A design for a Quartz Crystal Microbalance (QCM) setup for use with viscous liquids at temperatures of up to 300 °C is reported. The system response for iron and gold coated QCM crystals to two common lubricant base oils, polyalphaolefin and halocarbon, is reported, yielding results that are consistent with theoretical predictions that incorporate electrode nanoscale surface roughness into their analysis.

## I. INTRODUCTION

A Quartz Crystal Microbalance (QCM) consists of a thin single crystal of piezoelectric quartz sandwiched between a pair of metal electrodes. It is set into oscillation by applying an alternating electric field between the two electrodes at a frequency close to the crystal resonance frequency. The crystal can be cut at various angles, which allows a variety of properties (mode of oscillation, resonant frequency, temperature stability, etc.) to be precisely controlled. Historically, QCM has been employed for a wide range of scientific studies, and the number of applications continues to expand rapidly. The most common and early use of QCM was as a time standard, followed by its use as a mass sensing device^{1,2} by monitoring frequency shifts associated with mass loading of its electrodes. It has also been used in tribology to investigate the friction between surfaces and adsorbed films,^{3} sliding friction between interfaces,^{4} superconductivity dependent friction,^{5} magnetic friction,^{6,7} and frictional phase transition of materials,^{8} by monitoring shifts in both the frequency and quality factor.

QCM has also been increasingly utilized for a range of studies in liquid environments.^{9,10} These include electrochemistry,^{11} biochemistry,^{12} tribology,^{13} biomedical,^{14} polymer science,^{15} and rheology.^{16} The QCM technique has been commercialized for both vacuum metallization and liquid applications,^{17} and QCMs are commercially available for use in vacuum at temperatures as high as 450 °C.^{18} High temperature QCMs for use in liquid environments are, however, currently unavailable from a commercial supplier. Published reports of studies of QCM in liquid environments at high temperature are few in number and limited to low viscosity liquids.^{19} High temperature liquid-liquid phase-transitions, high temperature catalysis, density fluctuations at solid-liquid interfaces at high temperature, and high temperature lubrication are among the fields that would greatly benefit from a liquid QCM apparatus operational at high temperatures.

QCM is especially difficult for viscous fluid operation when both sides are immersed in the liquid. Wang *et al.*,^{19} for example, employed a QCM with both sides immersed in liquids at high temperatures to measure their viscosity and thermal degradation properties. But their study was limited to liquid viscosities below 0.025 P because the QCM ceased to oscillate for higher viscosity fluids. One sided measurements increase the range of viscosity of liquids to be studied but also increase the complexity of the holder employed to house the QCM. As will be described herein, we have extended the applicable viscosity range of high temperature liquid QCM through the use of appropriate materials and exposure of only side of the QCM to the liquid. Notably, we have been able to work at elevated temperatures with high viscosity oils, up to 0.47 P, more than one order magnitude above the viscosity range of Wang *et al.*^{19} study.

## II. LIQUID QCM HOLDER DESIGN FOR HIGH TEMPERATURE OPERATION

Commercially available holders^{20–23} for 1 in. QCMs for operation in liquid environments are routinely fabricated from Teflon or Kynar, with a maximum recommended operating temperature of up to 110 °C. An example of one such holder, a Stanford Research System (SRS) QCM holder^{20} and its components are depicted in Figure 1(a). The QCM is held in place between two O-rings, which provide a flexible platform for the QCM to oscillate while also preventing the liquid from coming in contact with the outer regions of the QCM and/or the back contact electrode. Virtually all commercial (proprietary) and open source (non-proprietary) holders that are presently available are similar in design with two O-rings on either side of QCM, differing only in the materials used and overall size of the holder. The high temperature holder design described herein is similar to all of these, but utilizes materials that can withstand higher temperature levels. As depicted in Figure 1(b), these include a 1 in. stainless steel ultra-Torr vacuum fitting (Swagelok Company, Part No. SS-16-UT-6) and two spring contact pins (Everett Charles Tech., E-S-F test spring probe) with connection wires cemented to the fitting by means of high temperature cement (OMEGA Engineering, Inc., Part No. CC HIGH TEMP). These pins provide flexible electrical connections to the surface electrodes of QCM and can be readily insulated from the metal housing with ceramic beads, as depicted in the figure.

