Surfactant Assembly

The fluctuations in size of aggregates, exchange rate of surfactants between those in solution versus those in the aggregate, as well as the rate of aggregate formation, are all assumed to be fast in comparison with experimental time scales. The scheme as laid out by Israelachvili (1992) and Israelachvili et al. (1976) will be followed.

The first step in the scheme of any model is the distinction between monomers, dimers, trimers, and so on, in solution, all in equilibrium with one another. We denote an aggregate containing a number of n monomers as an aggregate with size n, or in short as an n-mer.

The second step is assume chemical equilibrium and hence equate the chemical potential of the surfactant as monomer with the chemical potential of the surfactant within an n-mer. This leads to the following expression for all aggregates of size N, M >1 (Israelachvili 1992)

μ1 = μ10 + kT ln X₁ = μM = μM0 + 1/M kT ln(X_M/M) = μN = μN0 + 1/N kT ln(X_N/N) (9.5) where μ1 denotes the chemical potential of the surfactants as monomer, that is, the surfactant in solution, M and N denote different sizes of the aggregate, X_N the mole fraction of surfactants that are present in aggregates of size n = N, μ_N the chemical potential of a surfactant within an aggregate of size N, and μ_N⁰ the self free energy of that the use of the term kT ln XN is justified, and the factorization of 1/N ensures that the contribution of each surfactant molecule within an aggregate of size N is only counted once. We note that mass balance yields that X₁+ X₂ +…+ X_N = X_total, the total

XN = N{X1exp[ μ10 – μN0 N (9.6) As the total mole fraction cannot exceed one, there appears automatically an upper boundary for X₁, also known as a critical aggregation concentration, or critical mi-celle concentration, Xc which is given by

X_c = exp – (μ10 – μN0)/kT (9.7) Until now we have not assumed anything about the type of surfactant nor about the morphology of the aggregate. For that matter, the above is equally valid for 156

mole fraction of surfactant in solution. Equation (9.5) yields

the surfactant within an aggregate of size N. Here one assumes a dilute solution so

]/kT}

Assembly of Structures in Foods

proteins. If one wants to advance with the description above, one needs to include molecular information of the surfactant in order to calculate the interaction potential between surfactants within an aggregate of size n, to subsequently arrive at an expre-ssion for the chemical potential of a surfactant in that aggregate of size N, that is, μ_N⁰. Such an expression should in principle also contain information about the shape of the aggregate, as the aggregate shape also influences the distances between the hydrophilic and hydrophobic parts of the surfactants and thus their chemical poten-tial. It should be realized at this point that the following complicating factor exists in general. If one wants to arrive at a size distribution of aggregates of a particular surfactant, on the basis of an equilibrium approach as above, one must include the shape of the aggregate in order to calculate the chemical potential. However, it is this shape itself that also should in principle be part of the minimisation scheme of the free energy of the overall system. In fact, one should minimise the free energy of the system on the basis of a size distribution and morphology of the system simultane-ously. Setting aside for a moment this complication in the variational calculus, one may set an a priori shape at first and put forward hydrophilic and hydrophobic inter-actions within such an aggregate and determine, for example, the average size as a function of experimental parameters.

Israelachvili et al. (1976) argue that despite the intrinsic difficulties of incorporat-ing all of the hydrophilic interaction contributions, the contribution to the chemical potential per surfactant within an aggregate of size N by the repulsive interaction can be assumed to adapt a simple form:

μN0

hydrophillic ~ constant D e²/ε a (9.8)

which is based on modelling the energy contribution as the energy of a capacitor with a charge per unit area of e/a, and separation D of the planes (resulting from the double layer of charge with thickness D), where the ε is the relative dielectric con-shown that this 1/a dependence explains micelle sizes and critical micelle concentra-tions satisfactorily.

