Different size populations

Colloidal crystals . At the end of Section 2.1.4, there is a brief account of regular, crystal-like structures formed spontaneously by two differently sized populations of hard (polymeric) spheres, typically near 0.5 nm in diameter, depositing out of a colloidal solution. Binary superlattices of composition AB2 and ABn are found. Experiment has allowed phase diagrams to be constructed, showing the crystal structures formed for a fixed radius ratio of the two populations but for variable volume fractions in solution of the two populations, and a computer simulation (Eldridge et al. 1995) has been used to examine how nearly theory and experiment match up. The agreement is not bad, but there are some unexpected differences from which lessons were learned. 

A method and Worksheet for comparing correlation coefficients for different size populations at user selected confidence levels. 

Testing Correlation for Different Size Populations Are two correlation coefficients (rl and r2 different based on a difference in the number of observations for each (N) ... 

Figure 3.4 shows the experimental results, using laser techniques, to follow the evolution of floe sizes over time. The primary data has been evaluated, using image analysis techniques, and the result is also shown in the figure as the evolution of the number concentration of floes of different sizes ( population ) over time. It can be seen that the number of small floes decrease, and larger floes form, over time. 

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

The energy laws of Bond, Kick, and Rittinger relate to grinding from some average feed size to some product size but do not take into account the behavior of different sizes of particles in the mill. Computer simulation, based on population-balance models [Bass, Z. Angew. Math. Phys., 5(4), 283 (1954)], traces the breakage of each size of particle as a function of grinding time. Furthermore, the simu-... 

Because there has been no recent analysis of the relationship between the concentration of total suspended particulate matter in the air of cities with populations of different sizes, we are forced to use data for the decade... 

Lest I leave the erroneous impression here that colloid science, in spite of the impossibility of defining it, is not a vigorous branch of research, I shall conclude by explaining that in the last few years, an entire subspeciality has sprung up around the topic of colloidal (pseudo-) crystals. These are regular arrays that are formed when a suspension (sol) of polymeric (e.g., latex) spheres around half a micrometre in diameter is allowed to settle out under gravity. The suspension can include spheres of one size only, or there may be two populations of different sizes, and the radius ratio as well as the quantity proportions of the two sizes are both controllable variables. Crystals such as AB2, AB4 and AB13 can form (Bartlett et al. 1992, Bartlett and van... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

Biocides are added to the wet end process to prevent slime formation. Introduction of neutral or alkaline sizing instead of acidic papermaking, the closed water circuits, and the increasing proportions of recycled paper have required changes in biocide types in order to control different microbial populations. 

Factors such as assay variations, age, and prostate gland size are known to affect cutoff values. Also, free to total PSA cutoffs are influenced by the sensitivity and specificity values chosen, the reflex range for total PSA used, differences in free PSA assays, and differences in populations studied. Different PSA values are considered due to differences in cutoffs in different assays. Studies have shown that the comparison of a chemiluminescent free PSA showed a 25 percent difference in values. These types of variations suggest a need for standardization (9,29). 

In general, different sized particles may have different cycle time distributions and different mass deposition distributions in the spray zone. One approach would be to use small discrete size distribution increments and then to apply Eq. (12) to each size fraction. Inherent in this approach is the assumption that each particle size fraction acts independently. This assumption may not be valid, especially if different particles take different circulation paths within the bed. From the population balance approach, Randolf and Larson (1988) have suggested the use of an effective growth diffusivity coefficient to account for random fluctuations in growth rate. Thus Eq. (6) would be modified to give ... 

In the case of polymer blends, the fractionated crystallization phenomenon that has been widely reported for many polymer systems can not be attributed to simple size effects. For instance, in Fig. 1, one could argue that the different exotherms originated in the crystallization of different droplet populations that have diverse average diameters. This cannot be the case, since the droplet distribution is monomodal and a smooth variation in heat... 

The uptake of oleate by pre-added vesicles, and in particular the matrix effect, permits regulation of the growth of the size and the number of particles, and in this way it is possible to tackle a series of novel questions. One such question is is there a difference in the rate of uptake of fresh surfactant between two vesicle populations of different sizes ... 

Chen, Z., Schols, H. A., Voragen, A. J. G. (2004). Differently sized granules from acetylated potato and sweet potato starches differ in the acetyl substitution pattern of their amylose populations. Carbohydr. Polym., 56, 219-226. 

Three doubly spin-labelled [2]catenanes with different sizes were studied by 4-pulse DEER.52 The experimental distribution of interspin distances was compared with a theoretical pair-correlation function computed based on geometrical constraints. In chloroform solution the medium and large catenanes were close to fully expanded, but in glassy o-terphenyl they were partially collapsed. For the smaller catenane there was a higher population of shorter interspin distances, which was attributed to interactions between unsaturated sections of the molecule. 

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