Ornstein-Zernike Formula Research Articles

Molecular Model for Critical Opalescence of Carbon Dioxide

Taking into consideration the fact that the variance of the density of critical CO2 is small, a molecular model of critical CO2 is formulated for the first time. Based on it, the intensity of Rayleigh light scattering is calculated. The expression for the Rayleigh scattering intensity differs from the Ornstein–Zernike formula by a more complex dependence on the scattering angle and on the diffraction parameter. The scattering process shows the features necessary for the critical opalescence. For the first time, the contribution of thermal motion of molecules to Rayleigh light scattering on a critical medium is explicitly reckoned.

Small-angle scattering and morphologies of ultra-flexible microemulsions.

The phase diagrams of ternary mixtures of partly miscible solvents containing a hydrotropic co-solvent exhibit a variable miscibility gap and one critical point. This work investigates the entire monophasic region far from and near to the miscibility gap in octan-1-ol/ethanol/water, for which ultra-flexible micro-emulsions (UFMEs) are observed by small-angle scattering techniques. SWAXS (combined small- and wide-angle X-ray scattering) allows the elucidation of these types of structure. Three distinct areas can be identified in the phase diagram, with scattering data resembling those from direct, bicontinuous and reverse local structures. These UFMEs are far more polydisperse than their surfactant-based counterparts. Water-rich and solvent-rich domains are only delimited by a small excess of hydrotrope, instead of a well defined surfactant layer of fixed area per molecule. It is shown that all scattering spectra obtained for the nanostructured compositions can be modelled by a simple unified analytical model composed of two uncorrelated contributions. The main one is the Ornstein-Zernike formula for composition fluctuations which gives information about the pseudo-phase domain size. The second is a Lorentzian that captures the structure of at least one of the coexisting pseudo-phases. No Porod law can be measured in the SAXS domain. The proposed expression gives access to two characteristic sizes as well as one inter-aggregate distance.

Nonequilibrium dynamic correlation-length scaling method

The finite-size scaling method in the equilibrium Monte Carlo(MC) simulations and the finite-time scaling method in the nonequilibrium-relaxation simulations are compromised. MC time data of various physical quantities are scaled by the MC time data of the dynamic correlation length, which corresponds to changing the system size in the finite-size scaling method. This scaling method is tested in the three-dimensional ferromagnetic Ising spin model and in the three dimensional $\pm J$ Ising spin-glass model. The transition temperature and the critical exponents, $\eta$ and $\nu$, are obtained by the nonequilibrium relaxation data of the susceptibility and the dynamic correlation length apart from the dynamic exponent. We also comment on the definition of the dynamic correlation length in the nonequilibrium relaxation process. The Ornstein-Zernike formula is not always appropriate.

Statistical mechanical description of liquid systems in an electric field

We formulate the statistical mechanical description of liquid systems for both polarizable and polar systems in an electric field in the E ensemble, which is the pendant to the thermodynamic description in terms of the free energy at constant potential. The contribution of the electric field to the configurational integral Q[over ]_{N}(E) in the E ensemble is given in an exact form as a factor in the integrand of Q[over ]_{N}(E) . We calculate the contribution of the electric field to the Ornstein-Zernike formula for the scattering function in the E ensemble. As an application, we determine the field-induced shift of the critical temperature for polarizable and polar liquids and show that the shift is upward for polarizable liquids and downward for polar liquids.

Small-angle x-ray scattering of supercritical fluid Hg: Bi-impurity effect

We have carried out small-angle x-ray scattering and x-ray transmission measurements of supercritical fluid Hg-0.2%Bi and pure Hg systems at SPirng-8 in Japan. We have obtained the static structure factors S(Q) in the low-momentum transfer region (0.5< Q <3 nm−1) for both samples at temperatures and pressures up to 1600 °C and 200 MPa. A Bi-impurity effect can be seen in both S(Q) and density deduced from x-ray transmission, only in the high temperature and high pressure region; the sign of the phase separation is seen near the critical density for Hg-0.2%Bi sample. We have analyzed S(Q) by Ornstein-Zernike formula and deduced the structure factors in the long-wavelength limit S(0), correlation lengths ξ, and corresponding short-range correlation lengths R = ξ≡/√S(0). Temperature variation of R is very similar to that of the density in each sample. This experimental fact hints that the variation of R is highly dependent on its metallic nature because the density is the most relevant parameter for the electronic features in fluid Hg and its impurity systems.

Pair correlations in a multicomponent fluid placed in a porous medium

We consider a multicomponent fluid placed in a porous medium. The Ornstein—Zernike approximation is used to calculate the pair correlation functions for density fluctuations in the mixture components. We show that light scattering in the neighborhood of the critical state of the system is determined (in the single-scattering approximation) by the commonly known Ornstein—Zernike formula. We investigate the shift in the critical parameters due to the porous medium.

Small angle X-ray scattering measurements for supercritical fluid metals using synchrotron radiation

Small angle X-ray scattering measurements for supercritical fluid metals such as selenium and mercury have been carried out using synchrotron radiation at SPring-8 to study density fluctuations near the liquid–vapor critical points. The absolute value of the scattering intensity was estimated from measurements on compressed He gas, a method developed using the in-house X-ray source at Hiroshima University. The observed scattering intensity of fluid selenium increases when one approaches the isochore of the critical density with decreasing pressure. The correlation length of the density fluctuations and the long wave length limit of the structure factor, S(0), are deduced to be 16.5 Å and 31 at 1680 °C and 459 bar using the Ornstein–Zernike formula. Based on these data, the density fluctuations in fluid selenium are discussed.

