Molecular Modes Elucidate the Nuclear Magnetic Resonance Relaxation of Viscous Fluids.

The Bloembergen, Purcell, and Pound (BPP) theory of nuclear magnetic resonance (NMR) relaxation in fluids dating back to 1948 continues to be the linchpin in interpreting NMR relaxation data in applications ranging from characterizing fluids in porous media to medical imaging (MRI). The BPP theory is founded on assuming molecules are hard spheres with 1H-1H dipole pairs reorienting randomly; assumptions that are severe in light of modern understanding of liquids. Nevertheless, it is intriguing to this day that the BPP theory was consistent with the original experimental data for glycerol, a hydrogen-bonding molecular fluid for which the hard-sphere-rigid-dipole assumption is inapplicable. To better understand this incongruity, atomistic molecular simulations are used to compute 1H NMR T1 relaxation dispersion (i.e., frequency dependence) in two contrasting cases: glycerol, and a (non hydrogen-bonding) viscosity standard. At high viscosities, simulations predict distinct functional forms of T1 for glycerol compared to the viscosity standard, in agreement with modern measurements, yet both in contrast to BPP theory. The cause of these departures from BPP theory is elucidated, without assuming any relaxation models and without any free parameters, by decomposing the simulated T1 response into dynamic molecular modes for both intramolecular and intermolecular interactions. The decomposition into dynamic molecular modes provides an alternative framework to understand the physics of NMR relaxation for viscous fluids.

Theory and modeling of molecular modes in the NMR relaxation of fluids.

Traditional theories of the nuclear magnetic resonance (NMR) autocorrelation function for intra-molecular dipole pairs assume a single-exponential decay, yet the calculated autocorrelation of realistic systems displays a rich, multi-exponential behavior, resulting in anomalous NMR relaxation dispersion (i.e., frequency dependence). We develop an approach to model and interpret the multi-exponential intra-molecular autocorrelation using simple, physical models within a rigorous statistical mechanical development that encompasses both rotational diffusion and translational diffusion in the same framework. We recast the problem of evaluating the autocorrelation in terms of averaging over a diffusion propagator whose evolution is described by a Fokker-Planck equation. The time-independent part admits an eigenfunction expansion, allowing us to write the propagator as a sum over modes. Each mode has a spatial part that depends on the specified eigenfunction and a temporal part that depends on the corresponding eigenvalue (i.e., correlation time) with a simple, exponential decay. The spatial part is a probability distribution of the dipole pair, analogous to the stationary states of a quantum harmonic oscillator. Drawing inspiration from the idea of inherent structures in liquids, we interpret each of the spatial contributions as a specific molecular mode. These modes can be used to model and predict the NMR dipole-dipole relaxation dispersion of fluids by incorporating phenomena on the molecular level. We validate our statistical mechanical description of the distribution in molecular modes with molecular dynamics simulations interpreted without any relaxation models or adjustable parameters: the most important poles in the Padé-Laplace transform of the simulated autocorrelation agree with the eigenvalues predicted by the theory.

Effect of Nanoconfinement on NMR Relaxation of Heptane in Kerogen from Molecular Simulations and Measurements.

Kerogen-rich shale reservoirs will play a key role during the energy transition, yet the effects of nanoconfinement on the NMR relaxation of hydrocarbons in kerogen are poorly understood. We use atomistic MD simulations to investigate the effects of nanoconfinement on the 1H NMR relaxation times T1 and T2 of heptane in kerogen. In the case of T1, we discover the important role of confinement in reducing T1 by ∼3 orders of magnitude from that of bulk heptane, in agreement with measurements of heptane dissolved in kerogen from the Kimmeridge Shale, without any models or free parameters. In the case of T2, we discover that confinement breaks spatial isotropy and gives rise to residual dipolar coupling which reduces T2 by ∼5 orders of magnitude from the value for bulk heptane. We use the simulated T2 to calibrate the surface relaxivity and thence predict the pore-size distribution of the organic nanopores in kerogen, without additional experimental data.

Hydration Free Energies of Polypeptides from Popular Implicit Solvent Models versus All-Atom Simulation Results Based on Molecular Quasichemical Theory.

