Nonideal Free Energy Research Articles

Unconventional Charge Transport in MgCr2O4 and Implications for Battery Intercalation Hosts.

Ion transport in solid-state cathode materials prescribes a fundamental limit to the rates batteries can operate; therefore, an accurate understanding of ion transport is a critical missing piece to enable new battery technologies, such as magnesium batteries. Based on our conventional understanding of lithium-ion materials, MgCr2O4 is a promising magnesium-ion cathode material given its high capacity, high voltage against an Mg anode, and acceptable computed diffusion barriers. Electrochemical examinations of MgCr2O4, however, reveal significant energetic limitations. Motivated by these disparate observations; herein, we examine long-range ion transport by electrically polarizing dense pellets of MgCr2O4. Our conventional understanding of ion transport in battery cathode materials, e.g., Nernst-Einstein conduction, cannot explain the measured response since it neglects frictional interactions between mobile species and their nonideal free energies. We propose an extended theory that incorporates these interactions and reduces to the Nernst-Einstein conduction under dilute conditions. This theory describes the measured response, and we report the first study of long-range ion transport behavior in MgCr2O4. We conclusively show that the Mg chemical diffusivity is comparable to lithium-ion electrode materials, whereas the total conductivity is rate-limiting. Given these differences, energy storage in MgCr2O4 is limited by particle-scale voltage drops, unlike lithium-ion particles that are limited by concentration gradients. Future materials design efforts should consider the interspecies interactions described in this extended theory, particularly with respect to multivalent-ion systems and their resultant effects on continuum transport properties.

The equation of state and ionization equilibrium of dense aluminum plasma with conductivity verification

The equation of state, ionization equilibrium, and conductivity are the most important parameters for investigation of dense plasma. The equation of state is calculated with the non-ideal effects taken into consideration. The electron chemical potential and pressure, which are commonly used thermodynamic quantities, are calculated by the non-ideal free energy and compared with results of a semi-empirical equation of state based on Thomas-Fermi-Kirzhnits model. The lowering of ionization potential, which is a crucial factor in the calculation of non-ideal Saha equation, is settled according to the non-ideal free energy. The full coupled non-ideal Saha equation is applied to describe the ionization equilibrium of dense plasma. The conductivity calculated by the Lee-More-Desjarlais model combined with non-ideal Saha equation is compared with experimental data. It provides a possible approach to verify the accuracy of the equation of state and ionization equilibrium.

Extending the diffuse layer model of surface acidity behaviour: III. Estimating bound site activity coefficients

Although detailed thermodynamic analyses of the 2-pK diffuse layer surface complexation model generally specify bound site activity coefficients for the purpose of accounting for those non-ideal excess free energies contributing to bound site electrochemical potentials, in application these terms are ignored based on one or more of the following assumptions: (1) bound site activity coefficients cancel out in the mass action quotients; (2) bound sites display ideal behaviour; and/or (3) these energies are already included in the exponential Boltzmann terms. In this work it is demonstrated that the bound site charging energy terms discussed in the two previous papers in this series have both conceptual and computational analogies to the charging energy contribution to the activity coefficients obtained from the Debye–Huckel Limiting Law. On high charge density colloidal particles at constant counterion condensation (τ), these charging energies can be related to the surface potential (ψ) by: ΔGcharging = (1 – τ)Fψ (where F is the Faraday constant). If one assumes a maximum practical accuracy of ± 10% in experimental estimates of ψ, then it is suggested that charging energies are likely to be experimentally indiscernible under conditions where τ > 0.9. These findings support the historical practice of ignoring bound site activity coefficients with the 2-pK diffuse layer surface complexation model in the following situations: for spherical particles with a radius ≥ 0.1 μm at ionic strengths ≥ 0.001 M (1 : 1), and for spherical particles with a radius >0.01 μm at an ionic strength >0.1 M (1 : 1). In contrast, charging energies (and non-ideal behaviour) are predicted to be significant at all charge densities and ionic strengths for spherical particles with a radius of 0.001 μm.

