1 Introduction

To improve energy-saving performance and reduce environmental impact, automotive lubricants exhibiting higher performance and conforming to more stringent specifications than ever before should be selected. To improve energy-saving performance, lubricant viscosity must be lowered [1, 2]. Zero sulfated ash, phosphorous, and sulfur (SAPS) lubricants are desirable owing to their environmental friendliness [3, 4]. In the future, lubricating oil is expected to be used in cooling motors in electric vehicles. Therefore, suppressing metal corrosion and providing insulation is necessary [5]. Polymer additives have attracted considerable attention to meet these requirements.

Polymer additives have been used as viscosity index improvers for decades [6,7,8]. Viscosity index improvers are used to control the temperature dependence of lubricant viscosity and maintain an appropriate viscosity range within the expected operating temperature. This is necessary to maintain an oil film under hydrodynamic lubrication conditions. Additionally, according to previous reports, polymer additives adsorbed on the surface contribute to oil film retention during hydrodynamic lubrication and to low friction during mixed and boundary lubrication [9,10,11,12,13,14,15,16,17]. Use of a lubricating oil with low viscosity for improving energy efficiency results in decreased load capacity and narrowed sliding gaps. This facilitates the transition from hydrodynamic to mixed and boundary lubrication. Therefore, the friction-reduction properties of polymer-adsorption films have attracted considerable attention. Simultaneously, polymers have a high degree of freedom in their molecular structure; therefore, the development of SAPS-free and non-corrosive additives with controlled insulating properties is expected.

Understanding the lubrication mechanisms is essential for the design of highly functional polymer additives. However, since the polymer adsorption film is expected to be thin (< 100 nm), the gap in which it predominantly contributes to friction reduction is < 100 nm. Thus, to elucidate the lubrication mechanism, an approach that captures nanoscale phenomena is required. Yamashita et al. verified the friction properties of polyalkylmethacrylate (PAMA) as a friction modifier (FM) in the temperature range 23–100 °C using neutron reflectometry (NR) measurements, a mini-traction machine (MTM) tribometer, and atomic force microscopy (AFM). The results showed that the adsorbed layer of PAMA contributed to low friction even at high temperatures [17]. Song et al. used NR measurements and vertical ellipsometric microscopy (VEM) to verify the effect of the molecular weight and polarity of PAMA on the conformation of films adsorbed on metal surfaces and their ability to form adsorbed films. Their results indicated that owing to an increase in the polarity of the polymer, the first and second layers may be strongly and loosely adsorbed on the surface, respectively [18, 19].

Although the thickness and friction properties of PAMA-adsorbed films can be investigated using VEM, NR measurements, and AFM, quantifying their mechanical properties is necessary for the development of polymer additives. The mechanical properties of polymer adsorption films, polymer brush films, and confined oil films have been investigated using surface force apparatus (SFA), and many unique properties specific to nano-thicknesses have been reported [20,21,22,23,24,25,26]. It is assumed that the mechanical properties of the PAMA-adsorbed films are similar; however, they are not yet fully understood. In particular, a method for measuring temperature dependence has not been established. In our previous study, we successfully measured the shear viscoelasticity of lubricants (base oil only) sheared in nanogaps using an originally developed measurement method: the fiber wobbling method (FWM) [27]. This study aimed to extend the FWM and measure the shear viscoelasticity of polymer-added lubricants. In addition, a heating stage applicable to FWM measurements was established to clarify the temperature dependence of viscoelasticity. PAMA, whose effects on oil film retention and friction reduction have been reported in previous studies, was used as the sample in this study.

