A
Pulsed Laser Viscometer:Accurate High-Speed Sensing Technique for Viscosity
and Surface Tension of Liquids
Yuji
Nagasaka
1✉
Emailyuji_nagasaka@keio.jp
Kazunori
Shibata
1
Yoshihiro
Taguchi
1
1
A
Department of System Design Engineering
Keio University
3-14-1 Hiyoshi
223-8522
Yokohama
Japan
Yuji Nagasaka1*, Kazunori Shibata1, Yoshihiro Taguchi1
1Department of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Yokohama 223–8522, Japan
*Corresponding author E-mail: yuji_nagasaka@keio.jp
A
Abstract
The pulsed laser viscometer (PLV) measures viscosity and surface tension using nanoscale liquid surface deformation generated by two-beam interference of a pulsed heating laser and detects damping oscillation from the first-order diffracted intensity of a probing laser. PLV features (1) non-contact and in situ measurement, (2) high spatial resolution of 10 to 100 µm, (3) high time resolution of microseconds to milliseconds, (4) minute sample volume of microliters to milliliters, and (5) a wide viscosity range of 0.1 to 104 mPa·s. We established the PLV theory by solving the Navier‒Stokes‒Thermoelastic equation for volumetric heating of a liquid by a weakly absorbed pulsed laser. A new PLV experimental apparatus was developed using a pulsed YAG laser with a wavelength of 1064 nm as the heating laser, and experiments used six liquid samples (acetone, toluene, water, ethanol, 2-propanol, and 1-hexanol) at room temperature to verify that the instrument operates according to the theory. Measured viscosities agree with the literature values with an average deviation of 3.2%. Measured surface tension agrees with the literature values with an average deviation of 6.2%. Uncertainties are estimated at 2% for viscosity and 3% for surface tension. The temperature rise of the sample during the experiment was less than 0.01 K. Furthermore, eight other thermophysical properties required in the PLV theory (such as absorption coefficient and volume expansion coefficient) only shift the signal waveform vertically, and the influence of viscosity and surface tension on the curve-fitting results remains dominant.
Keywords
Grating excitation techniques
Optical methods
Pulsed laser viscometer
Surface tension
Viscosity
Transport properties
A
1 Introduction
Numerous types of viscometers, including oscillating-body, vibrating, and falling-body viscometers, are well established, and some are regarded as standard viscometers [1, 2]. Users can choose among them depending on the viscosity range of their sample and required uncertainty. These conventional viscometers are quite useful for measuring a wide range of liquid viscosities with the required accuracy, but they are generally unsuitable for non-contact, small-sample, high-speed measurements for processes where viscosity changes dramatically and rapidly. Numerous possible examples of such processes span a wide variety of industries. The viscosity of liquid foods, for example, is a fundamental thermophysical property in food industries because it is an essential physical parameter in the design and control of processing plants and is useful in evaluating food quality [3, 4]. Another example occurs in the manufacture of highly functional polymeric films such as optical films. To precisely control and optimize the microstructure of highly functional polymer-solvent films, it is essential to understand surface properties such as viscosity and surface tension during dynamic drying in real time [5–7]. To fabricate transparent and conductive films (TCFs) from single-wall carbon nanotubes (SWCNTs), viscosity and surface tension play crucial roles in uniform TCF formation during coating [8].
New viscometers that fill these demands can be regarded as an extension of the concept of “thermophysical properties sensing,” defined broadly by the following essential characteristics: noninvasiveness (mainly optical methods), high temporal and spatial resolution, 2D or 3D distribution measurements, and small sample volume measurement [9]. Many studies focus on non-contact viscometers that detect the dynamics of capillary waves on the liquid surface. For example, Rhim et al. [10] measured the viscosity and surface tension of molten zirconium up to about 2200 K by detecting the characteristic oscillation frequency and the decay time of an electrostatically levitated liquid zirconium drop. The drop oscillation was externally induced by superimposing a small sinusoidal electric field on the levitation field. Another non-contact (optical) technique for measuring liquid viscosity and surface tension involves detection of spontaneous capillary waves (ripplons) caused by thermal fluctuations. After Katyl and Ingard initiated this surface laser-light scattering (SLLS) technique [11, 12], it has been applied to a wide variety of liquids, including polymer solutions, liquid crystals [13], high-temperature molten silicon, and lithium niobate [14–16], as well as to the precise measurement of toluene [17]. Additionally, a high-precision instrument for measuring surface tension, viscosity, and surface viscoelasticity with tunable wavelength selection was reported [18]. However, when the viscosity of the liquid is relatively high and the surface tension relatively small, this technique is somewhat limited because the spontaneous capillary waves are overdamped under such conditions [19].
