Anal. Chem. 2003, 75, 479-485
Probing the Vapor-Liquid Phase Behaviors of Near-Critical and Supercritical Fluids Using a Shear Mode Piezoelectric Sensor Robert M. Oag, Peter J. King,* and Christopher J. Mellor
School of Physics & Astronomy, University of Nottingham, Nottingham, NG7 2RD, U.K. Michael W. George, Jie Ke, and Martyn Poliakoff
School of Chemistry, University of Nottingham, Nottingham, NG7 2RD, U.K.
With the rapidly expanding industrial and research applications of near-critical and supercritical technology there is a pressing need for a simple and inexpensive sensor that may be used to determine the phase coexistence regions of fluid mixtures and to establish whether a fluid system is below, at, or above, a critical point. Mechanically vibrating AT-cut quartz plates may be used to determine the product of the fluid density and viscosity of a fluid in which it is immersed, through measurement of the impedance minimum of the electrical equivalent circuit or of the corresponding frequency. The densityviscosity product changes abruptly between fluid phases and rapidly along the isotherm corresponding to the critical temperature, enabling such a plate to act as a sensor of these fluid features. We consider the limitations and linearity of such a sensor and its behavior when a liquid-gas meniscus crosses its surface. We demonstrate for the first time the effective use of an AT-cut quartz sensor in mapping the phase behavior of fluids, using measurements made on carbon dioxide and ethane for calibration and then investigating an ethane-carbon dioxide mixture. The advantages of this experimental approach are that (i) piezoelectric sensors are available for operation up to 1000 °C and at extremely high pressures and (ii) the measurement of the densityviscosity product of supercritical fluids is inherently simpler than traditional techniques for determining phase behavior. Supercritical fluids are becoming increasingly attractive as environmentally acceptable solvents for chemical reactions1 and material processing.2 Applications range from particle processing and drug delivery to selective reactions1,3 and large-scale chemical manufacture.4 In many of these applications, it is important whether the supercritical fluid and other components are in a * To whom correspondence should be addressed. E-mail:- P.J.King@ nottingham.ac.uk. (1) Oakes, R. S.; Clifford, A. A.; Rayner, C. M. J. Chem. Soc., Perkin Trans 1 2001, 917. (2) Eckert, C. A.; Knutson, B. L.; Debendetti, P. G. Nature 1996, 383, 313. (3) Jessop, P. G.; Leitner, W. Chemical Synthesis Using Supercritical Fluids; WileyVCH: Weinheim, 1999. 10.1021/ac020322g CCC: $25.00 Published on Web 01/04/2003
© 2003 American Chemical Society
single phase or multiphase condition because the phase behavior can frequently affect the outcome of processes and reactions.5 As a result, the study of phase behavior is becoming more important. Traditionally, phase equilibria have been studied by view cells, incorporating windows that allow direct observation of the fluid inside.6 Such measurements are difficult to automate and are frequently subjective. There is therefore a need for simple, reliable, and inexpensive sensors for monitoring phase equilibria in the near-critical and supercritical region. In this paper, we describe a simple shear mode piezoelectric sensor that has the potential of use up to very high temperatures and pressures and also in aggressive environments. Mechanically oscillating piezoelectric crystals may be used to measure the properties of a surrounding fluid either through shifts in their resonant frequencies or through measurements of the damping of resonance.7,8 Crystals oriented to exhibit oscillations of thickness, such as X-cut quartz, may be used to measure acoustic impedance, the product of the density, and the sound velocity of the adjacent fluid. However, the damping of the oscillations may be so severe that measurement is extremely difficult. This usually limits use of thickness mode sensors to gases rather than to conventional liquids. Piezoelectric crystals oriented to