Initiation of Sliding of an Elastic Contact at a Nanometer Scale Under a Scanning Force Microscope Probe

Abstract

A contact between a scanning force microscope Si₃N₄ probe and a flat surface of similar material is created. The low roughness of surfaces and their mechanical properties allow generating a nanoscopic elastic contact governs by an extended Hertz theory (DMT theory). The behavior of the initiation of sliding is investigated by submitting the contact to lateral sinusoidal displacements whose amplitude increases from zero to a few nanometers. The lateral force generated by the displacement is analyzed by a lock-in amplifier and the in-phase and out-of-phase components are recorded as a function of the displacement amplitude. Experimental results are compared to the Mindlin and Savkoor theories, which describe the initiation of sliding of macroscopic elastic contact. A relatively good agreement between our experiments and these theories is observed. For our particular experimental conditions, i.e., Si₃N₄ probe sliding on a similar material, the Mindlin’s model gives a slightly better agreement than the Savkoor’s model. This study shows that macroscopic concepts remain valid at the nanoscale, at least for the particular case studied here.

Appendix: Conditions for using DMT Theory and Amontons’s Law in a Nanoscale Contact

We seek to establish a contact that complies with both Hertz theory and Amontons’s law theories with a SFM probe. It is thus necessary that some assumptions are checked:

• 1. The contact must be perfectly elastic without plastic deformation. For that, it is necessary that the hardness of surfaces is large compared to the contact pressure.

• 2. Surfaces must be sufficiently flat to not modify the Hertzian geometry of the contact. The Hertz theory is based on surfaces without roughness that is difficult to valid at a nanometer scale. It is thus necessary to check that the surface roughness can be neglected. We propose to compare the roughness of the surfaces of the theoretical contact area with the theoretical indentation depth (Fig. 7). If the roughness of the theoretical contact area is weak compared to the theoretical indentation depth, the surface asperities are deformed elastically and that the contact is governed by the Hertz theory. On the other hand, if the roughness is large compared to the theoretical indentation depth, it is difficult to imagine that the surface asperities are deformed elastically to form a single asperity contact: The contact will thus be done between the asperities of the two surfaces. The Hertz law can thus be validated.

• 3. The presence of a capillary meniscus generates the presence of adhesive force that can form an adhesive neck. This neck increases the contact area and generates a contact that complies with the JKR theory. It is thus necessary to check that the elastic deformation is negligible compared to the range of the capillary forces.

• 4. Lastly, the Mindlin theory is based on a linear relation between the friction force and the normal force (Amontons’s law). It is thus necessary to check that the friction force is proportional to the normal force.

Fig. 7

Comparison between the indentation depth and the contact roughness. The theoretical indentation is calculated for a contact between a sphere and a flat surface and is compared with the total roughness in the same area. If the total roughness is well lower than the indentation depth, it becomes obvious that the asperities are flattened out and that the contact could be considered as a single asperity contact. At the opposite, if the total roughness is high compared to the indentation depth, the asperities are not flattened out and the contact should be considered as a polyasperities contact

1.1 Elasticity of the Contact

In order to confirm the elastic nature of the contact, the theoretical contact stress is compared to the hardness of the sample. If one supposes that the contact complies with the Hertz theory, the required force F _lim to have a maximal contact stress equal to the hardness H is:

$$ F_{{\lim }} = \frac{{\pi ^{3} H^{3} R^{2} }} {{6E^{\ast 2} }} $$

(22)

If we take the measured mechanical properties of the silicon nitride (H = 25 GPa, E* = 75.5 GPa taking ν = 0.27 [23]) and the nominal probe radius (R = 26 nm), the required force to generate plastic flow is equal to 9,600 nN. This force is well superior to the highest force used during the experiments and the hypothesis that the contact remains fully elastic is valid.

1.2 Validity of the Hertzian Theory for Rough Surfaces

We propose to compare the theoretical indentation depth with the roughness of the theoretical contact area. Indeed, roughness parameters are highly dependent on the considered area [43]. For large considered area, the roughness parameters like the Arithmetic Roughness R _A and the Mean Square Roughness R _MS are generally constant. For low considered area, the roughness parameter follows a power relationship to the considered area. The measurement of the surface roughness in the contact area is particularly difficult due to its very small value. The theoretical contact radius is of the order of a few nanometers. At this scale, it is difficult to measure accurately the topography for two reasons: first, the probe radius is big compared to the considered area leading to an artifact known as dilation [38]; second, for low roughness the noise-to-signal ratio becomes high. These two phenomena do not allow measuring the true topography of the surfaces. Nevertheless, we propose to estimate the roughness of the contact area by extrapolating the roughness measured at larger scales.

A topographic image of the silicon nitride surface was made using a sharp probe (TESP-NCL, Nanosensors, Switzerland) in order to reduce the dilation phenomenon. The so-called Soft Tapping imaging mode was used and servo loop parameters were fixed as low as possible to optimize the noise-to-signal ratio.