2 Shear Viscoelastic Measurement by Using FWM

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 [28, 29]. Specifically, an optical fiber probe with a spherical tip was used as the shear-force-detecting probe. The probe material was quartz glass and a laser was used to melt the tip to form a spherical shape. The surface of the sphere, which served as the sliding surface, was smooth, with a root mean surface roughness of approximately 0.2 nm [28]. 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 [28]. We controlled the gap between the probe and substrate, where the oil was confined and sheared, using a piezo stage. The resolution of the piezo stage was 0.1 nm. However, as described in the next section, the control accuracy of the shear gap was approximately 1 nm, owing to the accuracy of the gap measurement method using optical interferometry. To measure the viscoelasticity of the lubricating oil, a drop of oil was placed on the substrate and the probe tip was completely immersed in the drop (see inset in Fig. 1). The volume of the droplet was adjusted such that the probe tip sphere was just immersed (a few microliters). Because the tip was completely immersed in oil, the state of contact between the probe and sample was easily reproducible. The probe was sinusoidally vibrated using a piezo-actuator, and the amplitude and phase of the probe tip vibration were measured. The shear viscoelasticity of the lubricant sheared between the probe tip and substrate was quantified from the rate of decrease in the amplitude and phase delay relative to the focus vibration of the probe. The volume of the oil droplets was measured using a micropipette to an accuracy of 0.1 µl. Slight variations in the droplet volume had no significant effect on the measurement results because the shear force applied to the probe occurred predominantly in the narrow gap between the probe and substrate. Furthermore, the inertial force of the oil was negligible compared to the measured shear force because the amplitude of the vibration was less than 100 nm. As a preliminary validation in the development of FWM, we intentionally reduced the volume of the oil droplet to a level at which the tip of the probe was half-immersed (this was not the case in the actual measurement) and verified the difference from the fully immersed case. As a result, we confirmed that the difference in viscosity measured in the nanogap was less than a few percent.

Fig. 1
figure 1

Equipment configuration of the fiber wobbling method (FWM)

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To determine the viscoelasticity from the measured values (amplitude and phase changes), we used the mechanical model shown in Fig. 2. In our previous study, we confirmed that an optical fiber probe can be assumed to be a one-degree-of-freedom vibration system comprising of a mass (effective mass: \(m\)), spring (spring constant: \({k}_{p}\)), and damper (damping coefficient: \({c}_{p}\)), if the oscillation frequency is below the resonance frequency of the first-order mode [29]. 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:

$$c = \frac{{a_{0} }}{{X{\upomega }}}\left[ {\left( {k_{p} - m{\upomega }^{2} } \right)\tan {\Delta }\delta + c_{p} {\upomega }} \right]\sqrt {\frac{{k_{p}^{2} + c_{p}^{2} {\upomega }^{2} }}{{\left( {1 + \tan^{2} {\Delta }\delta } \right)\left[ {\left( {k_{p} - m{\upomega }^{2} } \right)^{2} + c_{p}^{2} {\upomega }^{2} } \right]}}} - c_{p}$$
(1)
$$k = \frac{{a_{0} }}{X}\left( {k_{p} - m{\upomega }^{2} - c_{p} {\upomega }\tan {\Delta }\delta } \right)\sqrt {\frac{{k_{p}^{2} + c_{p}^{2} {\upomega }^{2} }}{{\left( {1 + \tan^{2} {\Delta }\delta } \right)\left[ {\left( {k_{p} - m{\upomega }^{2} } \right)^{2} + c_{p}^{2} {\upomega }^{2} } \right]}}} - \left( {k_{p} - m{\upomega }^{2} } \right)$$
(2)

where \({a}_{0}\) and \(\omega\) are amplitude and frequency of the forced oscillation by the piezo actuator, respectively. \(X\) and \(\Delta \delta\) 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 Eqs. (1) and (2) include the effects of geometry, such as the contact area between the sample, probe tip, and sliding gap. In the measurement system in this study, the probe tip sphere was completely immersed in a liquid, in which case the geometric parameter \(\Omega\) is expressed as follows [29]:

$$\Omega =6\uppi r\left[\frac{8}{15}\text{ln}\left(\frac{r}{h}\right)+E\right]$$
(3)

where \(r\) is the radius of the spherical tip and \(h\) is the minimum gap between the tip and substrate. \(E\) is the correction term obtained by Goldman et al. by comparing the results obtained from Eq. (3) with those obtained from numerical calculations (\(E=0.9588\)) [29]. Strictly speaking, the area where the probe tip sphere was connected to the cylindrical portion was not in contact with the liquid, and there was a difference between the geometric parameter model and experiment. However, in our previous study, we confirmed that the viscosity at a sufficiently wide shear gap measured using the above-mentioned procedure agreed with that measured using a general-purpose rotational viscometer [27, 28]. Therefore, we consider the influence of this difference as negligible.

