Eisenbergs Theory

The elastic work in cluster formation is related to the energy changes in two different kinds of chain deformations. The first kind is the work of stretching a polymer chain from a distance Xq to X, the distance corresponding to a cluster size of N, The other kind is the work of contracting a chain from a distance of Xq to 0. Accordingly, the elastic work per chain in cluster formation is given by 

The electrostatic energy per ion pair released upon cluster collapse is a function of the geometry of the cluster and the dielectric constant s of the medium. It is given by 

The cluster is not infinitely stable. At some temperature the cluster decomposes, and at that temperature the elastic forces and the electrostatic forces just balance each other. Since the elastic force was calculated per chain and the electrostatic force per ion pair, Wch can be set equal to W at thus 

Tc can be determined experimentally and A calculated for any particular cluster geometry, N and X can be calculated from Eq. (4). 

The Eisenberg theory indicates that at low ion concentration, as the distance between multiplets increases, the interaction between multiplets is expected to decrease to the point where the elastic forces become too large to be overcome below that concentration clustering would not be expected. This is consistent with the experimental findings. This theory has been applied well to both ethylene and styrene ionomers which contain carboxylate groups. Hash-imoto et have recently shown that the Eisenberg theory 

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It should be mentioned that the Eisenberg theory is somewhat approximate and should be regarded only as demonstrating feasibility rather than predicting true cluster geometries. The theory has been consulted extensively in many recent models proposed for Nafion membranes (see Section II.4). 

Ion clusters are commonly observed in the ionized forms of the perfluorinated membranes. The size of the clusters appears to be larger for sulfonate than for carboxylate membranes." " The size increases in the order Na, and Cs" and decreases with increasing number of functional groups per chain and with increasing temperature.As in the case of ethylene ionomers, the perfluorinated carboxylic acid membranes do not form ion clusters, at least in the dry state." The electrostatic interaction may be too weak to form ionic clusters. These observations are expected according to the Eisenberg theory (see Section II.2). 

These results provide qualitative support for the validity of the ring-chain structural model for vitreous selenium implied by the Tobolsky-Eisenberg theory (1,2) of equilibrium polymerization. The establishment... 

Lewis Carroll proposed a solution that did not satisfy some of modern scientists. There exists a lot of attempts to improve the problem statement (Eisenberg and Sullivan, 1996 Falk and Samuel-Cahn, 2001 Guy, 1993 Portnoy, 1994) reduction from infinite plane to a bounded set, to a compact symmetric space, etc. But the elimination of paradox destroys the essence of Carroll s problem. If we follow the paradox and try to give a meaning to "points are taken at random on an infinite plane" then we replace cr-additivity of the probability measure by finite-additivity and come to the applied probability theory for finite-additive probabilities. Of course, this theory for abstract probability spaces would be too poor, and some additional geometric and algebraic structures are necessary to build rich enough theory. 

Gee s theory was unified by Toboesky and Eisenberg (31) and further improved by Tobolsky et al. (52) who removed the original restriction demanding an multiple numbers of 8 S atoms in each chain. This theory demands the existence of a transition (floor) temperature and excellently accounts for its sharpness as determined from Gee s results. A similar situation is observed in liquid selenium (32). 

Among other approaches, a theory for intermolecular interactions in dilute block copolymer solutions was presented by Kimura and Kurata (1981). They considered the association of diblock and triblock copolymers in solvents of varying quality. The second and third virial coefficients were determined using a mean field potential based on the segmental distribution function for a polymer chain in solution. A model for micellization of block copolymers in solution, based on the thermodynamics of associating multicomponent mixtures, was presented by Gao and Eisenberg (1993). The polydispersity of the block copolymer and its influence on micellization was a particular focus of this work. For block copolymers below the cmc, a collapsed spherical conformation was assumed. Interactions of the collapsed spheres were then described by the Hamaker equation, with an interaction energy proportional to the radius of the spheres. 

J. M. Eisenberg and W. Greiner, Nuclear Theory vol. 2, North-Holland, Amsterdam, 1970. 

Free Volume Theory. Free volume theory suggests that the glass transition temperature is observed for polymers when their viscosity approaches that of their liquid state. Following a derivation based on the Doolittle expression for polymer viscosity (r ) as a function of free volume (Eisenberg, 1984)... 

