Study of Transport Phenomena in Chromatographic Columns by Pulsed Field Gradient NMR

Detailed analysis of the effective and intra-particle diffusion coefficient of proteins at elevated pressure in columns packed with wide-pore core-shell particles

To determine the efficiency that can be obtained in a packed-bed liquid-chromatography column for a particular analyte, a correct determination of the molecular and effective diffusion coefficients (Dm and Deff) of the analyte is required. The latter is usually obtained via peak parking experiments wherein the flow is stopped. As a result, the column pressure rapidly dissipates and the measurement is essentially conducted at ambient pressure. This is problematic for analytes whose retention depends on pressure, such as proteins and potentially other large (dipolar) molecules. In that case, a conventional peak parking experiment is expected to lead to large errors in Deff. To obtain a better estimate ofDeff, the present study reports on the use of a set-up enabling peak parking measurements under pressurized conditions. This approach allowed us to report, for the first time, Deff for proteins at elevated pressure under retained conditions. First, Deff was determined at a (average) pressure of about 105 bar for a set of proteins with varying size, namely: bradykinin, insulin, lysozyme, β-lactoglobulin, and carbonic anhydrase in a column packed with 400 Å core-shell particles. The obtained data were then compared to those of several small analytes: acetophenone, propiophenone, benzophenone, valerophenone, and hexanophenone. A clear trend between Deff and analyte size was observed. The set-up was then used to determine Deff of bradykinin and lysozyme at variable (average) pressures ranging from 28 bar to 430 bar. These experiments showed a decrease in intra-particle and surface diffusion with pressure, which was larger for lysozyme than bradykinin. The data show that pressurized peak parking experiments are vital to correctly determine Deff when the analyte retention varies significantly with pressure.

Limits to flow detection in phase contrast MRI

Pulsed gradient spin echo (PGSE) complex signal behavior becomes dominated by attenuation rather than oscillation when displacements due to flow are similar or less than diffusive displacements. In this "slow-flow" regime, the optimal displacement encoding parameter q for phase contrast velocimetry depends on the diffusive length scale q s l o w = 1 / l D = 1 / 2 D Δ rather than the velocity encoding parameter v _enc = π/(qΔ). The minimum detectable mean velocity using the difference between the phase at +q _slow and -q _slow is 〈 v m i n 〉 = 1 / SNR D / Δ . These theories are then validated and applied to MRI by performing PGSE echo planar imaging experiments on water flowing through a column with a bulk region and a beadpack region at controlled flow rates. Velocities as slow as 6 μm/s are detected with velocimetry. Theories, MRI experimental protocols, and validation on a controlled phantom help to bridge the gap between porous media NMR and pre-clinical phase contrast and diffusion MRI.

Characterizing dispersivity and stagnation in porous media using NMR flow propagators

Low-field nuclear magnetic resonance (NMR) displacement probability distributions (flow propagators) are presented for water flowing through heterogeneous porous materials. Four sedimentary rocks have been chosen as example systems: Dolostone, Bentheimer sandstone, Berea sandstone, and Indiana limestone (in order of decreasing permeability). The fluid displacement is characterized by pre-asymptotic Stokes' flow and so the probability distributions are bimodal, with peaks corresponding to stagnant fluid in dead-end pores and flowing fluid in the connected porosity. Cut-off Gaussian functions are used to fit the flowing and stagnant peaks independently. An effective dispersivity length scale Lv (also known as the mixing length scale) is estimated by fitting the portion of the probability distribution corresponding to the flowing fluid. For the relatively homogeneous Bentheimer sandstone, the ratio of effective dispersivity length scale to effective transport diameter dt is Lv/dt≈16, which is an order of magnitude larger than for randomly packed glass beads where Lv/dt≈1.8. We compare these dispersivity parameters to similar values extracted from a cumulant analysis of the entire propagator. Fitting a cut-off Gaussian avoids the usual complications of analyzing dispersion in the presence of the ubiquitous stagnant fluid, and results in a clear demonstration of the influence of long-range heterogeneities on the dispersivity for flow in real sedimentary rocks.

