Scattering function intermediate

From the theory of neutron scattering [62], S JQ, CO) may be written as the Fourier transfonn of a time correlation function, the intermediate scattering function, hJQ, t) ... 

Having demonstrated that our simulation reproduces the neutron data reasonably well, we may critically evaluate the models used to interpret the data. For the models to be analytically tractable, it is generally assumed that the center-of-mass and internal motions are decoupled so that the total intermediate scattering function can be written as a product of the expression for the center-of-mass motion and that for the internal motions. We have confirmed the validity of the decoupling assumption over a wide range of Q (data not shown). In the next two sections we take a closer look at our simulation to see to what extent the dynamics is consistent with models used to describe the dynamics. We discuss the motion of the center of mass in the next section and the internal dynamics of the hydrocarbon chains in Section IV.F. 

Figure 4. Intermediate scattering function ( c(t F(k,t) and dynamic structure factor (right), S(k,(o), computed from MCY with and without three-body corrections.

The bond fluctuation model not only provides a good description of the diffusion of polymer chains as a whole, but also the internal dynamics of chains on length scales in between the coil size and the length of effective bonds. This is seen from an analysis of the normalized intermediate coherent scattering function S(q,t)/S(q,0) of single chains ... 

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] ... 

The Fourier transform of this quantity, the dynamic structure factor S(q, ffi), is measured directly by experiment. The structural relaxation time, or a-relaxation time, of a liquid is generally defined as the time required for the intermediate coherent scattering function at the momentum transfer of the amorphous halo to decay to about 30% i.e., S( ah,xa) = 0.3. 

Figure 10 Intermediate incoherent scattering function for the bead-spring model at T = 0.48 for different values of momentum transfer given in the legend.

Figure 11 MCT P-scaling for the amplitudes of the von Schweidler laws fitting the plateau decay in the incoherent intermediate scattering function for a R value smaller than the position of the amorphous halo, q = 3.0, at the amorphous halo, q = 6.9, and at the first minimum, q = 9.5. Also shown with filled squares is the P time scale. All quantities are taken to the inverse power of their predicted temperature dependence such that linear laws intersecting the abscissa at Tc should result. 

Figure 12 Test of the factorization theorem of MCT for the intermediate coherent scattering function for the bead-spring model and a range of -values indicated in the Figure. Data taken from Ref. 132 with permission.

In the discussion on the dynamics in the bead-spring model, we have observed that the position of the amorphous halo marks the relevant local length scale in the melt structure, and it is also central to the MCT treatment of the dynamics. The structural relaxation time in the super-cooled melt is best defined as the time it takes density correlations of this wave number (i.e., the coherent intermediate scattering function) to decay. In simulations one typically uses the time it takes S(q, t) to decay to a value of 0.3 (or 0.1 for larger (/-values). The temperature dependence of this relaxation time scale, which is shown in Figure 20, provides us with a first assessment of the glass transition... 

Figure 20 Temperature dependence of the a-relaxation time scale for PB. The time is defined as the time it takes for the incoherent (circles) or coherent (squares) intermediate scattering function at a momentum transfer given by the position of the amorphous halo (q — 1.4A-1) to decay to a value of 0.3. The full line is a fit using a VF law with the Vogel-Fulcher temperature T0 fixed to a value obtained from the temperature dependence of the dielectric a relaxation in PB. The dashed line is a superposition of two Arrhenius laws (see text).

Figure 21 Coherent intermediate scattering functions at the position of the amorphous halo versus time scaled by the a time, which is the time it takes the scattering function to decay by 70%. The thick gray line shows that the a-process can be fitted with a Kohlrausch-Williams-Watts (KWW) law.

Figure 22 von Schweidler fits (dotted lines) to the plateau decay of the coherent intermediate scattering function in the temperature interval 198-253 K. 

To proceed further we introduce the intermediate scattering function as the Fourier transform of S(Q,< ) ... 

The intermediate scattering function directly depends on the (time-dependent) atomic positions ... 

Fig. 4.1 a Typical time evolution of a given correlation function in a glass-forming system for different temperatures (T >T2>...>T ), b Molecular dynamics simulation results [105] for the time decay of different correlation functions in polyisoprene at 363 K normalized dynamic structure factor at the first static structure factor maximum solid thick line)y intermediate incoherent scattering function of the hydrogens solid thin line), dipole-dipole correlation function dashed line) and second order orientational correlation function of three different C-H bonds measurable by NMR dashed-dotted lines)... 

