Particle expansion

Figure 2. Effect of equilibration time on carboxylic particle expansion at pH = 12.5 as determined by photon correlation spectroscopy. The confidence interval on d is at the 90% probability level. The initial particle size is given by d0 (acrylic latex, 2% AA(ll), d0 = 1120 A).

A quantitative comparison of particle expansion determined by the three methods is given in Table I. The particle diamete of the standard acrylic latex was determined by PCS to be 1120 A. This value was used in the calculation of the increase in particle radius at maximum expansion in each case. The sedimentation method yielded the largest increase in radius, 302 A, followed by the viscometric value of 240 K. Possibly the shear involved in the latter method resulted in a partial collapse of the surface layer. The value determined by PCS was found to be approximately half that determined by sedimentation. Since the PCS determination is presumed to be free of particle interactions at a concentration of 5 X 10 4%, we must conclude that the other two methods (at 1% solids) exhibit such interactions. As a result, the charged particles settle slower (19) and yield a higher viscosity than in the absence of these (repulsive) interactions. 

In addition, maximum expansion occurred at pH = 12.5 in the PCS experiments compared with approximately 10.5 observed in the sedimentation and viscometry experiments. Since PCS is carried out at much lower particle concentrations, interactions between the charged particles at the higher concentrations are probably involved. Similar comparisons with non-expanding carboxylic latex particles were carried out in an effort to separate interparticle charge effects from true particle expansion in interpreting apparent particle sizes determined by hydrodynamic methods. 

Sedimentation studies of the same latex showed a pronounced increase in hydrodynamic size with pH so that particle expansion must occur by a layer of very low density. 

The bottle test results in Figure 5.12 show that the onset and rate of particle expansion represented by the solution viscosity changes depended on temperature and time. Figure 5.13 shows the change in resistance factor (RF) with the test time for a 2250 ppm BrightWater solution in a 3.5 darcy silica sand at 85°C (185°F). 

A simple way to implement n-particle space truncation is to use the uncorrelated wave function (which as noted above is a very substantial fraction of the exact wave function) to classify terms in the n-particle space. If we consider the Hartree-Fock determinant, for example, we can construct all CSFs in the full n-particle space by successively exciting one, two,.., electrons from the occupied Hartree-Fock MOs to unoccupied MOs. For cases in which a multiconfigurational zeroth-order wave function is required, the same formal classification can be applied. Since only singly and doubly excited CSFs can interact with the zeroth-order wave func tion via the Hamiltonian in Eq. (1), it is natural to truncate the n-particle expansion at this level, at least as a first approximation. We thus obtain single and double excitations from Hartree-Fock (denoted SDCI) or its multiconfigurational reference analog, multireference Cl (MRCI). 

Particle expansion. This is effective for strongly absorbing particles on transparent substrates. 

Figure 4. Dependences of the extinction A of gold bispheres in water on the number of multipoles included in the single-particle expansions. Calculations by the T-matrix method for randomly oriented bispheres with touching component spheres (hd = 0 ) and those separated by distances hd = 0.5 nm and hd = 1 nm. Particle diameter equals d = 5 nm the wavelength in vacuum equals A = 515 nm (a) and 550 nm (b).

For systems with four or more particles, expansions of die correlation factors are generally employed. The expansion most frequently used is that of Sack [10] ... 

Fig. 23 Propylene polymerization with a silica-supported metallocene/MAO catalyst, (a) Plot of polymerization rate against time. Electron microscope images of particles at stages of (b) prepolymerization, (c, d) particle fragmentation, and (e) particle expansion. See text fm details...

It could also be demonstrated that a very rapid passing of these stages occurs when the polymerization is carried out in liquid propylene [45 7] this means that the industrially important bulk polymerization can also be exactly described by this polymer growth and particle expansion model [41-43, 48]. 

Figure 3.14 Schematic polymer growth and particle expansion from experimental analysis (reproduced from Ref. [60]).

The mechanism of particle expansion in spray drying of foods. AICHE Symp. Ser. Food Process Engineering 78(218) ... 

Although we may defer the introduction of double-group symmetry by performing the correlated calculations first, we still need to take the symmetry of the spin-orbit operator into account in the selection of the -particle expansion space. 

Bassett, B.R., and Hoy, K.L., The Expansion Characteristics of Carboxylic Emulsion Polymers. I. Particle Expansion Bet-ermined by Sedimentation in Polymer Colloids II. 

Figure 11. Characteristic expansion behavior for a soft acrylic latex containing 1% acrylic acid, as measured by sedimentation technique, showing effect of electrolyte on particle expansion. (Bassett and Hoy, Ref. 32)...

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