exhibit shearing motion, such as AT-cut quartz, may be used to measure the product of the density (F) and the viscosity (η) of the surrounding fluid. In this case, the damping due to the fluid is less severe and these crystals have found widespread application in the investigation of sugars,7,9,10 oils,11 alcohols,12 glycols,12-15 salts,8,16 polymers,10 and DNA from mammals and fish.17-19 (4) Adams, D. Nature 2000, 407, 938. (5) Smail, F. R.; Gray, W. K.; Hitzler, M. G.; Ross, S. K.; Poliakoff, M. J. Am. Chem. Soc. 1999, 121, 10711. (6) McHugh, M. A.; Krukonis, V. J. Supercritical Fluid Extraction; ButterworthHeinmann: Boston, MA, 1994. (7) Lin, Z.; Yip, C. M.; Joseph, I. S.; Ward, M. D. Anal. Chem. 1993, 65, 15461551. (8) Shana, Z. A.; Josse, F. Anal. Chem. 1994, 66, 1955-1964. (9) Lau, O.; Shao, B.; Zhang, W. Anal. Chim. Acta 1995, 312, 217-222. (10) Stoyanov, P. G.; Grimes, C. A. Sens. Actuators 2000, 80, 8-14. (11) Loiselle, K. T.; Grimes, C. A. Rev. Sci. Instrum. 2000, 71, 1441-1446. (12) Muramatsu, H.; Tamiya, E.; Karube, I. Anal. Chem. 1988, 60, 2141-2146. (13) Mao, Y.; Wei, W.; Zhang, J.; Li, Y. J. Appl. Polym. Sci. 2001, 82, 63-69. (14) Martin, S. J.; Granstaff, V. E.; Frye, G. C. Anal. Chem. 1991, 63, 22722281.
Analytical Chemistry, Vol. 75, No. 3, February 1, 2003 479
Figure 1. Sensor geometry and mounting (right) together with the simplified sensor and fluid configuration used in our derivation of the sensor sensitivity and linearity (left). The transverse surface motion is indicated by arrows.
While most of these applications have involved measurements made at atmospheric pressure and over limited ranges of temperature, totally immersed piezoelectric sensors may be operated at extremely high pressures. Many piezoelectric materials also function over a very wide range of temperature. For example, quartz may be used to temperatures exceeding 300 °C, lithium niobate to temperatures exceeding 500 °C, and aluminum nitride to temperatures in excess of 1000 °C.20 The viscosity changes abruptly between fluid phases and the density usually does so also, making piezoelectric shear mode sensors very suitable for determining the phase behavior and critical points of a very wide range of fluid systems. In this paper, we examine the limits to the linear operating range of shear mode sensors and demonstrate the successful use of AT-cut quartz transducers in investigations of the phase behavior of carbon dioxide, ethane, and carbon dioxide-ethane mixtures. SHEAR-MODE SENSOR BEHAVIOR A shear mode piezoelectric sensor consists of a thin plate with parallel sides, upon which are deposited thin metal coatings used to apply voltages across the plate. Figure 1 shows a common commercially available geometry. Upon the application of an electric field, the plate changes shape through the piezoelectric effect. By choosing a suitable crystallographic orientation for the plate, the motion of the surfaces may be made to lie within their own plane. Figure 1 shows the plate undergoing shear. A sinusoidal waveform of a suitable frequency will excite the plate into mechanical resonance, and this is reflected in the electrical impedance measured between the two metal coatings.21,22 If the plate is then immersed in a liquid, this will affect both the (15) Teston, F.; Feuillard, G.; Tessier, L.; Tran Hu Hue, L. P.; Lethiecq, M. J. Appl. Phys 2000, 87, 689-694. (16) Okajima, T.; Sakurai, H.; Oyama, N.; Tokuda, K.; Ohsaka, T. Electrochim. Acta 1993, 38, 747-756. (17) Wu, Y.; Yi, L.; Xie, Q.; Zhang, Y.; Yin, F.; Yao, S. Talanta 2001, 54, 263270. (18) Wu, Y.; Zhou, A.; Xie, Q.; Cai, Y.; Yao, S. Microchem. J. 2000, 65, 67-74. (19) He, D.; Xie, Q.; Peng, H.; Wei, W.; Nie, L.; Yao, S. Enzyme Microb. Technol. 2001, 29, 84-89. (20) Asher, R. C. Ultrasonic Sensors for Chemical and Process Plant; Institute of Physics Publishing: Bristol, U.K., 1997. (21) Berlincourt, D. A.; Curran, D. R.; Jaffe, H. In Physical Acoustics; Mason, W. P., Ed.; Academic Press: New York, 1964; Vol. IA. (22) Mason, W. P. In Physical Acoustics; Mason, W. P., Ed.; Academic Press: New York, 1964; Vol. IA.
