Microrheology of Newtonian fluids using microcantilever

Preprint ‐ The Second IASTED International Conference on Nanotechnology and Applications, NANA 2010,

November 1 ‐ 3, 2010, Cambridge, USA, pp. 540‐546. (doi : 10.2316/P2010.707‐018).

MICRORHEOLOGY OF NEWTONIAN FLUIDS USING MICROCANTILEVER

M. Youssry, B. Caillard, C. Ayela, C. Pellet, I. Dufour Université de Bordeaux, IMS Laboratory

351 Cous de la Libération, 33405 Talence, France

[email protected]

ABSTRACT The vibrating silicon-based microcantilever proved to be microrheometer. One calibration step and non iterative calculations are enough to investigate the viscosity and density of fluids over higher frequency than that obtained by mechanical rheometers. The selection of a cantilever with higher aspect ratio (thickness/width) results in highly accurate measurements. Moreover, combination of the results of more than one cantilever broadens the frequency measurement limits. KEYWORDS MEMS; Microrheology; vibrating microcantilever; high-frequency viscosity; Newtonian fluids. 1. Introduction Microrheology is the study of the viscoelastic properties of soft materials on micrometer scales and how materials store and dissipate mechanical energy as a function of length scale [1,2]. Recently, several complementary techniques have been developed to measure viscoelastic properties of soft materials on micrometer scales. Such methods provide new prospects for the characterization of local inhomogeneities in materials, for studies of material properties in small samples, including living cells, surface viscoelasticity, shear flow effect and for high frequency rheology [2]. Among these techniques, particle tracking microrheology, diffusing wave spectroscopy, laser tracking, and magnetic tweezers have been extensively used during the past decade. However, these techniques have some drawbacks due to a possible interaction between the probe particles and the material under investigation and limited measurements due to the complexity of materials [3]. Since Atomic force microscope (AFM) has been used to study the dynamic of biological materials qualitatively [4], considerable attention has been devoted to AFM as a probe to study the viscoelasticity of complex fluids quantitatively [5,6]. With AFM, the attachment of a colloidal sphere to the cantilever of the microscope provides more accurate measurement of the hydrodynamic interaction between the particle and the probe [5]. These rheological functions can be subsequently modeled with standard rheological theory and compared with bulk measurements. Based on this approach, Belmiloud and coauthors proposed an

analytical model based on a simplified hydrodynamic function [7] to estimate the material properties using the vibrating phenomena of silicon-based microcantilever [8]. Under the circumstances of incompressible Newtonian fluids using a microcantilever with a negligible thickness, the authors could for some extent estimate with good accuracy the viscosity and density of silicon oils over large frequency range. This non-destructive method offers some advantages over the mechanical rheometer: a) studying the high-frequency viscoelasticity since large forces and torques are required in the rheometers and b) minimization of sample volumes where few microliters are enough. In the current study, we use the silicon-based microcantilevers with different geometries to probe density and viscosity of some Newtonian fluids with known viscosities and densities (silicon oils and dodecane) over large frequency range in order to examine the validity of our novel approach. In addition, the variation of the parameters of the hydrodynamic function with Reynolds number (Re) and consequently on the accuracy of measurements have been studied. The optimization of cantilever geometry is discussed. The complex fluids are involved in many industrial applications including cosmetics, pharmaceutics, and food technology and processing [9]. Non-destructive in-situ probe for viscoelasticity is highly required to investigate the microstructure of complex fluids in microliters volume scale. There is thus a wide range of fields that will benefit from new rheological probes driving the current research.

2. Experimental 2.1 Materials Silicone oil with viscosity range from 10 to 500 cP, at 25°C, and dodecane were purchased from Sigmaaldrich.

2.2 Experimental setup Arrays of rectangular silicon cantilevers were provided by ESIEE Engineering, Paris, France (www.esiee.fr). A series of cantilevers were used for measurements, namely



D4, D6, E5 and E6, having different geometries as shown in Table 1.

Table 1. The geometries (L=length, b=width, h=thickness) and resonance frequency in both air (measured) and vacuum

(calculated using Eq. 7) of the cantilevers. Reynolds number (Re) ranges corresponding to the experimental frequency limits

for each fluid and cantilever, where2

2f

Re f b .

