Slotted-Plate Device To Measure the Yield Stress of Suspensions

Ind. Eng. Chem. Res. 2002, 41, 6375-6382

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Slotted-Plate Device To Measure the Yield Stress of Suspensions: Finite Element Analysis Lixuan Zhu and Daniel De Kee* Chemical Engineering Department and Tulane Institute for Macromolecular Engineering and Science, Tulane University, New Orleans, Louisiana 70118

A slotted-plate technique has been used to measure the yield stress of suspensions. The wall slip effect is eliminated by opening a number of slots on the plate, which is dragged through the suspension at a constant speed. The yield stress is obtained by determining the point deviating from linearity on the force versus time curve. The suspension filling the slots is assumed to move with the plate. Shearing with the bulk suspension occurs at the slot edges. The preyielding stress evolution is investigated using a finite element method. Stress-strain instead of stressshear rate material models are used. Both solid and slotted plates are simulated. At the yielding point, the stress is uniformly distributed along the solid-plate surface or along the slot edge, supporting the assumptions put forward toward the yield stress determinations. That is to say, this technique is suitable for yield stress measurements. Introduction Most multiphase systems such as suspensions exhibit yield stress behavior. The interactions between the particles in the material preclude flow of a system which is subjected to a stress below its yield stress value. Therefore, prior to yielding, suspensions behave like solids and only deform to a finite deformation which is proportional to the applied stress. When the applied stress is larger than the yield stress, the static structures in the material disintegrate and the material begins to flow. To measure this yield stress value, many techniques have been employed and, frequently, different values were obtained for the same material sample.1,2 Traditional plate-plate or cone-plate rheometers are used to measure the shear stress versus shear rate relationship. The experimental data are extrapolated to zero shear rate to obtain the yield stress value.3 Such yield stress results therefore rely on the mathematical models describing the shear stress versus shear rate relationship. One may also encounter serious wall slip and sample evaporation problems with rheometer measurements, particularly when dealing with concentrated suspensions.4 The data obtained at low shear rate are often not accurate and not reproducible. Direct yield stress measurements, such as the creep test and the strain ramp test, were developed to avoid these problems. A direct measurement progresses from a static state to a flowing state.5 For a strain ramp test, a constant strain rate is applied to the material and the stress in the material is measured. At first, when the strain is small, the material displays elastic properties and the stress versus strain relationship is linear. When the strain reaches a critical value, the material structures begin to degrade and the slope of stress versus strain curve decreases: the curve gradually levels off. This is the plastic region of the material where yielding has occurred. The stress is then almost constant, no longer depending on the strain, and the material is flowing. Direct yield stress measurement techniques currently used are the vane geometry and the plate geometry. The * Corresponding author. E-mail: [email protected].

vane device allows the material to yield in itself rather than at the interface between the device and the material.6,7 It assumes that yielding occurs on a virtual cylinder defined by the blades of vane and that the stress is evenly distributed on this virtual cylinder. Some finite element analysis work was performed to verify these assumptions.8-10 The simulations used shear rate dependent fluid mechanics models such as the power law, the Bingham model, the Casson model, and the Herschel-Bulkley model. That is, the simulations were based on shearing instead of on elastic deformation and therefore did not describe the actual yielding procedure but the shearing following sample yielding. The other feature in these simulations was that most of the computed yield stress values in the simulations were on the order of 102 Pa, reflecting the success of the vane method when dealing with high concentration suspensions. The simulation results confirmed that the assumptions were approximately valid in the high yield stress/shear rate dependent model situation. The plate method to measure yield stress was reported prior to the vane method but attracted less attention.11,12 This method involves vertically dragging a flat plate through a suspension, subjecting the sample to an excursion from the static to the flowing states. The force exerted on the plate is measured, and this allows for the calculation of the yield stress. The yielding of the suspension occurs evenly in the vicinity of the plate surface and is not restricted to the assumptions associated with the vane method. However, at the plate/ sample interface, when dealing with high concentration suspensions, there may exist severe wall slip effects. The measured stress is then not necessarily the yield stress of the suspension but reflects the friction stress between the plate and the suspension. A modified plate device avoids this problem by opening a number of slots on the plate.13 So far, no numerical analysis has been performed to verify the assumptions associated with the (slotted) plate geometry. In this paper, a finite element analysis method was used to simulate the material deformation and yielding with the plate geometry. Solid mechanics material models were used to describe the actual yielding pro-

