Low-frequency dielectric response of a periodic array of charged spheres in an electrolyte solution: The simple cubic lattice.

We study the low-frequency dielectric response of highly charged spheres arranged in a cubic lattice and immersed in an electrolyte solution. We focus on the influence of the out-of-phase current in the regime where the ionic charge is neutral. We consider the case where the charged spheres have no surface conductance and no frequency-dependent surface capacitance. Hence, the frequency dispersion of the dielectric constant is dominated by the effect of neutral currents outside the electric double layer. In the thin double-layer limit, we use Fixman's boundary condition at the outer surface of the double layer to capture the interaction between the electric field and the flow of the ions. For periodic conditions, we combine the methods developed by Lord Rayleigh for understanding the electric conduction across rectangularly arranged obstacles and by Korringa, Kohn, and Rostoker for the electronic band structure computation. When the charged spheres occupy a very small volume fraction, smaller than 1%, our solution becomes consistent with the Maxwell Garnett mixing formula together with the single-particle polarization response, as expected, because interparticle interactions become less prominent in the dilute limit. By contrast, the interparticle interaction greatly alters the dielectric response even when charged spheres occupy only 2% of the volume. We found that the characteristic frequency shifts to a higher value compared to that derived from the single-particle polarization response. At the same time, the low-frequency dielectric enhancement, a signature of charged spheres immersed in an electrolyte, becomes less prominent for the periodic array of charged spheres. Our results imply that the signature of the dielectric response of a system consisting of densely packed charged spheres immersed in an electrolyte can differ drastically from a dilute suspension.

High-frequency dielectric response of a spheroidal particle with a thin double layer.

This paper addresses the dielectric response of a charged, oblate spheroidal particle immersed in an electrolyte. Analytic solutions are obtained for the polarization coefficient in the thin double layer limit for frequencies in the megahertz to gigahertz range. Two different distributions of the equilibrium surface charge density in the double layer are considered. The first is when the surface charge density is uniform. The second is a particular nonuniform distribution with more charges along the flat surface of the spheroid than along the edges. For this latter case, an exact solution for the polarization coefficients is obtained. In these solutions for the polarization coefficients, to a good approximation, the particles act as if they have conductivity in addition to permittivity, with different values for the conductivity depending on whether the applied electric field is parallel or perpendicular to the flat surface of the particle. When it is parallel to the flat surface, the apparent conductivity of the particle is roughly proportional to total charge on the particle and, hence, does not appear to depend on the details of the surface charge density. When the electric field is perpendicular to the flat surface, the apparent conductivity decreases as the particle becomes more platy and as the charge along the edges decreases. The effect on the dielectric dispersion of a dilute mixture of these charged spheroids in an electrolyte is also calculated. When their symmetry axes are aligned parallel to the applied electric field, the charges on the spheroids reduce the dielectric dispersion, while for spheroids aligned in the other direction, the charges increase both the conductivity and the dielectric dispersion.

Low frequency complex dielectric (conductivity) response of dilute clay suspensions: Modeling and experiments.

In this work, we establish an effective medium model to describe the low-frequency complex dielectric (conductivity) dispersion of dilute clay suspensions. We use previously obtained low-frequency polarization coefficients for a charged oblate spheroidal particle immersed in an electrolyte as the building block for the Maxwell Garnett mixing formula to model the dilute clay suspension. The complex conductivity phase dispersion exhibits a near-resonance peak when the clay grains have a narrow size distribution. The peak frequency is associated with the size distribution as well as the shape of clay grains and is often referred to as the characteristic frequency. In contrast, if the size of the clay grains has a broad distribution, the phase peak is broadened and can disappear into the background of the canonical phase response of the brine. To benchmark our model, the low-frequency dispersion of the complex conductivity of dilute clay suspensions is measured using a four-point impedance measurement, which can be reliably calibrated in the frequency range between 0.1 Hz and 10 kHz. By using a minimal number of fitting parameters when reliable information is available as input for the model and carefully examining the issue of potential over-fitting, we found that our model can be used to fit the measured dispersion of the complex conductivity with reasonable parameters. The good match between the modeled and experimental complex conductivity dispersion allows us to argue that our simplified model captures the essential physics for describing the low-frequency dispersion of the complex conductivity of dilute clay suspensions.

