Etienne Lemaire
email: etienne.lemaire@u-bordeaux1.fr

Benjamin Caillard

Mohamed Youssry

Isabelle Dufour

High frequency viscoelastic measurements of fluids based on microcantilever sensing: New modeling and experimental issues

Keywords: microcantilever, sensor, viscoelasticity, modeling, measurements, microrheology

teaching and research institutions in France or abroad, or from public or private research centers.

INTRODUCTION

Exploring in-situ microrheology is of fundamental interest and is widely used in many applications, such as viscoelastic monitoring of screen-printing ink and small-volume texture monitoring of commercial foams or gels. This paper presents a method for the improvement of viscoelastic property extraction for complex fluids using a microcantilever as a dynamic rheometer oscillating in the kiloHertz domain [START_REF] Agoston | Evaluation of a vibrating micromachined cantilever sensor for measuring the viscosity of complex organic liquids[END_REF]. This frequency range is difficult to explore with classical dynamic rheometers due to inertial limitations. MEMS rheometers [START_REF] Christopher | Development of a MEMS based dynamic rheometer[END_REF][START_REF] Mather | Liquid viscoelasticity probed by a mesoscale piezoelectric bimorph cantilever[END_REF] (as well as other mechanical resonators [START_REF] Fritz | Characterizing complex fluids with high frequency rheology using torsional resonators at multiple frequencies[END_REF][START_REF] Martin | Characterization of a quartz crystal microbalance with simultaneous mass and liquid loading[END_REF][START_REF] Jakoby | Miniaturized sensors for the viscosity and density of liquidsperformance and issues[END_REF]) are good candidates as high-frequency rheometers (for determining elastic, G', and viscous, G", shear moduli vs. frequency) by characterizing complex fluids at the microscale by using very small amounts of liquid. In this context, dynamic rheology is performed by exploring the use of small-scale devices and short characteristic times; this yields viscoelastic property measurements on downscaled specimen sizes not obtainable with traditional instruments, thereby forming the basis of "microrheology".

The concept of microrheology dates from 1922, when Freundlich and Seifriz used magnetic particles to study the elasticity of gelatin gels [START_REF] Freundlich | Ueber die elastizität von solen und gelen[END_REF]. Over the last two decades, rapid developments have occurred in this field [START_REF] Breedveld | Microrheology as a tool for high-throughput screening[END_REF][9][START_REF] Ortega | Interfacial microrheology: Particle tracking and related techniques[END_REF][START_REF] Cicuta | Microrheology: A Review of the Method and Applications[END_REF]. Renewed interest in microrheology has been increased by virtue of the technological innovations in colloidal engineering, light scattering, position sensitive detection and video microscopy. Currently there are two main categories of microrheology: active microrheology which involves the active manipulation of the probe particle by external forces (e.g., optical and magnetic tweezers and atomic force microscopy (AFM)) and passive microrheology which relies on the Brownian motion (thermal fluctuation) of the probe particles (e.g. video (VPT) and laser (LPT) particle tracking). Diffusing wave spectroscopy (DWS) is also a common microrheological technique widely used to characterize complex mixtures. As with other optical methods, DWS is based on the mean square displacement measurement which characterizes Brownian motion and permits the calculation of an equivalent microscopic and local shear modulus.

Microcantilevers vibrating in fluids are strongly influenced by the surrounding fluid properties [START_REF] Kirstein | The influence of a viscous fluid on the vibration dynamics of scanning near-field optical microscopy fiber probes and atomic force microscopy cantilevers[END_REF]; thus, the out-of-plane transverse deflection response may be mathematically linked to the rheological properties of the surrounding liquid [START_REF] Belmiloud | Rheological behavior probed by vibrating microcantilevers[END_REF]. Based on previous work [START_REF] Youssry | A straightforward determination of fluid viscosity and density using microcantilevers: from experimental data to analytical expressions[END_REF][START_REF] Dufour | The Microcantilever: A Versatile Tool for Measuring the Rheological Properties of Complex Fluids[END_REF], we present here a considerably enhanced analytical model in order to (1) identify highly sensitive parameters in the determination method; (2) eliminate a misleading simplifying hypothesis in the solution; and

(3) produce accurate rheograms in the case of non-Newtonian fluids. We also discuss the accuracy of the proposed method in order to examine the validity of the high-frequency rheograms produced and to demonstrate proof of concept. Using this new modeling approach, microcantilevers can probe the high-frequency viscoelastic behavior of fluids, especially the viscous characteristics, over a large frequency bandwidth around the resonant frequency. One of the main advantages of our technique is a faster response time than macrorheometers or other microrheological techniques.

I. MODELING OF MICROCANTILEVER BEHAVIOR IN COMPLEX FLUIDS

The Euler-Bernoulli differential equation (Eq. 1) governing the out-of-plane bending vibration of a microcantilever in a liquid medium is used to model the sensing situation: ( , ) ( , ) ( , ) ( , ) ( , ) ( ) ( )

L ext wx EI j g x w x g x m w x F x L x ( 1
)
where E is the Young Modulus, I is the second moment of area of the rectangular cross section, w(x, ) is the vibration amplitude at position x along the length axis, depending on the radial frequency of the vibration, L is the beam length and the position of the free-end of the cantilever, m L is the cantilever mass per unit length, F ext ( ) is the applied force amplitude at the cantilever free-end, is the Dirac function and g 1 and g 2 are, respectively, functions of the dissipative and inertial part of the hydrodynamic force per unit length, F hydro , exerted by the fluid on the cantilever. Assuming that g 1 and g 2 are independent of x [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF], an exact analytical solution of Eq. 1 exists for the free-end cantilever deflection, w(L, ). This solution gives a frequency-dependent response partially influenced by the rheological properties of the fluid:

33 0 3 sinh( ) cos( ) cosh( ) sin( ) ( , ) 1 cosh( ) cos( ) ext F AL AL AL AL wL k A L AL AL ( 2
)
where

2 4 21 ( ) ( ) / L A m g jg EI , (3)
and k 0 is the cantilever stiffness. In Eqs. 2-3, the hydrodynamic properties of the surrounding fluid are contained in the terms g 1 and g 2 that are related to the imaginary and real parts of F hydro . However, Eqs. 2-3 do not allow the inverse problem to be solved, meaning that g 1 and g 2 cannot be expressed explicitly as functions of the different parameters and measurement data ( w(L, ), E, I, L, k 0 , F ext , m L ). Nevertheless, these analytical equations may be used to fit the experimental mechanical spectra of microcantilevers measured in known fluids. To do so, final rheological properties must be linked to g 1 and g 2 , respectively, depending on the viscous and inertial parts of the hydrodynamic force:

2 2 1 ( ) ( , ) ( ) ( , ) hydro F g w x j g w x (4)
The differential equation solution supposes that the viscosity η (hidden in the terms g 1 and g 2 ) is a real number. Viscoelastic characterization probed by microcantilevers is based on the assumption that the fluid viscosity can have both a real and an imaginary part. Consequently, the constant real term corresponding to viscosity is in this case replaced by a complex number * accounting for both the viscous and elastic behavior of viscoelastic fluids. (This is classic within the context of rheological modeling [START_REF] Levine | Response function of a sphere in a viscoelastic two-fluid medium[END_REF][START_REF] Gittes | Microscopic viscoelasicity: Shear moduli of soft materials determined from thermal fluctuations[END_REF].) This notation is equivalent to considering a complex shear modulus G* for a viscoelastic fluid instead of an imaginary shear modulus. (See details in APPENDIX A). Moreover, a fluid is rheologically defined by a mass density f , and values G' and G" which are, respectively, the elastic (real) part and the viscous (imaginary) part of the complex shear modulus G* at each frequency. A Newtonian fluid such as water has no elasticity (G' = 0 Pa) and a viscosity with a frequency-independent real part (η = 10 -3 Pa.s = 1 cP, in the case of water). This defines a viscous modulus G" that linearly increases with frequency (G" = η ). A fluid is considered as complex if it does not satisfy those two conditions.

In the case of thin rectangular microcantilevers, an exact equation of the hydrodynamic force has been developed, first by Sader [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF] and then fitted by Maali et al. [START_REF] Maali | Hydrodynamic of oscillating atomic force microscopy cantilevers in viscous fluid[END_REF] in order to express separately the real and imaginary parts of the hydrodynamic force in terms of four constant coefficients (a 1 , a 2 , b 1 , b 2 ) whose values are dependent on the Reynolds number range. The accuracy of the fitting expressions are very good, provided that the appropriate Reynolds number range restriction has been verified. Based on those works, analytical equations of g 1 and g 2 as functions of the shear modulus were established [START_REF] Belmiloud | Vibrating microcantilever used as viscosimeter and microrheometer[END_REF]. Those expressions are recalled in Eqs. 5-7 (due to physical considerations detailed in APPENDIX A, a 1 = 1 and a 2 = b 1 ):

22 1 '' ' '' ' g DG B G G G (5) 22 2 ' ' '' ' C DG B G G G g ( 6
)
where

2 1 2 42 22 f f b b B b C b D ( 7
)
and b is the cantilever width. Using Eqs. 2-7 and considering a fluid with a known rheological behavior and a microcantilever with known geometry and material, it is possible to calculate the free-end deflection of the microstructure at each frequency and to compare it with experimental measurements. This also means that we are able to simulate experimental data.

II. INVERSE PROBLEM: METHOD FOR DETERMINATION OF RHEOLOGICAL FLUID PROPERTIES

A. From g 1 , g 2 to G', G"

With the expression of the hydrodynamic force from Sader [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF] it is impossible to dissociate analytically the real part from the imaginary part, whereas it is possible with the approximation developed by Maali et al. [START_REF] Maali | Hydrodynamic of oscillating atomic force microscopy cantilevers in viscous fluid[END_REF]. Then, based on Eqs. 5-7, the elastic and viscous shear moduli can be expressed. (Details are presented in APPENDIX B.) The analytical expressions are

2 22 2 1 2 1 2 " 2 4 2 2 g B B B G g C g g C D D D DD (8) 2 2 1 1" ' " BG G g C D g DG (9)
The calculation of G' and G" for any radial frequency can then be performed provided that g 1 and g 2 can be estimated with enough accuracy from the spectrum measurements. The calculation of the terms g 1 and g 2 as a function of both the amplitude and phase of the cantilever free-end deflection measurements is the purpose of the next section.