Analysis of Phase Separation in Compressible Polymer Blends and Block Copolymers

For the analysis of phase-separation behavior of compressible polymer blends and block copolymer melts, an equation-of-state (EOS) model is combined with the compressible random-phase approximation (RPA) formalism. The EOS model to be employed is obtained from extending an off-lattice model, recently given for interpreting compression response of pure polymers to pressure by Cho and Sanchez (CS), to polymer blends. The free energy for the CS model consists of an ideal free energy of Gaussian chains and a nonideal free energy that represents the excluded volume effects and the attractive interactions in given blend systems. This nonideal free energy yields the RPA interaction fields, with which monomer−monomer correlation functions for polymer blends or block copolymers are calculated to determine the condition of phase separation. It is shown that the theory can predict not only macrophase separation in some polymer blends but also microphase separation in the corresponding block copolymer melts. Especial...

Excess free energy, enthalpy and entropy of surfactant-surfactant mixed micelle formation

Enthalpies of dilution and osmotic coefficients of sodium decylsulfate (NaDeS)-dodecyldimethylamine oxide (DDAO) mixtures in water were determined at 298 and 310 K, respectively. From the enthalpies of dilution, the apparent and then the partial molar relative enthalpies of the surfactant mixtures were calculated. From the osmotic coefficients, calculated at 298 K, the non-ideal free energies were derived. The latter were combined with the partial molar relative enthalpies to obtain the non-ideal entropies. From the apparent molar properties, using a previously reported approach, the excess thermodynamic properties for the surfactant-surfactant mixed micelle formation in water were evaluated as functions of the mixture composition at some total micellized concentration. In the whole range of the mixture composition, the excess free energy is negative, indicating that the mixed micelle formation is favoured with respect to that of pure micelles. This process is governed by the enthalpy and/or the entropy, depending on the mixture composition. The effect of the alkyl chain length was also studied by comparing the present results to those of the sodium dodecylsulfate-DDAO mixture.

Thermodynamics of aqueous micellar solutions of selected permeatoxins at 298.15 and 313.15 K

Thermodynamic properties of aqueous solutions of N-dodecyl- N-ethylpiperidinium chloride (DMePC) and N-dodecyloxymethylene- N-methylpiperidinium chloride (DOMePC) at 298.15 and 313.15 K are reported for the first time. These compounds show biological activity and influence ion transport through membranes. Apparent and partial molar enthalpies and volumes have been obtained from the experimental data and expressed as a function of molalities, assuming infinite dilution as the standard state. The critical micelle concentrations at 313.15 K have been obtained from osmotic measurements, allowing also the evaluation of solution non-ideal free energies and entropies. The changes in thermodynamic properties for the micellization process have been derived, assuming a pseudo phase transition approach. The trends of the enthalpic curves of DMePC with DOMePC and also of DMePC with those of N-dodecyl- N-methylmorpholinium chloride (DMeMC) previously reported have been compared. It has been stated that the substitution of a methylene group by an oxygen atom, either in the polar part of an amphiphile or in its chain in the vicinity of the polar head group, has a small effect on the thermodynamic properties and micellization process of the amphiphile. A similar effect was also observed in the case of substitution of the first two methylene groups of n-alkyltrimethylammonium chlorides by the CH 2COO group. The above behaviour is explained as the result of micelle hydration at the depth of the first two methylene groups.

Energetics of sodium dodecylsulfate-dodecyldimethylamine oxide mixed micelle formation

Enthalpies of dilution and osmotic coefficients of the sodium dodecyl-sulfate (NaDS)-dodecyldimethylamine oxide (DDAO) mixtures in water have been measured at 25 and 37°C, respectively. From the enthalpies of dilution the apparent molar relative enthalpies LΦ were calculated. The change of the LΦ vs. total molality mt profiles with the mole fraction reflects the variation of the ionic character of the mixed micelles. From the osmotic coefficients the nonideal free energy G2ni were calculated. By combining G2ni with the partial molar relative enthalpies, the nonideal entropies TS2ni were determined. At a given mole fraction, G2ni and TS2ni values are decreasing and increasing respectively, tending to become constant at high mt. The excess properties for the mixed micelle formation were evaluated as a function of the mixture composition at some mt. The profiles are compared with those obtained from thermodynamics of binary liquid mixtures and the regular solution theory.