D4 D6 E5 E6 L (µm) 4000 6000 5000 6000 b (µm 600 600 1000 1000 h (µm) 100 110 109 43 fr (Hz) 8639 4212 6020 1649 f0,vac (Hz) 8652 4218 6033 1657 h/b 0.167 0.183 0.109 0.043 Fluid Re range

Dodecane 1120-1630

406-914

1130-4510

71-1410

10 cP 106-247

68-181

94-848 16-173

20 cP 91-142 34-90 47-424 9-157 50 cP 21-67 11-34 13-157 3-63 100 cP 10-33 5-16 6-79 2-31 500 cP 1-5 1-3 --- ---

The silicon chip containing the cantilever was fixed on a support inbetween a magnet (for the electromagnetic actuation of the cantilever) in the lower and Teflon vessel (to put the fluid around the cantilever) on the upper side as shown in Fig. 1. Few hundreds of microliters were enough to completely immerse the cantilever in the fluid. This cell is then connected to a Scanning Laser Doppler Vibrometry (MSA-500 Micro-System Analyzer from Polytech). Vibrometer is powerful to characterize quickly out-of-plane and in-plane dynamic parameters of microstructures without contact up to several MHz. For out-of-plane measurement the vibrometer principle is based on the use of a two beam laser interferometer that measures the frequency (or phase) difference between an internal reference laser beam and a measurement beam. By applying a periodic excitation to the cantilever, the backscattered laser light interferes with the reference beam. This interference is recorded by a photodetector, and then decodes providing a voltage which is proportional to the velocity of the vibration parallel to the measurement beam. The voltage is digitized and processed as vibrometer signal. For measurement in liquid media, the signals were collected after averaging 100 runs. All the measurements have been done in a clean

room, in which temperature and humidity were fixed at 19.4°C and 40%, respectively. Thus, the estimation of the viscosity and density at 19.4°C was necessary since the values provided by Sigmaaldrich are at 25°C. The viscosities were determined using mechanical rheometer supported by Peltier system for temperature control (±0.1°C) (Table 2).

Figure 1. Schematic representation summarizing the steps of the data treatment and showing the experimental tools.

3. Theory For a vibrating cantilever immersed in a Newtonian incompressible fluid, the drag force per unit length exerted on the surface of the cantilever can be obtained by (the underlined functions correspond to the functions in the frequency domain):

2 2

1 22 ( ) 4 ( )

fluidfF j f g f w g f w (1)

where w is the amplitude of vibration, f is the frequency

and, 1( )g f and

2( )g f are related to the imaginary and real

parts of the hydrodynamic function f , respectively:

2 2

12

f

i

b fg f

(2a)

2

24

f

r

bg f

(2b)

where i

f and r

f are, respectively, the imaginary

and real parts of the hydrodynamic function, f

is the

fluid density and b is the cantilever width.



For a rectangular cantilever with negligible thickness, Maali et al. [7] simplified the hydrodynamic function of ribbon geometry proposed by Sader [10] to be:

1 2ra a

b

(3a)

2

1 2ib b

b b

(3b)

where is the thickness of the thin viscous layer

surrounding the cantilever in which the velocity has dropped by a factor of 1/e (e=2.72), and can be given by

/f

f

[11]. The parameters a1=1.0553,

a2=3.7997, b1=3.8018 and b2=2.7364 were estimated by Maali et al over a wide range of Reynolds number (1-103) [7].

If higher order modes doesn’t interfere with the first order mode, the ratio of the deflection, at the free-end of the

cantilever, ,w L f , to the excitation force at the free end

,exc

F L f which is defined as the transfer

function H ( L, f ) can be written as:

0

2

0 0

HH( L, f )

f f1 2 j

f f

(4)

where H0 is the static value of the transfer function, f0 and ξ are, respectively, the eigen frequency and the damping

coefficient in the fluid which both depend on the terms

1( )g f and

2( )g f . The imaginary and real parts of the

normalized transfer function with respect to the static

value (0

( ) / sini

H f HH and0

( ) / cosr

H f HH ,

respectively) can be obtained experimentally at each frequency, knowing the module and its phase .

