Multiple Particles

In this section we extend the null-field method to the case of an arbitrary nrnnber of particles by using the translation properties of the vector spherical wave functions. Taking into account the geometric restriction that the particles do not overlap in space, we derive the expression of the transition matrix for restricted values of translations. Our treatment closely follows the original derivation given by Peterson and Strom [187,188]. 

---

Another concern is the effect of multiple particle sizes. In general, the presence of fine particles will affect the requirements of suspen-... 

Hill, P.J. and Ng, K.M., 1996b. Statistics of multiple particle breakage. American Institution of Chemical Engineers Journal, 42, 1600-1608. 

Cornish(128) considered the minimum possible value of the Nusselt number in a multiple particle system. By regarding an individual particle as a source and the remote fluid as a sink, it was shown that values of Nusselt number less than 2 may then be obtained. In a fluidised system, however, the inter-particle fluid is usually regarded as the sink and under these circumstances the theoretical lower limit of 2 for the Nusselt number applies. Zabrodsky 1 291 has also discussed the fallacy of Cornish s argument. 

Understanding the behavior of single particles is, however, a solid foundation upon which to build knowledge of multiple-particle systems. In addition, the literature on single particles is already so extensive that it warrants a book of its own. 

If we have one type of adsorbate the adsorption energy per adsorbate can again be written as eqn. (53) provided that the summation extends not just over pair interactions, but also contains multiple-particle interactions. The coefficients c stand for the number of such interactions per adsorbate. 

A major advantage of the expansion of the adsorption energy in terms of l 5s> pair, and multiple-particle interactions is that the adsorption energy depends linearly on the interactions parameters. This means that a determination of these parameters involves only equations of linear algebra. 

There are various ways in which people have done this. We assume that we model the lateral interactions with pair and multiple-particle interactions as in eqn. (53). This expansion has the advantage that the energy of a system depends linearly on the parameters of our model, and the equations determining these parameters will also be linear and easy to solve. Let s suppose that we have done calculations on N tr different adlayer structures and that we have obtained... 

The emulsion polymerization of vinyl hexanoate has been studied to determine the effect of chain transfer on the polymerization kinetics of a water-insoluble monomer. Both unseeded and seeded runs were made. For unseeded polymerizations, the dependence of particle concentration on soap is much higher than Smith-Ewart predictions, indicating multiple particle formation per radical because of chain transfer. Once the particles have formed, the kinetics are much like those of styrene. The lower water solubility of vinyl hexanoate when compared with styrene apparently negates its increased chain transfer, since the monomer radicals cannot diffuse out of the particles. 

As gene carriers are internalized by endocytosis, the motion characteristics inside the cell resembles the movement of the endosomal compartments within the cell and the formed vesicles are transported along the microtubule network [38]. Suh et al. [41] quantified the transport of individual internalized polyplexes by multiple-particle tracking and showed that the intracellular transport characteristics of polyplexes depend on spatial location and time posttransfection. Within 30 min, polyplexes accumulated around the nucleus. An average of the transport modes over a 22.5 h period after transfection showed that the largest fraction of polyplexes with active transport was found in the peripheral region of the cells whereas polyplexes close to the nucleus were largely diffusive and subdiffusive. Disruption of the microtubule network by nocodazole completely inhibits active transport and also the perinuclear accumulation of polyplexes [37, 40, 47]. 

Suh J, Wirtz D, Hanes J (2004) Real-time intracellular transport of gene nanocarriers studied by multiple particle tracking. Biotechnol Prog 20 598-602... 

Evident progress in studies of liquids has been achieved up to now with the use of computer simulations and of the models based on analytical theory. These methods provide different information and are mutually complementary. The first method employs rather rigorous potential functions and yields usually a chaotic picture of the multiple-particle trajectories but has not been able to give, as far as we know, a satisfactory description of the wideband spectra. The analytical theory is based on a phenomenological consideration (which possibly gives more regular trajectories of the particles than arise in reality ) in terms of a potential well. It can be tractable only if the profile of such a well is rather... 

The particle tracking-based advective control model described in this chapter is capable of solving two- and three-dimensional plume capture problems with multiple candidate wells and multiple particles in confined and unconfined aquifers. The formulation provides a direct approach to solving plume control problems and uses both forward and reverse particle tracking to exploit numerical characteristics of the two reference frames. 

Features of PEPT of particular benefit to engineering studies include the fact that the actual particles of interest may be used as tracers, rather than dissimilar materials of unknown behaviour, and that y-rays are sufficiently penetrating that location is unimpaired by the presence of metal walls, for example. In recent developments, the minimum size of particles which can be tracked has been reduced to approximately 60 pm. It is now possible to track multiple particles, to determine particle rotation and to track motion within real industrial equipment by use of a mobile modular positron camera. These developments are described later. 

The correction for primary coincidence is derived through Poisson probabilities of finding multiple particles in different parts of the sensing zone simultaneously. This yields the following relationship between true count N and observed count n ... 

FIGURE 15 Computer simulation indicates multiple particle formation paths exist in bench scale spray dryer particle formation rate and residence time dependent upon flow streamline. 

Another concern is the effect of multiple particle sizes. In general, the presence of fine particles will affect the requirements of suspension of larger particles. The fine particles act largely as a potential viscosity-increasing agent and give a similar result to what would happen if the viscosity of the continuous phase were increased. 

The need to overcome intracellular barriers may require improved methods of determining rate-limiting steps for specific gene carriers, which will lead to the rational design of new carriers. To this end, our lab uses multiple-particle tracking (MPT) to investigate quantitatively the motion of nanometersized DNA delivery vehicles [170],... 

Steps taken to understand the intracellular barriers to gene delivery have led to the rational modification and improvement of carriers. Quantitative methods, such as multiple-particle tracking, to assess to the intracellular transport of gene carriers promise to add valuable insight that may ultimately lead to nonviral carriers rivaling the efficiencies seen in viral systems. 

---