Cubic Array Of Spheres Research Articles

Structure and viscoelasticity of matched asymmetric diblock and triblock copolymers in the cylinder and sphere microstructures

A polystyrene–polyisoprene (PS–PI) diblock copolymer (10,000–50,000 g/mol) and a matched PS&ndashPI–PS triblock (10,000–100,000–10,000 g/mol) were employed to study the effect of chain architecture on the rheological response of ordered block copolymer melts. Both samples adopt hexagonal microstructures with PS cylinders embedded in a PI matrix; on further heating, an order–order transition (OOT) into a cubic array of spheres takes place prior to the order–disorder transition. Each morphology was verified by SAXS and TEM. Interestingly, at the OOT the low-frequency elastic modulus of the diblock increased abruptly, whereas that of the triblock decreased. In contrast, the modulus of the cubic phase was roughly independent of chain architecture. Chain relaxation parallel and perpendicular to the cylinders was probed by measuring the elastic modulus of a macroscopically aligned sample in directions parallel G and perpendicular (G) to the cylinder orientation. For both materials G < G < G where G is the elastic modulus of a randomly oriented sample. This result is attributed to the ability of the unentangled PS blocks to move along the direction of the cylinder axis, and thus relax the stress in the PI matrix in the parallel alignment. In each of the three cylindrical orientations the triblock had a larger modulus than the diblock, which is attributed to the presence of bridging PI blocks that connect distinct PS domains. About 20° below the OOT G showed a distinct change in its temperature dependence, which, coupled with SAXS measurements, is indicative of the onset of an undulation in the cylinder diameter that presages the pinching off of cylinders into spheres, as recently predicted by theory. The use of oriented samples also permitted SAXS confirmation of an approximate epitaxial relationship between the cylinder and the sphere unit cells, although a distinct change in the location of the structure factor maximum, q*, is noted at the OOT. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 2811–2823, 1997

Structure and viscoelasticity of matched asymmetric diblock and triblock copolymers in the cylinder and sphere microstructures

A polystyrene–polyisoprene (PS–PI) diblock copolymer (10,000–50,000 g/mol) and a matched PS&ndashPI–PS triblock (10,000–100,000–10,000 g/mol) were employed to study the effect of chain architecture on the rheological response of ordered block copolymer melts. Both samples adopt hexagonal microstructures with PS cylinders embedded in a PI matrix; on further heating, an order–order transition (OOT) into a cubic array of spheres takes place prior to the order–disorder transition. Each morphology was verified by SAXS and TEM. Interestingly, at the OOT the low-frequency elastic modulus of the diblock increased abruptly, whereas that of the triblock decreased. In contrast, the modulus of the cubic phase was roughly independent of chain architecture. Chain relaxation parallel and perpendicular to the cylinders was probed by measuring the elastic modulus of a macroscopically aligned sample in directions parallel G and perpendicular (G) to the cylinder orientation. For both materials G < G < G where G is the elastic modulus of a randomly oriented sample. This result is attributed to the ability of the unentangled PS blocks to move along the direction of the cylinder axis, and thus relax the stress in the PI matrix in the parallel alignment. In each of the three cylindrical orientations the triblock had a larger modulus than the diblock, which is attributed to the presence of bridging PI blocks that connect distinct PS domains. About 20° below the OOT G showed a distinct change in its temperature dependence, which, coupled with SAXS measurements, is indicative of the onset of an undulation in the cylinder diameter that presages the pinching off of cylinders into spheres, as recently predicted by theory. The use of oriented samples also permitted SAXS confirmation of an approximate epitaxial relationship between the cylinder and the sphere unit cells, although a distinct change in the location of the structure factor maximum, q*, is noted at the OOT. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 2811–2823, 1997

Limit Process Expansions and Homogenization

The use of limit process expansions and asymptotic matching is illustrated for a simple problem of homogenization. Rayleigh’s problem of calculating the effective thermal conductivity $k_{eff} $ of a regular cubic array of spheres of conductivity $k_I $ in a matrix of conductivity $k_M $ is considered for small volume fractions $f_I $. An inner expansion is found near each sphere. This is matched to an outer expansion in which the sphere shrinks to a singularity. Rayleigh’s result that the expansion for $k_{eff} $ contains terms $f_I ,f_I^2 ,f_I^3 ,f_I^4 ,f^{13/3} $ is reconstructed. In addition an independent verification of Rayleigh’s summability method for an infinite array of dipoles is given. Comments are made on related problems and the random case where similar summability difficulties appear.

