Quantitative Viscosity Mapping Using Fluorescence Lifetime Measurements

Thioflavin-T (T3516) and IGEPAL CO-520 (238643) from Sigma-Aldrich were used as received. They are referred to as ThT and IGEPAL in this work, respectively. The properties of IGEPAL are shown in Table 1.

The model lubricant in this work consists of IGEPAL doped with 0.6–1.7 mM ThT. Details on its choice can be found in [22]. High dye concentration is required due to the ultra-thin films studied in this work and the dyes’ low quantum yield. ThT was dissolved in IGEPAL at 55 °C using a magnetic stirrer for one hour. The solutions were then filtered using a 1 μm filter (514-4027 Syringe filters, Acrodisc^®, glass fibre VWR) to remove any undissolved impurities in the dye.

In this work, an optically accessible high-pressure rheometer based on a circular point contact is used [21]. This is created by a ball loaded on a flat disc. Both the speeds of ball and of the disc, \(U_{\text{ball}}\) and \(U_{\text{disc}}\), can be controlled independently to regulate the entrainment speed \(U_{\text{e}}\) (Eq. 1), and the slide roll ratio SRR (Eq. 2) simultaneously. As \(U_{\text{e}}\) increases, more lubricant is drawn into the contact, and the lubricant film increases in thickness.

For an EHD point contact, its radius \(a\) (Eq. 3) is controlled by the applied load \(N\), effective radius \(R\) and the combined elastic modulus \(E^{*}\). The contact pressure varies with radial position \(r\), and the pressure distribution \(P\left( r \right)\) is commonly approximated by a Hertzian pressure distribution (Eq. 4). The maximum contact pressure \(P_{max}\) occurs at the centre of the contact and is defined by Eq. 5, and the average pressure \(P_{\text{a}}\) can be expressed by Eq. 6. The lubricant viscosity under EHD conditions affects both average and local shear stress within the contact. Viscosity and hence shear stress is a function of pressure, temperature and shear rate (for non-Newtonian liquids). The effect of pressure \(P\) on viscosity \(\eta\) can be described by the Barus relationship, where \(\eta_{0}\) and \(\alpha\) are the low pressure (bulk) viscosity and the pressure–viscosity coefficient, respectively (see Eq. 7). The Barus equation deviates from measurements at high pressure [24]. However, the peak pressures used in this work are less than 0.5 GPa. Hence, it is adequate in fitting our rheological data, as confirmed by the linearity of the pressure–fluorescence lifetime relationship for ThT in IGEPAL up to 600 MPa as shown in Fig. 1a.

The point contact is chosen for this work as it allows the shear rate and the pressure to be controlled across the range of engineering interest. It also makes for easier comparison with results from friction and film thickness measurements. With independent control of entrainment speed and SRR, shear rates experienced by the lubricant film can be varied while maintaining a constant film thickness. This allows the effect of shear rate on both the local and average/apparent viscosity within the contact to be studied.

Transparent flat surfaces are used in this study to allow the optical access necessary for film thickness and fluorescence lifetime measurements. The surfaces chosen are a glass ball and a glass disc for rheological measurements. For film thickness measurements, where a reflective surface is required for interferometry, a steel (AISI52100) ball and chromium/silica coated glass disc are used. The properties of the rubbing surfaces, the range of contact pressures, shear rates and film thicknesses used are shown in Table 2. The contact pressures applied are chosen to prevent IGEPAL from shear thinning excessively.

To examine the local viscosity of an EHD lubricant in situ, fluorescence lifetime measurements are applied using the optically accessible point contact-based rheometer discussed above. The rheometer is mounted over an inverted microscope. The ball is loaded against the flat surface using a dead load. It is moved using an automated microscope stage which allows the local viscosity to be mapped with a high spatial resolution of 12 µm. Details of the set-up for fluorescence lifetime measurements are discussed in Sect. 2.3.2.

