Torben Smith Sørensen

Electric impedance of cellulose acetate membranes and a composite membrane at different salt concentrations

Impedance measurements have been performed on dense and asymmetric cellulose acetate membranes cast in our laboratory. The frequency range was from 1 mHz to 65 kHz. Different concentrations of NaCl in water at 25°C were used. Two different potentiostats and different types of measurement cells were used and compared. The repeatability of the time constants was within ±7% except at extreme dilutions. A hybrid RC model is proposed, where the resistance (R) is calculated from the Donnan distribution of the ions and the ratios of membrane diffusion coefficients found in earlier work. The diffusion constant of Na+ is determined as a new parameter, taking the fixed charge of the membrane from previous data. DNA+ is of the order of 10−8–10−9 cm2/s. At low NaCl concentrations the diffusion coefficient of Na+ decreases, probably due to electrostatic binding to the unshielded glucuronic acid groups. The capacity (C) exhibited a marked increase with concentration, most drastically seen in the case of asymmetric membranes. This feature is explained using a theory of Trukhan combined with the integration method of Bruggemann. The effect arises from the dynamically created double layer generated by the applied field, when ions are confined in spherical alveoles. Alveolar sizes of about 70 ± 30 A are determined for the dense membranes. For the asymmetric we determine 2000 ± 1000 A, but the alveoles here seem to form a (fractal?) hierarchy of sizes. This is reflected in the lower value of the Cole-Cole α-parameter, especially at high NaCl concentrations. Heat curing of an asymmetric membrane only influenced the resistance, indicating a tightening of the pores in the skin layer. In contrast to the Trukhan-Bruggemann theory, the Maxwell-Wagner-Sillars theory is not able to account for the observed large variations in the membrane capacitances with concentration. A commercial composite membrane for seawater desalination (Fluid Systems) shows two relaxations corresponding to a skin layer and a support layer. The capacitance of the polysulfone support layer is determined by the Trukhan-Bruggemann theory, too.

Chemical and hydrodynamical analysis of stability of a spherical interface

Abstract The hydrodynamical and chemical stability of deformation of the interface of a spherical drop suspended in an infinite amount of another immiscible liquid is investigated by the methods of linear, hydrodynamical stability theory. The two bulk fluids are homogeneous and continuous throughout. A general determinantal dispersion relation is evaluated between the complex frequency of the perturbation and the number characterising the surface harmonic normal mode of perturbation in the case of an arbitrary number of fluctuating and reacting species on the interface. The coupling between chemical reactions, surface diffusion, and hydrodynamics is effected by the interfacial through the equation of state of the interface and by convection motions on the interface. Surface shear and dilatational viscosity are taken into account assuming the surface fluid to be Newtonian. The stability of a stationary state of the interfacial chemical reaction with the bulk fluids in hydrodynamical rest with regard to small perturbations in the surface concentrations and in the velocity of the fluid is then studied. The fluxes from the bulk fluids to the interface remain constant, or otherwise they are perturbed proportional to the fluctuations in the surface concentrations. The case of one fluctuating species and small drop radii is treated in detail. The necessary condition for the system to be unstable is that the surface chemical reaction is unstable itself. In addition, the coefficient of autocatalysis of the surface reactions has to exceed a threshold value composed by the quenching effects of surface diffusion and of the bulk and surface viscosities. In cases with more than one fluctuating species there exist possibilities for the total system to be unstable even for stable surface reactions. The present theory is an extension of the theory of oscillations of a viscous drop due to capillary forces. It is thought to be an introduction to the study of “kicking drops” and motile events connected with the deformation of the biological cell membrane.

Monte Carlo simulations of single-ion chemical potentials. Preliminary results for the restricted primitive model

Abstract Widoms formula has been used to calculate activity coefficients for the restricted primitive model of electrolyte systems in Monte Carlo simulations based on the Metropolis sampling procedure. The agreement with literature data based on other approaches is found to be very satisfactory. The method can be used to evaluate single-ion chemical potentials.

