Approximate analytic solution of the nonlinear Poisson–Boltzmann equation for spherical colloidal particles immersed in a general electrolyte solution

By means of polynomial approximation and iteration procedure using linearization solution as initial solution, the non-linear Poisson–Boltzmann equation describing a spherical colloidal particle immersed in an arbitrary valence and mixed electrolyte solution is solved analytically, and analytical expressions for electrical potential distribution ψ(r) and surface charge density/surface potential relationship (σ/ψ ₀) are acquired. The σ/ψ ₀ expression performs very well for the entire range of a reduced colloidal radius x ₀ only if a reduced surface potential y ₀ is lower than 15.5; particularly, in the case of x ₀ >10, the σ/ψ ₀ expression applies over the entire range of y ₀, and maximum of the absolute value of percent relative error (PRE) is not larger than 2 if one does not come across a parameter combination as of an extreme case of x ₀?<?0.9 and y ₀?>?15.5. The ψ(r) expression outperforms the linearization solution greatly and performs very well for the domain of x ₀?≤?1.5 and y ₀?≤?4.5 with maximum of the absolute value of PRE not larger than 3, whereas that of the linearization solution may stand 22 for the same parameter range. It is concluded that the present σ/ψ ₀ expression is the first one valid for entire range of x ₀, given that y ₀ is not unreasonably high, and the present paper proposes, to some extent, a “universal” way for solving nonlinear differential equations.     

Dimensionality control of coupled scattering equations using partitioning techniques

A method of variable reduction of the dimensionality of the coupled equations for inelastic scattering is presented, based upon a projection operator P with a restricted range of orbital angular momentum states. For rotational states in the range O?j ?j^* and total angular momentum large, the coupled equations have dimensionality (j^* + 1) ? N ?(j^* + 1)², where the value of N is controlled by the choice of P. This is in contrast to conventional partitioning techniques which utilize further restrictions on the important molecular rotational states. The equations for the P subspace and its complementary Q subspace are decoupled by an approximation on the equation of motion of Qψ_scat. Information about scattering into the Q subspace is retained, within this degree of approximation, and is reintroduced at the end of the computation with little additional labor. The theory is developed in terms of atom-rigid-rotor scattering, although addition of vibrational modes would not in any way interfere with the basic techniques used.     

Coexistence curves of critical nonaqueous microemulsions of (AOT + DMA + nonane) and (AOT + DMA + octane)

The coexistence curves of (T, n), (T, ϕ) and (T, ψ) (n, ϕ, ψ are the refractive index, the volume fraction, and the effective volume fraction, respectively) for two ternary nonaqueous microemulsion systems have been determined at constant pressure and a constant molar ratio ω of DMA to AOT. The critical exponent β and the critical amplitude B have been deduced, and the values of β were consistent with the theory of 3D-Ising in a region sufficiently close to the critical temperature. The results show that the coexistence curves have been successfully described by the Wegner equation and the expression for the diameter.     

Spin polarization and cross sections of electrons elastically scattered from heavy alkaline-earth atoms

Semi-relativistic approach is employed to compute the differential and integrated cross sections, spin polarizationP and the spin polarization parametersT andU for the scattering of electrons from barium and strontium atoms in the energy range from 2.0–300 eV. The projectile-target interaction is represented both by real and complex optical potential in the solution of Dirac equation for the scattered electrons. The real optical potential includes the static, a parameter free correlation polarization and a modified semi classical exchange potentials. The complex optical potential is constructed by adding a model absorption potential as its imaginary part to the real optical potential.     

Nonlinear response of the surface electrostatic potential formed at metal oxide/electrolyte interfaces. A Monte Carlo simulation study

An analysis of surface potential nonlinearity (ψ₀) at metal oxide/electrolyte interfaces is presented. By using grand canonical Monte Carlo simulations of a simple lattice model of an interface, we show that a correlation exists between ionic strength, as well as surface site densities, and the non-Nernstian response of a metal-oxide electrode. We propose two approaches to deal with the ψ₀-nonlinearity: one based on perturbative expansion of the Gibbs free energy and another based on the assumption of the pH dependence of surface potential slope. The theoretical analysis based on our new potential form gives excellent performance in extreme pH regions, where classical formulae for ψ₀ are unjustified. The new formula is general and independent of any underlying assumptions. For this reason, it can be directly applied to experimental surface potential measurements, including those for individual surfaces of single crystals, as we present for data reported by Kallay and Preo?anin [6].     