All tests were undertaken at an ambient temperature of 26 ± 1 °C, except the calibration which was undertaken at 21 °C. A SRR from 0 to 50% was applied. Fluorescence lifetime measurements show no correlation between SRR and viscosity. As the temperature rise in the contact depends on the SRR, at a given pressure, this indicates that the temperature rise due to shear heating in the present experiments is small, and its effect on the viscosity is within the statistical uncertainty of the measurements.

Once ThT in IGEPAL is excited by the pulsed laser, the fluorescence intensity decays with time. The decay curves are fitted to find the characteristic decay time using a stretch exponential fit as shown in Eq. 8, where \(I\) is the intensity at time \(t\), \(I_{0}\) is the intensity at \(t = 0\), \(\tau\) is the characteristic decay time (i.e. ThT fluorescence lifetime \(\tau_{\text{ThT}}\)) and \(n\) is the stretch factor. A stretch exponential fit is commonly used to fit decay curves from molecular rotors [26, 31]. The stretch factor accounts for any other quenching mechanisms not listed in Sect. 2.3.1 [32]. In bulk conditions \(n\) is between 0.68 and 0.75. As the solvent becomes more viscous and fluorescence emission becomes the dominant relaxation process, \(n\) increases and tends to unity. For each decay curve, the maximum intensity value is identified and is taken as \(I_{0}\). Data are then fitted from \(t = 0.11 - 6.11\;{\text{ns}}\) after \(I_{0}\).

A pressure–viscosity relationship for IGEPAL-CO520 is needed to obtain the local viscosity of the lubricant in an EHD contact with fluorescence lifetime measurements. Bair measured the lubricant viscosity using a falling cylinder viscometer at elevated pressure at 20 and 25 °C [23]. A falling cylinder viscometer involves determining the terminal velocity of a sinker as it falls under gravity through the test fluid. The sinker calibration was carried out against values of Tris(2-ethylhexyl) trimellitate (TOTM) from the literature [33]. The improved Yasutomi model [34] was applied to estimate the viscosity of the working fluid at elevated pressures from 21 to 36 °C. These estimates are referred to as high-pressure rheological estimates in this work. For simplicity, the results of the model were fitted with the single exponential Barus equation within the experimental applied pressure range. The pressure–viscosity relationship for IGEPAL at 21 °C is shown in Fig. 1b. Together with the pressure–ThT fluorescence lifetime relationship we had obtained with the pressure cell [22] (Fig. 1a), a relationship between IGEPAL viscosity \(\eta_{\text{IGEPAL}}\) and ThT fluorescence lifetime \(\tau_{\text{ThT}}\) is established (Fig. 1c). It is fitted as a mono-exponential relationship and is used to estimate the local viscosity of lubricant in the EHD regime.

To determine if the link between Fig. 1a and b is valid in a contact as well as in the pressure cell, it is required that the lubricant is Newtonian. Previous results [22] and friction measurements (presented in Sect. 4) suggest that shear effects on IGEPAL are small for peak pressures below 400 MPa and slide roll ratios below 50% for an entrainment speed of 100 mms⁻¹, within which the test conditions of this work fell. Note, once a molecular rotor fluorescence lifetime–lubricant viscosity relationship is established, the relationship can be applied to obtain local viscosity in a contact even if shear thinning occurs [22].

The local \(\tau_{\text{ThT}}\) is measured across a region just larger than the contact using a grid size of 18 × 18 points (Fig. 2a). The \(\tau_{\text{ThT}}\) map can be converted to a local \(\eta_{\text{IGEPAL}}\) map using the calibration detailed in Sect. 2.5 and is shown in Fig. 2b. From Fig. 2b, the data points within Hertzian contact area are extracted. Data points within the contact are selected based on the calculated Hertzian contact area (Eq. 3) and a threshold lifetime value.

