Free Surface Effects on Normal Stress Measurements in Cone and Plate Flow. David C. Venerus 1 INTRODUCTION

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1 Free Surface Effects on Normal Stress Measurements in Cone and Plate Flow David C. Venerus Department of Chemical Engineering and Center of Excellence in Polymer Science and Engineering, Illinois Institute of Technology, Chicago, Illinois 60616, USA Fax: x Received: , Final version: Abstract: The effects of free surface shape on normal stress difference measurements in cone and plate flow are investigated. The analysis shows that the stress field is significantly altered by deviations of the free surface from an ideal (spherical) shape. For the cone and partitioned plate technique, it is shown how modest deviation from a spherical free surface shape can lead to errors of roughly 10% in the measured normal stress differences. Zusammenfassung: Der Einfluß der freien Oberfläche auf die Messung der ersten Normalspannungsdifferenz in der Kegel-Platte Geometrie wird diskutiert. Die Analyse zeigt, dass das Spannungsfeld erheblich durch Abweichungen von der Oberflächenidealform beeinflusst werden kann. Im Fall einer geteilten Kegel-Platte Anordnung können schon moderate Abweichung von der spherischen freien Oberfläche zu einem Fehler von 10% des Normalspannungsdifferenzwertes führen. Résumé: L' influence de la forme de la surface libre sur les mesures de différences de contraintes normales dans le cadre d' un écoulement cône-plan est étudiée. Cette analyse révèle que le champ des contraintes est fortement altéré lorsque la surface libre s'éloigne de sa forme idéale (sphérique). Concernant la technique de cône-plan partionné, il est montré comment de légères deviations par rapport à une surface libre sphérique peuvent conduire à des erreurs d' environ 10% sur la mesure des différences de contraintes principales. Key words: normal stresses, free surface, Cone and Plate 1 INTRODUCTION Cone-and-plate flow is widely used to study the rheological behavior of complex fluids. In most cone-and-plate rheometers, one of the fixtures, say the cone, is rotated and the torque and axial force are measured on the stationary plate, or vice-versa. The primary advantage of cone-andplate flow is that the shear rate g is approximately uniform within the fluid sample. Hence, unlike torsional flow between parallel disks, or pressure-driven flow in a capillary, the shear stress at a given shear rate s(g ) can be obtained from a single measurement, even for fluids displaying highly non-linear rheological behavior. In addition, the first normal stress difference N 1 (g ) can be obtained from a single axial force measurement. If the radial distribution of stress on the plate is measured, both N 1 (g ) and the second normal stress difference N 2 (g ) can be obtained. There have been numerous analyses of cone and plate flow and the assumptions used that allow for the measurements described above to be made [1-4]. In this note, we examine the effects of the free surface between the test fluid and the surrounding ambient gas on measurements of N 1 in a cone-and-plate rheometer. In particular, we focus on the cone and partitioned plate technique used by Meissner et al. [5] and more recently by Schweizer [6]. Appl. Rheol. 17 (2007)

2 Figure 1: Schematic of cone and partitioned plate geometry. Right side shows spherical free surface and left side shows bulged free surface. 2 ANALYSIS OF CONE AND PLATE FLOW We consider the steady, isothermal flow of an isotropic, viscoelastic liquid with constant density r between a cone rotating with angular velocity W and stationary plate as shown in Figure 1. The cone angle is a, which is typically in the range ; larger values of a are used to minimize transducer compliance effects on axial force measurements. The surrounding gas is assumed to be inviscid and have uniform pressure p 0. For this analysis, we shall assume inertial and gravitational effects are negligible and that there is symmetry about the z - axis (q = 0). The velocity field in spherical coordinates is assumed to have the form which satisfies the continuity equation. The rate of strain tensor for the assumed velocity field in Eq. 1 has the form (1) For this flow the extra stress tensor for has the form (5) The shear stress and two normal stress differences are given by (6) (7) (8) The r -, q - and f - components of the equations of motion for this flow are, respectively, (9) (10) where the shear rate is given by (2) (3) (11) From Eq. 1, the velocity boundary conditions can be expressed as follows (12) (13) The stress tensor p can be expressed as the sum of isotropic pressure p and extra stress tensor t contributions The f - component of the equations of motion, Eq. 11, can be integrated immediately to give (4) (14)

Figure 2: Images of free surface in cone and plate geometry: a) (left) free surface of polymer solution after excess fluid is trimmed from edge of fixtures; b) free surface of polymer melt after

