Viscous fingering in packed chromatographic columns: linear stability analysis

When a fluid is displaced by a less viscous one in a porous medium, a hydrodynamic instability appears leading to the formation of some kind of fingers of the upstream fluid invading the downstream one, hence the name "viscous fingering" (VF) given to this instability. In a LC column, such an instability is likely to appear at that of the two interfaces between the sample and the eluent which exhibits an unfavorable viscosity contrast. It leads to distorted peak shapes and contributes to peak broadening. This phenomenon has been observed for long in SEC and more recently in RPLC on elution peak shapes as well as with various methods of in-column visualization. A simplistic LC column model is described to explain the origin of the VF instability and its characteristics. The general principles for analyzing hydrodynamic instabilities are described and the results of the linear stability analysis performed by Tan and Homsy [C.T. Tan, G.M. Homsy, Phys. Fluids 29 (1986) 3549 [1]], at the onset of the VF phenomenon for a step interface between two fluids are here applied to typical operating conditions encountered in analytical LC. The most probable growth rate and wavelength (linked to the finger width) of the instability are given in terms of particle size and solute diffusion coefficient, with particular emphasis on the role of the carrier velocity. Previously published qualitative observations about VF in chromatography are examined and interpreted at the light of this theory. The role of the column geometry on the development of the instability, the possible sources of noise or fluctuations triggering the instability, and the various experimental situations in which a significant viscosity contrast is encountered in LC are discussed.     

Visualising viscous fingering in chromatography columns: high viscosity solute plug

The interface between two fluids that have different viscosities and are percolating through a porous bed is unstable. Sooner or later, a flow instability termed viscous fingering (VF) develops. This phenomenon is important in chromatography because the solute plug does not have the same viscosity as the mobile phase. Because the sample is often much more viscous than the mobile phase, it is the interface at the rear of the sample band that is usually unstable. This situation is frequent in many modes of chromatography, e.g., in preparative and in multidimensional chromatography, in size exclusion chromatography, in frontal analysis, and in other physicochemical measurements (e.g., determination of adsorption isotherms and of mass transfer parameters). When the solute plug is more viscous than the mobile phase, we observed that the solute band compressed. When the viscosity contrast increased up to 0.30 cP, fingers appeared to trail behind the solute plug. The development of fingers then became more substantial as the viscosity contrast increased. To avoid effects associated with VF, the mobile phase and the solute plug should have nearly the same viscosity.     

Viscous fingering induced flow instability in multidimensional liquid chromatography

Viscous fingering is a flow instability phenomenon that results in the destabilisation of the interface between two fluids of differing viscosities. The destabilised interface results in a complex mixing of the two fluids in a pattern that resembles fingers. The conditions that enhance this type of flow instability can be found in coupled chromatographic separation systems, even when the solvents used in each of the separation stages have seemingly similar chemical and physical properties (other than viscosity). For example, the viscosities of acetonitrile and methanol are sufficiently different that instability at the interface between these two solvents can be established and viscous fingering results. In coupled chromatographic systems, the volume of solvent transported from one separation dimension to the second often exceeds the injection volume by two or more orders of magnitude. As a consequence, viscous fingering may occur, when otherwise following the injection of normal analytical size injection plugs viscous fingering would not occur. The findings in this study illustrate the onset of viscous fingering in emulated coupled chromatographic systems and show the importance of correct solvent selection for optimum separation performance.     