The parameter R _MS is then computed as a function of the considered area: The image is cut up in square part of decreasing size. For each part of the image, the R _MS is computed. Then, for all parts of the image of similar size, the median values of the R _MS are calculated and plotted as a function of the considered area.

As the height of the surface follows a Gaussian distribution, six times the RMS roughness corresponds to 99.7% of the distribution and could be considered as a good approximation of the value of the total roughness R _T (difference between the highest and the lowest point). Then, it is judicious to compare this value to the theoretical indentation depth h of a contact of same area A. The indentation depth is equal for a Hertzian contact to:

$$ h = \frac{{a^{2} }} {{2R}} = \frac{A} {{2\pi R}} $$

(23)

Figure 8 shows on the same graph the estimated roughness R _T = 6 R _MS and the theoretical indentation depth h of a contact area A as a function of its contact radius a (a = (A/π)^1/2). The indentation depth is superior to the roughness for a limit value of 2.7 nm. This value corresponds to an applied force equal to about 100 nN. The probability to generate a single asperity contact is superior to 1/2 for a force superior to 100 nN.

Fig. 8

Comparison between the theoretical indentation depth (black line) and the average total roughness by means of six times the RMS roughness (gray line) in a contact of area A as a function of the contact radius a. The roughness is computed from the inset SFM image. The theoretical indentation depth becomes superior to the total roughness for contact radius equals to approximately 2.7 nm. This contact radius is theoretically achieves for a total applied force close to 100 nN

Obviously, due to the very small-scale value of the contact area, the estimation of the roughness in this area is poorly accurate and underestimated by the high noise-to-signal ratio. For forces superior to 100 nN, the probability to have a single asperity contact superior to 1/2 is concluded.

1.3 Capillary Adhesion in an Elastic Contact

The Hertz theory showed its robustness in many experimental cases [44] including at the nanometer scale [2–7]. However, in some experimental cases, the adhesion force cannot be neglected. In ambient air, the presence of water generates the formation of a capillary meniscus around contacts [45–47], the main adhesion force is due to the attractive force generated by this capillary meniscus.

As a function of the ratio of the elastic strain with the range of the adhesion force, the JKR theory [47] or the DMT theory [48] governs the contact. Maugis [49] has studied the transition between the DMT and JKR theories using a Dugdale model [50] for which the depression due to adhesion σ₀ is constant if the distance between surfaces is lower than a threshold value and null beyond that. Maugis introduces a dimensional parameter λ:

$$ \lambda = 2\sigma _{0} = \root{3}\of{\frac{{9R}} {{16\pi wE^{\ast 2} }}} $$

(24)

and shows that the DMT and JKR theories are valid if λ < 0.1 and λ > 3, respectively. The Dudgale model is perfectly adapted to capillary condensation. If one modifies the parameter of Maugis to the case of capillary condensation (σ₀ = ΔP = −γ_LV/r _K and w = 2γ_LV), one obtains:

$$ \lambda = - \frac{1} {{r_{{\text{K}}} }}\root{3}\of{\frac{{9R\gamma ^{2}_{{{\text{LV}}}} }} {{4\pi E^{\ast 2} }}} $$

(25)

In the case of capillary adhesion, the numerical applications show that the Maugis criterion is close to zero for current relative humidity (30–60%) and current SFM probes (R = 10–40 nm). Exception occurs for very low Young’s modulus (E < 1 GPa) for which the Maugis or JKR theories are more adapted.

During experiments, the humidity was typically between 30 and 60% leading to a Kelvin’s radius between −1 and −2.3 nm, the nominal probe radius 20 nm and the combined Young’s modulus equal to 75.5 GPa. The calculation of the Maugis parameter λ applied to a capillary force has been done using Eq. 25 and a value between 0.010 and 0.024 has been found. This value shows us without ambiguity that the DMT theory is valid in our experimental conditions.

1.4 Validity of the Amontons’s Law

The relationship between the friction force and the normal force is measured by means of the present method with displacement half-amplitude well higher than the contact radius (d* ≫ D). At these displacement half-amplitudes, the value of the half-amplitude of the first harmonic of the lateral force signal is close to 4/π the value of the friction force [25, 28, 29]. The friction force by means of the half-amplitude of the lateral force divided by π/4 is proportional to the normal force (Fig. 9). The friction force is not equal to zero at snap-off. This phenomenon is due to the oscillating displacements that generate a premature snap-off: The adhesive force was measured to be well lower when the probe is submitted to a lateral oscillating displacement. The Amontons’s law is then valid in our experiments.

Fig. 9

Dynamic friction force as a function of the applied load. The friction force (by means of the half-amplitude of the lateral force divided by π/4) is proportional to the normal force but not equal to zero at snap-off. This phenomenon is due to the oscillating displacement that generates a premature snap-off