## 1 Introduction
### Fundamental Thermodynamic Principles for Effective Dispersion of Carbon Nanotubes
In this case, the surface area will be:  
\[A=N\cdot\pi d^{2}_{aggregate}, \tag{3}\]  
where \(N\)--the number of spherical MWCNT aggregates; and \(d\)--the diameter of the MWCNT aggregate, nm. The total number of spherical MWCNT aggregates in the \"carbon nanotubes-hydrocarbon plasticizer\" system can be determined by the total mass of MWCNTs (\(m_{total}\)) according to the following equation:  
\[N=\frac{m_{total}}{m_{aggregate}}, \tag{4}\]  
The mass of a spherical aggregate (\(m_{aggregate}\)) can be found through the volume of the aggregate (\(V_{aggregate}\), cm\({}^{3}\)) and the density of MWCNTs (\(\rho_{MWCNT}\)):  
\[m_{aggregate}=\rho_{MWCNT}\cdot V_{aggregate}=\rho_{MWCNT}\cdot\frac{\pi d^{3 }_{aggregate}}{6}, \tag{5}\]  
So, the formula for determining the total surface area of interaction (3) takes the following form:\[A=\frac{6m_{total}}{\pi\rho_{MWCNT}-d_{aggregate}^{3}}.\pi d_{aggregate}^{2}=\frac{6m_{MWCNT}}{ \rho_{MWCNT}d_{aggregate}}, \tag{6}\]  
The change in the total surface area of MWCNT interaction in the system with a hydrocarbon plasticizer (\(\Delta A\)) is calculated as the difference between the total surface area before (\(A_{1}\)) and after ultrasonic dispersion (\(A_{2}\)). According to the statistical definition of entropy (Boltzmann's principle), [PERSON] in 1877 introduced the relationship between entropy and the probability of a given state of a system as \(S=k_{B}\cdot\ln\Omega\), where \(k_{B}\)--Boltzmann constant, \(\Omega\)--number of microscopic states corresponding to the macroscopic state of the system. The total entropy of a system is an additive quantity, meaning it equals the sum of the entropies of all its components. In our case, the entropic contribution (\(\Delta S\)) is associated with changes in the order of the system. During the dispersion of aggregated MWCNTs in a hydrocarbon plasticizer, the system's disorder (entropy) increases, which positively influences the process (\(\Delta S>0\)). An increase in entropy contributes to a decrease in free energy, as higher disorder makes the process thermodynamically more favorable. The separation (deagglomeration) and even distribution of MWCNTs in the plasticizer lead to an increase in system entropy. Therefore, the necessary condition for successful dispersion is \(\Delta G=\Delta H-T\Delta S<0\), where the dispersion process will be spontaneous with a negative or small positive enthalpy (\(\Delta H\)) and a large increase in entropy (\(\Delta S\)). Thus, the primary task is to minimize enthalpy by improving the interaction between MWCNTs and the hydrocarbon plasticizer (in our case, by using affinity and ultrasound) and maximizing \(\Delta S\) (by breaking down MWCNT aggregates). To calculate the entropy change (\(\Delta S\)) in the system during MWCNT dispersion in the hydrocarbon plasticizer, it is necessary to account for the change in particle order and distribution within the system. The total entropy change in the considered system, associated with the breakdown of nanotube aggregates and their more uniform distribution, can be calculated as the sum of these two contributions:  
\[A\Delta S=\Delta S_{mixing}+\Delta S_{disordering}, \tag{7}\]  
where \(\Delta S_{mixing}\)--the mixing entropy, a component of entropy related to the mixing of two phases--carbon nanotubes and the hydrocarbon plasticizer--and \(\Delta S_{disordering}\)--the disordering entropy, related to the breakdown of MWCNT aggregates and the increase in the number of possible microstates of the system. If carbon nanotubes are evenly dispersed in the hydrocarbon plasticizer, the system can be described in terms of the ideal mixing of two components.