Constant gradient PFG sequence and automated cumulant analysis for quantifying dispersion in flow through porous media

This paper describes a new variant of established stimulated echo pulse sequences, and an analytical method for determining diffusion or dispersion coefficients for Gaussian or non-Gaussian displacement distributions. The unipolar displacement encoding PFGSTE sequence uses trapezoidal gradient pulses of equal amplitude g and equal ramp rates throughout while sampling positive and negative halves of q-space. Usefully, the equal gradient amplitudes and gradient ramp rates help to reduce the impact of experimental artefacts caused by residual amplifier transients, eddy currents, or ferromagnetic hysteresis in components of the NMR magnet. The pulse sequence was validated with measurements of diffusion in water and of dispersion in flow through a packing of spheres. The analytical method introduced here permits the robust determination of the variance of non-Gaussian, dispersive displacement distributions. The noise sensitivity of the analytical method is shown to be negligible, using a demonstration experiment with a non-Gaussian longitudinal displacement distribution, measured on flow through a packing of mono-sized spheres.

Accounting for adsorption and desorption in lattice Boltzmann simulations

We report a Lattice-Boltzmann scheme that accounts for adsorption and desorption in the calculation of mesoscale dynamical properties of tracers in media of arbitrary complexity. Lattice Boltzmann simulations made it possible to solve numerically the coupled Navier-Stokes equations of fluid dynamics and Nernst-Planck equations of electrokinetics in complex, heterogeneous media. With the moment propagation scheme, it became possible to extract the effective diffusion and dispersion coefficients of tracers, or solutes, of any charge, e.g., in porous media. Nevertheless, the dynamical properties of tracers depend on the tracer-surface affinity, which is not purely electrostatic and also includes a species-specific contribution. In order to capture this important feature, we introduce specific adsorption and desorption processes in a lattice Boltzmann scheme through a modified moment propagation algorithm, in which tracers may adsorb and desorb from surfaces through kinetic reaction rates. The method is validated on exact results for pure diffusion and diffusion-advection in Poiseuille flows in a simple geometry. We finally illustrate the importance of taking such processes into account in the time-dependent diffusion coefficient in a more complex porous medium.

Flow characteristics and exchange in complex biological systems as observed by pulsed-field-gradient magnetic-resonance imaging

Water flow through model porous media was studied in the presence of surface relaxation, internal magnetic field inhomogeneities and exchange with stagnant water pools with different relaxation behavior, demonstrating how the apparent flow parameters average velocity, volume flow and flow conducting area in these situations depend on the observation time. To investigate the water exchange process a two component biological model system consisting of water flowing through a biofilm reactor (column packed with methanogenic granular sludge beads) was used, before and after a heat treatment to introduce exchange. We show that correction of the stagnant fluid signal amplitude for relaxation at increasing observation time using the observed relaxation times reveals exchange between the two fractions in the system. Further it is demonstrated how this exchange can be quantified.

Mass-Transfer Kinetics in a Shell Packing Material for Chromatography

A shell particle consists of a solid, nonporous core that is surrounded with a shell of a porous solid having essentially the same physicochemical properties as those of the conventional porous particles used as packing media in chromatography. The diameter of the solid core and the thickness of its shell or the external diameter of the particle characterizes the chromatographic properties of the packing material. The potential advantage of this particle structure would be the shorter average path length experienced by solute molecules during their diffusion across the particles of packing material when they are retained. Compounds having slow pore diffusion would exhibit higher efficiencies on columns packed with shell than with conventional, fully porous particles. Using columns packed with Halo, a new type of porous silica shell particles, we assess the gain achieved with this principle for peptides of moderate molecular weights and for small proteins.

