Micromechanical transient sensor for measuring viscosity and density of a fluid

[0002] This application is related to pending U.S. patent application Ser. No. 09/042,601 filed Mar. 16, 1998; and, Ser. No. 09/281,032 filed Mar. 30, 1998; and, Ser. No. 09/281,256 filed Mar. 30, 1998.

[0004] 1. Field of the Invention

[0005] This invention relates to apparatus and methods for the accurate determination of viscosity, density or both of very small volumes of fluids.

[0006] 2. Prior Art

[0007] The atomic force microscope (AFM) was first demonstrated by Binnig and co-workers at IBM in Switzerland. In the AFM, the tip of a flexible cantilever stylus is rastered over the surface of a sample and the movement of the tip of the cantilever is monitored as a measure of minute forces characteristic of surfaces at the atomic level. Demonstration of this principle led to rapid development of microcantilevers [(Albrecht et al., J. Vac. Sci. Technol,8, 3386 (1990)]. The concept of micromechanical and microelectromechanical detection devices has been developed for a number of analytical uses. Wachter et al., U.S. Pat. No. 5,445,008 describes the use of vibrated microcantilevers having a chemical coating as a detector for the presence of specific chemical entities. Thundat, et al. in Appliance Manufacturer, April1997, 57 (1997) and Microscale Thermophysical Engineering1, 185 (1997) describe developments of the microelectromechanical sensors (MEMS) for the measurement of chemicals and physical phenomena including the use of sensors to determine concentrations of glycerol in water based on viscosity.

[0008] U.S. Pat. No. 5,719,342 to Thundat et al., addresses additional methods for analysis using MEMS devices particularly directed to induced stress in the microcantilever.

[0009] U.S. Pat. No. 5,130,257 to Bear et al., discloses a viscosity sensor fabricated using a surface transverse wave device for use in the measurement of viscosity.

[0010] Oden et al., Appl. Phys. Lett.,68, 3814 (1996) discloses method for the measurement of viscosity using microfabricated cantilevers in a confined medium. The frequency of vibration of an isolated vibrating cantilever is measured in different solutions.

[0011] U.S. Pat. No. 5,494,639 to Grzegorzewski discloses a disposable biosensor which uses a vibrating member beneath a cell to accurately measure blood coagulation time as a function of viscosity.

[0012] The foregoing methods are attempts to find an alternative to the instruments most currently used to measure viscosity of both liquids and gases. These are instruments which perform the analysis by a comparison with a “control fluid.” For this reason, measurements are still routinely done using instruments such as the Redwood viscometer, the Couette or rotational concentric-cylinder viscometer (MacMichael or Stormer viscometer), the Rotating Sphere viscometer, the Sayboult Falling Body viscometer, the Vibrating String viscosity meter and the thickness-shear mode resonators. All of these methods require comparatively large volumes and bulky equipment. The need remains for a small reliable and inexpensive instrument which can measure the viscosity or specific density of small amounts of a liquid or gas and which can be used in difficult-to-access locations.

[0013] It is an object of this invention to provide a microminiaturized instrument for the measurement of viscosity and density in liquids and gases. It is a further object of this invention to provide such an instrument which is reliable and inexpensive in comparison with other instrumentation available.

[0014] It is a further object to provide a means for profiling the viscosity in a non-inhomogeneous fluid.

[0015] It is a further object of this invention to provide means for detecting viscosity and density of very small sample volumes of fluids; to determine viscosity and density of fluids in small confined spaces; to determine kinetics of chemical, biological, and physical reactions by measuring viscosity and density; and, to determine the completeness of chemical, biological, and physical reactions by measuring the viscosity and density.

[0016] These and other objects are met by using microcantilevers having typical dimensions of 50-200 μm in length, 10-40 μm in width, and 0.3-3 μm in thickness. When such a microcantilever is driven into resonance, the resonance frequency, amplitude and Q-value varies with the viscosity and the density of the medium surrounding the cantilever. More particularly, according to this invention, the cantilever is excited into resonance by an external means for a finite amount of time until a stable resonance amplitude is established. After that period, the external excitation is switched off and the transient behavior (decay) of the vibration is recorded. The resonance frequency, amplitude and Q factor are determined from the transient spectrum. The decay in amplitude of vibration can be used to determine the density of the material. Numerous methods for measuring the vibration of the cantilever are available and a multiplicity of cantilevers can be used for the determination of a wide range of densities.