Shear flow of periodic arrays of particle clusters: a boundary-element method

The boundary-element method is used to solve Stokes equations for periodic arrays of force-free and torque-free rigid particles. Simple cubic arrays of spheres, spheroids, cubes, and clusters of spheres are subjected to a bulk simple shearing flow. The effective volume-averaged stress tensor for the suspension and the detailed velocity and stress fields throughout the Newtonian suspending fluid are calculated. We find that even crude meshes give very good volume-averaged results, but fine meshes are required to track local minima and maxima in the stress field. For simple cubic arrays of spheres, the boundary-element results are in excellent agreement with the analytical viscosity predictions of Nunan &amp; Keller (1984). Even at the highest concentration of solids studied, no significant normal stress differences were observed, in agreement with Nunan &amp; Keller's results (1984). Up to moderate concentrations of particles, the volume-averaged properties of the suspension display only a weak dependence on the particle geometry. Suspensions of spheroids and cubes behave approximately as suspensions of spheres on the average despite large differences in the local micromechanics of stress and velocity fields. Simple cubic arrays of clusters of spheres tend to behave on a macroscopic level as a cubic array of spheres whose effective volume fraction is about 150% of the total volume fraction of the spheres in the clusters.

Grain impingement during recrystallization

The spherical computer model based on a cubic array of spheres growing at the same rate with the same radius is an obvious oversimplification of recrystallization reactions. However, it does provide calculable impingement geometry, and it yields results that are in agreement with experimental results. The model supports the use of the parabolic relation in Eq. [1], although it emphasizes the empirical nature of the relation and demonstrates limitations of the relation of f = 0 and f = 1. The results generated by the model further indicate that computer models of regular geometric solids can provide useful information on grain impingement during recrystallization reactions.

High Reynolds number flow through cubic arrays of spheres Steady-state solution and transition to turbulence

A five-mode truncation of the Navier—Stokes equation for incompressible flow through a cubic array of spheres is analyzed for its stationary and time-dependent solutions. Under the imposed symmetry requirements, the stationary solutions accurately depict the appearance of a vortex pair, and the volume-averaged velocity is in excellent agreement with the Ergun equation up to Re = 250. At the point the deviation from empirical data begins, the unique stationary solution destabilizes via a Hopf bifurcation point. A numerical analysis of the resulting periodic solution shows that for a certain range of imposed pressure gradient the system exhibits chaotic behavior, approached through a period-doubling subharmonic bifurcation sequence.

Hydrodynamic properties of macromolecular complexes. V. Improved calculation of rotational diffusion coefficient and intrinsic viscosity

AbstractIn our previous calculations of rotational diffusion coefficients and intrinsic viscosities of macromolecular complexes modeled by arrays of spherical subunits [J. G. de la Torre &amp; V. A. Bloomfield, Biopolymers 16, 1765, 1779 (1977); 17, 1605 (1978)], results were poor when the dominant subunit was located near the center of frictional resistance. A simple means of correcting this flaw, which gives satisfactorily accurate results with little increase in computation time, is to replace the single large subunit with an array of smaller ones. We have examined trigonal bipyramidal, octahedral, and cubic arrays of spheres whose radii were chosen to give the same total volume or the same rotational diffusion coefficient as the parent sphere. These all give similar results, so the details of the modeling are not important. Results obtained using this stratagem are in much better agreement with the theories of Perrin and Simha for short prolate ellipsoids of revolution, and with experimental measurements of rotational diffusion coefficients of T‐even bacteriophage without fibers or with fibers retracted. We have also extended our previous calculations to consider phage with various numbers of fibers attached.

Exact modelling of cubic lattice permittivity and conductivity

A PROBLEM of practical importance in a wide variety of situations is that of calculating the permittivity (or equivalently, the conductivity) of simple cubic arrays of spheres embedded in another material. Applications occur in, for example, the analysis of optical properties of inhomogeneous films1, dielectric dispersion in mixtures2, the behaviour of artificial dielectrics3, the study of lunar soil samples4 and in vision research5. Although the problem is an old one, and has attracted the attention of some of the great names of physics (for example, J. C. Maxwell6 and Lord Rayleigh7) it seems that a completely satisfactory theory has not existed until the present time. This is shown by the fact that in the most critical case, that of perfectly conducting spheres, none of the existing theories give the expected divergence at the critical volume fraction (fc = π/6) at which the spheres touch. We have extended Lord Rayleigh's treatment and present a theory which gives the correct behaviour for all volume fractions.