Calculating the hydration free energy of a macromolecule in all-atom simulations has long remained a challenge, necessitating the use of models wherein the effect of the solvent is captured without explicit account of solvent degrees of freedom. This situation has changed with developments in the molecular quasi-chemical theory (QCT)─an approach that enables calculation of the hydration free energy of macromolecules within all-atom simulations at the same resolution as is possible for small molecular solutes. The theory also provides a rigorous and physically transparent framework to conceptualize and model interactions in molecular solutions and thus provides a convenient framework to investigate the assumptions in implicit solvent models. In this study, we compare the results using molecular QCT versus predictions from EEF1, ABSINTH, and GB/SA implicit solvent models for polyglycine and polyalanine solutes covering a range of chain lengths and conformations. The hydration free energies or the differences in hydration free energies between conformers obtained from the implicit solvent models do not agree with explicit solvent results, with the deviations being largest for the group additive EEF1 and ABSINTH models. GB/SA does better in capturing the qualitative trends seen in explicit solvent results. Analysis founded on QCT reveals the critical importance of the cooperativity of hydration that is inherent in the hydrophilic and hydrophobic contributions to hydration─physics that is not well captured in additive models but somewhat better accounted for by means of a dielectric in the GB/SA approach.

Thermal and concentration effects on 1H NMR relaxation of Gd3+-aqua using MD simulations and measurements.

Gadolinium-based contrast agents are key in clinical MRI for enhancing the longitudinal NMR relativity (r1) of hydrogen nuclei (1H) in water and improving the contrast among different tissues. The importance of MRI in clinical practice cannot be gainsaid, yet the interpretation of MRI relies on models with severe assumptions, reflecting a poor understanding of the molecular-scale relaxation processes. In a step towards building a clearer understanding of the relaxation processes, here we investigate thermal and concentration effects on r1 of the Gd3+-aqua complex using both semi-classical molecular dynamics (MD) simulations and measurements. We follow the MD simulation approach recently introduced by [Singer et al., Phys. Chem. Chem. Phys., 2021, 23, 20974], in which no NMR relaxation model or free-parameter is assumed to predict r1, thereby bringing new insights into the physics of r1 on a molecular scale. We expand the autocorrelation function G(t) in terms of molecular modes and determine the thermal activation energies of the two largest modes, both of which are consistent with the range of literature values for rotational diffusion. We also determine the activation energies for translational diffusion and low-field electron-spin relaxation, both of which are consistent with the literature. Furthermore, we validate the MD simulations at human body temperature and concentrations of the paramagnetic ion used in clinical MRI, and we quantify the uncertainties in both simulations and measurements.

Block copolymer self-assembly: Melt and solution by molecular density functional theory.

The self-assembly of block copolymer melts and solutions with two-dimensional density inhomogeneity is studied using modified inhomogeneous statistical associating fluid theory (iSAFT). A real-space combinatorial screening method under density functional theory formalism is proposed and used to map out the phase diagram of block copolymer melts including order-disorder transitions and order-order transitions. The predicted phase diagram agrees well with molecular dynamics simulation and self-consistent field theory. The compressibility effect on order-disorder transition temperature for block copolymer melts is modeled using iSAFT. The pressure induced temperature change by theory has a similar trend to experimental studies. Then, the lyotropic and thermotropic self-assembly phase behavior of block copolymer solutions is investigated. Detailed density distributions by iSAFT provide insight into the lyotropic properties of the block copolymer solutions at the molecular level. The effect of the block copolymer molecular architecture is studied by comparing block copolymers with different molecular packing parameters. Block copolymer solutions in the inverted hexagonal phase are predicted by theory for the block copolymer having a large molecular packing parameter. Finally, solvent selectivity is studied by modeling the block copolymers in a neutral good solvent. The enhanced local solvent concentration predicted by theory explains the reason for fewer ordered phases found in experiments.

Predicting 1H NMR relaxation in Gd3+-aqua using molecular dynamics simulations.

Atomistic molecular dynamics simulations are used to predict 1H NMR T1 relaxation of water from paramagnetic Gd3+ ions in solution at 25 °C. Simulations of the T1 relaxivity dispersion function r1 computed from the Gd3+-1H dipole-dipole autocorrelation function agree within ≃8% of measurements in the range f0 ≃ 5 ↔ 500 MHz, without any adjustable parameters in the interpretation of the simulations, and without any relaxation models. The simulation results are discussed in the context of the Solomon-Bloembergen-Morgan inner-sphere relaxation model, and the Hwang-Freed outer-sphere relaxation model. Below f0 ≲ 5 MHz, the simulation overestimates r1 compared to measurements, which is used to estimate the zero-field electron-spin relaxation time. The simulations show potential for predicting r1 at high frequencies in chelated Gd3+ contrast-agents used for clinical MRI.