Consequently, 1( )g f

and

2( )g f

in Eqs. 2 can be

calculated at each frequency by:

2

0,0

12 sinvac

L

fHg m

H f (5a)

2

0,0

21 cos 1vac

L

fHg m

H f

(5b)

where L

m ( )m L is the mass per unit length of the

cantilever, and 0,vac

f is the resonance frequency of the

cantilever in vacuum.

Thus, the determination of the two parts of the drag force allows one to extract analytically at each frequency the viscosity and density, i.e., the fluid’s rheological properties [8]:

2

2 1 2 2 1 2 1 2 2 1 2 1 2 1 1 2

2 21 1 2 2 1

2 (2 ) 2 8

( )f

f g a b a b a g a f g b a g f g g a b

b f a a b a b

(6a)

22

1 2 1 1 2 2 1 2 1 2 1 1 2

2

1 2 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 1 1 2

2 2 8

2 (2 ) 2 8

a f g b g a f g b a g f g g a b

a b a b f g a b a b a g a f g b a g f g g a b

(6b)

3. Results and discussion

Because of the thickness of the cantilever is not well-known, one calibration test for the microcantilever is necessary to estimate the cantilever thickness using the

measurement of the resonance frequency in air (r

f ). The

thickness of cantilever (h) is then calculated using the equation:



2

0

0 , 22 12

vac

h Ef

L

(7)

taking into account the relation between the resonance frequency in air and in vacuum:

2

0 ,

2

1

11

2

L

vac r

g

mf f

Q

(8)

where L is the length of the cantilever and Q the quality factor of the cantilever in air, E and ρ are, respectively, the Young’s modulus (1.7×1011 Pa) and the density (2330 kg.m-3) of the cantilever, and λ0=1.875.

For each cantilever used in this study, r

f was measured

using the vibrometer then the h values were calculated. The data are shown in Table 1. It is worth to mention that these cantilevers have been manufactured using etching technology which may result in less uniform thickness than with silicon-on-insulator (SOI) process.

Figure 2. The amplitude and phase spectra of D4 in fluids.

Figure 2 shows, as an example, the effect of fluids on the amplitude and phase spectra of the silicon-based microcantilever D4. As the fluid viscosity increases, the resonance frequency decreases and the peaks become broader until it vanishes at 500 cP silicone oil. Based on these experimental data, the variation of viscosity and density over frequency can be estimated as follows:

1. On fitting the normalized amplitude spectra by Eq. 4 over the experimental frequency range, the static value of the transfer function (H0) can be obtained for each fluid.

2. The parameters a1, a2, b1, and b2 are obtained over a Reynolds number range using the hydrodynamic function (Eq. 3).

3. Then, the treatment of the experimental data (normalized amplitude and phase) with Eqs. 5 and 6 yields the estimation of both viscosity and density over a relatively large frequency range depending on the cantilever geometry and fluid viscosity as discussed below.

Table 2. The mean values of viscosity η (cP) and density ρ (kg/m3) and their % error of dodecane and silicon oils using different

microcantilevers and different parameters.

can

tile

ver

Fluids (1) η (2) ρ

Maali’s parameters Modified parameters Maali’s param. Modified param.

η ρ η ρ %error η %error ρ

%error η %error ρ

D4

Dodecane 1.67 750 1.89 686 1.26 714 13.2 -8.5 -24.5 -4.8 10 cP 10.6

1000

12.78 855 10.61 875 20.6 -14.5 0.1 -12.5 20 cP 22.3 27.49 884 25.81 885 23.3 -11.6 15.7 -11.5 50 cP 54.2 72.59 870 70.74 866 33.9 -13.0 30.5 -13.4 100 cP 111 163.09 829 167.04 815 46.9 -17.1 50.5 -18.5 500 cP 553 1162 491 1146.50 498 110 -50.9 107.3 -50.2

D6

Dodecane 1.67 750 2.13 765 1.62 786 27.5 2.0 -3.0 4.8 10 cP 10.6

1000

15.3 984 14.45 984 44.3 -1.6 36.3 -1.6 20 cP 22.3 32.9 979 31.78 978 47.5 -2.1 45.5 -2.2 50 cP 54.2 83.8 956 85.38 942 54.6 -4.4 57.5 -5.8 100 cP 111 207.8 892 211.52 884 87.2 -10.8 90.6 -11.6



