Diffusion in nanoporous phases: size dependence and levitation effect

Self-diffusivity, D, of diffusants in widely differing mediums such as liquids (e.g., solution), porous solids (e.g., guests in zeolites), or ions in polar solvents exhibit strong size dependence. We discuss the nature of the size dependence observed in these systems. Altogether, different theoretical approaches have been proposed to understand the nature of size dependence of D not only across these widely differing systems but even in just one medium or class of systems such as, for example, ions in polar solvents. But molecular dynamics investigations in the past decade have shown that the size dependence of self-diffusion in guest-porous solids could have origins in the mutual cancellation of forces that occurs when the size of the diffusant is comparable to the size of the void. The effect leading to the maximum in D is known as the levitation effect (LE). Such a cancellation is a consequence of symmetry. This effect exists in all porous solids irrespective of the geometrical and topological details of the pore network provided by the solid. Recent studies show that the levitation effect and size-dependent diffusivity maximum exists for uncharged solutes in solvents. One of the consequences of this is the breakdown in the Stokes-Einstein relationship over a certain range of solute-solvent size ratio. Experimental measurements of ionic conductivity over the past hundred years have found the existence of a size-dependent diffusivity maximum leading to violation of the Walden's rule for ions in polar solvents. Molecular dynamics simulations and experimental data suggest that even this maximum has its origin in LE. Simulation studies of impurity atom diffusion in close-packed solids as well as ions in superionic and other solids suggest the existence of a size-dependent diffusivity maximum in these materials as well. The levitation effect is a universal effect leading to a maximum in diffusivity of a diffusant in a variety of condensed matter phases. The only condition for its existence appears to be the presence of van der Waals or electrostatic interactions.     

Diffusion of linear aliphatic esters in polyethylene as a function of chain length of the esters,pressure and temperature

Diffusion of linear aliphatic mono- and diesters (C _N ) havingN main chain atoms (N=13–68) in bulk medium-density polyethylene (MDPE) has been studied under hydrostatic pressures up to 2500 bar at temperatures between 60°C and 125°C. Three triglycerides, phenyl stearate, and p-aminoazobenzene (pAAB, 80°C) as the diffusants and low-density (LDPE) and high-density (HDPE) polyethylenes as polyethylene substrate were used for comparison. Diffusion coefficientD was determined from concentration distribution of the diffusants through stacked PE sheets as substrate. Regarding the linear esters at 90°C, the relationshipD N holds at constant pressures. Under the atmospheric pressure, became –2.10 in accordance with de Gennes's proposal (1971)D N ^–2 as well as with the experimental results reported by Klein and Briscoe (1979) forN larger than 30.D's for the glycerides deviate from the relationshipD N ^–2 toward the smaller values by comparison at the sameN. The exponent is pressure-dependent. It decreases with increasing pressure according to =–2.10–0.000942P, whereP is measured by the unit of bar. Plots of lnD vsP for all the diffusants show linear relationships with negative slopes, from which activation volume for the diffusion V was calculated. At 90°C, V increases slowly with increasingN and increasingV _Ki, the intrinsic molecular volume of the diffusant, from 39.3 cm³/mol for ethyl caprate (C ₁₃,V _Ki=136 cm³/mol) to 76.8 cm³/mol for behenyl behenate (C₄₅,V _Ki=466 cm³/mol). Observed V s are explainable on the basis of the reptation mode of the chain molecule diffusion. V s for C₂₅ and C₄₅ are found to increase with increasing degree of crystallinity where MDPE, heat-treated MDPE, LDPE, and HDPE were used. The results obtained by varying temperature are as follows. V for C₄₅ was always found to be larger than C₂₅. Both decreased linearly with increasing temperature, giving two linear lines with different slopes whose extensions intersected at 132°C, the melting point of the MDPE, where the difference in V disappeared. The apparent activation energiesE _Ds for the diffusion of C₂₅ and C₄₅ increased linearly with increasing pressure, whose slopes are explainable according toE _D=E ₀+PV [1-(dln V /dlnT) _P ].     