The centre of the contact was determined to find the contact area and to allow the calculation of the local pressure in the contact. The centre of the contact is identified as the point with the highest viscosity. This point is located based on a polynomial fit of suitably smoothed lifetime viscometry data. The contact area is then identified as a circular area around the centre thus located. This estimate of the contact area agrees well with the size of the experimental contacts observed using optical interference. This gives confidence that the actual contact pressure is the same as the Hertzian pressure. Local pressure can then be estimated with Hertzian pressure distribution (Eq. 4). Thus, the spatial distributions of \(\tau_{\text{ThT}}\) in a contact allow us to relate local \(\eta_{\text{IGEPAL}}\) to local pressure. Note smoothing of \(\eta_{\text{IGEPAL}}\) data is performed only to find the centre of the contact. No smoothing procedure has been applied for results reported in this work.

The local \(\tau_{\text{ThT}}\) distribution as shown in Fig. 2a shows an increase in \(\tau_{\text{ThT}}\) at the inlet of the contact. This is consistent with the inlet pressure rise commonly observed in an EHD contact. This inlet pressure rise region can extend up to half the contact width behind the contact and to a pressure of up to 150 MPa [9]. The corresponding viscosity rise at the inlet is better illustrated by plotting local \(\tau_{\text{ThT}}\) along the parallel and orthogonal directions to the flow, passing through the centre of the contact, as shown in Fig. 3. It shows that \(\tau_{\text{ThT}}\) at the inlet (solid line with circles, for position ≥170 μm) is approximately 0.2 ns higher than the rest of the bulk. This corresponds to an increase in viscosity from approximately 0.2–0.4 Pa s and is higher than the viscosity at the outlet and outside the contact orthogonal to the flow direction (dash line with crosses for position ≥170 μm). This suggests that the pressure at the inlet is about 50–100 MPa. Note Fig. 3 is an average across 3 grid points, so the value obtained is an underestimate of the actual \(\tau_{\text{ThT}}\), due to the circular nature of the contact.

Average EHD lubricant viscosity is commonly estimated using friction measurements combined with film thickness measurements. Friction is measured using a PCS Instruments Mini Traction Machine (MTM2). This friction data are used to determine the average shear stress \(\sigma\). The MTM2 measures the traction force \(T\), and from this and the applied normal load \(N\), the traction coefficient \(\mu\) is calculated. Along with the contact radius, the shear stress can be calculated as shown in Eq. 9.

Optical interferometry [35] is used to measure the film thickness of the working fluid using a PCS Instruments EHD2 ultra-thin film measurement system. SLIM (spacer layer imaging method) [36] is used to determine the degree of homogeneity of the film thickness throughout the contact (Fig. 4). Using a steel–glass contact, a relationship for IGEPAL film thickness as a function of entrainment speed is obtained. EHD central film thickness of a circular contact \(h_{c}\) can also be estimated using the Hamrock–Dowson relationship, as shown in Eq. 10 [37] where \(U\) is the dimensionless speed parameter, \(G\) is the dimensionless material parameter and \(W\) is the dimensionless load parameter. These parameters contain either test conditions (load or speed), material properties or both. The measured results from film thickness measurements match estimates from Eq. 10 well (not shown).

IGEPAL film thickness for the conditions used for the fluorescence lifetime and friction measurements can be estimated from results obtained with a glass–steel contact by film thickness measurements. To account for the material change and varying test loads in these two measurements, adjustments are made by dividing out material and test condition factors for film thickness measurements on both sides of Eq. 10 and then factoring in the correct values for fluorescence lifetime and friction measurements. Comparison between the estimated film thickness for glass–steel contact and glass–glass contact shows negligible difference. All quoted IGEPAL film thickness for fluorescence lifetime measurements are based on interferometry measurements with glass–steel contacts and adjusted for load and material properties as described. The shear rate \(\dot{\gamma }\) is determined from this film thickness \(h_{c}\), the entrainment speed \(U_{e}\) and the percentage slide roll ratio SRR. This is shown in Eq. 11.

With the average shear stress and average shear rate obtained from friction and film thickness measurements, respectively, an average or apparent viscosity is calculated using Eq. 12.