Fig. 2
figure 2

Mechanical model used in shear viscoelasticity measurement by FWM

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We calculated the complex viscosity as the shear viscoelasticity from c and k using \(\Omega\). The complex viscosity is expressed as \(\eta^{*} = \eta^{\prime} - {\text{i}}\eta^{\prime\prime}\), where the real part \(\eta^{\prime}\) represents viscosity, and the imaginary part \(\eta^{\prime\prime}\) represents elasticity (\(\text{i}\) is the imaginary unit). The relationship between these parameters is expressed as follows:

$$\eta {\prime}=\frac{c}{\Omega }$$
(4)
$$\eta {\prime}{\prime}=\frac{k/\upomega }{\Omega }$$
(5)

From the measured values X and Δδ and using Eqs. (1)–(5), 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. [28, 29] 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 the 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 [30] 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 3a 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. In the solid contact defined here, 1–2 molecules of the base oil may remain in the gaps. These are known as the hard wall layers [25, 26]. However, their thickness is generally of the order of 1 nm, which is approximately the same as the surface roughness of the sliding surfaces in this study. Therefore, the hard wall layer and solid contact could not be strictly distinguished in our experimental system. Therefore, for simplicity, the term “solid contact” is used. The bending point was also 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. 3b. 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 developed optical interferometer was calibrated using a spectral interferometric displacement meter (SI-F01, Keyence) with a guaranteed accuracy. The displacement detection limit of our interferometer, defined as a signal-to-noise ratio of 1, was 1.03 nm. Figure 4 shows the adsorption film thicknesses of PAMA with different molecular weights (20,000, 60,000, and 90,000). The details of the lubricant samples containing PAMA are described in Sect. 4.1. The PAMA used in this study is a linear molecule (Fig. 10). When the molecular weight is 20,000 (20 k), the length of the main chain is 20–30 nm and the chain diameter is approximately 2–3 nm. The thickness of the adsorbed film was approximately 3–4 nm, suggesting that the PAMA molecules were flattened and adsorbed onto the substrate surface. The adsorption film becomes thicker as the molecular weight increases because the degree of freedom of the molecular morphology increases. These results were in good agreement with those of previous studies that used a vertical ellipsometric microscope [18].

Fig. 3
figure 3

Interference intensities measured by optical interferometry when the gap between the probe and substrate was narrowed; the sample in the gap was a base oil only and b lubricating oil with a polymer additive. c Relationship between the gap and displacement of the piezo stage in the case of a lubricant containing a polymer additive calculated from the measurement results in (b)

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Fig. 4
figure 4

Thickness of adsorbed films of PAMA polymer additives with different molecular weights (20,000, 60,000, and 90,000) estimated from optical interferometry measurements

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3 Development and Evaluation of Heating Stage for FWM

3.1 Requirements and Concept of the Heating Stage

To measure the temperature dependence of shear viscoelasticity, a sample heating stage is essential. However, a general-purpose heating stage cannot be used for experiments in which the nanogap between the probe and the substrate surface needs to be controlled. This is because the surface of the heating stage expands as the temperature increases, causing the surface position of the sample to fluctuate. For example, a 10 mm-thick steel plate with a linear expansion coefficient of 10 × 10–6 °C−1 expands by 10 nm with a temperature rise of 0.1 °C. Temperature fluctuations of 0.1 °C can easily occur in a normal experimental environment with a slight air flow or convection caused by heating.