B. Improvement of the inverse problem: from deflection to g 1 , g 2

The calculation of g 1 and g 2 from the amplitude and phase of the cantilever free-end deflection has been previously done by Belmiloud et al. [START_REF] Belmiloud | Rheological behavior probed by vibrating microcantilevers[END_REF]. At that time, the authors have considered that the analytical solution (Eq. 2) of the differential equation (Eq. 1) can be approximated by a second order mechanical low-pass filter with the transfer function H considered as follows:

0 2 2 1 2 0, 0, () ( ) ( ) 11 ext L vac L vac H w Hf F g g j m m
(10) 0,vac is the cantilever's fundamental radial eigenfrequency in vacuum. Eq. 10 is almost equivalent to the Taylor series development of Eq. 2 as a function of the nondimensional term AL. However, assuming the non-dimensional term Z = (AL) 4 , it is also almost equivalent to the ratio of two polynomial functions of Z, with the numerator order equal to 0 and the denominator order equal to 2; this corresponds to taking only the first resonant mode into account. In order to improve the accuracy of the approximating function to Eq. 2, orders were added to the numerator and the denominator by expanding the Taylor series to higher orders. Examples of these results are presented in Fig. 1. As the Taylor series order is increased, the exact transfer function is approached more and more accurately.

Fig. 1 -Approximation of the amplitude of the transfer function (Eq. 2) for a 1440x285x20 µm 3 silicon microcantilever with numerator and denominator orders respectively equal to 0 and 2 (blue), 1 and 2 (dashed green), 2 and 2 (dotted red), 3 and 3 (dotted light blue) which correspond to Taylor series of increasing orders. The reference curve (Eq. 2 is shown in purple)

An optimum has been searched for in order to obtain both high accuracy and a quite simple expression of the transfer function H so that Z (hence A, g 1 , g 2 , G' and G") can be expressed analytically as a function of H. The Taylor series development order has been fixed to obtain numerator and denominator orders of three. This process led to the following approximate equation, which will be used in this paper instead of Eq. 10: 32 00 32 3 0 4 ( , ) sinh( ) cos( ) cosh( )sin( ) 2554051500 311850 315 3 3 1 cosh( ) cos( )

4 37424200 1260 3 exc Z Z Z k w L k H AL AL AL AL Z Z Z H F A EI AL AL (11)
This is a quotient of third order polynomials in Z with H as a parameter. (H 0 is the static value of H). Each one of the three solutions (the analytical expressions are detailed in APPENDIX C1) corresponds to a physical solution over different frequency ranges. As an illustration, Fig. 2 shows g 1 obtained with each of the solutions and with the solution of Eq. 10 for a specific complex fluid 1 and a specific cantilever whose first-and second-mode resonant frequencies are equal to f 0 and f 1 ; solution 3 is valid over a larger frequency range, until 10 kHz in the simulated case. Then, using the appropriate solution depending on the considered frequency range, the hydrodynamic behavior can be calculated up to 30 kHz, that is, up to f 1 . Furthermore, results obtained with Eq. 11 are significantly more accurate than those obtained with Eq. 10, even at low frequency. Similar results are obtained for g 2 and presented in APPENDIX C1. 1 The simulated complex fluid corresponds to data extracted from literature, fitted on Buchanan et al.'s results. The G' and G" are fitted from a 4wt% CTAB solution, characterized in [START_REF] Buchanan | Highfrequency microrheology of wormlike micelles[END_REF]. By joining the different solutions Z 1 , Z 2 and Z 3 of this upgraded Taylor approximation, the viscous (g 1 ) and inertial (g 2 ) parts of the hydrodynamic force are now estimated from the free-end cantilever deflection amplitude and phase with a better accuracy and over a larger frequency range than in previous works. By using Eqs. 8 and 9, it is now possible to analytically calculate the complex shear modulus as a function of the free-end cantilever deflection. This method leads to accurate rheograms of complex fluids over a large range of frequency from 0 Hz to the secondmode resonant frequency f 1 , whereas it was previously limited to the first-mode resonant frequency f 0 , and quite inaccurately.

Further improvements in the solution of Eq. 11 are presented in APPENDIX C2 that will be used in the following sections, thus ensuring high accuracy even at frequencies slightly above f 1 .

C. From simulated experimental data to rheograms: discussion on accuracy

Influence of the approximation made during the inverse problem

In this part, experimental data are generated by simulation from input rheological properties of fluids and geometrical and material properties of cantilevers (see part I), and rheograms are produced with Eqs. 8-10 (previous work) or 11 (this work). The produced rheograms differ from the original ones only due to the approximation assumed with the inverse problem. Nevertheless, the error introduced by this approximation can be estimated.

In Figs. 3 and4, the geometry of the silicon microcantilever is 1440x285x20 µm 3 , and the fluid used is water (with G" = η where η =10 -3 Pa.s, and G'=10 -11 Pa instead of 0); in Fig. 5, the microcantilever is the same, but the fluid is a complex fluid which corresponds to data extracted from the literature, fitted on Buchanan et al.'s results (the G' and G" are fitted from a 4wt% CTAB solution, characterized in [START_REF] Buchanan | Highfrequency microrheology of wormlike micelles[END_REF]).

The determination of G" (Figs. 3 and5) is accurate from 10 Hz to the second-mode resonant frequency f 1 for both fluids by using Eq. 11. In Fig. 3, it also clearly appears that using Eq. 10 leads to inaccuracy over the entire frequency range considered.

The determination of G' of the complex fluid is also accurate (Fig. 5), whereas it is more troublesome in the case of water (Fig. 4). The interpretation is that the approximation becomes an issue for very low values of G'. The validity of the inverse model is demonstrated with simulation data for G' and G" higher than 1 Pa. More specifically, it means that even for Newtonian fluids, the calculated value of G' will be non-zero. Fig. 3 -Viscous shear modulus (G") calculated from two of the solutions (Z 2 :•, Z 3 : ) of Eq. 11, from the solution of Eq. 10 (+) and expected (-) for a 1440x285x20 µm 3 silicon microcantilever immersed in water. Nevertheless this systematic error does not matter as it will be shown in part II.C.2 and in part III. Its influence on the accuracy of the produced rheograms is negligible when compared to the error related to model parameter uncertainties and to the experimental signal to noise ratio.

Influence of model parameters

As all of the geometrical and physical parameters are not known precisely, the results obtained with simulated experimental data do not validate the measurements results. For instance, in Fig. 6, the influence of an error on H 0 , corresponding to the static deflection of the cantilever normalized by the applied actuation voltage (the applied actuation force is proportional to the applied actuation voltage), is tested. Undoubtedly, the G' estimation is strongly dependent on the value of H 0 . The estimation of the viscous modulus (G") is also dependent on this parameter in the same proportions.

Fig. 6 -Elastic modulus (G') calculated from the third solution (Z 3 ) expressed in Eq. C11, with respectively -3% (red dotted line), -2% (green line), -1% (blue dotted line), 0% (light blue cross), 1% (magenta dotted line), 2% (orange line), 3% (brown dotted line) of error on H 0 and the expected function for a 1440x285x20 µm 3 silicon microcantilever immersed in a complex fluid (-).

In Table 1, a set of measured values of H 0 are presented for three cantilevers of different geometries. Those values strongly differ from one cantilever to another depending on manufacturing considerations, such as the position of the structure on the silicon wafer. In addition, the low frequency noise of the vibrometer used for the measurement of the microcantilever free-end deflection makes a direct measurement impossible and an optical profilometer or any optical measurement technique would not satisfy the required accuracy (in practice the applied actuation voltage is set to 0.1V, so according to Table 1 the absolute value of the deflection is at the nanoscale and a resolution of less than one percent is required). Therefore, it becomes difficult to measure those values experimentally with an accuracy better than 1%.

f 0 3% 2% 1% -1% -2% 0% Cantilever name Cantilever geometry H 0, min (m.V -1 ) H 0, mean (m.V -1 )
H 0, max (m.V -1 ) LL 2810x100x20 µm 3 1.8 10 -8 2.33 10 -7 3.0 10 -7 LH 1440x285x20 µm 3 1.0 10 -8 2.956 10 -8 5.0 10 -8 A 500x100x20 µm 3 1.5 10 -11 9.377 10 -10 1.3 10 -9

Table 1 -Measured values of H 0 in water for the three cantilevers of different geometries

A fitting and iterative method has been developed in order to determine precisely H 0 (the complete calculation method is not detailed in this paper). Parametrical errors (like the one concerning H 0 ) remain a problem, especially because few reference measurements exist for the probed complex fluids at the investigated frequency range. In this context, it is difficult to calibrate those parameters or to correct the results with a simple selfcalibration. Nevertheless, Fig. 6 shows that whatever the residual parametrical error on H 0 , the values of G' (and G") calculated at the resonant frequency is "exact". Consequently, in the experimental rheograms the more accurate measurements of G' and G" are obtained at the first resonant frequency. At other frequencies, an error on H 0 or one of the other sensitive parameters (not detailed in this article) will strongly affect both the value and the slope of G' and G" vs. frequency.

III. DETERMINATION OF RHEOLOGICAL PROPERTIES OF FLUIDS USING MICROCANTILEVER MEASUREMENTS

A. Experimental set-up and procedure

In order to characterize the viscosity of various fluids, silicon microcantilevers have been designed and fabricated [22]. Three different geometries have been used for the measurements (specimens 'LL', 'LH' and 'A'). They all have the same thickness (SOI wafer) but different lengths and widths, as presented in Table 1. An SEM image of a microcantilever can be seen in Fig. 7C, while the experimental set-up is summarized in Fig. 7A. The dynamic deflection (amplitude and phase) of the cantilevers has been measured optically using a laser vibrometer (MSA500, Polytec). The optical vibrometer is used in this paper because it provides accurate results, characterizing the dynamic behavior of the microstructure without any coupling or signal distortion issues, over a wide range of frequency. Silicon microcantilevers are actuated in the liquid medium (see Fig. 7B) by the electromagnetic Laplace force resulting from the interaction between an external magnetic field generated by a magnet and an electrical current generated by an external electrical voltage in a gold conductive path designed on the surface of the microcantilevers.