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mechanical resonant frequency and the mechanical damping, both reflected in the electrical impedance. The liquid properties may be determined either through following the frequency at which a resonant impedance minimum occurs (frequency shift mode) or by measuring the magnitude of the electrical impedance minimum itself (impedance minimum mode). The impedance maximum found at slightly higher frequencies is usually large and provides a less satisfactory measure of the fluid properties. A number of authors have obtained expressions for the change of impedance on resonance and the change in resonant frequency when such a plate is immersed in a fluid.7,9,19 We will use the following model to study the behavior of a shear mode piezoelectric sensor and to determine the range of fluid parameters for which the behavior is expected to be linear. Let us consider a sensor of density Fs, immersed in a fluid of density F and viscosity η (Figure 1). We suppose that the sensor lies between x ) d/2 and x ) -d/2. The elastic constant of the sensor material appropriate to mechanical waves propagating in the x direction, with displacements in the z direction, is Cs. In this derivation, we assume that the mass loading of the thin metal coatings may be ignored. We now consider the application of a time-dependent electric displacement D ) Doeiωt applied in the x direction. This establishes left-to-right and right-to-left traveling mechanical waves within the sensor of the form ei(ωt-kx) and ei(ωt+kx), respectively, where the wave vector k ) ω(Fs/Cs)1/2 and the mechanical displacements are in the transverse direction (see Figure 1). D also establishes transverse mechanical disturbances of the form eiωt+λ within the left-hand fluid and ei(ωt-λx) within the right-hand fluid, where λ ) (1 + i)(Fω/2η1/2. Consideration of the boundary conditions at x ) d/2 and x ) -d/2 leads to an expression for the total mechanical stress in the sensor, S, of the form
S)-
ikhD0(e-ikx + eikx)eiωt 1 2 [ωγ(1 + i) sin(θ) - iR cos(θ)]
(1)
Here h is the piezoelectric coupling constant appropriate to the plate, γ ) (Fηω/2)1/2, R ) kCs and θ ) kd/2 ) ω(d/2)(Fs/Cs)1/2. Expression 1 may be used to obtain the electrical impedance of the sensor through the relationship between S, D, the electric field E, and the inverse dielectric constant β 22,23
E ) -hS + βD
(2)
The voltage, V, between the electrodes may be obtained by integrating E between x ) -d/2 and x ) +d/2. The corresponding electrical current, I, is equal to iωADoeiωt, where A is the active area of the sensor. The electrical impedance, Z ) V/I, may then be written as
Z)
(
1 K2 1iωC0 θ[cot(θ) - g - ig]
)
(3)
where we have noted that A/(βd) ) C0, the interelectrode capacitance of the sensor and K2 ) h2/βCs, the electromechanical (23) Cady, W. G. Piezoelectricity; Dover Publications: New York, 1964.
