Measurement of dynamic properties of small volumes of fluid using MEMS

Available online at www.sciencedirect.com

Sensors and Actuators B 130 (2008) 701–706

Measurement of dynamic properties of small volumes of fluid using MEMS David Cheneler a,∗ , Michael C.L. Ward a , Michael J. Adams b , Zhibing Zhang b a

Department of Mechanical Engineering, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK b Department of Chemical Engineering, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

Received 22 August 2007; received in revised form 8 October 2007; accepted 18 October 2007 Available online 3 December 2007

Abstract A model based on the response of a micro-rheometer which permits the measurement of the linear viscoelastic properties of small volumes of a fluid is described. The configuration involves a liquid being contained within a capillary bridge between two flat smooth parallel platens that are actuated sinusoidally using a compliant MEMS device. Approximate closed-form equations are derived to analyse the data taking account of both the capillary forces and those arising from viscoelastic flow. The approximate theory is compared to a full numerical simulation of the response of the MEMS rheometer and the validity is discussed. © 2007 Elsevier B.V. All rights reserved. Keywords: Squeeze flow; Capillary bridges; MEMS; Viscoelastic properties; Complex fluid

1. Introduction There is often a need to measure the properties of liquid. Occasionally the available volume of the liquid of interest may be sufficiently small as to render conventional methods of rheometry such as cone and plate rheometry [1], stormer viscometry [2] or falling ball viscometry [3] inappropriate. Consequently, there is a growing interest in the use of MEMS devices to measure the required properties, especially with an aim of encouraging high throughput. These devices include pressure sensors [4], optical tweezers [5] and micro-particle image velocimetry [6] amongst others [7–11]. The current paper examines the potential of employing MEMS for making a rheometer based on squeeze flow, which is a convenient configuration for this technology. In particular, the design and analysis of a device based on sinusoidal oscillation for viscoelastic fluids in the linear strain region will be described. As will be seen, the linearisation of the model requires the use of small amplitudes,

∗

Corresponding author. E-mail address: [email protected] (D. Cheneler).

0925-4005/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.snb.2007.10.031

which precludes the use of traditional rheometers making this model particularly suitable for use with micro system technology. The analysis of squeeze flow rheometry is an area of continuing development in order to apply the method to fluids with more complex constitutive behaviour [12–18] than those that exhibit simple Newtonian flow. Generally steady rather than oscillatory flow has been considered although there are notable exceptions [14,16,19]. If the platens are not fully immersed so that the liquid is contained as a discrete bridge, it is necessary to consider the influence of the capillary forces that may be important for small viscosities and platen displacement velocities. Also if the micro-rheometer is based on a compliant oscillating device, the mechanical properties of the device need to be taken into account. These aspects have been neglected in previous work and will be considered here. 2. The micro-rheometer There are many possible different configurations that could be employed for a micro-rheometer [6–12]. It is not the purpose of this paper to suggest another alternative but rather to

702

D. Cheneler et al. / Sensors and Actuators B 130 (2008) 701–706

(<∼250 Hz) or the viscosities are high [15]. The oscillatory force (Fv ) arising from the viscoelastic response is given as [14]: Fv =

3πη∗ R4 iω eiωt 2h3

(1)

where η∗ = η − Fig. 1. Schematic of liquid bridge within the MEMS device.