Four different liquids have been characterized over a large frequency range (10 3 -10 5 Hz) with a controlled room temperature of 19°C: two Newtonian fluids, deionized water and a 20-cP calibrated silicone oil (provided by Sigma Aldrich). In addition two complex viscoelastic fluids, mixtures of cetyltrimethylammonium bromide (CTAB) and polyacrilamide (PAM), were tested.

Two different solutions of PAM were prepared by adding 50mM and 100mM of PAM, respectively, to 10mM of NaCl. CTAB was concentrated as follow: 100mM of surfactant added to 40mM of NaSal.

The same protocol has been used for the three cantilever geometries. First the cantilever deflection spectrum was measured in air (the eigenfrequency in vacuum needed for the rheogram calculation was deduced from the measurement in air). Then the cantilever resonant spectrum was acquired in deionized water. Finally the measurements in the complex mixture or in the silicone oil were obtained. Each presented experimental spectrum was obtained in less than one minute by averaging 100 spectra. In these conditions, for a given cantilever in a given fluid, repeatability was excellent provided that temperature does not change and that the concentration of the mixtures does not fluctuate due to, e.g., evaporation. This is guaranteed in our set-up by a room temperature control and by performing all the measurements in a short period of time after exposing a new sample to air. The measurements obtained with two identically designed cantilevers are slightly different due to small differences caused by the manufacturing process. This is the reason why a calibration procedure in water has been developed. Different sets of cantilevers were used for each complex fluid. The silicone oil was also characterized using other cantilevers for cleaning considerations. (Oil leaves many traces on the cantilever surface that are difficult to clean in water-based solutions.)

B. Newtonian Fluids

Calibration in water

Even though very few reference measurements exist over the investigated frequency range, many rheologists (see acknowledgments) believe that there is no single physical reason why water should deviate from Newtonian behavior: to our knowledge, it is the best existing reference fluid (it could be suspected that some Newtonian calibration fluids composed of large molecules might partially behave as non-Newtonian fluids at high frequency when considered at very small scales, see Fig. 10). Therefore, measurements in deionized water should lead to a very small value of G' and a constant viscosity (versus frequency) close to 1 cP, depending only on temperature.

A calibration method based on these expected measurements in water is used to calibrate the model parameters identified as crucial for the accuracy of the experimental shear modulus determination. This method depends on the cantilever's geometric parameters (h, b, L), the cantilever mass per unit length (m L ) and H 0 , and is based on minimizing the mismatch between the experimental and the simulated spectra obtained using the method developed in parts I and II, assuming specified variations in these five parameters of a few percent around their nominal values. The nominal values of h, b and L are taken directly from the cantilever mask design values. The nominal parameter m L is calculated using the cantilever mask design values and the actual cantilever density, taking into account that it is made of layers of different materials (Si, SiO 2 , Au, Ti). The nominal value of H 0 comes from the above-mentioned fitting and iterative method and is not detailed in this paper.

Fig. 8 presents the three microcantilevers' deflection measurements when the devices are immersed in water and the results of the best simulation cases; as can be seen, the comparisons are excellent. The results of the calibration parameters corresponding to the three cantilevers (LL, LH and A) immersed in water are presented in Table 2. The variations of parameters are deemed to be physically reasonable.

Fig. 8 -Real (reH) and imaginary (imH) parts of experimental and simulated freeend deflection complex modulus (H) for the three geometries of cantilevers immersed in water: 1) LL; 2) LH; 3) A. The rheogram presented in Fig. 9 illustrates the Newtonian behavior of water obtained with the three cantilever spectrum measurements. Referring to the curve illustrating the influence of H 0 (cf. Fig. 6), both G' and G" are calculated with an error which modifies the slope of those rheograms; the same influence is observed for the other model parameters. This is particularly observable in the case of cantilever A. The fact discussed in the simulation description (section II-C) that we are not able to accurately measure the elastic shear modulus under 1 Pa is confirmed: the behavior of G' appears to be clearly inconsistent. However the behavior of G" (and therefore the viscosity) is very close to the expected behavior in terms of the quantitative values and the continuity between the measurements made using different microcantilevers.

More than one decade separates both portions of the complex modulus; thus, G' can be considered negligible as compared to G". In the opinion of the authors, this is sufficient to confirm the Newtonian behavior of water.

Rheogram of a 20cP silicone oil

The model parameter values obtained after calibration in water for each cantilever were kept unchanged during measurements achieved in a 20 cP silicone oil. The corresponding obtained rheogram is presented in Fig. 10. The behavior of G" is as expected because there is continuity between the measurements obtained with the classical rheometer at low frequencies and the measurements made with the different cantilevers. The obtained viscosity value of 21.3 cP, is in agreement with the expected viscosity at 19°C for this silicone oil (measurement made with a cone/plate rheometer with a Peltier temperature control).

More than one decade of difference is systematically obtained between G' and the larger value G", which is deemed to be large enough for indicating Newtonian behavior. This is particularly true at the resonant frequency, where the model parameter influence is the weakest: the value of G' is more than two decades lower than the value of G". According to the G" high frequency slope, it can be suspected that the fluid become complex at very high frequency, i.e., in the megahertz range.

The results obtained with silicone oil and water using three microcantilever geometries (LL LH, A) demonstrate that the viscous modulus G" determination between 10 -1 -10 4 Pa is possible in the extended range of frequencies 10 3 -10 5 Hz.

C. Complex Fluids

Characterization of CTAB and PAM, prepared as mentioned above, are presented in Figs. 11 and12, respectively. They are supposed to be complex fluids behaving according to linear viscoelasticity with our experimental set-up, a point that has been carefully checked in a previous work [START_REF] Youssry | Onchip characterization of the viscoelasticity of complex fluids using microcantilevers[END_REF]. The limit to the linear regime depends on the amplitude of the applied pressure due to the cantilever motion and therefore on the amplitude of the driving voltage. In our case, the pressure that defines the limit between the linear and non-linear viscoelastic regimes, has been measured around 1 Pa at 10 kHz for the CTAB mixture. This corresponds to approximately 800 mV for the LH cantilevers whereas all of the presented measurements have been performed at 100 mV.

Fig. 11 presents our results for G' and G" of CTAB obtained at 19°C both with cantilevers and with a cone-plate rheometer as well as published [START_REF] Galvan-Miyoshi | Diffusing wave spectroscopy in Maxwellian fluids[END_REF] DWS reference measurements with the same mixture obtained at 22°C. As expected, unlike Newtonian fluids, G' is no longer negligible. Except for the G' measurements obtained with the LL cantilevers (G' LL ), all sets of measurements of G' and G" clearly follow the same trends until 100 kHz, given the temperature difference. The inconsistent results obtained for G' LL are most likely due to some cantilever inhomogeneities or to non-ideal clamping, effects which are not yet taken into account in our model. Nevertheless, except for G' LL , the cantilever-based shear modulus measurements seem to corroborate and extend the DWS reference measurements up to 100 kHz. For the two concentrations of PAM (Fig. 12), very encouraging results are obtained for the viscous part G", both in terms of its order of magnitude, qualitative trend and concentrationdependence. On the contrary, results for the elastic modulus G' are more bothersome. Below 5 kHz, these results are inconsistent, even though one could consider the elastic part to be much smaller than the viscous part. Above 5 kHz, G' exhibits a counter-intuitive concentration-dependent behavior: the elastic part seems to be decreasing with increasing concentration whereas the low- frequency behavior up to 100 Hz (e.g. measured in [START_REF] Darwiche | Microrheomètre sur puce pour l'étude de l'écoulement d'un liquide proche d'une surface solide[END_REF]) suggests the opposite. This surprising result is under investigation but at the moment we consider the obtained elastic value as inconsistent in this precise case due to a parametrical error, as will be discussed in the next section.

D. Discussion

In this paper we have demonstrated proof of concept for a new method for microrheological measurements but have also pointed out some limitations of the proposed method. The positive aspects are primarily related to the determination of G": an accurate measurement of the viscous part G" is obtained showing good continuity between the results obtained with the different cantilevers. This would not be possible without improvements made to the modeling effort as documented herein. The limitations mainly concern the determination of G': at present the method appears to yield valid results only for high values of G' and gives inconsistent results for smaller values (below 100 Pa), thus warranting further physical verification an/or model improvement.

For small values of G' (that is, for Newtonian or near-Newtonian fluids), one part of the problem may stem from the linearization performed in solving the inverse problem, as shown in Fig. 4. The other source of problem, and perhaps the more significant, is related to experimental issues, since Figs. 9 and 10 show larger discrepancies for experimental G' values than for their simulated G' counterparts when comparisons are made to the expected results. For higher values of G' (that is, for non-Newtonian fluids), linearization yields no problem (see Fig. 5) but major issues continue to appear with experimental results (see Fig. 12). It is worth mentioning that classical rheometers also have physical artifacts that lead to non-zero measured values for G' of Newtonian fluids of up to 10 -2 Pa (see Fig. 10). In the case of microcantilevers using the proposed method this phenomenon still exists with increased anomalous behavior in G' being exhibited at higher frequency. The reason why G' is problematic whereas G" is not requires further investigation. It could possibly be due to the very high sensitivity of the calculation to the system parameters and measurement data ( w(L, ), b, h, L, H 0 , m L ) and the uncertainty in these quantities. In this case, a very small error during the calibration procedure, due to phase or amplitude of deflection measurement errors, could lead to a large error in property estimation. This is indicated by Fig. 6 and by similar figures that could be presented for the other parameters. For example, we have preliminary evidence that any experimental phase offset may be critical for G' determination. These problems could also be due to the proposed model itself.