500 cP 553 1839.6 347 1773.6 362 232.7 -65.3 220.7 -63.8

E5

Dodecane 1.67 750 2.6 576 1.71 599 55.7 -23.3 2.4 -20.0 10 cP 10.6

1000

18.4 818 15.1 838 73.6 -18.2 42.5 -16.2 20 cP 22.3 39.5 739 36.9 741 77.0 -26.1 65.5 -25.9 50 cP 54.2 78.7 767 75.6 766 45.2 -23.3 39.5 -23.4 100 cP 111 174.3 746 174.2 739 57.0 -25.4 57.0 -26.1

E6

Dodecane 1.67 750 5.66 276 5.6 272 239.0 -63.2 235 -63.7 10 cP 10.6

1000

10.7 417 10.3 417 0.9 -58.4 -2.8 -58.4 20 cP 22.3 21.7 347 21.7 343 -2.7 -56.4 -2.7 -56.7 50 cP 54.2 64 262 65 259 18.0 -73.8 20.0 -74.1 100 cP 111 271 209 266 213 144.0 -79.1 139.6 -78.7

(1) The 10-500 cP values correspond to the viscosities of silicon oils at 25°C, and used as names only for identification. (2) The actual values of viscosities estimated using mechanical rheometer at 19.4°C.

Recently, Dufour et al. presented, in details, a new method to calculate the mean values of viscosity and density of Newtonian fluids using vibrating cantilever [12]. Based on this method, we calculated here the mean values of viscosity and density of the examined fluids. The values are depicted in Table 2 for four cantilever geometries. As can be seen, two different sets of parameters (a1, a2, b1, and b2), namely Maali’s and modified parameters, were used separately to calculate the mean values of viscosity and density. Maali et al. have used these parameters as a result of fitting the hydrodynamic function (Eqs. 3) over a large Re range (1-103) [7], however, we divided this relatively large range into four different Re ranges. For each range, the parameters were estimated by fitting the hydrodynamic function (Eqs. 3), as indicated in Table 3.

Table 3. The Maali’s and modified parameters over varied Re ranges.

Re range a1 a2 b1 b2

100-103 1.0553 3.7997 3.8018 2.7364 Maali’s param.

100-101 1.0800 3.7410 3.7230 2.8800 Modified param.

101-102 1.0560 3.8740 4.0120 1.5870 102-103 1.0230 4.3280 4.5340 -5.3700 103-104 1.0080 4.9410 5.1630 -31.240

It is obvious how the modified parameters improve the results through decreasing the % error in both viscosity and density data compared to Maali’s parameters, especially for D4 and D6. This will be more obvious on comparing the effect of these parameters on the viscosity and density-frequency curves as discussed below. As can be seen in Figure 3, the variation of both viscosity and density of various fluids over high

frequency (3-8 kHz) is well estimated with a good accuracy. Using the appropriate parameters (a1, a2, b1, and b2), high accuracy and less scattered frequency sweep curves have been obtained. This is more evident on using the so-called modified parameters which are estimated for each Re range specific for the experimental frequency limits for each fluid. Relatively high scattered curves have



been obtained on using Maali’s parameters (which were estimated over a broad Re range) as shown in Fig. 3b.

Figure 3. Variation of viscosity η (top) and density ρ (bottom) of fluids over frequency f using the

microcantilever D4, considering: a) the modified parameters, and b) Maali’s parameters.

The same results have been obtained using the microcantilever D6 as shown in Figure 4. However, the viscosity and density have been estimated over a relatively low frequency range (1-4 kHz). Interestingly, on combining the results of both D4 and D6, we could obtain a frequency dependent viscosity and density curves over larger frequency range. Eventually, an obvious independency of viscosity on the frequency, that is characteristics for Newtonian fluids, has been approved over higher frequency range using vibrating microcantilevers. Such high frequency range could not be achieved using conventional mechanical rheometers due to torque limits.

