1.20.5: Surfactants and Micelles: Mixed

In many industrial and commercial applications, mixed surfactant systems are used [1]. An extensive literature examines the properties of these systems [2-8].

A given aqueous solution contains two surfactants \(\mathrm{X}\) and \(\mathrm{Y}\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The critical micellar concentrations are \(\mathrm{cmc}_{\mathrm{X}}^{0}\) and \(\mathrm{cmc}_{\mathrm{Y}}^{0}\). For a solution containing both surfactants \(\mathrm{X}\) and \(\mathrm{Y}\), the critical micellar concentration of the mixed surfactant is cmc(mix). Here we use a pseudo-separate phase model for the micelles. The system under consideration comprises \(\mathrm{n}_{\mathrm{X}}^{0}\) and \(\mathrm{n}_{\mathrm{Y}}^{0}\) moles of the two surfactants. [Here the superscript ‘zero’ refers to the composition of the solution as prepared using the two pure surfactants.] A property \(\mathrm{r}\) is the ratio of the concentration of surfactant \(\mathrm{Y}\) to the total concentration of the two surfactants in the solution. Thus,

\[\mathrm{r}=\frac{\mathrm{n}_{\mathrm{Y}}^{0}}{\mathrm{n}_{\mathrm{X}}^{0}+\mathrm{n}_{\mathrm{Y}}^{0}}=\frac{\mathrm{c}_{\mathrm{Y}}^{0}}{\mathrm{c}_{\mathrm{X}}^{0}+\mathrm{c}_{\mathrm{Y}}^{0}} \nonumber \]

Here \(\mathrm{c}_{\mathrm{X}}^{0}\) and \(\mathrm{c}_{\mathrm{Y}}^{0}\) are the concentrations of the two surfactants in solution. We define a model system where cmc(mix) is a linear function of the property \(\mathrm{r}\); equation (b).

\[c m c(\operatorname{mix})=\left(\mathrm{cmc}_{\mathrm{Y}}^{0}-\mathrm{cmc}_{\mathrm{X}}^{0}\right) \, \mathrm{r}+\mathrm{cmc}_{\mathrm{X}}^{0} \nonumber \]

Or,

\[\mathrm{cmc}(\mathrm{mix})=\mathrm{r} \, \mathrm{cmc}_{\mathrm{Y}}^{0}+\mathrm{cmc}_{\mathrm{X}}^{0} \,(1-\mathrm{r}) \nonumber \]

Hence the critical micellar concentration of the two surfactants in a given solution, \(\mathrm{cmc}_{\mathrm{X}}\) and \(\mathrm{cmc}_{\mathrm{Y}}\), depend on parameter \(\mathrm{r}\).

\[\mathrm{cmc}_{\mathrm{Y}}=\mathrm{cmc}_{\mathrm{Y}}^{0} \, \mathrm{r} \nonumber \]

\[\mathrm{cmc}_{\mathrm{X}}=\mathrm{cmc}_{\mathrm{X}}^{0} \,(1-\mathrm{r}) \nonumber \]

Clearly in the absence of surfactant \(\mathrm{Y}\), micelles are not formed by surfactant \(\mathrm{X}\) until the concentration exceeds \(\mathrm{cmc}_{\mathrm{X}}^{0}\). If surfactant \(\mathrm{Y}\) is added to the solution, the \(\mathrm{cmc}_{\mathrm{X}}\) changes. In other words the properties of surfactants \(\mathrm{X}\) and \(\mathrm{Y}\) in a given solution are linked. We anticipate that for real system cmc(mix) is a function of \(\mathrm{c}_{\mathrm{X}}^{0}\) and \(\mathrm{c}_{\mathrm{Y}}^{0}\) so that cmc(mix) is a function of ratio \(\mathrm{r}\) and a quantity \(\theta\). The latter takes account of surfactant-surfactant interactions in the micellar pseudophase. Then

\[\mathrm{cmc}(\operatorname{mix})=\mathrm{r} \,\left(\mathrm{cmc}_{\mathrm{Y}}^{0}-\mathrm{cmc}_{\mathrm{X}}^{0}\right) \, \exp [-\theta \,(1-\mathrm{r})]+\mathrm{cmc}_{\mathrm{X}}^{0} \nonumber \]