10.1021/ie010606b CCC: $22.00 © 2002 American Chemical Society Published on Web 07/30/2002

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Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002

Material Models The sample suspension is assumed to be homogeneous, isotropic, and incompressible from the macroscopic point of view. The suspension particle size is very small [O(10-6 m)] in comparison to the slot dimension [O(10-3 m)], so that the sample could be considered to be a continuum. We further assumed that the suspension in the slots retains the bulk structure. The threedimensional rigid network (due to interactions between particles) deforms elastically when the applied stress is smaller than the material yield stress. The general continuum mechanics governing equations are

F Figure 1. Schematic diagram of the plate device: (1) balance; (2) wire; (3) container; (4) sample; (5) slotted plate; (6) platform.

(

)

(1)

σij ) -Pδij + 2µ˘ ij

(2)

1 ˘ ij ) (ui, j + uj, i) 2

(3)

∂ui + ujui, j ) Ffi + σij, j ∂t

where ui is the velocity component in the xi direction, fi is the component of volume force, σij is the component of the stress tensor, P is the hydrostatic pressure, µ is the viscosity, and ˘ ij is the strain rate. The continuity equation for the conservation of mass for an incompressible fluid is

ui,i ) 0 Figure 2. Sketch of a slotted plate.

cedure. The uniform stress distribution along the plate surface and the assumptions associated with the slotted plate idea were investigated.

For a yield stress material, some empirical models were proposed to describe the shear stress versus shear rate relationship.14,15 The Herschel-Bulkley shear rate dependent model is often used in the literature. The model is given by

γ˘ ) 0 Experimental Section Figure 1 shows the plate device including a linear motion platform driven by a step motor and a balance connected to a computer for data analysis. The speed of the platform varies from 0.003 to 3.0 mm/min and can be continually changed through a controlling computer. The plate hangs from the balance through a very thin stainless steel wire and thread. The sample is loaded into a beaker, which is put on the platform. The plate is immersed in the suspension, and the force exerted on the plate was measured using the balance when the platform is lowered. The whole system is kept at a constant temperature. A series of slotted plates were used for yield stress measurements. Such a slotted plate is inserted in the suspension, and the suspension fills the slots. It was assumed that when the plate moved, the suspension in the slots remained static relative to the plate. That is, the suspension in the slots moves with the plate. The suspension deformed at the slot edges until a static yield stress σb value was reached, after which the suspension began to flow. The plate/suspension interface could be associated with a stress σs < σb due to the slip effect. With a slot area Sb and a plate area Ss, the area ratio β ) Sb/Ss was varied, and the yield stress σb was determined from the total force vs β curve. Figure 2 shows a sketch of such a slotted plate and of the σs and σb distribution along the plate surface.

(4)

σ ) σy + kγ˘ n

if σ < σy if σ > σy

(5a) (5b)

where σ is the applied stress, σy is the yield stress, γ˘ is the measured shear rate, and k and n are model parameters. For n ) 1, one recovers the Bingham model. However, there exists indeterminacy of the shear stress and shear rate when σ < σy, and a simple way to solve this problem is to assume a critical shear rate below which the effective viscosity is considered to be a constant10,16

σ ) µ0γ˘ σ ) σy + kγ˘ n

if γ˘ < γ˘ c if γ˘ > γc

(6a) (6b)

where γ˘ c is the critical shear rate and µ0 is the low shear viscosity. This modification involving a shear rate dependent Herschel-Bulkley model does not describe the yield stress properties all that well in a strain ramp test. This is not surprising because the independent variable is not a shear rate. Also, a material with yield stress has an infinite viscosity at zero shear rate, but the modified model is associated with a constant (finite) viscosity at low shear rate. In this work, suspension yielding behavior was studied using solid mechanics.17 The simplest possibility involves a combination of Hooke’s law and Newton’s law to describe the motion before and after yielding. That

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Figure 4. Finite element meshes: (a) solid plate; (b) enlarged slotted plate with only one slot.