Two Viton GLT fluorocarbon O-rings (Anchor Rubber Products, Part No. M83485/1-116) with excellent operating temperature range and fluid compatibility were chosen to sandwich the QCM in-between the ferrule and the cap and also restrict the liquid to only one surface electrode side of the QCM, referred to herein as the “sensing” electrode. A cylindrical tube is welded onto the cap that holds the sample liquid in contact with the surface electrode. The QCM holder and all of its components except the O-rings are rated for temperatures in excess of 573°C, the quartz phase transition temperature beyond which the crystal will no longer function as an oscillator. The temperature range is in practice however limited by the O-rings to 300 °C.

## III. APPARATUS DESIGN

To explore the capabilities of the apparatus, QCM measurements were recorded between room temperature and elevated temperatures for QCMs immersed in two different oils, namely, halocarbon oil (Halocarbon Products Corporation, Halocarbon-4.2 oil) and polyalphaolefin (PAO)-6 (Chevron Phillips Chemical Company, Synfluid®PAO 6 cSt). Poly-alpha-olefins are polymers of alpha-olefin, an alkene with carbon-carbon double bond at the *α*-carbon atom. Low molecular weight PAOs are commonly used synthetic lubricants. They are viscous liquids even at low temperatures and have high flash and fire points, which makes them useful over wide temperature range. Halocarbon oils are the low molecular weight polymers of chlorotrifluoroethyene (CTFE) with chemical formula Cl–(C2F3Cl)_{n}–Cl. Chemical inertness and non-flammability of halocarbon oils make them excellent lubricants for a wide number of applications. Both of these oils are commercially available in different viscosity grades. The halocarbon oil used here has a viscosity of 4.2 cS at 37.8 °C, and the PAO oil has viscosity 5.9 cS at 100 °C. They are most often abbreviated as Halocarbon 4.2 and PAO-6. The properties of PAO-6 had been studied in the literature.^{24} The densities and viscosities of these oils are tabulated in Sec. IV in Table II.

1 in. diameter AT-cut(transverse shear mode, type A Temperature compensated), 5 MHz crystals with either a 0.5 in. diameter iron sensing electrode (INFICON, Inc., Part No. 149252-1) or alternatively a gold sensing electrode (FILTECH Part No. Q1001) were employed for the measurements. The QCM with the iron sensing electrode was cut for a zero temperature coefficient of 90 °C while the QCM with the gold sensing electrode had a zero temperature coefficient closer to 60 °C. Roughness characteristics of these QCM surfaces are reported in Section V.

A schematic diagram of the experimental setup is shown in Figure 2. The holder with liquid on the top of QCM was enclosed in a vented chamber with connections to a Resistance Temperature Detector (RTD) and QCM electrodes via feedthroughs. The vent was connected to a U-shaped open ended hose (not shown) that was cooled in a water bath to condense the oil vapors. A Pt 100 Ω RTD (OMEGA Engineering, Inc., Part No. RTD-1-F3102-36-G) was cemented on the holder to measure the temperature.

The chamber itself was housed within a temperature controlled oven. Connection wires from the contact electrodes were connected to a SRS QCM 25 crystal oscillator^{20} circuit controlled by a SRS QCM 100 control unit.^{20} The frequency and conductance of the crystal were measured by a frequency counter (HP 53181A Frequency Counter) and multimeter (TEKTRONIX, INC., Keithley 2000 Series), respectively, and connected to the output terminals of the QCM 100. The resistance of the RTD was measured by a second multimeter. The temperature controller was set to 300 °C, and the system was heated progressively while data were recorded, with ∼3 h elapsing between room temperature and 300 °C. A LabView program (National instrument, LabVIEW 7.1) recorded the frequency and conductance of the crystal and resistance of the RTD at 5 s intervals. The conductance, initially measured in volts, was converted to motional resistance of the QCM using the relation^{20} $R=104\u2212V/5\u221275$, where *V* is the conductance in volts and *R* is the motional resistance of QCM in Ohms. The electrical resistance of the RTD was calibrated to temperature using the Callendar-Van Dusen equation.^{25} Experimentally measured changes in the frequency and motional resistance of the QCM were next compared to theories for the response of a QCM to a liquid environment, as described in Secs. IV–VII.