consider the relevance of squeeze flow theory to micro-systems technology. Consequently, here a generic device will be considered that comprises a fixed lower platen and a parallel rigid upper platen with a known mass being supported by an elastic beam or spring of known stiffness (see Fig. 1). It will be assumed that the upper platen is electrostatically actuated with a sinusoidal force of known frequency and magnitude. The device will also have set operating parameters. It will comprise of electrostatic plates of approximate area 80 mm2 and gap 2 ␮m. It is assumed that the oscillation amplitude is in the range 25–250 nm and this can be measured with a resolution of 1 in 100. It is also assumed that the phase difference can be measured between 0 and π/2 rad with an accuracy of 1◦ ; the alternating voltage varies between 0.001 and 10 V with a 1 in 100 resolution and the frequency varies between 0.1 and 100 rad/s with a 1 in 100 resolution. This specification will allow an estimate to be made of the range of liquid properties that could be measured and also the associated errors. 3. Theory It is assumed that the liquid is accommodated as a pendular bridge between a fixed lower platen and a sinusoidally modulated upper platen (see Fig. 1). In a method similar to that employed by Bell et al. [14], the dynamic properties of the liquid can be calculated by measuring the amplitude of the oscillation of the upper platen and the phase difference between the force and displacement of this platen. The geometry of the liquid bridge as shown schematically in Fig. 1 is dependent on the surface tension, the contact angle and the volume of the liquid; it is assumed that the diameters of both platens are greater than the contact diameter of the liquid bridge. The meniscus curvature, ρ, is a function of the gap between the platens since the hydrostatic pressure difference, p, must be constant as defined by the Laplace–Young equation (see Eq. (3)). Viscoelastic and capillary forces act between the platens. Viscoelastic forces act only when the bridge is in motion as in the case of sinusoidal squeeze flow. Generally if the radius of the liquid bridge is at least 10 times greater than the gap, the lubrication approximation can be applied [14]. This requires that the liquid flows predominantly parallel to the surface of the platens with any elongational or transient flows being neglected. This corresponds to a radial pressure driven flow, which is analogous to Poiseuille flow in a tube such that only a shear velocity field exists. For most practical cases fluid inertial effects may be ignored as generally the frequencies of interest are low

iG ω

(2)

Here η* and η are the complex and dynamic viscosities of the fluid, ω is the angular frequency of oscillation G is the storage modulus of the liquid, h is the current gap between the platens and t is the time. Due to the curvature of the bridge profile, the radius of the bridge varies along the z-axis. However, it is assumed in the derivation of Eq. (1) that the bridge is cylindrical [14]. Here, to simplify the calculations for the viscoelastic force, the bridge radius, R, will be taken as the average value which is equivalent to the radius of a cylinder of the same volume. The viscoelastic force acts to resist motion during either the approach or separation of the platens. Strictly the above equations apply to a fixed value of the bridge radius. In the current scheme, this radius will vary sinusoidally. However, the assumption will be made that the amplitude of oscillation will be small compared to the gap so that the error involved will be small. The total capillary force, FC , is the sum of that due to surface tension and that associated with the pressure difference arising from the curvature across the liquid/vapour interface as given in Eq. (3). It cannot be calculated analytically except in a few special cases (for example cylindrical geometries and flat planes [21]) and may be written as [20]:   1 1 FC = 2πRN γlv − πR2N γlv (3) − RN ρ where γ lv is the surface tension of the liquid, RN is the neck radius and ρ is the other principal radius of curvature of the liquid bridge. It is assumed that to a close approximation, the total force exerted by the liquid may be given by the sum of the viscoelastic and capillary forces. The response of the MEMS device is the sum of the fluid forces (Eqs. (1) and (3)) and that arising from the compliance of the device. Thus the complete response is as follows: m

d2 z 3πη∗ R4 dz + kz + 2πRN γlv + dt 2 2h3 dt   1 1 2 −πRN γlv − = F0 sin ωt RN ρ

(4)

where z is the axial coordinate with the origin at the surface of the lower platen (see Fig. 1), F0 is the driving force, k is the stiffness of the MEMS device and m is the mass of the upper platen. Since the current value of ρ has to be obtained numerically [21], this equation cannot be solved analytically. Even if this were not the case, the radius, height and the curvature of the bridge are nonlinear functions of time, which precludes an analytical solution

