*3.4. Diffusion Coefficient*

The aggregation pattern of each molecule significantly affects the microstructure of the oil–water emulsion system. The nanoparticulate NGOs are spontaneously transported from the aqueous phase to the oil–water interface, and the temperature resistance and diffusion rates of the NGOs are analyzed by comparing the mean square displacement (MSD), as shown in Equation (8), at the three temperatures of 300 K, 330 K, and 360 K set by the model. The mean square displacement can be characterized by a diffusion coefficient (*D*) related to the simulation time [27]:

$$D = \frac{1}{6N} \lim\_{t \to \infty} \frac{d}{dt} \sum\_{i=1}^{N} \left\{ \left[ r\_i(t) - r\_i(0) \right]^2 \right\} \tag{8}$$

where *D* is the diffusion coefficient of the molecule, *N* is a molecular term of diffusion in the system, *ri*(*t*) is the position of the molecule at the moment, and the differential term is the ratio of mean square displacement to time.

The calculation shows that the mean squared displacement MSD is 7.43 Å at 300 K, 11.98 Å at 330 K, and 18.64 Å at 360 K. As the temperature increases, the NGO diffusion coefficient becomes larger. These results mainly originate from the interaction of the nanoparticle NGOs with the oil phase. The interaction of the surface-oxidized GO molecules with the aqueous phase was much larger than that with the oil phase, and the interaction with the oil phase was greatly enhanced by the surface-grafted cetylamine, so the transportability of the NGOs along the *Z*-axis was much larger than that of the nanoparticles in the X and Y directions. The results show that the surface-modified alkylamine graphene oxide is highly susceptible to aggregation towards the oil–water interface.

The NGOs move under the combined action of water and oil [28]. As can be seen from Figure 8, the higher the temperature, the greater the slope of the nanoparticle dynamic diffusion curve. It can be observed that the free energy of the mixed-phase is increased and the relative intermolecular displacement rate is expanded under the action of a high temperature, while the NGOs are found to have a good temperature resistance according to the molecular equilibrium conformation.

**Figure 8.** MSD curves for NGOs at different temperatures.

## *3.5. Interfacial Tension*

The oil phase is divided from the water phase by NGOs, forming a clear oil–water interface. Conventional surfactants can be used to improve recovery by reducing the surface tension at the oil–water interface. Numerous experiments demonstrated that modified 2D nanomaterials can significantly improve the recovery rate of cores. In this paper, the interfacial tensions at the oil–water interface, and at the oil−NGO−water interface at three temperatures of 300, 330, and 360 K, were calculated based on Equation (9) [29], and the results are shown in Figure 9:

$$\gamma = \frac{1}{2} L\_z \left[ p\_{zz} - \frac{1}{2} (p\_{xx} + p\_{yy}) \right] \tag{9}$$

where *Lz* denotes the length of the system in the *z*-axis direction. *pxx*, *pyy*, *pzz* are denoted by the pressure tensor in the *x*, *y*, and *z* directions, respectively.

**Figure 9.** (**1**) Change in interfacial tension without and (**2**) with the addition of NGOs.

In order to compare the effect of conventional surfactants with modified graphene oxide by reducing interfacial tension, the oil–water surface tension of three systems (disodium laureth sulfosuccinate DLS, disodium cocoate monoethanolamide sulfosuccinate DMSS and modified graphene oxide) was measured. The results are shown in Table 6, and the modified graphene oxide was the most effective in reducing interfacial tension.

**Table 6.** Comparison of different surfactants for reducing interfacial tension (mN·m<sup>−</sup>1).


From Table 7, the interfacial tension in both systems decreases with increasing temperature, and the interfacial tension of the system without the addition of NGOs decreases by 7.2 mN·m−<sup>1</sup> with increasing temperature. The analysis showed that the modified graphene oxide could still significantly reduce the interfacial tension between oil and water at 360 K, showing an excellent temperature resistance. The decrease in interfacial tension was observed at all three temperatures with the addition of NGOs, as shown in Table 7, and it was found that the NGOs could be excellent surfactant substitutes.

**Table 7.** Variation of interfacial tension (mN·m<sup>−</sup>1) at different temperatures.


In the process of tertiary oil recovery, the remaining oil is mainly subject to the combined effect of pressure gradient force, surface tension, cohesive force [30], oil drops and oil films, which are the main methods of maintaining oil in the pore space. By using interfacial tension as an important parameter to describe the nature of the oil–water interface, this paper analyzes the force of oil droplets and oil films in the nanopore space to obtain the mechanism of tertiary oil recovery to improve the recovery factor.

The cohesive force: When there is relative motion between the oil droplets and the solid surface, a force that blocks this motion occurs, and a force of this nature is known as the cohesive force. Pressure gradient force is the constant velocity in the flow through a small orifice compared with the value of change in pressure per unit time. Surface tension can be considered as the contraction force acting on the interface of a unit length of liquid.

It can be observed from Figure 10 that, after NGOs were added, the surface tension (orange line in Figure 10) at the oil drops and oil films in the remaining oil becomes less intense, resistance to the three recovery processes and the kinetic energy required at the injection end decrease, and the remaining oil is more easily displaced.

**Figure 10.** Vector diagram of forces on oil droplets and oil film: (**a**) Force on oil droplets, (**b**) oil film stresses.