## IV. LIQUID QCM THEORY

The resonance frequency of a Quartz crystal depends on the parameters specific to the environment that is in contact with its surface electrode. The overall response of a QCM frequency can be summarized by the equation

where $\delta f$ is the total shift in frequency with respect to the frequency of an unloaded crystal at a reference temperature, and the subscripts $M,L,T,$ and *S* represent the contribution of mass loading, liquid loading, temperature, and stress on the crystal due to external factors, respectively.

The frequency shift due to a rigidly attached mass on QCM is given by the Sauerbrey equation^{1}

where $fo$ is the starting frequency of unloaded crystal, $\delta m/A$ is the mass deposited per unit area, $\mu q=2.947\xd71011gcm\u22121s\u22122$ is the shear modulus of quartz, and $\rho q=2.648gcm\u22123$ is the density of quartz. This equation is the main basis for the use of QCM as a mass sensor in vacuum applications.

Kanazawa and Gordon^{9} described the viscous effect of a Newtonian liquid on QCM frequency with the equation

where $\rho L$ and $\eta L$ are the density and viscosity of the liquid.

Equations (2) and (3) are for one side of QCM used as a sensor. For measurements recorded with both sides immersed, the right side of the equations must be multiplied by a factor 2.

Martin *et al.*^{26} studied the effect of simultaneous mass and liquid loading on QCM and demonstrated that (2) and (3) can be added linearly to obtain the combined effect so long as the mass is not slipping on the surface electrode. They also derived the equation for the change in motional resistance of QCM due to energy loss by liquid damping as

where $Co$ is the static capacitance between electrodes on opposite sides of QCM and $\mathit{K}02=7.74\xd710\u22123$ is the electromechanical coupling factor for AT cut quartz.

The frequency of a crystal changes with the temperature and $\delta fT$ in (1) represents the temperature dependence of the crystal, which can be expressed as a polynomial^{27} of degree 3 as

where *T* is the temperature and $ai\u2032s$ are the temperature coefficients.

The values of these coefficients depend on dimension of the crystal, shape and size of the electrodes, and on the angles at which the crystal is cut. It is therefore important to choose suitably cut crystals on the basis of intended operating temperature range.

In practice, the frequency shifts observed in liquid environment are larger than those predicted by theory, an effect generally attributed to roughness of the surface electrode.^{28} This effect can be minimized by using a polished crystal, but cannot be completely neglected because in practice no surface has perfectly zero roughness.^{29,30} Martin *et al.*^{28} referred to this effect as a “non-laminar contribution” and observed it to be measurable even for surfaces with root mean square (rms) roughness below 10 nm. Their model therefore included an additional “*ρ*” term to account for this effect in linear combination with the $(\rho L\eta L)1/2$ term for liquid loading. They reported this contribution to be on the order of 10% for surfaces with rms roughness 3 nm, and increasing with increasing surface roughness.

Daikhin and Urbakh also extensively studied the effect of surface roughness on the response of QCMs immersed in liquids.^{31,32} Their model accounted for the multiscale nature of surface roughness using a set of two parameters: the root mean square (rms) roughness $(\sigma )$ and the lateral correlation length $(\xi )$. Based on these two parameters and the liquid velocity decay length $\delta L=(\eta L/\pi f\rho L)$^{1/2}, they predicted the roughness contribution to the shift in resonance frequency in two limiting cases of surface morphology: “slight” and “strong” roughnesses. Johannsmann^{33} calculated the roughness induced frequency shift in the “slight roughness” regime of Daikhin and Urbakh model for a typical liquid (with $\rho L$ = 1 g cm^{−3} and $\eta L$ = 0.01 g cm^{−1} s^{−1}) to be −13.5 Hz for a QCM of resonance frequency 5 MHz and a typical gold surface (σ ≈ 3 nm and ξ ≈ 10 nm). This is equivalent to a 2% correction to the frequency shift associated with a hypothetically flat surface.

The final term $\delta fS$ in Eq. (1) is a frequency shift resulting from stress on the crystal. A general term “stress” is used here that includes all possible factors such as hydrostatic pressure^{34} of liquid and mechanical stress. Mounting crystal vertically and/or maintaining same depth of liquid can eliminate the effect of hydrostatic pressure. The stress can also be caused mechanically by the holder or cell. Frequency shifts due to stress on the crystal by an added mass layer have also been reported in some scientific studies.^{35,36} The effects are expected to be very minimal for the present work.

