Cone-plate geometry

Experiments by Muller et al. [17] on the lamellar phase of a lyotropic system (an LMW surfactant) under shear suggest that multilamellar vesicles develop via an intermediate state for which one finds a distribution of director orientations in the plane perpendicular to the flow direction. These results are compatible with an undulation instability of the type proposed here, since undulations lead to such a distribution of director orientations. Furthermore, Noirez [25] found in shear experiment on a smectic A liquid crystalline polymer in a cone-plate geometry that the layer thickness reduces slightly with increasing shear. This result is compatible with the model presented here as well. 

From Equations 3C.5 and 3C.6, it follows that the shear stress in a cone-plate geometry is essentially uniform. 

The cone-plate geometry is widely used in rheological measurements of viscoelastic fluids. The fluid is placed between a plate of radius and a cone of the same radius. The angle, a, between the cone and the plate is usually smaller than 3° (see Fig. 13.19). 

In cone-plate geometry, the velocity field of a simple shearing flow has the following components in a spherical coordinate system (2,23,29) ... 

Important disadvantages of this geometry are evaporation and free boundary effects for polymer solutions prepared with volatile solvents. Moreover, measurements are restricted to relatively low shear rates because polymer melts and other fluids will not stay in the gap at high rotational speeds. The cone-plate geometry is not recommended for measuring the viscosity of multiphase systems because in some cases domain sizes may be of the same order of magnitude as the gap size. 

Oscillation experiments are performed in a polymer melt with cone-plate geometry, obtaining G and G" = 10 and 10 N/m, respectively, at CD = 10 rad/s. If in this region G and G" are linear functions of frequency, estimate the force per unit area tending to separate the cone and the upper plate when the lower plate rotates at an angular velocity O = 10 " rad/s. The cone-plate angle is or = 2°. 

The flow curves can be established for different concentrations and different molar masses of HA samples, and at different temperatures for a better insight into the molecular properties of polymers. Fig. (14) shows results of a series of rheological tests of HA polymers with different molar masses at different concentrations. Fig. (14, left panel) shows the flow curves for three different HA samples with the Mw values of 850 kDa, 600 kDa, and 400 kDa. Fig. (14, right panel) exhibits the flow curves for an HA sample at four different concentrations ranging from 0.11% to 2.16%. The flow curves are obtained by using an AR 2000 stress-controlled rheometer from TA Instruments (New Castle, DE, USA). A cone/plate geometry is used. The rotor was made of the acrylic material, 4 cm of diameter and 1° of cone angle. The measurements were performed at 20 °C. 

Fig. 4 Reprinted from [96], (a) Schematics of the confocal rheoscope of the Edinburgh group [96]. The top arrow marks translation of the rheometer head to adjust the geometry gap, the horizontal arrow indicates translation of the arm supporting the objective to image at different radial positions r. (b) Close up of the central part of the rheoscope, similar to the cone-plate imaging system of Derks [111] except that in the latter the lower plate can also be rotated, while in the former the microscope objective radial position r can be varied, (c) Gap profile of a 1° cone-plate geometry, measured in the confocal rheoscope with fluorescent particles coated on both surfaces...

Rheological measurements were performed in shear using a stress controlled rheometer (Carri-Med CSL 100) operating in cone-plate geometry. Each sample is submitted successively to a first frequency sweep in range 10 3-40 Hz under 3% strain, to a creep and recovery test, and finally to a second frequency sweep identical to the first one. The dynamical strain amplitude (3%) and the value of the creep stress (chosen so as to keep the maximum strain below 10%) were set in order to remain within the linear viscoelasticity domain. Creep and creep recovery were recorded during 20 h and 80 h, respectively, times which allowed the steady state to be reached in all cases. A fresh sample was used for each solvent/temperature combination. 

In practice, the cone is often truncated by a small amount which avoids contact with the cone tip (which might become worn) and the plate (which might become indented). Figure 15b shows the schematic diagram of a truncated cone-plate geometry. A truncated cone also facilitates tests on suspensions (Barnes et al., 1989). If Ri < 0.2R, the torque is reduced by less than 1%. The total torque is reduced by much less than 1% because the parallel plate section near the axis will contribute to the torque (Whorlow, 1980). 

For cone-plate geometry, the major errors are the edge and end effects which arise from the fact that the geometry has finite dimensions and a fracturing effect (Walters, 1975). 

Viscometers with cylinder and cone/plate geometries can also be employed. The cylinder viscometers are easier to use and provide more reproducible results. Cone and plate systems can be used to investigate the hardening behavior of paints. The system can easily be cleaned and only a small amount of sample is required. High velocity gradients can be achieved with small cone angles. The potential uses of cone and plate systems are limited for several reasons and they cannot be used with dispersions [9.6], [9.7]. 

Rheological measurements of the silica suspensions were performed using a Paar Physica MCR300 rheometer with a cone-plate geometry. 

---