Diffusion of linear polyisoprene molecules into polyisoprene networks

Diffusion and equilibrium absorption of polyisoprene liquids into crosslinked samples of cis-polyisoprene (natural rubber) have been studied by direct observation of volume swelling. Natural rubber was crosslinked in the form of fine threads, about 40 ± 20 μm in diameter, using a gaseous reaction with SO₂ and H₂S (the Peachey process). An optical microscope was used to observe the relatively rapid absorption of linear high-molecular-weight polyisoprene liquids by these fine threads. From the kinetics of absorption, values of the self-diffusion coefficient D_s of polyisoprene were estimated. They ranged from 10^?16 m²/s to 10^?12 m²/s, depending upon molecular weight, and varied with molecular weight approximately as M^?2 over the entire range employed, from 1,000 to 60,000 g/mol, i.e., both above and below the entanglement molecular weight. Amounts of polymer absorbed at equilibrium varied widely, depending upon the degree of crosslinking of the host material and the molecular weight of the absorbing liquid. They were in reasonable agreement in all cases with simple swelling theory, with the heat of mixing equated to zero.     

Separation of mixtures at nano length scales: blow torch and levitation effect

A new conceptual basis for the separation of multicomponent molecular mixtures is proposed. A separation method where different components of the mixtures are driven in opposite directions is realized by a judicious combination of two effects, viz., levitation and blow torch effects. Monte Carlo simulations of two Lennard-Jones binary mixtures with different-sized components are shown to be separated well if at least one of the components lies in the anomalous regime and the others lie in the linear regime. A separation factor of 10(8) is obtained on nano length scales as compared to 10(3), obtainable through conventional methods of separation on macrolength scales.     

Evidence in support of levitation effect as the reason for size dependence of ionic conductivity in water: a molecular dynamics simulation

We report extensive molecular dynamics simulations of (i) model ions in water at high concentrations as a function of the size and charge of the ion as well as (ii) realistic simulation of Cl- and Br- ions at low concentrations in water at room temperature. We also analyze existing experimental data in light of the results obtained here. The halide ion simulations have been carried out using the interaction potentials of Koneshan et al. (J. Phys. Chem. B 1998, 102, 4193). We compute structural and dynamical properties of ions in water and explore their variation with size and charge of the ion. We find that ions of certain intermediate sizes exhibit a maximum in self-diffusivity in agreement with previous experimental measurements and computer simulations. We analyze molecular dynamics trajectories in light of the previous understanding of the levitation effect (LE) and the recent suggestion that ionic conductivity has its origin in LE (J. Phys. Chem. B 2005, 109, 8120). We report the distribution of void and neck radii that exist amidst water. Our analysis suggests that the ion with maximum self-diffusivity is characterized by a lower activation energy and a single-exponential decay of F(s)(k,t). The behavior of these and other related quantities of the ion with maximum self-diffusivity are characteristic of the anomalous regime of the LE. The simulation results of Br- and Cl- ions in water also yield results in agreement with the predictions of LE. A plot of experimental conductivity data in the literature for alkali ions in water by Kay and Evans (J. Phys. Chem. 1966, 70, 2325) also yields a lower activation energy for the ion with maximum conductivity in excellent agreement with the LE. To the best of our knowledge, none of the existing theories predict a lower activation energy for the ion with maximum conductivity.     

The rotational partition function for linear molecules

An important function in the statistical treatment of a gas of linear molecules is This sum is convenient to use mainly when α is large and alternate expressions, generally asymptotic expansions, are often used when α is small. In this paper, the sum is evaluated to yield a single expression that is valid for large and small values of α. The expression is composed of three terms, each of which involves the theta functions of Jacobi. One term is in the form of an integral, but is small relative to the other two and easily evaluated by numerical means. The expression is readily differentiated and can be used for the general evaluation of the rotational partition function for gases of linear molecules at all temperatures. This revised version was published online in July 2006 with corrections to the Cover Date.     

