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PubMed2007

Solvent release upon ion association from entropy data. II.

When a cation and an anion associate, the charge on the product is lower than that on the individual ions and solvent is released from their solvation shells to the bulk solvent. This release occurs when the associate is a solvent-shared or contact ion pair or an inner-type complex. The measurable molar entropy change involved is considered to be made up of four contributions: translational, rotational, electrostatic, and desolvation entropies. The former three can be calculated from the properties of the ions and solvents involved; hence, the fourth is obtained by difference. The release of solvent molecules from the crystalline frozen solvent to the liquid on melting is analogous to the solvent release from translational immobilization in the solvation shells of the ions. The molar entropy of melting of the solvent is used to estimate the amount of solvent released in the association process.

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PubMed1996

Nonlinear enthalpy-entropy compensation for the solubility of phenacetin in dioxane-water solvent mixtures.

The solubility of phenacetin was determined at five temperatures in solvent mixtures of aprotic-amphiprotic mixtures of dioxane and water. Enthalpy-entropy compensation analysis is used to study the effect of changing polarity of the medium on the solute. The apparent heats of solution and free energies of solution are nonlinear functions of the cosolvent (dioxane) ratio. The free energy curve goes through a minimum at 80-90% dioxane in water, whereas the apparent heat of solution displays a maximum at low cosolvent ratio (40% dioxane) and a minimum at high cosolvent ratio (90% dioxane). A plot of delta H against delta G shows a nonlinear compensation effect. Two different mechanisms (entropy and enthalpy) are suggested to be the driving forces to increase solubility. These two mechanisms can be related to the nonlinearity of the compensation effect. The slope changes from positive to negative at 40% dioxane. The overall nonlinear function can also be considered as two linear relationships that intersect at 40% dioxane. The results support the usefulness of enthalpy-entropy compensation analysis for a better understanding of the solubility of drugs in aqueous mixtures as related to the role of cosolvents.

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PubMed2000

Aryl phosphate complexation by cationic cyclodextrins. An enthalpic advantage for guanidinium over ammonium and unusual enthalpy-entropy compensation.

The enthalpies and entropies for phosphate complexation by ammonium and guanidinium groups have been compared using cationic cyclodextrins. Aryl phosphate binding by guanidinium hosts is associated with more favorable enthalpies and less favorable entropies, consistent with the idea that guanidinium-phosphate interactions are stronger but more directional than ammonium-phosphate interactions. The slope of an enthalpy-entropy plot suggests that complexation occurs with surprisingly small changes in the order of these systems.

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PubMed2000

Configurational entropy and diffusivity of supercooled water

As a liquid approaches the glass transition, its properties are dominated by local potential minima in its energy landscape. The liquid experiences localized vibrations in the basins of attraction surrounding the minima, and rearranges via relatively infrequent inter-basin jumps. As a result, the liquid dynamics at low temperature are related to the system's exploration of its own configuration space. The 'thermodynamic approach' to the glass transition considers the reduction in configuration space explored as the system cools, and predicts that the configurational entropy (a measure of the number of local potential energy minima sampled by the liquid) is related to the diffusion constant. Here we report a stringent test of the thermodynamic approach for liquid water (a convenient system to study because of an anomalous pressure dependence in the diffusion constant). We calculate the configurational entropy at points spanning a large region of the temperature-density plane, using a model that reproduces the dynamical anomalies of liquid water. We find that the thermodynamic approach can be used to understand the characteristic dynamic anomalies, and that the diffusive dynamics are governed by the configurational entropy. Our results indicate that the thermodynamic approach might be extended to predict the dynamical behaviour of supercooled liquids in general.

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PubMed1996

Enlarged scaling ranges for the KS-entropy and the information dimension.

Numerical estimates of the Kolmogorov-Sinai entropy based on a finite amount of data decay towards zero in the relevant limits. Rewriting differences of block entropies as averages over decay rates, and ignoring all parts of the sample where these rates are uncomputable because of the lack of neighbours, yields improved entropy estimates. In the same way, the scaling range for estimates of the information dimension can be extended considerably. The improvement is demonstrated for experimental data. (c) 1996 American Institute of Physics.