coupling. The dimensionless parameter g is given by
g)
x
Fηω 2CsFs
(4)
g is the ratio of the “viscous impedance” of the fluid, γ, to the acoustic impedance of the sensor material, (CsFs)1/2. When g , K2, that is, for very low damping due to the fluid, the frequency dependence of g may be ignored in eq 3, since the shift in resonant frequency due to damping is very small. It is then straightforward to show that |Z| exhibits a sharp maximum close to a frequency of ωmax ) (π/d)(Cs/Fs)1/2 (corresponding to θ ) π/2) and that |Z| exhibits a sharp minimum at a slightly lower frequency, ωmin, given by
ωmin ) Fθo - Fg
(5)
where F ) (2/d)(Cs/Fs) and θo is the lowest finite solution to K2 tan(θ) ) θ. For K2 , 1, θo ≈ π/2 - 2K2/π. At ωmin, the value of |Z| is given by
|Z|min )
θo
g ωminCoK2
(6)
Results equivalent to eqs 5 and 6 have been previously published for the low damping limit.12,14 It is the dependence of |Z|min and ωmin on g and thus on the product (Fη)1/2 upon which the two modes of operation of the sensor, the “impedance minimum mode” and the “frequency shift mode”, depend. The factor within the parentheses of eq 6 relating |Z|min to g represents the linear sensitivity of the sensor. It should be noted from eq 6 that, in impedance minimum mode, the sensor is more sensitive for lower values of the electromagnetic coupling. In the very low damping regime and for frequencies in the region of the impedance maximum and minimum, it is well known that the impedance Z may be thought of as resulting from a simple electrical circuit consisting of a capacitance Co placed in parallel with a series combination of a capacitor, an inductor, and a resistor.21,22 The latter represents the damping due to the fluid. Sensor Linearity. For larger values of g, the resonant frequency shifts are sufficiently appreciable that the effects of its dependence upon ω1/2 (eq 4) cannot be ignored. ωmin and |Z|min then deviate substantially from the simple expressions given in eqs 5 and 6, and the sensor may become appreciably nonlinear in g. The simple electrical equivalent circuit, used by many authors, becomes invalid. In this situation, the expression for Z given in eq 3 becomes difficult to treat analytically, and to assess the onset of nonlinearity, numerical techniques become appropriate. We first considered an electromechanical coupling K2 ) 0.020 that corresponds to AT-cut quartz.22,23 We find that as g is increased toward and through g ) K2 ) 0.020, |Z|min saturates, tending toward a fixed value close to that corresponding to the impedance of Co. Even at far lower values of g, considerable deviations from the predictions of eq 6 are to be noted. By g ) 0.002 the estimate of eq 6 is 2% too high, by g ) 0.004 it is 10%
Figure 2. Schematic diagram of the arrangement used to measure the sensor properties under computer control.
high, while at g ) 0.008 it is 30% high. By g ) 0.02 the deviation from linearity is somewhat over 100%. As g is increased, the resonant minimum shifts, broadens, and becomes shallower. Since it is superimposed on an impedance corresponding to Co, the minimum in |Z| eventually disappears as g is increased. While the minimum may be recovered by the addition of an additional parallel inductor (see Figure 2), the sensor is extremely nonlinear for values of g for which this is necessary. Our numerical calculations based on eq 3 show that the estimates for the frequency shift provided by eq 5 are 1% too high by g ) 0.0002, 10% too high by g ) 0.0015, and 30% in error by g ) 0.004. As the value of g corresponding to the disappearance of the minimum (g ) 0.15) is approached, the deviation from linearity rises to 72% and then declines. It is interesting to note that, for water at 20 °C, g ) 0.000 44. For such a value of g the sensor lies well within 1% linearity when operated in impedance minimum mode. This is not, however, the case for operation in frequency shift mode. While the frequency shift mode is in many ways convenient, nonlinearities appear at far lower values of g than for the impedance minimum mode of operation. Corrections to the Estimated Sensor Sensitivity. The model that we have discussed requires a number of corrections. Practical sensors usually have an active metallized area in the center of a more extended plate (see Figure 1). The mechanical shear will extend beyond the boundaries of the coated area for an effective distance of ∼d. This increases the active area of the sensor and thus decreases the magnitude of Z by a numerical factor. The effects of this correction on the linear sensitivity will be treated in detail below. The surface roughness found in commercial sensors may also act to change Z by a small numerical factor. Neither effect is expected to provide low-order corrections to the frequency shift of eq 5. Mass loading due to the metallization and the electrical fringing at the edges of the coating may each introduce small numerical corrections to Z and to ωmin. While these effects can modify the sensitivity, none of them are expected to substantially alter our discussion of the onset of nonlinearity given above. If such a sensor were to be operated over a very wide range of pressures and particularly over a very wide range of temperAnalytical Chemistry, Vol. 75, No. 3, February 1, 2003