Low-frequency dielectric response of charged oblate spheroidal particles immersed in an electrolyte.

We study the low-frequency polarization response of a surface-charged oblate spheroidal particle immersed in an electrolyte solution. Because the charged spheroid attracts counterions which form the electric double layer around the particle, using usual boundary conditions at the interface between the particle and electrolyte can be quite complicated and challenging. Hence, we generalize Fixman's boundary conditions, originally derived for spherical particles, to the case of the charged oblate spheroid. Given two different counterion distributions in the thin electric double-layer limit, we obtain analytic expressions for the polarization coefficients to the first nontrivial order in frequency. We find that the polarization response normal to the symmetry axis depends on the total amount of charge carried by the oblate spheroid while that parallel to the symmetry axis is suppressed when there is less charge on the edge of the spheroid. We further study the overall dielectric response for a dilute suspension of charged spheroids. We find that the dielectric enhancement at low frequency, which is driven by the presence of a large ζ potential (surface charge), is suppressed by high ion concentrations in the electrolyte and depends on the size of the suspended particles. In addition, spheroids with higher aspect ratios will also lead to a stronger dielectric enhancement due to the combination of the electric double layer and textural effects. The characteristic frequency associated with the dielectric enhancement scales inversely with the square of the particle size, the major radius of the spheroid, and it has a weak dependence on the shape of spheroids.

Dispersion of T1 and T2 nuclear magnetic resonance relaxation in crude oils.

Crude oils, which are complex mixtures of hydrocarbons, can be characterized by nuclear magnetic resonance diffusion and relaxation methods to yield physical properties and chemical compositions. In particular, the field dependence, or dispersion, of T1 relaxation can be used to investigate the presence and dynamics of asphaltenes, the large molecules primarily responsible for the high viscosity in heavy crudes. However, the T2 relaxation dispersion of crude oils, which provides additional insight when measured alongside T1, has yet to be investigated systematically. Here we present the field dependence of T1-T2 correlations of several crude oils with disparate densities. While asphaltene and resin-containing crude oils exhibit significant T1 dispersion, minimal T2 dispersion is seen in all oils. This contrasting behavior between T1 and T2 cannot result from random molecular motions, and thus, we attribute our dispersion results to highly correlated molecular dynamics in asphaltene-containing crude oils.

A more accurate estimate of T2 distribution from direct analysis of NMR measurements.

In the past decade, low-field NMR relaxation and diffusion measurements in grossly inhomogeneous fields have been used to characterize pore size distribution of porous media. Estimation of these distributions from the measured magnetization data plays a central role in the inference of insitu petro-physical and fluid properties such as porosity, permeability, and hydrocarbon viscosity. In general, inversion of the relaxation and/or diffusion distribution from NMR data is a non-unique and ill-conditioned problem. It is often solved in the literature by finding the smoothest relaxation distribution that fits the measured data by use of regularization. In this paper, estimation of these distributions is further constrained by linear functionals of the measurement that can be directly estimated from the measured data. These linear functionals include Mellin, Fourier-Mellin, and exponential Haar transforms that provide moments, porosity, and tapered areas of the distribution, respectively. The addition of these linear constraints provides more accurate estimates of the distribution in terms of a reduction in bias and variance in the estimates. The resulting distribution is also more stable in that it is less sensitive to regularization. Benchmarking of this algorithm on simulated data sets shows a reduction of artefacts often seen in the distributions and, in some cases, there is an increase of resolution in the features of the T(2) distribution. This algorithm can be applied to data obtained from a variety of pulse sequences including CPMG, inversion and saturation recovery and diffusion editing, as well as pulse sequences often deployed down-hole.

Estimation of petrophysical and fluid properties using integral transforms in nuclear magnetic resonance.

