Thermodynamics of relaxation methods

See article on "Thermodynamics of Relaxation Methods" in this volume. 

The formalism of relaxation methods demands stationary var iation of an external thermodynamic parameter at frequencies near the relaxation frequency of the equilibrium or a fast jump of this parameter (fast compared with the relaxation rate). In the case of pressure as external parameter all published techniques apply pressure changes at a rate at which the system is no longer isothermal, butisentropic Instead of eqn. (1) we have to write... 

We assume that the unbinding reaction takes place on a time scale long ( ompared to the relaxation times of all other degrees of freedom of the system, so that the friction coefficient can be considered independent of time. This condition is difficult to satisfy on the time scales achievable in MD simulations. It is, however, the most favorable case for the reconstruction of energy landscapes without the assumption of thermodynamic reversibility, which is central in the majority of established methods for calculating free energies from simulations (McCammon and Harvey, 1987 Elber, 1996) (for applications and discussion of free energy calculation methods see also the chapters by Helms and McCammon, Hermans et al., and Mark et al. in this volume). 

A large programme utilizing temperature-jump relaxation methods for the study of tautomerism in aqueous solution has led the Dubois group to determine the kinetic and thermodynamic parameters of the equilibrium (130a) (130b) (78T2259). The tautomeric... 

T, 0.1 MPa) condition to be calculated, t is then known for any P and T corresponding to an ambient pressure value of TV. This method is quite powerful because it allows determination of relaxation times for any thermodynamic condition T, P, or V) provided only that the same value of T had been measured at ambient pressure. 

The use of computer simulations to study internal motions and thermodynamic properties is receiving increased attention. One important use of the method is to provide a more fundamental understanding of the molecular information contained in various kinds of experiments on these complex systems. In the first part of this paper we review recent work in our laboratory concerned with the use of computer simulations for the interpretation of experimental probes of molecular structure and dynamics of proteins and nucleic acids. The interplay between computer simulations and three experimental techniques is emphasized (1) nuclear magnetic resonance relaxation spectroscopy, (2) refinement of macro-molecular x-ray structures, and (3) vibrational spectroscopy. The treatment of solvent effects in biopolymer simulations is a difficult problem. It is not possible to study systematically the effect of solvent conditions, e.g. added salt concentration, on biopolymer properties by means of simulations alone. In the last part of the paper we review a more analytical approach we have developed to study polyelectrolyte properties of solvated biopolymers. The results are compared with computer simulations. 

Consideration of the thermodynamics of a representative reaction coordinate reveals a number of interesting aspects of the equilibrium (Fig. 5). Because the complex is in spin equilibrium, AG° x 0. Only complexes which fulfill this condition can be studied by the Raman laser temperature-jump or ultrasonic relaxation methods, because these methods require perturbation of an equilibrium with appreciable concentrations of both species present. The photoperturbation technique does not suffer from this limitation and can be used to examine complexes with a larger driving force, i.e., AG° 0. In such cases, however, AG° is difficult to measure and will generally be unknown. 

For a measure of amount of water relevant to stability concerns, vapor pressure, or its related thermodynamic parameters, is more relevant. Determination of vapor pressure uses methods developed from thermodynamic roots, though if the product is not at true equilibrium, the measured quantity is not a thermodynamic descriptor of the product, although it is still a measure of a product characteristic. Water mobilities are often inferred from spectroscopic measurements of relaxational phenomena. Many workers attempt to identify different classes of water characteristic of different ranges of water content and water partial vapor pressure. Spectroscopic measurements, too, are often interpreted in terms of populations of water molecules with similar characteristics. 

Heat capacity measurements give information complementary to magnetic susceptibility and magnetization. This complementarity arises naturally from thermodynamics. This technique has also become much more available as instrumentation based on the relaxation method has been marketed widely in the past few years. The heat capacity diverges at Tc and thus provides a very precise measure of Tc, as seen in Figure 12. The shape of the anomaly resembles the Greek... 

Table 14. Thermodynamic parameters for the complex formation of macrotetrolide compounds with sodium and potassium determined by microcalorimetry and relaxation methods at 298 °K (Table from Ref.315,))... 

In all relaxation methods a system at equilibrium is perturbed by changing one of the thermodynamic variables which govern the equilibrium. Provided the perturbation is rapid and leads to a significant change in the concentrations of the reactants, the system may be observed to relax to a new equilibrium position with a rate determined by the fundamental rate constants of the steps and by the equilibrium concentrations of the components. 

The present chapter gives also detailed introduction to a large number of experimental methods, suitable for studying dynamic interfacial tensions. The methods are discussed in terms of the available time window. There are methods which complement each other such that a time interval from less than 100 microseconds up to hours and days of adsorption time can be covered (about ten orders of magnitude). The relaxation methods, also suitable for detecting the adsorption mechanism of surfactant s adsorption provide in addition the dilational rheology of interfacial layers. It is discussed that in particular these dilational rheological studies are most informative in respect to adsorption mechanisms, as the data interpretation includes the thermodynamic model as well as the adsorption dynamics. 

The information on the structure of electrolyte solutions provided by thermwlynamic and transport properties on the one hand and by spectroscopic, relaxation and kinetic investigations on the other, complement one another with regard to the chemical model. Thermodynamic and transport properties provide the distance parameter R, the overall association constant Ka, and the activity coeffident y linked to it. No direct information can be achieved on the structure of the region a g r R and possible regions a Rj Rj. .. R. This problem, however, can be solved by modem spectroscopic and relaxation methods. 

The amplitudes of chemical relaxation processes are determined by the equilibrium concentrations (and strictly speaking, associated activity coefficients) and by thermodynamic variables appropriate for the particular perturbation method used. Thus, for example, an analysis of the amplitudes of relaxation processes associated with temperature jump measurements can lead to determination of the equilibrium constants and enthalpies associated with the mechanism under study. As might be anticipated from our previous discussion, the relaxation amplitudes are determined by normal mode thermodynamic variables which are linear combinations of the thermodynamic variables associated with the individual steps in the mechanism. The formal analysis of relaxation amplitudes has been developed in considerable detail [2, 5,7],... 

---