Materials are not infinitely separable, and the intrinsic properties of a particle in the internal length range will affect the mechanical properties of the microstructure. Interface effects and surface effects are ignored at the macroscopic level, but they play important roles at the microscopic level (Chiu and Chen, 2011; Li and Mi, 2019; Lu and Guo, 2019; Wang and Zhang, 2019). Soil shows a typical multi-scale property; that is, the stress and deformation of soil are different in each scale space. A continuous soil slice at the macroscopic level is shown in Figure 1A, which demonstrates that the stress distribution in the soil slice is nearly uniform under the load. However, at the microscopic level, where the research focus is a single soil particle, contact surface discontinuity causes the stress to appear abruptly on the contact surfaces of the particles. In Figure 1B, the cross-scale deformation behavior of the foundation from the macro-level to the micro-level is illustrated. At the macroscopic level, the foundation produced continuous compressive deformation under uniform load, whereas at the microscopic level, the soil particles showed an obvious rotational effect. Microscopically, the particles appeared in individual forms and the contact between particles was highlighted. However, macroscopically, the number of particles in a soil slice is huge and the individuality of particles is weakened; therefore, particle groups are treated as a continuous medium and the contact between particles is no longer considered. In general, the effects of interface properties between particles on soil properties change at different scales.

For soil particles, when particle size decreases, the specific surface area of particles and the proportion of charges on the surface of particles increase gradually; thus, the ability to adsorb ions and water molecules increases. As a result, the adsorbed water and the electric double layer formed on the surface of the particles are more obvious, which in turn affects the contact surface between the particles. At the same time, a typical soil mixture contains clay minerals and non-clay minerals. Most clay minerals have excess negative charges on the surface of particles due to isomorphous replacement. The negative charges adsorb the polar water molecules and cations to form an electric double layer (Figure 2). Clay particles tend to be smaller in size and have a larger specific surface area; therefore, the volume of adsorbed water and the electric double layer on the surface of particles is considerably larger than the volume of the particles. When clay particles are in contact, the contacts occur mainly between the water films on the particle’s surface. For non-clay minerals, the ability to adsorb ions and water molecules is very weak because the excess charges on the surface of the non-clay minerals are less; the adsorbed water on the surface of such mineral particles is so thin that the contact between particles is mainly direct contact between mineral crystals. For soil materials, both particle size and mineral composition influence the properties of the particle surface, thereby affecting the contact surface between particles, as shown in Figure 3. Therefore, as a granular medium, soil particles also exhibit the surface effects and interface effects.

The surface and interface effects of soil particles affect the soil microstructure (Guo and Yu, 2019) and thereby affect the soil engineering properties at the macroscopic level, which is the basis for the soil-scale phenomenon. For example, particle size affects the occlusion between sand particles; increased coarse particle content leads to increased microstructure rolling and increased internal friction angle of the soil at the macroscopic level; the electric double layer (DEL) and the adsorbed water on the surface of clay particles result in a cohesive effect among the particles with increased cohesive force and increased clay mineral content; the balance between the microscopic force and the gravity of the particles affects the formation of the microscopic and macroscopic structure of the soil; at the same time, the mineral components have different adsorption water states on the surface, which will also directly affect the state of the liquid flowing over its surface, thereby affecting the permeability characteristics of the soil.

The surface state of a particle is largely determined by the particle’s ability to adsorb water or other ions, and this ability is called the particle’s surface energy. This adsorption energy is mainly manifested as the van der Waals force between particles and the electric field force generated by the excess charges on the particle surface. At the micro-level, the forces with high size factor power, such as gravity and inertial force, are relatively small; while the forces with low size factor power, such as surface tension, van der Waals force, and electric field force, are relatively large, as shown in Figure 4. The surface and contact interface features of the particles are produced mainly by the direct action of these forces. Among these forces, gravity, van der Waals force, and Coulomb force can be used to characterize both the particle size and the mineral composition of soil particles.

