1.20.3: Surfactants and Micelles: Non-Ionics
The properties of aqueous solutions containing non-ionic surfactants [1-6] can be described using two models.
Model Phase Equilibrium: Dry Micelle
We envisage a non-ionic surfactant \(\mathrm{X}\). When chemical substance \(\mathrm{X}\) is added to \(\mathrm{n}_{1}\) moles of water (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)), solute \(\mathrm{X}\) exists as a simple solute \(\mathrm{X}(\mathrm{aq})\) until the concentration of solute \(\mathrm{X}\), \(\mathrm{c}_{\mathrm{X}}\) reaches a characteristic concentration \(\mathrm{cmc}_{\mathrm{X}}\) when a trace amount of the micellar phase appears. Each micelle comprises \(\mathrm{n}\) molecules of surfactant \(\mathrm{X}\). The equilibrium between monomer surfactant \(\mathrm{X}(\aq) and surfactant in the micelles is described by the following equation.
\[\mu_{\mathrm{X}}^{\mathrm{eq}}(\mathrm{aq})=\mu_{\mathrm{X}}^{*}(\mathrm{mic}) \nonumber \]
If \(\mathrm{X}(\mathrm{aq})\) is a typical neutral solute in aqueous solution \(\mu_{\mathrm{x}}^{\mathrm{eq}}(\mathrm{aq})\) is related to the cmc of \(\mathrm{X}(\mathrm{aq})\) in solution at the point where only a trace amount of micellar phase exists. Hence,
\[\mu_{\mathrm{X}}^{0}(\mathrm{aq} ; \mathrm{c}-\mathrm{scale})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{cmc} \, \mathrm{y}_{\mathrm{X}} / \mathrm{c}_{\mathrm{r}}\right)=\mu_{\mathrm{X}}^{\mathrm{*}}(\mathrm{mic}) \nonumber \]
Here \(\mathrm{y}_{\mathrm{X}}\) is the solute activity coefficient for \(\mathrm{X}(\mathrm{aq})\) taking account of solute-solute interactions in the aqueous solution. Therefore, by definition,
\[\Delta_{\text {mic }} \mathrm{G}^{0}(\mathrm{aq})=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\text {mic }}^{0}\right)=\mu_{\mathrm{X}}^{*}(\mathrm{mic})-\mu_{\mathrm{x}}^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale }) \nonumber \]
Equilibrium constant \(\mathrm{K}_{\text {mic }}^{0}\) describes the phase equilibrium involving surfactant X in aqueous solution and micellar phase.
\[\Delta_{\text {mic }} \mathrm{G}^{0}(\mathrm{aq})=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{mic}}^{0}\right)=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{cmc} \, \mathrm{y}_{\mathrm{X}} / \mathrm{c}_{\mathrm{r}}\right) \nonumber \]
If \(\mathrm{X}(\mathrm{aq})\) is a neutral solute and the cmc is low, a useful approximation sets \(\mathrm{y}_{\mathrm{X}}\) at unity. Therefore \(\Delta_{\text {mic }} \mathrm{G}^{0}(\mathrm{aq})\) is the standard increase in Gibbs energy when one mole of surfactant \(\mathrm{X}(\mathrm{aq})\) forms one mole of \(\mathrm{X}\) in the micellar phase. Combination of equations (c) and (d) yields equation (e).
\[\mathrm{K}_{\text {mic }}^{0}=\left(\mathrm{cmc} / \mathrm{c}_{\mathrm{r}}\right)^{-1} \nonumber \]
Here \(\mathrm{K}_{\text {mic }}^{0}\) describes the equilibrium between surfactant in the micellar phase and the aqueous solution. A famous equation suggested by Harkin relates the cmc to the number of carbon atoms in the alkyl chain, \(\mathrm{n}_{\mathrm{C}}\); equation (f)
\[\log \left(\mathrm{cmc} / \mathrm{c}_{\mathrm{r}}\right)=\mathrm{A}-\mathrm{B} \, \mathrm{n}_{\mathrm{C}} \nonumber \]
With equation (e),
\[\log \left(\mathrm{K}_{\text {mic }}^{0}\right)=A-\left(B \, n_{C}\right) \nonumber \]
The above analysis is also used for ionic surfactants if it can be assumed the degree of counter ion binding by the micelles is small, the thermodynamic properties of the solution are ideal and the aggregation number is high.