To overcome the difficulties of the SLLS technique described above, a promising alternative is to replace weak random fluctuations with strong, coherent excitation via a laser-induced grating—that is, forced Brillouin scattering. The first forced Brillouin scattering experiment was performed by Terazima’s research group in 1999 [20, 21]. They generated a capillary wave by crossing two excitation beams of the third harmonics of a Nd:YAG pulsed laser and observed a strong overdamped wave from the 1-hexanol surface in the presence of added light-absorbing materials at large wave numbers. Subsequently, they clarified the mechanism underlying capillary wave generation by laser excitation by solving the phenomenological equations for the case of strong surface heating and compared the observed signals with the theoretically calculated waveforms [22]. Meanwhile, our group independently began developing a new non-contact technique for measuring a wide range of viscosities by observing laser-induced capillary waves (LiCW) generated by the two-beam interference of a pulsed carbon dioxide laser [23]. By using a far-infrared wavelength laser beam (10.6 µm) to excite the IR absorption bands of a liquid sample, capillary waves can be induced in pure liquids without adding any light-absorbing substances (dyes). The signals detected from several liquid samples (e.g., acetone, toluene, 1-hexanol, ethylene glycol, JS1000, JS14000) over the viscosity range of 0.33–7080 mPa·s were in excellent agreement with theory [24]. The time required to acquire a single waveform ranges from tens of microseconds to one millisecond, extremely short compared to conventional viscometers. We also developed a technique to solve the inverse problem of extracting unknown viscosity and surface tension values from the detected waveforms [25] and measured the apparent viscosity of milk fermenting into yogurt. The relative increase in apparent viscosity was more than 100-fold within a few hours [26].
The series of experimental studies on the LiCW technique using the pulsed carbon dioxide laser demonstrated that LiCW is potentially a practical viscometer. Although LiCW offers excellent performance characteristics, its accuracy as a viscometer is compromised by strong surface pulse heating. The temperature rise can be several degrees to several tens of degrees, which may not be acceptable for accurate viscosity measurement. This limitation arises because the temperature dependence of liquid viscosity cannot be neglected; for example, (1/η)(∂η/∂T) is approximately 2% for water at 20°C. This can cause serious problems, particularly in biological fluids such as blood and protein solutions, due to thermal denaturation. Therefore, we developed an improved LiCW technique for volumetric heating using a pulsed YAG laser whose wavelength (1064 nm) exhibits fairly weak absorption in various liquids, including human whole blood, and the temperature increase is less than 0.01°C [27, 28]. In this paper, we redefine this volumetrically heated LiCW as the Pulsed Laser Viscometer (PLV) and present appropriate working equations to theoretically calculate the behavior of surface displacement to determine the viscosity and surface tension of liquids with an accuracy of within a few percent. We also developed a PLV instrument and experimentally verified the theoretical calculations with the measurements of six Newtonian liquids at room temperature.
PLV allows simultaneous non-contact measurement of viscosity and surface tension using a single instrument without changing the measurement parameters: (1) a very short time (µs – ms), (2) high spatial resolution (10–100 µm), (3) small sample volume (µl – ml), (4) small temperature rise (< 0.01 K), and (5) a wide viscosity range (10− 1 to 104 mPa·s). All these properties characterize the PLV as an innovative and accurate viscometer for sensing thermophysical properties. It should be noted that PLV is a grating excitation technique (GET) [29, 30] along with Forced Rayleigh Scattering (FRS: thermal conductivity) [31, 32] and Soret Forced Rayleigh Scattering (SFRS: mass diffusion coefficient and Soret coefficient) [33]. “Grating Excitation Techniques” refers to methods for measuring transport properties by generating temperature, concentration, and velocity gradients through two-beam interference.
2 Principle of the PLV
Figure 1 illustrates the principle behind the PLV, which enables measurement of both the viscosity and surface tension of a liquid sample. In this method, the liquid is initially at a uniform temperature, and pulsed heating laser beams of equal intensity and equal wavelength intersect on a weakly absorbing liquid surface at a crossing angle θ. This configuration generates an energy distribution within the volume-grating interference region with a grating period Λ (10 µm to 100 µm in the present experimental setup). At this instant, the liquid sample is heated not only on the surface but also in the depth direction, depending on the optical absorption length (= 1/α, α : absorption coefficient), which ranges from 10− 3 to 10− 1 m, since the wavelength of the pulsed heating laser is selected to be weakly absorbed within the sample volume, as depicted in Fig. 2.
Fig. 1
Principle of pulsed laser viscometer (PLV)
Fig. 2
Volumetric heating model
The diameter of the heating laser beam is about 5 mm at the liquid surface in the present experimental setup; therefore, the heating volume is approximated as a cylinder with a diameter of 5 mm and a length of 1 cm (as a typical example of α = 10 m− 1), within which approximately 100 lines of lattice heating structure are present in the case of Λ = 50 µm. The pulse energy distribution within the interference volume of the sample, Eh(x,z), generated by two pulse heating laser beams of equal intensity Eh (J·m− 2), can be written as [33]:
1
2
where k denotes the wavenumber of the interference pattern and λh is the wavelength of the pulsed heating laser. At the end of the nanosecond-order heating laser pulse, the corresponding initial spatially sinusoidal temperature distribution induced by the volume grating pattern is:
,
3
4
where ΔTP is the initial spatial temperature amplitude generated by short-pulse laser heating [29].
The temperature grating creates a spatially modulated displacement on the liquid surface driven by thermal expansion and the temperature dependence of surface tension. The spatially modulated displacement along the z-direction uz1(z, t) is on the order of nanometers and will be rigorously derived later in this section. The spatially periodic displacement produces a dynamic reflection grating that diffracts a probing laser beam with a different wavelength than the heating beams. The intensity of the first-order diffraction of the probing laser beam I1(t) (W·m− 2) is directly proportional to the square of the surface displacement, as expressed in the following equation:
.