[0017] FIG. 1 is a schematic diagram illustrating the components of this invention.

[0018] FIG. 2 is an illustration of the signal which is used to discern the viscosity and density of a fluid medium.

[0019] FIG. 3(a) shows a square wave signal used to excite a cantilever; FIG. 3(b) shows the corresponding transient response of a cantilever.

[0020] FIG. 4 is a trace of the transient response of a cantilever.

[0021] A MEMS device can be constructed using available or easily fabricated components. The microcantilevers used are commercially available from several sources, and may include those conventionally used in scanning force microscopy. Alternatively, the microcantilever may be prepared from various ceramics metals and semiconductor materials by methods well known to those familiar with the fabrication of microelectronic devices and according to the method of the aforementioned Albrecht et al. reference. Likewise, a mounting block with an inventive piezoelectric exciting device is commercially available, as is a mounting stage for the cantilever and laboratory base. Detection methods may be optical beam deflection, optical interference, capacitance, piezoresistance, or electron tunneling. On the basis of familiarity, we prefer optical methods such as those described in the Wachter et al. patent, although one or more of the other methods described above may be suitable for liquids with a high degree of opacity or density.

[0022] The technique involve exciting the cantilever into resonance frequency by an external means such as a piezoelectric crystal or by electrostatic means, or electromagnetic, or photoinduced, or the like as are known in the art. The microcantilevers can also be excited using a photothermal technique where the light source is used in a pulse mode to impart energy to the cantilever. Microcantilevers can also be excited using acoustic waves. It may take many pulses to excite the microcantilevers into resonance. Because the square pulses contain all the Fourier components, a cantilever of any natural resonance frequency can be excited using square pulse excitation. The form of the excitation amplitude can be periodic at or near the resonant frequency of the cantilever and can be pure harmonic, square, or other arbitrary waveform. A pure harmonic waveform to induce the maximum amplitude resonance of the cantilever may be used. An impulse or rectangular pulse can also be used. Once the cantilever is excited into resonance, the excitation source is switched off and the transient behavior of the microcantilever deflection is recorded. In the case of rectangular pulses, the transient response occurs at the beginning and at the end of the pulse. The cantilever resonance amplitude drops off exponentially with time. From the zero crossing of the deflection, the resonance frequency (crossing per unit time) can easily be found. The Q-factor can be determined from decay curve of the amplitude (from the logarithmic decrement).

[0023] The mathematical expression for viscosity and density of the fluid can be derived as follows. Let L, W, and T be the length, width, and thickness of the cantilever. The density of the cantilever is taken as ρ_c. The cantilever resonates in vacuum with a resonance frequency, ω_o(2πf^o) and a quality factor Q_o. The finite Q-factor in vacuum is mainly due to internal energy loss. However, when operated in a fluid, the resonance frequency and Q-factor change due to energy loss due to viscosity. The equation of motion of the cantilever in a medium can be written as: $\begin{array}{cc}\mathrm{EI}\frac{{\partial }^{2}u}{\partial x^2}+\left(b_0+b\right)\frac{\partial u}{\partial t}+\left(M_c+M_f\right)\frac{{\partial }^{2}u}{\partial t^2}=0& \left(1\right)\end{array}$

[0024] where E is the Young's modulus, I is the moment of inertia, b_ois the intrinsic damping of the cantilever, M_ois the mass per unit length of the cantilever, M_fus the induced mass (mass of fluid moving along with the cantilever per unit length), and u is the cantilever displacement at distance x from the fixed end of the cantilever.