Figure 3. Material models: (a) the simplest model without a plastic region; (b) the model with a plastic region; (c) relationship among stress, strain, and shear rate.

is to say,

σ ) Eγ

if γ < γc

(7a)

σ ) σy

if γ > γc

(7b)

where γ is the shear strain, γc is the yield shear strain, σ is the shear stress, and σy is the yield stress. When the applied strain rate is constant, the stress after yielding is also constant according to Newton’s law. Figure 3a illustrates this simple yielding procedure. The stress versus strain relationship is linear until the stress exceeds the yield stress, after which the stress remains

constant with increasing strain. Figure 3b shows a modified model by adding a transition region which represents the plastic behavior of the suspension. We assume the elastic modulus to be 200 Pa. The critical strain is 0.1, and the static yield stress is then 20 Pa. The maximum stress value in Figure 3b is 25 Pa. These parameters are in a range consistent with those obtained from experimental data on TiO2 suspensions.13 Although shear rate does affect the yield stress measurement results in the strain ramp tests, experiments showed that yield stress values obtained at low shear rate are almost constant.13,18 Figure 3c illustrate the relationships among shear stress, shear strain, and shear rate. In this material model, a fixed shear rate is used to represent the constant platform speed. Initially the sample is deformed to a critical strain value and then begins to flow, reflecting a realistic yielding procedure. The material model in Figure 3b was not analytically given. The transition region describes the fact that the suspension structure partially disintegrates, followed by permanent deformation. Both elastic and plastic strains accumulate as the material deforms in the post-yield region. To use the finite element analysis software ABAQUS, plastic region data were tabulated in the input file. These points (1-4 in Figure 3b) were arbitrarily chosen and used in the computation. This does not affect the simulation result because only the shape of the output reaction forces versus time and the shape of the input stress versus strain curves need to be compared. The absolute values are not required.

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Figure 5. Stress development in the material with the solid-plate geometry.


Geometry and Boundary Conditions To simplify the plate device simulation, a twodimensional geometry was considered. The side profile was used to represent the whole device. Two geometries (solid and slotted plates) were studied. Figure 4a refers to the full (solid) plate. This is a side view of the plate shown in Figure 1. The mesh represents the suspension in the beaker, and the thin blank area at the center of the suspension represents the plate. The suspension in the container was 90 mm high and 60 mm wide. The plate was 0.6 mm thick and 30 mm high and was centered in the material. The finite element analysis for this geometry was investigated using a mesh which consists of 1560 elements and 1664 nodes. The plate was supposed to have the same property as the suspension. That is to say, no slip is involved (σs ) σb). The boundary conditions at the wall and at the bottom of the container are

ux ) u y ) 0

(8a)

and on the surface of plate, we have

ux ) 0, uy ) 0.005 mm/s

(8b)

The upper surface of the suspension is considered to be free. Figure 4b refers to the slotted plate (one slot is shown). This is again a side view of the slotted plate sketched in Figure 2. To investigate the stress distribution in the slot, the area near the slot was enlarged. The suspension in the container was 50 mm high and 50 mm wide. The plate was 10 mm thick and 50 mm high with a slot 10 mm high. The mesh consists of 2100 elements and 2241 nodes. Here, we assumed no friction force between the plate and the suspension. That is to say, wall slip occurs at the surface of the plate (σs ) 0). The boundary conditions on the side wall of the container are

ux ) u y ) 0

(9a)

and on the plate surface, including the upper and lower edges of the slot, we have

ux ) 0, uy ) 0.02 mm/s

(9b)

The upper and lower surfaces of the suspension are considered to be free. Results and Discussion All computations were carried out on an IBM RS/6000 work station under a UNIX operating system. The PATRAN program was used to generate the geometry and the mesh as well as the input material parameters. The ABAQUS program was used to perform the computations and to provide the result analysis and graphic display. ABAQUS, used mostly for solid mechanics analysis, was adopted here to simulate the preyielding procedure of fluids with yield stress. This software is also able to deal with the plastic and viscous regions. The stress values shown in this work are all Von Mises stresses, the most commonly used yielding criteria. The reaction force on the plate boundary is actually the measured force using the balance in the experiment. Full-Plate Geometry. During an experiment, the plate moved upward in the suspension at a constant

Figure 6. Reaction force on the solid plate: (a) material model in Figure 3a; (b) material model in Figure 3b.