In the past decade, low-field NMR relaxation and diffusion measurements in grossly inhomogeneous fields have been used to characterize properties of porous media, e.g., porosity and permeability. Pulse sequences such as CPMG, inversion and saturation recovery as well as diffusion editing have been used to estimate distribution functions of relaxation times and diffusion. Linear functionals of these distribution functions have been used to predict petro-physical and fluid properties like permeability, viscosity, fluid typing, etc. This paper describes an analysis method using integral transforms to directly compute linear functionals of the distributions of relaxation times and diffusion without first computing the distributions from the measured magnetization data. Different linear functionals of the distribution function can be obtained by choosing appropriate kernels in the integral transforms. There are two significant advantages of this approach over the traditional algorithm involving inversion of the distribution function from the measured data. First, it is a direct linear transform of the data. Thus, in contrast to the traditional analysis which involves inversion of an ill-conditioned, non-linear problem, the estimates from this new method are more accurate. Second, the uncertainty in the linear functional can be obtained in a straight-forward manner as a function of the signal-to-noise ratio (SNR) in the measured data. We demonstrate the performance of this method on simulated data.

Structural relaxation in dense liquids composed of anisotropic particles.

We perform extensive molecular dynamics simulations of dense liquids composed of bidisperse dimer- and ellipse-shaped particles in two dimensions that interact via purely repulsive contact forces. We measure the structural relaxation times obtained from the long-time α decay of the self part of the intermediate scattering function for the translational and rotational degrees of freedom (DOF) as a function of packing fraction φ, temperature T, and aspect ratio α. We are able to collapse the packing-fraction and temperature-dependent structural relaxation times for disks, and dimers and ellipses over a wide range of α, onto a universal scaling function F(±)(|φ-φ(0)|,T,α), which is similar to that employed in previous studies of dense liquids composed of purely repulsive spherical particles in three dimensions. F(±) for both the translational and rotational DOF are characterized by the α-dependent scaling exponents µ and δ and packing fraction φ(0)(α) that signals the crossover in the scaling form F(±) from hard-particle dynamics to super-Arrhenius behavior for each aspect ratio. We find that the fragility of structural relaxation at φ(0), m(φ(0)), decreases monotonically with increasing aspect ratio for both ellipses and dimers. For α>α(p), where α(p) is the location of the peak in the packing fraction φ(J) at jamming onset, the rotational DOF are strongly coupled to the translational DOF, and the dynamic scaling exponents and φ(0) are similar for the rotational and translational DOF. For 1<α<α(p), the translational DOF become frozen at higher temperatures than the rotational DOF, and φ(0) for the rotational degrees of freedom increases above φ(J). Moreover, the results for the slow dynamics of dense liquids composed of dimer- and ellipse-shaped particles are qualitatively the same, despite the fact that zero-temperature static packings of dimers are isostatic, while static packings of ellipses are hypostatic. Thus, zero-temperature contact counting arguments do not apply to structural relaxation of dense liquids of anisotropic particles near the glass transition.

Continuous moment estimation of CPMG data using Mellin transform.

This paper provides a theoretical basis to directly estimate moments of transverse relaxation time T(2) from measured CPMG data in grossly inhomogeneous fields. These moments are obtained from Mellin transformation of the measured CPMG data. These moments are useful in computing petro-physical and fluid properties of hydrocarbons in porous media. Compared to the conventional method of estimating moments, the moments obtained from this method are more accurate and have a smaller variance. This method can also be used in other applications of NMR in inhomogeneous fields in characterizing fluids and porous media such as in the determination of hydrocarbon composition, estimation of model parameters describing relationship between fluid composition and measured NMR data, computation of error-bars on estimated parameters, as well as estimation of parameters and σ(lnT(2)) often used to characterize rocks. We demonstrate the performance of the method on simulated data.

Mellin transform of CPMG data.

This paper describes a new method for computing moments of the transverse relaxation time T(2) from measured CPMG data. This new method is based on Mellin transform of the measured data and its time-derivatives. The Mellin transform can also be used to compute the cumulant generating function of lnT(2). The moments of relaxation time T(2) and lnT(2) are related to petro-physical and fluid properties of hydrocarbons in porous media. The performance of the new algorithm is demonstrated on simulated data and compared to results from the traditional inverse Laplace transform. Analytical expressions are also derived for uncertainties in these moments in terms of the signal-to-noise ratio of the data.

Probing asphaltene aggregation in native crude oils with low-field NMR.

