# Relation of EDL Forces between Clay Particles Calculated by Different Methods
## 1 Introduction  
An electrical double layer (EDL) consisting of negative charges on the clay particle surface and the surrounding electrolyte ions in the electric fluid exists widely within natural materials including clay minerals in a watery environment [1; 2]. The electrostatic interaction between clay particles due to the overlapping EDLs plays a key role among a variety of engineering contexts, e.g., wellbore stability due to hydration of shale [3], swelling behavior of nuclear waste barrier materials [4], swelling capacity of expansive soil [5], aggregation of coal slurry [6], plasticity of treated clayy soil [7], reduction in soil erosion considering soil organic matter [8], effects of clay swelling/collapse on geological ultrafiltration [9], unfrozen water content in frozen soil [10], fluid-mineral interaction mechanisms underlying enhanced oil recovery [11], efficiency of tailing dewatering [12], and froth flotation of fine coal [13]. To determine the above electrostatic interaction quantitatively, three methods including direct measurement with atomic force microscopy [14; 15; 16] or surface forces apparatus [17], theoretical or numerical analysis based on the classical theory of Derjaguin, Landau, Verwey, and Overbeek (DLVO) [18; 19; 20; 21; 10; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34], and molecular dynamics (MD) simulations [4; 5; 9; 31; 35] have often been used in previous studies. Although some deviations from DLVO theory have been revealed by direct measurements and MD simulations [14; 15; 16; 17; 36; 37; 35], e.g., ion-ion correlations (short-range hydration interactionsand ion polarization are neglected), DLVO theory is still very reliable under most circumstances [17]. According to this theory, the electrostatic interaction between clay particles is the sum of the van der Waals force and EDL force [2; 33; 34]. Further, the EDL force is the sum of excess osmotic pressure (it was named the electrical double-layer force in [34], whereas osmotic pressure is used throughout this study to avoid possible confusion) and Maxwell's stress [23; 29; 30; 31; 33]. The osmotic pressure felt by a clay particle is exerted by the electrolyte ions in the EDL, and Maxwell's stress is the coulomb interaction between charges on the clay particle surface and the counter ions within the EDL [33]. Both of them can be estimated by the electric potential field described by the Poisson-Boltzmann equation. Although the shape of particles in natural materials including clay minerals may be complex, the material is usually simplified into two parallel infinite plates [8] or ideal spheres [13] based on which the EDL force can be calculated analytically. Scholars from geotechnical engineering have shown that many important mechanical properties of clay such as plasticity [38], compressibility [18; 19; 20], swelling [22; 23; 24], and shear strength [25; 26] are directly related to the EDL force between clay particles as well as the fabric, i.e., particle arrangement. A reasonable calculation of the EDL force is essential to the deep understanding of the mechanical behavior of clay [25; 38]. Early such studies in geotechnical engineering also utilized a simplified particle configuration to represent the actual clay particles system, e.g., [PERSON] calculated analytically the inter-particle repulsive force based on a model that includes two infinite parallel separate charged plates and indicated that the experimental compressibility of pure clay can be predicted quantitatively well [18]. To investigate the macro-mesoscopic links in mechanical behaviors of soils taking the particle arrangement into account explicitly, the discrete element method (DEM) has been developed into a powerful tool as it was introduced into the field of geotechnical engineering four decades ago [39]. DEM research on clay has made considerable progress in recent years [20; 21; 22; 24; 26; 27].