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atures, it would also be necessary to consider the variation of the electromechanical coupling and of the sensor dimensions with these parameters. Sensor Operation in Mixed Fluid Phases. If a sensor is mounted vertically within a fluid system there will be situations where different parts of the active sensor area are covered by distinct fluid phases. There may, for example, be an upper gas phase separated from a lower liquid phase by a meniscus that lies across the sensor surfaces. There may be distinct upper and lower liquid phases. It might be supposed that this situation may be dealt with in terms of two parallel sensors, one for each phase, the areas of the two sensors changing as the meniscus moves. There would then be two resonant minimums at distinct frequencies. As we shall demonstrate below, this does not occur. Rather, the strong coupling between different areas of the sensor maintains a single resonant minimum whatever the position of the meniscus, even in the case of liquid and gas phases. The effective damping of this single resonance is proportional to (Fη)1/2 integrated across the active sensor area. If, for example, a fraction, φ, of the active area is in liquid, and a fraction 1 - φ is in gas, then the sensor measures φ(Flηl)1/2 + (1 - φ) (Fgηg)1/2, where l refers to the liquid phase and g to the gas phase. The sensor effectively acts as a simple level detector. We now consider the use of quartz shear mode sensors, operated in impedance minimum mode, to characterize the liquid-vapor coexistence and critical point regions of a fluid phase diagram. In these particular experimental studies, the influence of temperature and pressure upon the sensor properties themselves (Cs, Fs, h, etc.) may be neglected. MEASUREMENT SYSTEM We have examined the phase diagrams of a number of fluids and fluid mixtures using 6-MHz AT-cut quartz sensors operated within a temperature-controlled pressure cell, principally using the impedance minimum mode. The vertically mounted sensors have circular central electrodes of diameter 4.3 mm. Figure 2 shows the circuit used to make electrical measurements. A computer-controlled pure sine wave of amplitude Vo is applied as shown. The voltage across the sensor is rectified by a temperaturemaintained germanium diode, and after smoothing by R and C, the resulting dc signal, V, is sampled by a Keithley 195A DMM and sent to the computer. The computer, PC, scans the frequency to finds the impedance minimum, recording both |Z|min and ωmin. The results of a calibration, made using a range of fixed resistors in place of the sensor, enables the computer to correct for the diode nonlinearities, etc., the measured voltage being converted to the corresponding |Z|min values to within (0.5% using piecewise polynomial fits. The optional inductor shown in Figure 2 may be used to largely cancel the sensor and associated circuit capacitances over the frequencies of operation. It was not found necessary to incorporate this component when taking the data presented here. The stainless steel pressure cell is contained inside a thermostated copper enclosure from which it is thermally insulated. This enclosure is held at a stable temperature ((0.02 °C) a few degrees below the temperatures of measurement. The insulation, combined with the use of heat shunts on major connections to the cell, ensures an extremely uniform temperature distribution across the cell. The temperature of the pressure cell is itself stabilized 482
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to high precision. A computer-controlled Wheatstone bridge is used, where one arm contains a number of thermistors connected in a parallel network and distributed about the surface of the cell. An analog two-term controller provides power to a number of distributed resistive heaters. This system is capable of a longterm temperature stability of the pressure cell of better than (2 mK. The temperature of the cell is measured independently using a calibrated platinum resistance thermometer, sensed by the computer through a digital ohmmeter. The pressure of the cell is controlled by pumping a secondary fluid from the syringe pump (Isco model 260D) into the top of the cell, where it acts against a piston, which in turn acts on the fluid being measured. The secondary fluid in the syringe pump is maintained at the temperature of the copper enclosure. The computer measures the pressure via a sensor attached to the cell. To enhance the homogeneity, the fluid under measurement is intermittently stirred, under computer control, by the use of a magnetically coupled “flea” located in the bottom of the cell. Before measurements were made, the cell was evacuated, using a rotary vacuum pump, and then filled with either carbon dioxide (99.995% purity), ethane (99.95% purity), or a 85%-15% carbon dioxide-ethane mixture. The mixture was prepared by filling two separate bombs with the individual components and weighing them while