Viscous fingering in a horizontal flow through a porous medium induced by chemical reactions under isothermal and adiabatic conditions

In this work we analyze the viscous fingering instability induced by an autocatalytic chemical reaction in a liquid flowing horizontally through a porous medium. We have analyzed the behavior of the system for isothermal as well as adiabatic conditions. The kinetics of the reaction is chosen so that the rate depends on the concentration of only a single species. Since the reaction is autocatalytic the system admits a traveling wave solution. For endothermic reactions the concentration wave and temperature wave are mirror images, whereas for an exothermic reaction they are similar or parallel. The viscosity of the fluid is assumed to depend strongly on the concentration of the product and temperature of the medium. The dependence of viscosity on concentration (decrease with concentration) can destabilize the traveling wave resulting in the formation of viscous fingers. We have performed a linear stability analysis to determine the stability of the base traveling wave solution. The stability predictions have been confirmed by nonlinear simulations of the governing equations based on a finite difference scheme. We observe that including the temperature dependency of viscosity stabilizes the flow for an endothermic reaction, i.e., regions which exhibited viscous fingering now demonstrate stable displacement. For exothermic systems, however, the system exhibits less stable behavior under adiabatic conditions, i.e., it is destabilized by both concentration and temperature dependencies of viscosity.     

Calculation of the effect of collisional broadening on high-resolution translational energy loss spectra

The collisional broadening of peaks in the spectra arising from translational energy spectroscopy (TES) studies is theoretically investigated. A numerical calculation, based on the TRIO matrix ion-beam transport computer programme, is used to simulate the collision event and its effects on the focussing properties of a number of ‘double-focussing’ instrument designs. The ion-optical models utilised include a commercial mass spectrometer and two novel high-resolution energy spectrometers (TESI and TESII), incorporating quadrupole and hexapole field lenses to focus the beam and reduce image aberrations. For a given design of spectrometer, peak broadening is evaluated in terms of the translational energy change suffered by the ion during collision and the angle through which it is scattered.     

Visualization and quantification of the onset and the extent of viscous fingering in micro-pillar array columns

New experimental data of the viscous fingering (VF) process have been generated by studying the VF process in perfectly ordered pillar array columns instead of in the traditionally employed packed bed columns. A detailed quantitative analysis of the contribution of VF to the observed band broadening could be made by following the injected species bands using a fluorescence microscope equipped with a CCD-camera. For a viscosity contrast of 0.16 cP, a plate height increase of about 1 μm can be observed, while for a contrast of respectively 0.5 cP and 1 cP, additional plate height contributions of the order of 5–20 μm were observed. Citing these values is however futile without noting that they also depend extremely strongly on the injection volume of injected sample. It was found that, for a given viscosity contrast of 0.314 cP, the maximal plate height increase varied between 0.5 μm and 18 μm if the injection volume was varied between 3.0 nl and 32.7 nl. These values furthermore also strongly vary with the distance along the column axis.     

Broadening of peaks eluted before the solvent in capillary GC Part II: The role of phase soaking

Summary Phase soaking is a solvent effect which tends to reconcentrate peaks eluted after and to broaden peaks eluted before the solvent. The principles of the phase soaking effect on peaks eluted before the solvent are discussed. The broadening effect is distinguished from the broadening effect occurring in the flooded column inlet by partial solvent trapping. It was found that in most cases broadening by partial solvent trapping strongly predominated over broadening by phase soaking. Phase soaking was noticeable only in the neighbourhood of the solvent peak.Phase soaking does not broaden peaks eluted before the solvent as much as it re-concentrates those eluted after it. The two phase soaking effects on the front and the rear of the solvent band (that is, of the soaked zone) differ strongly from each other.Peak broadening by phase soaking is negligible for non-trapped components, because such solutes start their chromatography before a significant quantity of solvent enters the column. Phase soaking only broadens solute bands which were retained by the solvent in the column inlet, that is, bands already broadened by partial solvent trapping.     

