Critical concentration thermodynamic

Below a critical concentration, c, in a thermodynamically good solvent, r 0 can be standardised against the overlap parameter c [r)]. However, for c>c, and in the case of a 0-solvent for parameter c-[r ]>0.7, r 0 is a function of the Bueche parameter, cMw The critical concentration c is found to be Mw and solvent independent, as predicted by Graessley. In the case of semi-dilute polymer solutions the relaxation time and slope in the linear region of the flow are found to be strongly influenced by the nature of polymer-solvent interactions. Taking this into account, it is possible to predict the shear viscosity and the critical shear rate at which shear-induced degradation occurs as a function of Mw c and the solvent power. 

Critical currents, 23 821-823 in superconducting, 23 819-825 Critical event (CE), 15 462 Critical failure, 26 982 Critical features, in separating nonideal liquid mixtures, 22 307 Critical fields, thermodynamic, 23 809-811 Critical flocculation concentration, 11 631 Critical item evaluation, for reliability, 26 991... 

The critical concentration at which the first micelle forms is called the critical micelle concentration, or CMC. As the concentration of block copolymer chains increases in the solution, more micelles are formed while the concentration of nonassociated chains, called unimers, remains constant and is equal to the value of the CMC. This ideal situation corresponds to a system at thermodynamic equilibrium. However, experimental investigations on the CMC have revealed that its value depends on the method used for its determination. Therefore, it seems more reasonable to define phenomenologically the CMC as the concentration at which a sufficient number of micelles is formed to be detected by a given method [16]. In practical terms, the CMC is often determined from plots of the surface tension as a function of the logarithm of the concentration. The CMC is then defined as the concentration at which the surface tension stops decreasing and reaches a plateau value. 

Figure 1. Nucleation and growth of actin filaments. Nucleation is shown here as a thermodynamically unfavored process, which in the presence of sufficient actin-ATP will undergo initial elongation to form small filament structures that subsequently elongate with rate constants that do not depend on filament length. Elongation proceeds until the monomeric actin (or G-actin) concentration equals the critical concentration for actin assembly.

The threshold concentration of monomer that must be exceeded for any observable polymer formation in a self-assembling system. In the context of Oosawa s condensation-equilibrium model for protein polymerization, the cooperativity of nucleation and the intrinsic thermodynamic instability of nuclei contribute to the sudden onset of polymer formation as the monomer concentration reaches and exceeds the critical concentration. Condensation-equilibrium processes that exhibit critical concentration behavior in vitro include F-actin formation from G-actin, microtubule self-assembly from tubulin, and fibril formation from amyloid P protein. Critical concentration behavior will also occur in indefinite isodesmic polymerization reactions that involve a stable template. One example is the elongation of microtubules from centrosomes, basal bodies, or axonemes. 

The effectiveness of a polymeric flow enhancer is influenced decisively by the state of solution and the solvation characteristics. In the case of polyelectrolytes, in particular, the chemical nature plays a significant role, e.g., it was found for poly(acrylamide)-coacrylate that a significant increase in effectiveness arises with the increasing number of ionic groups. It is therefore necessary to consider, for example, such factors as the question of critical concentration, polymer-polymer and polymer-solvent interactions, the thermodynamic quality of the solvent, the proportion of ionic molecular groups and their behavior in the presence of lower-molecular-weight charge carriers. 

Since they act as surfactants, copolymers are added in only small amounts, typically from a thousandth parts to a few hundredth parts. Theoretically, Leibler [30] showed that only 2% of a diblock copolymer may thermodynamically stabilize an 80%/20% incompatible blend with an optimum morphology (submicronic droplets). However, in practice kinetic control and micelle formation interfere in this best-case scenario. To a some extent, compatibilization increases with copolymer concentration [8,31,32], Beyond a critical concentration (critical micellar concentration cmc) little or no improvement is observed (moreover, for high amounts, the copolymer can act as a plasticizer). Copolymer molecular weight influence is similar to that of the concentration effect. For example, in a PS/PDMS system [8,31,32], when the copolymer molecular weight increases, domain size decreases to a certain extent. Hu et al. [31] correlated their experimental results with theoretical prediction of the Leibler s brush theory [30]. Leibler distinguishes two regimes to characterize the behaviour of the copolymer at the interface... 

