Free energy contributions

Finally, Newmann and co-workers [30] (see also Ref. 31) have argued that while free energy contributions may not be strictly additive as in Eq. IV-11, there should, in principle, be an equation of state relating the work of adhesion to the separate liquid surface tensions such as... 

If 7 = 0, AT = 0, regardless of particle size. This is not expected, however, since chains emerging from a crystal face either make a highly constrained about-face and reenter the crystal or meander off into the liquid from a highly constrained attachment to the solid. In either case, a free-energy contribution is inescapable. 

Combining Eqs. (12)-(14) yields Eq. (7) directly. Although such a free energy decomposition is path-dependent [8], it provides a useful and rigorous framework for understanding the nature of solvation and for constructing suitable approximations to the nonpolar and electrostatic free energy contributions. 

In the following sections, we describe an implicit solvent model based on this free energy decomposition that is widely used in biophysics. It consists in representing the nonpolar free energy contributions on the basis of the solvent-accessible surface area... 

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

Modem understanding of the hydrophobic effect attributes it primarily to a decrease in the number of hydrogen bonds that can be achieved by the water molecules when they are near a nonpolar surface. This view is confirmed by computer simulations of nonpolar solutes in water [15]. To a first approximation, the magnimde of the free energy associated with the nonpolar contribution can thus be considered to be proportional to the number of solvent molecules in the first solvation shell. This idea leads to a convenient and attractive approximation that is used extensively in biophysical applications [9,16-18]. It consists in assuming that the nonpolar free energy contribution is directly related to the SASA [9],... 

The electrostatic free energy contribution in Eq. (14) may be expressed as a thennody-namic integration corresponding to a reversible process between two states of the system no solute-solvent electrostatic interactions (X = 0) and full electrostatic solute-solvent interactions (X = 1). The electrostatic free energy has a particularly simple form if the thermodynamic parameter X corresponds to a scaling of the solute charges, i.e., (X,... 

In Section III we described an approximation to the nonpolar free energy contribution based on the concept of the solvent-accessible surface area (SASA) [see Eq. (15)]. In the SASA/PB implicit solvent model, the nonpolar free energy contribution is complemented by a macroscopic continuum electrostatic calculation based on the PB equation, thus yielding an approximation to the total free energy, AVP = A different implicit... 

Excited state electron transfer also needs electronic interaction between the two partners and obeys the same rules as electron transfer between ground state molecules (Marcus equation and related quantum mechanical elaborations [ 14]), taking into account that the excited state energy can be used, to a first approximation, as an extra free energy contribution for the occurrence of both oxidation and reduction processes [8]. 

If it is assumed that the total free energy for the transfer of solute X from the gas phase to the stationary phase (with molecular interactions characteristic of Infinite dilution) is the linear sum of the individual free energy contributions to the transfer process then a general expression for the solution process, equation (2.11), can be written as follows... 

Now we assume that the inter and the intrapart are additive in a way that the inter chain interaction is given by the pairwise overlap of the ellipsoids of the different chain. Since each mass tensor corresponds to a density distribution we can describe the interchain free energy contribution of the pair ij ... 

FEP calculations for paths A, B and C were performed with a 40 ps equilibration run prior to the sampling for all points along the path. The free energy contributions were sampled for 20 ps for each point on the MEP. In all cases a time step of 2.0 fs was employed, maintaining a constant temperature of 300 K. The SHAKE [47] algorithm was used to constrain all bonds involving hydrogen atoms. 

We now generalize the model of Fig. 6 to account for the wave-vector dependence. There are three contributions to the free energy, F (a) density difference rfi between the baby nuclei and the amorphous background giving a free-energy contribution that is proportional to - ATx(r2(AT = - T) ... 

The interaction-energy contribution to the quantum correction is larger in magnitude than the free energy contribution, suggesting the entropic part is positive. 

SPT provides a conceptual basis relating the nonpolar free energy contribution to the solvent-exposed surface area. An attractive approximation is to ignore curvature effects and write... 

In fact at least this amount of strain energy is needed for the formation of the circular crack with radius q. The total free energy contribution due to the presence of the crack is... 

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