The limits of the separation power of unidimensional column liquid chromatography

The practical limit of the separation power of HPLC depends on time, money, and skill. That is it depends on the time available for the analysis, on the quality and performance of the pump and hardware and particularly on the maximum pressure at which the pump can deliver the mobile phase to the column, and on the temperature at which the column can be operated. It also depends on the properties of the packing material selected (e.g., its particle size, its pore geometry, and its connectivity) and on the packing method used since it affects the coefficients of the HETP equation. Finally, it depends on the thermal stability of the sample and the packing material. The complexity of the sample also plays an important role in that it determines whether the analysis should be made under isocratic, isothermal conditions, in gradient elution, in temperature programming, or with a combination of both types of programming. The various phenomena that affect column properties and separation performance are discussed. Past achievements suggest that columns providing efficiencies in excess of a million plates in less than 1 day are within the grasp of current technology. The possibility of further advances are considered.

Diffusion NMR spectroscopy in supramolecular and combinatorial chemistry: an old parameter--new insights

Intermolecular interactions in solution play an important role in molecular recognition, which lies at the heart of supramolecular and combinatorial chemistry. Diffusion NMR spectroscopy gives information over such interactions and has become the method of choice for simultaneously measuring diffusion coefficients of multicomponent systems. The diffusion coefficient reflects the effective size and shape of a molecular species. Applications of this technique include the estimation of association constants and mapping the intermolecular interactions in multicomponent systems as well as investigating aggregation, ion pairing, encapsulation, and the size and structure of labile systems. Diffusion NMR spectroscopy can also be used to virtually separate mixtures and screen for specific ligands of different receptors, and may assist in finding lead compounds.

Some perspectives on dispersion and the use of ensemble-averaged PGSE NMR

Magnetic resonance methodology has made a significant impact in helping us understand the physics of porous media. Among an important class of experiments is that set of techniques designed to measure fluid dispersion. This paper provides some background on some of the underlying physics of dispersion, and outlines some of the NMR approaches that have proven successful. The local and nonlocal dispersion tensors are described and the prospects for future NMR advances considered.

Axial and radial diffusion coefficients in a liquid chromatography column and bed heterogeneity

The axial and transverse diffusion coefficients of a band of iodine in a chromatographic column were measured optically as a function of time. It was found that the axial diffusion coefficient remains constant even when the edges of the sample band get close to the wall. By contrast, the radial diffusion coefficient decreases progressively with increasing time when the edges of the sample band leave the core region and begin to diffuse inside the wall region. The local axial and transverse diffusion coefficients of the band decrease from the column center toward the wall. Hence, the increase in local height equivalent to a theoretical plate observed in the region close to the wall must be explained by increasing mass transfer resistances and degree of heterogeneity of the bed.

Molecular mobility in fixed-bed reactors investigated by multiscale NMR techniques

The complex problem of a fixed-bed reactor consisting of catalytically active particles provides an exceptional opportunity of combining a wide range of NMR methods which have become available over time as tools to probe porous media. This work demonstrates the feasibility of different NMR techniques for the investigation of the intra- and interparticle pore space over length scales from nanometers up to centimeters. Many industrially relevant cracking reactions leave a coke residue on the inner surface of the porous catalyst particles so that the active sites become inaccessible to the reactants. Moreover, the pore space shrinks due to the formation of coke, thereby hindering molecular transport. The presence of the coke residue and its influence on the mobility of adsorbed fluid molecules are probed by 129Xe spectroscopy, NMR cryoporometry, relaxation dispersion measurements, and investigations of the reduced diffusivity in the intraporous space. The voids surrounding the random arrangement of catalyst pellets represent another pore space of much larger dimensions, the properties of which can be more directly investigated by mapping the fluid density and the velocity distribution from velocity-encoded imaging. Propagator representations averaged over large sample volumes are discussed and compared to velocity images obtained in selected axial slices of the reactor.

