Inuence of particle geometry on dispersion force Yifei Liuyiumy.swjtu.edu.cn Heping Xie and Cunbao Liyycunbao.liszu.edu.cn

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Inﬂuence of particle geometry on dispersion force

Yifei Liu∗

∗yﬂiu@my.swjtu.edu.cn

, Heping Xie, and Cunbao Li†

†cunbao.li@szu.edu.cn

Shenzhen Key Laboratory of Deep Underground Engineering Sciences and Green Energy,

Shenzhen University, Shenzhen 518060, China and

Guangdong Provincial Key Laboratory of Deep Earth

Sciences and Geothermal Energy Exploitation and Utilization,

College of Civil and Transportation Engineering,

Shenzhen University, Shenzhen 518060, China

Dong-Sheng Jeng

School of Engineering and Built Environment,

Griﬃth University Gold Coast Campus, Queensland 4222, Australia

Bo Nan Zhang

School of Nuclear Science and Technology,

Lanzhou University, Lanzhou 730000, China

(Dated: October 20, 2022)

arXiv:2210.10079v1 [cond-mat.mes-hall] 18 Oct 2022

Abstract

Dispersion forces (van der Waals force and Casimir force) originating from quantum ﬂuctuations

are crucial in the cohesion of microscale and nanoscale particles. In reality, these particles have a

variety of irregular shapes that diﬀer considerably from any idealized geometry. Previous experi-

ments have demonstrated that dispersion forces strongly depend on the geometry. Because of the

nonadditivity of these forces, commonly used numerical additive methods, such as the Hamaker

and Derjaguin approximations, are not suitable for calculations with complex geometries. More-

over, experimental studies are diﬃcult to identify the contributions of the dispersion force from

the many forces that constitute the cohesion. Therefore, no general law about the inﬂuence of

particle geometry on dispersion forces has been established. Thus, in this paper, the ﬂuctuat-

ing surface current (FSC) technique, an exact scattering theory-based nonadditive algorithm, was

used to study this inﬂuence. To characterize complex geometries, a data-adaptive spatial ﬁltering

method was introduced to perform scale decomposition, and descriptors at three observation levels

(global, local, and surface) were used. Based on the advanced geometric analyses and accurate

numerical calculations, the inﬂuence of multiscale surface ﬂuctuations on dispersion forces was de-

termined. Furthermore, a convenient formula for predicting the dispersion forces between particles

with complex shapes from the exact Lifshitz solution was established via multistage corrections.

Keywords: Particle geometry; Dispersion force; Spherical Empirical Mode Decomposition; Nonadditivity;

Fluctuating surface current algorithm.

I. INTRODUCTION

Cohesion plays a key role in determining the behaviors of particles and particle systems

ranging from the nanoscale to the microscale. The dispersion force is one of the main sources

of cohesion. In nearly dry and uncharged particle systems with particle sizes less than 10

micrometers, dispersion forces dominate [1–4].

In diﬀerent historical stages, the dispersion force has had various names. The earliest

phenomenological name of the dispersion force was the ’van der Waals force’ [5], followed by

the ’London force’ [6] and ’Casimir force’ [7]. However, in the 1950s, Lifshitz realized that

these forces all originated from quantum ﬂuctuations, and a uniﬁed theory was developed

[8]. This theory can be applied to reproduce the van der Waals force (vdW force) and

Casimir force as limiting cases of small and large separations [9]. In this study, we use the

general term ‘dispersion force’ to refer to these forces with the same origin [10, 11]. In reality,

almost all particles have various irregular shapes that diﬀer considerably from any idealized

geometry, including both natural and artiﬁcial particles. Moreover, it has been proven

both theoretically and experimentally that dispersion forces depend strongly on geometry,

and changes in dispersion forces caused by geometric changes can reach several orders of

magnitude. [12–15]. Therefore, it is of signiﬁcance to quantify the inﬂuence of particle

geometries on dispersion forces.

For objects with arbitrary shapes, including objects made from idealized materials, the

dispersion force is challenging to calculate analytically [16]. At present, the most eﬀective

methods for determining the inﬂuence of geometry on the dispersion force are physical exper-

iments and numerical calculations. Diﬀerent experimental approaches have been developed

in ﬁelds where these forces play important roles. Many experimental studies are based on

two main types of measuring equipment: the surface force apparatus (SFA) and atomic

force microscope (AFM). The SFA technique was developed by Israelachvili [17] and can

be applied to measure the force between two macroscopically curved surfaces over relatively

large areas with angstrom resolution. Some researchers have studied the eﬀect of surface

roughness on contact mechanics with SFA approaches [18–20]. However, because the mea-

surements are generally carried out under loading conditions, it is diﬃcult to extract the

contribution of dispersion force from the various forces that make up the surface force.

