## 5 Discussion
### Modeling  
The behavior of observed birefringence in Figure 3c resembles what is observed in dispersions of magnetic particles [21; 25; 29]. The usual explanation assumes that the system under investigation consists of individual optically anisotropic particles (both solid [22; 30; 31] or soft [25; 27; 28; 32; 33]) dispersed in a liquid carrier.  
Assuming that particles are diamagnetic (no permanent dipole moment), when the system is exposed to external magnetic field, the magnetic dipole is induced in the particles. Since particles are magnetically anisotropic, each dipole direction does not coincide with the direction of external field. Consequently, a torque (resulting from the interaction between induced magnetic dipole and the external magnetic field) on the particles arises and the particles are induced to orient along the magnetic field. Since each particle is optically anisotropic, the dispersion as a whole becomes optically anisotropic and so birefringent.  
It must be noticed that some similarities between the model and the system object of this work are present: acid-base adducts are inherently anisotropic due to the directional H-bond or the direction of DBP-to-PA proton transfer in the case of ion pairs. In addition, these structures are in liquid phase. The principle is schematically depicted in Scheme 2.  
**Scheme 2.** The principle of formation of charged species in the liquid state (ionic liquid) starting from the originally neutral molecules DBP and PA.  
As a result of this similarity the theory from dispersion of magnetic particles can be borrowed to be applied to the present data. Here, the dimensionless magnetic energy, \(U\), for a system composed of magnetically anisotropic objects (described by anisotropy in magnetic susceptibility (\(\Delta\chi\))) placed in the external magnetic field of induction \(B\), can be defined:  
\[U=\frac{\Delta\chi}{2k_{B}T}B^{2} \tag{4}\]  
where \(k_{B}\) is Boltzmann constant and \(T\) is the temperature. The knowledge of the magnetic energy \(U\) allows calculating the average orientation of the anisotropic particles with respect to the external magnetic field. This is formally given by the value of the orientational order parameter \(S\):  
\[S=\frac{1}{2}\big{(}3\big{\langle}\cos^{2}(\theta)\big{\rangle}-1\big{)} \tag{5}\]  
The average square of cosine of the orientation angle \(\theta\), \(<\cos^{2}\left(\theta\right)>\), can be calculated from:  
\[\left\langle\cos^{2}(\theta)\right\rangle=-\frac{1}{2U}+\frac{e^{U}}{\sqrt{\pi U }mfer(\sqrt{U})} \tag{6}\]where _merf_ stands for modified error function.  
Finally, birefringence dependence of magnetic field can be calculated from:  
\[\Delta n/\lambda=\left(\Delta n/\lambda\right)_{MAX}S \tag{7}\]  
where \((\Delta n/\lambda)_{MAX}\) is saturation birefringence observed for high magnetic energies and is proportional to the anisotropy in optical polarizability, \(\Delta\alpha\), of the individual objects.  
The fit with Equation 7 to experimental data is shown on Figure 3c as a continuous line. The data are satisfactorily reproduced. There are obvious differences between our system and the ordinary dispersion for which the model was developed; specifically, (i) in our system, no individual particle exists and (ii) separate micellar objects are not expected but, rather, only loosely connected self-aggregated aggregates can be imagined. Despite of these differences, the model seems to also possess some descriptive ability for such systems. It is surprising how well the model derived for simple liquid dispersion describes the behavior of our DPB/PA system even if it obviously does not fulfill the assumptions. This, in conclusion, suggests that the fact that acid-base adducts can be considered inherently anisotropic (due to the directional H-bond or the direction of DBP-to-PA proton transfer in the case of ion pairs) and dispersed in liquid phase, allows the magnetically-induced formation of structures, which can be, therefore, described as magnetically-deformable soft nanoparticles.