N,N,N-Alkyloctyldimethylammonium Chlorides in Water: A Thermodynamic Investigation

Specific conductivities, densities, heat capacities and enthalpies of dilution at 298 K and osmotic coefficients at 310 K were measured for N,N,N -octylbutyldimethylammonium, N,N,N -octylpentyldimethylammonium, and N,N -dioctyldimethylammonium chlorides in water as functions of concentration. From the specific conductivity data, the CMC and the degree of the counterion dissociation have been calculated. It is shown that additional CH 2 groups in the variable alkyl chain affect both CMC and β less than they do in the single chain surfactants. The partial molar volumes, heat capacities, relative enthalpies, nonideal free energies, and entropies at 298 K were derived as functions of the surfactant concentration. By increasing the tail of the variable alkyl chain, the apparent molar volume and heat capacity vs concentration curves are shifted regularly towards larger and smaller values, respectively. As well, in the micellar region, by increasing the alkyl chain length while the nonideal entropy vs log m/CMC curves are shifted regularly towards more positive values, those for free energy and enthalpy are not. The standard partial molar volumes and heat capacities linearly increase with the alkyl chain length slopes being slightly larger and smaller, respectively, than those for single chain surfactants. The partial molar volumes and heat capacities in the micellized form are also linearly related to the number of methylene groups with CH 2 contributions which are close and larger, respectively, than those for single chain surfactants. However, the heat capacity for the longer homologues deviates from the linearity. Since this peculiarity has been observed for most of the investigated properties, it has been ascribed to a different location of this chain in the micellar structure with respect to the shorter chains. In addition, thermodynamic properties of N,N,N-octylbutyldimethylammonium chloride are compared with those of N,N-octylmethylpiperidinium chloride which can be considered to be derived from the fusion of a methyl and the butyl groups in the N,N,N-octylbutyldimethylammonium chloride.

Thermodynamics of Micellar Systems: Volumes, Osmotic Coefficients, and Aggregation Numbers of a Homologous Series of 1,2,3-Alkane Triols in Water

Aqueous solutions of a homologous series of alkanetriols, from 1,2,3-hexanetriol to 1,2,3-nonanetriol, have been examined with different techniques, including density determination, freezing point depression, and vapor pressure osmometry. Their hydrophobic or micellar parameters were determined at different temperatures (0, 20, 30, and 40°C): volumes of monomers or micelles, volume changes during micelle formation, osmotic coefficient, activity of the solute and solvent, nonideal free energy of the solute, and relative chemical potential. 1,2,3-Hexanetriol does not form micelles and tends to form hydrophobic aggregates at 2.09 (20°C) and 1.80 (30°C) mol · dm -3. 1,2,3-Heptanetriol is a key compound since it is difficult to speak about clusters or real micellization at 1.03 (30°C) and 0.98 (40°C) mol · dm -3. 1,2,3-Octanetriol and 1,2,3-nonanetriol self associate in a manner similar to that of conventional surfactants. The CMCs respectively obtained are 0.370 to 0.320 mol · dm -3 (temperature range of 0 to 40°C) and 0.098 mol · dm -3 (40°C). Calculations using the phase separation model led to the following values for the aggregation numbers: 2.3 ± 0.2 for 1,2,3-hexanetriol, 7.6 ± 0.1 for 1,2,3-heptanetriol and 31.9 ± 0.1 for 1,2,3-octanetriol. The values directly obtained by freezing point depression (31.5 ± 1.8 ) or vapor pressure osmometry measurements (27.1 ± 0.9) at 40°C for 1,2,3-octanetriol show that the phase separation model is thus satisfactorily applied for this compound. The calculated free energy of the micelle formation agrees with published data for similar nonionic surfactants.