## V. SURFACE ROUGHNESS CHARACTERIZATION

Surface roughness characterizations of the QCM sample sensing electrodes were performed using an Atomic force microscope (AFM) (Asylum Research MFP 3D). Silicon tips were used to image the surfaces in a tapping mode in air. Figures 3(a) and 3(c) depict the AFM image of an iron QCM electrode and a height profile along a line drawn across it, respectively. Figures 3(b) and 3(d) are the corresponding images of the gold QCM sensing electrode. Although the lateral variation of the features on both surfaces is of same scale, the height of the features on iron surface is nearly ten times larger than those on the gold surface.

Height profiles of the surfaces were quantified by the rms roughness value,^{37} which is dependent on the size of the area sampled.^{38,39} For a self-affine fractal surface, the rms roughness increases with the lateral length of the sampled area as, $\sigma \u221d$ L^{H}, where H is the roughness exponent whose value lies between 0 and 1. Fractal surfaces are often characterized by fractal dimension D = 3 − H. Self-affine surfaces have an upper horizontal cut-off length (the lateral correlation length $(\xi )$) above which the rms roughness saturates towards a value of $\sigma s$ and no longer exhibits fractal scaling. The surface roughness parameters (D, ξ, and $\sigma s$) were obtained from the log$(\sigma )$ vs log(scan size) plot method as described by Krim and co-workers.^{38,39} A detailed comparison of the results obtained by this method to several other methods yielded roughness parameters within experimental error of each other.^{39}

Graphs of log$(\sigma )$ vs log(scan size) are presented in Figure 4, for (a) iron and (b) gold. Each data point represents the average of multiple locations on the surface. The standard errors on these average values are within the size of the data point. The slope of a linear fit in lower length scale gives the roughness exponent (H), and an exponential fit for the larger length scale gives the asymptotic value of the rms roughness, so-called the saturated rms roughness $(\sigma s)$. The lateral length where the linear fit intersects the saturated rms roughness is the correlation length $(\xi )$. The roughness parameters calculated for iron and gold QCMs are tabulated in Table I.

. | Roughness . | Fractal . | Saturated rms . | Correlation . |
---|---|---|---|---|

. | exponent (H) . | dimension (D) . | roughness $(\sigma s)$ (nm) . | length $(\xi )$ (nm) . |

Iron surface | 0.82 ± 0.02 | 2.18 ± 0.02 | 9.51 ± 0.08 | 97 ± 7 |

Gold surface | 0.85 ± 0.02 | 2.15 ± 0.02 | 1.10 ± 0.13 | 92 ± 7 |

. | Roughness . | Fractal . | Saturated rms . | Correlation . |
---|---|---|---|---|

. | exponent (H) . | dimension (D) . | roughness $(\sigma s)$ (nm) . | length $(\xi )$ (nm) . |

Iron surface | 0.82 ± 0.02 | 2.18 ± 0.02 | 9.51 ± 0.08 | 97 ± 7 |

Gold surface | 0.85 ± 0.02 | 2.15 ± 0.02 | 1.10 ± 0.13 | 92 ± 7 |

## VI. RESULT OF QCM MEASUREMENTS

In this section, we first present the temperature dependence of the frequency and motional resistance of an AT cut, 90 °C, 1 in. QCM with iron surface electrodes in three media: air, PAO-6, and halocarbon oil. We then compare the results with theoretical values for the frequency and motional resistance changes based on the viscosity and density of the liquids involved. The gold QCM response in halocarbon oil is next reported. All shifts depicted in the figures are measured with respect to their values for unloaded crystals at room temperature.