velocity and the suspension was dragged and deformed. Figure 5 illustrates the stress distribution in the suspension at different times. It is obvious that the suspension stress concentrates on the upper and lower edges of the plate (dark red in Figure 5) and spreads over the entire plate surface. That is, when the plate moved upward, the suspension was initially yielded in the vicinity of the upper and lower edges of the plate, which is consistent with intuitive prediction. At short times, the suspension was still in the elastic region and the stress profile in the suspension looks similar (the magnitudes of the stress values increased). A stress evolution was observed after the yield stress value was reached. The yielding region started at the plate corners, and spread toward the plate center, to generate a uniform stress on the plate surface. This stress distribution profile changed little with time because in the viscous region stress remains constant (see Figure 3a). The stress is concentrated in the vicinity of the plate and decreased substantially with the distance from the plate. Thus, the container size is not critical to the yield stress measurement as long as the plate thickness (0.6 mm) is small, compared to the container width (60 mm). In the actual experiment, the container width is larger than 60 mm. Figure 6a shows the reaction force on the plate in the vertical direction, corresponding to the material model illustrated in Figure 3a. This force is directly measured using a balance in the experiment, and the shape obtained is qualitatively similar to the situation portrayed in Figure 3a. The lack of a quantitative match is due to the fact that material yielding is a function of position and time; i.e., local yielding spreads to achieve complete yielding along the plate. Obtained experimen-

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Figure 7. Stress development in the material with the slotted-plate geometry.


tal results confirm this force versus time progression. That is to say, one can easily determine the yield stress from experimentally determined force-time curves. The obtained value corresponds to the information in Figure 6a, which in turn corresponds to the information in Figure 3a. Similarly, the reaction force computed and drawn in Figure 6b corresponds to the information contained in Figure 3b. The computational results are thus consistent with the experimental results, and the experimental results are reliable to determine yield stress values. Slotted-Plate Geometry. To verify the validity of the assumptions for the slotted-plate device, the suspension in the vicinity of a slot was investigated. To simplify the situation, the plate is considered to be completely smooth and no interaction between the plate surface and the sample is considered. The measured force is associated with the normal forces exerted on the slot, as shown by the arrows in Figure 2c. Figure 7 shows the stress distribution in the suspension at different times for a slotted-plate geometry. The suspension material model is the same as that in Figure 3a. As the plate moves upward and the suspension in the slot moves with the plate, the development of the stress in the suspension is shown for different times. The stress first concentrates at the corner of the slot and then spreads along the edge of the slot. The suspension in the slot experiences a much smaller stress value than the material at the edge. Once the yield value is reached, the suspension near the corner of the slot begins to flow (local secondary flow). This is followed by a prompt yielding of the suspension along the edge of the slot. After complete yielding is achieved, the stress distribution profile keeps changing and the stress spreads away from the plate to the container wall. This is because a smaller container was used for the convenience of illustrating the detailed stress distribution around the slot. In a real experiment, the container size is much larger and the stress distribution profile would be constant shortly after the yielding point is reached. During the yielding procedure, involving the suspension away from the slot, the stress value of the suspension in the slot never exceeded the yield stress value. That is, no yielding or flow occurred in the suspension in the slot. A sharp yielding profile was observed at the edge of the slot. This confirms our starting assumptions of the slotted-plate device. The sample in the slot moves with the plate without secondary flow in the slot and yields at the slot edges. Figure 8 shows the reaction force versus time on the slotted plate. Because the plate is supposed to be completely smooth, the reaction force on the plate surface is zero. Surfaces AA′ and BB′ (see Figure 2b) are subjected to forces transported from the bulk shear stress due to sample deformation. In the simulation, the slotted-plate velocity was 4 times that of the solid-plate simulation. This scaling is responsible for an increased slope in the reaction force versus time representation, achieving yield after ∼100 s. Parts a and b of Figure 8 correspond to the material models in parts a and b of Figure 3, respectively. Note that solid-plate simulation in Figure 8b achieved yield after ∼500 s. Figure 9 shows a typical force versus time experimental output. The force, which is the actual measured force exerted on the plate during the motion of the plate in the suspension, corresponds to the reaction force computed earlier. These two force profiles are qualita-

Figure 8. Reaction force on the slotted plate: (a) material model in Figure 3a; (b) material model in Figure 3b.

