Structure factor static

In a previous paper the Car-Parrinello (CP) technique was applied to the equimolar NaSn alloy [6]. In a further publication [7] we extended these investigations to a wide range of compositions ranging from 20% up to 80% of sodium and discussed the static structure factors and the behaviour of the Zintl anions (Sn ) in the molten alloys. 

The obtained static structure factors agree well with the experimental ones [4], all trends of the peak positions are reproduced correctly. There are only small deviations from the experiments (i) due to the pseudopotential (slighly too small bond lengths which correspond to slightly too large peak positions in the reciprocal lattice) and (ii) correct positions but a wrong trend in the heights of the prepeaks. For a detailed description see Ref. [7]. 

Thus, the calculation of 2(Q) requires the knowledge of the partial static structure factors SaP(Q, t) and the elements PaP(Q) of the mobility matrix p(Q), which itself depend on SaP(Q,0). 

Equations (125) and (126) explicitly show that in the initial slope approximation the elements of the generalized mobility matrix can be expressed only in terms of integrals over the corresponding partial static structure factor. Both equations are valid as long as one assumes a Gaussian distance distribution of the distances r between the monomers i on arm a and monomers j on arm p. 

Fig. 47a, b. Structure and dynamics of star-shaped polymers with different functionalities, a Kratky plot of the static structure factor (S(Q, 0) Q2 vs. Q Rg. b Q(Q)/Q3 vs. Q Rg, as derived from Eqs (94) and (123), assuming Rouse dynamics... 

The salient feature of the experimental results is the observation of a pronounced minimum in the Q(Q)/Q3 vs. z plot. It occurs at the same position, where the static structure factor in its Kratky representation exhibits its maximum. Furthermore, the reduced line width scales with the scaling variable z in the same way that the static structure factor does. Thus, the occurrence of the minimum is directly related to peculiarities of the star architecture. 

At higher Q, however, where the static structure factor reveals the asymptotic power law behavior S (Q, 0) Q 1/v, the assumption of ideal conformation clearly fails. In particular, this is evident for the core (sample 1) and shell contrast conditions (sample 2). 

Recently the wall-PRISM theory has been used to investigate the forces between hydrophobic surfaces immersed in polyelectrolyte solutions [98], Polyelectrolyte solutions display strong peaks at low wavevectors in the static structure factor, which is a manifestation of liquid-like order on long lengths-cales. Consequently, the force between surfaces confining polyelectrolyte solutions is an oscillatory function of their separation. The wall-PRISM theory predicts oscillatory forces in salt-free solutions with a period of oscillation that scales with concentration as p 1/3 and p 1/2 in dilute and semidilute solutions, respectively. This behavior is explained in terms of liquid-like ordering in the bulk solution which results in liquid-like layering when the solution is confined between surfaces. In the presence of added salt the theory predicts the possibility of a predominantly attractive force under some conditions. These predictions are in accord with available experiments [99,100]. 

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] ... 

For a polyelectrolyte chain that has non-Gaussian statistics, exact analytical expression for B is not feasible. To get some insight, we notice that the static structure factor has the limiting behavior. 

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)... 

Fig. 4.2 Static structure factor measured on fully deuterated 1,4-polybutadiene [123] at tbe temperatures indicated...

Fig. 4.3 Scaling representation of the spin-echo data at the first static structure factor peak Qmax- Different symbols correspond to different temperatures. Solid line is a KWW description (Eq. 4.8) of the master curve for 1,4-polybutadiene at Qmax=l-48 A L The scale r(T) is taken from a macroscopic viscosity measurement [130]. Inset Temperature dependence of the non-ergodicity parameter/(Q) near the lines through the points correspond to the MCT predictions (Eq. 4.37) (Reprinted with permission from [124]. Copyright 1988 The American Physical Society)...

Table 4.1 Parameters related to the structural relaxation for the polymers investigated by NSE glass transition temperature Tg, position of the first static structure factor peak Qmax> shape parameter magnitude considered to perform the scaling of the NSE data, and temperature dependence of the structural relaxation time...

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. 4.27 Q-dependence of the amplitude of the relative quasi-elastic contribution of the -process to the coherent scattering function S (Q)/S(Q) obtained for PB from the hopping model solid line) with dp=l.5 A. The static structure factor S(Q) at 160 K [123] is shown for comparison dashed-dotted line)...

Fig. 4.28 a Form factor associated to the ds-unit of PB, which is schematically represented in the inset, b and c show the Q-dependence of the amplitude of the relative quasi-elastic contribution of the j -process to the coherent scattering function obtained for rotations of the ds-unit around an axis through the centre of mass of the unit and through the main chain, respectively, for different angles 30° (empty diamond), 60° (filled diamond), 90° (empty triangle) and 120° (filled triangle). The static structure factor S(Q) at 160 K [123] is shown for comparison (dashed-dotted line) (Reprinted with permission from [133]. Copyright 1996 The American Physical Society)... 

Fig. 5.15 Momentum transfer dependence of the amplitude of the KWW functions describing the dynamic structure factor. For 390 K all the values obtained are shown (filled triangle)y while the dashed and solid lines represent the smoothed behaviour for 335 K and 365 K, respectively. The static structure factor is shown in arbitrary units for comparison (cross). (Reprinted with permission from [147]. Copyright 2002 The American Physical Society)...

Fig. 5.20 Temporal evolution of the dynamic structure factor of PVE at the first maximum of the static structure factor at 418 K. The solid line represents a KWW fit (p=0.5). Insert S(Q) of PVE measured at 320 K. (Reprinted with permission from [39]. Copyright 2004 EDP Sciences)...

This relation also holds between x°(Q) and (Q). From Eq. 6.9 the basic result of RPA for the static structure factor matrix immediately follows ... 

For both polymer systems the static structure factors were investigated using small angle neutron scattering and the results interpreted in terms of RPA theory. Figure 6.6 displays the temperature-dependent static structure factor obtained from a PE-PEE melt (sample IV). 

Fig. 6.6 Variation of the static structure factor S(Q) measured on hPE-dPEE diblock copolymer chains (sample IV) as a function of the wave number Q. Temperature closed star 393 K, closed circles 403 K, closed square 413 K, inverted triangle 423 K, closed star 433 K, open triangle 433 K, open circle 453 K, open square 463 K. Solid lines represent the fit with a two-component static RPA approach (Eq. 6.12). (Reprinted with permission from [44]. Copyright 1999 American Institute of Physics)...

In the low Q-regime RPA describes well the static structure factor for the short chain melt, where the ODT is sufficiently far away (kN 7). In the dynamics we would expect the diblock breathing mode to take over around QRg 2 (Q=0.04 A ). Instead, deviations from Rouse dynamics are already observed at Q values as high as QR =5. At QJ g=3 a crossover to a virtually Q-independent relaxation rate about four to five times faster than the predicted breathing mode is found. This phenomenon is only visible under h-d labelUng. Under single chain contrast (see below) these deviations from RPA are not seen. Thus, the observed fast relaxation mode must be associated with the block contrast. 

The anomalous static structure factor of the aggregates associated with a relatively small size polydispersity could produce the observed behaviour of /2K. The latter parameter is given by ... 

---