Should that be the case, the model could be generalized to take into account (i) internal losses in the beam, (ii) microscale temperature effects due to Joule heating in the gold path, (iii) effects of finite b/h and L/b ratios that are neglected in the present elementary beam theory model, and (iv) the inertial effects of the gold path at the free-end of our devices. Generalization of the model, however, would pose some technical challenges. For example, the inverse problem may no longer be solvable analytically (e.g., if the inertia of the gold is included) or addition system parameters may need to be incorporated (e.g., if the internal losses are modeled). Another strategy for overcoming the above-mentioned obstacles can be used. In lieu of attempting to determine G' and G" over a broad frequency range, the objective could be focused only on the properties at the eigenfrequency, in which case use can be made of a similar model [START_REF] Belmiloud | Rheological behavior probed by vibrating microcantilevers[END_REF]. Recently we achieved consistent measurement of G' and G" for opaque viscoelastic fluids using microstructures [START_REF] Lemaire | Comparison and experimental validation of two potential resonant viscosity sensors in the kilohertz range[END_REF]. Nevertheless the strategy we employed eliminates completely the advantage of the frequency-dependent analytical method developed in this work. A future analytical correction to the present model will be developed, based on some of the results provided by the model in the present paper. It will permit maintaining the generality of the analytical method developed here, which is valid for a wide range of cantilevers and fluids. Moreover, it will help to characterize accurately not only G" but also G' as a function of frequency in a very general context.

To conclude, the viscous modulus obtained with the present method is consistent with linear trends for Newtonian fluids as well as with reference data from the literature for complex fluids; however, such consistency has not yet been achieved for the elastic modulus results. Further improvements to the model are currently being pursued, but in its present form it shows promise for measuring the viscoelastic tendencies of fluids at high frequencies.

CONCLUSIONS AND FUTURE WORK

Here we have demonstrated the potential for microcantilevers to be used to probe the highfrequency viscous modulus of small volumes of Newtonian and non-Newtonian fluids covering a specific frequency range from 1 to 100 kHz. The analytical method has been improved in order to eliminate the previous error3 of the shear modulus estimation method as a function of the complex free-end deflection. In addition some parameters have been identified as crucial for the accuracy of the experimental shear modulus determination; they have been optimized as much as possible given the present state of the model. Finally, high-frequency rheograms of four fluids, Newtonian and non-Newtonian, have been produced: they partially validate the presented inverse analytical model and prove the feasibility of a microcantilever-based microrheometer. Those two points have been carefully demonstrated, highlighting the actual advantages and limitations of this characterization method.

One of the advantages of the proposed method is the very short computational time associated with the approach (on the order of seconds). Possible further improvements concerning the parameter determination methods and some modifications of the actual modeling have been noted. In particular, future improvements are needed to be able to measure accurately the elastic modulus G'. Then, the integration of arrays of cantilevers, simultaneously performing measurements, is another direction for future work. Finally an electrical characterization of the cantilever deflection strain could possibly lead to an embedded microrheometer that could overcome practical issues associated with making measurements on opaque fluids.

' '' ' 2 ' 1 () 2 ' '' ' r f f a b G G G G complex a b b b a b G G G (A7) 22 12 2 22 22 21 ' '' ' 2 '' 1 () 2 ' '' ' i f f b a G G G bG complex b b a b G G G (A8)
If the fluid is Newtonian (no elasticity): '' G jG with '' 0 G and G' = 0. Therefore

If the fluid has only elasticity (no viscosity):

' GG , '' 0 G . So:

1 2 2 1 22 2 ' 1 ( ) 2 ' 2 r f f G elasticity a b a b G b b (A12) 12 ' () i f b a G elasticity b ( A13
) In this case because there is no viscosity and, thus, no damping effect; consequently 0 i . We can deduce that: General case considering Eq. A11 and Eq. A14

U T U T U U T U U Z V i V V U V (C7) 22 2 2 2 2 22 1 1 2 1 1 3 1 1 1 1 1 3 9 3 9 13 2 2 3 2 U T U T U U T U U Z V i V V U V (C8)

C.2 Beyond Eq. 11

Until now, Eq. 11 has been approximated by a ratio of 3 rd -order polynomials, but it can be written exactly using an infinite sum since all the coefficients k p and q p can be calculated: The three solutions (Z 1 , Z 2 , Z 3 ) are the solutions of (C10) if the sum is truncated at p = 3: (C12) The difference between the exact solutions of Eq. C10 and Eqs. C6-C8 can be included into a term called Δ which is added to each known solution Z n : (C13) is a polynomial equation of since both Z 1|2|3 and α i are known. Due to the properties of Taylor series development, the role of Δ is very small at low frequencies, but increases with frequency. Also, the weight of parameters k i and q i (and consequently α i ) is decreasing while i increases. Therefore, at low frequencies, the terms α i and j can be neglected for the highest values of i and j. For example, if (C13) is truncated at i=6 and j=2 (and taking (C11) into account):

Z Z Z Z Z Z Z Z Z Z Z Z (C14)
For each of the three solutions Z 1 , Z 2 and Z 3 , Eq. C14 has two solutions, denoted by

Fig. 2 -

 2 Fig.2-Viscous dissipative term (g 1 ) respectively calculated from the three solutions (Z 1 :+, Z 2 : , Z 3 : ) of Eq. 11, from the solution of Eq. 10 (o) and expected (-) for a 1440x285x20 µm3 silicon microcantilever immersed into a simulated complex fluid 2 .

Fig. 4 -

 4 Fig. 4 -Elastic shear modulus (G') calculated from two of the solutions (Z 2 :•, Z 3 : ) of Eq. 11, from the solution of Eq. 10 (+) and expected (-) for a 1440x285x20 µm 3 silicon microcantilever immersed in water (G' = 0, simulated here with G'=10 -11 Pa).

Fig. 5 -

 5 Fig. 5 -Elastic (G': blue) and viscous (G": red) moduli calculated from the three solutions Z 1 , Z 2 and Z 3 of Eq. 11, and the expected functions which have been usedto generate the simulation data (-) in the case of a 1440x285x20 µm 3 silicon microcantilever immersed in a complex fluid.

Fig. 7 -

 7 Fig. 7 -Experimental set-up illustration. Method summary (A); view of a microcantilever within liquid recipient (B); an SEM photo of the silicon-etched device (C).

Table 2 -

 2 Model parameter variations around their nominal values obtained to optimally simulate the free-end cantilever deflection which corresponds to the one measured for each geometry (LL, LH or A) immersed in water.

Fig. 9 -

 9 Fig. 9 -High-frequency rheogram of water measured with three microcantilevers (LL, LH and A) and spanning the frequency range 10 3 -10 5 Hz using the inverse model (Eqs. 8-9 and 11).

Fig. 10 -

 10 Fig. 10 -Rheogram of a 20-cP silicone oil measured with a classical cone-plate rheometer and three microcantilevers (LL, LH and A) spanning the frequency range 10 3 -10 5 Hz using the inverse model (Eqs. 8-9 and 11).

Fig. 11 -

 11 Fig. 11 -Rheogram of CTAB over a [10 -2 -10 5 ] frequency range measured with a classical cone-plate rheometer, with three different cantilever geometries using the inverse model (Eqs. 8-9 and 11) and reported from DWS measurements in the literature.

Fig. 12 -

 12 Fig. 12 -High-frequency rheogram of PAM (two concentrations) measured with three cantilever geometries (LL, LH and A) and spanning the frequency range 10 3 -10 5 Hz using the inverse model (Eqs. 8-9 and 11).

  Considering the relation between the viscosity and the shear modulus G, Gj (A5) the expression of the hydrodynamic function becomes:

  The model of Maali et al[START_REF] Maali | Hydrodynamic of oscillating atomic force microscopy cantilevers in viscous fluid[END_REF] a very similar result for the value of a 1 as it was based on a fit of Sader's equations[START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF] Erreur ! Signet non défini. (a 1 = 1.0553, a 2 = 3.7997, b 1 = 3.8018 and b 2 = 2.7364 are the "Maali's parameters").

  of Maali et al[START_REF] Maali | Hydrodynamic of oscillating atomic force microscopy cantilevers in viscous fluid[END_REF] gives b 1 ≈ a 2 ).

Fig. C1 -

 C1 Fig.C1-Inertial term g 2 calculated from the three inverse solutions (Z 1 :+, Z 2 : Z 3 : ) of Eq.11 and the expected function which has been used to generate the simulation data.

b

  are more accurate solutions of Eq. 11, over a larger frequency range, than Z 1|2|3 .

  The solutions (Z 1 , Z 2 , and Z3) of Eq. 11 are expressed as follows:

	2 This is a polynomial equation in K" which can be written as 4 2 2 1 2 2 " 2 " " g K B B K g C D K D 1 1 1 H U nn 2 2 5 2 3 H U n	2 D B	1 g K " K	"	(C3) (B7) (C4)
	V	U	4 " U K		4 2 D B U	2 D B	2	g U	2	C K U	2 "	4 2 D B U	2 1 g	U	0	U T U	(B8) (C5)
	The positive solution of Eq. B8 is
	K	2 "	4 2 D B	2 D B	2	(	2 11 g C ZV	)	4 2 2 B 22 2 UT D 1 1 2 11 2 2 ( B g D 2 39 3 U U T VU	2	C	)	2	4	42 1 2 Bg D	(B9) (C6)
	22 2 2 2 Then using the K" definition we obtain 2 2 22 1 1 2 1 3 9 3 9 13 1 22 2 1 2 1 " 2 4 2 g B B B G g C g D D D DD 2 1 1 1 1 2 2 3 2	2	g 1	2	C	(B10)
						G	'	1 D	g	2		C	2 BG 1 g DG "	"	(B11)
	where
			B	1 22 b	b		f					C	4	f	b	2	D	2 b 2	(B12)

The simulated complex fluid corresponds to data extract from literature, fitted on Buchanan et al.'s results. The G' and G" are fitted from a 4wt% CTAB solution, characterized in[START_REF] Buchanan | Highfrequency microrheology of wormlike micelles[END_REF].