AFM can accurately measure the interaction force between two microparticles [21]. Mo-

hideen and Roy [22] ﬁrst precisely measured the dispersion force between a metal sphere and

plate with AFM. An accurate measurement involves strictly excluding the contributions of

forces other than the dispersion force, which is typically very diﬃcult [14]. Therefore, at

present, accurate measurements are limited to a few special geometries (sphere and plate)

[23, 24]. Although some studies have investigated particles with complex morphologies

[25, 26] with AFM, the environment was not strictly controlled, and the adhesion or co-

hesion was measured instead of the dispersion force. In addition, in the ﬁeld of chemical

engineering, the centrifugal method has been used to study the eﬀects of particle morphology

on cohesion and adhesion [27, 28]. However, the centrifugal method is a rougher technique

than SFA and AFM. Additionally, cohesion and adhesion were measured in this study as

opposed to the dispersion force. Recently, in microelectromechanical systems (MEMS), spe-

cial equipment has been developed to quantify the eﬀect of the boundary geometry on the

dispersion force [15, 29–32]. However, the research objects in these studies were artiﬁcial

plates with relatively regular surfaces (rectangular silicon gratings and corrugated plates)

rather than real particles with complex morphologies. Thus, in summary, previous experi-

mental studies mainly focused on how particle geometry inﬂuences adhesion or cohesion and

did not precisely determine the eﬀect of the dispersion force. Critical experiments on the

dispersion force are limited to a few special geometries and materials. Therefore, a general

law about the inﬂuence of geometry on the dispersion force has not yet been established due

to the lack of experimental data on real particles with complex morphologies.

Other eﬀective methods include numerical calculations. Thus far, two approximate al-

gorithms have been commonly applied because of their simplicity: pairwise summation ap-

proximation (PWS) and proximity force approximation (PFA) [33–35]. PWS, which is also

known as Hamaker summation [36], calculates the dispersion force between two macroscopic

bodies according to the pairwise summation of volumetric elements interacting through the

vdW-Casimir force, which can be established based on dipolar dispersive interactions. PFA,

which is also known as the Derjaguin approximation, models the interaction between nearby

surfaces as additive line-of-sight interactions between inﬁnitesimal, planar surface elements

(computed via the Lifshitz formula [37]). Both algorithms are based on the assumption of

additivity, with the force calculated by simply summing the force contributions of the surface

or dipole interactions. Unfortunately, a concise and convenient assumption always comes

at the expense of accuracy. In fact, the interaction between two dipoles/surfaces is inﬂu-

enced by nearby dipoles/surfaces, which is ignored in the additivity assumption. Moreover,

the multiple scattering caused by multibody interactions cannot be ignored for condensed

matter. The discrepancy between additive approximations and exact calculations is often

referred to as nonadditivity. A detailed explanation of nonadditivity can be found in the

review by Rodriguez et al. [14]. Nonadditivity considerably reduces the accuracy of PWS

and PFA techniques for handling condensed matter with complex morphology, which has

been conﬁrmed by several theoretical and experimental studies [13, 15, 38–40]. Therefore,

these two additivity methods cannot be applied to study the inﬂuence of geometry on the

dispersion forces of real particles.

Obviously, the interactions among each dipole are impossible to calculate except at the

subnanometer scale [40, 41]. As previously mentioned, dispersion forces originate from quan-

tum ﬂuctuations. In terms of ﬁeld ﬂuctuations, the issues caused by nonadditivity can be

addressed. This kind of algorithm originated in the 1950s [37] and was designed to work for

one special geometry. Since 2007 [42], dramatic progress has been made in this ﬁeld, and

general-purpose schemes for arbitrary materials and various geometric conﬁgurations have

been developed [43]. These schemes can be roughly divided into two categories according

to the choice of physical basis: fully quantum mechanical approaches and semiclassical ap-

proaches. The former include the path integral (or scattering) approach [42, 44] and the

lattice ﬁeld approach [16, 45], which are highly eﬃcient in certain geometries but not ﬂex-

ible enough for general geometries, especially geometries with sharp corners [43]. In the

semiclassical approach (also known as the stress tensor approach) [46, 47], the computation

of the ﬂuctuation force can be reduced to solving the classical electromagnetic scattering

problem according to the ﬂuctuation-dissipation theorem (FDT). Thus, many classical elec-

trodynamics methods can be applied after a few important modiﬁcations (detailed in reviews

Rodriguez et al. [14] and Johnson [48]). In addition to these two schemes, the ﬂuctuating

surface current (FSC) technique (which will be introduced in detail in Section §II B) devel-

oped by Reid et al. [49] can be regarded as a third scheme that combines the advantages

of the previous two approaches. In these nonadditivity algorithms, interactions among in-

stantaneous dipoles are not considered and are replaced by the global optical response of

the material, which can be described by macroscopic dielectric functions. Based on the

dielectric functions of the materials, the dispersion force can be accurately calculated by

considering the complex electromagnetic modes and surface scattering properties [33]. The

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InuenceofparticlegeometryondispersionforceYifeiLiuyiu@my.swjtu.edu.cn,HepingXie,andCunbaoLiyycunbao.li@szu.edu.cnShenzhenKeyLaboratoryofDeepUndergroundEngineeringSciencesandGreenEnergy,ShenzhenUniversity,Shenzhen518060,ChinaandGuangdongProvincialKeyLaboratoryofDeepEarthSciencesandGeothermalEnergyE...

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