Zernike Approach Research Articles

Integral equation study of soft-repulsive dimeric fluids

We study fluid structure and water-like anomalies of a system constituted by dimeric particles interacting via a purely repulsive core-softened potential by means of integral equation theories. In our model, dimers interact through a repulsive pair potential of inverse-power form with a softened repulsion strength. By employing the Ornstein–Zernike approach and the reference interaction site model (RISM) theory, we study the behavior of water-like anomalies upon progressively increasing the elongation λ of the dimers from the monomeric case () to the tangent configuration (). For each value of the elongation we consider two different values of the interaction potential, corresponding to one and two length scales, with the aim to provide a comprehensive description of the possible fluid scenarios of this model. Our theoretical results are systematically compared with already existing or newly generated Monte Carlo data: we find that theories and simulations agree in providing the picture of a fluid exhibiting density and structural anomalies for low values of λ and for both the two values of the interaction potential. Integral equation theories give accurate predictions for pressure and radial distribution functions, whereas the temperatures where anomalies occur are underestimated. Upon increasing the elongation, the RISM theory still predicts the existence of anomalies; the latter are no longer observed in simulations, since their development is likely precluded by the onset of crystallization. We discuss our results in terms of the reliability of integral equation theories in predicting the existence of water-like anomalies in core-softened fluids.

Wavefront measurement by phase retrieval through single defocus intensity patterns

An algorithm for wavefront measurement using a phase retrieval method is proposed. Only one defocus intensity pattern of the imaging optical system is needed for phase retrieval. Based on the extended Nijboer–Zernike (ENZ) approach, this algorithm draws on general inverse matrix theory, and a predictor–corrector method is also used to correct the linearization errors for medium-to-large aberrations. This method does not need multi-defocus intensity in the focal region, and it can exclude vibration during multi-intensity pattern capturing, making the whole process simple and brief. Some simulated results are presented which show this algorithm has good convergence against noise. A confirmatory experiment is carried out, and its results verifying this method are also given.

Thermodynamics of a long-range triangle-well fluid

The long-range triangle-well fluid has been studied using three different approaches: firstly, by an analytical equation of state obtained by a perturbation theory, secondly via a self-consistent integral equation theory, the so-called self-consistent Ornstein–Zernike approach (SCOZA) which is presently one of the most accurate liquid-state theories, and finally by Monte Carlo simulations. We present vapour–liquid phase diagrams and thermodynamic properties such as the internal energy and the pressure as a function of the density at different temperatures and for several values of the potential range. We assess the accuracy of the theoretical approaches by comparison with Monte Carlo simulations: the SCOZA method accurately predicts the thermodynamics of these systems and the first-order perturbation theory reproduces the overall thermodynamic behaviour for ranges greater than two molecular diameters except that it overestimates the critical point. The simplicity of the equation of state and the fact that it is analytical in the potential range makes it a good candidate to be used for calculating other thermodynamic properties and as an ingredient in more complex theoretical approaches.

Extended Nijboer–Zernike approach to aberration and birefringence retrieval in a high-numerical-aperture optical system

The judgment of the imaging quality of an optical system can be carried out by examining its through-focus intensity distribution. It has been shown in a previous paper that a scalar-wave analysis of the imaging process according to the extended Nijboer-Zernike theory allows the retrieval of the complex pupil function of the imaging system, including aberrations as well as transmission variations. However, the applicability of the scalar analysis is limited to systems with a numerical aperture (NA) value of the order of 0.60 or less; beyond these values polarization effects become significant. In this scalar retrieval method, the complex pupil function is represented by means of the coefficients of its expansion in a series involving the Zernike polynomials. This representation is highly efficient, in terms of number and magnitude of the required coefficients, and lends itself quite well to matching procedures in the focal region. This distinguishes the method from the retrieval schemes in the literature, which are normally not based on Zernike-type expansions, and rather rely on point-by-point matching procedures. In a previous paper [J. Opt. Soc. Am. A 20, 2281 (2003)] we have incorporated the extended Nijboer-Zernike approach into the Ignatowsky-Richards/Wolf formalism for the vectorial treatment of optical systems with high NA. In the present paper we further develop this approach by defining an appropriate set of functions that describe the energy density distribution in the focal region. Using this more refined analysis, we establish the set of equations that allow the retrieval of aberrations and birefringence from the intensity point-spread function in the focal volume for high-NA systems. It is shown that one needs four analyses of the intensity distribution in the image volume with different states of polarization in the entrance pupil. Only in this way will it be possible to retrieve the "vectorial" pupil function that includes the effects of birefringence induced by the imaging system. A first numerical test example is presented that illustrates the importance of using the vectorial approach and the correct NA value in the aberration retrieval scheme.

Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach

A recently found extension of the Nijboer–Zernike approach to optical aberrations allows analytic computation of the intensity point-spread function in the focal region of an optical system. Here, the exit pupil function is expanded as a Zernike series , with a complex coefficient of the single aberration , and the contribution of each of the terms of this series to the complex-amplitude point-spread function U in the focal region has an analytic form. This representation of U, and hence of the intensity point-spread function I=|U|2, is highly efficient since normally a pupil function is already accurately described by a few β's. In this paper, the inverse problem of retrieving the β's from a given intensity I in the focal region is studied. A computation scheme for solving this nonlinear estimation problem, under the assumption of small-to-medium–large aberrations, is proposed. The key step is to linearize the theoretical intensity, comprising the β's as unknowns, by deleting second-order terms, and to optimize the match between the linearized, theoretical intensity and the given intensity in the focal region. The special case of a small pure-phase aberration pupil function is separately considered. For general pupil functions (also containing amplitude errors) or larger pure-phase aberrations, the linearization error(s) cannot be ignored. By adopting a predictor–corrector approach, the effect of linearization can be eliminated iteratively and this yields accurate or even perfect retrieval of aberrations well beyond the diffraction limit. Although the method was developed for lithographic applications, with numerical apertures ≤ 0.60 and almost-ideal point sources, it has potential applications to more general light optics settings (microscopy, astronomy, …). The application range of the method is, furthermore, extended in this paper to cover the medium–high numerical aperture range (≤ 0.80) and to the case that the lateral size of the illumination source is comparable to the diffraction unit. The paper is explorative in nature and aims at illustrating the potential and key features of the methods, on the whole by showing results from simulations. Accordingly, little effort has been spent on addressing the many fundamental aspects of a mathematical, statistical and modelling nature; a list containing these fundamental issues is included at the end of the paper.

Ornstein−Zernike Random-Walk Approach for Polymers

A new theory of the thermodynamics and structure of polymer-chain fluids is developed. It is based on relating, in a novel way, polymer-chain structure with random-walk structure in the context of Ornstein−Zernike equations. It incorporates much of the interaction-site description of correlation used in reference-system interaction site model (RISM) theory but introduces several simplifying approximations in connecting the RISM equation to random-walk theory that departs in a significant way from the extension of RISM used in polymer RISM (PRISM) theory. Both continuum-space and lattice-gas versions of the theory are summarized, and the extension of the theory to a self-consistent Ornstein−Zernike approach (SCOZA) for polymers is given.

A simple SCOZA for simple fluids

A simple version of the self-consistent Ornstein–Zernike approach (SCOZA) is investigated quantitatively. The simplification consists of retaining thermodynamic self-consistency between the compressibility and energy routes to thermodynamics from the pair distribution function g( r). However, we do not require that g( r) be identically zero within the hard repulsive core of the models we consider here. The resulting theory lends itself to analysis of Hamiltonian models to which the earlier SCOZA formulation cannot be readily applied as well as providing a convenient means of analyzing simpler models. We choose the hard-core Yukawa fluid as an illustrative example to calibrate the accuracy of the new version. The results are compared with our earlier SCOZA and simulation results, and a suggestion is given to systematically improve the accuracy of the theory in the applications that warrant so doing.