A density functional analysis of the restricted orientation model of liquid crystals and its implications for theories of orientational ordering

We compare the predictions of various density functional approximation schemes as applied to the restricted-orientation, hard, rectangular-parallelepiped model of liquid crystals. These calculations illustrate some features of density functional calculations which may aid in understanding other approximate, mean-field theories of orientational ordering phase transitions and their interfaces. The isotropic–nematic coexistence curve of this model is apparently well described by the so-called third-order y expansion, and the analytical expression for the nonideal free energy derived from the simple y expansion thus provides a benchmark against which to compare the density functional approximation schemes. Using standard relations applicable to inhomogeneous fluids, we show how expansions about the bulk isotropic phase compare more favorably with the ‘‘exact’’ y-expansion results when truncated at third order than do second-order truncations. Thus this model behaves somewhat differently than other hard-particle models of fluids, notably the hard-sphere one. We also examine the possibility of expanding the free energy about the ordered phase to obtain the properties of a disordered phase. An expansion about local values of the density leads to the widely used smoothed-density approximation and a hierarchy of systematic extensions. The latter appear more stable than conventional Taylor expansions about bulk disordered phases.

Thermodynamic studies of octyltrimethylammonium chloride in water

Densities, heat capacities, enthalpies of dilution at 298 K and osmotic coefficients at 310 K of octyltrimethylammonium chloride were measured as functions of concentration. From the experimental data, the partial molar volumes, heat capacities, relative enthalpies, nonideal free energies and entropies at 298 K were derived as functions of concentration. A comparison between the above data and those of dodecyltrimethylammonium chloride reported in the literature shows that the increase of the alkyl chain length shifts the apparent molar volumevs. concentration curves towards greater values and the heat capacity, relative enthalpy and free energyvs. concentration curves towards smaller values. By assuming the pseudo-phase transition model the properties of micellization (ΔYm) were graphically evaluated. TheΔYm values of OTAC compared with those of DTAC are consistent with the increase of the hydrophobicity by increasing the alkyl chain length.

Thermodynamic properties of N-octyl-, N-decyl- and N-dodecylpyridinium chlorides in water

Densities, heat capacities and enthalpies of dilution at 25°C and osmotic coefficients at 37°C were measured for N-octyl-, N-decyl- and N-dodecyl-pyridinium chlorides in water over a wide concentration region. Conductivity measurements were performed in order to evaluate the cmc and the degree of counterion dissociation. Partial molar volumes, heat capacities, relative enthalpies and nonideal free energies and entropies at 25°C were derived from the experimental data as functions of the surfactant concentration. The changes with concentration of these properties are quite regular with the exception of the heat capacities which display anomalies at about 0.9, 0.25 and 0.12 mol-kg−1 for the octyl, decyl and dodecyl compounds, respectively. At these concentrations there were also changes in the slopes of the specific conductivity and of the product of the osmotic coefficients and the molality vs. concentration. These peculiarities can be ascribed to micelle structural transitions. The thermodynamic functions of micellization were graphically evaluated on the basis of the pseudo-phase transition model. These data have been compared to those for alkyltrimethylammonium bromides and alkylnicotinamide chlorides. It is shown that the introduction of the hydrophilic CONH2 group lowers the hydrophilic character of the pyridinium ring.

Theoretical Studies of the Isotropic-Nematic Interface

Abstract We discuss phenomenological and molecular theoretical approaches to the thermodynamics of isotropicnematic interfaces. Traditional square-gradient theories are inconsistent with the observed anisotropy of the isotropic-nematic surface tension unless the liquid crystal molecules possess up-down symmetry which they manifest as interfacial polar order. Using the Onsager functional for the nonideal free energy of hard spherocylinders, we show that for short spherocylinders, the fourth and higher-order terms in the gradient expansion of the order parameter profiles may also account for the anisotropy of the surface tension. However, these terms have a negligible effect on the isotropic-nematic interfaces of very long spherocylinders to which the Onsager approximation applies asymptotically.