5
Immediately after the generation of the initial grating structure on the liquid surface, the grating begins to propagate as surface waves. The temporal behavior of these waves, in the case of a Newtonian liquid, is primarily governed by shear viscosity (damping) and surface tension (restoration). Therefore, in essence we can determine both viscosity and surface tension by analyzing the temporal behavior of I1(t) and the grating period Λ.
The temperature change generated by the weakly absorbed pulse-heating laser ΔT(x, z, t) can be described using the two-dimensional heat conduction equation (Eq. 6), based on the following assumptions:
(1) The grating period Λ is much smaller than the depth of the liquid sample d (Λ/d < < 1).
(2) Λ is sufficiently smaller than the optical absorption length α− 1 at the heating laser wavelength (Λ/α−1 < < 1).
(3) The diameter of the heating laser beam 2Wh is much larger than the grating period Λ (Λ/2Wh < < 1), ensuring uniform energy density within the heated region.
(4) The absorption of the heating laser energy by the sample liquid is weak (1/α < d), corresponding to volumetric heating conditions.
.
6
Here, ρ (kg·m− 3) is the density of the sample, cp (J·kg− 1·K− 1) is the specific heat capacity at constant pressure, λ (W·m− 1·K− 1) is the thermal conductivity, 
, and
represents the delta function. The energy input from the pulsed heating laser is expressed by the second term on the right-hand side of Eq. 6, indicating that the sample volume is simultaneously heated sinusoidally in the x-direction and exponentially in the negative z-direction at time t = 0.
In carefully considering the hydrodynamic motion of the sample in the process of PLV measurement, the following assumptions can be used to theoretically model the laser-induced capillary wave for volumetric heating by a pulsed laser:
(1) The displacement of the generated wave is much smaller than its wavelength (uz1/Λ << 1), so that the motion of the induced capillary wave can be expressed by the Navier‒Stokes equation with the Stokes approximation, neglecting the convective and the gravitational terms.
(2) The stress and strain in the sample (within a very short time domain) caused by transient temperature variations can be described by the thermoelastic equations [34], with the shear modulus set to zero for liquid.
(3) The temperature rise generated by the interference of two pulsed laser beams is small, so that the thermophysical properties, especially viscosity and surface tension, are assumed to be constant.
Under these assumptions, the Navier‒Stokes‒Thermoelastic equation describing the PLV measurement with respect to the displacement vector u(x, z, t) (m) can be expressed as:
,
7
where ν (m2·s− 1) is the kinematic viscosity (= η/ρ: η (Pa·s) is the viscosity), VL (m·s− 1) is the velocity of longitudinal wave propagation in the sample, β (K− 1) is the volume thermal expansion coefficient, and τ (s) is the decay time constant of PLV. The elastic stress and thermal stress terms are described by the third term on the left-hand side and the first term on the right-hand side of Eq. 7, respectively. The second term on the right-hand side of Eq. 7 is the pulsed laser volumetric heating‒induced body force term, which is an essential contribution distinguishing it from the surface heating [22, 24] that expresses the volumetric heating effect.
The stress boundary conditions at the gas‒liquid interface are derived from the stress balance caused by viscous shear stress and by stresses present in the deformed interface: the stress due to the surface tension gradient, the thermoelastic stress, and the additional volumetric heating effect stress corresponding to the second term on the right-hand side of Eq. 7. Assuming that stresses in the gas phase are negligible compared with those in the liquid phase, the normal and tangential stress boundary conditions are given by: [35, 36]
,
8
9
where σ (N·m− 1) is the surface tension of the sample liquid.
For temperature boundary conditions, it is assumed that the temperature change far from the liquid surface is zero and that heat conduction from the liquid to the gas phase is negligible due to the low thermal conductivity of the gas.
10
11
It is also assumed that the velocity of the liquid is zero sufficiently far from the surface.
12
Assuming the sample liquid is initially at equilibrium, the initial conditions for surface displacement and temperature change are given by the following two equations:
13
14
Because the optical excitation in Eq. 1 is spatially sinusoidal in the x-direction, both the resulting temperature rise and liquid surface displacement are also sinusoidal in x. Thus, the temperature rise and surface displacement can be expressed by Eqs. 15‒17, with subscripts 0 and 1 denoting the background (uniform) and spatially modulated components, respectively [37].
15
16
17
The surface displacement of the laser-induced capillary wave under volumetric heating can be analytically obtained by solving Eqs. 6 and 7 with the boundary and initial conditions of Eqs. 8–14, using a Laplace transform following the method described in Ref. 22. Only the spatially modulated component (subscript 1) is detected experimentally as a change in diffracted light intensity, and its Laplace-transformed solution takes the following form:
,
18
19
where s is the Laplace parameter and b1 through b4 and b6 are given by:
.
20
21
22
23
24
Here, a (m2·s− 1) is the thermal diffusivity of the liquid.
These 10 coefficients, each multiplying an exponential function in Eqs. 18 and 19, are analytically determined from a system of 10 simultaneous equations, including the heat conduction equation (Eq. 6), the Navier‒Stokes‒Thermoelastic equation (Eq. 7), and boundary conditions for each component (x, z directions and 0, 1 modes) all expressed in Laplace-transformed form. The final analytical solutions are presented in Appendix B, and the corresponding FORTRAN 90 code is included in Appendix C.