Self-consistent Ornstein–Zernike Approach to Polymer Fluids

In this work we study Flory–Huggins lattice polymers using a self-consistent Ornstein–Zernike approach. The results are compared with those derived from classical mean-field theory. A simple continuum-space (i.e. off-lattice) extension of the Flory–Huggins model is also proposed and studied.

Yukawa sticky m-point model of associating fluid

The product-reactant Ornstein–Zernike approach, supplemented by the ideal network approximation, is formulated for the Yukawa sticky m-point (YSmP) model of associating fluid. The model is represented by the multicomponent mixture of the Yukawa hard spheres with m sticky points randomly located on the surface of each hard sphere. Extensions of the regular integral equation closures, which include polymer Percus–Yevick, polymer hypernetted chain and polymer mean spherical approximations, are presented. An analytical solution of the polymer mean spherical approximation is derived and closed form analytical expressions for the structure (contact value of the radial distribution function, structure factor) and thermodynamic (internal energy) properties of the YSmP model are obtained. Due to generality and flexibility of the model it can be used to study the properties of a number of different associating fluids, including water and aqueous solutions. By way of illustration liquid–gas phase diagrams for the model with m=0, 1, 2, 3, 4 are presented and discussed. Predictions of the theory for the liquid–gas phase diagram of the YS4P model with the parameters similar to those assumed in the frames of the statistical associating fluid theory to mimic water are in reasonably good agreement with the corresponding experimental data for water.

Adsorption of electrolyte in a templated hard-sphere matrix. Predictions of the continuum replica Ornstein–Zernike approach

Thermodynamic and structural properties of the primitive model of size asymmetric 1-1 electrolyte solutions in a disordered hard-sphere matrix environment mimicking a microporous adsorbent were studied. The influence of the hard-sphere template on the adsorption properties of the material was analyzed. The numerical results of our study follow from the application of the replica Ornstein–Zernike integral equations, complemented by the hypernetted chain closure. Two different electrolyte models were examined with parameters mimicking lithium and sodium chloride solutions. The excess chemical potential for adsorbed electrolyte in an untemplated and a templated hard-sphere matrix was studied. Steric effects due to matrix confinement are known to have substantial influence on the properties of the annealed electrolyte. Our results showed the hard-sphere template produces a distribution of obstacles which increases the adsorption of electrolyte compared to the untemplated matrix case. The results were explained with regard to the pair-distribution functions for the various interaction pairs.

Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions.

We assess the validity of an extended Nijboer-Zernike approach [J. Opt. Soc. Am. A 19, 849 (2002)], based on ecently found Bessel-series representations of diffraction integrals comprising an arbitrary aberration and a defocus part, for the computation of optical point-spread functions of circular, aberrated optical systems. These new series representations yield a flexible means to compute optical point-spread functions, both accurately and efficiently, under defocus and aberration conditions that seem to cover almost all cases of practical interest. Because of the analytical nature of the formulas, there are no discretization effects limiting the accuracy, as opposed to the more commonly used numerical packages based on strictly numerical integration methods. Instead, we have an easily managed criterion, expressed in the number of terms to be included in the Bessel-series representations, guaranteeing the desired accuracy. For this reason, the analytical method can also serve as a calibration tool for the numerically based methods. The analysis is not limited to pointlike objects but can also be used for extended objects under various illumination conditions. The calculation schemes are simple and permit one to trace the relative strength of the various interfering complex-amplitude terms that contribute to the final image intensity function.

Self-consistent Ornstein–Zernike approximation compared with Monte Carlo results for two-dimensional lattice gases

The self-consistent Ornstein–Zernike approach (SCOZA) is solved numerically for a lattice gas or Ising model on the simple square lattice in two dimensions. Interactions of varying range are considered, and the results are compared with corresponding simulation ones. We focus especially upon the location of the critical temperature Tc which is identified with the maximum of the specific heat. The maximum remains finite for the finite-sized simulation sample and also for SCOZA, which treats infinite lattices in two dimensions as though they were finite samples. We also investigate the influence of the precise form of the interaction, first using an interaction that extends the nearest-neighbor case in a simple way and then considering the square-well interactions used in the simulations. We find that the shift in Tc away from its mean-field value is governed primarily by the range of interaction. Other specific features of the interaction leave a smaller influence but are relevant to a quantitative comparison with simulations. The SCOZA yields accurate results, and the influence of the precise form of the attractive interaction plays a significant role in SCOZA’s success.

