Dynamic structure factor particles

Figure 18. (a) Response versus the dynamical structure factor for the binary mixture Lennard-Jones particles system in a quench from the initial temperature Ti = 0.8 to a final temperature T( = 0.25 and two waiting times t = 1024 (square) and = 16384 (circle). Dashed lines have slope l/Tf while thick hues have slope l/T (t ). (From Ref. 182.) (b) Integrated response function as a function of IS correlation, that is the correlation between different IS configurations for the ROM. The dashed fine has slope Tf = 5.0, where Tf is the final quench temperature, whereas the full lines are the prediction from Eq. (205) andF = F (T ) Teff(2") 0.694, Teff(2 ) 0.634, and 7 eff(2 ) 0.608. The dot-dash line is for t , = 2" drawn for comparison. (From Ref. 178.)... 

The function S(Q) in the above equation is known as the (interparticle) structure factor or static structure factor (the label static is used to differentiate S(Q) from its time-dependent version, known as the dynamic structure factor) and contains information on how the particles are spatially distributed in the dispersion. When there are no interparticle effects, S(Q) becomes unity and we recover Equation (60) from Equation (79). 

In the last chapter, equations were derived for the particle-scattering factor, the mean-square radius of gyration, the diffusion coefficient and the first cumulant of the dynamic structure factor. All these have the common feature that, for homopolymers at least, they can be written in the following form ... 

Calculational procedure of all the dynamic variables appearing in the above expressions—namely, the dynamic structure factor F(q,t) and its inertial part, Fo(q,t), and the self-dynamic structure factor Fs(q,t) and its inertial part, Fq (q, t) —is similar to that in three-dimensional systems, simply because the expressions for these quantities remains the same except for the terms that include the dimensionality. Cv(t) is calculated so that it is fully consistent with the frequency-dependent friction. In order to calculate either VACF or diffusion coefficient, we need the two-particle direct correlation function, c(x), and the radial distribution function, g(x). Here x denotes the separation between the centers of two LJ rods. In order to make the calculations robust, we have used the g(x) obtained from simulations. 

The energy loss function is related to the dynamical structural factor S(q, ft)),90 which describes the scattering of particles in a liquid, namely,... 

As particle concentration increases, particle interactions and multiple scattering invalidate Eq. (33). The cross terms (y /) in the static and dynamic structure factors. Eq. (29), no longer cancel out, and thus they lead to more complex relationships [l 15-119] for (l>(diffusive motion of interacting particles also becomes more complex, depending on colloidal and hydrodynamic interactions among the particles and their spatial configurations. DLS measurements of particle motion can provide information about suspension microstructure and particle interactions. 

The dynamic structure factor of two such PMMA-grafted SiO2 particles... 

By definition, the dynamic structure factor accounts for the correlations between the positions of the particles at different moments of time ... 

Without doing detailed quantitative analysis of the data, it can be stated that the polyion diffusion can be qualitatively described by two theoretical concepts. The first concept capable of qualitative explanation of the polyion diffusion is the concept based on considering polyions as interacting Brownian particles with direct interactions between polyions and hydrodynamic interactions. The short-time collective diffusion coefficient for a system of interacting Brownian particles treated by statistical mechanics is calculated from the first cumulant F of the dynamic structure factor S(q, t) as [15-17]... 

Elementary excitations also include single particle diffusive excitations beside quantized vibrations (i.e., molecular vibrations and vibrations of the crystal as a whole associated with phonons/magnons). Consider the incoherent dynamic structure factor 5snc(Q,(o), which is the Fourier transform pair of the time-dependent self-correlation function, compare... 

The experimentally measured quantity in a scattering experiment is the dynamic structure factor S(q, co), describing the probability for the photon to acquire a momentum transfer q = k — koandan energy transfer o) = )o — o). In 1954, Van Hove demonstrated that the dynamic structure factor, known as the scattering function, relates to the probability G (r, t) for finding any particle at position r and time t when the particle was at r = 0 and t = 0 before by a spatial and temporal Fourier transforms [89] ... 

Dynamic Structure Factor of a Diffusing Particle More often than not, the particles move according to the diffusion equation. The transition probability is given by Eq. 3.30. Then... 

So, one can consider the parameters r, Xi and Me to be equivalent. One of these parameters is introduced in each theory of polymer dynamics. Note that the correlation time is expressed through the two-particle correlation function and the dynamic structure factor of the system of the interacting Brownian particles in many-chain theories [54, 91]. 

Figure 4 is a comparison of the behaviour of these functions, along with W( ) for a particle moving through a typical liquid. Fourier transforming the Fs(Q, )s gives the corresponding forms for the (self) dynamic structure factors ... 

The time dependence of S(q,t) is thus entirely determined by a two-particle, two-time correlation function, the field correlation function (or dynamic structure factor) g q, t). Up to a normalization (which is of no consequence in this calculation because normalizations cannot affect the physical time dependence)... 

In order t account not only spatial but also temporary structure particle order correlation, in principle, it is possible to use the dynamic structural factor S (k,o), determined by the neutron scattering method. 

The decay of the structural correlations measured by the static structure factor can be studied by dynamic scattering techniques. From the simulations, the decay of structural correlations is determined most directly by calculating the coherent intermediate scattering function, which differs from Eq. [1] by a time shift in one of the particle positions as defined in Eq. [2] ... 

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