Total cross-sections for electron scattering from iron at 10–5000 eV using a model optical potential

We report calculations on the total (elastic plus inelastic) electron-scattering cross sections in the energy range 10–5000 eV. A model complex optical potential, composed of static, exchange, polarisation and absorption terms, is employed to describe the collision system at each electron energy. The Iron atom is described by Dirac-Hartree-Fock-Slater self-consistent charge density. The complex phase shifts are computed in a variable phase approach. The absorption cross sections are compared with the experimental results. The experimental absorption cross sections are obtained by adding the experimental ionisation cross sections and available experimental excitation cross sections for electron impact of the allowed transitions a⁵ D → (x,y,z)⁵ D ⁰, (w,y,z)⁵ P ⁰. We have good qualitative agreement between our results and the experimental results available below 200 eV. The Born-Bethe parameters are also calculated. Elastic differential cross-sections with and without absorption are also reported at a few selected energies.     

A conformational study of aromatic imide compounds. Part 2. Compounds containing diphenyl sulfide,diphenyl sulfone,and diphenylmethane moieties

An attempt was made to estimate the dihedral angles, φ, ψ, ω₁, and ω₂, of bis(4-hydroxyphthalimide)s (BHPI) and bis(phenylphthalimide)s (BPI) having diphenyl sulfide, diphenyl sulfone, or diphenylmethane linkages at the center of molecules using solid–state ¹³C CP/MAS NMR and ab initio nuclear shielding calculations. The TOSS and TOSS & DD pulse sequences were performed in the NMR measurements to obtain exact chemical shifts of each carbon. Total energies were calculated using the B3LYP/6-31G(d) level of theory, and shielding constants were calculated using the RHF/6-31G(d) level of theory for diphenyl sulfide, diphenyl sulfone, diphenylmethane with varying angles of φ, ψ from 0 to 180° at intervals of 10°. It was clarified that the –S– and –SO₂– linkages lead asymmetrical conformations with different ω₁ and ω₂ or with different φ and ψ for BHPIs and BPIs. In contrast, the compounds having –CH₂– linkages have symmetrical conformations. The dihedral angle of imide ring and phenylene ring (ω) are in the range of 40–90°, and the dihedral angles (φ,ψ) distribute in the stable regions of the energy surfaces ranging from 40 to 90°.     

A simple method for non-parametric estimation of the probability distribution function of diatomic molecules by electron diffraction

Treating the Debye intensity relationship as a linear Fredholm integral equation of the first kind, a method is developed for a non-parametric estimation of the probability distribution function P/r) for diatomic molecules from electron-diffraction data. Since the problem is an ill-posed one, Tikhonov's regularization procedure was used for the solution. The method was applied to iodine for which the non-parametric P/r) function is obtained. Based on this function the electron-diffraction parameters r_g, r_a and l_a are estimated by the linear least-squares method without a priori assumptions about the form of the vibrational potential. Approximating the potential by the Dunham expansion, the parameters r_e, ω_e, K₃ and K₄ are also estimated. The results are compared with those obtained from conventional analytical representation of an intensity function. Comparison is also made with spectroscopic data for iodine.     

Some simple remarks on variational bounds in Faddeev-type formalisms

A bound on the energy obtained from the differential equation forms of Faddeev-type equations is derived. The procedure is based on the standard variation principle for the Schrödinger equation plus the observation that the ordinary Schrödinger wavefunction |ψ(i)> equals the sum of the Faddeev-type “channel wavefunctions” |ψ_λ(i)>.     

Investigation of optical properties of amorphous Ge15Se85-xCux thin films using spectroscopic ellipsometry

Different compositions of amorphous Ge₁₅Se_85-xCu_x thin films were deposited onto glass substrates by the thermal evaporation technique. Their amorphous structural characteristics were studied by X-ray diffraction (XRD). The optical constants (n, k) of amorphous Ge₁₅Se_85-xCu_x thin films were obtained by fitting the ellipsometric parameters (ψ and Δ) data for the first time using three layers model system in the wavelength range 300–1100 nm. It was found that the refractive index, n, increases with the increase of Cu content. The possible optical transition in these films is found to be indirect transitions. The optical energy gap decreases linearly from 1.83 to 1.44 eV with increasing the Cu. The experimental transmittances spectrum can be simulated using the thickness and optical constants modeled by spectroscopic ellipsometry model.     