3 Curve-Fitting Method to Determine Viscosity and Surface Tension Using Inverse Laplace Transform (FILT) with Downhill Simplex Method
To determine the viscosity and surface tension of a sample liquid from a PLV experiment, experimental data for the change in diffracted light intensity versus time {ti, (I1)i} at a certain grating period Λ must be curve-fitted to the theoretical formula described in the previous section. The time dependence in the z-direction displacement of the liquid surface uz1(t), which is required for the fitting, can be calculated numerically from the Laplace space solution 
at z = 0 presented in Appendix B using the following FILT algorithm described in [38]
25
and
.
26
Here, i is the imaginary unit, fec(t, b) is an approximation of the target function f(t), 
is the Laplace-transformed function of f(t) and b is the parameter for approximation that represents the estimated precision of digits in the calculation result.
We adopted the multidimensional downhill simplex method [39, 40] as the curve-fitting technique because the function to be minimized is multivariable and its derivative is analytically difficult to calculate. This method requires only function evaluation, not the derivative of the function. The curve fitting involves finding the set of variables that minimizes the standard deviation between the experimental data set of time ti and voltage signal Vi corresponding to diffracted light intensity and the calculated value of uz12, which are viscosity η (or kinematic viscosity ν), surface tension σ, and Cf while other variables are held fixed (see Sect. 5.2). The standard deviation Sd that should be minimized can be expressed by the following equation.
27
Here, Cf (V·m− 2) means the conversion factor between uz12 and the output voltage V. For the programming language used in the curve fitting, we chose Fortran 90, which is considered relatively mature for optimizing code involving complex calculations.
4 Experimental apparatus
Fig. 3
Experimental setup for the pulsed laser viscometer (PLV). BPF: bandpass filter; BS: beam splitter; DMM: digital multimeter; DPO: digital phosphor oscilloscope; FI: Faraday isolator; HWP: half-wave plate; M: mirror; PC: personal computer; PMT: photomultiplier tube; PRT: platinum resistance thermometer; ||: horizontal to the optical bench surface; ⊥: vertical to the optical bench surface
The experimental setup of the pulsed laser viscometer (PLV) developed in the present study is shown in Fig. 3. A Q-switched pulsed Nd:YAG laser (Litron Lasers Ltd., Nano-L-200-10; λh = 1064 nm, pulse width 6 ns, maximum output energy 200 mJ and beam diameter (1/e2) 5 mm) is employed as a heating light source to generate a nanoscale sinusoidal displacement on the liquid surface with repetition rates from a single shot to 10 Hz. The nanosecond pulse width is short enough to instantaneously induce the capillary wave, because the photon‒phonon relaxation time is on the order of picoseconds and the oscillation period of the capillary wave is on the order of microseconds to milliseconds. The vertically polarized heating laser beam is divided into two beams of equal intensity using a non-polarized beam splitter (BS: CVI Melles Griot Ltd., BS1-1064-50-1525-45P), which is set up to cross at the liquid sample surface by means of six mirrors (M1 to M6, M4 tilts the laser beam vertically upward) to generate an optical interference zone. The adjustable range of the grating period is about 10 to 200 µm, corresponding to a beam crossing angle of about 6.1° to 0.3°. The probing laser is a continuous-wave solid-state laser (Coherent Ltd., CUBE 640-100C; λp = 641 nm, maximum output power 100 mW, and beam diameter (1/e2) 2 mm) whose output passes through a Faraday isolator (FI: Edmund Optics Ltd., 650NM) and a half-wave plate (HWP: Thorlabs Ltd., WPH05M-633) to ensure s-polarized probing light reflection from the liquid surface. A function synthesizer (NF Corporation, WF1946) is used to turn the CW probe laser on and off in synchronization with the pulses of the heating YAG laser.
The probing laser beam is redirected vertically upward by M7 and is incident on the liquid surface at about 60 degrees from the vertical using M8. The first-order diffracted beam is detected by a photomultiplier tube (PMT: Hamamatsu Photonics, R9110) through an iris (Thorlabs Ltd., SM1D12CZ: adjustable iris diameter from 0 mm to 12 mm) and an optical bandpass filter (BPF: Asahi Spectra Co. Ltd., MX0640). The output current signal from the PMT is amplified by a current-voltage amplifier (Hamamatsu Photonics, C9663), recorded by a digital phosphor oscilloscope (DPO: Tektronix Inc., DPO3034), and transferred to a personal computer. Mirrors M5, M6, M8, and M9 are placed on a breadboard fixed vertically on a horizontal optical table. All other optical systems are installed on the horizontal optical table.