[0025] The moment of inertia of a rectangular cantilever is given by: $\begin{array}{cc}I=\frac{{\mathrm{WT}}^3}{12}& \left(2\right)\end{array}$

[0026] where W is the width and T is the thickness of the cantilever. The intrinsic damping b_ocan be determined from the resonance characteristics of the cantilever in vacuum (w_oand Q_o. The intrinsic damping, b_o, can be written as: $\begin{array}{cc}{b}_c=\frac{{\omega }_o{\rho }_c\mathrm{WT}}{Q_o}& \left(3\right)\end{array}$

[0027] where w_oand Q_oare the resonance frequency and the quality factor, respectively of the cantilever in vacuum. The damping due to viscosity of the fluid is given by: $\begin{array}{cc}{b}_f=\frac{W}{T}{M}_f\left(\frac{\omega }{\sqrt{2R_K}}\right)& \left(4\right)\end{array}$

[0028] Where w is the angular frequency, η is the viscosity of the fluid and R_kis the Reynold's number of the fluid defined as: $\begin{array}{cc}{R}_K=W^2\left(\frac{\rho }{\eta }\right)\omega & \left(5\right)\end{array}$

[0029] The mass per unit length, M_c=p_cWT:

[0030] M_c=ρ_cWT  (6)

[0031] The induced M_fequals: $\begin{array}{cc}\left(\frac{W}{T}\right){\rho }_c\omega u\left(1+\frac{4}{\sqrt{2R_K}}\right)& \left(7\right)\end{array}$

[0032] Solution for the equation of motion is complex quantity. The angular resonance frequency is given by: $\begin{array}{cc}\omega =\left(\sqrt{\frac{{3.52}^2\mathrm{EI}}{\left(M_o+M\right)L^4}}\right)+i\left(\frac{b_o+b}{2\left(M_o+M\right)}\right)& \left(8\right)\end{array}$

[0033] Equation 8 is a complex number of form, X+iY.

[0034] The measured angular frequency is: $\begin{array}{cc}{\omega }_{\mathrm{real}}=\sqrt{X^2+Y^2}& \left(9\right)\end{array}$

[0035] And the full width at 2_−0.5of the maximum angular frequency is:

[0036] Δω=Y  (10)

[0037] The Q-factor can be expressed as,

[0038] Q=ω/Δω

[0039] defining M*=2(M_c+M_f) and b*=(b_v+b_f) in Eq.1,

[0040] $M^{*}=\frac{C}{{\omega }^2}\left(\frac{Q^2}{Q^2-1}\right)\mathrm{and}{b}^{*}=\frac{C}{\omega }\left(\frac{Q^2}{Q^2-1}\right)\mathrm{with}C={3.52}^2\mathrm{EI}/L^4$

[0041] Therefore, M_fand b_fcan now be written as:

[0042] M=M*/2−M_c  (11)

[0043] b_f=b*−b_c  (12)

[0044] Using Eq. 4, Reynold's number can now be expressed as: $\begin{array}{cc}{R}_K={\frac{1}{2}\left[\left(\frac{M_f}{b_f}\right)\frac{W\omega }{T}\right]}^2& \left(13\right)\end{array}$

[0045] Now R_xcan be calculated using the resonance frequency, Q-factor, and the amplitude of vibration in vacuum and in the medium. the length, width, and the thickness of the cantilever as well as the density of the cantilever material is known apriori. Once the Reynold's number and M_fare known, the density and the viscosity of the medium can be calculated as follows: $\begin{array}{cc}\rho =\frac{M_{f}T}{W\omega u\left(1+\frac{4}{\sqrt{2R_K}}\right)}& \left(14\right)\end{array}$

[0046] The viscosity of the medium is given by: $\begin{array}{cc}\eta =\frac{W^2{\rho }_f\omega }{R_K}& \left(15\right)\end{array}$

[0047] Initially the cantilever is calibrated in vacuum or in a fluid with known properties for determining its intrinsic resonance properties. Later the cantilever can be resonated in unknown fluids. Or the cantilever can be calibrated in vacuum and a fluid of known viscosity and density to determine the relative changes in viscosity and density observed with respect to a fluid of known viscosity and density.

[0048] Another advantage is that this technique is the speed by which measurements can be carried out. The typical time involved is in the range of sub milliseconds to sub microseconds.

[0049] As can be seen from the derivation of equation (15) the method has two inherent advantages. The cantilever can be recalibrated as needed by testing in a vacuum or a fluid of known properties. Any drift in the instrument can be immediately identified. In addition, different cantilevers can be compared to each other so that there is no instrumental error resulting from the fabrication of the devices. A second advantage is that, for a given device, comparisons between fluids can be readily made. As noted earlier, for most industrial uses, it is not the absolute viscosity but the viscosity relative to another fluid which is important. A third advantage is magnetic excitation. Magnetic excitation can be achieved by attaching a magnetic particle or evaporating a magnetic film such as Cr or Co or their alloys to the apex of the cantilever. For process control, this device quickly senses the direction and the magnitude of any change in viscosity. Since viscosity is affected by temperature, the device may be used to identify a process or operational change before a bulk viscosity or density change could be quickly measured.