Free energy functional expansion for inhomogeneous polymer blends

Density functional theory for inhomogeneous polymer systems is reformulated using the new ideal system of noninteracting Gaussian chains to replace the Flory–Huggins-like formulation of McMullen and Freed in which the polymer chains have unspecified connectivity. The price paid for introducing this more realistic ideal system is the fact that the density-field relation may only be inverted in powers of the density gradients, so the ideal free energy functional is obtained as a density gradient expansion. The relevant expansion parameter involves the radius of gyration of the polymer, as expected. However, the coefficient of the square gradient term (and those of higher gradients) involves the spatially varying density in the interface as postulated by de Gennes and first derived rigorously here. The nonideal free energy functional is treated by expansions about a homogeneous reference system, and the correlation functions are evaluated in the random phase approximation (RPA). Although truncations are made at second order, there are no difficulties in including higher order terms provided the RPA approximation is retained. The theory is formulated in general for compressible polymer systems, and the incompressible case follows as a special limiting situation. We also analyze the contribution from higher order terms in a traditional Landau-type free energy functional expansion for inhomogeneous polymer systems in which coefficients are evaluated in a homogeneous reference system. Despite the difference of the former coefficients from the de Gennes postulate, it is shown that this Landau expansion may be resummed to produce the identical functional that we obtain by rigorous density functional methods.

A density functional theory of polymer phase transitions and interfaces. II. Block copolymers

We extend our previous density functional theory of homopolymers to block copolymers. The constraints on the relative number densities of the different types of monomers comprising the block copolymers alter the ideal free energy compared to that of homopolymers and of polymer blends. As in our previous work, the second-order functional derivatives of the nonideal free energy with respect to monomer densities are simply related to monomer–monomer direct correlation functions. When applied to incompressible diblock copolymers, this formalism reduces to quasi-one-component form and reproduces the Landau theory of near symmetric diblock copolymers. For homogeneous liquids, we recover the Flory–Huggins ideal free energy of mixing of block copolymers. The present theory, however, permits the treatment of compressible systems and therefore of more strongly first-order microphase separations. It also provides a rigorous formulation for developing improved density functional models for block copolymer systems.

Isotropic-nematic interface of hard spherocylinders: Beyond the square-gradient approximation.

We numerically evaluate the inhomogeneous grand-potential functional for a planar interface separating coexisting isotropic and nematic phases of long, hard spherocylinders. This approach avoids the use of a second-order gradient expansion that could bias the theory's predictions about the orientation of the interface relative to the bulk nematic phase. As in Onsager's pioneering treatment of the bulk phases, we truncate the expansion of the nonideal free energy at second order in the density. For aspect ratios L/D (the length of the spherocylinders divided by their width) greater than about 10, the theory predicts that the nematic director should lie perpendicular to the interfacial normal in accord with a previous, square-gradient calculation. As the aspect ratio decreases, the interface broadens as does the range of the correlations between spherocylinders. For L/D10, we observe a nontrivial tilt of the director compared to the interfacial normal. We compare our findings to recent experimental and theoretical calculations of isotropic-nematic interfacial properties and briefly discuss the effects that the second-order truncation of the free energy may have on our results.