Phase behavior and percolation in mixed patchy colloids.

Patchy colloids can be modeled as hard spheres with directional conical association sites. A variety of physical phenomena have been discovered in the patchy colloid system due to its short range and directional interactions. In this work, we combined a cluster distribution theory with generalized Flory and Stockmayer percolation theory to investigate the interplay between phase behavior and percolation for a binary patchy colloid system. The binary patchy colloid system consists of solute molecules with spherically symmetric bonding sites and solvents with two singly bondable sites. Wertheim's first order thermodynamic perturbation theory (TPT1) has been widely applied to the patchy colloids system and it has been combined with percolation theory to study the percolation threshold. However, due to assumptions behind TPT1, it will lose accuracy for a system in which particles have multiple association sites or multiply bondable sites. A recently proposed cluster distribution theory accurately models association at sites that can form multiple bonds. In this work, we investigate the comparison among cluster distribution theory, TPT1, and Monte Carlo simulation for the bonding states of this binary system in which cluster distribution theory shows excellent agreement with Monte Carlo simulation, while TPT1 has a large deviation with the simulation. Cluster distribution theory was further combined with the Flory and Stockmayer percolation theory to investigate the interplay between phase behavior and percolation threshold. We found that the reduced density and the relative bonding strength of solvent-solvent association and solute-solvent association are key factors for the phase behavior and percolation. Percolation can form at low density and low temperature in the vapor phase of this binary system, where the star-like molecules with 12 long branches formed.

NMR 1H-1H Dipole Relaxation in Fluids: Relaxation of Individual 1H-1H Pairs versus Relaxation of Molecular Modes.

The intramolecular 1H NMR dipole-dipole relaxation of molecular fluids has traditionally been interpreted within the Bloembergen-Purcell-Pound (BPP) theory of NMR intramolecular relaxation. The BPP theory draws upon Debye's theory for describing the rotational diffusion of the 1H-1H pair and predicts a monoexponential decay of the 1H-1H dipole-dipole autocorrelation function between distinct spin pairs. Using molecular dynamics (MD) simulations, we show that for both n-heptane and water this is not the case. In particular, the autocorrelation function of individual 1H-1H intramolecular pairs itself evinces a rich stretched-exponential behavior, implying a distribution in rotational correlation times. However, for the high-symmetry molecule neopentane, the individual 1H-1H intramolecular pairs do conform to the BPP description, suggesting an important role of molecular symmetry in aiding agreement with the BPP model. The intermolecular autocorrelation functions for n-heptane, water, and neopentane also do not admit a monoexponential behavior of individual 1H-1H intermolecular pairs at distinct initial separations. We suggest expanding the autocorrelation function in terms of modes, provisionally termed molecular modes, that do have an exponential relaxation behavior. With care, the resulting Fredholm integral equation of the first kind can be inverted to recover the probability distribution of the molecular modes. The advantages and limitations of this approach are noted.

Elucidating the 1H NMR Relaxation Mechanism in Polydisperse Polymers and Bitumen Using Measurements, MD Simulations, and Models.

The mechanism behind the 1H nuclear magnetic resonance (NMR) frequency dependence of T1 and the viscosity dependence of T2 for polydisperse polymers and bitumen remains elusive. We elucidate the matter through NMR relaxation measurements of polydisperse polymers over an extended range of frequencies (f0 = 0.01-400 MHz) and viscosities (η = 385-102 000 cP) using T1 and T2 in static fields, T1 field-cycling relaxometry, and T1ρ in the rotating frame. We account for the anomalous behavior of the log-mean relaxation times T1LM ∝ f0 and T2LM ∝ (η/T)-1/2 with a phenomenological model of 1H-1H dipole-dipole relaxation, which includes a distribution in molecular correlation times and internal motions of the nonrigid polymer branches. We show that the model also accounts for the anomalous T1LM and T2LM in previously reported bitumen measurements. We find that molecular dynamics (MD) simulations of the T1 ∝ f0 dispersion and T2 of similar polymers simulated over a range of viscosities (η = 1-1000 cP) are in good agreement with measurements and the model. The T1 ∝ f0 dispersion at high viscosities agrees with previously reported MD simulations of heptane confined in a polymer matrix, which suggests a common NMR relaxation mechanism between viscous polydisperse fluids and fluids under nanoconfinement, without the need to invoke paramagnetism.