3 Figure 2: Images of free surface in cone and plate geometry: a) (left) free surface of polymer solution after excess fluid is trimmed from edge of fixtures; b) free surface of polymer melt after known mass of fluid is squeezed between fixtures. The torque exerted by the fluid on the inner portion of the plate can be determined by integrating the vector product of the position vector p and stress vector (15) where n is a unit vector normal to the plate surface directed into the liquid. Equation 14 for the shear stress can now be written as (16) where M = M. If we set R i = R and q = p/2 in Eq. 16, we recover the well-known result For constant g, Eqs. 9 and 10 can be integrated to obtain (20) To complete the analysis, the force balance at the free surface must be considered. The radial position of the surface is described by r = f(q) with unit normal vector n* to the surface directed into the gas. In the absence of mass transfer and gradients in interfacial tension, the r -, q - and f - components of the jump linear momentum balance at the gas/liquid interface, respectively, can be written as [1, 2] (17) Note that because the shear stress is not constant throughout the sample, as shown in Eq. (16), the magnitude of the torque exerted by the fluid on the cone is approximately a 2 larger than that given in Eq. (17). According to Eq. 3 the shear rate g, and, through Eqs. 7 and 8, N 1 are functions of q. Consequently, the r - and q - components of the equations of motion, Eqs. 9 and 10, are incompatible. If we assume that g is constant, which is valid for a << 1, then the solution of Eq. 3 subject to Eq. 12 is [4] Combination of Eqs. 13 and 18 gives the result (18) (19) (21) (22) (23) where H is the mean curvature of the surface and g is the interfacial tension. It is clear that Eqs. 22 and 23 are not compatible with the assumed stress and velocity fields unless n q */n r * = 0, which corresponds to a spherical gas/liquid interface with radius R [1, 2]. In practice, one of two methods is used to load the fluid between the cone and plate. In one method, an excess amount of fluid is loaded and forced to fill the gap by bringing the cone and plate together. Excess fluid is trimmed from the edge resulting in a spherical, or nearly spherical, interface with radius R as shown schematically on the right side of Figure 1. In the second method, a known mass m of fluid is centered on the plate which, when the cone and plate are brought together, partially fills the gap. This loading method tends to generate a bulged inter

4 face as shown in the left side of the Figure 1. Images of actual free surfaces produced by these two loading techniques are shown in Figure 2: a) from trimming excess fluid and b) squeezing fluid with known mass. We now examine the influence a non-spherical free surface has on the stress field in cone and plate flow. The additional stress components generated at the free surface would require the existence of more complex velocity and stress fields than those given in Eqs. 1 and 5, respectively. To keep the present analysis tractable, we assume that these perturbations to the base flow are confined to a relatively small region near the free surface. If we adopt this point of view, it is possible to proceed with the analysis by combining Eqs. 21 and 22 to obtain where (27) As shown in Figure 1, a is a radius of curvature of the free surface. The shape of the surface is controlled by the ratio a/r: a/r = 1 corresponds to a spherical surface (right side of Figure 1), and a/r = sin(a/2) corresponds to a bulged surface (left side of Figure 1). On the plate surface, the ratio of surface unit normal vector components is given by (28) and the mean curvature of the surface is given by (24) (29) Combination of Eqs. 20 and 24 gives The effective sample radius (where the fluid is in contact with the cone/plate) is obtained by (25) The actual shape of the free surface is usually not known and depends on the loading method, fluid properties and wetability of the cone and plate surfaces. Here, we assume the radial position of gas/liquid interface is given by (30) Equation 30 shows that a bulged free surface reduces the effective sample radius, by a factor of the order a 2 R/a, from the value obtained by assuming a spherical free surface. Substitution of Eqs. 28 and 29 in Eq. 25 and setting q = p/2 gives the following expression for the stress distribution on the plate: (26) (31)