The sample liquid is filled in a quartz petri dish with an outer diameter of 50 mm and a height of 8 mm. The sample volume required for the measurement is approximately 10 ml. Because the quartz glass cell transmits roughly 94% of the heating YAG laser light, holes must be drilled in the insulation and in the temperature-controlled sample bath directly below the cell to prevent unwanted reflection or absorption of the laser light. As shown in Fig. 3, the sample stage is basically a stainless steel hollow cylinder with an outer diameter of 150 mm and a height of 33 mm. However, the top surface of the stage is a truncated cone with an angle of approximately 43 degrees to maintain the optical path of the obliquely incident probing laser beam, and a hole with a diameter of 20 mm is drilled at the center to allow the heating laser beams to pass through. This stage is equipped with an inlet and an outlet for circulating water maintained at a constant temperature, and the volume of the constant-temperature water held is about 163 cm3. A 10-mm-thick glass fiber insulation material (thermal conductivity 0.08 W·m− 1·K− 1) is attached to the underside of the stage, and 3-mm-thick insulating tape is attached to the sides of the stage. A black-light-absorbing sheet with a thickness of 0.51 mm is attached to the surface of the truncated cone to absorb the scattered light from the heating and probing laser beams. A beam diffuser (Sigma Koki, BD-40) is installed approximately 42 mm from the bottom of the dish to scatter and absorb unwanted heating laser light energy that has passed through the sample and the dish. The entire temperature-controlled stage is mounted on a Z-axis stage, allowing the liquid level to be controlled with an accuracy of 0.01 mm.
Water from a thermostat bath (Tokyo Rikakikai, NCB-1200) flows into the stage to control the temperature of the sample. The stability of the thermostat bath temperature is ±0.1°C. The sample temperature is monitored using a thin, flexible platinum resistance thermometer (0.7 mm thickness and 3 mm width, 100 Ω) (Nestushin, Aichi, Japan, NFR-CF2-0305-50-100S-1-2000TF-A-4-Mukidashi) attached to the stage with Kapton tape as close as possible to the petri dish. The resistance is measured using a digital multimeter, and the temperature is determined with an uncertainty of ±0.1°C.
To obtain viscosity and surface tension measurements using PLV, it is necessary to accurately determine the grating period Λ separately from the time dependence of the first-order diffracted light intensity I1(t). After optical alignment, a black PMMA plate is placed at the same position as the liquid surface where the heating YAG laser beams intersect, and a laser pulse is then fired to imprint a diffraction pattern onto the plate. This procedure replicates the interference conditions used during actual measurement and produces visible fringes on the PMMA surface. The resulting fringe pattern is observed using a color 3D laser microscope (Keyence, DVK-9710, display resolution of 0.001 µm), and the average spacing of 10 fringes is used as the measured grating period.
5 Results and discussion
5.1 Experimental Results for Six Newtonian Liquids
A
To verify whether the developed PLV experimental apparatus is operating in accordance with the proposed PLV theory described in Sect. 2, the viscosity and surface tension of six pure Newtonian liquids listed in Table 1 were measured at room temperature and atmospheric pressure. All liquid samples except water were supplied by Wako Chemical Co., Ltd. with the stated purities listed in Table 1. The values of viscosity, surface tension, and density of the six liquids at 298 K and 101 kPa were taken from Ref. 2, except for 1-hexanol [41]. The PLV experimental signals and theoretical curve-fitting results for each liquid are shown in Figs. 4–9, respectively.Fig. 4
Comparison of PLV experiment with fitting curve for acetone at 16.2℃ (Λ = 52.6 µm)
Fig. 5
Comparison of PLV experiment with fitting curve for toluene at 22.3℃ (Λ = 52.6 µm)
Fig. 6
Comparison of PLV experiment with fitting curve for water at 22.9℃ (Λ = 52.6 µm)
Fig. 7
Comparison of PLV experiment with fitting curve for ethanol at 21.8℃ (Λ = 52.6 µm)
Fig. 8
Comparison of PLV experiment with fitting curve for 2-propanol at 22.0℃ (Λ = 52.6 µm)
Fig. 9
Comparison of PLV experiment with fitting curve for 1-hexanol at 24.3℃ (Λ = 52.6 µm)
A
In all experiments, the grating period Λ was 52.6 µm, the pulse energy of the heating laser pulse was 124 mJ, and the waveform was averaged 16 times. Simultaneous curve fitting was performed using the Downhill Simplex method described in Sect. 3, with viscosity η, surface tension σ, and signal strength conversion factor Cf in Eq. 27 as fitting parameters. In all calculations, the FILT approximation parameters in Eq. 25 were set as b = 4 and K = 10,000 (maximum iteration number) to ensure four-digit numerical accuracy. The fractional convergence tolerance for the Downhill Simplex method was set to 10− 6. The remaining eight thermophysical properties were set to the fixed values listed in Table 2, and the PLV time constant was determined by trial and error and used as a fixed value τ = 10− 7 s for all liquids. It should be noted that the vertical axis represents the voltage proportional to the diffracted light intensity and that the data are not normalized to the peak value. The influence of fixed physical property values on fitting is explained in the next section, but it is clear that numerical values with one significant digit are sufficient for all of them (calculations cannot be performed without input, but this does not affect the final result). Upon initial inspection, PLV appears to require many additional thermophysical parameters to evaluate viscosity and surface tension, which might seem disadvantageous; however, this has been demonstrated not to pose a practical problem.Fig. 10
Comparison of measured viscosity with reference values
Fig. 11
Comparison of measured surface tension with reference values
Figures 10 and 11 respectively compare viscosity and surface tension measured by PLV to reference values [2, 41]. In the viscosity range from approximately 0.3 mPa·s for acetone to approximately 5 mPa·s for 1-hexanol, the PLV experimental results and reference values were in agreement with an average deviation of 3.6%. In the surface tension range spanning 2-propanol (∼21 mN·m− 1) to water (∼72 mN·m− 1), the average deviation between PLV and reference values was 6.2%. By taking into account the following sources of uncertainty—measurement of the grating period (within 0.2%), decay time constant determination from curve fitting (1‒2%), and oscillation frequency determination from curve fitting (2‒3%)—we estimated that the relative expanded combined uncertainty of viscosity (with a coverage factor of k = 2) was 2%, and that of surface tension was 3%.