Therefore, we developed a heating stage that can suppress positional fluctuations of the sample substrate surface, as shown in Fig. 5a. An aluminum nitride heater was used for heating. A thermocouple was attached to the heater to monitor the heating temperature for PID control. Gallium (Ga) was interposed between the heater and the sample substrate on which the lubricant was dropped. The melting point of gallium is 27.96 °C. Thus, Ga was in liquid form when the lubricant was heated. Therefore, as shown in Fig. 5b, the thermal expansion of the heater did not affect the fluctuation of the substrate position owing to the deformation of liquid Ga. A quartz glass substrate was used owing to its extremely low thermal expansion coefficient. A stainless steel thin film was deposited by sputtering on one side of the surface, where the lubricant was dropped. The thickness of the stainless-steel film was < 100 nm; therefore, its effect on thermal expansion can be ignored. The surface of the substrate in contact with Ga was not coated because liquid Ga has high wettability to the glass surface, allowing uniform heating of the contact area. The size of the heating area could be adjusted by varying the amount of Ga. In addition, since the thermal conductivity of Ga is relatively high, the lubricant on the substrate was heated to almost the same temperature as that of the heater. Heat from the heater was also transferred to the fixtures below the heater and to those holding the substrate. The thermal expansion of these parts also changes the position of the substrate surface. To reduce this effect, we used fixtures made of Super Invar steel with heat insulators at the joints between the fixtures.

Fig. 5
figure 5

Schematic of the developed heating stage. a Gallium suppresses the effects of the thermal expansion of the heater with its flexible deformation to maintain the position of the substrate surface at a constant value. b Optical interference displacement meter for measuring the positional variation of the substrate surface

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3.2 Evaluation of the Developed Heating Stage

As shown in Fig. 5b, an optical interference displacement meter (SI-F01, Keyence) was used to measure the positional variation of the substrate surface during the heating stage developed in this study. The accuracy of the displacement meter was 1 nm. The temperature of the measuring surface at which the displacement meter guarantees the accuracy was 80 °C. Therefore, the heating temperatures were 40, 60, and 80 °C. Figure 6a–d show the results of the time variation of the substrate surface position. Heating commenced at t = 0 s. Figure 6b–d show enlarged views of the positional variations after reaching steady state (after t = 5000 s). The resolution of the displacement meter (1 nm) resulted in discrete measurements. The positive direction of the vertical axis in each figure indicates the direction in which the substrate surface approached the probe. As shown in Fig. 6a, immediately after heating, the stage surface position contracted in the direction opposite to that of thermal expansion. This shrinkage was due to the reduction in the volume of solid Ga as it liquefied upon heating (liquid Ga is one of the “anomalous liquids” that, like water, increases in volume as it transforms into a solid). After a certain time, the thermal expansion of the quartz glass substrate and fixture supporting the substrate became dominant, and the substrate surface was displaced in the direction of expansion. The magnitude of displacement due to thermal expansion of the developed stage was < 200 nm at 80 °C, which was < 1/50th of the displacement observed with the commercial Peltier temperature-controlled stages used in our previous study [27]. These results demonstrated the effectiveness of the heating method that used liquid Ga. In our shear viscoelasticity measurements, rather than the absolute value of the thermal expansion, the positional variation caused by maintaining the sample temperature constant under heating conditions affected the measurements. According to the results shown in Fig. 6b–d, after the thermal expansion reached a steady state, the position fluctuations were successfully suppressed to within ± 1.5 nm maximum. Although only discrete measurement results were obtained owing to limitations in the accuracy of the displacement meter, we calculated the standard deviation of the measured values. The results, summarized in Table 1, confirmed that the position fluctuation was independent of the heating temperature. Therefore, the position fluctuation was unlikely to be caused by thermal expansion and was considered a random noise in the displacement measurement. As described below, the maximum heating temperature for the shear viscoelasticity measurements was 100 °C. From the results presented in Table 1, the positional fluctuations of the stage surface were considered to be suppressed to ≤  ± 1.5 nm, even at 100 °C.