D. Cheneler et al. / Sensors and Actuators B 130 (2008) 701–706

of Eq. (4). However, an approximate solution to Eq. (3) can be derived and used to simplify Eq. (4). This involves treating the meridional curvature of the liquid bridge as the arc of a circle, which is known as the toroidal approximation. There has been considerable work on the use of the toroidal approximation for the simplification of the calculation of capillary forces in pendular liquid bridges between spherical bodies, (e.g. [21–25]). It has been shown that with minor corrections the accuracy can be improved. Relatively little attention has been given to the geometry of interest here (the flat-on-flat case) but notable exceptions include [12,13]. The solution to Eq. (3) using the toroidal approximation is given in Appendix A. It will be shown later that if the amplitude of the oscillations is sufficiently small the capillary force, as given by the toroidal approximation, can be taken to be linear. This means the approximate solution of Eq. (3) given by Eq. (A10) can be simplified to give: FC = kCap z + cCap

kCap cCap

¯ d(ζFCT (h)) = dh¯ ¯ − kCap h¯ = ζFCT (h)

arising from the capillary forces. The parameters ε and φ can be deduced by comparing Eqs. (9) and (10). The difference between the initial position of the upper platen and h0 is equal to the ¯ which can be determined by mean gap between the platens, h, equating the surface tension and the spring stiffness forces. It is ¯ is calculated in the at this gap that the mean bridge radius, R, same way as in Eq. (1). Eq. (9) can be rearranged thus: z=

F0 F0 ωr cos(ωt) + 2 B sin(ωt) 2 2 +ω r B + ω2 r 2 cCap − k − kCap B2

(6)

(11)

From this equation the dynamic properties of the liquid bridge can be found: G =

2h¯ 3 ((F0 /ε) cos φ + ω2 m − k + kCap ) ¯4 3πR

(12)

G =

2h¯ 3 F0 sin φ ¯4 3πεR

(13)

(5)

where the constants kCap and cCap are given by the following expressions:

703

where G = ωη and G is the loss modulus. Eq. (12) reduces to that given by Bell et al. [14] when the capillary force, the compliance and inertia of the MEMS device tends to zero.

(7)

¯ is where ζ is the scale factor as given in Appendix A and FCT (h) ¯ It can be seen that there the solution of Eq. (A7) evaluated at h. is a close agreement between the solution of Eq. (A10) and the linearised capillary force. This linear approximation becomes equal to Eq. (A10) as the oscillations tend to zero viz., as the maximum and minimum gap approach the mean value. Even with a closed-form solution for Eq. (3), it still remains to solve Eq. (4). The non-linearities already discussed may be relaxed by assuming that the amplitude of the sinusoidal displacement is small compared to the gap. Also this implies that the bridge radius and the gap between the platens can be assumed to be equal to their mean values and that the capillary force varies linearly with gap. This allows Eq. (4) to be simplified to

4. Validation and results The method described in the previous section for linearising the capillary and viscoelastic forces with respect to the gap between the platens has been validated by comparing the response of the liquid bridge to the full non-linear response of the bridge (see Fig. 2). This non-linear response was calculated numerically using a variable order Adams-BashforthMoulton PECE solver using MatLab v7.0 (The MathWorks, Inc.). In order to consider a specific case, it is assumed that the MEMS device consists of an upper platen of mass 0.1 mg supported by a spring of stiffness 2000 N/m, with an initial

¯ 4 dz d2 z 3πη∗ R + kz + kCap z + cCap = F0 sin ωt + (8) dt 2 2h¯ 3 dt ¯ and h¯ are the mean values of the bridge radius and the where R gap between the platens, respectively. The solution to Eq. (8) is as follows:   ωr   F02 cCap z= (9) − sin ωt + tan−1 2 2 2 k − kCap (B + ω r ) B

m

¯ 4 G /2h¯ 3 ) + k − kCap and where B = −ω2 m + (3πR ¯ 4 η /2h¯ 3 ). r = (3πR Eq. (9) has the form: z = ε sin(ωt + φ) + h0

(10)

where ε is the amplitude, φ is the phase difference and h0 is the static deviation of the upper platen from its initial position

Fig. 2. Comparison between full numerically calculated non-linear response and linearised response of MEMS device.