Nonlinear fingering dynamics of reaction-diffusion acidity fronts: self-similar scaling and influence of differential diffusion

Nonlinear interactions between chemical reactions and buoyancy-driven Rayleigh-Taylor instability of reaction-diffusion acidity fronts of the chlorite-tetrathionate (CT) reaction are studied theoretically in a vertical Hele-Shaw cell or a porous medium. To do so, we perform a numerical integration of a two-variable reaction-diffusion model of the CT system coupled through an advection term to Darcy's law ruling the evolution of the velocity field of the fluid. The fingering dynamics of these chemical fronts is characterized by the appearance of several fingers at onset. These fingers then undergo coarsening and eventually merge to form one single symmetric finger. We study this asymptotic dynamics as a function of the three dimensionless parameters of the problem, i.e., the Damkohler number Da, the diffusivity ratio delta of the two chemical species, and the Rayleigh number Ra constructed here on the basis of the width L(y) of the system. For moderate values of Ra, the asymptotic single finger is shown to have self-similar scaling properties while above a given value of Ra, which depends on the other values of the parameters, tip splitting comes into play. Increasing the difference of diffusivities of the two chemical species (i.e., increasing delta) leads to more efficient coarsening and smaller asymptotic fingers. Experimental procedures to verify our predictions are proposed.     

Characterization of voltage degradation in dynamic field gradient focusing

Dynamic field gradient focusing (DFGF) is an equilibrium gradient method that utilizes an electric field gradient to simultaneously separate and concentrate charged analytes based on their individual electrophoretic mobilities. This work describes the use of a 2-D nonlinear, numerical simulation to examine the impact of voltage loss from the electrodes to the separation channel, termed voltage degradation, and distortions in the electric field on the performance of DFGF. One of the design parameters that has a large impact on the degree of voltage degradation is the placement of the electrodes in relation to the separation channel. The simulation shows that a distance of about 3 mm from the electrodes to the separation channel gives the electric field profile with least amount of voltage degradation. The simulation was also used to describe the elution of focused protein peaks. The simulation shows that elution under constant electric field gradient gives better performance than elution through shallowing of the electric field. Qualitative agreement between the numerical simulation and experimental results is shown. The simulation also illustrates that the presence of a defocusing region at the cathodic end of the separation channel causes peak dispersion during elution. The numerical model is then used to design a system that does not suffer from a defocusing region. Peaks eluted under this design experienced no band broadening in our simulations. Preliminary experimental results using the redesigned chamber are shown.     

A multi-domain spectral method for non-Darcian mixed convection flow in a power-law fluid with viscous dissipation

The purpose of this study is to investigate non-Darcian mixed convection flow, heat and mass transfer in a non-Newtonian power-law fluid over a flat plate embedded in porous medium with suction and viscous dissipation and also is to demonstrate the application and utility of a recently developed multi-domain bivariate spectral quasi-linearisation method (MD-BSQLM) in finding the solutions of highly nonlinear differential equations. The flow is subject to, among other source terms, internal heat generation, thermal radiation and partial velocity slip. The coupled system of nonlinear partial differential equations are solved using a MD-BSQLM to find the fluid properties, the skin friction, as well as the heat and mass coefficients. We have presented selected results that give the significance of some system parameters on the fluid properties. This MD-BSQLM has not been used before in the literature to find the nature of the solutions of power-law fluids. Indeed, validation of this numerical method for general fluid flows, heat and mass transfer problems has not yet been done. This study presents the first opportunity to evaluate the accuracy and robustness of the MD-BSQLM in finding solutions of non-Newtonian fluids.     

Continuous-flow production of polymeric micelles in microreactors: experimental and computational analysis

The dynamics and stability of a thin, viscous film of volatile liquid flowing under the influence of gravity over a non-uniformly heated substrate are investigated using lubrication theory. Attention is focused on the regime in which evaporation balances the flow due to gravity. The film terminates above the heater at an apparent contact line, with a microscopically thin precursor film adsorbed due to the disjoining pressure. The film develops a weak thermocapillary ridge due to the Marangoni stress at the upstream edge of the heated region. As for spreading films, a more significant ridge is formed near the apparent contact line. For weak Marangoni effects, the film evolves to a steady profile. For stronger Marangoni effects, the film evolves to a time-periodic state. Results of a linear stability analysis reveal that the steady film is unstable to transverse perturbations above a critical value of the Marangoni parameter, leading to finger formation at the contact line. The streamwise extent of the fingers is limited by evaporation. The time-periodic profiles are always unstable, leading to the formation of periodically-oscillating fingers. For rectangular heaters, the film profiles after instability onset are consistent with images from published experimental studies.     