μN0=μN0

hydrophobic + μN0

hydrophillic = γ a + c/a + g (9.9)

where γ denotes the surface tension, a the area between hydrophobic groups versus water (a constant), and where the term g denotes constant surface area and/or bulk contributions (Israelachvili et al. 1972). Combining Equations (9.8) and (9.9) yields an optimum area, a0 = (c/γ)^1/2, for which the free energy is a minimum. This relates to a specific number of surfactant molecules per aggregate. Having a larger number of surfactants per aggregate yields a smaller area per surfactant, and, vice versa, having a smaller number leads to a larger area. Both cases imply a higher free energy, leading to a size distribution of micelles.

The optimal area a₀ can be used as a practical criterion for the shape of the mi-celle versus the size of the head group and tail of the surfactant, assuming that the

157

(Israelachvili 1992)

stant of the medium around the surfactant head group. Indeed, Tanford (1974) has

Including hydrophobic interactions leads to an overall chemical potential

E. van der Linden

chemical potential is not affected too much by the change of shape, that is, assuming that a₀ is not affected too much by the choice of aggregate shape. Considering a spherical micelle, with radius R, and denoting the volume of the hydrocarbon tail as v, and the surface area between hydrocarbon and water as a₀ then R = 3 v/a₀ . Realiz-ing that there is an upper limit to the extension of the hydrocarbon chain given by the maximum tail length l_c, a spherical micelle will only be formed when 3 v/a₀ < l_c, leading to the condition for spherical micelle formation as v/a₀l_c < 1/3 (Israelachvili et al. 1976) Similarly, one has for a cylinder the condition v/a₀l_c < 1/2 and for planar objects v/a0lc<1.

One may investigate further, now assuming a certain shape, again following Israelachvili et al. (1976), the effect of the number of surfactants on the chemical potential. Suppose one has energy of binding between monomers when within an aggregate, of magnitude α. In the case of a rodlike structure, the endpoints are un-bound, leading to (Israelachvili 1974)

N μN0 = –Nα kT + α kT = –(N – 1)α kT (9.10) or

μN0 = μ∞0 +α kT/N (9.11) where μ∞0

μN0 = μ∞0 +α kT/N ^1/2 (9.12) and for spherical aggregates (Israelachvili et al. 1976)

μN0 = μ∞0 +α kT/N^1/3 (9.13) Now we know how the chemical potential depends on size, one may answer how the

XN = N{X1exp[α]}^N.exp(–α) (9.14) for rod like structures, while for planar and spherical structures one has

X_N = N{X₁exp[α]}^N.exp(–αΝ ^p) (9.15) with p being 1/2 or 2/3 for discs and spheres, respectively. One can see from Equa-tion (9.15) that when α is of the order of 1, XN becomes negligible for larger N. In-deed, a separate phase is then formed consisting of infinitely large aggregates. This is the case for a constant α, and where μ_N⁰ is a constantly decreasing function of N, that 158

for planar aggregates (Israelachvili 1976)

is the energy of the surfactant within an infinite aggregate. Similarly,

rewrite Equation (9.6) as

aggregate size depends on the value of α? Using Equations (9.11)–(9.13) one may taking always into account the number of endpoints of the aggregate one may derive

Assembly of Structures in Foods

is, the chemical potential is smaller the larger the aggregate. In other words, it favours infinite aggregates. It is only when μN0 reaches a minimum value for finite N that one will end up with finite aggregates, and that one will have more of these aggregates as the concentration of surfactants increases. This miminum of the chemical potential can be addressed by Equation (9.9) from which an optimum head group area of the surfactant was deduced, and consequently an optimal number N per aggregate.

When the concentration becomes high enough to induce interaction between aggregates, thus influencing the chemical potential of the surfactants again, shape transitions may take place, as, for example, is the case of SDS exhibiting a spherical-to-rod transition at a certain concentration. The success of the model above can be attributed to an accurate enough description of the shape of the surfactant and the contributions to the chemical potential of the surfactant in various circumstances.

When α is large, but p = 1, that is, referring to Equation (9.14), X_N does not decrease rapidly to zero, that is, one may have large aggregates of finite size provided that the structure is rodlike.