5.2 Sensitivity Analysis of Eight Input Thermophysical Properties other than Viscosity and Surface Tension
A
As mentioned in the previous section, a quick look at the formulas for the PLV shows that many additional thermophysical property values must be supplied to determine viscosity and surface tension. This could be considered a drawback of the PLV method. Moreover, no data from the literature are available for these thermophysical properties, such as the temperature dependence of surface tension or the speed of sound, and some parameters, such as the absorption coefficient, may require separate measurements. To address this, we performed numerical calculations using the PLV theoretical equation and analyzed the effect (i.e., sensitivity) of these thermophysical property values on the viscosity and surface tension determined by curve fitting, as well as the accuracy or recommended values that the input values should satisfy. The results are summarized in Table 3.As the table shows, five thermophysical properties, namely absorption coefficient α, volume thermal expansion coefficient β, speed of sound VL, thermal diffusivity a, and thermal conductivity λ, only shift the signal waveform by a constant factor along the vertical axis. In other words, these parameters do not affect the fitting results. Therefore, it is considered sufficient for these input parameters to have significant figures of about one digit. This means, in practice, that by adjusting the conversion factor Cf of the function to be minimized in Eq. 27 in Sect. 3, it is possible to perform curve fitting that depends only on the viscosity and surface tension. In addition, the temperature dependence of surface tension ∂σ/∂T is not significant because the temperature rise is very small in PLV. However, an error will occur if some numerical value is not entered; therefore, for example, -0.1 mN·m− 1·K− 1 should be entered. The specific heat capacity at constant pressure cp is necessary only for calculating the absolute value of the temperature rise and does not play a role in the curve fitting. Although density ρ is not involved in the curve fitting, it is necessary to convert kinematic viscosity to viscosity, so data with an uncertainty of within 1% are required. For ordinary liquids, this is likely easy to obtain.
6 Conclusions
A novel high-precision PLV system was developed to simultaneously measure the viscosity and surface tension of liquids without altering measurement conditions: (1) microsecond to millisecond timescale, (2) 10– 100 µm spatial resolution, (3) microliter to milliliter sample volumes, (4) < 0.01 K temperature rise, and (5) a wide viscosity range (10− 1 to 104 mPa·s). The basic measurement principle of PLV is to generate a spatial sinusoidal displacement on the liquid surface on the order of nanometers by using two-beam interference of a pulsed YAG laser, which is weakly absorbed by liquids. The method then determines viscosity and surface tension by detecting the damped oscillation through first-order diffracted light intensity of the reflected probing laser. Until now, the mechanism of liquid surface displacement caused by weakly absorbed short-pulse laser heating has not been fully understood. However, by solving the newly derived Navier‒Stokes‒Thermoelastic equations presented here, we completed the working equation for PLV. We performed PLV measurements on six Newtonian liquids (acetone, toluene, water, ethanol, 2-propanol, and 1-hexanol) at room temperature and ambient pressure, determining the viscosity and surface tension by curve fitting the signal waveforms using the theoretical model. The measured viscosities agreed with literature values with an average deviation of 3.6%, while surface tension values agreed with literature values with an average deviation of 6.2%. The estimated uncertainties are 2% for viscosity and 3% for surface tension. Furthermore, numerical analysis using the theoretical formulas revealed that the eight additional thermophysical properties required for PLV curve fitting only shift the signal waveform by a constant coefficient along the vertical axis, without affecting the fitting results. Therefore, it is sufficient for these input parameters to have significant figures of approximately one digit. These results demonstrate significant potential for PLV as an accurate, innovative viscometer, and it is expected to be recognized as a next-generation international standard method for viscosity measurement in the future.
A
Acknowledgement
The authors thank Naohiro Takahashi for his assistance with the experiments.