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PubMed1997

Entropy conservation as h(T(&mgr;) ) approximately lambda(&mgr;) (+)d(&mgr;) in neurobiological dynamical systems.

That the topological entropy, h(T(&mgr;) ), of a C(1 M, of a surface, M, upon which invariant measure(s) &mgr; are concentrated, varies as the product of its average leading Lyapunov characteristic exponent, lambda(&mgr;), and the Hausdorff dimension of its support, d(&mgr;),was proven by Pesin [Russ. Math Surveys 32, 55-114 (1977)] for nonuniform partial hyperbolic systems and by Ledreppier and Young [Ergod. Theor. Dyn. Syst. 2, 109-123 (1982)], and Manning [Ergod. Theor. Dyn. Syst. 1, 451-459 (1981)] for uniformly hyperbolic (Axiom A) diffeomorphisms. When considered in conjunction with the post-Shannon information encoding theorems of Adler [Trans. Am. Math. Soc. 114, 309-319 (1965); Mem. Am. Math. Soc., No. 219 (1979)] and others, this suggests a way to differentiate equal entropy behaviors in systems with varying patterns of dynamical behaviors. Here we show this relation to be useful in the quantitative discrimination among the behaviors of abstract neuronal models and two real, finite time, partially and nonuniformly hyperbolic, brain-related dynamical systems. We observe a trade-off in finite time between two competing dynamical processes, jittery sticking (tending to increase d(&mgr;)) and convective escaping (more prominently incrementing lambda(&mgr;) (+)). In finite time systems, these changes in combination can statistically conserve the dynamical entropy, h(T(&mgr;) ), while altering the Levy characteristic exponent, alpha (describing the tail of the density distribution of observables, rho(x) approximately exp-gammamid R:xmid R:(alpha),1</=alpha</=2), and the Mandelbrot-Hurst exponent 0 0.5 implicates sequential correlations and H(*)<0.5 sequential anticorrelation. When the relation h(T(&mgr;) )=lambda(&mgr;) (+)d&mgr; fails, the way it does so provides information about the system. (c) 1997 American Institute of Physics.

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PubMed1997

Generalized entropies of chaotic maps and flows: A unified approach.

A thermodynamic study of nonlinear dynamical systems, based on the orbits' return times to the elements of a generating partition, is proposed. Its grand canonical nature makes it suitable for application to both maps and flows, including autonomous ones. When specialized to the evaluation of the generalized entropies K(q), this technique reproduces a well-known formula for the metric entropy K(1) and clarifies the relationship between a flow and the associated Poincare maps, beyond the straightforward case of periodically forced nonautonomous systems. Numerical estimates of the topological and metric entropy are presented for the Lorenz and Rossler systems. The analysis has been carried out exclusively by embedding scalar time series, ignoring any further knowledge about the systems, in order to illustrate its usefulness for experimental signals as well. Approximations to the generating partitions have been constructed by locating the unstable periodic orbits of the systems up to order 9. The results agree with independent estimates obtained from suitable averages of the local expansion rates along the unstable manifolds. (c) 1997 American Institute of Physics.

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PubMed1998

Entropy evolution for the Baker map.

Gibbs entropy is invariant for the Baker map. A Jordan basis spectral decomposition of the Baker Frobenius-Perron operator suggests that any initial density evolves to the stationary density that has maximal entropy. This entropy conundrum is resolved by considering the difference between weak and strong convergence. A binary representation is used to make these points transparent. (c) 1998 American Institute of Physics.

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PubMed2000

Entropy computing via integration over fractal measures.

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems, one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures. (c) 2000 American Institute of Physics.

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PubMed2004

Excess entropy scaling for the diffusion coefficient in expanded liquid metals.

Molecular-dynamics simulation is used to compute the pair correlation function and the velocity autocorrelation function of Cs and Rb along the liquid-vapor coexistence curve, from which the excess entropy S(ex) and the diffusion coefficient D are deduced. The numerical results of both physical properties are correlated and a scaling law between the excess entropy and the reduced diffusion coefficient D(*)(=D/D(0)) is investigated for different expressions of the reduction parameter D(0). The choice of thermodynamic states along the liquid--vapor coexistence curve gives us the possibility to extend the investigation of the relation between the reduced diffusion coefficient and the excess entropy over a wide area and to test the adequacy of the scaling law confidently.

