SELF-CONSISTENT MEAN-FIELD THEORY
In this theory, the space is divided into concentric shells and each shell is divided into lattice cells of equal size. The volume fraction constraint is applied where each lattice cell contains either surfactant segment or solvated ion or water. The micelle is considered to be located at the center of the lattice.
The volume fraction profile of the species is a function of the layer number only because the mean field approximation is applied within each layer. Then the possibility of obtaining radial concentration is coupled to the existence of radial distance dependent potentials. The species potential, us(i), is determined as the derivative of the free energy with respect to the species concentration and can be assumed to be composed of three contributions [17]:
us(i) = u'(i) + ¿bT£Xss(<^(i)> - tf) + qsV(i). (1)
s
The first term is a Lagrange parameter which follows from the constraint that each layer i is exactly filled or in other words that each lattice site i is on average filled exactly by one molecular unit:
(i) =1. (2)
The second term in Eq. (1) represents the short-range interactions that account for the various contacts of species in layer i, according to a Bragg-Williams approximation of random mixing. The summation runs over all the species s, Xss is Flory-Huggins interaction parameter for the s-s pair, kB is the Boltzmann's constant, T is the temperature. <s(i) is
the volume fraction of species s in layer i and <bs is the bulk volume fraction of species s. The angular brackets (} indicate an averaging of the volume fraction of species s over three adjacent layers, A,-1, X0, so that
«,(i)} = <Mi -1) + i) + Ai<Mi +1). (3)
The third term in Eq. (1) is the contribution from longrange Coulomb interactions where qs is the charge of species s and y the electrostatic potential of mean force in layer i. In a random mixing approximation of the short-range interaction, it is reasonable to let the electrostatic potential depends only on the radial distance and the potential of the mean force is related to the charge density through the Poisson equation:
eV2 y( i) = -p( i). (4)
Here e is the dielectric permittivity of the medium and assumed to be constant. V2 is the Laplacian and p is the charge density in the layer i and can be expressed as
p( i) = X q*<s( i).
(5)
If the species potential is known, then the free species distribution function Gs(i), which is the weighting factor for a free species of type s in layer i, is given by the Boltzmann factor
Gs (i) = exp (-us( i) /kB T).
(6)
If us(i) and Gs(i) are known, then the relative weight of all the possible conformations can be calculated, and hence also the volume fraction profiles can be evaluated by relation
<s( i ) = Gs(i )<b .
(7)
Aexc = A° + kBT ln
( V ) •
(8)
where V: is the volume of the micelle.
A necessary requirement for stable micelles at equilibrium is that the positive A° be balanced by the favorable entropy of mixing of the micelle, implying that Aexc = 0. Given that the equilibrium state and the segment distributions are determined. A° can be calculated from Eqs. (1), (4)-(6) and Eqs. (2), (14), (15) in [19].
The volume of the micelle is approximated by
V =
<P, i = i-
(9)
Here rp is the excess number of surfactant (polymer) segments in the cell and can be defined as
r/ Np
r p =
i = 1
XL' X<
LVs = 1
(10)
where ^ = npsJLii npsi is the number of sites in layer i-,
Li is the number of lattice sites in the layer i and Np is the number of segments. The total volume fraction of
surfactant (polymer), <pot, in the cell is the sum of the excess and bulk volume fraction, <p, according to
r
c = V+<p.
(11)
The micelle aggregation number Nagg is the excess number of surfactant (polymer) molecules in the cell given by
r
N = -p
agg N ' p
4. THE MODEL
(12)
For further details concerning the numerical procedure the reader is referred to the literature [18].
3. MICELLE FORMATION
The micellar solution is divided into cells; each of cells contains one micelle and its accompanying solution. The volume of the cell Vs is simply the inverse micellar number density. The excess free energy, Aexc, of cell consists two parts: one is the free energy indicated by A° and associated with the formation of a micelle fixed in space and contacting with the bulk solution. The second is the entropy of mixing of micelles and indicated by kBTln( V:/Vs). Thus,
In this paper, an individual micelle in aqueous solution is formed at the center of the lattice and is in equilibrium with specified bulk concentration of the components. The model contains five different species. They are uncharged tail segment, charged head segment, cation, anion and solvent. Consider an L x L x L array of lattice unit cells of equal volume completely occupied by these species. Each lattice site contains one surfactant (or polymer) segment, one solvated ion or solvent. It means
that the space is filled in layer i according to X<si = 1
i
where < si is the volume fraction of species s in layer i. The model surfactant (polymer) is represented as HxTy where H is the head unit and T is the tail unit. x is the number of the segments in the head unit and taken to be equal to 2 and 4 while the y is the number of the segments in the tail unit and taken to be equal to 4.
This model is suitable for describing the behavior of ionic surfactants such as C:EO„, where the EO = (-C-C-O-) makes up the head part and the alkyl chain forms the tail. We assume that a head group H in our model is approxi-
KOnnOHAHblH XyPHAn TOM 69 < 5 2007
дехс
50
40
30
20
10
20
0.0018
\
\
\
\
\
4
\
\
л
\
\
\
40
60
80
100
120
N
agg
Fig. 1. The excess free energy, Aexc, as a function of aggregation number, Nagg, for H4T4 (1) and H2T4 (2) systems. XHW = 0 and t = 0.
0.0014 0.0010 0.0006
0.0002
1
/
У
/
20 40 60 80 100 120 120
N
agg
Fig. 2. The bulk micelle volume fraction, 0p , vs the aggregation number, Nagg, for H4T4 (1) and H2T4 (2) systems. Xhw and t = 0.
2
1
2
mately equivalent to one EO unit and one tail T represents about 3 CH2 groups. The model H4T4 and H2T4 surfactants are then roughly equivalent to the real ionic surfactants, C12EO4 and C12EO2, respectively.
5. RESULTS AND DISCUSSIONS
In order to carry out the calculations, parameters describing the interaction among the species and the equilibrium state have to
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