[0050] Because cantilevers can be readily fabricated in one- or two-dimensional arrays, inhomogeneities of viscosity or density in the monitored fluid can be sensed. Individual cantilevers in the arrays sense physical properties in their immediate vicinity. For example, a one-dimensional array could be placed in a fluid stream, such as a pipe, to measure any inhomogeneities or gradients in viscosity or density of the fluid flow.

[0051] Microcantilevers can be excited and their movement measured in a variety of manners. Excitation may be by piezoelectric, piezoresistive methods, by capacitive, photothermal, electrostatic, or acoustic techniques. Movement of the cantilever is preferably measured by reflection of light from the tip of the microcantilever but may also be measured piezoelectricly, piezoresistively, and by the methods disclosed in U.S. Pat. Nos. 5,440,008 and 5,719,324.

[0052] An exemplary device for measuring viscosity and density is shown in FIG. 1. An oscillator circuit 1 generates a square wave signal 2 which energizes a device 4 such as a piezoelectric transducer to cause a cantilever 3 to resonate. The resonance is detected, in the example, by shinning a light from laser diode 5 onto the tip of cantilever 3. The reflected light impinges upon position sensitive diode 6. The signal from PSD 6 is amplified (7) and analyzed at amplitude measuring circuit 8. The display of the result form the measurement is shown in FIG. 2 as a plot of amplitude versus time for the decay of resonance when the stimulation is interrupted. FIG. 3 illustrates the relationship between the square wave signal and cantilever response.

[0053] The method of this invention may be used to determine many physical parameters and for process and quality control. The sensors are very small and may be located in remote and even hazardous locations. Viscosity/specific density provides a good indication of the completeness of a chemical reaction and provides a nearly instantaneous indication of when a process stream condition changes. Leaks, especially gas leaks, are quickly identified at the source.

[0054] The devices are also useful in kinetic studies, especially using shock tubes to study reaction process.

[0055] Because this method is based on relative changes in the successive amplitudes rather than absolute amplitudes, this method can be used for measuring viscosity and density in noisy environments. However, when used in severely interfering noise environments, a reference cantilever can be used to avoid transients caused by noises. For example, acoustic noises in the environments can produce transient motion of a cantilever. Using a reference cantilever the noise contribution from the transient can be filtered using wavelet or Fourier transform analysis. The same approach can be used for other interference such as noise introduced by temperature variation.

[0056] The invention will be described in terms of an example which illustrates the operation of the invention. The example is not limitative of the invention and modifications obvious to those with skill in this art are included within the scope of applicants' invention.

[0057] A microcantilever may be fabricated to specify length, width and thickness. The microcantilever is attached, at its base, to a piezoelectric device which is energized by a square wave generator. The microcantilever may be first characterized in a vacuum to determine the resonance frequency and Q-factor by observing the transient response of the cantilever. This is the calibration of the cantilever.

[0058] The chamber is then opened to admit a gas, pure nitrogen, and allowed to equilibrate at a pressure of 760 mm of Hg. The decay curve of the microcantilever is determined as before and plotted as illustrated in FIG. 4. It is noted that the absolute magnitude of the amplitude is not critical, only the decrement. The experiment was repeated using Argon and He. The longarithimic decrement of the amplitude σ is given by equation (16):

[0059] The value of Q can be determined from the decay curve of the amplitude. The value of the resonance frequency can be determined from the center crossing of the decay curve. The logarithmic decrement of the amplitude δ is given by $\begin{array}{cc}\delta =\left[\frac{A_{n+1}}{A_n}\right]& \left(16\right)\end{array}$

[0060] Where A(n+1) and An are successive amplitudes The Q factor is given as Q=π/δ.

[0061] Density and viscosity of some gases:

[0062] Experimental results obtained from transient techniques for density and viscosity closely follows published values. This technique was also extended to liquids.