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PubMed2004

Finite size scaling for the atomic Shannon-information entropy.

We have developed the finite size scaling method to treat the criticality of Shannon-information entropy for any given quantum Hamiltonian. This approach gives very accurate results for the critical parameters by using a systematic expansion in a finite basis set. To illustrate this approach we present a study to estimate the critical exponents of the Shannon-information entropy S approximately (lambda-lambda(c))(alpha(S) ), the electronic energy E approximately (lambda-lambda(c))(alpha(E) ), and the correlation length xi approximately mid R:lambda-lambda(c)mid R:(-nu) for atoms with the variable lambda=1/Z, which is the inverse of the nuclear charge Z. This was realized by approximating the multielectron atomic Hamiltonian with a one-electron model Hamiltonian. This model is very accurate for describing the electronic structure of the atoms near their critical points. For several atoms in their ground electronic states, we have found that the critical exponents (alpha(E),nu,alpha(S)) for He (Z=2), C (Z=6), N (Z=7), F (Z=9), and Ne (Z=10), respectively, are (1, 0, 0). At the critical points lambda(c)=1/Z(c), the bound state energies become absorbed or degenerate with continuum states and the entropies reach their maximum values, indicating a maximal delocalization of the electronic wave function.

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PubMed2004

Calculation of the aqueous solvation energy and entropy, as well as free energy, of simple polar solutes.

With the advent of more powerful computers, the question of calculating thermodynamic quantities, such as the energy and the entropy, in solute-solvent systems is revisited. The calculation of these thermodynamic quantitites was limited in the past by their slow convergence relative to the free energy. Using molecular dynamics simulations, the energy, entropy, and free energy of solvation of NMA and CH(3)NH(2), as well as their relative values, have been determined. Three different methods (the thermodynamic perturbation method, the thermodynamic integration method, and a finite-difference method) are compared. The finite difference method gives the best results and accurate values for the entropy and energy were obtained using a reasonable amount to computer time. The results suggest that a meaningful thermodynamic description of biomolecular processes can be realized with present methods and the available computer time.

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PubMed2004

Combinatorial entropy and phase diagram of partially ordered ice phases.

A close analytical estimate for the combinatorial entropy of partially ordered ice phases is presented. The expression obtained is very general, as it can be used for any ice phase obeying the Bernal-Fowler rules. The only input required is a number of crystallographic parameters, and the experimentally observed proton site occupancies. For fully disordered phases such as hexagonal ice, it recovers the result deduced by Pauling, while for fully ordered ice it is found to vanish. Although the space groups determined for ice I, VI, and VII require random proton site occupancies, it is found that such random allocation of protons does not necessarily imply random orientational disorder. The theoretical estimate for the combinatorial entropy is employed together with free energy calculations in order to obtain the phase diagram of ice from 0 to 10 GPa. Overall qualitative agreement with experiment is found for the TIP4P model of water. An accurate estimate of the combinatorial entropy is found to play an important role in determining the stability of partially ordered ice phases, such as ice III and ice V.

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PubMed2005

Modified Kelvin-Thomson equation considering ion-dipole interaction: comparison with observed ion-clustering enthalpies and entropies.