Diffusion coefficients as a monitor of reaction kinetics of biological molecules

For revealing spectrally silent dynamics in chemical reactions, a new method, the time-dependent diffusion coefficient, is presented. Principles and typical examples of this method, in particular applications to biologically related reactions, are reviewed. The pulsed laser induced transient grating signal of the photo-decomposition reaction of caged ATP showed that the diffusion coefficient increases gradually with time reflecting the molecular size decrease by the dissociation. Hence, this rate should be a direct measurement of the photo-dissociation rate of ATP from the caged state. In an application to a protein folding reaction, the time-development of the diffusion coefficient was observed during the folding reaction. This time dependence was interpreted in terms of the intermolecular interaction change; i.e., conversion from the intermolecular hydrogen bonding to intramolecular one. It was found that the change of the hydrogen bonding network occurred by the two state manner in entire refolding process of cytochrome c. The unique feature of this time-dependent diffusion coefficient method is discussed.     

Excluded volume effect of linear polymer molecules

An approximate closed expression for the excluded volume effect of linear polymer molecules is developed with the aid of a uniform expansion model of perturbed chains. The linear expansion factor α for the end-to-end distance is given by (α³ ? 1) + (3/8) (α⁵ ? α³) = (5/2)z where z is the excluded volume parameter. This equation is numerically close to the Ptitsyn equation in the ordinary range of α; i.e., for 1 ≤ α ≤ 2.     

Toward a global maximization of the molecular similarity function: Superposition of two molecules

A quantum similarity measure between two molecules is normally identified with the maximum value of the overlap of the corresponding molecular electron densities. The electron density overlap is a function of the mutual positioning of the compared molecules, requiring the measurement of similarity, a solution of a multiple-maxima problem. Collapsing the molecular electron densities into the nuclei provides the essential information toward a global maximization of the overlap similarity function, the maximization of which, in this limit case, appears to be related to the so-called assignment problem. Three levels of approach are then proposed for a global search scanning of the similarity function. In addition, atom—atom similarity Lorentzian potential functions are defined for a rapid completion of the function scanning. Performance is tested among these three levels of simplification and the Monte Carlo and simplex methods. Results reveal the present algorithms as accurate, rapid, and unbiased techniques for density-based molecular alignments. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 826–846, 1997     

Vibrational isotope effect of linear and planar molecules: Deuterated bromoethenes

Linear (planar) molecules A and B which are identical except for isotopic substitutions at the atomic sites are considered. Stretching (bending, out-of-plane) frequencies _k and normal modes _k of the isotopically perturbed molecule B are expressed in terms of stretching (bending, out-of-plane) frequencies _i and the corresponding normal modes _i of the unperturbed molecule A. Complete specification of the unperturbed normal modes is not required. All that is needed are stretching (bending, out-of-plane) amplitudes | _i of the normal modes _i at those sites that are affected by isotopic substitution. The rule which interlaces frequencies _k of molecule A with frequencies _i of molecule B is derived. Given two isotopic molecules A and B that differ by a single isotopic substitution at site , the inversion relation is derived. This relation expresses unperturbed stretching (bending, out-of-plane) amplitudes at the site in terms of stretching (bending, out-of-plane) frequencies of molecules A and B . As an example, out-of-plane vibrations of deuterated bromoethene were considered. In the simplest method 12 out-of-plane frequencies of four polydeuterated bromoethenes were calculated from 12 out-of-plane frequencies of bromoethene and three monodeuterated bromoethenes. Standard deviation of thus calculated frequencies from experimental frequencies is =2.74 cm^–1. In another method, 15 out-of-plane frequencies of four polydeuterated bromoethenes and selected monodeuterated bromoethene are calculated from 9 out-of-plane frequencies of bromoethene and the remaining two monodeuterated bromoethenes. Depending on which monodeuterated bromoethene is selected (1-, cis- or trans-), standard deviation of thus obtained frequencies from experimental frequencies is 1=2.84 cm^–1, c=2.96 cm^–1 and t=2.72 cm^–1.     