Thermodynamic and Kinetic Description of the Second Order Phase Transitions

A thermodynamic and kinetic description of phase transitions for the model of ferroelectrics on the basis of kinetic equations for the distribution function of values of the "order parameter", coordinates and time is discussed. For one-domain ferroelectrics the self-consistent approximation for the first moment is used. The kinetic equation is reduced to the relaxation Ginsburg–Landau equation. The susceptibility is governed by the Curie law and there is an increase in heat capacity. Calculations are carried out for one-domain and polydomain ferroelectrics. In the first case the self-consistent approximation for the first moment is used. In the second one the self-consistent approximation for the second moment is carried out. In the last case there is a jump in susceptibility. The heat capacity is governed by the Curie law. It is shown also that the Ornstein–Zernike formula is valid not for the space correlator of fluctuations, but only for the temporal spectral density of the space correlator at zero frequency. In the kinetic theory of the phase transition all physical characteristics at the critical point have finite values. Thus, the problem of "infinities" is absent.

Gelation Process of Poly(vinyl alcohol) As Studied by Small-Angle Neutron and Light Scattering

The authors report small-angle neutron scattering (SANS) and light scattering (LS) studies on poly(vinyl alcohol) (PVA) gels formed in a mixture of deuterated dimethyl sulfoxide (DMSO-d{sub 6}) and heavy water at 23 C. It was reported in a previous paper that the SANS intensity I(Q) of the PVA gels is well described by the Ornstein-Zernike (OZ) formula I(0)/(1 + {zeta}{sup 2}Q{sup 2}) and Porod`s law Q{sup {minus}4} for 0.01 {angstrom}{sup {minus}1} < Q < 0.035 {angstrom}{sup {minus}1} and 0.05 {angstrom}{sup {minus}1} < Q < 0.1 {angstrom}{sup {minus}1}, respectively. In this work, the authors extended the Q range down to 3 {times} 10{sup {minus}3} {angstrom}{sup {minus}1} and found that I(Q) turns up for Q < 8 {times} 10{sup {minus}3} {angstrom}{sup {minus}1} to deviate from the OZ formula. This upturn has been assigned to structure due to phase separation based on the results of SANS and LS measurements. In order to investigate the gelation process, time-resolved SANS measurements were carried out on the PVA solutions after quenching to 23 C from 100 C. It was found that the correlation length {zeta} evaluated by the OZ formula in a Q range of 0.01--0.035 {angstrom}{sup {minus}1} is dominated by concentration fluctuations in the earlymore » stage of the gelation before crystallization (t < 200 min) while, after the crystallization initiates, the average correlation distance between the nearest-neighboring crystallites becomes a dominant factor in {zeta}. Distance distribution function P(r) which is defined by inverse Fourier transformation of the scattering intensity was calculated to see the size and the distribution of the crystallites.« less

Universality of critical phenomena in complex fluids

A theory for static and dynamic light scattering from micellar and microemulsion systems near the critical point is given which incorporates the universality of critical phenomena in fluids and the finite size effect of the constituent particles in the system. This theory reduces to the Ornstein-Zernike formula for the static light scattering intensity and the Kawasaki mode-coupling result for the line width of the dynamic light scattering in the limit when the size of the particles is vanishingly small compared to the wavelength of the probing light. When this is not the case the critical dynamics shows considerable deviation from the mode-coupling theory and the order parameter fluctuation exhibits non-exponential relaxation. The main ingredient of the theory is the recognition of the fact that the physical particle cluster size distribution near the critical point is nearly identical to the standard percolation cluster size distribution characterised by universal values of the fractal dimension D = 2.49 and of the polydispersity index τ = 2.21. We analysed light scattering data from several micellar solutions and a microemulsion system to support the validity of this theory and establish firmly that the underlying critical phenomena are indeed universal.

Boundary of the macroscopic region above the critical point of iron

The results of a small angle neutron scattering experiment in iron just above the critical point are discussed in terms of the static scaling assumptions and of the Ornstein-Zernike relation. By means of suitable analysis of the scattering data, a rather well defined boundary line is obtained which indicates that the range of applicability of the Ornstein-Zernike formula for static fluctuations is restricted within the limits of the macroscopic region.

Onsager Relations and the Spectrum of Critical Opalescence

The necessity of extending the hydrodynamic equations near the critical point is investigated on the basis of Onsager's assumption concerning the regression of fluctuations. It is shown that it follows from the Onsager relations that in general the new equations are nonlocal. The most important coefficient corresponding to the inverse compressibility in the original equations is obtained from the Onsager relations and expressed in terms of the equilibrium pair correlation function. Fixman's extension of hydrodynamics follows when the pair correlation is given by the Ornstein—Zernike formula. The consequences for the spectrum of light scattering near the critical point are studied and the width and relative intensity of the central scattering peak are expressed in terms of the equilibrium pair correlation function.

Exact treatment of particle correlation functions and free energy

The exact integral equation for the pair distribution functions, published recently by J. M. J. van Leeuwen, J. Groeneveld and J. de Boer 1) is readily derived by previously published methods 2) 3). The relation of the new equation to the Ornstein-Zernike formula for the direct correlation function is discussed, and the importance of this relation for treatment of phase transitions is noted. Exact formulas for the Helmholtz free energy are presented. The convolution approximation, obtained by neglecting prototype graphs, and its relation to the Kirkwood superposition approximation and to the nodal expansion sequance are discussed.