An analytic solution of capillary rise restrained by gravity

We derive an analytic solution for the capillary rise of liquids in a cylindrical tube or a porous medium in terms of height h as a function of time t. The implicit t(h) solution by Washburn is the basis for these calculations and the Lambert W function is used for its mathematical rearrangement. The original equation is derived out of the 1D momentum conservation equation and features viscous and gravity terms. Thus our h(t) solution, as it includes the gravity term (hydrostatic pressure), enables the calculation of the liquid rise behavior for longer times than the classical Lucas-Washburn equation. Based on the new equation several parameters like the steady state time and the validity of the Lucas-Washburn equation are examined. The results are also discussed in dimensionless form.     

Non-exponential non-Arrhenius relaxation in the course of CO rebinding to heme proteins

The dispersive transport model for relaxation of photolyzed heme proteins has been improved to take into account the coupling of the ligand-heme geminate recombination and the non-Gaussian diffusive dynamics of conformational changes in heme proteins. Contrary to the earlier deterministic version of the model, the present more rigorous formulation is based on the stochastic approach to the problem. This implies that the time evolution of protein conformations should be described in terms of the transient distribution which satisfies the Smoluchowski-type differential equation with a time-dependent diffusion coefficient. The obtained analytical solution of this equation enables us to relate main kinetic parameters of the geminate recombination and quantities characterizing the ligand-heme interaction. The derived expressions demonstrate that the reaction barrier shifts with time towards higher values following the near-stretched exponential behavior in agreement with experiment. Such a behavior is governed by the non-exponential non-Arrhenius conformational relaxation. The latter process can be identified by the characteristics “footprint” left on the experimental rebinding curve and is shown to be responsible for some kinetically different phases of the ligand-heme geminate recombination observed within distinct temperature ranges.     

A mathematical model for stress-driven diffusion in polymers

A model for case II diffusion into polymers is presented. The addition of stress terms to the Fickian flux is used to produce the characteristics progressive front. The stress in turn obeys a concentration-dependent evolution equation. The model equations are analyzed in the limit of small diffusivity for the problem of penetration into a semiinfinite medium. Provided that the coefficient functions obey two monotonicity conditions, the solvent concentration profile is shown to have a steep front that progresses into the medium. The formulas governing the progression of the front are developed. After the front decays away, the long time behavior of the solution is shown to be a similarity solution as in Fickian diffusion. Two techniques for approximating the solvent concentration and the front position are presented. The first approximation method is a series expansion; formulas are given for the initial speed and deceleration of the front. The second approximation method uses a portion of the long time similarity solution to represent the short time solution behind the front.     

A perturbation theory without energy corrections

We employ logarithmic perturbation theory in a form that utilizes the force, instead of the potential, and yields equations for wavefunction‐correction terms without requiring any information of energy corrections at any stage. The knowledge of unperturbed eigenfunctions of other states is also unnecessary. The perturbed energy eigenvalue can be obtained as a series either by going back to the parent equation, or as an average value involving the perturbed state. The latter scheme applies to any other average property as well. Both ground and excited states of a few systems are chosen for demonstrative calculations. Influence of the nodal structure on the exponential function controlling spatial behavior of the probability density is discussed. Interrelations among specific correction terms are shown in the small‐ and large‐x regime; in the latter case, certain terms nicely sum up to yield the correct decay characteristics of the probability density as well. Relevance of the basic equation to an alternative, nonperturbative scheme of approximation is also outlined. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Evaluation of adsorption of nonionic surfactants blend at water/oil interfaces

The inherent biocompatibility of Span and Tween surfactants makes them an important class of nonionic emulsifiers that are employed extensively in emulsion and foam stabilization. The adsorption of Span-Tween blend at water/oil surface of emulsion has been investigated using a population balance model for the first time. Destability of emulsion was modeled by considering sedimentation, coalescence and interfacial coalescence terms in population balance equation (PBE). The terms of coalescence efficiency and interfacial coalescence time were considered as a function of surface coverage of droplets by surfactant molecules. The surface coverage at different surfactant concentrations was determined by minimization of difference between the model predictions and experimental average droplet sizes. After optimization, the surface coverage outputs were fitted with different adsorption isotherms to evaluate the adsorption behavior of Span-Tween surfactants blend at water/oil surface. The results show that Freundlich isotherm can predict the adsorption behavior of closer to the experimental observation. Moreover, fitted parameters imply the favorable adsorption of Span-Tween blend at water/oil interface.     