704

D. Cheneler et al. / Sensors and Actuators B 130 (2008) 701–706

Fig. 3. Capillary force as a function of gap between platens.

gap between the platens of 10 ␮m. The excitation force was taken to have a frequency of 10 rad/s and a magnitude of 10 mN. The liquid used as an example here is PDMS (polydimethylsiloxane) with a volume of 1 nL. This liquid has a surface tension of 21.5 mN/m [18] and a shear viscosity of 30 Pa s [14]. The storage and loss moduli at the frequency stated are ca. 20 and 300 Pa, respectively [14]. For this example, it is assumed that the surfaces of the platens have been treated to give a contact angle of 45◦ . As may be seen in Fig. 2, the linearised response is an excellent approximation to the numerical non-linear solution. In this case the mean error was ca. 0.05% with a standard deviation of 0.06%. It can also be seen in Fig. 2 that the mean gap h¯ is about 9.45 ␮m. The amplitude of the oscillations is ca. 0.2 ␮m which provides a measure of the maximum and minimum gaps. These extrema can be used to calculate the upper and lower limits of the capillary force as shown in Fig. 3. In Fig. 3 it has been shown that by comparing the linearised force with the exact solution of Eq. (3), as the amplitude tends to zero the capillary force can be approximated by a straight line of the form Eq. (5). In Fig. 4, the linearised response has been normalised and plotted as a function of time. The excitation force has also been normalised and plotted so that the

Fig. 4. A comparison between the normalised and linearised response and the normalised excitation force.

Fig. 5. The range of accurately measurable loss moduli.

phase difference, φ, between the force and the response is evident. From Eq. (9) the phase difference was calculated to be ∼0.73 rad which is consistent with the value in Fig. 4. This coupled with the amplitude of the oscillation and Eqs. (11) and (12) allow the storage and loss modulus to be calculated as 20 and 300 Pa, respectively. This is consistent with the values used in the numerical non-linear simulation with an error of ca. 0.01%. Given the operating parameters stated previously, the current method can be used to calculate the range of properties that can be measured within 10% of the actual value. The range of loss moduli that can be measured accurately is shown in Fig. 5. The maximum accurately measurable loss modulus for this particular device is 0.1 GPa and the minimum is 0.01 Pa with values being similar for the storage modulus. This range is sufficient for most applications and can be varied by changing the dynamic properties of the MEMS device. There are of course natural limits to the applicability of the theory including those already discussed in relation to the lubrication approximation. In addition, the fluid must be continuous and therefore the liquid bridge must be sufficiently large to contain a suitable number of molecules for continuum mechanics to apply [26]. The silicone oils considered above all had radii of gyration that were less than 20 nm and as the films were 10 ␮m thick the molecular dynamics are comparable to the bulk fluid and so the liquid can be treated as continuous [27]. Other limits depend on the MEMS device. It is possible to create very thin gaps (ca. 3 nm) [28] by growing an oxide on a silicon surface, depositing a layer of polysilicon and then removing the oxide. It is also possible to detect sub-angstrom displacements [29] and there are a number of actuators that may be utilised to achieve them [30]. Another consideration is that as the gap becomes small, the force required to actuate the upper platen increases exponentially. While these forces can be reduced by decreasing the volume of fluid, accurate volume control and

D. Cheneler et al. / Sensors and Actuators B 130 (2008) 701–706

705

positioning is not trivial and specialist technology is required [31]. The limits therefore become one of practicality. Whilst it is possible to measure the dynamic properties of very small volumes using small gaps and small amplitudes, the number, complexity and cost of processes required to manufacture the MEMS rheometer may be the limiting factor on for the volume of fluid. 5. Conclusion A scheme for analysing the data from a compliant MEMS device to measure the dynamic properties of a liquid contained within a capillary bridge has been proposed. It is much more convenient than the full non-linear analysis whilst still retaining accuracy. It is also particularly suited to MEMS as the volume of fluid and the amplitude of oscillations can be restricted to small values and yet still be measured accurately. Acknowledgements

Fig. 7. A comparison between the scaled dimensionless toroidal approximation of the surface tension force and the dimensionless numerical solution calculated in the same way as in Ref. [21] for various contact angles.

From Fig. 6 The University of Birmingham and Unilever Research & Development are acknowledged and thanked for their financial support.