Figure 9. Experimental force versus time measurement of a TiO2 suspension using the slotted plate.

tively similar and suggest that the simple material model chosen provides an acceptable description of the material behavior. Conclusions A finite element analysis showed that, in the measurement of the yield stress of suspensions using a plate device, the stress initially concentrates at the upper and lower edges of the plate and then spreads along the entire plate surface. Once a local stress reaches the sample yield stress value, secondary flow may occur

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at the plate corner (location of maximum stress). This is immediately followed by yielding along the entire plate surface. The stress distribution profile on the plate surface is uniform after the yielding point has been reached. The reaction force versus time curve has a shape similar to that of the material stress versus strain curve. The measurement of the force exerted on the plate represents the sample yielding behavior. One slot was considered to simulate a smooth slotted plate. When the plate moves upward, the suspension in the slot moves with the plate and the stress experienced by the suspension inside the slot is (much) less than the yield stress, indicating that no secondary flow occurs in the slot. The yielding of the suspension occurs (sharply) at the edge of the slot, supporting the assumption that the distribution of stresses σs and σb in Figure 2b is valid. The reaction force (experimentally measured force) can thus be used to compute the yield stress via this plate method. Literature Cited (1) Bird, R. B.; Dai, G. C.; Yarusso, B. J. The rheology and flow of viscoplastic materials. Rev. Chem. Eng. 1983, 1, 1. (2) Nguyen, Q. D.; Boger, D. V. Measuring the flow properties of yield stress fluids. Annu. Rev. Fluid Mech. 1992, 24, 47. (3) James, A. E.; Williams, D. J. A.; Williams, P. R. Direct measurement of static yield properties of cohensive suspensions. Rheol. Acta 1987, 26, 437. (4) Barnes, H. A. A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure. J. Non-Newtonian Fluid Mech. 1995, 56, 221. (5) Carreau, P. J.; De Kee, D.; Chhabra, R. P. Rheology of Polymeric Systems: Principles and Applications; Hanser: New York, 1997. (6) Nguyen, Q. D.; Boger, D. V. Yield stress measurement for

concentrated suspensions. J. Rheol. 1983, 27, 321. (7) Nguyen, Q. D.; Boger, D. V. Direct yield stress measurement with the vane method. J. Rheol. 1985, 29, 335. (8) Barnes, H. A.; Carnali, J. O. The vane-in-cup as a novel rheometer geometry for shear thinning and thixotropic materials. J. Rheol. 1990, 34, 841. (9) Keentok, M.; Milthorpe, J. F.; O’Donovan, E. J. On the shearing zone around rotating vanes in plastic liquids: theory and experiment. J. Non-Newtonian Fluid Mech. 1985, 17, 23. (10) Yan, J.; James, A. E. The yield surfave of viscoelastic and plastic fluids in a vane viscometer. J. Non-Newtonian Fluid Mech. 1997, 70, 237. (11) De Kee, D.; Turcotte, G.; Fildey, K.; Harrison, B. New method for the determination yield stress. J. Texture Stud. 1980, 10, 281. (12) De Kee, D.; Mohan, P.; Soong, D. S. Yield stress determination of styrene-butadiene-styrene triblock copolymer solutions. J. Macromol. Sci. Phys. 1986, 25, 153. (13) Zhu, L.; Sun, N.; Papadopoulos, K.; De Kee, D. A Slotted Plate Device for Measuring Static Yield Stress. J. Rheol. 2002, 45, 1011. (14) Oldroyd, J. G. Proc. Cambridge Philos. Soc. 1947, 43, 100. (15) Doraiswamy, D.; Mujumdar, A. N.; Tsao, I.; Beris, A. N.; Danforth, S. C.; Metzner, A. B. The Cox-Merz rule extended: A rheological model for concentrated suspensions and other materials with a yield stress. J. Rheol. 1991, 35, 647. (16) Beverly, C. R.; Tanner, R. I. Numerical Analysis of Extrude Swell in Viscoelastic Materials with Yield Stress. J. Rheol. 1989, 33, 989. (17) Christensen, G. Modelling the flow of fresh concrete: The slump test. Ph.D. Dissertation, Princeton University, Princeton, NJ, 1991. (18) Alderman, N. J.; Meeten, G. H.; Sherwood, J. D. Vane rheometry of bentonite gels. J. Non-Newtonian Fluid Mech. 1991, 39, 291.

Received for review July 16, 2001 Revised manuscript received May 23, 2002 Accepted June 6, 2002 IE010606B

Slotted-Plate Device To Measure the Yield Stress of Suspensions

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