### A. Iron QCM response in air

Figure 5 shows the temperature dependence of the iron QCM frequency and motional resistance in air. The motional resistance remained almost constant and the frequency has a zero temperature coefficient around 90 °C as expected from the manufacturer’s specification (INFICON Research Crystals © 2013). However, the frequency response over the total temperature range (25 °C–200 °C) exhibited a wide variation and a single polynomial fit is unable to correlate the measured values with good agreement. It was found that two polynomials for two temperature ranges: 25 °C–90 °C and 90 °C–200 °C best describe the frequency response. The polynomial fits, which are plotted in Figure 5 for two temperature ranges, are as follows:

### B. Iron QCM measurements in PAO-6 and halocarbon oil

The shifts in frequency and motional resistance of QCM for PAO-6 and halocarbon oil are shown in Figure 6. The measured frequency and motional resistance shifts have been corrected for the temperature dependence of QCM as obtained from the measurements in air. The theoretical curves are obtained using Equation (3) for frequency shift and Equation (4) for resistance shift. The values of density and viscosity used are given in Table II. The values with “*” signs were taken from the manufacturer’s specification data sheets^{40,41} and Kałdoński *et al.*^{24} result, and these were used to extrapolate other values. The densities were obtained by fitting a linear equation of form^{42} $\rho =A+B\xd7T$ and viscosities were extrapolated from ASTM D341-09.^{43}

. | PAO-6 . | Halocarbon-4.2 oil . | ||
---|---|---|---|---|

Temperature . | Density . | Kinematic viscosity . | Density . | Kinematic viscosity . |

(°C) . | (g/cm^{3})
. | (cS) . | (g/cm^{3})
. | (cS) . |

25 | 0.816* | 57.60 | 1.873 | 6.42 |

37.8 | 0.809 | 33.12 | 1.85* | 4.2* |

40 | 0.806* | 30.4* | 1.849 | 3.94 |

50 | 0.801 | 21.21 | 1.833 | 3.01 |

71.1 | 0.789 | 11.32 | 1.8* | 1.9* |

75 | 0.786 | 10.24 | 1.793 | 1.76 |

99 | 0.772 | 6.02 | 1.75* | 1.2* |

100 | 0.769* | 5.9* | 1.753 | 1.20 |

125 | 0.756 | 3.83 | 1.713 | 0.92 |

150 | 0.741 | 2.71 | 1.673 | 0.76 |

175 | 0.726 | 2.04 | 1.633 | 0.67 |

200 | 0.711 | 1.61 | 1.593 | 0.61 |

. | PAO-6 . | Halocarbon-4.2 oil . | ||
---|---|---|---|---|

Temperature . | Density . | Kinematic viscosity . | Density . | Kinematic viscosity . |

(°C) . | (g/cm^{3})
. | (cS) . | (g/cm^{3})
. | (cS) . |

25 | 0.816* | 57.60 | 1.873 | 6.42 |

37.8 | 0.809 | 33.12 | 1.85* | 4.2* |

40 | 0.806* | 30.4* | 1.849 | 3.94 |

50 | 0.801 | 21.21 | 1.833 | 3.01 |

71.1 | 0.789 | 11.32 | 1.8* | 1.9* |

75 | 0.786 | 10.24 | 1.793 | 1.76 |

99 | 0.772 | 6.02 | 1.75* | 1.2* |

100 | 0.769* | 5.9* | 1.753 | 1.20 |

125 | 0.756 | 3.83 | 1.713 | 0.92 |

150 | 0.741 | 2.71 | 1.673 | 0.76 |

175 | 0.726 | 2.04 | 1.633 | 0.67 |

200 | 0.711 | 1.61 | 1.593 | 0.61 |

For both oils, the measured values of motional resistance shifts are in good agreement with the theoretical values, which is to be expected since the calibration procedure used would mask potential surface roughness effects. The observed frequency shifts are however always slightly greater than the calculated values for both liquids, as discussed earlier in association with surface roughness effects.^{28,31,32} Hydrostatic pressure of the liquid could potentially contribute to the frequency shift but it is not in the direction observed^{34} and moreover is small in magnitude compared to the shifts due to liquid loading for the conditions studied.

A possible way of finding the surface roughness contribution for the frequency shift as proposed by Martin *et al.*^{28} is to include an additional term proportional to the density of the liquid. The relation between the frequency shift of the crystal and liquid parameters will then be given by

where the first term is for the laminar contribution for the smooth surface as given in Equation (3), the second term is the non-laminar contribution proportional to the density ($\rho L$) of the liquid, and *D* is a constant that depends on the roughness^{28} of the surface. Ethanol and deionized water were used for calibration to find the value of $D$ to be $155\xb123g\u22121cm3s\u22121$ for iron coated crystals, which yielded an approximately 22% contribution.

