1.3.8: Calorimetry- Titration Microcalorimetry- Micelle Deaggregation
Aliquots of a concentrated surfactant solution are injected into the sample cell of a titration microcalorimeter. The sample cell initially contains water(\(\lambda\)). As more surfactant solution is injected into the sample cell a stage is reached where the concentration of surfactant in the sample cells exceeds the critical micellar concentration, \(\mathrm{cmc}\). The magnitude of the recorded heat changes dramatically, leading to estimates of both the cmc and the enthalpy of micelle formation.
This calorimetric techniques has proved important in studies of ionic surfactants; e.g. hexadecyltrimethylammonium bromide (CTAB). For these surfactants the microcalorimeter signals a marked difference in recorded heats as the concentration of the surfactant changes from below to above the cmc. Titration microcalorimetric results for non-ionic surfactants are unfortunately not so readily interpreted. In addition to micelle formation, the monomers cluster in small aggregates below the cmc and the micelles cluster above the cmc.
Analysis
The volume of injected aliquot \(\operatorname{vinj}\) is significantly less than the volume of the sample cell. The amount of surfactant in each aliquot is \(\operatorname{ninj}\), the concentration of surfactant being \(\operatorname{cinj}[=\operatorname{ninj} / \text { vinj }]\). If \(\operatorname{cinj}\) is significantly above the \(\mathrm{cmc}\), the contribution of the surfactant to the enthalpy of the injected aliquot is \(\operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})\) where \(\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})\) is the contribution of one mole of monomer to the molar enthalpy of a micelle. If the concentration of solution in the sample cell is below the cmc, the contribution of each monomer to the enthalpy of the solution equals \(\mathrm{H}_{\mathrm{j}}^{0}(\text { mon })\). We concentrate attention on the contribution of the surfactant to the enthalpies of injected solution and the solution in the sample cell.
Enthalpy of the injected aliquot, \[\mathrm{H}(\mathrm{inj})=\operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic}) \nonumber \]
The contribution of the surfactant to the enthalpy of the solution in the sample cell at injection number I is given by equation (b).
Enthalpy of solution in the sample cell at injection number I, \[\mathrm{H}(\mathrm{I})=\mathrm{I} \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0} \text { (mon) } \nonumber \]
Enthalpy of solution in the sample cell at injection number (I+1), \[\mathrm{H}(\mathrm{I}+1)=(\mathrm{I}+1) \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0} \text { (mon) } \nonumber \]
Recorded heat \[\mathrm{q}=\mathrm{H}(!+1)-\mathrm{H}(\mathrm{I})-\mathrm{H}(\text { inj }) \nonumber \]
\[\text { or, }[\mathrm{q} / \text { ninj }]_{\text {lowl }}=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mon})-\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})=-\Delta_{\text {mic }} \mathrm{H}^{0} \nonumber \]
At high injection numbers the enthalpies of solution in the sample cell are \(\mathrm{I} \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})\) and \((\mathrm{I}+1) \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})\). \[[\mathrm{q} / \text { ninj }]_{\text {highl }}=(\mathrm{I}+1) \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})-\mathrm{I} \, \mathrm{H}_{\mathrm{j}}^{0}(\text { mic })-\mathrm{H}_{\mathrm{j}}^{0}(\text { mic })=0 \nonumber \]
At low injection numbers the recorded \((q/\operatorname{ninj}\)) is effectively the enthalpy of micelle formation. The recorded ratio \([q/\operatorname{ninj}]\) is effectively zero at high injection numbers, the switch in pattern of \((q/\operatorname{ninj}\)) from \(\Delta_{\operatorname{mic}} \mathrm{H}^{0}\) to zero marking the cmc of the surfactant.