We show that low-field proton nuclear magnetic resonance (NMR) relaxation and diffusion experiments can be used to study asphaltene aggregation directly in crude oils. Relaxation was found to be multiexponential, reflecting the composition of a complex fluid. Remarkably, the relaxation data for samples with different asphaltene concentrations can be collapsed onto each other by a simple rescaling of the time dimension with a concentration-dependent factor xi, whereas the observed diffusion behavior is unaffected by asphaltene concentration. We interpret this finding in terms of a theoretical model that explains the enhanced relaxation by the transitory entanglement of solvent hydrocarbons within asphaltene clusters and their subsequent slowed motion and diffusion within the cluster. We relate the measured scaling parameters xi to cluster sizes, which we find to be on the order of 2.2-4.4 nm for an effective sphere diameter. These sizes are in agreement with the typical values reported in the literature as well as with the small-angle X-ray scattering (SAXS) experiments performed on our samples.

Temperature and pressure dependence of the diffusion coefficients and NMR relaxation times of mixtures of alkanes.

Recently, it was shown (1, 2) that the diffusion coefficient and nuclear magnetic resonance (NMR) relaxation times of a molecule in a mixture of alkanes follow scaling laws in the chain length of the molecule and the mean chain length of the mixture. These relations can be used to determine the chain length distribution of crude oils from diffusion and relaxation measurements. Oil reservoirs are usually at elevated temperatures and pressures, so it is important to know how these scaling relations depend on temperature and pressure. In this paper, we obtain the relation between the molecular composition of mixtures of alkanes at elevated pressures and temperatures and their diffusion coefficients D(i) and relaxation times T(1i) and T(2i). Using properties of free volume theory and the behavior of the density of alkanes, we show that, for a large range of pressures, the diffusion coefficients and relaxation times depend on pressure and mean chain length of the mixture only through its density. We further show that the pressure effect can be taken into account in the power laws of refs 1 and 2 by a multiplicative prefactor that depends only on temperature and the free volumes of pure alkanes at the pressure of interest and a reference pressure. We also combine the scaling laws for D(i), T(1i), and T(2i) and the Arrhenius dependence on temperature to obtain the temperature dependence of the diffusion coefficients and relaxation times. We obtain good fits between the scaling relations and literature data. These scaling relations can be used to determine the composition of a mixture of alkanes from measurements of diffusion coefficients or relaxation times at elevated pressures and temperatures.

Dependence on chain length of NMR relaxation times in mixtures of alkanes.

Many naturally occurring fluids, such as crude oils, consist of a very large number of components. It is often of interest to determine the composition of the fluids in situ. Diffusion coefficients and nuclear magnetic resonance (NMR) relaxation times can be measured in situ and depend on the size of the molecules. It has been shown [D. E. Freed et al., Phys. Rev. Lett. 94, 067602 (2005)] that the diffusion coefficient of each component in a mixture of alkanes follows a scaling law in the chain length of that molecule and in the mean chain length of the mixture, and these relations were used to determine the chain length distribution of crude oils from NMR diffusion measurements. In this paper, the behavior of NMR relaxation times in mixtures of chain molecules is addressed. The author explains why one would expect scaling laws for the transverse and longitudinal relaxation times of mixtures of short chain molecules and mixtures of alkanes, in particular. It is shown how the power law dependence on the chain length can be calculated from the scaling laws for the translational diffusion coefficients. The author fits the literature data for NMR relaxation in binary mixtures of alkanes and finds that its dependence on chain length agrees with the theory. Lastly, it is shown how the scaling laws in the chain length and the mean chain length can be used to determine the chain length distribution in crude oils that are high in saturates. A good fit is obtained between the NMR-derived chain length distributions and the ones from gas chromatography.

Scaling laws for diffusion coefficients in mixtures of alkanes.

Natural fluids, such as crude oils, are often mixtures of a broad range of different molecules, and in situ measurement of their composition is highly desirable. Furthermore, the relationship between their composition and their physical properties has always been a challenge for such mixtures. We have analyzed diffusion in alkane mixtures to find a power law for the self-diffusion coefficient in terms of molecular sizes. We demonstrate that this power law can be used to obtain the molecular size distribution of crude oils using noninvasive measurements of diffusion distributions.