In this study, a relationship between gravity, van der Waals force, and Coulomb force was established. As introduced previously, these forces can be used to characterize the adsorption energy of the particle surface. Therefore, our energy multi-scale method involves calculation of the ratio of the micro-Coulomb force to gravity and the ratio of the micro-van der Waals force to gravity. The equations used for these calculations are expressed as follows: λ e = F e G = 1 4 π ε 0 q 1 q 2 r 2 / ρ g π d 3 6 = 3 q 1 q 2 2 π 2 ε 0 r 2 ρ g d 3 , (1) λ w = F w G = A d 24 H 2 / ρ g π d 3 6 = A 4 ρ g π d 2 H 2 , (2)

where λ _e and λ _w are the ratio of micro-Coulomb force to gravity and the ratio of micro-van der Waals force to gravity, respectively, F _e is Coulomb force, F _w is van der Waals force, G is gravity, ε ₀ is vacuum permittivity, q ₁ and q ₂ are particle charge, r is particle center spacing, ρ is the particle density, d is the particle diameter, A is the Hamaker constant, and H is the clear particle spacing. Soil particles exist mainly in a water environment, and the amount of charge associated with the particles can be obtained according to colloid and interface chemistry and electrostatic theory (Helmy, 1998; Missana and Adell, 2000): q = 2 π ε 0 ϕ 0 , (3)

where φ ₀ is the particle surface potential and the clay particle surface potential can be determined by the cation exchange capacity (CEC) method. For the convenience of calculation and the unification of measurement indicators, the adjustment coefficient α is used to combine λ _e and λ _w as one parameter, and this parameter is called the comprehensive ratio of micro-force to weight (CRFW). The expression of λ is as follows: λ = α λ e + 1 − α λ w = α F e G + 1 − α F w G . (4)

The adjustment coefficient α must be determined according to the properties of the material. Taking the common soil minerals montmorillonite and quartz as examples, the CRFW of these two mineral particles were calculated. When the value of the adjustment coefficient α was varied, the CRFW of montmorillonite and quartz particles also changed according to particle size; these data are displayed in Figure 5. As shown in the figure, when α is 0 the λ degenerates as λ_w and when α is 1 the λ degenerates as λ_e , respectively.

Figure 5 demonstrates that the relationships between λ and the particle sizes of quartz and montmorillonite were basically the same, and when particle size decreased, λ increased gradually. The specific surface area of particles is relative to particle size. As a result, the amount of negative charge on the particle surface increased and the surface energy increased; therefore, the thickness of the water film on the particle surface became larger and the interface between particles was mainly soft contact between water layers. In contrast, with increased particle size, the effect of adsorbed water on the particle surface decreased, and the contact between particles was mainly hard contact between mineral crystals. It is reasonable to use the CRFW to describe the variation of soil properties with particle size. At the same time, it can be seen from Figure 5 that, although the particle size of two different mineral particles is the same, their CRFW values can be very different. As demonstrated previously, compared with non-clay minerals, the isomorphic replacement of clay minerals is more obvious, and the negative charge density of particle surfaces is higher, so the surface effect is more obvious. This also shows the rationality of the formula for CRFW. In summary, the CRFW in the energy multi-scale method reflects the mechanism of property changes based on particle size and mineral composition at the micro-level.

The plasticity index of soil is important for engineering endeavors involving soils such as clay and silty clay. A large number of studies have found that the shear strength, permeability coefficient, and liquefaction index of plastic soils were directly or indirectly related to the plasticity index of soil. Furthermore, the plasticity index of plastic soil reflects the water adsorption capacity of the surface of the particles, which is a direct reflection of the surface energy of the particles. At the same time, the mineral composition and the diameter of the soil particle are the main factors in the calculation of the CRFW by the energy multi-scale method. Therefore, in this study, the surface potential and the diameter of the particles were tested first. Next, the samples with different mineral compositions and relative content, as well as different particle sizes, were prepared to assess the plasticity index of the soil. In terms of soil mineral composition, three typical clay minerals, montmorillonite, kaolinite, and illite, and two non-clay minerals, quartz and feldspar, were selected for these experiments. Due to the high purity of commercial soil minerals, customized commercial soil mineral powders were selected as test material. Samples with clay mineral content of 100%, 60%, 40%, 20%, and 0% were assessed, and the diameter of all mineral particles was 2 μm. Sample characteristics are summarized in Table 1.