Computation of forced convection in slow flow through ducts and packed beds—III. Heat and mass transfer in a simple cubic array of spheres

Temperature profiles and Nusselt numbers are calculated for slow three-dimensional flow through a simple cubic array of isothermal spheres. The problem is solved by a collocation method in a coordinate system which permits the surface temperature to be prescribed directly. Numerical tables and asymptotes of the calculated solutions are given. These results should be useful for planning and correlation of packed-bed experiments.

Computation of forced convection in slow flow through ducts and packed beds—IV. Convective boundary layers in cubic arrays of spheres

Detailed boundary-layer solutions are given for heat and mass transfer in forced convection through packed beds. The simple cubic and the dense cubic array of spheres are chosen as examples. Numerical results are given for uniform and catalytic surface conditions, and the uniform-accessibility assumption is quantitatively tested. Close agreement is found with the data of Karabelas et al.[1] for diffusion-controlled electrolysis in a dense cubic array.

Computation of forced convection in slow flow through ducts and packed beds—II velocity profile in a simple cubic array of spheres

The velocity profile is calculated for slow three-dimensional flow through a simple cubic array of spheres. The velocity is represented by stream-function expansions which directly satisfy the equation of continuity and the no-slip conditions. The problem is solved by a variational principle which gives a lower bound on the total flow rate. The computed bound is within 0·2 per cent of the experimental results of Martin et al.[6].

Flow pattern in a close packed cubic array of spheres near the critical Reynolds number

Profiles of pressure, velocity gradient, and local mass transfer rate on the surface of a sphere in a close packed cubic array are interpreted in terms of visual studies which determine the singularities of the limiting streamlines. Contact points are found to play an important role and studies of flow through a single layer of spheres are used to help determine their influence. On the basis of these experiments, a flow pattern is suggested for Reynolds numbers close to that for a transition to unsteady flow. Some of the problems associated with applying boundary layer theory are discussed.

Use of asymptotic relations to correlate mass transfer data in packed beds

The effect of Reynolds number, Grashof number and Schmidt number on mass transfer rates in packed beds is not yet established. It is suggested that an improved method for interpreting mass transfer results is to determine asymptotic relations for different ranges of variables. This approach is illustrated by presenting results of a study of mass transfer to one of the spheres in a close packed cubic array of spheres at Sc ≅ 1600. For pure laminar natural convection it is shown that ▪ For forced convection at low Reynolds numbers and for large Pe, ▪ For forced convection at Reynolds numbers close to the transition to unsteady flow ( Re c = 90–120), ▪

Visual studies of flow in a regular array of spheres

Studies have been made of the motion of dye streamers on the surface of a sphere in a close packed cubic array of spheres. At a Reynolds number of 82 a completely steady flow is observed with nine distinct regions of reverse flow. The same type flow pattern is also observed a Reynolds number of 200 where the flow is unsteady. The flow patterns are described by identifying the singular points of the limiting streamlines. Contact points were observed to be saddle points of these limiting streamlines.

On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres

Spatially periodic fundamental solutions of the Stokes equations of motion for a viscous fluid past a periodic array of obstacles are obtained by use of Fourier series. It is made clear that the divergence of the lattice sums pointed out by Burgers may be rescued by taking into account the presence of the mean pressure gradient. As an application of these solutions the force acting on any one of the small spheres forming a periodic array is considered. Cases for three special types of cubic lattice are investigated in detail. It is found that the ratios of the values of this force to that given by the Stokes formula for an isolated sphere are larger than 1 and do not differ so much among these three types provided that the volume concentration of the spheres is the same and small. The method is also applied to the two-dimensional flow past a square array of circular cylinders, and the drag on one of the cylinders is found to agree with that calculated by the use of elliptic functions.

Stress-Strain Relations and Vibrations of a Granular Medium

Abstract A differential stress-strain relation is derived for a medium composed of a face-centered cubic array of elastic spheres in contact. The stress-strain relation is based on the theory of elastic bodies in contact, and includes the effects of both normal and tangential components of contact forces. A description is given of an experiment performed as a test of the contact theories and the differential stress-strain relation derived from them. The experiment consists of a determination of wave velocities and the accompanying rates of energy dissipation in granular bars composed of face-centered cubic arrays of spheres. Experimental results indicate a close agreement between the theoretical and experimental values of wave velocity. However, as in previous experiments with single contacts, the rate of energy dissipation is found to be proportional to the square of the maximum tangential contact force rather than to the cube, as predicted by the theory for small amplitudes.