Equilibrium Dispersion Research Articles

Modelling the pH dependent retention and competitive adsorption of charged and ionizable solutes in mixed-mode and reversed-phase liquid chromatography

This study investigated the influence of pH on the retention of solutes using a mixed-mode column with carboxyl (-COOH) groups acting as weak cation exchanger bonded to the terminal of C18 ligands (C18-WCX column) and a traditional reversed-phase C18 column. First, a model based on electrostatic theory was derived and successfully used to predict the retention of charged solutes (charged, and ionizable) as a function of mobile phase pH on a C18-WCX column. While the Horváth model predicts the pH-dependent retention of ionizable solutes in reversed-phase liquid chromatography (RPLC) solely based on solute ionization, the developed model incorporates the concept of surface potential generated on the surface of the stationary phase and its variation with pH. To comprehensively understand the adsorption process, adsorption isotherms for these solutes were individually acquired on the C18-WCX and reversed-phase C18 columns. The adsorption isotherms followed the Langmuir model for the uncharged solute and the electrostatically modified Langmuir model for charged solutes. The elution profiles for the single components were calculated from these isotherms using the equilibrium dispersion column model and were found to be in close agreement with the experimental elution profiles. To enable modelling of two-component cases involving charged solute(s), a competitive adsorption isotherm model based on electrostatic theory was derived. This model was later successfully used to calculate the elution profiles of two components for scenarios involving (a) a C18 Column: two charged solutes, (b) a C18 Column: one charged and one uncharged solute, and (c) a C18-WCX Column: two charged solutes. The strong alignment between the experimental and calculated elution profiles in all three scenarios validated the developed competitive adsorption model.

Simulation of Fixed-Bed Chromatographic Processes Considering the Nonlinear Adsorption Isotherms.

This paper presents the numerical approximation of a nonlinear equilibrium-dispersive (ED) model of multicomponent mixtures for simulating single-column chromatographic processes. Using Danckwerts boundary conditions (DBCs), the ED is studied for both generalized and standard bi-Langmuir adsorption isotherms. Advection-diffusion partial differential equations are used to represent fixed-bed chromatographic processes. As the diffusion term is significantly weaker than the advection term, sophisticated numerical techniques must be applied for solving such model equations. In this study, the model equations are numerically solved by using the Runge-Kutta discontinuous Galerkin (RKDG) finite element method. The technique is designed to handle sudden changes (sharp discontinuities) in solutions and to produce highly accurate results. The method is tested with several case studies considering different parameters, and its results are compared with the high-resolution finite volume scheme. One-, two-, and three-component liquid chromatography elutions on fixed beds are among the case studies being considered. The dynamic model and its accompanying numerical case studies provide the initial step toward continuous monitoring, troubleshooting, and effectively controlling the chromatographic processes.

The Competitive Adsorption of Water and Methanol on a Hybrid Silica Stationary Phase in Supercritical Fluid Chromatography

This study investigates the adsorption of methanol, water, and their mixture in a hybrid silica stationary phase with supercritical carbon dioxide as a mobile phase in supercritical fluid chromatography (SFC). The adsorption isotherms of methanol and water were determined by two dynamic methods: the elution by characteristic point (ECP) method and the inverse method (IM). Both the single-component and competitive bi-Langmuir models were pre-selected for the inverse method. The initial parameters of the single-component isotherm for both methanol and water were estimated with the ECP method by fitting the experimental data to the bi-Langmuir isotherm model. Then, using the inverse method, we refined the single-component isotherm parameter values, which were then further used for determining the competitive isotherm of the methanol–water mixture. The elution profile of the (methanol–water) mixture sample was calculated by the equilibrium-dispersive (ED) model. The results indicated that there is a good agreement between the experimental band profile and the calculated band profile, which was obtained from the parameters of the competitive bi-Langmuir isotherm model, revealing a competition between methanol and water to reach the adsorption sites. Furthermore, the saturation capacity of the adsorption sites in the stationary phase decreased in the case of the mixture sample compared to those for the single-component sample.