Fig. 6
figure 6

Measurement results of the substrate surface position when temperatures are set to 40, 60, and 80 °C with the developed heating stage. a Measurement results immediately after heating until the surface position stabilized, and enlarged images of the measurement results at b 40 °C, c 60 °C, and d 80 °C when the surface position became constant

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Table 1 Standard deviation of measurements from Fig. 6b–d after the substrate position was stabilized
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An infrared camera was used to calibrate the heating temperature. Figure 7 shows the relationship between the heater and substrate surface temperatures measured by the thermocouples and infrared camera. This result was used as a calibration curve to set the desired substrate-surface temperature by appropriately adjusting the heater temperature. Because the amount of lubricant dropped onto the substrate was minute (less than 1 µL), we considered its temperature to be the same as the substrate surface temperature. We measured the temperature distribution on the substrate surface using an infrared camera when the heating temperature was set to 100 °C. The results are shown in Fig. 8a and b. Uniform heating of ± 0.5 °C was confirmed to have been achieved in an area with a diameter of approximately 2 mm around the heating center point. This area was sufficiently large for the FWM measurements.

Fig. 7
figure 7

Temperature calibration curve of the developed heating stage

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Fig. 8
figure 8

Temperature uniformity within the heating surface as assessed by an infrared camera: a infrared camera image and b temperature distribution at the center of the heating area when heated to 100 °C

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In the FWM measurements, our previous study empirically confirmed that the viscosity in the bulk state could be measured if the gap was set to approximately 500 nm or larger. We measured the temperature dependence of lubricant viscosity in the bulk state via FWM using the heating stage developed in this study, compared it with the results of a rotational viscometer, and found that they were generally in agreement (Fig. 9). This result validates the temperature dependence measurement of shear viscoelasticity using FWM.

Fig. 9
figure 9

Comparison of the temperature dependence of the viscosity of lubricating oils with PAMA added to mineral oil in the bulk state measured using FWM and a rotational viscometer

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4 Experiments

4.1 Samples

The lubricating oil was a base oil of Gr. III mineral oil with 2 wt% PAMA added to it. Three types of PAMA with molecular weights of 20,000, 60,000 and 90,000 were used. The molecular structures of main and side chains (R) of PAMA are shown in Fig. 10. Four types of side chains were included in R: methyl group (–CH3), 2-ethylhexyl methacrylate group (–C8H17: EH), lauryl methacrylate group (–C12H25: LA), and stearyl methacrylate group (–C18H37: ST), each in a proportion of 25 wt%. The ester bond to which R is attached has polarity and adsorbs on surfaces; the longer the hydrocarbon chain length of R, the lower the surface adsorption of PAMA owing to the steric hindrance of the adsorption of the ester bond on the surface. Thus, the shortest methyl chain resulted in the highest surface adsorption of PAMA. Song et al. quantified the polarity of PAMA with different R lengths by measuring the Hansen solubility parameter and further measured the adsorption film thickness using VEM. The results showed that shorter chain lengths resulted in higher polarity and improved the surface adsorption of PAMA [19]. Furthermore, Tagawa et al. found that the lowest friction coefficient was obtained when four types of R (methyl, EH, LA, and ST) were introduced at a ratio of 25%, similar to the PAMA used in this study [16]. It has also been experimentally confirmed that the same PAMA forms an adsorption layer on stainless-steel surfaces [18]. However, the adsorption was not very strong, and it was experimentally confirmed that the adsorbed layer was completely removed by ultrasonic cleaning with hexane. Surface adsorption was assumed to be the highest when all side chains were methyl groups; however, the solubility in the base oil was reduced. Hydrocarbon chains of different lengths (EH, LA, and ST) were introduced in addition to methyl groups to control the solubility of PAMA in the base oil. It was assumed that there is an optimum molecular structure that achieves the highest lubricity depending on the balance between surface adsorption and solubility in the base oil; however, the details have not yet been clarified. A quartz glass substrate with a sputtered stainless-steel thin film (thickness of approximately 60 nm) was used. The surface roughness was approximately 0.2 nm. After dropping the lubricating oil onto the substrate, it was subjected to a 24 h heating treatment at 80 °C to facilitate the surface adsorption of the polymer.

