Brownian Rods

A suspension of long rods with aspect ratio of 50 or more can only be considered dilute if its concentration is very low, less than 1 % by volume. The reason is that diluteness requires that the rods be able to rotate freely without being impeded by neighboring rods (Fig. 6-18a), and the volume that a single long rod can sweep out by rotation about its center of mass must be large, around. Thus, rod-rod interactions should be expected when the number concentration of rods, y, reaches a value proportional to 7. . Experimentally (Mori et al. 1982), found that the transition occurs at more than 30 times this estimate, apparently because a rod can easily dodge several other rods that invade its sphere of rotation. Thus 

The dimensionless constant turns out to be large (around 1350) for perfectly rigid rods (Teraoka and Hayakawa 1989 Bitsanis et al. 1990). The transitional concentration Vo from dilute to semidilute occurs when Dr departs from its dilute solution value D o according to Eq. (6-44), this occurs when 30, or so, as noted above. 

Measurements (Mori et al. 1982) of dynamic electric birefringence in solutions of rodlike poly(y-benzyl-L-glutamate) (PBLG) are in agreement with Eq. (6-44). 

Under flow, the orientation distribution of the rods becomes anisotropic, and the cage diameter increases such that a oc 1/ (sin(0)), where 9 is the angle of orientation of a cage Tod with respect to the trapped test rod (Doi and Edwards 1986). If u is the orientation of the test rod, then 

Here u x u l is the absolute value of the sine of the angle between u and u. Then, defining Dr as the orientation-dependent rotary diffusivity, we have oc oc 1/ (sin 9), or 

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Brownian rod-like objects of high aspect ratio are usually molecules, not colloidal particles. As exception is tobacco mosaic virus (TMV), which is a Brownian particle of length 300 nm and diameter 18 nm (Caspar 1963). For completeness, we shall discuss the theory of Brownian rod-like particles in this chapter, with the understanding that the theory for such particles is actually more relevant to long stiff molecules than to rod-like fibers. The behavior of non-Brownian fiber suspensions is covered in Section 6.3.2.2.------------------... 

Figure 6.21 Steady-state values of for non-Brownian rod-like particles as a function of...

Figure 6.23 The relative viscosity at steady state of suspensions of non-Brownian rod-like particles versus dimensionless concentration vL. Simulations for p = L/d = 16.9 ( ), 31.9 (A) Bibbo s (1987) experimental results for L/d = 16.9 (O). and 31.9 (A). (Reprinted from J Non-Newt Fluid Mech 54 405, Yamane et al. (1994), with kind permission from Elsevier Science - NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)...

The viscous and elastic properties of orientable particles, especially of long, rod-like particles, are sensitive to particle orientation. Rods that are small enough to be Brownian are usually stiff molecules true particles or fibers are typically many microns long, and hence non-Brownian. The steady-state viscosity of a suspension of Brownian rods is very shear-rate- and concentration-dependent, much more so than non-Brownian fiber suspensions. The existence of significant normal stress differences in non-Brownian fiber suspensions is not yet well understood. 

In order to determine the molecular weight of nylon-6 polymer in the nanocomposite, it was assumed that the viscosity of nanocomposite solution had been altered by the carbon nanotubes according to the theory of dilute Brownian rods. " The intrinsic viscosity of nylon-6 polymer in the nanocomposite was determined by excluding the effect of carbon nanotubes " ... 

In all microscopic methods, sample preparation is key. Powder particles are normally dispersed in a mounting medium on a glass slide. Allen [7] has recommended that the particles not be mixed using glass rods or metal spatulas, as this may lead to fracturing a small camel-hair brush is preferable. A variety of mounting fluids with different viscosities and refractive indices are available a more viscous fluid may be preferred to minimize Brownian motion of the particles. Care must be taken, however, that the refractive indices of sample and fluid do not coincide, as this will make the particles invisible. Selection of the appropriate mounting medium will also depend on the solubility of the analyte [9]. After the sample is well dispersed in the fluid, a cover slip is placed on top... 

An interesting aspect of Eq. (4.24) is that, even though F (t) and C (/) must represent genuine Brownian motions, / (/)may represent non-Brownian, even cyclic, motions of the antenna in the rod frame. Motion of the coordinates o>R t) and 2R(t) has been assumed to be statistically independent of the rod motions, but the nature of their trajectories has not yet been specified. 