For the surfactants \(\mathrm{X}\) and \(\mathrm{Y}\),

\[\mathrm{cmc}_{\mathrm{Y}}(\mathrm{mix})=\mathrm{r} \, \mathrm{cmc}_{\mathrm{Y}}^{0} \, \exp [-\theta \,(1-\mathrm{r})] \nonumber \]

and

\[\mathrm{cmc}_{\mathrm{x}}(\mathrm{mix})=\mathrm{cmc}_{\mathrm{x}}^{0} \,\{1-\mathrm{r} \, \exp [-\theta \,(1-\mathrm{r})]\} \nonumber \]

Therefore we envisage that the cmc of solutions containing a mixture of surfactants differs from that for model systems.

We turn attention to the enthalpies of mixed surfactant solutions. In the case of a mixed aqueous solution containing surfactants \(\mathrm{X}\) and \(\mathrm{Y}\), the partial molar enthalpies of the surfactants are anticipated to depend on their concentrations. We characterize a given system by a single enthalpic interaction parameter, \(\mathrm{h}(\text {int})\).

\[\mathrm{H}_{\mathrm{x}}(\mathrm{aq})=\mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\mathrm{h}(\text {int}) \,\left(\mathrm{c}_{\mathrm{x}} / \mathrm{c}_{\mathrm{r}}\right) \nonumber \]

\[\mathrm{H}_{\mathrm{Y}}(\mathrm{aq})=\mathrm{H}_{\mathrm{Y}}^{\infty}(\mathrm{aq})+\mathrm{h}(\text { int }) \,\left(\mathrm{c}_{\mathrm{Y}} / \mathrm{c}_{\mathrm{r}}\right) \nonumber \]

Here \(\mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})\) and \(\mathrm{H}_{\mathrm{Y}}^{\infty}(\mathrm{aq})\) are the ideal (infinite dilution) partial molar enthalpies of the two monomeric surfactants in aqueous solutions at defined \(\mathrm{T}\) and \(\mathrm{p}\).

The micellar pseudo-separate phase comprises two surfactants amounts \(\mathrm{n}_{X}(\text { mic })\) and \(\mathrm{n}_{Y}(\text { mic })\). The mole fractions \(\mathrm{x}_{\mathrm{X}}(\mathrm{mic})\) and \(\mathrm{x}_{\mathrm{X}}(\mathrm{mic}) \left[=\left(1-\mathrm{x}_{\mathrm{X}}(\mathrm{mic})\right]\) are given by equation (k).

\[\mathrm{x}_{\mathrm{X}}(\mathrm{mic})=\mathrm{n}_{\mathrm{X}}(\mathrm{mic}) /\left[\mathrm{n}_{\mathrm{x}}(\mathrm{mic})+\mathrm{n}_{\mathrm{Y}}(\mathrm{mic})\right]=1-\mathrm{x}_{\mathrm{Y}}(\mathrm{mic}) \nonumber \]

We relate the partial molar enthalpies \(\mathrm{H}_{\mathrm{X}}(\text {mic})\) and \(\mathrm{H}_{\mathrm{Y}}(\text {mic})\) in the mixed pseudo-separate phase to the molar enthalpies of surfactants \(\mathrm{X}\) and \(\mathrm{Y}\) in pure pseudo-separate micellar phases, \(\mathrm{H}_{\mathrm{X}}^{*}(\mathrm{mic})\) and \(\mathrm{H}_{\mathrm{Y}}^{*} \text { (mic) }\) using equations (l) and (m) where \(\mathrm{U}\) is a surfactant-surfactant interaction parameter.

\[\mathrm{H}_{\mathrm{X}}(\text { mic })=\mathrm{H}_{\mathrm{X}}^{*}(\mathrm{mic})+\left[1-\mathrm{x}_{\mathrm{X}}(\mathrm{mic})\right]^{2} \, \mathrm{U} \nonumber \]

\[\mathrm{H}_{\mathrm{Y}}(\mathrm{mic})=\mathrm{H}_{\mathrm{Y}}^{*}(\text { mic })+\left[\mathrm{x}_{\mathrm{X}}(\mathrm{mic})\right]^{2} \, \mathrm{U} \nonumber \]

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