On the apparent dispersion coefficient of the equilibrium dispersion model: An asymptotic analysis

To model chromatography, researchers have developed several approaches. These cover a broad range of applications and, depending on the assumptions adopted, have different levels of accuracy. In general, the most suitable modelling approach is the simplest that can describe a process with the desired accuracy. A model that often meets this criterion is the equilibrium dispersion model (EDM). This features one mass balance equation per analyte, including an axial dispersion term, and assumes the analyte concentrations in the mobile and stationary phases to be in local equilibrium. To account for the finite mass transfer rate between the phases, the model employs an apparent dispersion coefficient. Two expressions are available for this coefficient, one being used much more frequently than the other. In this paper, we aimed to clarify which one should be favoured. A desirable feature of simple models is that they can be derived from more general ones with appropriate physical assumptions and rigorous mathematical methods. Thus, to answer our research question, we derived the EDM from the more general pore diffusion model (POR), using an asymptotic method. The expression obtained for the apparent dispersion coefficient does agree with one of the two reported in the literature – the less frequently used. To test the validity of this expression, we simulated elution profiles using the two versions of the EDM and compared the results against those from the POR model. The simulations were conducted in the range where the POR and EDM models should be essentially equivalent, their results confirming the outcome of the asymptotic analysis. This work offers a solid theoretical grounding for the EDM, clarifies which formulation of the model is correct, and provides usable applicability conditions for the model.

Virial masses of late-type galaxies from the SDSS DR16

ABSTRACT Motivated by the challenges of calculating the dynamical masses of late-type galaxies (LTGs) and the enormous amount of data from the Sloan Digital Sky Survey (SDSS), we calculate virial masses of a sample of approximately 126 000 LTGs from the sixteenth data release of the SDSS. The virial mass estimations were made considering Newtonian mechanics, virial equilibrium and velocity dispersion from stars and gas. The procedure gave as a result seven mass estimations for each galaxy. The calculated masses were calibrated using a sample of spiral galaxies with velocity rotation curves. Considering the results from the calibration, we find that the correlation between virial and dynamical (rotation curve) masses is stronger for high inclination values. Therefore, the calibration relies more on the available data for higher inclination angle galaxies. We also show that if we have a heterogeneous sample of galaxies one must take into consideration the size and colour of these galaxies by using the following variables: Sersic index n, concentration index, and colour of the stars. For relatively smaller and bluer LTGs, the gas velocity dispersion provides a more consistent mass calculation, while for LTGs that are relatively larger and redder the stellar velocity dispersion provides a better correlated mass calculation.

Analysis and experimental demonstration of temperature step gradients in preparative liquid chromatography

The amount of substance adsorbed on solid surface depends on temperature. Therefore, the migration velocities of the solutes in a chromatographic column can be altered by introducing temperature gradients. Such gradients designed to change retention behaviours can be exploited to improve the separation performances in preparative chromatography. To describe key process features, we used analytical solutions of the equilibrium model with instant stepwise shift of temperature. To achieve a more realistic description, the equilibrium dispersion model was additionally applied to treat finite column efficiencies. The effect of temperature gradients was illustrated experimentally using two identical columns sequentially connected. Temperature of the second column was modulated by thermostats. Wide pulse injections of a single component led to instructive elution profiles in a preliminary investigation. The observations were found to be in qualitative agreement with predictions of the equilibrium dispersion model. Subsequently, the separation of a ternary model mixture was investigated considering a simple two-step temperature gradient. To support the quantitative analysis and to identify suitable switching and cycle times, the temperature dependencies of the Henry constants were determined by short pulse injections. A meaningful variation of the parameters of the temperature gradient is required for adjusting the cycle times, which is the time difference between two consecutive injections that needs to shorten. Decreasing this time is connected with a desirable increase in process productivity. The results achieved revealed that relatively simple to implement stepwise temperature gradients offer an option to improve and fine-tune the performance of repetitive batch chromatography.