The second piece of evidence in distinguishing rods in a magnetic field to those out of the magnetic field was the rotational diffusion coefficient of the rod. It was the rotational diffusion coefficient that revealed the effect that an applied magnetic field had on a nanorod moving non-Brownian outside a field (2000 ° /s) and in it (70 ° /s). 

Bitsanis et al. [122,123] simulated Brownian motion of rodlike polymers over the concentration range 5 < LV < 150, where L and c are the length and number concentration of the rod, respectively, with the intermolecular potential u given by... 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

The two contributions to the rate of rotation, li, of the rod are convection and Brownian diffusion. Unlike the elastic dumbbell, where the springs were allowed to deform by the flow, the fixed separation of the beads in the rigid dumbbell must be maintained. For that reason, the vector u can rotate, but it cannot stretch. This constraint is satisfied by ensur-... 

The molecular theory of Doi [63,166] has been successfully applied to the description of many nonlinear rheological phenomena in PLCs. This theory assumes an un-textured monodomain and describes the molecular scale orientation of rigid rod molecules subject to the combined influence of hydrodynamic and Brownian torques, along with a potential of interaction (a Maier-Saupe potential is used) to account for the tendency for nematic alignment of the molecules. This theory is able to predict shear thinning viscosity, as well as predictions of the Leslie viscosity coefficients used in the LE theory. The original calculations by Doi for this model employed a preaveraging approximation that was later... 

In the original Doi-Edwards model, retraction is assumed to occur infinitely fast. The stress tensor is then given by the elastic, or Brownian, stress for rigid rods [see Eq. (6-36)] ... 

The stress tensor for a semidilute solution of rods is given by Eq. (6-36), the formula for dilute solutions. However, if in a thought experiment one holds the shear rate fixed at a low value while increasing the concentration of rods from dilute to semidilute, the Brownian contribution to the stress will greatly increase, since the rotary diffusivity decreases according to (6-44). The viscous stress contribution, however, only increases in proportion to u. Thus, as Doi and Edwards (1986) argued, the ratio of viscous to Brownian stresses decreases as as the concentration increases in the semidilute regime. Hence, in the semidilute... 

As mentioned earlier, suspensions of particulate rods or fibers are almost always non-Brownian. Such fiber suspensions are important precursors to composite materials that use fiber inclusions as mechanical reinforcement agents or as modifiers of thermal, electrical, or dielectrical properties. A common example is that of glass-fiber-reinforced composites, in which the matrix is a thermoplastic or a thermosetting polymer (Darlington et al. 1977). Fiber suspensions are also important in the pulp and paper industry. These materials are often molded, cast, or coated in the liquid suspension state, and the flow properties of the suspension are therefore relevant to the final composite properties. Especially important is the distribution of fiber orientations, which controls transport properties in the composite. There have been many experimental and theoretical studies of the flow properties of fibrous suspensions, which have been reviewed by Ganani and Powell (1985) and by Zimsak et al. (1994). 

In fact, the fiber contribution to the shear viscosity of a fiber suspension at steady state is modest, at most. The reason is that, without Brownian motion, the fibers quickly rotate in a shear flow until they come to the flow direction in this orientation they contribute little to the viscosity. Of course, the finite aspect ratio of a fiber causes it to occasionally flip through an angle of n in its Jeffery orbit, during which it dissipates energy and contributes more substantially to the viscosity. The contribution of these rotations to the shear viscosity is proportional to the ensemble- or time-averaged quantity (u u ), where is the component of fiber orientation in the flow direction and Uy is the component in the shear gradient direction. Figure 6-21 shows as a function of vL for rods of aspect... 

First, we need to determine if the rods are Brownian or non-Bro wnian under the above conditions. So, we compute the rotational diffusivity under dilute conditions using Eq. (6-32b) for rods of aspect ratio p = L/d = 600 ... 

The first two terms in this expression for the stress tensor are elastic terms due to Brownian motion and the nematic potential, respectively. The last term is a purely viscous term produced by the drag of solvent as it flows past the rod-like molecules [see Eq. (6-36)]. is a drag coefficient, which for modestly concentrated solutions is predicted to follow the dilnte-solution formula (Doi and Edwards 1986 see Section 6.3.1.4) ... 

Chapter 19 addresses another important technological application, the polymerization of rod-like molecules results of Brownian dynamics simulations are compared to those obtained from approximate theories and experimental studies. 

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