Figure 4. Variation of viscosity η (top) and density ρ

(bottom) of fluids over frequency f using the microcantilever D6, considering: a) the modified

parameters, and b) Maali’s parameters.

The cantilever geometry plays an important role in the accuracy of measurements since the choice of the parameters depends on the Reynolds number

( 22

fRe f b ). Thus, examination of the results

obtained by various cantilevers having different aspect ratios (h/b) (Table 1) is necessary. As depicted in Table 2, relatively smaller % errors in the viscosity and density measurements have been obtained using both D6 and D4 than E5 and E6, i.e., the accuracy of measurements increases on increasing the aspect ratio (h/b).

4. Conclusion We show that the vibrating silicon-based microcantilevers can be used to measure the mean values of viscosity and density of Newtonian fluids and their variation over high frequency range. One calibration step and non iterative calculations are sufficient to probe those fluid properties over higher frequencies overcoming the limitations of mechanical rheometers. Combination of the results of two cantilevers enables us to extend the experimental frequency range. Moreover, the selection of a cantilever of relatively high aspect ratio (thickness/width) guarantees higher accuracy in the measurements. The relatively higher percent error in viscosity and density measurements of some fluids may be attributed to the non-uniform thickness of cantilever as a result of the manufacturing process. Moreover, the clamping of cantilever with the support may affect as well on the results, because it modifies the effective Young’s modulus and then the cantilever thickness estimation which affect the numerical results of both the viscosity and the density estimations. Both reasons need careful investigation to enhance the accuracy of measurements. Moreover, the fitting allowing the estimation of the Maali’s or modified parameters of the hydrodynamic function is based on a hydrodynamic function expression [10] which corresponds to the ribbon case (negligible thickness compared to the other cantilever dimensions). Then the same method has to be tested with new parameters obtained with a more accurate expression of the hydrodynamic function for the considered cantilever geometries.



The microcantilevers offer a fast relatively accurate in-situ tool to measure the viscosity and density of fluids in the microliters volume scale. These advantages are highly desired in various application fields such as pharmaceutical and food industries and biological fluids analysis.

Acknowledgments This work has been partially supported by the French National Agency (project n°ANR-08-NANO-004), by the CPER Pôle 4N Nanosciences en Aquitaine (GP 206-action 216/1) and by the Conseil Régional Aquitaine (project MicRhéo-Aquitaine 2009-1102001). We thank Dr. Annie Colin for her help in the rheological measurements. References [1] F.C. MacKintosh, C.F. Schmidt, Curr. Opin. Colloid Interface Sci. 4, 300 (1999).

[2] T.A. Waigh, Rep. Prog. Phys. 68, 685 (2005).

[3] G. Pesce, A.C. De Luca, G. Rusciano, P.A. Netti, S. Fusco, A. Sasso, J. Opt. A: Pure Appl. Opt. 11, 034016 (2009).

[4] C. Rotsch, K. Jacobson, M. Radmacher, Proc. Natl. Acad. Sci. U.S.A. 96, 921 (1999).

[5] R.E. Mahaffy, C.K. Shih, F.C. MacKintosh, and J. Käs, Phys. Rev. Lett. 85, 880 (2000).

[6] B. Du, O.K.C. Tsui, Q. Zhang, T. He, Langmuir 17, 3286 (2001).

[7] A. Maali, C. Hurth, R. Boisgard, C. Jai, T. Cohen-Bouhacina, J.-P. Aiméa, J. Appl. Phys. 97, 074907 (2005).

[8] N. Belmilouda, I. Dufour, A. Colin, L. Nicu, Appl. Phys. Lett. 92, 041907 (2008).

[9] R.G. Larson, The Structure and Dynamics of Complex Fluids (Oxford University Press, Oxford, UK, 1999)

[10] J. E. Sader, J. Appl. Phys. 84, 64 (1998).

[11] G. K. Batchelor, Fluid Dynamics (Cambridge University Press, Cambridge, UK, 1974).

[12] M. Youssry, N. Belmiloud, B. Caillard, C. Ayela, C. Pellet, I. Dufour. Procedia Engineering xx, xx, (2010).

Microrheology of Newtonian fluids using microcantilever

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