full. First the ethane was released into the evacuated cell, and then the CO2 was added. The bombs were then weighed again, and the mass of each substance vented into the cell was calculated. From this, the composition of the mixture was determined. To carry out measurements, the cell was electrically heated to the required temperature and automatic sensor measurements were made while the fluid pressure was very slowly changed, by inserting or withdrawing secondary fluid using the Isco pump. MEASUREMENT OF FLUID PHASE DIAGRAMS General Measurement Strategy. Equation 6 relates |Z|min to g by the linear sensitivity factor. In principle, each value of g can be calculated from the corresponding measured value of |Z|min on the basis of the active area of the sensor and the properties of quartz. In practice, however, the effective active area of the sensor is larger than the area of the quartz crystal covered by the exciting electrodes and an appreciable correction must be applied. There may be additional corrections due to other effects, but these are expected to be far smaller. A crude estimate of the correction factor for the active area may be obtained by extending the limits of the electrode area by about the plate thickness, d. This estimate is not sufficiently accurate for our purposes, yet a more accurate value for the correction is difficult to calculate without using a detailed theory involving the anisotropic elastic properties of the sensor crystal. Our strategy, therefore, has been to demonstrate the use of the sensor on pure carbon dioxide and ethane, for which accurate values of g may be obtained using data from the NIST databases.24,26 Thus, for each pressure and temperature for which we have measured |Z|min, we may obtain a value of g obtained via NIST and value of g via eq 6. A comparison enables us to estimate the correction factor, principally due to the active area of the (24) National Institute of Standards and Technology (NIST 12 Database), 1992. (25) Adkins, C. J. Equilibrium Thermodynamics, 3rd ed.; Cambridge University Press: New York, 1986. (26) National Institute of Standards and Technology (NIST 14 Database), 1992.
Figure 3. Experimental phase diagram of pure carbon dioxide, showing measurements along isotherms at temperatures of (A) 22.650, (B) 24.330, (C) 26.190, (D) 28.010, and (E) 31.400 °C. The scale for the parameter g, corresponding to the left-hand |Z|min scale, is shown on the right-hand side. The two scales have been related using eq 6 together with the correction of 42.6% for the active area provided the method described in the text. The critical point, shown as •, and the continuous coexistence line passing through it, have been taken from NIST data and superimposed using the g scale.
sensor, and to establish that, to within experimental error, it is the same for these two fluids. With the sensitivity of the sensor calibrated in this way, we have studied the phase behavior of a mixture of carbon dioxide and ethane for which the values of the viscosity available from NIST 14 are not as accurate as those for the pure fluid components.. Pure Carbon Dioxide. Measurements of |Z|min against pressure made upon pure carbon dioxide along five isotherms close to the critical point (7.834 MPa and 31.087 °C) are shown in Figure 3. Both in the main body of Figure 3 and in the expanded detail, the experimental data clearly indicate the positions of the (upper) bubble point line, the (lower) dew point line, and the coexistence region of the phase diagram in great clarity. Below the critical temperature, there are abrupt changes in gradient as the coexistence line is crossed and |Z|min changes very rapidly with pressure as the liquid-vapor interface crosses the sensor. We have compared the measured values of the sensitivity of our sensor with estimates of the sensitivity based upon eq 6. The measured values have been obtained from our data for |Z|min together with the corresponding values of g obtained from taking F and η from the NIST 12 database24 and using Cs ) 3.73 × 1010 kg/ms2 and Fs ) 2.65 × 103 kg/m3. The estimated values of the sensitivity have been obtained from eq 6, used K2 ) 0.020, and a calculated value for Co based on the known electrode area, dielectric constant, and plate thickness, but ignoring fringing. Averaged across the whole region of the phase diagram for which we have data, we find that the measured sensitivities lie 42.6% lower than those given by eq 6. This correction is consistent with the active area extending outside the coated electrode area, the
Figure 4. Experimental phase diagram of pure ethane, showing measurements along isotherms at temperatures of (A) 25.243, (B) 27.111,, (C) 29.128, and (D) 30.520 °C. The scale for the parameter g, corresponding to the left-hand |Z|min scale, is shown on the righthand side. The two scales have been related using eq 6 together with the correction of 42.8% for the active area of the sensor provided by the method described in the text. The critical point, shown as •, and the continuous coexistence line passing through it, have been taken from NIST 12 data and superimposed using the g scale.
