# Research Progress on the Influence of Thermo-Chemical Effects on the Swelling Pressure of Bentonite
## 4 Effect Mechanism of Swelling Properties
### Intergranular Stress  
At present, studies on the swelling mechanism of bentonite generally remain at the microscopic level. A few scholars have proposed a new formula to determine the effective stress in unsaturated soil from a macroscopic perspective, which can describe the stress state in unsaturated soil well [89; 90]. In 2014, [PERSON] [91] proposed a theory of intergranular stress acting between soil particles, which comprehensively characterized the physicochemical interactions between soil particles from a macroscopic perspective. [PERSON] [91] introduced the concept of equilibrium solution and pointed out that the measured pore pressure is not the same as the true pore pressure within the soil. For bentonite, when the soil is saturated with an aqueous solution, the magnitude of the water pressure within the pore space is not uniform due to the physicochemical interaction between pore solution and soil water. The equilibrium solution is defined as the water in a measurement vessel that has reached equilibrium with a pore solution without surface action of the soil particles. The soil is in equilibrium with the sensor, when measuring the pore pressure. In other words, we determine the pore pressure by measuring the pressure of the equilibrium solution, whereas the instrument measures only the potential energy of the water and not the mechanical pressure. [PERSON] [91] deduced the expression of true pore pressure considering that the water potential energy in the pore is the same as that in the container at an equilibrium state:  
\[p^{l}\ =\ p^{l}_{w}+\Pi \tag{4}\]  
where \(p^{l}_{w}\) is the actual pore water pressure in the soil, and there is a difference \(\Pi\) from the measured pore pressure \(p^{l}\); \(\Pi\) is called generalized osmotic pressure. Intergranular stress is defined as:  
\[\sigma^{\prime\prime}\ =\ \sigma-(p^{l}_{w}+\Pi) \tag{5}\]  
where \(\sigma^{\prime\prime}\) is the intergranular stress; \(\sigma\) is the total stress. This generalized osmotic pressure reflects the interaction between soil particles and pore water and is divided into two components: the Donnan osmotic pressure and the pressure derived from surface forces. The Donnan osmotic pressure relates to the concentration difference between the pore solution and the equilibrium solution. The second component of generalized osmotic pressure is the pressure caused by surface forces, including the surface tension of the interface, the electrostatic force, Van der Waals gravity, electric double-layer repulsion, etc. The equation is:  
\[\Pi\ =\ \Pi_{D}-\rho^{l_{\mathrm{H_{2}O}}}_{\oplus}\Omega^{l} \tag{6}\]  
In the equation, \(\rho^{l_{\mathrm{H_{2}O}}}_{\oplus}\) is the mass density of pure water, equal to 1.0 g/cm\({}^{3}\), \(\Pi_{\mathrm{D}}\) is the Donnan osmotic pressure, and \(\Omega^{l}\) is the surface potential energy caused by microscopic surface forces. Donnan osmotic pressure is the pressure caused by the pressure difference between the internal and the external concentration of the soil related to the porosity and solution concentration in the soil. It is expressed by:  
\[\Pi_{\mathrm{D}}\ =\ \frac{\mathrm{RT}\rho^{l_{\mathrm{H_{2}O}}}_{\oplus}}{ \mathrm{M_{H_{2}O}}}\ln\left(\frac{a^{l_{\mathrm{H_{2}O}}}_{\mathrm{A}}}{a^{l_ {\mathrm{H_{2}O}}}}\right) \tag{7}\]  
where \(\mathrm{M_{H_{2}O}}\) is the molar mass of water, equal to 18 g/mol, \(a^{l_{\mathrm{H_{2}O}}}_{\mathrm{A}}\) is the activity of water in the equilibrium solution, which is equal to the molar fraction of water in the equilibrium solution in the ideal solution, \(a^{l_{\mathrm{H_{2}O}}}\) is the activity of water in the soil pore solution, which is equal to the molar fraction of water in the soil pore solution in the ideal solution.