Micellar aggregates are considered in chapter 3 and a critical concentration is defined on the basis of a change in the shape of the size distribution of aggregates. This is followed by the examination, via a second order perturbation theory, of the phase behavior of a sterically stabilized non-aqueous colloidal dispersion containing free polymer molecules. This chapter is also concerned with the thermodynamic stability of microemulsions, which is treated via a new thermodynamic formalism. In addition, a molecular thermodynamics approach is suggested, which can predict the structural and compositional characteristics of microemulsions. Thermodynamic approaches similar to that used for microemulsions are applied to the phase transition in monolayers of insoluble surfactants and to lamellar liquid crystals. 

Summarizing, it can be seen that the reaction rate of a given number of atoms of a micro component decreases drastically with the spatial expansion of the system in which they are contained (i.e. the number of atoms per volume unit). Therefore, at critical concentrations, micro-microcomponent kinetics prevails largely over thermodynamics in a reaction. 

The critical concentration Cc for formation of foam and emulsion bilayers of Do(EO)22 are 4-10 6 mol dm 3 and 1.6 10 5 mol dm 3, respectively, and are in good correlation with the lowest concentrations, 2-31 O 6 mol dm 3 and 10 5 mol dm 3 [421] at which maximum filling of the surfactant adsorption monolayer is attained. It should also be noted that in the case of the emulsion bilayers, CMC < Ce which implies that it is not possible to obtain infinitely stable (i.e. with r = °°) bilayers of Do(EO)22 between two droplets of nonane under the described conditions. For this reason, it may be thought that thermodynamically stable nonane-in-water emulsions stabilised with Do(EO)22 do not exist. 

The determination of the binding energy of DMPC molecule in the foam bilayer was carried out using the experimental results for the temperature dependence of the critical concentration for formation of foam bilayer (Fig. 3.95) and the theory of Kashchiev-Exerowa (see Section 3.4.4.2). The concentrations Cc and Ce (Eq. (3.129)) are specific constants of each system which determine the ability of a foam bilayer to exist in a metastable state within the concentration range Cc< C < Ce. When C >Ce the foam bilayer is thermodynamically stable (there is no driving force for the whole nucleation process in the foam bilayer). It follows from the theory that the critical concentration of amphiphile molecules in the solution equals the equilibrium one (Cc = Ce) in the case of a missing metastable region when only thermodynamically stable foam bilayers are formed. As mentioned above, the DMPC foam... 

Above the critical concentration v the solution becomes metastable and separates into two phases — isotropic and anisotropic. The condition of thermodynamic equilibrium of the two phases corresponds to the equality of the chemical potentials of each of the components in each of the coexisting phases. The concentration corresponding to a complete transition to the anisotropic state, v, is 1.56 times as high as vf (see also... 

If this interpretation of the thermodynamics is correct, the solvent activities should be invariant at all concentrations within the critical region. The exact location of this region is given by zero equivalence of the first and second derivatives of the chemical potential (log ai) with respect to concentration. We have chosen to define this region, arbitrarily, as the region where solvent activity exceeds unity since the tedious calculation necessary to establish it exactly seemed unwarranted by the approximate nature of the model. The maximum resin concentration at which the activity initially exceeds unity is defined as the critical concentration. For the three systems under study, these concentrations are listed in Table IV. It follows that y in Equation 1 is given by 1/cnu where is the critical solvent volume fraction. 

Important features of the selective oxidation process are shown schematically in Figure 1. The slow growth rates of alumina and silica, illustrated in the plot of parabolic rate constants versus temperature at lower right, makes the formation of one of these oxides as a continuous surface layer necessary for long term oxidation protection. This requires that the protective oxide be more stable thermodynamically than the more rapidly growing oxides. The plot of standard free energy of formation as a function of temperature at lower left shows that the Ni-Al system satisfies this condition. Alumina is stable, relative to NiO, even when the activity of aluminum in the alloy is very low. However, when the Al concentration is low the alumina forms as internal oxide precipitates and is non-protective allowing an external layer of NiO to form (illustrated in the cartoon at top). Therefore, a critical concentration of Al exists above which out-... 

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