Chain length effects in the second virial coefficient of linear polymer molecules

A generalized perturbation theory is presented for the second virial coefficient of linear and branched polymer systems. Results have been computed for linear chains having two to five hundred statistical segments. These are found to differ significantly from the long-chain asymptotic results of Zimm. A semi-empirical modification of the Flory--Orofino theory is also suggested.     

Theoretical evolution of the third-order molecular polarizabilities as a function of chain length in thiophene and pyrrole oligomers

On the basis of INDO (intermediate neglect of differential overlap)/MRD -CI (multireference determinant–configuration interaction) calculations, the SOS (sum-over-states) formalism is applied to calculate the third-order polarizabilities, γ, in thiophene and pyrrole oligomers. For both types of oligomers, the chain-length dependence of γ can be divided into three regimes: Describing γ as a power law of the number of rings (N), we observe that the power value first strongly increases with N, than reaches a nearly constant value (power regime), and, finally, decreases toward one, indicating the appearance of a saturation regime. Very good agreement with the experimental evolution has been found in the case of the oligothiophenes. The comparison of the results obtained for the two types of oligomers indicates a lower polarizability in the oligopyrroles. © 1994 John Wiley & Sons, Inc.     

Rotational viscosity of fluids composed of linear molecules: an equilibrium molecular dynamics study

In this paper, we investigate the rotational viscosity for a chlorine fluid and for a fluid composed of small linear molecules by using equilibrium molecular dynamics simulations. The rotational viscosity is calculated over a large range of state points. It is found that the rotational viscosity is almost independent of temperature in the range studied here but exhibits a power-law dependency on density. The rotational viscosity also shows a power-law relationship with the molecular length, and the ratio between the shear and rotational viscosities approaches 0.5 for the longest molecule studied here. By changing the number of atoms or united atomic units per molecule and by keeping the molecule length fixed, we show that fluids composed of molecules which have a rodlike shape have a lower rotational viscosity. We argue that this phenomenon is due to the reduction in intermolecular connectivity, which leads to larger fluctuations around the values possessed by the fluid on average. The conclusions here can be extended to fluids composed of uniaxial molecules of arbitrary length.     

Technomimetic molecules: synthesis of a molecular wheelbarrow

A molecular analog of a wheelbarrow is synthesized following a strategy based on sequential double Knoevenagel and Diels-Alder reactions.     

Lennard-Jones sticks: a new model for linear molecules

We consider the anisotropic interaction between two line segments consisting of a homogeneous distribution of Lennard-Jones centers. The potential energy of such a pair cannot be expressed in closed form. However, we show that it may be approximated in a way that renders this intuitively appealing model competitive both for simulations and theory.     

Diffusion of small molecules inside a peptide hydrogel

A set of four phenylalanine analogues experiences diffusion retardation when transferred from phosphate-buffered saline into a peptide hydrogel of the same pH and ionic strength. The extent of retardation increases linearly with logP(oct), their lipophilicity.     

Some thermodynamic quantities of n-alkanes as a function of chain length

Summary Results from high pressure dilatometry onn-alkanes and linear polyethylene and literature data yield a linear relation between specific volume, entropy and enthalpy of fusion, the reciprocal melting temperature and 1/n, wheren denotes the number of C-atoms per molecule. Extrapolating towards infiniten one always obtains polyethylene data.The differences in the properties ofn-alkanes and polyethylene can be ascribed to the influence of the chain ends. Assuming entropy and enthalpy contributions from these chain ends as made probable by a molecular model one can quantitatively explain then-dependence of the above mentioned quantities including their pressure dependence.With 10 figures and 2 tables     

Diffusion of vibrationally excited molecules

A treatment is presented for the effect of intermolecular vibrational energy transfer on the diffusion coefficient of vibrationally excited molecules. An analytic treatment based on random walk statistics and a Monte Carlo type calculation have been performed. Both methods yield very similar results which correlate well with existing experimental studies. A hard sphere collision model is treated extensively with comparisons made to other internmolecular potentials. The results support the involvement of long range energy transfer in V → V interactions. The effect of temeprature on the diffusion coefficient of vibrationally excited molecules is calculated, with applications to the CO^*₂CO₂ system.