Complex permittivity of a conducting, dielectric layer containing arbitrary binary Nernst–Planck electrolytes with applications to polymer films and cellulose acetate membranes

The theory of Trukhan [Sov. Phys. Solid State (Engl. Transl.), 1963, 4, 2560] for the calculation of the complex permittivity of a conducting dielectric film containing a binary 1 : 1 electrolyte is critically reviewed. It is shown that the final result of Trukhan is correct, in spite of some errors and odd procedures in the original derivation. The method of Trukhan is changed to a more concise procedure which is better suited to generalisations, and the theory is generalised to binary electrolytes of any charge type. The dimensionless excess impedance (over and above the Maxwell–Wagner–Sillars impedance) and the complex relative permittivity are functions of the type of electrolyte, the dimensionless frequency, the ratio of the ionic diffusion coefficients and the film thickness scaled by the Debye length. The complex relative permittivity has a Debye-like relaxation, but for very different values of the two diffusion coefficients and for films which are thin compared with the Debye length, one can clearly distinguish two dielectric relaxations, one for each ion. The peak in the dielectric loss corresponding to the ion of higher valency, is dominating in thin films. When the film is thicker, (or when the electrolyte concentration is increased) the low-frequency peak diminishes relative to the high-frequency peak. The maximum in the loss tangent is positioned at higher frequencies than the maximum in the dielectric loss. Therefore it is often an advantage to fit the loss tangents to actual measurements, since measurements at low frequencies are generally more difficult. Two applications of the theory are treated: the low-frequency dielectric relaxation of a dry copolymer of vinylidene cyanide and vinyl acetate with conducting ions (T. Furukawa, M. Date, K. Nakajima, T. Kosaka and I. Seo, Jpn. J. Appl. Phys., 1986, 25, 1178) and a wet membrane of dense cellulose acetate (I. W. Plesner, B. Malmgren-Hansen and T. S. Sorensen, J. Chem. Soc., Faraday Trans., 1994, 90, 2381). The use of the so-called ‘constant-phase element’ is discussed and criticised. The constant-phase element leads to infinite dissipation at zero frequency in contrast with the generalised Trukhan theory (and common sense). The constant-phase element is just superficially fitting the data in a limited range of frequency and no physical information (e.g. diffusion coefficients) is obtained from the fitted parameters of such an element.

Monte Carlo simulations of single ion chemical potentials. Results for the unrestricted primitive model

Abstract Widoms formula has been used to calculate single-ion activity coefficients for the primitive model of electrolyte systems, with unequal ionic radii, in Monte Carlo simulations based on the Metropolis sampling procedure. The results are found to be in reasonable agreement with values obtained by the MSA theory.

Monte Carlo calculations of chemical potentials in ionic fluids by application of Widom's formula: Correction for finite-system effects

Abstract A method for correcting (single) ion chemical potentials obtained by Monte Carlo calculations using Widoms test particle formula is proposed. The method is shown to remove most of the considerable system-size dependence, which is found for these quantities.

Electrokinetic effects in charged capillary tubes

The electrokinetic phenomenological coefficients in a narrow tube filled with electrolyte solution and with a surface charge due to ionizable groups are calculated. The calculation holds for stationary, laminar flow of Newtonian fluids and (at least) within the domain of validity of the linearized Poisson–Boltzmann equation. Our formulae become identical to those given by Helmholtz in the limit of pores which are large compared with the Debye–Huckel length κ–1.The reciprocity relation of Onsager is shown by hydrodynamic and electrostatic arguments to be strictly fulfilled even for an arbitrary cross-section with an arbitrary charge distribution.

Improved Differential Evolution Based on Stochastic Ranking for Robust Layout Synthesis of MEMS Components

This paper introduces an improved differential evolution (DE) algorithm for robust layout synthesis of microelectromechanical system components subject to inherent geometric uncertainties. A case study of the layout synthesis of a comb-driven microresonator shows that the approach proposed in this paper can lead to design results that meet the target performance and are less sensitive to geometric uncertainties than the typical designs. It is also demonstrated that the algorithm proposed in this paper cannot only obtain better results than the standard DE algorithm but also outperform some other state-of-the-art algorithms in constrained optimization.