Figure 7 depicts the dependence of QCM frequency on $\rho \eta $ for PAO-6 and halocarbon oil for the range of temperatures studied. The variation of $\rho \eta $ arises primarily from changes in viscosity, as changes in density are much smaller than changes in viscosity. The theoretical line for a planar surface, obtained from Eq. (3), which is linearly proportional to $\rho \eta $ with a slope of $\u22127112g\u22121cm2s\u22121/2$ is depicted by the solid line in Fig. 7. A roughness correction term as described by Equation (7) would add a vertical offset to the solid line with nearly the same slope because the variation in density is small. However, the trend in observed values suggested a larger correction factor for higher viscosities. The liquid velocity decay length $\delta L$ = $(\eta L/\pi f\rho L)$^{1/2} increases with the viscosity yielding higher mass loading, and the roughness correction factor is observed to be scaling with it. The effect of roughness is therefore not captured solely by Equation (7) with a constant additive but is well described by a multiplicative constant of the form

The dashed line in Fig. 7 depicts Eq. (8) employing K = 1.22, a roughness correction reflecting the 22% calibrated value obtained for ethanol and deionized water. The dashed line fits the data well, yields a larger correction factor for higher viscosities, and is intuitively consistent with higher effective mass loading for the higher penetration depths associated with higher viscosity liquids suggested by Daikhin and Urbakh.^{31,32}

### C. Gold QCM measurements in halocarbon oil

We next compare the frequency and motional resistance responses of the gold QCM with their theoretical values in halocarbon oil based on the method of temperature compensation and roughness contribution described above.

Figure 8 shows the gold QCM response in air and halocarbon oil. The frequency response in air showed that the crystal has zero temperature coefficient at around 60 °C, and the two polynomials fit for two temperature ranges are

The measured frequency shifts in halocarbon oil are corrected for the temperature dependence of QCM. A 19% roughness contribution is obtained for the gold-coated crystals using Equation (7) followed by calibration with DI water and ethanol. This correction is applied to the theoretical values obtained from Equation (3) to get the corrected theoretical curve plotted in Figure 8(c) (dashed line). Figures 8(c) and 8(d) show an agreement between the theoretical values and the measured values of frequency and motional resistance shifts in halocarbon oil, except for temperatures close to 25 °C. This is potentially an artifact resulting from the fact that the oven heating rate is highest at room temperature, increasing the likelihood of a temperature lag between the thermometer and the liquid. Further studies would be required to definitively confirm the origin of this effect.

## VII. COMPARISON OF MODELS ON THE SURFACE ROUGHNESS EFFECT ON QCM

Consistent with prior literature reports, the frequency shifts observed here in liquid environments are larger than those theoretically predicted for a planar surface. We next comment on how theoretical predictions that incorporate surface roughness parameters compare to the present results.

The roughness contribution to shift in resonance frequency for iron and gold QCMs obtained on the basis of the approach of Martin *et al.* followed by the calibration with DI water and ethanol successfully explained the observed deviations in the frequency shift. This approach however does not account for the surface roughness parameters in the equations.

Daikhin and Urbakh^{31,32} accounting for the multiscale nature of surface roughness also predicted the roughness correction to the shift in resonance frequency of a QCM. Figures 3(c) and 3(d) showed that the gold and iron surfaces have features with smaller vertical height compared to their width. Both of the surfaces have features of width of about 100 nm, but the height of these features is within 6 nm for gold and 60 nm for iron surfaces. The calculated saturated rms roughness for both surfaces is smaller than their lateral correlation lengths. The values of $\sigma s/\xi $ are 0.012 for gold and 0.098 for iron surfaces. This suggested that both of these surfaces are closer to the “slight roughness” regime of Daikhin and Urbakh model defined by $\sigma s/\xi <<$1.

In this regime, the corrected shift in resonance frequency of a QCM is given by

and

The first term on the right hand side of Equations 10(a) and 10(b) is same as Equation (3), and the remaining terms represent the correction due to the roughness of the QCM surface. Both Equations 10(a) and 10(b) reduce to Equation (3) for $\sigma s$ = 0, a geometrically flat surface.