The Adam-Gibbs equation and the out-of-equilibrium alpha relaxation of glass forming systems

The temperature dependence of the alpha-relaxation time out of equilibrium has been investigated by means of dielectric relaxation in a series of fragile glass formers including several polymers. The influence of physical aging on this behavior has also been studied. The experimental results have been quantitatively compared with the predictions of the Adam-Gibbs equation. It has been found that, whereas for small molecule glass formers the experimental values of the apparent activation energy agree quite well with the prediction of the Adam-Gibbs equation, for polymers the experimental activation energy values are systematically higher. Moreover, whereas for small molecule glass formers the experimental values of the apparent activation energy remains essentially unaffected by physical aging, for polymers a pronounced reduction of the experimental apparent activation energy is observed. These results are found to be consistent with the Adam-Gibbs equation if a significant temperature variation of the configurational entropy in the investigated temperature range would occur for nonannealed polymers, being the possible variation hardly noticeable for the small molecules. With this assumption, all the obtained results would support the validity of the Adam-Gibbs equation for describing the temperature dependence of the time scale of the alpha-relaxation also out of equilibrium, at least for fragile glass formers.     

Ordinary differential equation for local accumulation time

Cell differentiation in a developing tissue is controlled by the concentration fields of signaling molecules called morphogens. Formation of these concentration fields can be described by the reaction-diffusion mechanism in which locally produced molecules diffuse through the patterned tissue and are degraded. The formation kinetics at a given point of the patterned tissue can be characterized by the local accumulation time, defined in terms of the local relaxation function. Here, we show that this time satisfies an ordinary differential equation. Using this equation one can straightforwardly determine the local accumulation time, i.e., without preliminary calculation of the relaxation function by solving the partial differential equation, as was done in previous studies. We derive this ordinary differential equation together with the accompanying boundary conditions and demonstrate that the earlier obtained results for the local accumulation time can be recovered by solving this equation.     

The significance of the surface condition in solutions to the diffusion equation: explaining “anomalous” sigmoidal, Case II, and Super Case II absorption behavior

Absorption into polymers is frequently described by the terms Fickian, sigmoidal (S-shaped), Case II, or Super Case II. This terminology is used to describe absorption that is respectively, linear with the square root of time, has a slight delay or S-shape with the square root of time, is linear with linear time, or increases more rapidly than with linear time. Solutions to the diffusion equation, Fick’s second law, that include a potentially significant surface condition are shown to reproduce all of these. Sigmoidal absorption results when the surface condition is moderately significant for either a constant diffusion coefficient or exponential diffusion coefficients. Exponential diffusion coefficients and a lower surface mass transfer coefficient result in Case II type behavior, with Super Case II behavior resulting when the surface condition becomes still more significant. The results reported here are supported by extensive experimental data with reasonable and verifiable values for the diffusion coefficients and surface mass transfer coefficients.     

Derivation of Martin-Hou Equation of State from Hard-particle Perturbation Theory

IntroductionIn the mid 1950's, an empirical equation of state was proposed by Martin and HouLI] abbreviated to the M--H equation. It is a more accurate equation of state applicable to real gasesincluding polar compounds, such as ammonia and water over a wide range of density and temperature. In 1981, the M--H equation was extended further to liquid domain by Hou et al. [2),and later, it was used to calculate the vapor--liquid equilibrium, liquid--liquid equilibrium andother thermodynamic p…     

Modeling of penetrant diffusion in glassy polymers with an integral sorption deborah number

A mathematical model was developed to explain the anomalous penetrant diffusion behavior in glassy polymers. The model equations were derived by using the linear irreversible thermodynamics theory and the kinematic relations in continuum mechanics, showing the coupling between the polymer mechanical behavior and penetrant transport. The Maxwell model was used as the stress–strain constitutive equation, from which the polymer relaxation time was defined. An integral sorption Deborah number was proposed as the ratio of the characteristic relaxation time in the glassy region to the characteristic diffusion time in the swollen region. With this definition, an integral sorption process was characterized by a single Deborah number and the controlling mechanism was identified in terms of the value of the Deborah number. The model equations were two coupled nonlinear differential equations. A finite difference method was developed for solving the model equations. Numerical simulation of integral sorption of penetrants in glassy polymers was performed. The simulation results show that (1) the present model can predict Case II transport behavior as well as the transition from Case II to Fickian diffusion and (2) the integral sorption Deborah number is a major parameter affecting the transition. © 1993 John Wiley & Sons, Inc.     