figure corresponding to an effective diameter for the active area of 5.1 mm. This corresponds to an area extending 0.4 mm beyond the metal coating, a distance just larger than the sensor thickness, d ) 0.3 mm. Having calibrated the mean sensitivity of our sensor against NIST data, we have provided a vertical scale for g in Figure 3 based on this calibration and have overlayed the NIST-based coexistence line upon the figure using this calibration. Following the calibration, the agreement between NIST and our data is too small to be shown in Figure 3, even in the expanded inset. However, we note systematic variations at the 1% level between the NIST data and our calibrated measurements in some parts of the phase diagram, almost all of the experimental data lying within (2% of the NIST data. These variations are not correlated with the magnitude of g in a simple manner. Pure Ethane. Similar measurements of |Z|min against pressure made upon pure ethane and along four isotherms just below the critical point (4.880 MPa and 32.278 °C) are shown in Figure 4. The coexistence region of the phase diagram between the bubble and dew point lines is again delineated in great detail. We have again compared the measured values of the sensitivity of our sensor, obtained using NIST 12 data, with estimates of the sensitivity based upon eq 6. It is found that, averaged across the whole area of the phase diagram that we have investigated, the measured sensitivity lies 42.8% lower than that given by eq 6, corresponding to an active diameter of 5.1 mm. This factor is not significantly different from the calibration based upon carbon dioxide. The g scale of Figure 4 has been provided using this calibration. The critical point and coexistence lines, obtained using NIST 12, have then been added to the diagram using this scale. Once again, we note systematic variations at the 1% level between NIST and the measurements in some parts of the phase diagram, almost all of the data lying within (2% of the predictions once the global 42.8% correction to the active area has been applied. Coexistence of Phases at the Sensor. |Z|min has been measured as a function of meniscus position in the coexistence Analytical Chemistry, Vol. 75, No. 3, February 1, 2003
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Figure 5. | Z |min versus the pressure cell volume for pure CO2 at 30.089 °C. As the meniscus crossed the sensor, | Z |min changes from the value corresponding to liquid immersion to vapor immersion.
regime in order to investigate the sensor behavior when both liquid and gas phases provide damping. We find a single resonant minimum, whatever the position of the two phases. Figure 5 shows data for carbon dioxide at 30.089 °C, |Z|min being plotted against the cell volume determined from the Isco pump movement. |Z|min drops from ∼120 Ω when the sensor is completely immersed in fluid to ∼65 Ω when it lies completely within gas. The shape of this curve may be compared to that predicted from the integrated damping hypothesis mentioned above, relating the change in cell volume to the change in meniscus position using the well-known coexistence regime “lever rule”.25 The maximum gradient of the curve of Figure 5, which occurs when the meniscus lies at the midpoint of the sensor, may be related to the active area of the sensor and the values of |Z|min for gas and liquid immersion (65 and 120 Ω). The data yield 5.3 mm, only slightly larger than the value obtained from calibration using data from carbon dioxide and from ethane. With this diameter, the fit between the data of Figure 5 and calculations based on the integrated damping are very good indeed, confirming that the sensor is measuring the mean damping which is proportional to φ(Flηl)1/2 + (1 φ)(Fgηg)1/2. Carbon Dioxide-Ethane Mixtures. Finally, we report measurements of the phase behavior of a 85.05%:14.95% CO2-ethane mixture. Measurements made on multicomponent mixtures provide important information about the phase behavior of such systems and also enable the progress of any chemical reactions or processes to be studied. Data for |Z|min against pressure are shown for three isotherms below the critical point and one above in Figure 6. The lower temperature data clearly delineate the coexistence region. For such a mixture, the pressure at the dew point is lower than the pressure at the bubble point corresponding to the same temperature. We have used the calibration factor of 42.7%, obtained from measurements on pure ethane and carbon dioxide, to place upon the right-hand side of Figure 6 the scale for g corresponding to |Z|min. While NIST 14 26 provides interpolated viscosity data for carbon dioxide-ethane mixtures, these data do not have the accuracy of that for either pure carbon dioxide or ethane. Nevertheless, it has been used to place the NIST-based coexistence line and critical point on Figure 6. The discrepancies are not evident on the scale of the diagram. However, comparison of our measurements and the NIST data would suggest a calibration factor of 44.5%, together with systematic discrepancies in some regions 2-3 times larger than those found for pure carbon dioxide and ethane. 484 Analytical Chemistry, Vol. 75, No. 3, February 1, 2003
Figure 6. Experimental phase diagram of a CO2-ethane mixture (85.05%: 14.95 wt %), showing measurements along isotherms at temperatures of (A) 25.243, (B) 27.111, (C) 29.128, and (D) 30.520 °C. The scale for the parameter g, corresponding to the left-hand |Z|min scale, is shown on the right-hand side. The two scales have been related using eq 6 together with the correction of 42.7% for the active area of the sensor obtained from measurements on pure carbon dioxide and pure ethane. Values for the critical point, shown as •, and the continuous coexistence line shown passing through it, have been taken from NIST 14 and superimposed on the diagram using the g scale.