Intramolecular dynamics: time evolution of superposition states in the regular and irregular spectrum

The time evolution of wavefunctions ψ(t) is compared in two systems, one with a regular energy spectrum and one with an irregular spectrum. The behavior of P(t) = {|[ψ(O)|ψ(y]}|² is found to be the same in both systems, P(t) decay is insensitive to the coupling between the degrees of freedom.     

Ab-initio study of the structural, linear and nonlinear optical properties of CdAl2Se4 defect-chalcopyrite

The complex density functional theory (DFT) calculations of structural, electronic, linear and nonlinear optical properties for the defect chalcopyrite CdAl₂Se₄ compound have been reported using the full potential linearized augmented plane wave (FP-LAPW) method as implemented in the WIEN2k code. We employed the Wu and Cohen generalized gradient approximation (GGA-WC), which is based on exchange-correlation energy optimization to calculate the total energy. Also we have used the Engel-Vosko GGA formalism, which optimizes the corresponding potential for band structure, density of states and the spectral features of the linear and nonlinear optical properties. This compound has a wide direct energy band gap of about 2.927 eV with both the valence band maximum and conduction band minimum located at the center of the Brillouin zone. The ground state quantities such as lattice parameters (a, c, x, y and z), bulk modulus B and its pressure derivative B′ are evaluated. We have calculated the frequency-dependent complex ε(ω), its zero-frequency limit ε₁(0), refractive index n(ω), birefringence Δn(ω), the reflectivity R(ω) and electron energy loss function L(ω). Calculations are reported for the frequency-dependent complex second-order nonlinear optical susceptibilities. We find opposite signs of the contributions of the 2ω and 1ω inter/intra-band to the imaginary part for the dominant component through the wide optical frequency range.     

A different approach to X-ray stress analysis

A different approach to X-ray stress analysis has been developed. At the outset, it must be noted that the material to be analyzed is assumed homogeneous and isotropic. If a sphere with radius r within a specimen is subjected to a state of stress, the sphere is deformed into an ellipsoid. The semi-axes of the ellipsoid have the values of (r + ε_x), (r + ε_y), and (r + ε_z), which are replaced by d_x, d_y, and d_z, or for the cubic case, a_x, a_y, and a_z. In this technique, at a particular ϕ angle (see Fig. 1), the two-theta position of a high angle (hkl) peak is determined at ψ angles of 0, 15, 30, and 45°. These measurements are repeated for 3 to 6 ϕ angles in steps of 30°. The d_ϕψ or a_ϕψ values are then determined from the peak positions. The data is then fitted to the general quadratic equation for an ellipsoid by the method of least squares. From the coefficients of the quadratic equation, the angle between the laboratory and the specimen coordinates (direction of the principle stress) can be determined. Applying the general rotation of axes equations to the quadratic, the equation of the ellipse in the x–y plane is determined. The a_x, a_y, and a_z values for the principal axes of the lattice parameter ellipsoid are then evaluated. It is then possible to determine the unstressed a₀ value from Hooke's Law using a_x, a_y, and a_z. The magnitude of the principal strains/stresses is then determined.     

Spatial correlation of atomic electrons: He

For any two-electron wavefunction whose angular dependence is given in terms of the spherical harmonics of the individual electrons and/or o₁₂, where o₁₂ is the interelectronic angle. a transformation is generated which reduces |ψ(r₁, r₂, o₂, o₁, o₁₂ )|² to |ψ(r₁, r₂, o₁₂)|². Geometric aspects of electron correlation are analyzed in terms of this resulting three-coordinate function, with specific application to a number of wave functions for doubly-excited helium. Angular correlation has been studied by integrating |ψ(r₁, r₂, o₁₂)|² over its radial dependence to yield p(o₁₂), the probability density function for the interelectronic angle. Trends of p(o₁₂) for doubly-excited states of the helium atom are related to a number of quantities including energies and autoionization widths. These trends can be rationalized in terms of a simple classical model. The full spatial correlation involving |ψ(r₁, r₂, o₁₂)|² is explored by the use of three-dimensional graphs for some of these states.     