As demonstrated in this paper, the previous error of the shear modulus estimation (developed in[START_REF] Youssry | A straightforward determination of fluid viscosity and density using microcantilevers: from experimental data to analytical expressions[END_REF]) was due to the use of a simplified second order transfer function (Eq.10) to estimate the complex free-end deflection.

r

Acknowledgements

The authors would like to thank Cédric Ayéla for the cantilever mask designs, and Liviu Nicu, Fabrice Mathieu and Daysuke Saya for the fabrication of the silicon cantilevers. They also would like to thank Yacine Amarouchène, Annie Colin and Hamid Kellay for useful discussions about microrheology. This work was partially supported by the CPER Pôle 4N Nanosciences en Aquitaine (GP-206-action 216/1) with the contribution of the Conseil Régional d'Aquitaine, the FEDER and the Ministry of Education and Research, by the French National Agency (MicRheo project n°ANR-08-NANO-004) and by the Conseil Régional d'Aquitaine (n°20091102001).

APPENDIX A: HYDRODYNAMIC FORCE FOR NEWTONIAN AND COMPLEX FLUIDS

The hydrodynamic force per unit length, F hydro , exerted by the fluid on the cantilever is composed of two terms (Eq. A1): one proportional to the microcantilever velocity (viscous coefficient g 1 depending on the radial frequency ) and one proportional to the microcantilever acceleration (inertial coefficient g 2 depending on the radial frequency ).

Furthermore, g 1 and g 2 can also be expressed as functions of both the real part r and the imaginary part i of the hydrodynamic function: (with R e the Reynolds number as defined by Sader [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF]).

The real and imaginary parts of the hydrodynamic function have been expressed by Maali et al. 20 :

These equations can be written as follow:

APPENDIX B: COMPLEX SHEAR MODULUS AS FUNCTION OF THE TERMS g 1 AND g 2

The objective is to invert Eqs. A19-A20 to obtain G' and G" as functions of g 1 and g 2 .

Eqs. A19-A20 can be expressed as follows:

The product of Eq. B1 and Eq. B2 gives

From Eq. B4, it follows that

Then using the definition of K':

Combining Eq. B5 and Eq. B3 yields

APPENDIX C: SOLUTIONS OF THE INVERSE PROBLEM

C.1 Solution of Eq. 11

Three solutions (Z 1 , Z 2 , Z 3 ) of Eq. 11 are obtained using a mathematical solver. Each analytical solution has been tested simulated data and they correspond to the physical case (each solution corresponding to a different frequency range).This can be seen in Fig. C1 which represents the inertial term (g 2 ) calculated using those three solutions.

Using the following notation, 1


Ahmad Darwiche

François Ingremeau

Yacine Amarouchene

Abdelhamid Maali

Isabelle Dufour

Hamid Kellay

Rheology of polymer solutions using colloidal-probe atomic force microscopy

Keywords: number(s): 83.80.Rs, 47.85.mf, 83.50.-v, 47.57.Ng

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Fluids such as polymer solutions, surfactant solutions, emulsions, or colloidal suspensions behave very differently from low molecular weight liquids such as water when a flow field is imposed. These differences are usually highlighted by the manner in which the viscosity of such solutions depends on the deformation rate and by the existence of a nonzero elastic modulus. This behavior is generally due to the presence of mesoscopic constituents that interact with each other and with the flow field. It is the complexity of these interactions that is responsible for the rich behavior of complex fluids. Different methods are used to characterize the behavior of these fluids subjected to a flow deformation. Rheological measurements turn out to be very sensitive to the nature of the fluid studied. The rheology of these fluids is a key issue in several domains of applications, such as food processing, materials processing, cosmetics, and lubrication, where the fluids are subjected to strong deformations. Conventional methods for measuring the rheological properties of a given fluid rely on the use of commercial rheometers using different geometries [START_REF] Barnes | An Introduction to Rheology[END_REF][START_REF] Morrison | Understanding Rheology[END_REF]. The classical rheometer [START_REF] Barnes | An Introduction to Rheology[END_REF][START_REF] Morrison | Understanding Rheology[END_REF] measures the viscosity of a fluid as a function of imposed shear stress or shear rate and provides information about the viscoelasticity (elastic and viscous moduli) of the probed fluid for different time scales. Despite the widespread use of this technique, it is limited by the performance of the device: a rheometer does not allow rheological measurements beyond a certain range of frequencies, for example, and demands large quantities of fluid for operation. To overcome the frequency-range limitation, alternative methods dubbed "microrheology" have been developed [START_REF] Brau | [END_REF]4]. These methods are well adapted to miniaturization, thereby allowing the characterization of a given material using very small volumes. The ability to characterize small volumes of fluid enabled these techniques to be widely used in the field of biocharacterization, for example [5,6]. The major drawback of such a technique is the long image and signal processing times needed to extract the rheological properties. Further developments also include the use of microelectromechanical systems (MEMS) to measure the local viscosity of fluids as well as their viscoelastic response in small volumes [7,8]. These usually rely on the use of miniature cantilevers that oscillate in the solution to be studied. The oscillations of the cantilever are sensitive to the ambient medium properties so quantities such as the viscosity of the solution or its elastic response can be extracted. The reduction in fluid volume accompanied by the reduction of the gaps used to create flow may pose a certain number of problems. When mesoscopic constituents are present (consider large molecular weight polymers in a liquid or colloids or nanoparticles in solution that introduce a characteristic size), if the gap used approaches a few characteristic sizes, the local concentration of the constituents may vary with solicitation. Recent calculations and observations have highlighted the fact that local concentrations of nanoparticles in small gaps may be much smaller than the bulk concentration, for example [9,10]. The role of confinement on the behavior of fluids is, therefore, essential not only for probing their properties at scales comparable to their constituents' dimensions but also for a better understanding of the results of different recent techniques.

In this paper, we use a colloidal-probe atomic force microscope to measure the rheological properties of confined polymer solutions between two surfaces: the surface of a sphere glued on the probe of a microcantilever and the flat surface holding the liquid. The principle of the technique is simple: if the sphere approaches the bottom surface at a constant velocity, the fluid exerts a hydrodynamic force on the sphere and this force gives direct information about the local viscosity of the fluid. This technique was used previously to study the hydrodynamic interaction between a fluid and a solid surface and to determine the boundary conditions for liquid flow near a solid surface [11][12][13][14][15][16].

The properties of the liquid near the solid surface play a major role in setting the forces acting on the sphere. In the 1960s, Brenner and others [17,18] have theoretically determined the hydrodynamic force on a colloidal particle flowing in a Newtonian fluid near a solid wall. This force depends on the separation distance between the sphere and the flat surface and on the viscosity of the fluid.

This hydrodynamic force has been tested explicitly by Honig and Ducker [19], who used colloidal-probe atomic force microscopy to study the hydrodynamic force exerted by the flow on a small sphere approaching a flat surface in a sucrose solution. Measurements of the viscosity of the solution, using the hydrodynamic force expression calculated by Brenner, turn out to be insensitive to the gap width as expected for a Newtonian fluid. It should be noted here that such a result may not be valid at gap widths comparable to the molecular dimensions of the fluid. Previous experiments [in surface force apparatus (SFA), for example, or using the colloidal probe] have observed ordering near solid surfaces for gap widths near five molecular sizes [20].

On the other hand, Bhattacharya and Blawzdziewicz [9] have carried out a theoretical analysis of the hydrodynamic force exerted on a colloidal particle approaching a solid surface in a dilute solution of smaller, nonadsorbing, spherical particles. The authors found that as the particle-flat surface gap width diminished, the viscosity in the gap region varied. At large separations with respect to the small particle diameter, the viscosity asymptotically approached that of the solution. At smaller distances, the viscosity is very close to the solvent viscosity. Specifically, the viscosity in the gap region was less than the solution viscosity at separation distances as large as 20 particle diameters. The local concentration of the small particles diminished as the gap between the sphere and the flat surface decreases. This calculation has been tested by G. K. James and J. Y. Walz [10], who used the colloidal-probe force microscopy technique to measure the hydrodynamic forces exerted on a 30-μm-diameter sphere being moved toward or away from a flat surface in an aqueous solution of nanoparticles (22 nm diameter). They have shown, for large separation distances, that the measured viscosity of this confined solution is close to the viscosity measured using classical rheology. At small separation distances, they found, using the hydrodynamic force of Brenner, that the viscosity was smaller than that measured using classical rheology and close to that of the solvent. The viscosity measured did not depend on the velocity, and the changes in the effective viscosity extended to large (at least 20-nanoparticle diameters) separation distances. These results are consistent with the predictions of Bhattacharya and Blawzdziewicz [9]. Does such a phenomenology persist when the colloids are replaced by deformable objects such as polymers? Deformation of polymers in high shear rate regions or depletion of such molecules from such regions (as may occur near boundaries) has been observed before [21] and may affect the hydrodynamic resistance experienced by a sphere approaching a solid surface. In this paper, we examine this issue in detail.

In the experiments presented here, we have used the colloidal-probe atomic force microscopy technique to measure the force exerted on a sphere, as it approached a flat solid surface at different velocities, in a large molecular weight polymer solution. The separation distance between the sphere and the flat surface was varied from about 10 μm down to a fraction of a micrometer. These separations span a range of distances from about 50 molecular diameters down to less than 1 molecular diameter. Confinement effects were, therefore, probed over a wide range of distances with respect to the polymer molecular dimensions. The diameter of the (probe) sphere was much larger than the molecular dimensions and was in the range of 115 to 125 μm. Through the hydrodynamic force exerted on the sphere, we deduce the viscosity of the fluid in the gap as a function of the velocity, the distance D separating the sphere from the flat surface, and the shear rate . γ within this gap. We systematically compared the measurements made by colloidal-probe atomic force microscopy to measurements using classical rheology. Our main results indicate that the measured viscosity is systematically smaller than the viscosity measured using a classical rheometer when the separation distance decreases. The effect is found even at distances as large as 20 times the radius of gyration of the polymer used. Depletion of polymer molecules in the gap region, most probably due to the high shear rates in the gap [21], is a probable reason for such a decrease of the viscosity of the solution. Complementary measurements using the Brownian motion of small particles in the gap region confirm these observations.