DISCUSSION AND CONCLUSIONS We have developed a shear mode piezoelectric sensor for monitoring the vapor-liquid phase behavior and determining the dew point lines, bubble point lines, and coexistence regions of fluids and fluid mixtures. The critical points themselves may be either estimated from the extrapolation of data taken at lower temperatures or obtained accurately by a detailed search of isotherms considering the existence of gradient discontinuities. The sensors are robust and are of low cost. They may be operated over very wide ranges of temperature and pressure. Their small size makes them attractive for use in industrial chemical processes, as they can easily be inserted into a reaction vessel. These sensors will have applications in any process where it is important to maintain supercritical conditions. One possibility would be to position one sensor at the bottom of a pressure vessel and another at the top. If phase separation occurs, then the difference in density and viscosity will show as a difference in the measured resistance from the sensors. We have presented the theory behind such a sensor and extended it to accommodate the nonlinearity of such a device when operated in impedance minimum and in frequency shift modes. The former mode of operation has a wider range of linearity and is therefore relatively easy to calibrate. The simple technique used here to determine impedance is sufficient for an accuracy of (0.5% following modest temperature stabilization of the diode detector and calibration against known resistors. Greater accuracy could be obtained by the use of a computer-controlled precision impedance bridge. We have demonstrated the performance of these sensors by applying them to CO2, ethane, and a CO2-ethane mixture, enabling the dew point and bubble point lines to be clearly identified in each case. The experimental data have been compared with data from the NIST databases. Comparison between the calculated and experimental sensitivities yields a correction factor of 43.7 ( 0.5% for both CO2 and ethane, an effect consistent
with the mechanical vibrations extending beyond the electrode area and thus increasing the active area of the device. After applying this standard correction, the measurements give good overall agreement with NIST data over the areas of the phase diagrams which we have investigated. We have also addressed experimentally the situation where part of the sensor is within a gas phase and part within a liquid phase. We have shown that the sensor offers a single resonant minimum whatever the position of the phase boundary and that the impedance minimum is consistent with the integrated damping across the active surface area. This suggests that these sensors may find wide application as level detectors, particularly if a geometry offering a more linear dependence upon level, such as rectangular, were used. While the sensor impedance is a measure of the product of fluid density and viscosity, the viscosity itself may be obtained from knowledge of the density. In our own experiments, the density is available from the known mass of fluid and the volume
of the cell, which itself is known from the position of the piston. For some purposes where knowledge of the viscosity-density product will not suffice, it may be convenient to determine viscosity in this way. ACKNOWLEDGMENT We are grateful to the EPRSC (GR/R02863) and to the Paul Instrument Fund of the Royal Society of London for financial support throughout this project. We are also indebted to the staff of the electronics and the engineering workshops at the School of Physics & Astronomy, for the prompt design and construction of many of the apparatus components. We also thank Mr. K. Stanley, of the School of Chemistry engineering workshop, for the construction of the pressure cell and piston. Received for review May 13, 2002. Accepted September 23, 2002. AC020322G
Analytical Chemistry, Vol. 75, No. 3, February 1, 2003
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