The values of $\delta L$ for the liquids studied in the temperature range are in the range 0.2-2 $\mu m$ for a 5 MHz QCM. The value of $\xi /\delta L$ is in the range: 0.045 < $\xi /\delta L$ < 0.45, which is suited for Equation 10(a). The predicted corrections to the shift in resonance frequency calculated from Equation 10(a) for the liquids (properties in Table II) and QCMs (roughness parameters in Table I) used in this work were calculated to be in the range of −0.03 Hz to −13.14 Hz. These figures are consistent with other theoretical^{33} and experimental^{44} results for the “slight roughness” cases. But, these corrections are small and not sufficient to account the deviations in frequency shift observed in this work. The plausible reason could be that these surfaces may not strictly be in the “slight roughness” regime and the framework under which the corrections are evaluated may not be well suited for these surfaces. A study by Rechendorff *et al.*^{44} revealed that the roughness correction could be in the order of several hundreds of Hertz for remarkably rough surfaces, and the “strong roughness” regime of this model had successfully explained this correction. The expected corrections of hundreds of Hertz for this work are possibly resolved by introducing an intermediate regime between “slight” and “strong” roughness. In particular, the regime that an interfacial layer is in contact with both the quartz crystal and the bulk liquid, and the frequency shift is caused by the inertial motion of the liquid trapped by the inhomogeneity in the interfacial layer. In this particular case, the layer induced frequency shift is given by

where L is the thickness and $\xi H$ is the local permeability of the interfacial layer. L − $\xi H$ is the effective thickness of the liquid film rigidly coupled to the oscillating surface. (See Ref. 32 for details.)

Equation (11) is same as the Sauerbery equation^{1} (Eq. (2)) and reveals the shift in frequency is proportional to the density of the liquid with the effective thickness of liquid layer assumed constant. A calculation of L − $\xi H$, using the value of $\delta fLayer$ to be hundreds of Hertz (order of correction expected in this work), yields the value: 9.43 nm < (L − $\xi H)$ < 21 nm, for the liquids studied. These are within the range associated with the surface characteristics of the QCMs, and this approach can likely explain the observed roughness correction. It has been observed that this “simple additivity model”^{44} with a rigidly coupled liquid layer provides a better description in the cases of slight to intermediate roughness.

Another notable point is the fact that the roughness corrections for the iron and gold QCMs do not differ significantly from each other, even though their saturated rms roughness values are in ratio of about 9:1 and correlation length is nearly equal. The Daikhin and Urbakh model prediction for the roughness correction is quadratic in $\sigma s$ for the “slight roughness” regime, which means that the correction for iron QCM would be significantly larger than that observed for the gold QCM if the surfaces were in this regime. This behavior was not however observed. The ratio of needed corrections for iron and gold QCMs is only 1.2:1. This can possibly be explained by considering the fractal characteristic, as these two surfaces do not differ much in fractal dimension, which is not addressed in the Daikhin and Urbakh model. Including the fractal characteristic together with the rms roughness and correlation length in the roughness correction analysis would be an interesting point for future investigation.

## VIII. CONCLUSION

A QCM has been successfully employed to study lubricant oils at high temperature up to 200°C in a setup comprised of materials that can withstand 300°C. The resonant frequency of QCM varies with temperature, and it was found that two separate polynomials for two temperature ranges are required to model this variation. The measured frequency shifts for PAO-6 and halocarbon oil were greater than the values predicted by liquid loading formula suggesting a contribution of the surface roughness of the QCM electrode. The roughness contribution to the frequency shift was calculated to be 19%–22% for the crystals used. The results were compared to various theoretical models incorporating surface roughness effects and are in reasonable agreement. Models that incorporate fractal scaling characteristics of the surfaces would be highly useful to further illuminate the role of surface roughness in QCM response. This study meanwhile provides a straightforward solution for extending the liquid QCM technique to various high temperature liquid studies.

## ACKNOWLEDGMENTS

This work was supported by Eastman Chemical Company and National Science Foundation No. DMR1535082. We are thankful to Colin K. Curtis for useful discussions and to Halocarbon Products Corporation and Chevron Phillips Chemical Company for supplying the oil samples.