Appendix A: List of Symbols
a : thermal diffusivity (m2·s− 1)
b
parameter of the FILT approximation
c
p : specific heat capacity at constant pressure (J·kg− 1·K− 1)
Cf
signal strength conversion factor (V·m− 2)
d : depth of sample (m)
E
h : pulse energy per unit area of heating pulse laser (fluence) (J·m− 2)
E
h(x,z) : fluence distribution within interference volume of sample generated by two pulse heating laser beams of equal intensity (J·m− 2)
i
imaginary unit
I
1 : intensity of the first-order diffracted beam (W·m− 2)
k : = 2π/Λ wave number of grating vector (m− 1)
K : parameter of approximation for FILT, maximum iteration number
t : time (s)
T : temperature of sample (K)
ΔT : spatial temperature amplitude (K)
ΔTP : initial spatial temperature amplitude generated by pulse laser heating (K)
u(x, z) : displacement vector (m)
u
z1 : z-direction spatially modulated component of the liquid surface displacement (m)
V : output voltage from photomultiplier tube (V)
V
L : speed of sound (m·s− 1)
W
h : spot size radius of the TEM00 heating laser beam (FW, 1/e2) (m)
Greek
α : absorption coefficient of sample at the heating laser wavelength (m− 1)
β : volume thermal expansion coefficient (K− 1)
δ(t) : delta function
η : viscosity (Pa·s)
θ : crossing angle of heating laser beams (rad)
λ : thermal conductivity (W·m− 1·K− 1)
λ
h : wavelength of heating laser (m)
λ
p : wavelength of probing laser (m)
Λ : grating period (m)
ν : kinematic viscosity (m2·s− 1)
ρ : density (kg·m− 3)
σ : surface tension (N·m− 1)
τ : decay time constant of PLV (s)
Appendix B: Coefficients in Eq. 19
The coefficients are very space-consuming, so only those needed to calculate uz1 are listed.
A1
A2
A3
A4
A5
where
A6
A7
A8
A9
(A10)
A11
A12
(A13)
Appendix C: Fortran 90 Code for Coefficients in Eq. 19
To facilitate implementation and reduce errors, the Fortran 90 code for calculating the coefficients in Eq. 19 is provided below.
A_3 -> A’’’
kei -> k
ss -> s
TC ->λ
A_3 = (b1*Eh*VL**2.*Beta*(b1*(b1 - b6)*(b1 + b6)*kei*ss**2. - (b1 - b2)*(b1 + b2)*(ss**2. + kei**2.*VL**2.)*Alpha - (-1. + b1**2.)*kei**2.*ss*(b1**3.*kei - b1*b6**2.*kei - b1**2.*Alpha + b2**2.*Alpha)*Nu))/((b1 - b2)*(b1 + b2)*(b1 - b6)*(b1 + b6)*kei*ss*TC*(-ss + (-1. + b1**2.)*kei**2.*Nu)*(-ss**2. + (-1. + b1**2.)*kei**2.*VL**2. + (-1. + b1**2.)*kei**2.*ss*Nu))C_3 = (b1**2.*Eh*VL**2.*Beta)/((b1**2. - b2**2.)*TC*(-ss**2. + (-1. + b2**2.)*kei**2.*VL**2. + (-1. + b2**2.)*kei**2.*ss*Nu))
C_3 = (b1**2.*Eh*VL**2.*Beta)/((b1**2. - b2**2.)*TC*(-ss**2. + (-1. + b2**2.)*kei**2.*VL**2. + (-1. + b2**2.)*kei**2.*ss*Nu))
J_3 = -((b1**2.*Eh*VL**2.*Alpha*Beta*(ss**2. + kei**2.*VL**2. - (-1. + b6**2.)*kei**2.*ss*Nu))/(b6*(-b1**2. + b6**2.)*kei*ss*TC*(-ss + (-1. + b6**2.)*kei**2.*Nu)*(-ss**2. + (-1. + b6**2.)*kei**2.*VL**2. + (-1. + b6**2.)*kei**2.*ss*Nu)))
C71 = (b3*kei**2.*VL**2.)/(-ss**2. - kei**2.*VL**2. - kei**2.*ss*Nu + b3**2.*kei**2.*ss*Nu)
C72 = -(((-ss**2. + b4**2.*kei**2.*VL**2. - kei**2.*ss*Nu + b4**2.*kei**2.*ss*Nu))/(b4*kei**2.*VL**2.)))
M11 = (ss*Eta*(ss**2. + (1. + b3**2.)*kei**2.*VL**2. - (-1. + b3**2.)*kei**2.*ss*Nu))/(b3*kei*VL**2.)
M12 = b4*kei*ss*Eta*(1. + (kei**2.*VL**2.)/(-ss**2. + b4**2.*kei**2.*VL**2. + (-1. + b4**2.)*kei**2.*ss*Nu))
M3 = -((b1*Eh*(dSdT/(b2*(b1 + b2)) + (2.*b1*ss*VL**2.*((-1. + b1)*(1. + b1)*(b1 - b6)*(b1 + b6)*kei + b1*(-b1**2. + b2**2.)*Alpha)*Beta*Eta)/((-1. + b1)*(1. + b1)*(b1 - b2)*(b1 + b2)*(b1 - b6)*(b1 + b6)*((-1. + b1**2.)*kei**2.*VL**2. + ss*(-ss + (-1. + b1**2.)*kei**2.*Nu))) - (2.*b1*kei*ss*VL**2.*Beta*Eta)/((b1 - b2)*(b1 + b2)*((-1. + b2**2.)*kei**2.*VL**2. + ss*(-ss + (-1. + b2**2.)*kei**2.*Nu))) + (VL**2.*Alpha*Beta*Eta*((1. + b1**2.)/(ss - b1**2.*ss + (-1. + b1**2.)**2.*kei**2.*Nu) + (b1*(1. + b6**2.))/(b6*(-1. + b6**2.)*(ss - (-1. + b6**2.)*kei**2.*Nu))))/((b1 - b6)*(b1 + b6)) + (2.*b1*b6*ss*VL**2.*Alpha*Beta*Eta)/((b1 - b6)*(b1 + b6)*(-1. + b6**2.)*((-1. + b6**2.)*kei**2.*VL**2. + ss*(-ss + (-1. + b6**2.)*kei**2.*Nu)))))/TC)
M21 = (b3*ss*(2.*Eta*(ss**2. + kei**2.*VL**2. - (-1. + b3**2.)*kei**2.*ss*Nu) + VL**2.*(ss - (-1. + b3**2.)*kei**2.*Nu)*Rho) + kei*(ss**2. + kei**2.*VL**2. - (-1. + b3**2.)*kei**2.*ss*Nu)*Sigma)/(b3*kei*VL**2.)