Research progress on the correlation between properties of nanoparticles and their dispersion states in polymer matrix

AbstractWhen fabricating polymer nanocomposites (PNCs), it is difficult for nanoparticles to disperse stably in polymer matrix since their high‐surface energy can cause them to attract each other. In this paper, we review the effects of nanoparticle size, surface chemical properties, and thermodynamic parameters on the dispersion state of nanoparticles (including inorganic and organic particles) in PNCs and describe some key measures to promote the nanoparticles dispersion. The ideal nanoparticles dispersion in polymer melt is available when the particle size is smaller than the rotation radius of polymer chains and the particle surface is highly compatible or has interactions with the matrix. In polymer solution, establishing proper thermodynamic parameters is necessary to ensure the balance interactions among particles, solvents, and polymers, leading to the successful preparation of PNCs with ideal structure. These results could supply theoretical suggestions for the design of PNCs. Moreover, the equilibrium dispersion may be unavailable without reasonable kinetic parameters, even if the dispersion process is thermodynamically favorable. Therefore, more studies are required on both thermodynamics and kinetics of the dispersion process.

Microfluidic osmotic compression of a charge-stabilized colloidal dispersion: Equation of state and collective diffusion coefficient.

We show, using a model coupling mass transport and liquid theory calculations for a charge-stabilized colloidal dispersion, that diffusion significantly limits measurement times of its equation of state (EOS), osmotic pressure vs composition, using the osmotic compression technique. Following this result, we present a microfluidic chip allowing one to measure the entire EOS of a charged dispersion at the nanoliter scale in a few hours. We also show that time-resolved analyses of relaxation to equilibrium in this microfluidic experiment lead to direct estimates of the collective diffusion coefficient of the dispersion in Donnan equilibrium with a salt reservoir.

Thermodynamic interpretation of the drift and noise of gradient baselines in reversed-phase liquid chromatography using mobile phase additives

The drift and noise of acetonitrile-water gradient baselines (5-95%, v/v, 5 min linear gradient) in reversed phase liquid chromatography (RPLC) are recorded at a wavelength of 215 nm using 0.1% trifluoroacetic acid (TFA) as the mobile phase additive, a 4.6 mm × 150 mm 5 μm Symmetry-C18 RPLC column, and an Arc system (low-pressure gradient proportioning valve or GPV, pump with a stroke volume of either 66 or 132 μL, no mixer) as the LC instrument. These observations are predicted from solid-liquid adsorption thermodynamics which requires the measurement of the excess adsorption isotherm of acetonitrile from water onto the RPLC column and of the variation of the Henry’s constant of TFA as a function of the volume fraction of acetonitrile in the bulk mobile phase. The incomplete mixing of the acetonitrile and water packets delivered by the low-pressure GPV is represented by a sinusoidal perturbation of the programmed volume fraction of acetonitrile during the entire gradient. The variation of the TFA absorbance at 215 nm with increasing acetonitrile concentration is measured in order to transform TFA concentration into the observable absorbance unit. The drift and noise of the gradient baseline are calculated by solving numerically (Rouchon method) the equilibrium-dispersive (ED) mass balance equations of acetonitrile and TFA.The agreement between the calculated and observed gradient baselines is very good as the proposed model of chromatography accurately accounts for the displacement of TFA between stationary and mobile phases (early excess and late deficit of TFA concentration relative to 0.1%) and for the frequency (equal to the ratio of the applied flow rate to the stroke volume) and the amplitude of the periodic noise recorded during the gradient. From a practical viewpoint, the drift of the gradient baseline can be minimized by maximizing the ratio of the gradient volume to the hold-up volume ( > 10) and/or by minimizing the retention factor of the mobile phase additive in the water-rich eluent (k < 0.2). The reduction of the noise amplitude below 0.1 mAU as requested by the pharmaceutical industry imposes the ratio of the flow rate to the stroke volume of the pump to be larger than 1 Hz. This opens avenues towards the development of new GPV, pump, and mixers in order to mix efficiently the solvent packets delivered by conventional LC instrument.