Structure–dynamics relationships of the α‐relaxation in flexible copolyesters during crystallization as revealed by real‐time methods

The evolution of the α‐relaxation during an isothermal crystallization process of a series of flexible copolyesters of hydroxybutyrate (HB) and hydroxyvalerate (HV) has been followed in real‐time by wide‐angle X‐ray scattering and dielectric complex permittivity measurements. The change of the dielectric parameters with crystallization time can be phenomenologically described in terms of the Havriliak‐Negami equation. The dielectric strength follows a sigmoidal‐shaped pattern similar to that shown by the crystallinity. A reduction of the overall mobility with crystallization time of the polymeric chains in the amorphous phase has been observed. This slowing down effect depends on the HV molar content. The influence of the chain flexibility on the crystalline‐induced restriction has been discussed in the light of similar studies carried out with more rigid polymers. Dielectric experiments suggest that the progressive immobilization of polymer segments as crystallization proceeds cannot be exclusively associated with the amount of crystalline material. Differences in microstructure, depending on the HV molar content, seem to be responsible for the observed behavior. The progressive broadening and symmetrization of the α‐relaxation with increasing crystallization time has been explained as due to a restriction of the large‐scale motions of the polymeric chains, as the material is being filled in with crystals. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 37–49, 1999     

The PVT Properties Calculation of Water by BWR Equation

The classical theory of the equation of state first formulated by van der Waals1 has shown that a universal relation among pressure, volume, and temperature exists. In spite of the large number of equations of state that have appeared during the last century, very few people seems to have developed an equation with a reasonable number of empirical constants, and few constants have a clear physical significance. The successful empirical development is the virial type equation of state. It is…     

Solving the adsorption integral equation with Langmuir-kernel and the influence of temperature on the stability of the solution

Adsorption energy distribution functions can be calculated from measured adsorption isotherms by solving the adsorption integral equation. In this context, it is common practice to use general regularization methods, which are independent of the kernel of the adsorption integral equation, but do not permit error estimation. In order to overcome this disadvantage, we present in this paper a solution theory which is tailor-made for the Langmuir kernel of the adsorption integral equation. The presented theory by means of differentiation and Fourier series is the basis for a regularization method with explicit terms for error amplification. By means of simple and complicated adsorption energy distribution functions we show for ideal gas adsorption isotherms without measurement error that reliable distribution functions can be obtained from the isotherms. Furthermore we show how the stability of the solution depends on temperature.     

Comment on "The surface potential of a spherical colloid particle: functional theoretical approach"

In a work published in this journal by Z.W. Wang, G.Z. Li, D.R. Guan, X.Z. Yi, and A.J. Lou [J. Colloid Interface Sci. 246 (2002) 302], an iterative method for the determination of the potential around a colloidal particle is presented. It is claimed that successive terms of the iteration series converge to the exact solution of the Poisson-Boltzmann equation. This claim seems to be unfounded when the analytical expressions of the iteration terms are compared with well established numerical data.     

Constitutive equations for elastomers

General relations were derived by expanding the strain-energy density function in terms of the invariants of the deformation tensor. Some constitutive equations obtained by keeping a third term in the expansion in addition to the two terms retained in the Mooney-Rivlin equation were tested in the light of currently available experimental data. It is shown that by the retention of the third term the upswing in the Mooney stress at low values of λ^?1 is successfully predicted, and the stress–strain behavior can be described with excellent accuracy up to break, even in carbon black-filled rubber which is notoriously difficult to describe by the Mooney-Rivlin equation.     