The average viscosity of the EHD lubricant can also be estimated using the viscosity map obtained with fluorescence lifetime measurements (see Sect. 2.6). The local \(\eta_{\text{IGEPAL}}\) in a viscosity map is an average through-thickness viscosity value at a particular position in a contact. Assuming the local viscosity at every position contributes equally to the average viscosity of the contact, and \(J\) is number of data obtained and these points distribute evenly within the contact (as in the case of Fig. 2), the average in-contact viscosity \(\eta_{\text{a}}\) is then the mean of the local viscosity values obtained from the map (Eq. 13). Eq. 13 is valid if the film thickness, hence the shear rate, within the contact region is reasonably homogeneous. The homogeneity of the film thickness in the contact is verified with interferometry (Fig. 4). Average viscosities obtained from friction and fluorescence lifetime measurements are compared to high-pressure rheological measurements.

By assuming a Hertzian pressure distribution, the relationship between local IGEPAL viscosity and local pressure can be obtained. Figure 5 shows the local \(\eta_{\text{IGEPAL}}\) from 21 viscosity maps (324 data points each) taken from contacts with different peak pressures and SRR ranging from 0 to 50%. All the data collapses onto one master curve. This confirms that the effect of shear thinning is minimal across all test conditions. The scatter of the data points in the master curve is due to variations in temperature across experiments. The scatter in a single experimental set is actually low as can be seen in Fig. 6, supporting the assumption that the variation in temperature throughout each mapping experiment is small and that the change in experimental temperature is over a relatively long timescale. The relationship can be described by a mono-exponential fit (solid line, Fig. 5), suggesting it follows a Barus-type pressure–viscosity relationship. This is expected as the fluid viscosity can reasonably be estimated by the Barus equation in this pressure range and minimal shear thinning is expected in all of our experimental conditions. The result is, however, non-trivial. Should local shear thinning occur, which is promoted by high pressure, one would expect deviations to occur at high pressures.

Results in Fig. 5 are below the expected values at a testing temperature of 26 °C as based on high-pressure rheological estimates (dashed line, Fig. 5). They can be accounted for by a temperature range of approximately 29–36 °C. On average the results are best fitted by expected viscosity at 33 °C, suggesting either (a) shear thinning has occurred giving rise to lower than expected viscosity or (b) the contact was hotter than the ambient temperature. Since shear thinning is negligible, except potentially at the centre of the contact where pressure is high, in our study, this discrepancy is likely to be due to contact temperature rise from two possible sources. Firstly, there could be shear heating. Shear heating is more prevalent at high SRR and high pressure. As the data obtained from different pressures and SRR collapses onto one master curve, shear heating is unlikely to be significant. The second possibility is another heat source affecting temperature in the contact. It has been found that the bearing where the transparent disc sits heats up significantly as it rotates due to frictional heating within the bearing. Thus, the disc may heat up as a result. To confirm this for individual data sets, ThT fluorescence lifetime in IGEPAL around the contact (excluding the inlet) was averaged and used to estimate the average low pressure viscosity and hence the temperature of IGEPAL before entrainment. Results of one of the individual data sets are shown in Fig. 6. The estimated IGEPAL temperature before entrainment for this particular test was about 34 °C. As shown in Fig. 6, this set of data fits high-pressure rheological estimates at 33.6 °C. It thus confirms that while shear heating is negligible in the contact, the lubricant is hotter than expected before it was entrained into the contact and temperature variations across tests may give rise to the spread of the data in Fig. 5. From Fig. 6, it can be seen that for local pressures greater than approximately 280 MPa localised shear thinning occurs. This is due to errors in surface speeds causing shear thinning at high-pressure locations. Note with respect to a point contact, absolute pure rolling is very difficult to achieve as small errors in surface speeds can lead to significant shear rates. For example, 1% SRR under our conditions would provide a shear rate of approximately 6 × 10³ s⁻¹. This will be discussed in detail in Sect. 4.2.

Since all data sets collapse into one master curve (see Fig. 5), for clarity the remaining discussion will be presented by using data obtained at a peak pressure of 380 MPa and 0 SRR (Fig. 6). The observation discussed below applies to all data set in this work.