Thermodynamic properties of N-octyl- and N-dodecylnicotinamide chlorides in water

Densities, heat capacities and enthalpies of dilution at 25°C and osmotic coefficients at 37°C were measured for N-octyl- and N-dodecylnicotinamide chlorides in water over an extended concentration region. Partial molar volumes, heat capacities, relative enthalpies and nonideal free energies and entropies at 25°C were derived as a function of the surfactant concentration. For both surfactants, plots of volumes, enthalpies and free energies vs. concentration are regular whereas those of heat capacities and entropies present anomalies at about 0.8 and 0.1m for the octyl and dodecyl compounds, respectively. Changes in the slope of a plot of osmotic coefficients times molality vs. molality were also observed at these same concentrations. These peculiarities are ascribed to micelle structural transitions. The nonideal free energies do not seem to depend on the alkyl chain length when they are plotted vs. m/Ccmc. Also, a plot of the nonideal free energy vs. logm/Ccmc is roughly independent of the nature of the surfactant because of the constant activity of surfactants in micellar solutions. Nonideal free energies, enthalpies and entropies have been calculated at 15 and 35°C. At each concentration the nonideal free energy is temperature independent as a result of a compensatory effect between enthalpy and entropy. The thermodynamic functions of micellization were graphically evaluated on the basis of the pseudo-phase transition model. These data suggest that the nicotinamide group possesses less hydrophilic character than the ammonium group.

Thermodynamic properties and conductivities of some dodecylsurfactants in water

Densities, heat capacities, enthalpies of dilution, osmotic coefficients and conductivities are reported for dodecylamine hydrochloride, dodecyldimethylammonium and dodecyltrimethylammonium chloride in water over a wide range of concentration. The last two properties were also measured for dodecyltrimethylammonium bromide. From the thermodynamic data partial molar volumes, heat capacities and relative enthalpies and nonideal free energies and entropies were derived as a function of the surfactant concentration. The cmc's and degree of counterion dissociation were also calculated from the transport properties. It is shown that the trends of volumes, enthalpies, free energies and entropies are quite regular whereas heat capacities present maxima and minima at concentrations which depend on the nature of surfactants. Corresponding changes were observed in the osmotic coefficients and specific conductivities. The thermodynamic functions of micellization were evaluated on the basis of the pseudo-phase transition model. Finally, the effects of the introduction of methyl groups in the hydrophilic moiety of the surfactant and of the nature of the counterion on the thermodynamic properties of monomers and micelles are examined.

Ion hydration co-sphere interactions in the double-layer and ionic solutions

Two-dimensional concentrations of adsorbed ions in double-layers at charged interfaces, especially when appreciable specific adsorption obtains, are equivalent to quite substantial (1–4 M ) three-dimensional concentrations in regular electrolyte solutions. Under such conditions, ion-specific Gurney co-sphere overlap interactions give an important contribution to the non-ideal free energy of electrolytes in solution. It is proposed that similar interaction effects arise two-dimensionally in double-layers, giving rise to a contribution to the lateral interaction energy in monolayer arrays of ions. Three types of calculations are described by which these interaction effects can be evaluated. One is applied to some recent data on tetrapropylammonium ion adsorption at Hg, where hydrophobic interactions arise. Related problems concerned with solvent dipole orientation in the inner layer, when appreciable surface concentrations of hydrated ions are present, are referred to. The probable role of field-gradient/quadrupole interactions is noted.

Ion hydration co-sphere interactions in the double-layer and ionic solutions

Two-dimensional concentrations of adsorbed ions in double-layers at charged interfaces, especially when appreciable specific adsorption obtains, are equivalent to quite substantial (1–4 M) three-dimensional concentrations in regular electrolyte solutions. Under such conditions, ion-specific Gurney co-sphere overlap interactions give an important contribution to the non-ideal free energy of electrolytes in solution. It is proposed that similar interaction effects arise two-dimensionally in double-layers, giving rise to a contribution to the lateral interaction energy in monolayer arrays of ions. Three types of calculations are described by which these interaction effects can be evaluated. One is applied to some recent data on tetrapropylammonium ion adsorption at Hg, where hydrophobic interactions arise. Related problems concerned with solvent dipole orientation in the inner layer, when appreciable surface concentrations of hydrated ions are present, are referred to. The probable role of field-gradient/quadrupole interactions is noted.