Self-assembly and phase behavior of mixed patchy colloids with any bonding site geometry: theory and simulation.

Patchy colloids and associating fluids have attracted continued interest due to the interesting phase behavior and self-assembly in solution. The ability to fabricate patchy colloids with multiple attractive surface patches of different number, size, shape, and relative location makes patchy colloids a good candidate as building blocks to form complex advanced materials. However, a theory that clearly relates the self-assembled structures that form based on the anisotropic interactions has been missing. Although Wertheim's theory in the form of the SAFT model is widely used to predict self-assembly and phase behavior in solution, SAFT does not include multibody correlations necessary to model any shape of association site or sites that can form multiple bonds. We have recently developed a new theory for associating colloids that naturally incorporates multibody correlations based on a cluster distribution approach due to Bansal, Asthagiri, Marshall, and Chapman (BAMC). In this paper, we extended the cluster distribution theory to predict the thermodynamic properties and phase behavior of binary systems consisting of anisotropic particles with any geometry of bonding site. In particular, we consider self-assembly of Janus particles, Saturn particles, and ternary particles mixed with solvent colloids that have two directional patchy sites. Good agreement between theoretical predictions and molecular simulation is shown for self-assembly, thermodynamic properties in this system. Re-entrant phase behavior has been investigated and low density gels is predicted.

Critical Role of Confinement in the NMR Surface Relaxation and Diffusion of n-Heptane in a Polymer Matrix Revealed by MD Simulations.

The mechanism behind the NMR surface-relaxation times (T1S,2S) and the large T1S/T2S ratio of light hydrocarbons confined in the nanopores of kerogen remains poorly understood and consequently has engendered much debate. Toward bringing a molecular-scale resolution to this problem, we present molecular dynamics (MD) simulations of 1H NMR relaxation and diffusion of n-heptane in a polymer matrix. The high-viscosity polymer is a model for kerogen and bitumen that provides an organic "surface" for heptane. Diffusion of n-heptane shows a power-law dependence on the concentration of n-heptane (ϕC7) in the polymer matrix, consistent with Archie's model of tortuosity. We calculate the autocorrelation function G(t) for 1H-1H dipole-dipole interactions of n-heptane in the polymer matrix and use this to generate the NMR frequency (f0) dependence of T1S,2S as a function of ϕC7. We find that increasing molecular confinement increases the correlation time, which decreases the surface-relaxation times for n-heptane in the polymer matrix. For weak confinement (ϕC7 > 50 vol %), we find that T1S/T2S ≃ 1. Under strong confinement (ϕC7 ≲ 50 vol %), we find that T1S/T2S ≳ 4 increases with decreasing ϕC7 and that the dispersion relation T1S ∝ f0 is consistent with previously reported measurements of polydisperse polymers and bitumen. Such frequency dependence in bitumen has been previously attributed to paramagnetism; instead, our studies suggests that 1H-1H dipole-dipole interactions enhanced by organic nanopore confinement dominate the NMR response in saturated organic-rich shales.

Modeling Lower Critical Solution Temperature Behavior of Associating Dendrimers Using Density Functional Theory.

We study the phase behavior of associating dendrimers in explicit solvents using classical density functional theory. The existence of association enables uptake of solvent inside the dendrimer even for unfavorable Lennard-Jones interaction between the solvent and dendrimer. Depending on the distributions of associating sites, the dendrimer conformation can be either dense-core or dense-shell. The conformation of the associating dendrimer is greatly affected by the temperature. Due to the interplay between association interaction and Lennard-Jones attractions, we find the lower critical solution temperature (LCST) behavior of dendrimer conformation and study how it changes as the dendrimer size or solvent size changes. The dendrimer in our study displays no LCST behavior at low generations, and it has a maximum LCST at G4. Moreover, increasing the solvent chain length decreases the LCST. For solvents with self-association, the competition between solvent-solvent association and solvent-dendrimer association also tends to reduce the LCST. Qualitatively consistent with experiments, our results provide insight into the molecular mechanism of the LCST behavior of associating dendrimers.

Bond cooperativity and ring formation in hydrogen fluoride thermodynamic properties: A two-density formalism framework.