Non-Ionic Surfactant: Phase Equilibrium: Wet Micelle
The aqueous phase comprises an aqueous solution of solute \(\mathrm{X}\), \(\mathrm{X}(\mathrm{aq})\). The micellar phase comprises both water and surfactant \(\mathrm{X}\) such that the mole fraction of surfactant in the micellar phase equals \(\mathbf{X}_{\mathrm{X}}^{\mathrm{eq}}\). We treat the micellar phase using the procedures used to describe the properties of a binary liquid mixture. For the micellar phase the chemical potential of \(\mathrm{X}\) is given by the following equation.
\[\mu_{\mathrm{x}}(\mathrm{mic})=\mu_{\mathrm{x}}^{*}(\text { mic })+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{x}} \, \mathrm{f}_{\mathrm{x}}\right)^{\mathrm{eq}} \nonumber \]
where
\[\operatorname{limit}\left(x_{x} \rightarrow 1\right) f_{x}=1 \text { at all } \mathrm{T} \text { and } \mathrm{p} \nonumber \]
But at equilibrium for a system containing a trace of the micellar phase,
\[\mu_{\mathrm{x}}^{\mathrm{eq}}(\mathrm{mic})=\mu_{\mathrm{x}}^{\mathrm{eq}}(\mathrm{aq}) \nonumber \]
Then,
\[\mu_{\mathrm{X}}^{\circ}(\text { mic })+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{X}} \, \mathrm{f}_{\mathrm{X}}\right)^{\mathrm{cq}}=\mu_{\mathrm{X}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{cmc}_{\mathrm{x}} \, \mathrm{y}_{\mathrm{X}} / \mathrm{c}_{\mathrm{r}}\right) \nonumber \]
By definition
\[\Delta_{\text {mic }} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\text {mic }}^{0}\right)=\mu_{\mathrm{x}}^{*}(\mathrm{mic})-\mu_{\mathrm{x}}^{0}(\mathrm{aq}) \nonumber \]
Then,
\[\mathrm{K}_{\text {mic }}^{0}=\left[\mathrm{x}_{\mathrm{X}} \, \mathrm{f}_{\mathrm{X}}\right]^{\mathrm{eq}} /\left[\mathrm{cmc} / \mathrm{c}_{\mathrm{r}}\right] \nonumber \]
If the micelle is only ‘damp’ rather than wet , a reasonable assumption sets \(\mathrm{f}_{\mathrm{X}}\) equal to unity although it is not obvious how \(\mathrm{x}_{\mathrm{X}}\) might be determined.
Non-Ionic Surfactant: Mass Action Model: Dry Micelle
Micelle formation is described as an equilibrium between \(\mathrm{X}(\mathrm{aq})\) as a solute in aqueous solution and a micellar aggregate in aqueous solution formed by n molecules of the monomer \(\mathrm{X}(\mathrm{aq})\). Then at the point where micelles are first formed, the following equilibrium is established.
\[\mathrm{nX}(\mathrm{aq}) \Leftrightarrow \mathrm{X}_{\mathrm{n}}(\mathrm{aq}) \nonumber \]
The total amount of surfactant in the system equals \(\mathrm{N}(\mathrm{X} ; \mathrm{aq})+\mathrm{n} \, \mathrm{N}\left(\mathrm{X}_{\mathrm{n}} ; \mathrm{mic}\right)\) where \(\mathrm{N}(\mathrm{X} ; \mathrm{aq})\) is the amount of monomer surfactant and where \(\mathrm{N}\left(\mathrm{X}_{\mathrm{n}} ; \mathrm{mic}\right)\) is the total amount of micelles , each micelle containing n surfactant molecules. But for the micellar aggregate \(\mathrm{X}_{\mathrm{n}}(\mathrm{aq})\) treated as a single solute,
\[\mu\left(\mathrm{X}_{\mathrm{n}} ; \mathrm{aq}\right)=\mu^{0}\left(\mathrm{X}_{\mathrm{n}} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{c}\left(\mathrm{X}_{\mathrm{n}}\right) \, \mathrm{y}\left(\mathrm{X}_{\mathrm{n}}\right) / \mathrm{c}_{\mathrm{r}}\right] \nonumber \]
Here \(c\left(X_{n}\right)\left[=N\left(X_{n} ; a q\right) / V\right.\) where \(\mathrm{V}\) is the volume of the system] is the concentration of micelles in the system, activity coefficient \(\mathrm{y}\left(\mathrm{X}_{\mathrm{n}}\right)\).The latter can be assumed to be unity if there are no micelle-micelle interactions and no micelle-monomer interactions in the aqueous system. Although this approach seems similar to that used to describe chemical equilibria, the procedure has problems in the context of determining \(\mathrm{c}\left(\mathrm{X}_{\mathrm{n}}\right)\).
Footnotes
[1] J. E. Desnoyers, G. Caron, R. DeLisi, D. Roberts, A. Roux and G. Perron, J. Phys. Chem.,1983, 87 , 1397.
[2] G. Olofsson, J.Phys.Chem.,1985, 89 ,1473.
[3] J. E. Desnoyers, Pure Appl.Chem.,1982, 54 ,1469.
[4] M. J. Blandamer, J. M. Permann, J. Kevelam, H. A. van Doren, R. M. Kellogg and J. B. F. N. Engberts, Langmuir, 1999, 15 ,2009.
[5] M. J. Blandamer, K. Bijma and J. B. F. N. Engberts, Langmuir, 1998, 14 ,79.
[6] M. J. Blandamer, B. Briggs, P. M. Cullis, J. B. F. N. Engberts and J. Kevelam, Phys.Chem.Chem.Phys.,2000, 2 ,4369.