5 which is the expression used to obtain N 1 from measurements of F(R i ) as a function of the ratio R/R i [5. 6]. Setting R i = R in Eq. 35 gives the well-known relation between N 1 and the total force on the plate: Figure 3: Normalized axial force on inner portion of plate with radius R i as a function of normalized sample radius R. Solid line shows ideal result with Ca = 0 and a/r = 1. Symbols show measured values for Ca = 0, a = 0.1 and different values of a/r: a( ), 2a(Á). For the case of a spherical surface (a/r = 1) and no interfacial tension (g = 0), we recover the wellknown result (32) which has been used to obtain N 1 from measurements of the radial stress profile [7-9]. The net force (excluding the force from the surrounding gas) exerted on the inner portion of the plate by the fluid can be computed from 3 RESULTS AND DISCUSSION (36) The analysis presented above shows that a nonspherical free surface affects the stress field in cone and plate flow. The reason for this can be seen in Eq. 24, which shows that both t rr and t qq are involved in the balance of the isotropic part of the stress tensor. This alteration of the stress field leads to additional terms in the measured forces used to obtain N 1. As noted above, these effects imply the existence of an additional component of the extra stress tensor t rq, which, in turn, would generate a secondary flow. The main result of the analysis in the previous section is Eq. 34 which, when divided by a characteristic modulus for the fluid G N, can be written as which, after substitution of Eq. 31, gives (33) (34) where F = F. For the case of a spherical surface (a/r = 1) and no interfacial tension (g = 0), we obtain from Eq. 34 (35) (37) where Y = - N 2 /N 1 and Ca = 2g/R i G N. From Eq. 37, it is clear that interfacial tension affects the measured value of the intercept (N 1 ). If, as is often the case, the sample radius R is varied for a single value of R i, interfacial tension would also affect the slope (Y). For polymer melts, Ca ~ 10-5, so the errors introduced by interfacial tension would be negligible. However, for polymer solutions, Ca ~ 10-3 or larger, so interfacial tension could lead to errors for a bulged free surface. From this point on, we assume interfacial tension can be neglected. To examine the effect of a non-spherical free surface, we set a = 1/10, N 1 = 1 and Y = 1/4,

6 which represent conditions for a typical experiment. Figure 3 shows measurements of the axial force on the inner portion of the plate for two values of the ratio a/r, which controls the shape of the free surface. As shown in this figure, deviations from a spherical free surface (decreasing a/r) lead to errors in the measured intercept from which N 1 is obtained. An error in N 1 also leads to an error in the measured value of Y. For example, for a/r = a (squares in Figure 3), the error in N 1 is approximately 13 %, which leads to an error of approximately 11 % in Y. As noted earlier, the shape of the free surface (a/r) is not known and therefore, the example used above is only for illustrative purposes. It is also possible, in contrast to the example above, that the shape of the free surface (a/r) is a function of the sample size (R/R i ). This would directly affect the measured slope leading to an additional source of error in Y. It should also be noted that larger relative errors would be observed for larger values of a and Y. CONCLUSIONS The effects of free surface shape on normal stress difference measurements using the cone and partitioned plate technique have been investigated. The analysis presented here shows that modest deviations from a spherical free surface can lead to errors on the order of 10 % in measured values of first normal stress difference N 1 and ratio of normal stress differences N 2 /N 1. These errors result from both interfacial tension and the modification of the normal stresses involved in the force balance at the free surface. This modification of the force balance also gives rise to an additional shear stress that would induce a secondary flow. Other possible sources of error, not considered here, are the dynamic nature of the free surface shape and sample flow in the gap between the inner and outer portions of the plate. ACKNOWLEDGMENT The author thanks Mr. Wei-Hsun Yeh for the images in Figure 2. REFERENCES [1] Adams N, Lodge AS: A cone-and-plate and parallel-plate pressure distribution apparatus for determining normal stress differences in steady shear flow, Phil. Trans. Roy. Soc. Lond. A256 (1964) [2] Slattery JC: Analysis of the finite cone-plate viscometer and of the finite parallel plate viscometer, J. Appl. Poly. Sci. 8 (1964) [3] Walters K: Rheometry, Halsted Press, Wiley NY(1975). [4] Huilgol RR: Continuum Mechanics of Viscoelastic Liquids, Halsted Press, Wiley, New York (1975). [5] Meissner J, Garbella RW, Hostettler J: Measuring Normal Stress Differences in Polymer Melt Shear Flow, J. Rheol. 33 (1989) [6] Schweizer: Measurement of the first and second normal stress differences in a polystyrene melt with a cone and partitioned plate tool, Rheol. Acta 41 (2002) [7] Christensen EB, Leppard WR: Steady-State and Oscillatory Flow Properties of Polymer Solutions, J. Rheol. 18 (1974) [8] Magda JJ, Baek SG: Concentrated Entangled and Semidilute Entangled Polystyrene Solutions and the 2nd Normal Stress Difference, Poly. 35 (1994) [9] Baek SG, Magda JJ: Monolithic rheometer plate fabricated using silicon micromachining technology and containing miniature pressure sensors for N 1 measurements, J. Rheol. 47 (2003)

Non-linear Viscoelasticity FINITE STRAIN EFFECTS IN SOLIDS

Non-linear Viscoelasticity FINITE STRAIN EFFECTS IN SOLIDS FINITE STRAIN EFFECTS IN SOLIDS Consider an elastic solid in shear: Shear Stress σ(γ) = Gγ If we apply a shear in the opposite direction: Shear Stress σ( γ) = Gγ = σ(γ) This means that the shear stress