NMR visualization of displacement correlations for flow in porous media

The temporal correlations of velocities for both water and a water-glycerol mixture flowing through a random packings of monodisperse spherical particles have been investigated using two-dimensional nuclear magnetic resonance methods. By combining various flow rates, fluid viscosities, and bead sizes, a wide range of flow parameters has been covered, the dimensionless Peclet number ranging from 100 to 100 000. The velocity exchange spectroscopy (VEXSY) technique has been employed to measure the correlation between velocities during two intervals separated from each other by a mixing time tau(m). This time is made both large and small compared with the time constant tau(c), required for a fluid element possessing the average flow velocity to cover a distance equal to the characteristic size in the system, the bead diameter. The two-dimensional conditional probability of displacement resulting from the VEXSY method reveals the existence of different "subensembles" of molecules, including a slow moving pool whose displacement is dominated by Brownian motion, an intermediate ensemble whose velocities change little over the mixing time, and a fast flowing ensemble which loses correlation due to mechanical dispersion. We find that that the approach to asymptotic dispersion, as tau(c)/tau(m) increases, depends strongly on the Peclet number, the deviation of the velocity autocorrelation function from a monoexponential Ornstein-Uhlenbeck process becoming more pronounced with increasing Peclet number.

Spectrally resolved velocity exchange spectroscopy of two-phase flow

The Velocity EXchange SpectroscopY (VEXSY) technique, which provides a means to correlate macroscopic molecular displacements measured during two intervals separated by a variable mixing period, has been applied for the first time to a system of two-phase flow. The chemical shift difference between water and methyl protons has been exploited to simultaneously determine the probability of displacements, or propagator, of both components in a water/silicone oil mixture flowing through a glass bead pack. The joint two-time probability densities as well as the conditional probabilities of velocities show a clearly distinct dispersion behaviour of both fluids which is a consequence of the different wetting properties of the fluids with respect to the glass surface of the bead pack.

Numerical simulation and measurement of liquid hold-up in biporous media containing discrete stagnant zones

We have studied hydrodynamic dispersion in single-phase incompressible liquid flow through a fixed bed made of spherical, permeable (porous) particles. The observed behaviour was contrasted with the corresponding fluid dynamics in a random packing of impermeable (non-porous) spheres with an interparticle void fraction of 0.37. Experimental data were obtained in the laminar flow regime by pulsed field gradient nuclear magnetic resonance and were complemented by numerical simulations employing a hierarchical transport model with a discrete (lattice Boltzmann) interparticle flow field. Finite-size effects in the simulation associated with the spatial discretization of support particles or dimension and boundaries of the bed were minimized and the simulation results are in reasonable agreement with experiment.

Axial development and radial non-uniformity of flow in packed columns

Flow inhomogeneity and axial development in low-pressure chromatographic columns have been studied by magnetic resonance imaging velocimetry. The columns studied included (a) an 11.7-mm I.D. column packed with either 50 microm diameter porous polyacrylamide, or 99 or 780 microm diameter impermeable polystyrene beads, and (b) a 5-mm I.D. column commercially packed with 10 microm polymeric beads. The packing methods included gravity settling, slurry packing, ultrasonication, and dry packing with vibration. The magnetic resonance method used averaged apparent fluid velocity over both column cross-sections and fluid displacements greater than one particle diameter and hence permits assessment of macroscopic flow non-uniformities. The results confirm that now non-uniformities induced by the conical distributor of the 11.7-mm I.D. column or the presence of voids at the column entrance relax on a length scale of the column radius. All of the 11.7-mm I.D. columns examined exhibit near wall channeling within a few particle diameters of the wall. The origins of this behavior are demonstrated by imaging of the radial dependence of the local porosity for a column packed with 780 microm beads. Columns packed with the 99-microm beads exhibit reduced flow in a region extending from ten to three-to-five particle diameters from the wall. This velocity reduction is consistent with a reduced porosity of 0.35 in this region as compared to approximately 0.43 in the bulk of the column. Ultrasonicated and dry-packed columns exhibit enhanced flow in a region located between approximately eight and 20 particle diameters from the wall. This enhancement maybe caused by packing density inhomogeneity and/or particle size segregation caused by vibration during the packing process. No significant non-uniformities on length scales of 20 microm or greater were observed in the commercially packed column packed with 10 microm particles.