Asymmetric particles

The size of a spherical particle is readily expressed in terms of its diameter. With asymmetrical particles, an equivalent spherical diameter is used to relate the size of the particle to the diameter of a perfect sphere having the same surface area (surface diameter, ds), the same volume (volume diameter, dv), or the same observed area in its most stable plane (projected diameter, dp) [46], The size may also be expressed using the Stokes diameter, dst, which describes an equivalent sphere undergoing sedimentation at the same rate as the sample particle. Obviously, the type of diameter reflects the method and equipment employed in determining the particle size. Since any collection of particles is usually polydisperse (as opposed to a monodisperse sample in which particles are fairly uniform in size), it is necessary to know not only the mean size of the particles, but also the particle size distribution. 

The scattering models employed in data processing invariably involve the assumption of particle sphericity. Size data obtained from the analysis of suspensions of asymmetrical particles using laser diffraction tend to be somewhat more ambiguous than those obtained by electronic particle counting, where the solid volumes of the particles are detected. 

At Oak Ridge, the focus was to develop specific-sequence DNA to improve the diffraction quality of NCP crystals. The positioning of the DNA on the histone core has to be precise so that all the NCPs are identical. A project was undertaken to understand the DNA sequence effects on nucleosome phasing [25]. Second, a DNA palindrome was developed to extend the two-fold symmetry of the histone core to the DNA. The objective was to eliminate the two-fold disorder caused by the indeterminacy of packing of an asymmetric particle into the crystal lattice. A palindrome based on one-half of the primary candidate sequence was constructed and methods were developed to produce the palindrome fragment in large quantities for reconstitution of NCPs. 

The matrix (13.21) is not restricted, however, to collections of particles each of which is congruent with its mirror image it applies equally well to any medium that is invariant under rotation and reflection, which includes the possibility of mirror asymmetric particles each of which is paired with its image in the appropriate orientation. 

Linear polarization peaking in the visible Asymmetric particles aligned in galactic magnetic fields... 

In dark-field microscopy, the particles are only a blur no details are distinguishable at all. Some rough indication of the symmetry of the particles is afforded by the twinkling that accompanies the rotation of asymmetrical particles, but this is a highly subjective observation. However, the technique does permit the rate of particle diffusion to be observed. We see in Chapter 2 how to relate this information to particle size and shape. The number of particles per unit volume may also be determined by direct count once the area and depth of the illuminated field have been calibrated. This is an important technique for the study of coagulation kinetics, a topic we discuss in Chapter 13. 

Asymmetry as well as solvation can cause a friction factor to have a value other than /0. Next let us consider the ratio///, which, according to Equation (14), accounts for the effect of particle asymmetry on the friction factor. We saw in Section 1.5 that ellipsoids of revolution are reasonable models for many asymmetric particles. 

As the force applied to the surface of the gel is increased, however, a point is ultimately reached —the yield value —at which the network begins to break apart and the system begins to flow (curve 1 in Fig. 4.4a). Increasing the rate of shear may result in further deflocculation, in which case the apparent viscosity would decrease further with increased shear. Highly asymmetrical particles can form volume-filling networks at low concentrations and are thus especially well suited to display these phenomena. 

For highly asymmetrical particles, the probability of collision is greater than that predicted for identical particles. This may be understood by noting that the diffusion coefficient is most influenced by the smaller dimensions of the particles (therefore increased), and the target radius is most influenced by the longer dimension (also increased, relative to the case of symmetrical particles) see Equations (24) and (42). 

When the liquid, solution or lyophobic colloidal suspension contains asymmetric particles or when it is too concentrated, other methods must be applied to measure the viscosity. This is for instance the case with clay suspensions. In the past the viscosity of clay suspensions was measured by means of a bucket with a hole in it. The bucket was filled with clay suspension and after the stopper had been removed from the hole, the time required by the volume to drain was measured as a function of e.g. the volume and composition. Later mechanical methods were applied. One of them is based on the principle that a metal cylinder or disc, suspended from a torsion thread, is exposed to a certain resistance when you rotate it in the solution or suspension. Before the measurement the cylinder or disc is turned 360° anti-clockwise and then released. After having revolved over a certain angle, the cylinder or disc will change its direction of rotation. The rotation angle is a measure for the viscosity. 

