Composition fluctuation factor

The partial structure factors for binary (Bhatia and Thorton, 1970) and multicomponent (Bhatia and Ratti, 1977) liquids have been expressed in terms of fluctuation correlation factors, which at zero wave number are related to the thermodynamic properties. An associated solution model in the limits of nearly complete association or nearly complete dissociation has been used to illustrate the composition dependence of the composition-fluctuation factor at zero wave number, Scc(0). For a binary liquid this is inversely proportional to the second derivative of the Gibbs energy of mixing with respect to atom fraction. 

In the remainder of this section it is desired to obtain the relative, constant-pressure heat capacity of the liquid at x=j and the concentration fluctuation factor for all compositions. Since the latter equation is complicated, it is not written out in full here. This has been done in Eqs. (37)-(45) of the paper by Liao et al. (1982) for the special case that 14 = 34 = 0 and / l3 is the only nonzero cubic interaction term, i.e., the version of the model applied here to the Ga-Sb and In-Sb binaries. Bhatia and Hargrove (1974) have given equations for the composition fluctuation factor at zero wave number for the special cases of complete association or dissociation and only quadratic interaction coefficients. 

The composition fluctuation factor of Eq. (95) is shown in Fig. 6 as a function of the atomic fraction of Sb in the liquid at 525°C. The minimum is due primarily to the assumed presence of a InSb species whose mole fraction is... 

Fig. 6. Composition fluctuation factor at zero wave number for In-Sb melt at 525 C.

The calculated composition fluctuation factor at zero wave number for the liquid at 709.2°C is shown in Fig. 10. For comparison the uppermost curve shows the same quantity for an ideal solution of species Ga and Sb,... 

Fig. 10. Composition fluctuation factor at zero wave number for Ga-Sb melt at 709.2°C. Curve 1 is for the parameters established here, whereas curve 2 is for a completely dissociated, ideal solution and curve 3 for a completely associated ideal solution. 

Fig. 18. Calculated composition fluctuation factor at zero wave number in the liquid phase versus atom fraction . The highest curve at 0.28 atom fraction is for Cd-Te at 1092°C. The other curve is for Hg-Te at 670°C.

Detector sensitivity is one of the most important properties of the detector. The problem is to distinguish between the actual component and artifact caused by the pressure fluctuation, bubble, compositional fluctuation, etc. If the peaks are fairly large, one has no problem in distinguishing them however, the smaller the peaks, the more important that the baseline be smooth, free of noise and drift. Baseline noise is the short time variation of the baseline from a straight line. Noise is normally measured "peak-to-peak" i.e., the distance from the top of one such small peak to the bottom of the next. Noise is the factor which limits detector sensitivity. In trace analysis, the operator must be able to distinguish between noise spikes and component peaks. For qualitative purposes, signal/noise ratio is limited by 3. For quantitative purposes, signal/noise ratio should be at least 10. This ensures correct quantification of the trace amounts with less than 2% variance. The baseline should deviate as little as possible from a horizontal line. It is usually measured for a specified time, e.g., 1/2 hour or one hour and called drift. Drift usually associated to the detector heat-up in the first hour after power-on. 

Fig. Z4 (a) Temperature ramp at a frequency a> = lOrads (strain amplitude A = 2%) for a nearly symmetric PEP-PEE diblock with Mn = 8.1 X 104gmol l, heating from the lamellar phase into the disordered phase. The order-disorder transition occurs at 291 1 °C, the grey band indicates the experimental uncertainty on the ODT (Rosedale and Bates 1990). (b) Dynamic elastic shear modulus as a function of reduced frequency (here aT is the time-temperature superposition shift factor) for a nearly symmetric PEP-PEE diblock with Mn = 5.0 X 1O g mol A Shift factors were determined by concurrently superimposing G and G"for w > and w > " respectively. The filled and open symbols correspond to the ordered and disordered states respectively. The temperature dependence of G (m < oi c) for 96 < T/°C 135 derives from the effects of composition fluctuations in the disordered state (Rosedale and Bates 1990). (c) G vs. G"for a PS-PI diblock with /PS = 0.83 (forming a BCC phase) (O) 110°C (A) 115°C ( ) 120°C (V) 125°C ( ) 130°C (A) 135°C ( ) 140°C ( ) 145°C. The ODT occurs at about 130°C (Han et at. 1995).

Fig. 15. Inverse maximum of the collective structure factor of composition fluctuations, N/S k 0), as a function of the incompatibility, x - Symbols correspond to Monte Carlo simulations of the bond fluctuation model, the dashed curve presents the results of a finite-size scaling analysis of simulation data in the vicinity of the critical point, and the straight, solid line indicates the prediction of the Flory-Huggins theory. The critical incompatibility, XcN = 2 predicted by the Flory-Huggins theory and that obtained from Monte Carlo simulations of the bond fluctuation model M 240, N = 64, p = 1/16 and = 25.12) are indicated by arrows. The left inset compares the phase diagram obtained from simulations with the prediction of the Flory-Huggins theory (c.f. (47)). The right inset depicts the compositions at coexistence such that the mean field theory predicts them to fall onto a straight line. Prom Muller [78]...

The real derivation of the mean-square fluctuation is obtained from an average over all magnitudes of the composition fluctuation with the corresponding Boltzmann factor exp( — SF/kT) ... 

This method is quite reliable for repetitive analyses involving stable matrices. However, if the matrix composition fluctuates, then this disturbs the equilibrium factors and reduces the precision of the result, by consequence of an irregular calibration. In this case cartridge-base extraction would be preferred. 

The influence of the chain connectivity on the dynamics of the composition fluctuations does not only influence two-point correlation functions like the global structure factor but it is also visible in the time evolution of composition profiles in the vicinity of a surface [109]. 

The RPA is a mean field approximation that neglects contributions from thermal composition fluctuations and that assumes the chain conformations to be unperturbed Gaussian chains. The last assumption becomes visible from the Debye form factor in the first two terms, which for Vp, = are in accordance with Eq. 7, while the third term involves the FH interaction parameter. 

Fig. 4 Structure factor of binary blend dPB/PS in Zimm representation at different temperatures (top) and pressures (middle), and polymer concentrations (bottom), with the other parameters constant. From the fitted straight lines the susceptibility and correlation length is evaluated. The increase of scattering is caused by stronger thermal composition fluctuations when approaching the critical point...

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