Ion exchange and membrane potentials in cellulose acetate membranes separating solutions of mixed electrolytes

The properties of (almost) symmetrical cellulose acetate membranes are studied by measuring the electromotive force (EMF) of concentration cells with the membrane as separator. One or two 1:1 salts with a common anion (Cl−) were used in different concentrations on each side of the membrane and AgAgCl were used as electrodes. The concentration(s) on one side of the membrane is fixed, whereas one concentration is varied on the other side. A theory is developed based upon ideal Nernst—Planck transport equations in the membrane, Donnan equilibrium at the membrane interfaces and Nernst equilibrium at electrodes. Furthermore, Hendersons continuous mixture hypothesis is used in the case of two different electrolytes. Comparing measurements and theory for pure salts (NaCl and KCl), the fixed charge of the membrane divided by the intrinsic binding constant for the salt can be evaluated. At low variable concentrations, however, considerable deviations from the one-electrolyte theory are observed. The reason is the competition of the H+ ion from the autoprotolysis of water with the Na+ or K+ ion. The measured values of EMF are therefore highly sensitive to pH at low variable concentrations. The variations with pH (pH between 5 and 8) are well described by the theory assuming unaltered fixed charge and a binding constant for H+ to the membrane which is around 50 times as great as the binding constant for Na+ or K+. The H+ ion is distributed almost evenly between aqueous phase and membrane phase, whereas K+ and Na+ are sterically excluded from the membrane. Comparison between theory and experiment further shows that half of the negative fixed charge is titrated at pH 4 and almost all has been titrated at pH 3. This indicates that the fixed charge in cellulose acetate membranes is caused by carboxylic acid groups (glucuronic acids) as has been reported also in earlier work. A certain asymmetry in the EMF measurements with pure NaCl was observed at high variable concentrations (up to 10 mV), when the membrane sides were changed. The asymmetry is ascribed to the evaporation asymmetry during preparation of the membrane. No asymmetry is observed at low or intermediate variable concentrations.

Error in the Debye–Hückel approximation for dilute primitive model electrolytes with Bjerrum parameters of 2 and ca. 6.8 investigated by Monte Carlo methods. Excess energy, Helmholtz free energy, heat capacity and Widom activity coefficients corrected for neutralising background

Monte Carlo (MC) simulations have been performed for primitive model electrolytes with a Bjerrum parameter B= 2 for five values of κa in the region ca. 0.022–0.11. Also, an extremely dilute 2 : 2 electrolyte (B= 6.8116 and κa ca. 0.0276) has been investigated. Between five and eight million configurations have been used in each simulation, and the number of ions in each simulation (N) was varied between 32 and 1728. The universal scaling of the results using the ratio of the Debye length to the half period of the periodic boundary conditions (minimum image cut-off distance), which was found for dilute systems with B= 1, 1.546 and 1.681 in an earlier paper, is found to hold also for the present simulations. In this way, precise extrapolations of excess energies (Eex/NkT), excess electrostatic Helmholtz free energies and excess heat capacities can be found.By means of an analytic correction to the Widom test particle method for simulation of activity coefficients most of the variation, with the number of ions (N), of the excess chemical potentials may be removed. The trick is to introduce a homogeneous neutralising background. The N-dependence left scales exactly like Eex/NkT, so that precise extrapolated values for the excess chemical potentials can be found.The deviations from the Debye–Huckel values increase with increasing B(at fixed κa). With B= 2, all the thermodynamic quantities except the excess heat capacity are situated between the Debye–Huckel law and the Debye–Huckel limiting law. With B= 6.8116 and κa≈ 0.0276, however, all quantities are on the ‘wrong’ side of the limiting law, and deviations from the Debye–Huckel values are very large, e.g. ca. 40% for Eex/NkT and 400–500% for the excess heat capacity. These deviations are well accounted for by the DHX theory, using the Debye–Huckel electric potential around a central ion as potential of mean force.