Fig. 10
figure 10

Chemical structure of PAMA

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4.2 Experimental Procedure

A drop of lubricant was placed on the substrate and the tip of the probe was inserted into the lubricant. The lubricant volume was adjusted so that only the spherical portion of the probe tip was completely immersed. The frequency and amplitude of the vibrations were set to 1 kHz and 50 nm, respectively. The probe excitation frequency was set sufficiently lower than the resonance frequency of the probe (10 kHz or higher) [29]. The initial shear gap was set to approximately 1 μm, which was reduced at a constant rate (approximately 10 nm/s). We measured the oscillation amplitude and phase delay of the probe tip during the process by synchronous detection using a lock-in amplifier and recorded them on a personal computer (PC). The output signal of the optical interferometer was simultaneously recorded on the PC. The complex viscosity and sliding gap of the samples were determined from these measurements, and the gap dependence of the shear viscoelasticity was evaluated. In the temperature dependence measurements, the temperatures were set to 25 (room temperature), 40, 60, 80, and 100 °C. Shear viscoelasticity measurements at each temperature were conducted using the abovementioned procedure after the respective temperatures had reached a constant value and the thermal expansion of the substrate surface had stabilized. The temperature dependence of shear viscoelasticity for each gap was obtained by plotting the measured values of the same gap against temperature.

4.3 Results and Discussion

4.3.1 Gap Dependence of Shear Viscoelasticity

The viscosity (\({\eta }{\prime}\)) and elasticity (\({\eta }^{{\prime}{\prime}}\)) measured at various temperatures for lubricating oil samples containing PAMA with molecular weights of 20,000, 60,000 and 90,000 are shown in Figs. 11, 12 and 13, respectively. Viscoelasticity measurements were performed several times, and reproducibility was confirmed. The plotted data in Figs. 11, 12, and 13 represent the averages of five or more measurements. When we calculated the variation in the measured values as the standard deviation and expressed them as a percentage of the mean value; the values were 8.8%, 4.9%, and 3.9% for molecular weights of 20 k, 60 k and 90 k, respectively. As the variation was predominantly due to the random noise component, the differences in these values for each sample were considered accidental. As explained in Sect. 2.2, according to the gap measurements using optical interferometry, when the lubricating oil contained PAMA, the minimum achievable gap was not zero (direct contact between the probe and substrate did not occur). The minimum gap corresponded to the thickness of the remaining PAMA adsorption film. The shear viscoelasticity data represent measurements conducted from a gap size of 100 nm to the minimum achievable gap.

Fig. 11
figure 11

Shear-gap dependence of a viscosity and b elasticity measured at various heating temperatures for lubricants containing PAMA with a molecular weight of 20,000

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Fig. 12
figure 12

Shear-gap dependence of a viscosity and b elasticity measured at various heating temperatures for lubricants containing PAMA with a molecular weight of 60,000. For comparison, the measurement results of the base oil at room temperature (25 °C) are indicated by the solid lines

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Fig. 13
figure 13

Shear-gap dependence of a viscosity and b elasticity measured at various heating temperatures for lubricants containing PAMA with a molecular weight of 90,000