ρ=

h 2 cos θ

(A3)

and

Appendix A. Appendix A The geometry used in the toroidal approximation is shown in Fig. 6. The profile of the bridge (denoted by a thick black line in Fig. 6) can be represented as   2 h h x(z) = Rc + tan(θ) − − z2 (A1) 2 2 cos θ where Rc is the contact line radius and the volume may be written

RN = Rc +

h h tan(θ) − 2 2 cos θ

Substituting Eqs. (A1), (A3) and (A4) into Eq. (A2) gives  h  2 V = π(RN + ρ − ρ2 − z2 ) dz (A5) 0

From Eq. (A5) the neck radius, RN , may be obtained for any volume, V, and any contact angle, θ, as a function of the gap between the platens, h. This may be written in the following form:

⎞ ⎛ ⎛ ⎞  

 2  2

2A 2 4V 1 hA ⎝ hA h 1 h ρ h ⎠+ ⎠ RN =  2 + ρ2 − ρ2 − − + − 3ρ2 − ⎝ρ − 2 cos θ 4 cos2 θ 2 πh 12 2 2 h as:



V =

h

π(x(z)2 )dz

(A2)

0

(A4)

(A6)

where A = arcsin(h/2ρ). The toroidal approximation of the solution to Eq. (3) is given thus:   1 1 2 FCT = 2πRN γlv − πRN γlv (A7) − RN ρ Now Eq. (3) can be solved. However, as Eqs. (A3) and (A6) are approximate, a scaling factor needs to be applied in order to recover the accuracy of the full solution to Eq. (3). The capillary force is first non-dimensionalised thus: ∗ FCT =

Fig. 6. The geometry of the liquid bridge as represented by the toroidal approximation.

FCT γlv a

where a is a characteristic length given by √ 3 a= V

(A8)

(A9)

706

D. Cheneler et al. / Sensors and Actuators B 130 (2008) 701–706

Then a scaling factor is introduced:   √ FCT 3 ∗ FCT = 0.695 π − sin( θ) (A10) γlv a   √ 3 where ζ = 0.695 π − sin( θ) is an empirical factor that improves the accuracy for all contact angles and volumes (see Fig. 7). This expression was obtained by comparing the exact solutions of the Young–Laplace equation and those of the toroidal approximation, which shows that ζ is a function of √ sin θ. The complete function was derived numerically. References [1] S. Raha, et al., Cone and plate rheometer for polymer melts, J. Phys. E: Sci. Instrum. 1 (1968) 1113–1115. [2] K. Nishimura, Viscosity of fluidized snow, Cold Reg. Sci. Technol. 24 (2) (1996) 117–127. [3] S. Feng, A.L. Graham, P.T. Reardon, J. Abbott, L. Mondy, Improving falling ball test for viscosity determination, J. Fluids Eng. 128 (2006) 157– 163. [4] S. Baek, J.J. Magda, Monolithic rheometer plate fabricated using silicon micromachining technology and containing miniature pressure sensors for N1 and N2 measurements, J. Rheol. 47 (5) (2003) 11–17. [5] A. Buosciolo, G. Pesce, A. Sasso, New calibration method for position detector for simultaneous measurements of force constants and local viscosity in optical tweezers, Opt. Commun. 230 (4–6) (2004) 357– 368. [6] D.M. Curtin, D.T. Newport, M.R. Davies, Utilising ␮-PIV and pressure measurements to determine the viscosity of a DNA solution in a microchannel, Exp. Therm. Fluid Sci. 30 (8) (2006) 843–852. [7] J.J. Crassous, R. R´egisser, M. Ballauff, Characterization of the viscoelastic behaviour of complex fluids using the piezoelastic axial vibrator, J. Rheol. 49 (4) (2005) 851–863. [8] H. See, J.S. Field, B. Pfister, The response of electrorheological fluid under oscillatory squeeze flow, J. Non-Newton. Fluid Mech. 84 (2/3) (1999) 149–158. [9] J.A. Odell, S.P. Carrington, Extensional flow oscillatory rheometry, J. NonNewton. Fluid Mech. 137 (1–3) (2006) 110–120. [10] J. Krayer, S. Tatic-Lucic, S. Neti, Micro-arheometer: high throughput system for measuring of viscoelastic properties of single biological cells, Sens. Actuators B: Chem. 118 (1/2) (2006) 20–27. [11] A.R.H. Goodwin, A.D. Fitt, K.A. Ronaldson, et al., A vibrating plate fabricated by the methods of microelectromechanical systems (MEMS) for the simultaneous measurement of density and viscosity: results for argon at temperatures between 323 and 423 K at pressures up to 68 MPa, Int. J. Thermophys. 27 (6) (2006) 1650–1676.