A direct correlation function for mesomorphic polymers and its application to the ‘smectic’ phase of isotactic polpropylene

A general model to calculate the structure of a smectic to nematic-like order in polymer systems is proposed. The model is limited to well-aligned molecules with respect to each other. It is especially suited for systems where the packing between neighboring chains is described by liquid-like disorder with axial registration in the chain-axis direction. In such a case the diffraction pattern from the oriented structure exhibits broad liquid-like reflections on the equatorial but sharper reflections on the layer lines. An Ornstein–Zernike approach is adopted to calculate the inter-chain packing and inter-chain registration, where the direct correlation function of packing and registration are decoupled. The model is applied to quantify the structure of the smectic phase of isotactic polypropylene (i-PP).

Molecular Ornstein–Zernike approach to the solvent effects on solute electronic structures in solution

A new approach to ab initio electronic structure calculations of solute molecules in solution is presented. Combined with the molecular Ornstein–Zernike (MOZ) integral equation theory for polyatomic liquids, solute electronic wave function and solvent distribution around a solute are determined in a self-consistent manner. The hypernetted chain approximation is employed for solving the MOZ equation. In order to describe the short-range solute–solvent interactions, the effective potential operating solute electron is placed on a solute molecule, which is determined by a least-squares fitting to ab initio exchange repulsion/charge transfer energies. The present method, referred to as the MOZ self-consistent-field (SCF) method, is applied to a solute H2O molecule in water solvent. The solvent shift for the vertical excitation to the nπ* state of H2CO in aqueous solution is also examined. The results obtained by the MOZ-SCF calculations are compared with those by the reference interaction site model-SCF theory and the polarizable continuum model.

SCOZA critical exponents and scaling in three dimensions

The critical behavior of a self-consistent Ornstein–Zernike approach (SCOZA) that describes the pair correlation function and thermodynamics of a classical fluid, lattice gas, or Ising model is analyzed in three dimensions below the critical temperature, complementing our earlier analysis of the supercritical behavior. The SCOZA subcritical exponents describing the coexistence curve, susceptibility (compressibility), and specific heat are obtained analytically ( β=7/20, γ′=7/5, α′=−1/10 ). These are in remarkable agreement with the exact values ( β≈0.326, γ′≈1.24, α′≈0.11 ) considering that the SCOZA uses no renormalization group concepts. The scaling behavior that describes the singular parts of the thermodynamic functions as the critical point is approached is also analyzed. The subcritical scaling behavior in the SCOZA is somewhat less simple than that expected in an exact theory, involving two scaling functions rather than one.

Structural and thermodynamic properties of freely-jointed hard-sphere rings and chains

In this paper, we employ the product-reactant Ornstein–Zernike approach (PROZA) developed by the authors to investigate the structural and thermodynamic properties of freely-jointed hard-sphere ring fluids. Using an M×m component sticky two-point (S2P) model and specifying an appropriate association rule between various species, the associating monomers will form M rings with each ring composed of m beads in the complete-association limit. Applying the PROZA to such a Hamiltonian and considering the limit of complete association, we are able to derive analytical expressions for the average monomer–monomer radial distribution function (RDF) as well as its intermolecular and intramolecular contributions and a closed form of the compressibility pressure. To test the theory, we also perform Monte Carlo simulations for the freely-jointed hard-sphere ring model over a wide range of densities and ring sizes. Compared to the simulation results, we find that the predictions of the PROZA for the compressibility factor of flexible ring melts are quantitatively accurate and the average monomer–monomer RDF g(r) is in excellent agreement with the simulation data over a wide range of densities that includes the polymer-melt regime. Based upon such a comparison as well as theoretical considerations, we conclude that ring-size independence of g(r) is a quantitatively accurate approximation and also that the g(r) of rings will be a good approximation for melts of long chains. Finally, we find that we must go beyond our PROZA framework in order to accurately obtain the separate intramolecular and intermolecular parts of g(r), for which we give a quantitatively satisfactory recipe.