The relationship between local IGEPAL viscosity and local pressure based on a single viscosity map (shown in Fig. 2b) is shown in Fig. 6. The inset in Fig. 6 shows the same data on a log-linear scale. The relationship can be described by a mono-exponential fit as discussed earlier (solid line, Fig. 6). Hence, the local viscosity of the fluid can be estimated by using a Hertzian pressure distribution and the Barus equation.

The average viscosity of the EHD lubricant is estimated based on ThT lifetime measurements as described in Sect. 2.7. The contact area was estimated in two ways (see Sect. 2.6). The average viscosity calculated with both area estimates (triangle and asterisk, Fig. 6) gives similar results. One would expect that the average viscosity and average pressure are related directly. In fact that is how average or nominal viscosity is calculated from a friction test, with an average shear stress calculated from the coefficient of friction. Plotting estimated average viscosity against average pressure Fig. 6 (asterisk and triangle respectively) shows they fall on the same exponential relationship fitted for the local viscosity data (circles). This observation supports the assumption mentioned in Sect. 2.7 that the thickness of the lubricant film, i.e. shear rate, in our EHD contact is homogeneous. More importantly, it suggests that the local velocity profiles within the contact obey Couette flow and are uniform. Under this condition, average measurements, such as friction measurements, can potentially be used to characterise the average viscosity of EHD lubricants. The validity of the homogeneity assumption can be assessed by studying the distribution of film thickness. The majority of the film will usually be made up by a flat central region; however, it is best to verify this using SLIM imaging. Similar procedures have been applied to estimate average IGEPAL viscosity for tests conducted at different peak pressures, SRR and entrainment speeds. Taking into consideration temperature variations among tests, average viscosities estimated at various average pressure from ThT fluorescence lifetime measurement match the relationship shown in Fig. 5 relatively well (not shown). This suggests that provided the homogeneity assumption holds, local viscosity measurements of this kind can be used to estimate average viscosity for a wide range of pressure with only a small number of experiments. This is particularly useful when the peak pressure that can be achieved by a point contact is higher than that can be achieved by conventional rheological techniques.

For conventional lubricants, they are commonly assumed to exhibit Couette flow in EHD contacts. In a contact where the flow profile is constant, should the EHD lubricant exhibit other types of flow such as Poiseuille’s flow, plug flow or shear banding, an apparent viscosity will be obtained from friction measurement, and it would not be equivalent to the average viscosity of the liquid. In cases where flow heterogeneity in an EHD contact is observed (see polybutene [16] and 5P4E [17]), using an average value across a contact may not be appropriate.

The average viscosity of IGEPAL obtained from fluorescence lifetime measurement shown in Fig. 6 supports that average measurements, such as the combination of film thickness and friction measurements can potentially be used to measure the average viscosity of an EHD lubricant. To check the validity of this combined approach, the average viscosity of IGEPAL is estimated at various average Hertzian pressures and SRR. The results obtained at average pressure of 185–276 MPa and for SRR = 5 and 50% are shown in Fig. 7 (circles and pluses, respectively). In comparison with expected viscosities from high-pressure rheological estimates, the estimated viscosity from combined friction and film thickness measurements at 26 °C and SRR = 5% matches the high-pressure rheological estimates at 23–24 °C. This is likely to be due to temperature fluctuations and results in approximately a 25% increase in viscosity. It should be noted at these temperatures small fluctuations in temperature can lead to large changes in viscosity (see Table 3) Also on Fig. 7, the average viscosity of IGEPAL obtained with ThT fluorescence lifetime measurements in a similar average pressure range and the same entrainment speed under pure rolling is also presented (triangles). They fit high-pressure rheological estimates at approximately 33 °C well. As discussed in Sect. 3.1, the higher than expected bulk IGEPAL temperature in ThT fluorescence lifetime measurements is the reason for the discrepancy in average IGEPAL viscosity obtained from combined friction/thickness measurements and ThT fluorescence lifetime measurements. The maximum viscosity (open squares) is also presented to show the slightly lower than expected viscosity at the centre of the contact which is likely to be due to a shear effect.