The coulomb condensate of the nonlinear Poisson-Boltzmann equation: A unified theory

A general form of the one-dimensional nonlinear Poisson-Boltzmann (PB) equation for a simple electrolyte is studied, namely r^?m(d/dr{r^mdψ^(m)/dr} = sinh ψ^(m), where r and ψ^(m) are dimensionless distance and potential respectively, and m = 2, 1 and 0 correspond to the spherical, cylindrical and planar equations respectively. Since analytic solutions are not available for this general PB equation, its properties are here studied via suitable upper bounds, both relative and absolute, on its solutions. When the final results are specialized to the sphere (m = 2) and cylinder (m = 1) cases, there are recovered the σ-funcfion condensate theorems for a point charge and line charge respectively, previously proven by the author. Since the point-charge and line-charge condensates are hereby shown to have a common origin, it is suggested that they both be called simply “Coulomb condensates”, in recognition of their physical origin.     

The reflectivity spectra of ZnXP2 (X=Si, Ge, and Sn) compounds

Full Potential Augmented Plane Wave plus local orbital method ( FAPW+lo) calculations were performed for ZnSiP₂, ZnGeP₂, and ZnSnP₂ in the chalcopyrite structure in order to investigate the optical properties and to show the origin of the different optical transitions and their correspondence in the band structure. It is found that the most important features of the band gap is pseudo-direct for ZnSiP₂, indirect for ZnGeP₂, and direct for ZnSnP₂. Then the contribution of the different transitions peaks are analyzed from the imaginary part of the dielectric function and the reflectivity spectra.     

Modeling of interface mobility in the description of flow-induced coalescence in immiscible polymer blends

A new semiempirical equation has been proposed for the rate of approach of highly flattened droplets, applicable in the whole range of the ratios, p, of viscosity of the dispersed droplets and matrix. The equation is utilized to calculate the probability, P _c, that the droplet collision induced by shear or extensional flow is followed by the droplet fusion for systems with Newtonian droplets and Newtonian or viscoelastic matrix. The comparison of the results of these calculations with available experimental data and with the calculation using the trajectory analysis shows that the proposed model of the matrix drainage provides more reasonable results than the broadly applied model of partially mobile interface.     

Evaluation of the parameters of the Bender equation of state for low acentric factor fluids and carbon dioxide

Volumetric properties of several low acentric factor fluids (Ar, CH₄, C₂H₆, Kr, N₂, Ne, O₂, Xe) as well as CO₂ are modeled using the Bender equation of state. This equation is a linear function of 19 adjustable parameters, which are evaluated from properties data, using a linear numerical procedure. The validity of the EOS is tested by calculating the Joule-Thomson inversion curve. A simple model is in particular used to correlate the inversion properties predicted by the Bender equation, expressed in term of reduced pressure as a function of reduced temperatures ranging from 0.8 to 6. The simple correlation reproduces accurately the used data. We employ data on state behaviour ρ(P,T) of homogeneous fluid phases, vapour-liquid equilibrium, second virial coefficient and the coordinates of the critical point.     

Über den Einfluß der elektrostatischen Wechselwirkungen auf die α-Helixbildung von basischen Poly-[α-aminosäuren]

A theoretical treatment of the electrostatic interactions influencing the conformation of poly-[α-aminoacids] containing ionogenic side-chains is reported. Using the Gouy-Chapman-equation for the relation between the density of surface chargeσ _e and surfacepotentialψ the latter one will be determined on the surface of anα-helix of a basic poly-[α-aminoacid] in an aqueous electrolyte solution. As it could be shown,ψ decreases with the electrolyte concentrationc. The equilibrium constants for the propagation step of theα-helix-formation is expressed as a function ofψ and a parameterK ₀ containingΔS₀ andΔH₀ dependent onc, considering the electrostatic energy of neighboured basic groups. The temperature at the mid-point of the transitionT _m is a function of the reciproce ionic strengthI. In a solution of an 1-1-electrolyte,T _m depends linearly on 1/√c. The experimental data for Poly-[L-lysine] are in a very good agreement with the straight line given by this relation. However, in the case of a 1∶1 copolymer ofL-lysine andL-leucine the calculated values are much lower than the measured ones, obviously due to neglecting the hydrophobic interactions of the leucine side chains. Furthermore, a relation for the change of theα-helix-contentf _H with 1/T is given. In this case, in the well-known relation (?f _H/? (1/T))=?ΔH⁰/4Rσ ^1/2, the cooperative parameterσ is substituted by a modified parameterΣ depending on the ionic strength. According to this equation and agreeing with the experimental results,T _m increases with the ionic strength, because the electrostatic potentialψ at theα-helix-surface decreases. However, in contrast to reality, the inclination of thef _H-T-curves increases withI orc, respectively. This may be due to an increase ofσ with the electrolyte concentration.