II. MATERIALS AND METHODS

On a silicon nitride rectangular cantilever (Veeco ORC8) of 200 μm in length and 40 μm in width, we glued a borosilicate sphere of diameter ranging from 115 to 125 μm (GL0186B/106-MO-Sci Corporation) using an epoxy glue. The advantage of using such large-sized spheres is to increase the hydrodynamic interaction and to minimize the contribution of the hydrodynamic drag force due to the cantilever itself. A small droplet of the studied liquid was deposited on a hydrophilic mica surface. The cantilever and the sphere were immersed in the droplet. Before each experiment, the mica substrate was cleaved, ensuring a clean and smooth surface. The sample (mica surface) was fixed on a piezoelectric stage (Nano-T225, City Labs Inc., Madison, WI, USA), which has a displacement of 5 μm/V.

The displacement of the piezoelectric system is controlled by a function generator using triangular waveforms with frequencies in the range of 0.01-1 Hz and a peak-to-peak amplitude of 8 V to move the fluid sample toward the probe and away from it. During the displacement of the sample, the deflection of the cantilever due to the hydrodynamic force exerted by the displacement of the fluid on the solid surface toward the probe is detected using a photodetector. An acquisition system is used to acquire the electrical signal from the photodetector.

We studied different fluids in this configuration: water-NaCl at 10 mM, an aqueous solution of polyethylene glycol [PEG; from Fluka Chemika Cat. No. 81300 Lot No. 408095/1; M w = 20 000 g/mol (16 000-24 000 g/mol), radius of gyration R g = 4 nm, and polymer overlap concentration c * = 124 000 ppm; the Zimm relaxation time is τ zimm = 1.6.10 -8 s, the concentration used is 18 300 ppm or c/c * = 0.15] and an aqueous solution of a large molecular weight polymer polyacrylamide (PAA; Polyscience Cat. No. 18522, Lot No. 518644; M w = 18 × 10 6 g/mol, R g = 300 nm) at different concentrations. The polymer overlap concentration c * for PAA is about 200 ppm [22]. The estimated Zimm relaxation time of this polymer is 6.8 ms. We examined concentrations both below and above c* for the PAA solution. Rheological measurements were performed in a controlled stress rheometer (AR1000, TA Instruments) in cone-plate geometry (radius 30 mm, cone angle 1 • ). The truncated gap (the smallest distance between the cone and the plate) is 15 μm. This is larger than the range of gaps explored using the AFM.

A. Hydrodynamic force

Our experiments make explicit use of the hydrodynamic force exerted on a sphere as it approaches a flat solid surface. This problem has been solved in the 1960s by Brenner, and we here recall the main result. For a solid sphere approaching a flat solid surface with a velocity V , the hydrodynamic force acting on the sphere is given by Brenner's expression [17]:

F = -6πηRV λ, (1)
where

λ = 4 3 sinh(α) ∞ n=1 n(n + 1) (2n -1)(2n + 3) × 2 sinh[(2n + 1)α] + (2n + 1) sinh(2α) 4 sinh 2 [(n + 1 2 )α] -(2n + 1) 2 sinh 2 (α) -1 (2)
and

α = cosh -1 1 + D R . (3)
In Eqs. ( 1)-( 3), R is the sphere radius, D is the separation distance between the surface of the sphere and the mica surface, and η is the liquid viscosity.

In the case where D is very small compared to the sphere radius (D R), λ = R D and the force expression reduces to Taylor's form:

F = - 6πηR 2 D V . ( 4
)
On the other hand, when the distance D is very large compared to the sphere radius (D R), the force expression simply reduces to the Stokes's form:

F = -6πηRV . ( 5
)
These expressions assume no slip boundary conditions on both the sphere surface and the flat surface.

B. Cantilever calibration

The measurements carried out here are based on recording the deflection of the cantilever with the sphere attached as the bottom flat surface is approached toward the sphere or retracted from it. The hydrodynamic force exerted by the fluid on the sphere causes the cantilever deflection. In order to extract the fluid characteristics, the cantilever properties and contact position with the flat surface need to be determined. The calibration of the cantilever deflection was performed in situ by making use of the cantilever deflection measured by the photodiodes as the mica substrate approached the sphere in a solution of water [23]. The addition of salt to the water reduces electrostatic forces between the mica surface and the sphere.

To convert the photodiode voltage to deflection in nanometers and get the contact position (D = 0) we have followed the procedure described in Refs. [19,24]. For each realization, we determine the slope of the measured photodiode voltage versus the piezo displacement in the constant compliance zone when the sphere and the substrate are in contact. This gives the cantilever deflection z c . Extrapolation of the constant compliance zone data to zero intercepts the displacement axis at the contact position (D = 0), allowing us to determine the latter with a precision of roughly 5 nm. This procedure is explained in Fig. 1(a). In order to obtain the actual separation distance, the cantilever deflection was added to the piezo displacement. The actual velocity (during the approach or withdrawal phases) is obtained from the full sphere-substrate distance by calculating its time derivative:

V = dD dt . ( 6
)
A typical example of the measurement of the cantilever deflection resulting from the hydrodynamic force acting on the sphere is presented in Fig. 1(b). The deflection of the cantilever is positive when the mica surface approaches the sphere. On the other hand, when the mica surface is withdrawn from the sphere surface, the deflection is negative. The approach phase (A) and the withdrawal phase are similar in shape down to about 400-nm separation as shown in the inset to Fig. 1(b). This inset displays the sum of the deflections [upon approach (A) and upon withdrawal (W)], which is zero down to 400 nm, showing that the withdrawal and approach phases are symmetric for distances larger than 400 nm. The difference between the deflections is also shown and deviations due to an asymmetry between the two phases also appears at distances smaller than 400 nm. This asymmetry between the approach and withdrawal is due to differences in the approach and withdrawal velocities at these distances.

The hydrodynamic force is related to the measured cantilever deflection by F h = kz c . From Eq. ( 4), which is valid for D R, we have

z c = 6πηR 2 kD dD dt . ( 7
)
To determine the stiffness k of the cantilever with a sphere attached, we plot

( dD dt )
z c versus D. This variation is linear with a slope given by k 6πηR 2 , as shown in Fig. 1(c). The stiffness is obtained from the value of the slope, for a known viscosity and known radius, giving k = 0.20 ± 0.01 N/m.

III. VISCOSITY MEASUREMENTS

For the salt water solution, the only force that acts on the sphere during the approach or withdrawal of the solid surface is the viscous hydrodynamic force. For a polymer solution there could be additional forces (depletion forces, elastic forces, etc.) that may result from the confinement and the shearing of the solution. We note that while the viscous force changes direction if the velocity changes direction, the additional forces do not. To extract the effective viscosity of the studied liquids from the measured force, we proceed as follows. during the displacement of the mica surface (surface carrying the liquid). For the approach, the deflection is positive and increases as D decreases. For the withdrawal phase, the deflection is negative. The inset shows a close up at small distances D. Note that the approach and withdrawal signals are not symmetric for distances below 400 nm, as shown by the nonzero value of the sum of deflections and by the deviation of the half difference between the deflections measured upon approach and upon withdrawal. This asymmetry at small distances is due to differences in velocity V between the approach and the withdrawal phases at similar gap heights. (c) Variation of the velocity V divided by the cantilever displacement versus the distance D. The confined liquid between the borosilicate sphere and the flat mica surface is salt water. The solid line is the fitting curve using the theoretical expression of Eq. (7). The fit goes through the origin, indicating no slip, and gives the value of the stiffness k.

We consider that the force acting on the sphere has two components, a viscous force F ν and an additional force F e . When the mica surface moves toward the sphere glued on the cantilever, the projection of the force gives

F ν + F e = kZ a . ( 8
)
On the other hand, when the mica surface moves away from the sphere, the projection of the force gives

-F ν + F e = kZ r . ( 9
)
Here, Z a and Z r are the cantilever deflections during the approach and withdrawal phases. While the viscous force changes sign, the additional forces do not. The sum and subtraction of Eqs. ( 12) and (13) gives

2F e = k(Z a + Z r ), (10)
2F ν = k(Z a -Z r ). ( 11
)
From the viscous force, and using the hydrodynamic force discussed above, we can extract the effective viscosity of the solution:

η = k(Z a -Z r ) 12πRV λ . ( 12
)

IV. RESULTS AND DISCUSSION

As explained above, the measurement of the deflection of the cantilever as a function of the distance D allows us to obtain the viscous hydrodynamic force from an analysis of the approach and withdrawal phases. We have used two polymer solutions: a PEG solution of low molecular weight and exhibiting little elasticity and a PAA solution of high molecular weight with an important elastic response. In Figs. 2(a) and 2(b), we show the measured deflection versus the distance between the sphere and the flat surface for the PEG and the PAA solutions, respectively. In the case of the PEG solution [Fig. 2(a)], we observe that the deflection of the cantilever is symmetric between the approach and withdrawal phases. Any asymmetry arises at distances smaller than 400 nm as seen in Fig. 1(b). As expected, the only forces acting on the sphere are of viscous origin and no sign of an elastic response or additional forces is observed. However, these deflections are not symmetric, even at large distances, in the case of PAA solutions as shown in Fig. 2(b). The difference between both cases is related mostly to the presence of a strong elastic component for PAA. We cannot rule out the presence of additional forces, however. In order to unravel the information conveyed by the variation of the cantilever deflection versus distance D, we represent, as shown in Fig. 3(a), the sum and the difference of the deflections for PEG solutions using two different velocities. The sum of deflections is close to zero for the PEG solution as expected. In this case, the force exerted on the sphere surface is solely the viscous force, which is low for large separations and increases markedly for low separations and for higher velocities as expected. In Fig. 3(b), we plot the sum and the difference of the deflections measured for the PAA solution. We find that the deflections' sum is not zero but positive and increases as the separation distance decreases. This confirms the fact that the polyacrylamide solution is a viscoelastic liquid with a strong elastic component 062601-4 (Color online) Measurement of the cantilever deflection during the approach and withdrawal of the mica surface. (a) In the case of Newtonian PEG solutions, symmetrical deflections between the approach and withdrawal of the cantilever were observed. (b) In the case of non-Newtonian polyacrylamide solutions, asymmetrical cantilever deflections have been observed. Two different concentrations (300 and 100 ppm, corresponding to 1 and 0.5 for c/c*) are used for a velocity of 79 μm/s. This velocity refers to that of the piezoelectric stage used to move the mica surface and not to the relative velocity between the mica surface and the sphere.

giving rise to a positive normal force. Possible depletion forces, which are on the other hand attractive, cannot be ruled out even though their magnitude was estimated to be much smaller than the measured forces at the examined separations. The forces exerted on the surface of the sphere in this case are both elastic and viscous forces, which increase as the separation distance decreases and as the polymer concentration increases.

