Tumbling predictions

The NO2 molecule offers an example which illustrates this point. The spectrum of N02 molecules rigidly held on MgO at —196° is characterized by gxx = 2.005, gyv = 1.991, and gzz = 2.002 (29). If this molecule were rapidly tumbling, one would expect a value of Qa.v — 1 999. The spectrum of NO2 absorbed in a 13X molecular sieve indicates an isotropic gzv = 2.003 (.80), which is within experimental error of the predicted value for NO2 on MgO. The hyperfine constants confirm that NO2 is rapidly tumbling or undergoing a significant libration about some equilibrium position in the molecular sieve (81). 

Methyl radicals formed on a silica gel surface are apparently less mobile and less stable than on porous glass (56, 57). The spectral intensity is noticeably reduced if the samples are heated to —130° for 5 min. The line shape is not symmetric, and the linewidth is a function of the nuclear spin quantum number. Hence, the amplitude of the derivative spectrum does not follow the binomial distribution 1 3 3 1 which would be expected for a rapidly tumbling molecule. A quantitative comparison of the spectrum with that predicted by relaxation theory has indicated a tumbling frequency of 2 X 107 and 1.3 X 107 sec-1 for CHr and CD3-, respectively (57). 

There are currently no mathematical techniques to predict blending behavior of granular components without prior experimental work. Therefore, blending studies start with a small scale, try-it-and-see approach. The first portion of this chapter is concerned with the following typical problem a 5-ft - capacity tumble blender filled to 50% of capacity and run at 15 rpm for 15 minutes produces the desired mixture homogeneity. What conditions... 

To ensure that specifications established for critical product quality attributes are met in a large-scale operation, the formulation and manufacturing process developed in the laboratory must be transferred to production and validated. It is necessary to start with a small scale in pharmaceutical research and development. Unfortunately, small-scale mixers necessary during the early development phase will not necessarily have the same characteristics as a commercial-scale mixer. Currently no mathematical techniques exist to predict the blending behavior of multicomponent solid mixtures therefore, experimental work to ensure the proper scale-up and transfer to the production facility is required. Consider the following process parameters for a tumbling blender during scale-up trials ... 

Figure 10.10 Tumbling parameter X versus reduced temperature Tr = T/Tj i for the various liquid crystals listed. The broken line is the exact prediction of the Smoluchowski equation for large aspect ratio p, and the solid and dot-dashed lines are from the approximate expression (10-24) with p large and p = 5. (From Archer and Larson, reprinted with permission from J. Chem. Phys. 103 3108, Copyright 1995, American Institute of Physics.)...

The damping of the stress oscillations presumably arises from a gradual loss of spatial coherence in the phase of the tumbling orbit across the sample. In a plate-and-plate rheometer, the strain is linearly dependent on the radial distance from the axis of rotation. As a result, the gap-averaged director orientation varies as a function of radial position in the sample. When this source of inhomogeneity in the tumbling orbit accounted for by integrating the torque contributions predicted by Eq. (10-31) over... 

From slow-shear-rate solutions of the Smoluchowski equation, Eq. (11-3), with the Onsager potential, Semenov (1987) and Kuzuu and Doi (1983, 1984) computed the theoretical Leslie-Ericksen viscosities. They predicted that ai/a2 < 0 (i.e., tumbling behavior) for all concentrations in the nematic state. The ratio jai is directly related to the tumbling parameter X by X = (1 -h a3/a2)/(l — aj/aa). Note the tumbling parameter X is not to be confused with the persistence length Xp.) Thus, X < I whenever ai/a2 < 0. As discussed in Section 10.2.4.1, an approximate solution of Eq. (11-3) predicts that for long, thin, stiff molecules, X is related to the second and fourth moments Sa and S4 of the molecular orientational distribution function (Stepanov 1983 Kroger and Sellers 1995 Archer and Larson 1995) ... 

Figure 11.18 Predictions of the tumbling parameter A as a function of reduced concentration C/C2 from the Smoluchowski equation for hard rods with the Onsager potential. The exact result from the spherical-harmonic expansion is shown, compared to approximate results from an analytic formula and from the perturbation expansion of Kuzuu and Doi. The open circles (O) are estimates from the periods of shear stress oscillations in transient shearing flows for PEG solutions (see Walker et al. 1995), and the closed circle ( ) is from a direct conoscopic measurement of Muller et al. (1994).

As the shear rate increases, the numerical solutions of the Smoluchowski equation (11-3) begin to show deviations from the predictions of the simple Ericksen theory. Tn particular, the scalar order parameter S begins to oscillate during the tumbling motion of the director (for a discussion of tumbling, see Sections 11.4.4 and 10.2.6). The maxima in the order parameter occur when the director is in the first and third quadrants of the deformation plane i.e., 0 < 9 — nn < it j2, where n is an integer. Minima of S occur in the second and fourth quadrants. The amplitude of the oscillations in S increases as y increases, until S is reduced to only 0.25 or so over part of the tumbling cycle. 

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