Ion distributions near a liquid-liquid interface
Luo, G., Malkova, S., Yoon, J. et al., Science 311, 216–218 (2006).
Gouy-Chapman theory, Poisson-Boltzmann equation, mean potential
This paper brings light to a fundamental approximation of the Debye-Huckel and Gouy-Chapman theories, which can be limiting when describing some real systems. In the Poisson-Boltzmann equation, the energy <math>E_i</math> is the energy of ion <math>i</math> relative to the bulk phase. In the Debye-Huckel and Gouy-Chapman theories, it is assumed that this energy is due only to the electrostatic energy. That is, these theories do not take into account changes in energy due to liquid structure. The actual energy, including the liquid structure contribution, can be expressed as <math>E_i(z) = e_i(z) \phi_i(z) + f_i(z)</math>, where <math>f_i(z)</math> is the free energy profile of ion <math>i</math>.
X-ray scattering, one of the few experimental techniques that can measure ion distributions in solutions near interfaces, has shown that the Gouy-Chapman theory fails for many systems, suggesting that liquid structure must be taken into account. The authors of the paper performed X-ray ion-distribution measurements at a liquid-liquid interface between an aqueous solution of hydrophilic ions and a polar organic solution of hydrophobic ions. In particular, solutions of tetrabutylalammonium tetraphenylborate (TBATPB) in nitrobenzene and tatrabutylammonium bromide (TBABr) in purified water were prepared. In equilibrium, the ions distribute across the two phases such that the potential at each ion is equal in both phases. By adjusting the concentration of TBABr, the electric potential across the interface can be varied.
The ion distribution measurements agree with Gouy-Chapman theory for the lowest concentrations of TBABr, but increasingly differs from the prediction as the concentration increases, reaching a difference of 25 standard deviations at the highest concentration. To better describe the data, the authors include liquid structure in the calculation by using the correct energy in the Poisson-Boltzmann equation, <math>E_i(z) = e_i(z) \phi_i(z) + f_i(z)</math>. A model for the free energy profile <math>f_i(z)</math> was found by calculating the potential of the mean force through molecular dynamics (MD) simulations. Though the true energy should take into account ion-ion interaction, the authors did not take this into account due to computational restrictions. Figure 1 shows the X-ray reflectivity data with Gouy-Chapman predictions as well as the new predictions.
The agreement between the authors' model and the data suggest that liquid structure does indeed change the ion distributions. Furthermore, the agreement also indicates that ion-ion interaction contributions to the energy are negligible in this case. Thus, this work provides a way to practically take into account the liquid structure in calculating ion distributions near interfaces, improving upon the Gouy-Chapman model.