Average viscosity measurements based on combined friction/film thickness measurements show a reduction in average viscosity as SRR (i.e. shear rate) increases from 5 (solid circles, Fig. 7) to 50% (pluses Fig. 7). In fact, a reduction in viscosity of less than 30% was seen when friction measurements were conducted at an average contact pressure below 250 MPa and that the viscosity where this shear thinning started is 15–20 Pa·s, after which it started to drop (results not shown). This is contrary to results from ThT fluorescence lifetime measurements where no obvious shear effect was detected for average viscosity under these test conditions (see Fig. 7). The different phenomena observed between the two experiments may be due to differences in actual contact temperature (see Sect. 3.1), which was up to around 10 °C. This means the bulk IGEPAL viscosity for the ThT fluorescence lifetime measurements is lower than that of friction measurements. As a result, a high normal pressure or a higher shear rate/SRR is required to reach the critical shear stress for shear thinning to occur in our fluorescence lifetime measurements. This is supported by results obtained above 325 MPa (shown in Fig. 5), where the experimental fit of the data (red solid line) starts to deviate from expected viscosity from high-pressure rheological estimates at 33 °C (dot dashed line). This suggests shear thinning at a local pressure above approximately 300 MPa during our ThT fluorescence lifetime measurements.

The ability to obtain local viscosity information from fluorescence lifetime measurements is highlighted in Fig. 7, where local shear thinning is observed. The triangles and squares in Fig. 7 correspond to the average viscosity and local maximum viscosity from 4 sets of experiments under pure rolling conditions. All average viscosities match well with the high-pressure rheological estimates, while all maximum viscosities obtained at position with maximum pressure are lower than expected. This local shear thinning at low shear rates at high-pressure locations is not apparent in combined friction and film thickness measurements as the average pressure was less than 280 MPa although significant shear thinning is observed at average pressure of about 300 MPa (not shown). As the maximum local pressure approaches 400 MPa, local shear thinning at even relatively low shear rates is expected. Hence, local viscosity measurements have successfully predicted conditions where shear thinning will occur despite the low average pressure used in these tests.

With viscosity being directly linked to shear stress by shear rate, fluorescence lifetime measurements provide an opportunity to map the local shear stress in the contact. If the film thickness is known, which can easily be measured using optical interferometry or estimated using the Hamrock–Dowson equation, the shear rate can be calculated by Eq. 10. Local shear stress can then be estimated by simply multiplying the local viscosity by the shear rate (see Eq. 11).

The shear stress map of an EHD contact at a peak (average) pressure of 342 (230) MPa and shear rate of 1.71 × 10⁵ s⁻¹ is shown in Fig. 8. In this case, the maximum (average) shear stress is 3.5 (2.3) MPa. From friction measurements under the similar conditions (30% SRR point used), the maximum (average) shear stress was 2.4 (1.6), 3.4 (2.3) and 4.1 (2.7) MPa at peak (average) pressure of 328 (218 MPa), 367 MPa (244 MPa) and 384 MPa (256 MPa), respectively. Using an exponential fit, the peak (average) shear stress at peak (average) pressure of 342 (230) MPa is estimated to be 2.7 (1.8) MPa. The discrepancy between the two estimates is likely due to differences in test temperatures. As discussed in Sect. 3.1, the temperature for fluorescence lifetime measurement was higher than that for friction measurements. This means the viscosity for the former tests was lower than that of the latter (see Fig. 7). The consequence, as described in 4.2, is that significant shear thinning occurs during friction measurements under these conditions, while during fluorescence lifetime measurements IGEPAL behaves close to Newtonian (see Fig. 7). The amount of shear thinning may be sufficient to render a lower shear stress in friction experiments. The need for interpolation also further increases the complexity of comparing the two measurements, where the pressure and temperature are not the same.