Separation of eicosapentaenoic acid and docosahexaenoic acid by three-zone simulated moving bed chromatography

Eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA) are essential fatty acids for human body, which are widely used in the field of healthy food and medicine. Meanwhile, there are some differences in their physiological functions, such as “scavenger for blood vessel” of EPA and “brain protector” of DHA. In order to make full use of EPA and DHA, it is necessary to prepare their high-purity component. In this paper, EPA and DHA were separated and purified by three-zone simulated moving bed (SMB) chromatography with C18 used as stationary phase and ethanol-water as mobile phase. For the single column experiment, a separation unit of SMB, the effects of the ratio of ethanol to water, pH value and temperature on the separation were investigated. The equilibrium dispersion (ED) model was used to obtain the adsorption parameters of EPA and DHA by inverse method and genetic algorithm, and the accuracy of the adsorption parameters was verified by fitting the overloaded elution curves under different conditions. Based on the acquired nonlinear adsorption isotherms the complete separation region was found according to triangle theory. The effects of sample concentration, flow ratios of adsorption zone and rectification zone, and column distribution mode of SMB on the separation were investigated. Under the optimized SMB conditions, the experimental result was that without regard to the other components, the chromatographic purity and recovery values of EPA and DHA exceeded 99% with the productivity of 4.15 g/L/h, and the solvent consumption of 1.11 L/g.

Quasiclassical method for calculating the density of states of ultracold collision complexes

We derive a quasiclassical expression for the density of states (DOS) of an arbitrary, ultracold, $N$-atom collision complex, for a general potential energy surface (PES). We establish the accuracy of our quasiclassical method by comparing to exact quantum results for the K$_2$-Rb and NaK-NaK systems, with isotropic model PESs. Next, we calculate the DOS for an accurate NaK-NaK PES to be 0.124~$\mu$K$^{-1}$, with an associated Rice-Ramsperger-Kassel-Marcus (RRKM) sticking time of 6.0~$\mu$s. We extrapolate the DOS and sticking times to all other polar bialkali-bialkali collision complexes by scaling with atomic masses, equilibrium bond lengths, dissociation energies, and dispersion coefficients. The sticking times calculated here are two to three orders of magnitude shorter than those reported by Mayle et al. [Phys. Rev. A 85, 062712 (2012)]. We estimate dispersion coefficients and collision rates between molecules and complexes. We find that the sticking-amplified three-body loss mechanism is not likely the cause of the losses observed in the experiments.

Modeling the competitive adsorption of sample solvent and solute in supercritical fluid chromatography

Uncommon retention behavior of a series of n-alkylbenzene homologues as well as the effect of different sample solvents on chromatographic efficiency were studied in supercritical fluid chromatography. After testing various columns, an alkylamide stationary phase was selected for detailed studies. The results showed that even a small amount of methanol originating only from the sample, overloaded the column and competitive adsorption was induced between the analytes and the sample solvent for adsorption on the stationary phase. This was indicated by the changes in column efficiency, retention and peak widths. The concentration of the analytes in the sample was negligible compared to the amount of methanol – but their adsorption was influenced by the solvent – while the adsorption of methanol remained unaffected by the n-alkylbenzenes. First, the competition was described by determining the single-component adsorption isotherms for both the analytes and their solvent, then competitive isotherms were calculated. Based on the peak profiles, bi-Langmuir and competitive bi-Langmuir isotherms were assumed. The solvent effect was modeled by a numerical method created in-house where the differential mass balance equation given by the equilibrium–dispersive (ED) model was integrated using the Rouchon algorithm. The experimental observations were confirmed by in silico experiments and additional cases involving two hypothetical analytes were studied as well.