Equations of state of freely jointed hard-sphere chain fluids: Numerical results

We continue our series of studies in which the equations of state (EOS) are derived based on the product-reactant Ornstein–Zernike approach (PROZA) and first-order thermodynamic perturbation theory (TPT1). These include two compressibility EOS, two virial EOS, and one TPT1 EOS (TPT1-D) that uses the structural information of the dimer fluid as input. In this study, we carry out the numerical implementation for these five EOS and compare their numerical results as well as those obtained from Attard’s EOS and GF-D (generalized Flory-dimer) EOS with computer simulation results for the corresponding chain models over a wide range of densities and chain length. The comparison shows that our compressibility EOS, GF-D, and TPT1-D are in quantitative agreement with simulation results, and TPT1-D is the best among various EOS according to its average absolute deviation (AAD). On the basis of a comparison of limited data, our virial EOS appears to be superior to the predictions of Attard’s approximate virial EOS and the approximate virial EOS derived by Schweizer and Curro in the context of the PRISM approach; all of them are only qualitatively accurate. The degree of accuracy predicted by our compressibility EOS is comparable to that of GF-D EOS, and both of them overestimate the compressibility factor at low densities and underestimate it at high densities. The compressibility factor of a polydisperse homonuclear chain system is also investigated in this work via our compressibility EOS; the numerical results are identical to those of a monodisperse system with the same chain length.

Critical properties of the self-consistent Ornstein–Zernike approximation for three-dimensional lattice gases with varying range of interaction

The self-consistent Ornstein–Zernike approach (SCOZA) is solved numerically, and its properties in the critical region are investigated for the lattice gas or Ising model in three dimensions. We especially investigate how critical properties depend upon the inverse range of interaction. We find effective critical indices that depend upon this range. However, the SCOZA does not fulfill scaling. Nevertheless, comparing with experimental results for fluids and magnets we find good agreement. Away from the critical point we find that SCOZA yields deviations from scaling that seem similar to experiments.

Interaction between macroparticles in a simple model system of a nonpolar liquid containing trace amounts of water

We report results of numerical analyses on the macroparticle interactions immersed in a simple model system of nonpolar liquid containing trace amounts of water. The singlet Ornstein–Zernike approach with the reference hypernetted-chain closures is employed. Particles of component 1 (water) are characterized by strong attractive interaction among them, those of component 2 (nonpolar liquid) are hard spheres, and particles of different components interact through hard-sphere potential. The mole fraction of component 1 x1 is very small. Beyond x1=x1P, the mixture cannot exist, even in the metastable state with a single phase. Some affinity is considered only between the macroparticle surface and component 1. When the affinity ξ (negative ξ implies repulsion) is increased with fixing x1 (at a value significantly smaller than x1P) and the macroparticle diameter dM, the macroparticle interaction φMM shifts to the lower (more attractive) side and eventually becomes extremely long-ranged and divergent. For larger x1, the divergence occurs at lower ξ. Whenever φMM becomes divergent, the reduced density profile of component 1 near the surface also becomes extremely long-ranged and divergent. The effects of dM on φMM is also analyzed. At the stability limit (x1→x1P), the divergences occur irrespective of ξ and dM, which is consistent with the recent prediction [Attard et al., Phys. Rev. A 45, 7621 (1992)].

Adaptive focusing in scattering media through sound-speed inhomogeneities: The van Cittert Zernike approach and focusing criterion

An original formalism that allows a thorough understanding of cross-correlation-based phase aberration correction techniques in a scattering medium is developed. This formalism is based on the analysis of the second-order statistics of the pressure field scattered by a random distribution of scatterers. One of the major interests of this analysis is the ability it provides to evaluate and monitor the convergence of phase aberration correction techniques. This is achieved by the computation of a single parameter C that serves as a focusing criterion. The theoretical developments are validated experimentally.