Free energy potentials

Fig. 7. The effective free-energy potentials for retraction of the free end of arms in a mon-odisperse star polymer melt. The upper curve assumes no constraint-release, the lower two curves take the dynamic dilution approximation with the assumptions (Ball-...

Fig.13 Schematic view of a typical setup of single molecule pulling experiments and an illustration of the generic free energy potential G(z) [87]. The x-position denominates the cantilever and the z-position denominates the cantilever tip to which one end of DNA molecule is attached. Reprinted with permission...

T he extension of the Smoluchowski (I) theory for rates of aggrega-tion of colloidal particles by Fuchs (2) requires a detailed knowledge of the free energy potential curves (see review in Verwey and Overbeek (3)). In the case of charged particles, these potentials have been taken... 

For measurements between crossed mica cylinders coated with phospholipid bilayers in water, see J. Marra andj. Israelachvili, "Direct measurements of forces between phosphatidylcholine and phosphatidylethanolamine bilayers in aqueous electrolyte solutions," Biochemistry, 24, 4608-18 (1985). Interpretation in terms of expressions for layered structures and the connection to direct measurements between bilayers in water is given in V. A. Parsegian, "Reconciliation of van der Waals force measurements between phosphatidylcholine bilayers in water and between bilayer-coated mica surfaces," Langmuir, 9, 3625-8 (1993). The bilayer-bilayer interactions are reported in E. A. Evans and M. Metcalfe, "Free energy potential for aggregation of giant, neutral lipid bilayer vesicles by van der Waals attraction," Biophys. J., 46, 423-6 (1984). 

The constant KV) which is called the ionic product of water, may be computed from equations (V-44) and (V-46), if partial molal free energies (potentials) of formation of all the reaction components in the corresponding standard states are known. For such standard states we select both the state of a hypothetical ideal solution with molal concentration of hydrogen and hydroxyl ions equalling unity and the state of hypothetical, absolutely undissociatcd pure water. Since in actual diluted solutions the activity of undissociated water hardly differs from the activity in its standard state, aji2 in the equation (V-51) may be considered as equalling unity so that then Km = K . The following expression is valid for a temperature of 25° C ... 

Figure 18 Free-energy potentials corresponding to the VE theory of dephasing. The equilibrium cavity radius is larger in v = 1 than in v = 0 by and amount <5r (see Fig. 17). As a result, the energy of the vibrational transition depends on the cavity radius. At thermal equilibrium, shear fluctuations in the liquid will cause fluctuations in the radius, which in turn cause fluctuations in the transition frequency a). (Adapted from Ref. 8.)...

Consider the energy balance of a nucleating (or condensing) drop. As the droplet (or embryo) is formed, its surface free energy goes from 0 to nrd2y, where d is the diameter of the drop and y is the liquid surface tension. If the free-energy potential per molecule is < >a in the vapor... 

Hi = fi° + RT In Xi + RT In 7, chemical standard free energy potential potential potential of mixing free energy... 

To see this, we can try to minimize the free energy and, in this section, any energy, free energy, potential, or chemical potential will be expressed in thermal units. Consequently, the free energy reads... 

Measurement of Free-Energy Potentials and Theoretical Concepts... 

Recent experimental advances have made quantitation of weak membrane adhesion possible in concentrated solutions of macromolecules. We report direct measurements of the free energy potential for adhesion of phospholipid Dilayers in solutions of two plasma proteins (fibrinogen and albumin) over a wide range of volume fraction (Q-0.1). Tne results are consistent with a thermodynamic model for adhesion based on depletion of macromolecules from the contact zone. 

Mechanical equilibrium is established when small reductions in free energy due to formation of adhesive contact just balance small increases in mechanical work of deformation of the vesicle (8,11). Tnis variational statement leads to a direct relation between the free energy potential for adhesion and the suction pressure applied to tne adherent vesicle,... 

Free Energy Potentials for Lipid Bilayer Adnesion... 

Clearly, our results for adhesion of lipid bilayers in fibrinogen and albumin solutions are consistent with the (non-adsorption) depletion type of assembly process. This deduction is based on (i) the null observation that no fiuorescently labelled material was detected in the gap between bilayers, (ii) the continuous increase of the free energy potential with concentration even for fairly large volume fractions, and (iii) the transfer of adherent vesicle pairs with subsequent separation which showed that adhesion energy depended only on the composition of the medium exterior to the gap but not the gap composition. Similar results have been obtained for adhesion of lipid oilayers in solutions of high molecular weight dextran polymers (Figure 4, J ). Hence, we have chosen to carefully examine (non-adsorption) depletion-based theories in conjunction with these experiments. 

The coefficient in the last variation is the free energy potential for creation of contact area at constant gap thickness, composition, and concentration profile. 

We have used this, mean-field approach to successfully predict the free energy potential for bilayer-bilayer adhesion in dextran polymer solutions (the theoretical results are snown in Fig. 4 along with the data, The correlation is excellent, based... 

For each proton transported out of the matrix across the inner membrane and into the inner membrane space of a mitochondrion, how much free energy potential is generated across the inner membrane ... 

Fredrickson and Binder [9] further improved this theory to describe the kinetics of the ordering process. Their concentration dependent free energy potential shows two side minima, which have the same depth as the middle one at the microphase separation transition temperature Tmst- Therefore, they presume a coexistence of the disordered and the lamellar phase at Tmot- As the temperature is further lowered, these side minima become dominant and the transition comes to completion. For a supercooled material, they expect after a completion time an Avrami-type ordering transformation with an exponent of 4 equivalent to spherically growing droplets of ordered material. This characteristic time corresponds to the time to form stable droplets of ordered material plus the time needed for the structures to grow to a size that they can be detected by the used technique. 

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