Relationship between contact size and static friction: An approach for rigid crystalline surfaces

10.1007/s40544-019-0352-9.F001 Fig. 1Commensurate contact between two identical crystalline surfaces. The lattices on two contact surfaces are shown in black dots (lower), red dots (upper) and blue dots (overlapped), respectively. (a) The trivial commensurate case in square lattice. In this case, the orientation is exactly the same (θ = 0°). The grey square area is the smallest repeat unit. (b, c) A nontrivial commensurate case in square/hexagonal lattice. The grey square area in the center is obviously repeated periodically (attention: the smallest repeat unit is the smaller grey square area shown in the upper left corner). The case of (a, b, c) = (3, 4, 5)/(3, 5, 7) is shown here.

10.1007/s40544-019-0352-9.F002 Fig. 2Simulated systems. The top view of two contact surfaces is shown here. The lower surface is colored in red and the upper surface is colored in green. (a-c) Cases in square lattice (both surfaces are <100> planes of the FCC lattice), (d, e) cases in hexagonal lattice (both surfaces are <0001> planes of the HCP lattice). For each system, both surfaces contain three layers of atoms. The detailed size of the upper surface is listed in Table S2 (in the Electronic Supplementary Material (ESM)). The size of the lower surface is slighted larger to ensure that the minimum distance between two adjacent upper surfaces under periodic boundary conditions is greater than the cutoff distance 2.5σ.

10.1007/s40544-019-0352-9.F003 Fig. 3ΔE_unit(θ) versus M of commensurate systems with a single unit cell. ΔE_unit(θ) is the corrugation of per-atom potential energy and M is number of atoms in contact area. Various sizes of commensurate systems with a single unit cell (the details are listed in Table S1 in the ESM) are calculated. Evidently, there is a linear relationship in double logarithmic plots, which means ΔE_unit(θ) $\propto$ N^C.

10.1007/s40544-019-0352-9.F004 Fig. 4ΔE(θ, A, S) of commensurate systems with FCC lattice. r is ratio of ΔE(θ, A, S) to ΔE_unit(θ), N is number of atoms in contact area, and M = N/c is number of minimum repeat units in contact area. (a-c) show the effects of shape and size of the contact area, (d, e) show the effect of the size c of the unit cell, and (f) shows the effect of misfit angle θ. Obviously, when N becomes larger, r gradually converges to 1 in all cases. Moreover, for irregular shapes, large c and smaller θ (θ ≤ 10°) lead to slower rate of convergence. The rate is inversely proportional to c and is more sensitive to shape and θ.

10.1007/s40544-019-0352-9.F005 Fig. 5Geometric configuration of the contact surfaces. The PES between two crystalline surfaces E = F(x, y) is a superposition of a series of PESs between one atom and the crystalline surface E = ΣE_i = Σf_i(x, y). Due to the periodicity of the crystalline surface, these superposed PESs are the same functions but with difference phases (f_i(x, y) = f(x+Δx_i, y+Δy_i) and we term (Δx_i, Δy_i) as the phase for atom i). One example is shown in (a), which is a commensurate contact with misfit angle θ = sin^-1(3⁄5) = 36.87° and size A = c × c = 5 × 5 (in reduced unit). The left side of (a) is the top view of the contact surfaces and the atoms are colored by black (lower), red (upper), and blue (overlapped), respectively. The right side of (a) shows the distribution of the phases. There are only c = 5 different phases (shown in red dots) and they can correspond to c adjacent atoms on a straight line. (b) shows an incommensurate contact with θ = 45° in the same manner. Evidently, the phases are distributed on a series of lines parallel to the X axis. (c) shows phase distributions of two incommensurate contact with the same misfit angle, similar size but different shapes. Here θ = 28° is very closed to θ = sin^-1(8/17) = 28.07°, which is a commensurate contact with c = 17. Therefore, the points in the phase space are distributed into 17 clusters when the size is not too large. But for points in each cluster, the distribution is square when the shape of the contact area is square, and fan-shaped when the shape of the contact area is circle. Obviously, the latter is more biased.

10.1007/s40544-019-0352-9.F006 Fig. 6ΔE(θ, A, S) of incommensurate systems with FCC lattice. (a-c) are double logarithmic plots between ΔE(θ, A, S) and N with different misfit angles and shapes. Clearly, there are ΔE(θ, A, S) $\propto$ N^C in all cases. (d) shows the distribution of the exponent C. In most cases, C is around -1 for regular shapes (square and rectangle) and C is around -0.75 for irregular shapes (circle). In very few cases of regular shapes, C is around -0.5. (e) shows double logarithmic plots between ΔE(θ, A, S) and N in three specific cases (θ = 30°/45° in FCC lattice and θ = 30° in HCP lattice) with C ≈ -1/2. (f) shows the distribution of the exponent C when any one of sinθ, cosθ, and tanθ is rational. Clearly, the expected cases of C = -1/2 for regular shapes or C = -1/4 for irregular shapes do not occur.

10.1007/s40544-019-0352-9.F007 Fig. 7Comparisons between commensurate contact and incommensurate contact with small difference in θ. ΔE_unit(θ) versus N is shown for commensurate contact and ΔE(θ, A, S) versus N is shown for incommensurate contact. (a) is the comparison between θ = sin^-1(3⁄5) = 36.870° (commensurate) and θ = 37° (incommensurate). (b) is the comparison between θ = sin^-1(39⁄89) = 25.989° (commensurate) and θ = 26° (incommensurate). When N is small, only in the case of A, there is a plateau of ΔE(θ, A, S) in incommensurate contacts and the plateau is closed to ΔE_unit(θ) of the commensurate contact. When N is large enough, in both cases there is a significant difference in ΔE(θ, A, S) between commensurate and incommensurate contact.