The classical Kelvin-Thomson (CKT) equation does not consider the interaction of condensing molecules with the ions and hence is not able to treat polar and nonpolar molecules differently. The ion-clustering enthalpy and entropy changes predicted by CKT equation for small ions are known to be significantly less negative than those observed. In this paper, we derive a modified Kelvin-Thomson (MKT) equation, which considers the effect of dipole-ion interaction, by taking into account the kinetic energy change of condensing polar ligands as they approach the ions or the extra energy needed for dipole molecules to escape from the ion cluster. The clustering enthalpies and entropies for protonated clusters (H(+)L(n), with L=H(2)O, NH(3), CH(3)OH, and C(5)H(5)N) are calculated based on MKT equation and compared with experimental data. Our calculations indicate that enthalpy values given by MKT equation are in very good agreement with experimental results for small ions (n< or =5) of all four species investigated. MKT predictions appear to be consistent with observed enthalpies for CH(3)OH at n> or =6 and for H(2)O at n=14-25, however, MKT equation cannot reproduce the observed discontinuous transition in enthalpy changes at n=6 for NH(3) and at n=7-13 for H(2)O which is probably associated with the formation of inner shell. The stepwise entropy changes for small ions are 5-15 cal mol(-1) K(-1) more negative when the effect of dipole-ion interaction is considered, which suggests that the ordered structure of the cluster ions can somewhat be accounted for by the dipole-ion interaction term.

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PubMed2005

Characteristic features of Shannon information entropy of confined atoms.

The Shannon information entropy of 1-normalized electron density in position and momentum space Sr and Sp, and the sum ST, respectively, are reported for the ground-state H, He+, Li2+, H-, He, Li+, Li, and B atoms confined inside an impenetrable spherical boundary defined by radius R. We find new characteristic features in ST denoted by well-defined minimum and maximum as a function of confinement. The results are analyzed in the background of the irreducible lower bound stipulated by the entropy uncertainty principle [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)]. The spherical confinement model leads to the ST values which satisfy the lower bound up to the limits of extreme confinements with the interesting new result displaying regions over which a set of upper and lower bounds to the information entropy sum can be locally prescribed. Similar calculations on the H atom in 2s excited states are presented and their novel characteristics are discussed.

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PubMed2006

Entropy theory of polymer glass formation revisited. I. General formulation.

A generalized entropy theory of glass formation is developed by merging the lattice cluster theory for the thermodynamics of semiflexible polymer melts at constant pressure with the Adam-Gibbs relation between the structural relaxation time and the configurational entropy. Since experimental studies have suggested that the relative rigidity of the chain backbone and the side groups is an essential parameter governing the nature of glass formation in polymers, we incorporate this rigidity disparity parameter, along with monomer structure, into our new theoretical description of the polymer fluid thermodynamics. Our entropy theory is compared with alternative theories that describe the rate of structural relaxation in glass-forming liquids in terms of an activated rate process.

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PubMed2006

Influence of the reaction mechanism on the time course of the entropy production during reversible polymerization.

The paper shows the influence of the reaction mechanism on the time course of the entropy production sigma in a closed system during reversible polymerization. We consider two different reaction mechanisms with (a) being a polycondensation and (b) a chain growth mechanism. For both mechanisms explicit expressions for the entropy production sigma as a function of time t are derived. To demonstrate the application of these general expressions we consider two different polymerization experiments where the reaction starts (i) from monomer molecules polymerizing to a defined number average chain length x(n,eq) and (ii) from monodisperse polymer molecules reacting with each other under the constraint that x(n) is the same at the beginning and the end of the reaction. In both cases we treat the system to be ideal and describe the kinetics of the reversible polymerization reactions using two kinetic constants for the forward and backward reactions, respectively. Under these assumptions the difference in the curvature of the entropy production sigma between a polycondensation and a chain growth mechanism is only marginal if the reaction starts from monomer molecules polymerizing to a defined number average chain length x(n,eq).

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PubMed2006

Free energy, entropy and volume of activation for electron transfer reactions in a polar solvent.

A continuum theory with account of cavity size fluctuations is employed to study free energy, volume and entropy of activation for nonadiabatic electron transfer (ET) reactions in polar solvents. By using a two-sphere cavity description, model calculations are performed for charge separation and recombination processes in acetonitrile under ambient conditions. It is found that the cavity size at the transition state varies with the free energy of reaction as well as with the thermodynamic conditions. In contrast to the Marcus theory predictions, the volume and entropy of activation show a monotonic behavior with the free energy of reaction and a strong correlation with each other. For example, for a given ET process, the volume and entropy of activation have the same sign. Their values for the charge separation and recombination processes are opposite in sign. These findings are in good qualitative agreement with measurements.

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