Equation (12) can be used to calculate the viscosity of the polymer solutions versus the gap height D. Specifically, for PEG, which has a constant viscosity over a large range of shear rates as measured with a Rheometer and for which the elastic part is basically absent, we find that the viscosity FIG. 3. Sum and difference of deflections: (a) Newtonian liquid is PEG at 18 300 ppm at different velocities of the mica surface and (b) non-Newtonian liquid PAA solution at 300 and 100 ppm for a velocity of 79 μm/s. This velocity refers to that of the piezoelectric stage and not to the relative velocity between the surface and the sphere. curves obtained during the approach and the withdrawal phases are superimposed. The value of the viscosity of the PEG solution shown in Fig. 4(a) turns out to be constant and independent of the distance D. These results are consistent with the rheology measurements for which the viscosity is independent of shear rate giving η = 0.007 Pa s, as shown in Fig. 4(b). Note that the molecular size of the PEG molecules is well below the heights used.

As mentioned above, the PAA solution has elastic (and perhaps additional forces) and viscous contributions to the measured force. We will focus only on the viscous part obtained using Eq. (12). Since these solutions are strongly shear-thinning, we first consider the variation of the viscosity versus shear rate and estimate the shear rate as This estimate considers a volume πD 3 around the center of the gap and assumes Poiseuille flow. In Fig. 5, we plot the viscosity of the polyacrylamide solution at a concentration of 200 ppm (parts per million by weight) or a value of c/c* of 1 extracted from the force measurements versus the estimated shear rate. The black curve represents the measurements obtained with a conventional method (a rheometer in the cone plate geometry) and the three other curves are obtained with the colloidal-probe AFM at different velocities of the piezoelectric stage.

From the rheology measurements, the viscosity is roughly constant at low shear rates and decreases with shear for rates higher than about 50 s -1 , which corresponds to the inverse of the relaxation time of the solution. The viscosity measured with the AFM probe does not follow this variation and the measurements seem to be below the rheology curve and reach values close to the solvent viscosity for high shear rates. The two viscosities seem to be close only at low shear rates or large distances D. In order to explore this discrepancy with respect to conventional measurements even further, we compare data using different velocities of the piezoelectric stage. In principle, and if the viscosity depends only on the shear rate, the results obtained with the three different velocities used should superimpose. It is clear from Fig. 5 that the three viscosity curves deduced from the measurement of the cantilever deflection are offset relative to each other. The different realizations show that the viscosity of a polyacrylamide solution decreases when the shear rate increases but they do not collapse versus shear rate as they should. All three curves are also offset from the rheology measurements. This behavior shows that the decrease of the viscosity observed here is not solely due to the shear thinning character of the solution. If this were the case, the viscosity curves should collapse versus shear rate for the three different velocities used and should agree with the rheology measurements.

On the other hand, if the viscosity is examined versus the distance D between sphere and surface, a surprising result emerges. Figure 6(a) shows that the curves of the viscosity at a concentration of 200 ppm for different piezoelectric stage velocities are superimposed when plotted versus the distance D. This shows that the viscosity is independent of the displacement velocity of the mica surface. This is reminiscent of the results mentioned above on nanoparticle suspensions, where the effective viscosity of the solution decreases with decreasing gap height due to a lowering of the suspension concentration in the gap region [9,10]. Note that for large gaps the viscosity is close to that of the bulk solution at low shear rates but the decrease persists at distances as high as 6 μm (about 10 molecular diameters). Our results also explore heights comparable to and even smaller than the molecular dimensions of the polymers. For such small dimensions where the gap can hardly accommodate more than one molecule, it is probably not surprising to find small viscosities.

Other concentrations show a similar behavior. The decrease in viscosity when the distance D between the two surfaces decreases is obtained for a 100 ppm solution or a value of c/c* of 0.5 as shown in Fig. 6(b) for different displacement velocities. A summary of these results is shown in Fig. 7. The measured viscosity depends on the concentration of polymer and on the distance between the surface and the sphere but is independent of the velocity. As the concentration increases, the viscosity increases as expected. For large distances, the velocities (79 μm/s, 39.5 μm/s, and 7.9 μm/s) versus the distance D. Note that the different curves collapse onto a single curve. Note also that the large distance measurements are close to the low shear rate viscosity of the solution, which is near 4 mPa s for the 200 ppm solution and near 2 mPa s for the 100 ppm solution. The velocity refers to that of the piezoelectric stage and not to the relative velocity between the surface and the sphere. extracted viscosity follows the viscosity measured with the rheometer. The inset to this figure shows the low shear rate viscosity from rheology measurements and that extracted at large distances. The two viscosities follow the same trend with values close to each other for the different concentrations examined. As the distance decreases, however, the viscosity shows a large decrease and reaches values close to the solvent viscosity for small distances.

The distance D is, therefore, an important parameter that controls the viscosity variation. The shear rate as estimated from the simple expression above does not seem to be a relevant parameter to collapse the viscosity data. Shear thinning alone is, therefore, not sufficient to explain the variation of the viscosity measured here. Furthermore, the fact that the viscosity decreases to values as small as that of the solvent cannot be explained by shear thinning alone since the rheology measurements indicate that at high shear rates (above 1000 s -1 ) the viscosity remains higher than that of the solvent.

One possible explanation for such a decrease of the viscosity for small distances D (below about 5 μm) is depletion of polymers from the gap. This has been observed and predicted for colloidal solutions where the lowering of the particle concentration in the gap region is responsible for a lower viscosity. This effect has been noted for gap heights as large as 20 particle diameters. For the case of polymer solutions, it has been noted before that large shear rates induce migration of polymers away from the solid boundaries where the shear is high [21]. Such a depletion layer has been observed experimentally in different experiments [21] and its thickness L is predicted to scale as L ≈ R g Wi 2/3 . Here, Wi is the so called Weissenberg number given by Wi = . γ τ r , where τ r is the relaxation time of the polymer solution. Typical depletion layer thickness is, therefore, comparable or higher than the radius of gyration R g of the molecules for Wi greater than 1. The radius of gyration of the PAA molecules used here is 0.3 μm. We may, therefore, expect that at small distances and, therefore, relatively high shear rates, both near the surface of the sphere as well as on the flat surface, a depletion layer almost devoid of polymers exists that renders the hydrodynamic force acting on the sphere smaller than expected. This effect will give rise to a viscosity that is smaller than that at the macroscopic scale measured with the Rheometer. Note that for the PEG solution, the radius of gyration is estimated at 4 nm [25], and the relaxation time of the solution is estimated to be smaller 062601-7 than 0.1 ms, so depletion effects may arise only at very small distances of order a few nanometers, so it is not surprising that the measured viscosity in this case remains close to that of the solution for gap heights larger than 80 nm.

Further checks of such a hypothesis follow through measurements of the Brownian motion of small colloidal particles in the gap between the sphere and the surface. These additional experiments explore the Brownian motion of small particles in the gap between the sphere and the surface and, therefore, probe the local viscosity of the solution within the gap. If depletion does occur, we may expect enhanced diffusion with respect to the bulk polymer case and, therefore, a lower effective viscosity.

Fluorescent nanoparticles of diameter 200 nm were used in the solution. The concentration of particles is kept extremely small so no particular effects due to the presence of particles in solution are expected. The same large sphere diameter was used for this experiment, but instead of mica as the bottom surface, a microscope glass slide was used. The concentration of polymer (PAA) was fixed to 500 ppm (c/c *= 2.5). The sphere was approached toward the surface at a velocity of 180 μm/s down to a distance D of 4 μm from the surface. The movement was then stopped and the particles tracked using an epi-fluorescence set up with a microscope objective of 63× magnification. A sensitive camera working at 120 frames per second was used to image the particles and their Brownian motion in the gap between the sphere and glass slide.

The mean-square displacement of the nanoparticles was calculated from a time trace of the particles' trajectories as 1/2 [x(t + τ )x(t)] 2 + [y(t + τ )y(t)] 2 . The brackets refer to an average over time t and over different particles. Different positions from the center of the gap were examined and different times after cessation of the flow were analyzed.

