Mechanics of Deformation under Traction and Friction of a Micrometric Monolithic MoS₂ Particle in Comparison with those of an Agglomerate of Nanometric MoS₂ Particles

Abstract

Tribology of small inorganic nanoparticles in suspension in a liquid lubricant is often impaired because these particles agglomerate even when organic dispersants are used. In this paper we use lateral force microscopy to study the deformation mechanism and dissipation under traction of two extreme configurations (1) a large MoS₂ particle (~20 μm width) of about 1 μm height and (2) an agglomerate (~20 μm width), constituting 50 nm MoS₂ crystallites, of about 1 μm height. The agglomerate records a friction coefficient which is about 5–7 times that of monolithic particle. The paper examines the mechanisms of material removal for both the particles using continuum modeling and microscopy and infers that while the agglomerate response to traction can be accounted for by the bulk mechanical properties of the material, intralayer and interlayer basal planar slips determine the friction and wear of monolithic particles. The results provide a rationale for selection of layered particles, for suspension in liquid lubricants.

Appendix: Calculation of Mode II Stress Intensity Factor

We model a particle being scratched by a conical indenter of β half cone angle under a normal load P (Fig. 12) we assume that slip occurs in a direction parallel to the scratch direction, as a mode II fracture. We situate a crack of length ‘2c’ in the slip plane at a variable depth δ and determine the δ* where K _II = K _IIc. Noting that there is a compressive stress σ_z which is a monotonic function of the normal load which decays in moving normally, and which acts normal to the crack length, we expect δ* to vary as a function of normal load.

Fig. 12

Configuration of a ‘2c’ length elliptical crack in the particle bulk

We have estimated K _II for sliding of the layers under both compressive and shear stress assuming a pre existing flaw of 0.1 μm at the interface. The stress intensity factor is calculated assuming the pre-existing crack is under a pure shear (σ = 0) and the interfacial shear strength of MoS₂ = 25 MPa [42].

For a conical indenter, Normal stress is given by,

$$ \sigma_{\text{z}} = - {\frac{{p_{\text{m}} }}{2}}\left[1n\left(1 + {\frac{1}{{\zeta^{2} }}}\right) + {\frac{2}{{1 + \zeta^{2} }}}\right] $$

(21)

where, p _m and $\zeta$ are defined in Eqs. 6 and 8, respectively.

Let us assume a crack of length ‘2c’ at the interface of two layers under the action of a uniform normal compressive stress σ_z and a uniform shear stress τ. The mode II stress intensity factor, K _II, for propagation of this crack can be written [43]

$$ K_{\text{II}} = {\frac{1}{{\sqrt {\pi c} }}}\left[{\frac{\pi }{2}}\tau c - {\frac{k - 1}{k + 1}}\sigma_{\text{z}} c\right] $$

(22)

where, (a) k = 3 − 4 v, for plane strain and (b) k = (3 − v)/(1 + v), for plane stress.

Figure 13 shows that, with increase in normal load, we can shear more material from peak of the particle, where K _II becomes equal to K _IIC. However, the figure gives only a qualitative estimate of the stress intensity factors and this validates the experimental finding of material removal qualitatively (Fig. 5).

Fig. 13

K_II as a function of depth at different loads. K_II reaches K_IIC (dotted line) at a much higher depth as the normal load is increased. Inset: Theoretical graph showing the depth of material for K_II = K_IIC as a function of normal load