[12] H. Matsuoka, S. Matsumoto, S. Fukui, Dynamic meniscus models for MEMS elements, Microsyst Tech. Micro Nanosyst. Inf. Stor. Proc. Syst. 11 (8–10) (2005) 1132–1137. [13] M. Meurisse, M. Querry, Squeeze effects in a flat liquid bridge between parallel solid surfaces, J. Tribol. 128 (2006) 575–584. [14] D. Bell, D.M. Binding, K. Walters, The oscillatory squeeze flow rheometer: comprehensive theory and a new experimental facility, Rheol. Acta 46 (1) (2006) 111–121. [15] M.T. Shaw, W.J. MacKnight, Introduction to Polymer Viscoelasticity, 3rd Ed., Wiley, 2005, ISBN 978-0-471-74045-2. [16] B. Debbaut, K. Thomas, Simulation and analysis of oscillatory squeeze flow, J. Non-Newton. Fluid Mech. 124 (2004) 77–91. [17] M.J. Adams, B. Edmondson, D.G. Caughey, R. Yahya, An experimental and theoretical study of the squeeze-film deformation and flow of elastoplastic fluids, J. Non-Newton. Fluid Mech. 51 (1994) 61–78. [18] E.C. Cua, M.T. Shaw, Using creeping squeeze flow to obtain lowfrequency linear viscoelastic properties: low-shear rate measurements on polydimethylsiloxane, J. Rheol. 46 (2002) 817–830. [19] S. Sakai, Improvements of an oscillatory squeezing flow rheometer for small elasticity measurements of liquids, Rheol. Acta 44 (2004) 16–28. [20] C.D. Willett, M.J. Adams, S.A. Johnson, J.P.K. Seville, Capillary bridges between two spherical bodies, Langmuir 16 (2000) 9396–9405. [21] G. Lian, C. Thornton, M.J. Adams, A theoretical study of the liquid bridge forces between two rigid spherical bodies, J. Colloid Interf. Sci. 161 (1993) 138–147. [22] F.M. Orr, L.E. Scriven, A.P. Rivas, Pendular rings between solids—meniscus properties and capillary force, J. Fluid Mech. 67 (1975) 723–742. [23] S.J.R. Simons, J.P.K. Seville, M.J. Adams, An analysis of the rupture energy of pendular liquid bridges, Chem. Eng. Sci. 49 (1994) 2331–2339. [24] R.A.J. Fisher, On the capillary forces in an ideal soil; correction of formulae given by W.B. Haines, J. Agric. Sci. 16 (1926) 492–505. [25] C. Willett, S. Johnson, M.J. Adams, J.P.K. Seville, Pendular Capillary Bridges Handbook of Powder Technology, vol. 2, Elsevier, 2007, ISBN 0-444-51871-1 (Chapter 28). [26] T.M. Squires, S.R. Quake, Microfluidics: fluid physics at the nanolitre scale, Rev. Mod. Phys. 77 (2005) 977–1026. [27] L. Hartmann, F. Kremer, P. Pouret, L. Leger, Molecular dynamics in grafted layers of poly(dimethylsiloxane), J. Chem. Phys. 118 (13) (2003) 6052–6058. [28] Y. Chen, C. Lee, J. Hwu, Ultra-thin gate oxides prepared by alternating current anodization of silicon followed by rapid thermal anneal, Solid-State Electron. 45 (9) (2001) 1531–1536. [29] H. Yamaguchi, S. Miyashita, Y. Hirayama, InAs/AlGaSb heterostructure displacement sensors for MEMS/NEMS applications, J. Cryst. Growth 251 (1–4) (2003) 556–655. [30] Y. Chen, Y. Huang, C. Kuo, S. Chang, Investigation of design parameters for droplet generators driven by piezoelectric actuators, Int. J. Mech. Sci. 49 (6) (2007) 733–740.