2.1 Principle of the viscoelastic measurement
A schematic of FWM is shown in Fig. 1. FWM can quantitatively measure the shear viscoelasticity of a liquid sample in a nanogap [21, 22]. Specifically, an optical fiber probe with a spherical tip was used as the shear-force-detecting probe. The diameter and length of the tip were approximately 200 µm and 2 mm, respectively. The probe was placed vertically on the sample substrate and vibrated in the in-plane direction by using a piezoelectric actuator. The probe tip sheared lubricating oil on the substrate. The shear force acting on the probe was measured by detecting fiber deflection using an optical technique [21]. We controlled the gap between the probe and substrate, where the oil was confined and sheared, using a piezo stage with a resolution of 0.1 nm. For viscoelastic measurements, we oscillated the probe sinusoidally and measured the amplitude and phase changes of the oscillations at the probe tip.
To determine the viscoelasticity from the measured values (amplitude and phase changes), we used the mechanical model shown in Fig. 2: The optical fiber probe was assumed to be a one-degree-of-freedom vibration system consisting of a mass (effective mass: \(m\)), spring (spring constant: \({k}_{p}\)), and damper (damping coefficient: \({c}_{p}\)). The sample was a Voigt model consisting of a parallel-connected spring (spring constant: \(k)\) and damper (damping coefficient: \(c)\). By solving the equation of motion, we obtained \(c\) and \(k\) as follows:
$$c=\frac{{a}_{0}}{X{\omega }}\left[\left({k}_{p}-m{{\omega }}^{2}\right)\text{tan}{\Delta }\delta +{c}_{p}{\omega }\right]\sqrt{\frac{{k}_{p}^{2}+{c}_{p}^{2}{{\omega }}^{2}}{\left(1+{\text{tan}}^{2}{\Delta }\delta \right)\left[{\left({k}_{p}-m{{\omega }}^{2}\right)}^{2}+{c}_{p}^{2}{{\omega }}^{2}\right]}}-{c}_{p}$$
1
,
$$k=\frac{{a}_{0}}{X}\left({k}_{p}-m{{\omega }}^{2}-{c}_{p}{\omega }\text{tan}{\Delta }\delta \right)\sqrt{\frac{{k}_{p}^{2}+{c}_{p}^{2}{{\omega }}^{2}}{\left(1+{\text{tan}}^{2}{\Delta }\delta \right)\left[{\left({k}_{p}-m{{\omega }}^{2}\right)}^{2}+{c}_{p}^{2}{{\omega }}^{2}\right]}}-\left({k}_{p}-m{{\omega }}^{2}\right)$$
2
,
where \({a}_{0}\) and \(\omega\) are amplitude and frequency of the forced oscillation by the piezo actuator, respectively. \(X\) and \(\varDelta \delta\) are the measured values of oscillation amplitude and phase shift, respectively. All other parameters (except \(X\) and \({\Delta }\delta\)) were determined prior to the shear viscoelasticity measurements. The values of 𝑐 and 𝑘 calculated using equations (1) and (2) include the effects of geometry, such as the contact area between the sample, probe tip, and sliding gap. In this study, we calculated the complex viscosity as the shear viscoelasticity. The complex viscosity is expressed as \({\eta }^{*}=\eta {\prime }-i\eta {\prime }{\prime }\), where the real part \(\eta {\prime }\)represents viscosity, and the imaginary part \(\eta {\prime }{\prime }\)represents elasticity. The relationship between these parameters is expressed as follows:
$$\eta {\prime }=\frac{c}{{\Omega }}$$
4
,
$$\eta {\prime }{\prime }=\frac{k/{\omega }}{{\Omega }}$$
5
,
where Ω is a geometric parameter, expressed as follows:
$${\Omega }=6{\pi }r\left[\frac{8}{15}\text{ln}\left(\frac{r}{h}\right)+E\right]$$
6
,
where \(r\) is the radius of the spherical tip and \(h\) is the minimum gap between the tip and the substrate. From the measured values 𝑋 and Δ𝛿 and using equations (2)–(6), we can obtain \(\eta {\prime }\) and \(\eta {\prime }{\prime }\). For simplicity, we referred to the former and latter as viscosity and elasticity, respectively. Please refer to ref. [21, 22] for the details of the FWM measurements.
2.2 Gap measurement by interferometry integrated in FWM
In previous FWM measurements, we experimentally determined the point at which a solid contact between the probe and substrate surface was made (solid contact point) and obtained the gap width as the displacement of the piezo stage from the solid contact point, which is considered the origin of the gap. This gap determination method was effective when the liquid sample consisted only of the base oil. Since lubricant oil containing adsorptive polymer was used in this study, a polymer adsorption layer was assumed to have formed on the surface, obstructing the detection of solid contact. Therefore, we introduced an optical interferometer [23] into the FWM to measure the gap between the probe tip and substrate. Specifically, as shown in Fig. 1, a laser beam was introduced into the probe from the end opposite to the spherical tip. The lights reflected from the end face of the tip ball and substrate served as the reference and object lights, respectively, and the gap was determined by measuring the intensity of their optical interference.
Figure 3(a) shows the relationship between the gap and optical interference intensity measured for the lubricant without polymer additives. The interference intensity varied sinusoidally as the gap changed and showed a discontinuous bending point at the point where solid contact was made. This is because owing to the solid contact, the gap does not change even if the piezo stage is displaced. The same phenomenon was observed for the polymer adsorption film on the solid surface when the sample was a lubricant containing a polymer additive (PAMA, molecular weight of 60,000), as shown in Fig. 3(b). However, the gap at which the bending point was measured had a finite value (non-zero value). This value was considered to be the thickness of the polymer adsorption film that was not discharged from the gap. Therefore, optical interferometry can be used to identify the gaps and estimate the thickness of the adsorbed film. The gap measurement accuracy was approximately 1.03 nm based on the standard deviation of the measured optical interference intensity. Figure 4 shows the adsorption film thicknesses of PAMA with different molecular weights. These results were in good agreement with those of previous studies that used a vertical ellipsometric microscope [18].