M22 = kei*VL**2.*(-Rho + (b4*kei**2.*(2.*b4*ss*Eta + b4*VL**2.*Rho + kei*Sigma))/(-ss**2. + b4**2.*kei**2.*VL**2. + (-1. + b4**2.)*kei**2.*ss*Nu))
M4 = -((b1*(b2**2. + b6**2. + b1*(b2 + b6))*Eh*VL**2.*Beta*Rho)/(b2*(b1 + b2)*b6*(b1 + b6)*kei*TC)) - (b1**2.*Eh*kei**2.*VL**6.*Alpha*Beta*Rho)/((b1**2. - b6**2.)*ss*TC*(-ss + (-1. + b6**2.)*kei**2.*Nu)*(-ss**2. + (-1. + b6**2.)*kei**2.*VL**2. + (-1. + b6**2.)*kei**2.*ss*Nu)) - (b1**2.*Eh*VL**2.*Alpha*Beta*(ss**2. + kei**2.*VL**2. - (-1. + b6**2.)*kei**2.*ss*Nu)*(2.*b6*ss*Eta + b6*VL**2.*Rho + kei*Sigma))/ (b6*(-b1**2. + b6**2.)*ss*TC*(-ss + (-1. + b6**2.)*kei**2.*Nu)*(-ss**2. + (-1. + b6**2.)*kei**2.*VL**2. + (-1. + b6**2.)*kei**2.*ss*Nu)) - (b1**2.*Eh*kei*VL**2.*Beta*(-(VL**2.*Rho) + b2*(2.*b2*ss*Eta + b2*VL**2.*Rho + kei*Sigma)))/ (b2*(-b1**2. + b2**2.)*TC*(-ss**2. + (-1. + b2**2.)*kei**2.*VL**2. + (-1. + b2**2.)*kei**2.*ss*Nu)) - (b1*Eh*VL**2.*Beta*(-(b6**2.*kei*ss*VL**2.*(ss + kei**2.*Nu)*Rho) + b1**6.*kei**3.*ss*Nu*(2.*ss*Eta + VL**2.*Rho) - b2**2.*kei*Alpha*(ss**2. + kei**2.*VL**2. + kei**2.*ss*Nu)*Sigma + b1**5.*kei**2.*ss*Nu*(-2.*ss*Alpha*Eta - VL**2.*Alpha*Rho + kei**2.*Sigma) + b1*ss*(b2**2.*Alpha*(-2.*Eta*(ss**2. + kei**2.*VL**2. + kei**2.*ss*Nu) - VL**2.*(ss + kei**2.*Nu)*Rho) + b6**2.*kei**2.*(ss + kei**2.*Nu)*Sigma) + b1**3.*ss*(2.*Alpha*Eta*(ss**2. + kei**2.*VL**2. + (1. + b2**2.)*kei**2.*ss*Nu) + VL**2.*Alpha*(ss + (1. + b2**2.)*kei**2.*Nu)*Rho - kei**2.*(ss + (1. + b6**2.)*kei**2.*Nu)*Sigma) + b1**2.*kei*(ss*(2.*b6**2.*ss*Eta*(ss + kei**2.*Nu) + VL**2.*((1. + b6**2.)*ss + (1. + 2.*b6**2.)*kei**2.*Nu)*Rho) + Alpha*(ss**2. + kei**2.*VL**2. + (1. + b2**2.)*kei**2.*ss*Nu)*Sigma) - b1**4.*kei*ss*(2.*ss**2.*Eta + 2.*(1. + b6**2.)*kei**2.*ss*Eta*Nu + ss*VL**2.*Rho + kei**2.*Nu*((2. + b6**2.)*VL**2.*Rho + Alpha*Sigma))))/ ((b1 - b2)*(b1 + b2)*(b1 - b6)*(b1 + b6)*ss*TC*(-ss + (-1. + b1**2.)*kei**2.*Nu)*(-ss**2. + (-1. + b1**2.)*kei**2.*VL**2. + (-1. + b1**2.)*kei**2.*ss*Nu))
E_3 = (M22*M3 - M12*M4)/(M11*M22 - M12*M21)/C71
G_3 = (M11*M4 - M21*M3)/(M11*M22 - M12*M21)/C72
uz1_hat = A_3*dexp(b1*kei*z) + C_3*cdexp(b2*kei*z) + E_3*cdexp(b3*kei*z) + G_3*cdexp(b4*kei*z) + J_3*cdexp(b6*kei*z)
A
A
Author Contribution
Nagasaka wrote the main manuscript text. Shibata conducted calculation and experiment. Taguchi managed the research project.
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