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As the gap narrowed to less than a few dozen nanometers, both \({\eta }{\prime}\) and \({\eta }^{{\prime}{\prime}}\) increased rapidly. This was attributed to the adsorption of PAMA films on the substrate surface. Surface adsorption of PAMA can occur on both the glass surface of the probe tip and stainless-steel coated substrate surface. In the present study, it was not possible to distinguish between the adsorption on either. Systematic experiments with different coating materials on the probe tip surface and substrate could elucidate the material-dependent relationship between surface adsorption and mechanical properties and should be the subject of our future study. In Fig. 12, \({\eta }{\prime}\) and \({\eta }^{{\prime}{\prime}}\) of the base oil measured at room temperature are indicated by solid lines for comparison. Although \({\eta }{\prime}\) and \({\eta }^{{\prime}{\prime}}\) of the base oil alone increased in the nanogap, the degree of increase was considerably lesser than that in the lubricant with PAMA. A comparison of the results around a gap of 100 nm showed that the viscosity of the PAMA-containing oil was higher than that of the base oil. This was not an increase in viscosity due to the adsorption film but a thickening effect of the polymers in the oil, which was also observed in the bulk state. Our measurement system does not allow separate measurement of the viscoelasticity under these two conditions. However, as the gap narrows to the nanometer scale, polymers that are not adsorbed on the surface are easily ejected from the gap; therefore, the adsorption layer is considered dominant in the gap. For the lubricant with PAMA, an increase in shear viscoelasticity was observed in almost the same gap region at all heating temperatures. This indicated that the adsorbed film did not desorb from the surface at 100 °C. Fatty acid-based friction modifiers do not form adsorption layers at temperatures above ˃70 °C. In contrast, the high adsorbability of PAMA may be one of the reasons for its friction-reducing effects at high temperatures [18]. PAMA added to lubricating oil in the bulk state swells as the temperature increases. In contrast, the gap range where the shear viscoelasticity of the adsorbed film was measured was independent of the temperature. This suggests that swelling of the polymer chain was suppressed by surface adsorption when heated. Comparing the results for PAMA with different molecular weights, an increase in viscoelasticity was observed over a larger gap with higher molecular weights. This result indicates that the thickness of the adsorption film increases as the molecular weight increases. The density of the film may not be significantly dependent on molecular weight. However, when adsorbed films of different thicknesses are sheared within the same shear gap, thicker films should have a higher polymer density. This may explain the higher viscoelasticity values measured for PAMA with higher molecular weights.

4.3.2 Temperature Dependence of Shear Viscosity

Figure 14 shows a plot of viscosity \({\eta }{\prime}\) versus temperature at representative gaps from the measurements shown in Figs. 11, 12, and 13. The temperature dependence of the viscosity was almost the same for the gaps of 100–20 nm, whereas it decreased for gaps < 10 nm. In terms of lubrication design, a decrease in temperature dependence indicated an increase in the viscosity index. Figure 15 shows the results of the viscosity index calculated from the results of Fig. 14. The results showed that the viscosity index increased by a factor of three for a gap of ≤ 10 nm. The viscosity index improvement effect of PAMA in the bulk state has been interpreted as the swelling of molecules and an increase in flow resistance with increasing temperature, which resists the decrease in viscosity of the base oil [7]. As mentioned previously, Figs. 11, 12, and 13 show that the PAMA adsorbed on the surface did not swell as the temperature increased. In other words, we considered that the viscosity index improvement in the nanogaps was caused by a mechanism different from that in the bulk state. Since the elasticity was dominant at gaps of a few nanometers, most of the PAMA was possibly adsorbed on the surface and not involved in the flow. In this case, the base oil contained in PAMA was responsible for the viscous flow and it flowed in the network of the adsorbed polymer chains. Even when the main chain was constrained, the thermal fluctuations of the side chains increased with increasing temperature. This may increase the resistance of the oil flowing between the polymer chain networks, resulting in an improvement in the viscosity index.

Fig. 14
figure 14

Temperature dependence of viscosity for lubricants with PAMA with a molecular weight of a 20,000, b 60,000 and c 90,000 at typical gaps of 5, 10, 20, 30, and 100 nm

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Fig. 15
figure 15

Gap dependence of the viscosity index for lubricants with PAMA

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4.3.3 Temperature Dependence of Shear Elasticity

The temperature dependence of elasticity at gaps of 5 and 10 nm is shown in Fig. 16. For all molecular weights, the elastic modulus increased in proportion to temperature. This result suggests that the measured elastic modulus is an entropic elasticity (rubber elasticity) [31]. As described in Sect. 4.1, the molecular structure of PAMA is such that the ether-bonded portion to which the methyl groups are attached is adsorptive to the surface. Therefore, the macromotion of the adsorbed PAMA molecule was constrained; however, side chains other than the methyl group and the surrounding main chain could move locally. When sheared by a probe, the main chains that can move may be stretched by shear, possibly resulting in entropic elasticity. In other words, the PAMA chain adsorption points may act as crosslinking points for bulk rubber. When comparing the temperature dependence of the elastic modulus at a gap of 5 nm between PAMAs of different molecular weights, the slope of the graph increased with increasing molecular weight. The adsorbed film was thicker for PAMA with a higher molecular weight, and both viscosity and elasticity were higher at the nanogap. In other words, for the same gap of 5 nm, a larger molecular weight was considered to result in a higher density of polymer chains. Based on the theory of entropic elasticity, the larger the density of polymer chains, the larger the temperature dependence of elasticity. This is consistent with the interpretation that the measured elastic modulus is based on the entropic elasticity.