Adapted numerical modelling of the Belousov–Zhabotinsky reaction

Adapted numerical schemes for the integration of differential equations generating periodic wavefronts have reported benefits in terms of accuracy and stability. This work is focused on differential equations modelling chemical phenomena which are characterized by an oscillatory dynamics. The adaptation is carried out through the exponential fitting technique, which is specially suitable to follow the apriori known qualitative behavior of the solution. In particular, we have merged this strategy with the information coming from existing theoretical studies and especially the observation of time series. Numerical tests will be provided to show the effectiveness of this problem-oriented approach.     

Application of an improved simplified perturbed hard chain theory equation of state for pure compounds and binary asymmetric mixtures

The simplified perturbed hard chain theory equation of state is modified using recently developed repulsive and attractive terms. Both terms meet the low density and close-packed density boundary conditions and are in reasonable agreement with molecular simulation data. The modified equation of state accurately predicts pure component properties including saturated vapor volume, liquid density, vapor pressure and enthalpy of vaporization. Compared with the original equation, the modified one predicts physical properties more accurately. However, the improvement of predicting enthalpy of vaporization is marginal. The two equations are tested for predicting the phase behavior of binary asymmetric mixtures. It is shown that the difference in predicting the phase behavior is not appreciable.     

Semiclassical initial value series representation in the continuum limit: application to vibrational relaxation

A recently formulated continuum limit semiclassical initial value series representation (SCIVR) of the quantum dynamics of dissipative systems is applied to the study of vibrational relaxation of model harmonic and anharmonic oscillator systems. As is well known, the classical dynamics of dissipative systems may be described in terms of a generalized Langevin equation. The continuum limit SCIVR uses the Langevin trajectories as input, albeit with a quantum noise rather than a classical noise. Combining this development with the forward-backward form of the prefactor-free propagator leads to a tractable scheme for computing quantum thermal correlation functions. Here we present the first implementation of this continuum limit SCIVR series method to study two model problems of vibrational relaxation. Simulations of the dissipative harmonic oscillator system over a wide range of parameters demonstrate that at most only the first two terms in the SCIVR series are needed for convergence of the correlation function. The methodology is then applied to the vibrational relaxation of a dissipative Morse oscillator. Here, too, the SCIVR series converges rapidly as the first two terms are sufficient to provide the quantum mechanical relaxation with an estimated accuracy on the order of a few percent. The results in this case are compared with computations obtained using the classical Wigner approximation for the relaxation dynamics.     

Revision of a multiparameter equation of state to improve the representation in the critical region: application to water

We have developed a crossover formalism for the thermodynamic surface of pure fluids, which can be applied to any multiparameter equation of state. This procedure has been used to incorporate scaling law behavior into a representation of the thermodynamic properties of water and steam developed by Pruss and Wagner (PW EOS) and adopted recently by the International Association for the Properties of Water and Steam. Our revision to this equation retains most of the functional form and coefficients of the PW EOS, but replaces two of the terms with a crossover representation of scaling law behavior. In order to develop this model, we first developed a new crossover formulation for steam in the critical region, and second, we have incorporated universal crossover functions into the original PW EOS. In the modified form, the PW equation of state reproduces the scaling laws down to dimensionless temperatures τ=10⁻⁷. Far from the critical point the equations practically coincide.     

Scaled particle theory for hard sphere pairs. I. Mathematical structure

We develop an extension of the original Reiss-Frisch-Lebowitz scaled particle theory that can serve as a predictive method for the hard sphere pair correlation function g(r). The reversible cavity creation work is analyzed both for a single spherical cavity of arbitrary size, as well as for a pair of identical such spherical cavities with variable center-to-center separation. These quantities lead directly to a prediction of g(r). Smooth connection conditions have been identified between the small-cavity situation where the work can be exactly and completely expressed in terms of g(r), and the large-cavity regime where macroscopic properties become relevant. Closure conditions emerge which produce a nonlinear integral equation that must be satisfied by the pair correlation function. This integral equation has a structure which straightforwardly generates a solution that is a power series in density. The results of this series replicate the exact second and third virial coefficients for the hard sphere system via the contact value of the pair correlation function. The predicted fourth virial coefficient is approximately 0.6% lower than the known exact value. Detailed numerical analysis of the nonlinear integral equation has been deferred to the subsequent paper.