In such a Gaussian case the intermediate scattering function is entirely determined by the mean squared displacement of the atom (r (t)) ... 

In the heterogeneous scenario, and in parallel with the procedure followed for developing Eq. 4.16, we may consider a distribution of local diffusivities associated with the different regions of the sample. The resulting intermediate scattering function can be expressed as ... 

Fig. 4.15 Momentum transfer (Q)-dependence of the characteristic time r(Q) of the a-relaxation obtained from the slow decay of the incoherent intermediate scattering function of the main chain protons in PI (O) (MD-simulations). The solid lines through the points show the Q-dependencies of z(Q) indicated. The estimated error bars are shown for two Q-values. The Q-dependence of the value of the non-Gaussian parameter at r(Q) is also included (filled triangle) as well as the static structure factor S(Q) on the linear scale in arbitrary units. The horizontal shadowed area marks the range of the characteristic times t mr- The values of the structural relaxation time and are indicated by the dashed-dotted and dotted lines, respectively (see the text for the definitions of the timescales). The temperature is 363 K in all cases. (Reprinted with permission from [105]. Copyright 2002 The American Physical Society)...

Fig. 6.18 Normalized intermediate scattering function from centre-labelled 18-arm PI solutions. Collective corresponds to a 4.85% solution of labelled stars, whereas the self data stem from a solution of 1% labelled and 16.6% non-labelled stars. Note the maximum Fourier time of 350 ns (A=1.9 nm), which was obtained at the INI 5 in the case of these strong scattering samples. (Reprinted with permission from [304]. Copyright 2002 Springer)...

The experiments on alkali iodides, PEOx-Nal or PEOx-Lil [316-318] were performed on PEO chains of 23 or 182 (-CH2-CH2-O-) monomers and Orion ratios between 15 and 50. The incoherent scattering from protonated polymers was measured using INI 1C, which yields the intermediate scattering function of the self-correlation. The experiments were performed in the homogeneous liquid phase where the added salt is completely dissolved and no crystalline aggregates coexist with the solution, i.e. at temperatures around 70 °C. 

A fully deuterated powder of the photosynthetic protein C-phycocyanin (CPC) has been investigated by NSE in a similar (Q,t)-range [335]. Here the coherent scattering conveying the pair correlation Spair(Q,t)/SpaiAQ) in terms of the intermediate scattering function was obtained and compared to molecular dynamics simulations. As illustrated in Fig. 6.31 the salient features of the NSE data were quantitatively and qualitatively reproduced by the MD results [348]. 

Fig. 6.31 Normalised intermediate scattering function from C-phycocyanin (CPC) obtained by spin-echo [335] compared to a full MD simulation (solid line) exhibiting a good quantitative matching. In contrast the MD results from simplified treatments as from protein without solvent (long dash-short dash /me), with point-like residues (Cpt-atoms) (dashed line) or coarse grained harmonic model (dash-dotted line) show similar slopes but deviate in particular in terms of the amplitude of initial decay. The latter deviation are (partly) explained by the employed technique of Fourier transformation. (Reprinted with permission from [348]. Copyright 2002 Elsevier)...

The intermediate scattering functions can be expressed in terms of the normalized properties Ulk and U2k ... 

Fig. 43. Intermediate scattering functions for the C.M. motion of a CO molecule from the modified Stockmayer simulation.

The intermediate scattering function for an isotropic system is given by... 

Figure 2. A pictorial representation of the mode coupling theory scheme for the calculation of the time-dependent friction (f) on a tagged molecule at time t. The rest of the notation is as follows Fs(q,t), self-scattering function F(q,t), intermediate scattering function D, self-diffusion coefficient t]s(t), time-dependnet shear viscosity Cu(q,t), longitudinal current correlation function C q,t), longitudinal current correlation functioa...

In calculating the contribution from the density mode given by Eq. (132), the required solvent dynamical variables are F(q, t), the intermediate scattering function, and F0(q, t), the short-time part of the intermediate scattering function. 

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