Competition in Designing Pandora's Boxes

We revisit the classic sequential-search model by Weitzman (1979) in which an agent is presented with a number of boxes containing uncertain prizes, from which he selects one prize after sequentially opening the boxes and learning the prize contained within. In our model, each box is owned by an independent and ex-ante identical designer, whose objective is to get the agent to select the prize of his box. Each designer persuades the agent to pick his prize by designing the prize distribution of his box, subject to the constraints that (i) the expected value of the distribution is a constant, and (ii) the size of prize realizations is bounded. We fully characterize all symmetric equilibria, and show that they have a simple linear structure. Moreover, we find that a decrease in the agent's search cost mitigates competition, thus benefitting the designers. Furthermore, when competition is mild, all symmetric equilibria are in mixed strategies. By interpreting prize distributions as firms' product-design choices, this finding suggests that equilibrium dispersion in product designs can arise in a sequential-search setting with ex-ante identical firms and consumers.

Dissipative instabilities in a partially ionised prominence plasma slab

This study deals with the dissipative instability that appears in a compressible partially ionised plasma slab embedded in a uniform magnetic field, modelling the state of the plasma in solar prominences. In the partially ionised plasma, the dominant dissipative effect is the Cowling resistivity. The regions outside the slab (modelling the solar corona) are fully ionised, and the dominant mechanism of dissipation is viscosity. Analytical solutions to the extended magnetohydrodynamic (MHD) equations are found inside and outside of the slab and solutions are matched at the boundaries of the slab. The dispersion relation is derived and solutions are found analytically in the slender slab limit, while the conditions necessary for the appearance of the instability is investigated numerically for the entire parameter space. Our study is focussed on the effect of the compressibility on the generation and evolution of instabilities. We find that compressibility reduces the threshold of the equilibrium flow, where waves can be unstable, to a level that is comparable to the internal cusp speed, which is of the same order of flow speeds that are currently observed in solar prominences. Our study addresses only the slow waves, as these are the most likely perturbations to become unstable, however the time-scales of the instability are found to be rather large ranging from 105–107 s. It is determined that the instability threshold is further influenced by the concentration of neutrals and the strength of the viscosity of the corona. Interestingly, these two latter aspects have opposite effects. Our numerical analysis shows that the interplay between the equilibrium flow, neutrals and dispersion can change considerably the nature of waves. Despite employing a simple model, our study confirms the necessity of consideration of neutrals when discussing the stability of prominences under solar conditions.

Price Dispersion in Dynamic Equilibrium

In product markets, there exists substantial dispersion in prices for transactions of physically identical goods, and incumbent sellers sell at higher prices than entrants. This study develops a theory of dynamic pricing that explains these facts as results from the same fundamental friction: Buyers are imperfectly aware of which sellers are operating, and the degree of awareness about a seller is endogenous. The equilibrium is unique and efficient, and features randomized pricing strategies where incumbents post higher prices than entrants. If buyers' memory depreciation is low, then the equilibrium of the industry tends to approximate perfectly competitive conditions over time.

Linearly implicit-explicit schemes for the equilibrium dispersive model of chromatography

Numerical schemes for the nonlinear equilibrium dispersive (ED) model for chromatographic processes with adsorption isotherms of Langmuir type are proposed. This model consists of a system of nonlinear, convection-dominated partial differential equations. The nonlinear convection gives rise to sharp moving transitions between concentrations of different solute components. This property calls for numerical methods with shock capturing capabilities. Based on results by Donat, Guerrero and Mulet (Appl. Numer. Math. 123 (2018) 22–42), conservative shock capturing numerical schemes can be designed for this chromatography model. Since explicit schemes for diffusion problems can pose severe stability restrictions on the time step, the novel schemes treat diffusion implicitly and convection explicitly. To avoid the need to solve the nonlinear systems appearing in the implicit treatment of the nonlinear diffusion, second-order linearly implicit-explicit Runge–Kutta schemes (LIMEX-RK schemes) are employed. Numerical experiments demonstrate that the schemes produce accurate numerical solutions with the same stability restrictions as in the purely hyperbolic case.

