Ensemble techniques

Two ensemble techniques, both based on light scattering (LS), are often used to characterize polymer emulsions. The first, usually referred to as laser diffraction (LD), is based on classical, or static (i.e. time averaged) LS. The second, usually referred to as dynamic light scattering (DLS), is based on fluctuations in the scattered light intensity due to Brownian motion of the suspended particles. While both of these techniques yield information on the polydispersity of the sample (i.e. the range of particle sizes contained therein), they are 

According to classical (Mie) LS theory, both the magnitude and angular dependence of the intensity of light scattered by particles depend on their size. If the particles are very small compared to the laser wavelength (in the liquid), X, there is no appreciable angular dependence. If the particles grow to approximately of A., there is measurable enhancement 

The angular pattern of scattering also depends on the shape of the particles. However, in the simplifying case of uniform (isotropic), nonadsorbing and noninteracting particles. 

Large particles larger than the wavelength of light 

Rayleigh region. The ratio of the intensity scattered at angle 0 to that of the incident beam, I/Iq, can be described as a sum of vertically and horizontally polarized intensity ratios (AUen, 1968), 

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Panagiotopoulos A Z 1995 Gibbs ensemble techniques Observation, Prediction and Simuiation of Phase Transitions in Oompiex Fiuids ed M Baus, L F Rull and J-P Ryckaert, vol 460 NATO ASi Series O (Dordrecht Kluwer) pp 463-501... 

Although the assumption that all apatite platelets are of equal thickness may not be well justified, it provides a useful approximation to the real situation. In fact, such assumption is inevitable for an ensemble technique like NMR. It is anticipated that HARDSHIP may be well suited to determine the relative thickness of apatite crystallites in different bone or dentin samples. We note in passing that C-REDOR, which is a hetero-nuclear recoupling technique with active suppression of homonuclear dipolar interaction,42 43 can be used to probe for the size of nanoparticles embedded in polymer matrix,44 provided that the nanoparticles do not... 

Ensemble techniques that process information produced simultaneously by many particles and require the use of complex mathematical algorithms to invert the data... 

If a fiVT ensemble simulation can be turned into a ( quasi ) NPT ensemble-type simulation (e.g., a pseudo- FT ensemble), the inverse transformation (a pseudo-NPT ensemble) is also possible. The key relationship for a pseudo-NPT ensemble technique is Eq. (5.1) [78]. Such a reverse strategy can be practical only if molecular insertion and deletion moves can be performed efficiently for the system under study (e.g., by expanded ensemble moves for polymeric fluids). Replacing volume moves by particle insertions can be advantageous for polymeric and other materials that require simulation of a large system (due to the sluggishness of volume moves for mechanical equilibration of the system) such an advantage has been clearly demonstrated for a test system of dense, athermal chains [78]. 

Fractionating (ensemble) techniques include the two steps of fractionation and detection. The former can either result in a physical separation of the different size classes or in the depletion of coarse or fine particles in the measurement zone. In the case of colloidal suspensions, the fractionating effect is usually related to the mobility of the particles (e.g. settling velocity). The detection system monitors the fractionation process and, thus, serves for evaluating the class frequencies. It frequently employs the phase shift, extinction, or scattering of some radiation (e.g. X-rays). The applied detection system determines the type of quantity in which the size fractions are intrinsically weighted (e.g. extinction of X-rays is mass proportional Q. ... 

The immediate result of a spectroscopic ensemble technique is a signal spec-tmm, i.e. the variation of the measured signal g over the spectral parameter (time, space or frequency). Each size fraction v possesses a characteristic spectrum k/s ), which in general covers the whole spectral range. Assuming that each size fraction contributes independently and linearly to the measured signal spectmm, the determination of the size distribution requires the inversion of a linear integral equation (Fredholm type) ... 

As usual, a problem arises when the Gibbs ensemble technique is applied to a system consisting of pol5mieric molecules. In particular, when a chain molecule is removed from one box and an attempt is made to insert it into the other box (the particle swap move described above), the new configuration almost always... 

An exciting development, finally, is the real-space and real-time exploration of physical and chemical phenomena such as molecular dynamics, elecbonics, and reactivity. Using STM, information of dynamic processes is obtained at the level of single molecules and in some cases new aspects of reaction mechanisms are revealed which remain unexplored with conventional ensemble techniques. 

Within each interval, we use an expanded ensemble technique to explore the sampling points and configurations between neighboring intervals are swapped via replica-exchange Monte Carlo moves [118]. The partition fimction, takes the form ... 

The Gibbs-ensemble technique was introduced by Panagio-topoulos as an efficient tool to simulate vapor-liquid phase equilibria. In the Gibbs-ensemble scheme simulations of the liquid and vapor phases are carried out in parallel. Monte Carlo rules which allow for changes in the number of particles and the volume ensure that the two boxes are in thermodynamic... 

Siepmann et al. have used the combination of the Gibbs-ensemble technique and configurational-bias Monte-Carlo to simulate vapor-liquid equilibria of the n-alkanes at conditions where experiments are not (yet) feasible. Phase diagrams are very sensitive to the choice of interaction potentials. Most available models for alkanes have been obtained by fitting simulation data to experimental properties of the liquid under standard conditions. In Figure 6 the vapor- liquid curve of octane as predicted by some of these models is compared with experimental data. This figures shows that the models... 

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