## 1 Introduction
### Fundamental Thermodynamic Principles for Effective Dispersion of Carbon Nanotubes  
Due to the high viscosity of bituminous binder, the introduction and dispersion of multi walled carbon nanotubes (MWCNTs) is often performed in a less viscous medium, typically a hydrocarbon plasticizer [26; 42]. The process of dispersing carbon nanotubes in a hydrocarbon plasticizer can be analyzed using the fundamental equation of Gibbs free energy change (\(\Delta G\)). The first mention of this equation was presented in the work \"On the Equilibrium of Heterogeneous Substances\", published in two parts in 1876 and 1878. This equation remains foundational for understanding physical processes and phase transitions. In general, the equation for Gibbs free energy change is as follows:  
\[\Delta G=\Delta H-T\Delta S, \tag{1}\]  
where \(\Delta G\)--the change in Gibbs free energy; \(\Delta H\)--the change in the system's enthalpy (related to the thermal effects during the interaction between MWCNTs and the hydrocarbon plasticizer); \(T\)--the absolute temperature of the system in Kelvin; and \(\Delta S\)--the change in system entropy. For successful dispersion of carbon nanotubes in a plasticizer, the Gibbs free energy change must be negative (\(\Delta G<0\)). In classical thermodynamics, the enthalpic contribution is considered as the total energy of bonds and interactions within the system (\(\Delta H=U+PV\), where \(U\)--internal energy of the system, \(P\)--pressure, \(V\)--volume). However, for dispersed systems with multi-walled carbon nanotubes (MWCNTs) subjected to ultrasonic dispersion, it is necessary to account for specific interactions between the nanotube surface and plasticizer molecules. Therefore, in our case, the enthalpic contribution (\(\Delta H\)) reflects the energy of interfacial interactions between carbon nanotubes and plasticizer molecules, as well as the change in the contact area of the particles. A positive \(\Delta H\) indicates that dispersion requires energy input, which is usually observed when there are strong intermolecular forces within the MWCNTs (e.g., van der Waals forces) that hinder their separation (deagglomeration). To improve dispersion in our case, we use less viscous, compatible hydrocarbon plasticizers and ultrasonic dispersion, which further increases the system temperature, helping to reduce intermolecular forces and thus decrease \(\Delta H\). Therefore, the equation linking enthalpy change to intermolecular interactions can be presented as follows:  
\[\Delta H=\gamma_{CP}\cdot\Delta A, \tag{2}\]  
where \(\gamma_{CP}\) --the surface (interfacial) tension in the \"carbon nanotubes-plasticizer\" system, N/m; and \(\Delta A\)--the change in the interaction surface area due to the ultrasonic dispersion of MWCNTs in the hydrocarbon plasticizer. In this case, the enthalpy component is evaluated through the change in surface energy at the phase boundary. The lower the interfacial tension in the \"MWCNT-hydrocarbon plasticizer\" system, the easier the dispersion process, as the energy barrier required for efficient breakdown of MWCNT aggregates decreases. The surface area of interaction between MWCNTs and the plasticizer can be calculated based on the geometric characteristics of multi-walled carbon nanotubes (MWCNTs). Generally, CNTs are cylindrical structures, and their surface area includes the lateral surface area and the area of the tube ends. However, due to the action of van der Waals forces, MWCNTs tend to be highly entangled and aggregated. Therefore, when measuring the dimensions of MWCNTs in a hydrocarbon plasticizer using a laser diffractometer, reliable data on the sizes of spherical aggregates, rather than individual nanotubes, may be obtained. This is because the diffractometer software assumes a spherical shape for the light-scattering particles when calculating the size distribution. Thus, the surface area of interaction will be determined for an ensemble (aggregate) of MWCNTs, assuming uniform distribution of plasticizer molecules across the surface of the MWCNTs.