Fig. 16
figure 16

Temperature dependence of elasticity for lubricants with PAMA with a molecular weight of a 60,000 and b 90,000 at typical gaps of 5, 10, and 100 nm

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5 Conclusion

This study was aimed at clarifying the mechanical properties (shear viscoelasticity) of polymer-added lubricants sheared in nanogaps. In particular, since polymer additives are generally used to improve the viscosity index, investigating the temperature dependence of the shear viscoelasticity is necessary. We introduced optical interferometry into the FWM developed in our previous study to simultaneously measure shear viscoelasticity and nanogaps. Furthermore, we developed a heating stage that can suppress the maximum fluctuation of the substrate surface position to ≤  ± 1.5 nm owing to the thermal expansion of the heater, and we succeeded in measuring the temperature dependence of the shear viscoelasticity in nanogaps. The shear viscoelasticity was measured for lubricants containing 2 wt% PAMA with different molecular weights, and the following findings were obtained:

  • An adsorption layer of PAMA was formed on the solid surface, and its effect on shear viscoelasticity was dominant below a shear gap of a few tens of nanometers.

  • The adsorbed layer of PAMA stably existed on the surface without desorption even when heated to 100 °C.

  • The viscosity index of the nanogaps was approximately three times higher than that of the bulk state.

  • The elasticity observed in the nanogaps increased with increasing temperature. This suggested the physical origin of the elasticity to be entropic elasticity (rubber elasticity).

These results provide fundamental information for designing polymer additives for lubricating oils. In particular, the improvement in the viscosity index and the appearance of entropic elasticity due to confinement in nanogaps are, to the best of our knowledge, the first findings. Based on the results of this study, two mechanisms can be assumed by which polymer adsorption films improve lubricity. The first is the prevention of direct contact between solid surfaces. The mechanical properties of the adsorbent film are several times more viscous and elastic than those of the base oil; therefore, the friction force increases compared to the fluid lubrication state. However, the adsorption film is unlikely to be ejected from the gap, thus reducing the increase in frictional force owing to the direct contact between solid surfaces and contributing to lower friction under boundary lubrication conditions. The second possible mechanism involves the occurrence of slip in the polymer adsorption films. Polymeric materials generally exhibit a certain relaxation times. When sheared at a shear rate faster than the inverse of the relaxation time, the film behaves as a rigid solid film and is not deformed, and slip can occur between the film and sliding surfaces. In this case, friction is governed by the intermolecular interaction between the adsorbed film surface and sliding surface, whereas the resistance force due to the mechanical deformation of the film itself does not directly contribute to the frictional force. In other words, a molecular structure that reduces intermolecular interactions improves lubricity. A previous study by Tagawa et al. showed that a mixture of four different side chains was optimal for the PAMA [16]. Although the mechanism for this has not been clarified, it is possible that the mixing ratio achieved an optimum balance between adsorption on the surface and easy slip on the surface.

Our future research target is to clarify the quantitative effect of mechanical properties (shear viscoelasticity) unique to nanogaps on the frictional properties. In particular, an increase in elasticity is unique to polymers and is expected to have a load-carrying effect in the vertical direction. However, it is not necessarily effective in lowering the coefficient of friction as it also increases the shear resistance. In addition, both the polymer and fluidity of the base oil and its interaction with the polymer are assumed to be involved in the friction characteristics of the siding in nanogaps. In other words, the compatibility between the polymer additives and the base oil is also considered an important factor in lubrication design. The measurement methods and findings presented in this study are expected to serve as a breakthrough in the development of polymer additives in the future.