Implicit–Explicit WENO scheme for the equilibrium dispersive model of chromatography

Chromatographic processes can be modeled by nonlinear, convection-dominated partial differential equations, together with nonlinear relations: the adsorption isotherms. In this paper we consider the nonlinear equilibrium dispersive (ED) model with adsorption isotherms of Langmuir type. We show that, in this case, the ED model can be written as a system of conservation laws when the dispersion coefficient vanishes. We also show that the function that relates the conserved variables and the physically observed concentrations of the components in the mixture is one to one and it admits a global inverse, which cannot be given explicitly, but can be adequately computed.As a result, fully conservative numerical schemes can be designed for the ED model in chromatography. We propose a Weighted-Essentially-non-Oscillatory second order IMEX scheme and describe the numerical issues involved in its application. Through a series of numerical experiments, we show that our scheme gives accurate numerical solutions which capture the sharp discontinuities present in the chromatographic fronts, with the same stability restrictions as in the purely hyperbolic case.

Search intensity, wage dispersion and the minimum wage

We study a labor market where employers post wages and workers simultaneously choose the number of applications they send out. Firms offer the job to a worker at random; workers with multiple offers pick the best one. If the application costs are sufficiently low, workers contact multiple firms and there is wage dispersion in equilibrium. The number of applications workers send out is excessive from a welfare perspective due to a rent seeking externality. A mandatory minimum wage increases the mean and reduces the variance of the wage distribution. The net effect on welfare is ambiguous.

Using the fundamentals of adsorption to understand peak distortion due to strong solvent effect in hydrophilic interaction chromatography

The peak distortion observed in hydrophilic interaction chromatography (HILIC) may be caused by the sample diluent to mobile phase mismatch. The United States Pharmacopeia (USP) method for organic impurities in cetirizine HCl tablets calls for such a mismatch, having a higher concentration of strong solvent in the sample diluent than in the mobile phase. A significant peak deformation is reported for cetirizine (a second-generation antihistamine) when it is purified on a Ethylene Bridged Hybrid (BEH) HILIC column (4.6mm×100mm, 2.5μm particles) using an acetonitrile–water eluent mixture and a sample diluent containing 7% and 9% water (in volume), respectively. The mechanism and physical origin of such peak distortion are related to (1) the diluent-to-eluent excess of water that propagates along the column at a velocity similar to that of the analyte, (2) the significant drop of the Henry's constant of the analyte upon increasing water concentration in the eluent, (3) the sample volume injected, and (4) to the pre-column sample dilution factor that depends on the characteristics of the LC instrument used.This proposed mechanism is validated from the calculation of the concentration profiles of cetirizine and water by using the equilibrium-dispersive (ED) model of chromatography. The observed distortion of cetirizine peaks is successfully predicted from the measurement of (1) the excess adsorption isotherm of water from acetonitrile onto the BEH HILIC adsorbent, (2) the retention factor of cetirizine as a function of the volume fraction (7, 8, and 9%) of water in the mobile phase, and (3) of the pre-column sample dispersion related to the instrument used (HPLC or UHPLC). The results of the calculations enables the user to anticipate the impacts of the diluent-to-eluent mismatch in water content, the injection volume, the analyte retention under infinite dilution, and of the pre-column sample dispersion on the amplitude of peak distortion in HILIC. Appropriate and permitted alterations of the USP method are then suggested based on a sound physico-chemical approach.

Determination of adsorption and kinetic parameters for methyl oleate (biodiesel) esterification reaction catalyzed by Amberlyst 15 resin

In this work, adsorption equilibrium constants, dispersion coefficients, and kinetic parameters were determined for the reversible esterification reaction of oleic acid with methanol, producing methyl oleate (biodiesel) and water. The reaction was carried out at room temperature in the presence of Amberlyst 15 resin, which acts both as an adsorbent and catalyst in an HPLC column, which served as a packed bed reactor. A quasi‐homogenous kinetic model coupled with a linear adsorption isotherm was explored. The elution profiles of the reactant and products were experimentally monitored and then compared with a mathematical model. The adsorption and kinetic parameters were determined by minimizing the error between the experimentally obtained elution curves and the model‐predicted values using the genetic algorithm optimization technique. Further experiments were conducted under varying conditions to establish the validity of the obtained model parameters. It was also found that the system is not affected by internal and external mass transfer resistances. The mathematical model predicted the experimental outcome quite accurately.