The frictional coefficient of an asymmetric particle depends on its orientation. At low velocities such particles are in a state of random orientation through accidental disturbances, and the resistance of the liquid to their motion can be expressed in terms of a frictional coefficient averaged over all possible orientations. For particles of equal volume the frictional coefficient increases with increasing asymmetry. This is because, although the resistance of the liquid is reduced when the asymmetric particle is end-on to the direction of flow, it is increased to a greater extent with side-on orientations, so that on average there is an increase in resistance. The frictional coefficient is also increased by particle solvation. 

For a system containing spherical particles, D = RT/6rrr]aNA - i.e. D oc 1/m1, where m is the particle mass. For systems containing asymmetric particles, D is correspondingly smaller (see Table 2.3). Since D = k77/, the ratio D/D0 (where D is the experimental diffusion coefficient and D0 is the diffusion coefficient of a system containing the equivalent unsolvated spheres) is equal to the... 

Asymmetry (asymmetric particles give a flashing effect, owing to different scattering intensities for different orientations). 

In contrast to osmotic pressure, light-scattering measurements become easier as the particle size increases. For spherical particles the upper limit of applicability of the Debye equation is a particle diameter of c. A/20 (i.e. 20-25 nm for A0 600 nm or Awater 450 nm or a relative molecular mass of the order of 10 ). For asymmetric particles this upper limit is lower. However, by modification of the theory, much larger particles can also be studied by light scattering methods. For polydispersed systems a mass-average relative molecular mass is given. 

For most pure liquids and for many solutions and dispersions, t) is a well-defined quantity for a given temperature and pressure which is independent of other solutions and dispersions, especially if concentrated and if the particles are asymmetric and/or aggregated deviations from Newtonian flow are observed. The main causes of non-Newtonian flow are the formation of a structure throughout the system and orientation of asymmetric particles caused by the velocity gradient. 

The capillary method is simple to operate and precise (c. 0.01-0.1 per cent) in its results, but suffers from the disadvantage that the rate of shear varies from zero at the centre of the capillary to a maximum (which decreases throughout the determination) at the wall. Thus, with asymmetric particles a viscosity determination in an Ostwald viscometer could cover various states of orientation and the measured viscosity, although reproducible, would have little theoretical significance. 

Shear-thinning is particularly common to systems containing asymmetric particles. Asymmetric particles disturb the flow lines to a greater extent when they are randomly orientated at low-velocity gradients than when they have been aligned at high-velocity gradients. In addition, particle interaction and solvent immobilisation are favoured when conditions of random orientation prevail. 

For asymmetric particles several other models are available [53,90]. An example is the equation for the settling of a circular disk, settling broadside on ... 

Asymmetric particles, such as ellipsoids or discs, do not generally fall vertically, but tend to drift to the side. Thin, flat, triangular laminae fall edgeways unless equilateral. Few particles possess high symmetry and small local features exert an orienting influence. 

The virial expansion has enjoyed greater appeal, especially as applied to lyotropic systems. Onsager s classic theory rests on analysis of the second virial coefficient for very long rodlike particles. It was the first to show that a solution of hard, asymmetric particles such as long rods should separate into two phases above a threshold concentration that depends on the axial ratio of the particles. One of these phases should be anisotropic (nematic), the other completely isotropic. The former is predicted to be somewhat more concentrated than the latter, but it is the alignment (albeit imperfect) of the solute molecules that is the predominent distinction. The phase separation is a consequence of shape asymmetry alone intervention of intermolecular attractive forces is not required. 

An approach based on the virial expansion suffers from the difficulty of evaluating higher coefficients for highly asymmetric particles and from the non-convergence of the virial series at the concentrations required for formation of a stable nematic phase Lattice methods therefore take precedence over the virial expansion as a basis for quantitative treatment of the liquid crystalline state. 

Lower values were obtained when the presistence length was estimated by the beginning of a sharp kink on theqsp/c vs. c plot (c is the concentration). This kink indicates the overlapping of the rotation spheres of asymmetric particles. The persistence length for PBA was found to be equal to 180-240 A, and for PPTA, 150-180 A %... 

For asymmetric particle populations the log normal distribution is one of the most commonly used distributions. 

The production of birefringence in certain normally isotropic liquids through the action of shearing stress has long been known. Since the work of Freundlich and his collaborators, beginning about 1915. the phenomenon has been recognized as arising from the orientation of asymmetric particles. Only recently, however, has the theory been developed adequately to permit calculation of molecular dimensions from the observed measurements. No attempt will be made to review all the earlier developments here I have discussed some of thein in two other reviews which have appeared in recent years [26), [29). 

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