Figure 8 displays the mean-square displacement of these nanoparticles in the bulk polymer solution, below the sphere in the polymer solution, and in the reference salt water solution. Note that the mean-square displacement varies linearly with time τ as expected for Brownian motion in all three cases. The slope of this variation gives the diffusion constant D SE , which is given by k B T 6πηR p , where k B is the Boltzmann constant, T is the temperature, η is the fluid viscosity, and R p is the nanoparticle radius. The slope extracted from the water case gives a value of the viscosity of 1.3 ± 0.4 mPa s slightly higher but close to that expected for water. Similarly, the slope for the polymer solution gives a value of 7.5 ± 3 mPa s, which is slightly lower than the viscosity of the solution measured at low shear rates (we expect 10 mPa s). The value of the slope for the polymer solution confined between the sphere and the surface right after cessation of the flow is, however, higher than that of the solution itself. This value corresponds to a viscosity of 3 ± 1 mPa s and is, therefore, smaller than the value of the viscosity of the polymer solution measured in the rheology experiments or in the Brownian diffusion experiments. This result is consistent with the fact that the AFM probe measurements suggest a smaller value for the viscosity of the solution at a similar gap height. For a gap height of 4 μm, the probe experiments give a value of 5 mPa s, while that from the Brownian motion gives a value of 3 mPa s. Both values are smaller than those measured for much larger gaps. FIG. 8. (Color online) Mean-square displacement (in one dimension) of 200-nm-diameter spheres in bulk water, bulk polymer solution, and for a confined polymer solution between a sphere and a flat surface. The sphere's displacement was at a velocity of 180 μm/s. The measurements started after the sphere's displacement was stopped at a distance of 4 μm from the flat surface. The measured diffusion constant is given. The inset shows the meansquare displacement of one particle, in the gap between the sphere and the surface for two different time periods. Note that for later times after cessation of the sphere's movement toward the flat surface, the diffusion constant decreases, indicating a higher viscosity.

A further examination of these results was carried out by studying the mean-square displacement at different times after cessation of the shear, i.e., after the sphere was stopped. These results are shown in the inset of Fig. 8. The diffusion constant was found to decrease slowly with time, indicating that the observed depletion at early times slowly disappears after the cessation of the flow as the polymer solution relaxes to its equilibrium concentration in the gap region.

The inset of the upper graph of Fig. 9 shows a typical velocity profile versus the distance to the center r (the points are the velocities measured using particle tracking of 200 nm particles at 3 μm from the flat plate for D = 4 μm). The line is the expected velocity profile assuming Poiseuille flow in the gap in the same conditions as the measurements. The reasonable agreement between the measurements and the expected variation allows us to estimate, in this configuration, the shear rate at the wall. Another estimate of the shear rate in the gap can also be obtained by dividing the velocity of the tracked particles by the distance to the wall; this estimate is necessarily lower than the calculated shear rate at the wall using the assumption of a Poiseuille velocity profile in the gap as indirectly suggested by the inset of the upper graph of Fig. 9. These shear values are then multiplied by the relaxation time of the solution (0.1 s) [26] to obtain the Weissenberg number, which is plotted in the upper graph of Fig. 9. The difference between the data points obtained by dividing the velocity by the distance to the wall to obtain the shear rate (solid squares) and the solid line that uses Poiseuille flow in the gap clearly indicates that dividing the velocity of the particles by the distance to the wall underestimates the shear rate but may also indicate deviations from Poiseuille flow, which are difficult to observe due to measurement errors. In The lower graph displays the diffusion coefficient for the small particles (obtained by tracking several particles) in the gap region for different distances from the center of the gap. Note that near the center, these diffusion constants are high, indicating smaller viscosities for the polymer solution. As the distance to the center increases, the diffusion constant decreases, indicating higher viscosities. The inset in the upper graph shows the variation of the velocity versus distance r from the center of the gap along with the expected profile assuming Poiseuille flow. These measurements and the calculated profile are for D = 4 μm, V = 180 μm/s, and at 3 μm from the flat surface. The main graph shows the estimated Weissenberg number versus distance from center using the expected profile as well as the measured velocities. Note that high Weissenberg numbers are obtained for distances where the viscosity is low, indicating the importance of polymer migration and depletion. addition to this, we have measured the diffusion constant of the nanoparticles at different distances r from the center of the gap. The result is shown in the bottom panel of Fig. 9. Note that farther away from the center, the diffusion constant is small, albeit still slightly higher than the bulk value. When the distance to the center decreases and, therefore, the gap height decreases, the diffusion constant increases, indicating a lower viscosity and, therefore, a smaller mean concentration of polymers. The shear rates are high where the diffusion constants are significantly higher than those measured for large gaps. These shear rates give Weissenberg numbers of the order of 10 in a large area extending up to 20 μm from the center of the gap (see Fig. 9, upper panel). These results suggest depletion layers of thickness greater than the molecular diameter. These measurements point to the importance of shear in the observed deviations from expected behavior based on rheology measurements of the viscosity and to the importance of shear-induced migration of polymers.

Note that it is not confinement alone that gives the trend in diffusion constant [27]: confinement generally reduces the diffusion constant, while here it increases as the gap height decreases. These results, therefore, comfort our AFM measurements in the sense that the approach of the sphere toward the flat surface may induce depletion of polymers within the gap, causing the viscous force to be smaller than expected and giving Brownian probes a higher diffusion coefficient.

V. CONCLUSIONS

We have used colloidal-probe atomic force microscopy to study the viscosity of a class of complex fluids: large molecular weight polymer solutions. The method relies on the use of the deflection of a microcantilever to deduce the viscosity of a liquid confined between a sphere glued on this cantilever and a flat surface holding the liquid sample. The viscosity obtained for a Newtonian fluid is constant and independent of shear rate and separation distance between the sphere and the surface as expected. The large molecular weight polymer solutions, however, exhibit viscoelastic behavior, which is strongly dependent on the gap height. The viscosity measured with this scheme decreases with the decrease of the distance between sphere and surface. This effect is independent of the relative velocity between the flat surface and the probe sphere. These measurements are confirmed by using local viscosity probes in the gap region and are reminiscent of previous results on colloidal suspensions. Depletion of polymers in the gap region due to shear-induced migration is a probable cause for the variation of the viscosity observed here.

FIG. 1 .

 1 FIG. 1. (a) Cantilever calibration. Note the linear region giving the calibration of the photodetector voltage. Once this calibration is carried out and the voltage converted to a distance z c , extrapolation of this linear zone to zero (indicated by the solid line) gives the contact position D = 0. (b) Cantilever deflection due to the hydrodynamic force versus the distance D between the surfaces (mica and sphere)during the displacement of the mica surface (surface carrying the liquid). For the approach, the deflection is positive and increases as D decreases. For the withdrawal phase, the deflection is negative. The inset shows a close up at small distances D. Note that the approach and withdrawal signals are not symmetric for distances below 400 nm, as shown by the nonzero value of the sum of deflections and by the deviation of the half difference between the deflections measured upon approach and upon withdrawal. This asymmetry at small distances is due to differences in velocity V between the approach and the withdrawal phases at similar gap heights. (c) Variation of the velocity V divided by the cantilever displacement versus the distance D. The confined liquid between the borosilicate sphere and the flat mica surface is salt water. The solid line is the fitting curve using the theoretical expression of Eq.(7). The fit goes through the origin, indicating no slip, and gives the value of the stiffness k.

  FIG. 2. (Color online)Measurement of the cantilever deflection during the approach and withdrawal of the mica surface. (a) In the case of Newtonian PEG solutions, symmetrical deflections between the approach and withdrawal of the cantilever were observed. (b) In the case of non-Newtonian polyacrylamide solutions, asymmetrical cantilever deflections have been observed. Two different concentrations (300 and 100 ppm, corresponding to 1 and 0.5 for c/c*) are used for a velocity of 79 μm/s. This velocity refers to that of the piezoelectric stage used to move the mica surface and not to the relative velocity between the mica surface and the sphere.

FIG. 4 .

 4 FIG. 4. (a) Viscosity of the PEG solution versus gap height D. (b) Viscosity of the PEG solution using rheology measurements in a cone plate geometry. Note that the two determinations give a similar result: a constant viscosity of about 7 mPa s.

FIG. 5 .

 5 FIG.5. Viscosity of the PAA solution (200 ppm, c/c* = 1) as a function of shear rate using different velocities. The results from AFM measurements are compared to rheological measurements. Note the differences between the measurements for different velocities and with the rheological measurements. The velocity refers to that of the piezoelectric stage and not to the relative velocity between the surface and the sphere.

FIG. 6 .

 6 FIG.6. (a) Viscosity of a PAA (200 ppm) solution at different velocities (79 μm/s, 39.5 μm/s, and 7.9 μm/s) versus the distance D. (b) Viscosity of a PAA (100 ppm, c/c* = 0.5) solution at different velocities (79 μm/s, 39.5 μm/s, and 7.9 μm/s) versus the distance D. Note that the different curves collapse onto a single curve. Note also that the large distance measurements are close to the low shear rate viscosity of the solution, which is near 4 mPa s for the 200 ppm solution and near 2 mPa s for the 100 ppm solution. The velocity refers to that of the piezoelectric stage and not to the relative velocity between the surface and the sphere.

FIG. 7 .

 7 FIG. 7. (Color online)Viscosity of the PAA solution at different concentrations (500, 300, 200, and 100 ppm, corresponding to c/c* values of 2.5, 1.5, 1, and 0.5) for a velocity of 39.5 μm/s. This velocity refers to that of the piezoelectric stage and not to the relative velocity between the surface and the sphere. The inset shows the zero shear viscosity from rheology measurements and the AFM measurements at large separation D.

  FIG.9. The lower graph displays the diffusion coefficient for the small particles (obtained by tracking several particles) in the gap region for different distances from the center of the gap. Note that near the center, these diffusion constants are high, indicating smaller viscosities for the polymer solution. As the distance to the center increases, the diffusion constant decreases, indicating higher viscosities. The inset in the upper graph shows the variation of the velocity versus distance r from the center of the gap along with the expected profile assuming Poiseuille flow. These measurements and the calculated profile are for D = 4 μm, V = 180 μm/s, and at 3 μm from the flat surface. The main graph shows the estimated Weissenberg number versus distance from center using the expected profile as well as the measured velocities. Note that high Weissenberg numbers are obtained for distances where the viscosity is low, indicating the importance of polymer migration and depletion.

ACKNOWLEDGMENTS

This work was partially supported by the CPER Pôle 4N Nanosciences en Aquitaine (GP-206-action 216/1) with the contribution of the Conseil Régional d'Aquitaine, the FEDER and the Ministry of Education and Research, by the French National Agency (MicRheo Project No. ANR-08-NANO-004) and by the Conseil Régional d'Aquitaine (Project No. 20091102001).