Theoretical calculation of the retention enthalpy effect on the viscous heat dissipation band broadening in high performance liquid chromatography columns with a fixed wall temperature

The problem of the additional band broadening arising from the trans-column temperature profile under conditions of significant viscous heat dissipation has been extended by including the retention enthalpy effect. This effect can significantly aggravate the trans-column migration velocity profile of retained species as compared to the already known zero-retention enthalpy case. An approximate analytical solution for the accompanying additional band broadening could be established. The solution is valid in nearly all practically relevant cases and shows how the additional band broadening varies as a function of the generated heat, the retention coefficient and the retention enthalpy. Surveying all possible HPLC operating conditions, it was found that the inclusion of the retention enthalpy effect leads to significantly larger viscous heating plate height contributions than those predicted by the zero-retention enthalpy solution (up to a factor of 10 or more).     

Rayleigh-Taylor instabilities in reaction-diffusion systems inside Hele-Shaw cell modified by the action of temperature

The influence of temperature in the buoyancy driven Rayleigh-Taylor instability of reaction-diffusion fronts is investigated experimentally in Hele-Shaw cells. The acid autocatalysis of chlorite-tetrathionate reaction coupled to molecular diffusion yields exothermic planar reaction-diffusion fronts separating two miscible reactant and product solutions. The resulting chemical front moves downwards invading the fresh reactants, leaving the products of the reaction behind it. The density of the product solution is higher than the reactant solution; hence, the traveling front is buoyantly unstable and develops density fingers in time (Rayleigh-Taylor instability) when the products are above the reactants. The kinetic constant of a chemical reaction varies due to thermal effects. This may stabilize the exothermic descending front when temperature is increased, so that the mixing zone decreases, modifying the fingering patterns, until it almost disappears. The authors study the influence of the temperature variation on the instability pattern figure observed in the chlorite-tetrathionate reaction for long times, corresponding to the nonlinear regime.     

A comparative study of jet formation in nozzle‐ and nozzle‐less centrifugal spinning systems

Centrifugal spinning, a recently developed approach for ultra‐fine fiber production, has attracted much attention as compared with the electrospinning, due to its high yield, no solution polarity and high‐voltage electrostatic field requirements, etc. In this study, the jet formation process and spinning parameters on jet path are explored and compared in nozzle‐ and nozzle‐less centrifugal spinning systems. For nozzle‐less centrifugal spinning, fingers are formed at the front of thin liquid film due to the theory of Rayleigh–Taylor instability. We find that the lower solution concentration and higher rotational speed favor the formation of thinner and longer fingers. Then, the critical angular velocity and initial jet velocity for nozzle‐/nozzle‐less centrifugal spinning are obtained in accordance with the balance of centrifugal force, viscous force, and surface tension. When jet leaves the spinneret, it will undergo a series of motions including necking and whipping processes, and then, a steady spiral jet path is formed with its radius getting tighter. Finally, we experimentally study the effect of rotational speed and solution concentration on jet path, which shows that the higher rotational speed results in a larger radius of jet path while the solution concentration has little effect on it. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2014 , 52, 1547–1559     

Determination of extracolumn effects in recycle gas chromatography with capillary columns

Extracolumn band broadening in multidimensional systems utilizing flow switching is clearly undesirable. In certain cases, i.e. capillary recycle gas chromatography, the success of an experiment is contingent on the minimization of pre- and post-column dispersion of the peaks. Knowledge of these sources of peak distortion is necessary to optimize the experimental design. A system that extracts statistical parameters from real chromatographic peaks is discussed and used to evaluate band broadening in a capillary recycle experiment.     