In this work, we develop a thermodynamic perturbation theory using a two-density formalism framework to model the bond cooperativity effect for associating hard sphere and Lennard-Jones fluids. The theory predictions are compared with Monte Carlo simulation results and they are in excellent agreement. We incorporate bond angle dependent ring formation into the theory to calculate hydrogen fluoride thermodynamic properties. The liquid density and vapor pressure obtained by the theory are in good agreement with the experimental data. Comparing the thermo-physical properties of hydrogen fluoride calculated by this theory with previous studies reveals the importance of bond angle dependent ring formation and cooperative hydrogen bonding to capture its anomalous behavior especially in the vapor phase. The cooperativity ratio obtained in our model is close to the values reported by previous quantum studies.

Competitive Sorption of CO2 with Gas Mixtures in Nanoporous Shale for Enhanced Gas Recovery from Density Functional Theory.

CO2 competitive sorption with shale gas under various conditions from simple to complex pore characteristics is studied using a molecular density functional theory (DFT) that reduces to perturbed chain-statistical associating fluid theory in the bulk fluid region. The DFT model is first verified by grand canonical Monte Carlo simulation in graphite slit pores for pure and binary component systems at different temperatures, pressures, pore sizes, and bulk gas compositions for methane/ethane with CO2. Then, the model is utilized in multicomponent systems that include CH4, C2H6, and C3+ components of different compositions. It is shown that the selectivity of CO2 decreases with increases in temperature, pressure, nanopore size, and average molecular weight of shale gas. Extending the model to more realistic situations, we consider the impact of water present in the pore and consider the effect of permeation of fluid molecules into the kerogen that forms the pore walls. The water-graphite interaction is calibrated with contact angle from molecular simulation data from the literature. The kerogen pore model prediction of gas absolute sorption is compared with experimental and molecular simulation values in the literature. It is shown that the presence of water reduces the CO2 adsorption but improves the CO2 selectivity. The dissolution of gases into the kerogen matrix also leads to the increase in CO2 selectivity. The effect of kerogen type and maturity on the gas sorption amount and CO2 selectivity is also studied. The associated mechanisms are discussed to provide fundamental understanding for gas recovery by CO2.

A cluster size distribution theory to study the thermodynamics and phase behavior of multi-bonding single site solutes in patchy colloidal mixtures.

We study binary mixtures of multi-bonding single site solute particles in a solvent comprising patchy colloid particles. The particles in the mixture interact by very short-ranged attraction and hard-sphere repulsion. The attractive patch on the solute can bond with multiple solvent particles, whereas the patch on the solvent is restricted to bond only once. From a quasi-chemical analysis of association, in the hard-sphere reference we develop an accurate multi-body correlation information for the distribution of solvent particles over the patch region of the solute. We use this information within Wertheim's multi-density formalism to develop a cluster size distribution theory that is capable of capturing the physics of multi-body association for any geometry of association sites on the solute. We use this general framework to study a mixture containing Janus solutes and one- or two-patch solvent particles over a range of concentration of the solute and association strengths. We find that a mixture of two-patch solvent (with both patches of the same kind) and multi-bonding solutes with different patch geometries can have a vapor-liquid equilibrium, although the pure components themselves cannot phase separate. The liquid state occurs at very low densities, forming a so-called empty liquid. For the relative association strengths studied in this work, we observe that the vapor-liquid coexistence curve broadens as the concentration of the patchy solvent particles in the liquid phase is increased. The pressure-composition phase equilibrium curves show negative azeotropes for these mixtures. We also observe that, for these mixtures, as the size of the patch on the solute particles is decreased, the critical temperature and the critical packing fraction decreases.

Density functional study of dendrimer molecules in solvents of varying quality.

Modified inhomogeneous statistical associating fluid theory (iSAFT) density functional theory is extended to dendrimer molecules in solvents of varying quality. The detailed structures of isolated dendrimers in implicit solvent are calculated and have a semi-quantitative agreement with simulation results available in the literature. The dendrimers form dense-core structures under all conditions, while their radius of gyration follows different scaling laws. Factors that affect the quality of the solvent are systematically studied in the explicit solvent case. It is found that the solvent size, density, chemical affinity and temperature all play a role in determining a solvent to be good or poor. New molecular dynamics simulations are performed to validate the iSAFT results. Our results provide insight into the phase behavior of dendrimer solutions as well as guidance in practical applications.