The dried mineral powders were mixed evenly according to the designed mass ratio, then a small amount of water was added to wet the mineral particles of the soil; the samples were left to stand overnight. Finally, the liquid and plastic limits of the aforementioned soil samples were measured using a liquid–plastic limit tester, and the plasticity index of each group of samples was calculated using the liquid and plastic limit data.

After the liquid–plastic test, the plasticity index (I _p) of each sample was assessed, as shown in Figure 6; the I _p of all soil samples increased with increased clay mineral content, but the three clay mineral samples had substantially different I _p values despite having the same particle size and content.

According to the energy multi-scale method, the ratio of micro-Coulomb force to gravity and the ratio of micro-van der Waals force to gravity can be calculated using equations 1 and 2 and the elements from the literature (Huang, 2018). When d = 2 μm, λ _e, and λ _w of the three materials could be calculated and is shown in Table 2; the λ _e values of montmorillonite and illite were 6.91 and 2.1 times the value of kaolinite, respectively, whereas the λ _w values of montmorillonite and illite were 0.73 and 0.79 times that of kaolinite, respectively, and the I _P values of montmorillonite and illite were 4.8 and 1.68 times that of kaolinite, respectively. The difference in λ _w between these minerals was not large. All samples were composed of clay minerals; therefore, the adjustment coefficient α could be set to 1 for the test. As showed in Figure 5, the λ_e is bigger than λ_w for montmorillonite , while λ_e is samller than λ_w for Quartz ,within a certain particle size range. These data indicate that, in clay, the Coulomb force of particles has a greater influence on the particle surface adsorption energy, and the contribution value of λ _e to λ is larger for clay.

As described previously, the surface state and interface state are the basis of stress transfer and deformation coordination, and the surface state and contact state of the particles at the microscopic level determine the internal structure and engineering properties of the soil. The adsorbed water and the electric double layer are mainly caused by the negative charges on the surface of the particles. The more charges on the particle surface, the stronger the surface adsorption energy of the particle, and the relatively thicker the adsorbed water layer. For clay minerals, the charge density of the particles is high, and the particle size is small, so the thickness of the adsorbed water layer on the surface of the particles is relatively large for a particle with such a tiny diameter. Currently, the properties of the interface between particles are generally determined by the adsorbed water layer. Therefore, many engineering properties of clay soils are affected by the adsorbed water layer. The plasticity index of clay minerals in this study was typical, and the plasticity index refers to the relative water content of plastic soil from the plastic state to the flowing state. It is an important index for measurement of surface adsorption capacity of mineral particles to free water in the soil, and it is also a typical measure of performance of the microscopic adsorption energy of mineral particles at the macroscopic level. The surface charge density of clay mineral particles such as montmorillonite is largest, illite has the second largest, and kaolinite has the smallest, so the plasticity index of each of the three minerals decreases accordingly, as shown in Figure 6. As shown in Table 2, λ _e of montmorillonite is the largest, that of illite is the second largest, and that of kaolinite is the smallest. These data indicate that the energy multi-scale method quantitatively reflects the adsorption capacity of the microscopic surface of particles. The adsorbed water layer discussed in this paper is relatively thick compared to the particle diameter. When the particle size is large, even if the adsorbed water layer on the surface of the particle is of the same thickness as that of a smaller particle, the relative thickness can be easily ignored compared to the huge particle size. Therefore, using the energy multi-scale method, both λ _e and λ _w decreased with increased particle size, as shown in Figure 5. In addition, we assessed the plasticity index of each of the three clay minerals when the λ _e parameter was controlled. First, according to Eq. 1 and Eq. 2, the relationship curves between the λ _e and particle size of each of the three materials were calculated, as shown in Figure 7.

The particle diameters of montmorillonite, illite, and kaolinite were set to 20 μm, 14 μm, and 8 μm, respectively, and the λ _e of each of the three materials were caculated from the fitted equation in Figure 7 and the results were shoud in Table 3.

Figure 8 indicates that the plasticity indices of these three clay minerals, calculated using the energy multi-scale method, are very close to each other and nearly on the same straight line. Therefore, there are benefits to use of this method to classify clay soil samples, as the parameter λ in the energy multi-scale method accounts for the influence of mineral composition and particle size on the microscopic surface and interface states of the particles.