Determination and Use of Cumulants of the Peak Broadening Function in Steric Exclusion Chromatography

Abstract

A procedure is developed which allows to deduce from recycle measurements the central moments and cumulants of the molecular mass distribution of polymers and the broadening function by the use of properties of statistical distributions. Results of an experimental application of the method are presented.

Equations for the calculation of true molecular mass averages from the elution curve are extended for the cases of nonlinear calibration curves and nonlinear broadening.     

Visualization of viscous fingering in high-performance liquid chromatographic columns. Influence of the header design

Using an on-column visualization technique, band profiles of solutes migrating along an HPLC column were studied. The study showed that, under conditions where viscous fingering is prevalent, the design of the inlet header has little influence on the outcome of the viscous fingers. Two types of headers were studied. The first contained a small diameter inlet frit, which localized the majority of the sample in or near the central region of the column. The second header contained a wide frit and produced a more uniform radial distribution of the sample. In both cases, the extent of viscous fingering was essentially the same.     

Highly efficient luminescent organic clusters with quantum dot-like properties

Organic clusters with uniform structure and size have been synthesized and studied. The quantum confinements on organic arms [chains shorter than exciton Bohr radii (aB)] have been confirmed by the discrete energy levels from PLE, narrow PL lines, size effect from UV, and spherical shape from TEM, as well as Raman peaks red-shifting and broadening. The results indicate the materials have QD-like properties.     

Chromatographic characterization of molecularly imprinted polymers

Recent efforts in the investigation of chromatographic characterization of molecularly imprinted polymers (MIPs) have focused mainly on the nature of heterogeneous binding sites. More data on the thermodynamics than on the kinetic features of MIP columns have been published. The present article addresses the sources of peak broadening and tailing, which are the main drawbacks often associated with imprinted polymers in chromatography for practical applications. With use of the theory of nonlinear chromatography, the peak properties of a MIP column, including the retention and peak broadening and tailing, can be well interpreted. Efforts to improve chromatographic efficiency using MIPs prepared by approaches different from the conventional method, including covalent imprinting and the format of uniformly sized spherical microbeads, are reviewed and discussed. This review leads to the conclusion that nonlinear chromatography theory is useful for characterizing chromatographic features of MIP columns, since a MIP is essentially an affinity-based chromatographic stationary phase. We expect more theoretical and experimental studies on the kinetic aspects of MIP columns, especially the factors influencing the apparent rate constant, as well as the analysis of the influences of mobile-phase composition on the chromatographic performance. In addition to revealing the affinity interaction by molecular recognition, slow nonspecific interactions which may be inherited from the imperfect imprinting and may be involved in the rebinding of the template to MIPs also need to be characterized. Figure The peak broadening and tailing associated often with molecularly imprinted polymers (MIPs) in column chromatography for practical applications can be well characterized by the theory of nonlinear chromatography.     

Observations of instability, hysteresis, and oscillation in low-Reynolds-number flow past polymer gels

We perform a set of experiments to study the nonlinear nature of an instability that arises in low-Reynolds-number flow past polymer gels. A layer of a viscous liquid is placed on a polydimethylsiloxane (PDMS) gel in a parallel-plate rheometer which is operated in stress-controlled mode. As the shear stress on the top plate increases, the apparent viscosity stays relatively constant until a transition stress where it sharply increases. If the stress is held at a level slightly above the transition stress, the apparent viscosity oscillates with time. If the stress is increased to a value above the transition stress and then decreased back to zero, the apparent viscosity shows hysteretic behavior. If the stress is instead decreased to a constant value and held there, the apparent viscosity is different from its pretransition value and exhibits sustained oscillations. This can happen even if the stress is held at values below the transition stress. Our observations suggest that the instability studied here is subcritical and leads to a flow that is oscillatory and far from viscometric. The phenomena reported here may be useful in applications such as microfluidics, membrane separations, and polymer processing. They may also provide insight into the rheological behavior of complex fluids that undergo flow-induced gelation.