## 1 Introduction
### Fundamental Thermodynamic Principles for Effective Dispersion of Carbon Nanotubes
The mixing entropy formula for ideal solutions can be used (at this stage, we will disregard any specific interactions that might lead to deviations from ideal behavior):  
\[\Delta S_{mixing}=-nR(x_{1}\ln x_{1}+x_{2}\ln x_{2}), \tag{8}\]  
where \(n\)--the total number of moles of particles in the system (approximately for MWCNTs and plasticizer); \(R\) is the universal gas constant, 8.314 J/(mol\(\cdot\)K); and \(x_{1}\) and \(x_{2}\)--the mole fractions of MWCNTs and plasticizer, respectively. The mole fraction \(x_{1}\) can be found from the known percentage of MWCNT content:  
\[x_{1}=\frac{n_{MWCNT}}{n_{MWCNT}+n_{plasticizer}};\ x_{2}=1-x_{1}, \tag{9}\]  
where \(n_{MWCNT}\) and \(n_{plasticizer}\)--the moles of carbon nanotubes and plasticizer, respectively, which can be calculated using the following equation:  
\[n_{MWCNT}=\frac{m_{MWCNT}}{M_{MWCNT}};\ n_{plasticizer}=\frac{m_{plasticizer }}{M_{plasticizer}}, \tag{10}\]where \(m_{MWCNT}\) and \(m_{plasticizer}\)--the masses of carbon nanotubes and hydrocarbon plasticizer, \(M_{MWCNT}\)--the molecular mass of multi walled carbon nanotubes (it may be difficult to determine due to the structure of the nanotubes, but it can be assumed to be close to the molecular mass of carbon atoms considering the number of atoms in the nanotube), and \(M_{plasticizer}\)--the molecular mass of the hydrocarbon plasticizer. From the molecular-statistical perspective, the process of dispersing carbon nanotubes can be viewed as a transition from aggregated nanotubes (an ordered state) to dispersed nanotubes (a more disordered state). This effect can be described using a statistical approach by analyzing the change in the number of possible microstates of the system. Suppose that before the disruption of multi walled carbon nanotube (MWCNT) aggregates, the system has \(W_{1}\) states, and after dispersion, it has \(W_{2}\) states. In this case, the change in entropy can be evaluated through the logarithm of the number of microstates (\(W\)) before and after ultrasonic dispersion:  
\[\Delta H=\gamma_{CP}\cdot\Delta A, \tag{2}\]0  
where \(k_{B}\)--Boltzmann's constant, 1.38\(\cdot\)10\({}^{-23}\) J/K, and \(W_{1}\) and \(W_{2}\)--the number of microscopic states of the system before and after ultrasonic dispersion, respectively. The number of microscopic states, \(W\), represents the various possible configurations of the system. To estimate \(W_{1}\), one must consider the degree of aggregation of the MWCNTs. In the aggregated state, carbon nanotubes are grouped into \"bundles\" or \"clusters\", so the number of possible states is limited by the number of such aggregates. This can approximately be assessed as one state or several possible states depending on the structure of the aggregates. In the dispersed state (\(W_{2}\)), after the aggregates are disrupted, the number of possible configurations increases significantly, as the CNTs can occupy various positions and orientations in the hydrocarbon plasticizer. \(W_{2}\) can be estimated using parameters such as the number of nanotubes (\(N\)) and the volume of the plasticizer available for the movement of the nanotubes. Alternatively, it can be calculated based on the density and concentration of MWCNTs in the plasticizer. The number of microscopic states \(W_{2}\) can be calculated using the following equation:  
\[\Delta H=\gamma_{CP}\cdot\Delta A, \tag{2}\]1  
where \(V_{eff}\)--the effective volume occupied by the dispersed carbon nanotubes (which can be calculated through the diameter of the dispersed MWCNTs, assuming the sphericity of the particles from the disrupted aggregate), \(V\)--the volume of the plasticizer, and \(N\) is the number of carbon nanotubes (disrupted MWCNT aggregates) in the system. The greater the difference between \(W_{1}\) and \(W_{2}\), the larger the positive change in entropy, which corresponds to stronger dispersion of the MWCNTs in the plasticizer system.