The theoretical tools for the interpretation of solution experiments are the thermodynamics of multicomponent solutions (Casassa and Eisenberg, 1964 Eisenberg, 1976, 1990) and the theory of small-... 

This equation suggests that the above discussion is reasonable. That is, application of Eisenberg s theory is valid for carboxymethylated wood containing metal ions. Accordingly, the mobility of carboxymethylate cellulose in carboxymethylated wood appears to depend on three factors electric charge, amount, and radius. The relationship is represented by Eq. (8). This result shows that Eisenberg s theory is valid for chemically modified wood containing metal ions. 

M. Eisenberg, W. Greiner, Microscopic Theory of the Nucleus, North-Holland, Amsterdam, 1972 H. Frauenfelder, E. Henley, Subatomic Physics, Prentice-Hall, Englewood Cliffs, NJ, 1974... 

In this work we used polystyrene-based ionomers.-Since there is no crystallinity in this type of ionomer, only the effect of ionic interactions has been observed. Eisenberg et al. reported that for styrene-methacrylic acid ionomers, the position of the high inflection point in the stress relaxation master curve could be approximately predicted from the classical theory of rubber elasticity, assuming that each ion pah-acts as a crosslink up to ca. 6 mol %. Above 6 mol %, the deviation of data points from the calculated curve is very large. For sulfonated polystyrene ionomers, the inflection point in stress relaxation master curves and the rubbery plateau region in dynamic mechanical data seemed to follow the classical rubber theory at low ion content. Therefore, it is generally concluded that polystyrene-based ionomers with low ion content show a crosslinking effect due to multiplet formation. More... 

For another synthetic ( conventional") vinyl polyelectrolyte, sodium polyacrylate, Takahashi and Nagasawa (42) have measured the intrinsic viscosity of a number of fractions under G conditions (1.056 N NaBr at 15° C) and report 11.0 as the characteristic ratio value of that polymer [using the value 0 — 2.87 10 for the universal parameter in equation (1)]. Both the polyvinylsulfonate and polyacrylate samples used by Eisenberg and Woodside and by Takahashi and Nagasawa were however of the conventional type, and thus composed of chains of unknown tacticity. This fact makes it more difficult to compare the experimental values with those dedudble from theories based on... 

The molecular theory of structural change from low-temperature liquid water to that of ice Ih structure is yet incomplete. Eisenberg and Kauzmann (1969) suggested that the observed temperature dependence of the molar volume of water can be explained qualitatively based on two competing effects (1) continuation of the strengthening of the four-coordination of molecules keeps the ice form ih an open structure (2) the expected decrease in all the anhar-monic vibrations will lead to a decrease in the ice volume. Probably, it is the dominance of the first effect that keeps ice Ih in a low density state. 

However, this model also predicts a continuous increase in the polymer content with temperature resulting in 48% at 250 °C and 68% at 350 °C while the best analytical data show that the maximum value for Sqo in high-purity sulfur melts is 40% [93]. In addition, the equilibrium reactions between rings other than Ss and the polymer as well as the substantial polymer content at temperatures below 157 °C are neglected in this model. Therefore, the polymerization theory by Tobolsky and Eisenberg as well as its slightly modified versions [42, 69, 130-132] are also unsatisfactory. 

Kincaid, B.M. and Eisenberger, P. "Synchrotron radiation studies of the /C-edge photoabsorption spectra of Kr, Br2 and GeCI4 a comparison of theory and experiment." Phys. Rev. Lett. 34 1361-1364 1975. 

Considerable progress has been made in the solution theory of poly-electrolytes. However, for the condensed-phase analogs of polyelectrolytes, ionomers, this is not the case. Eisenberg (1) has put forth an initial theory of ionomer structure that contains conceptual formalisms of general use. His theory has been consulted extensively in the work reported here. Ponomarev and Ionova (2) have attempted to construct a sophisticated statistical mechanical model to describe the thermodynamics of ionomers. Recently, Gierke (3) has described a theory of ion transport in the Nafion ionomer based on a specific molecular organization. 

Kincaid BM, Eisenberger P (1975) Synchrotron radiation study of the X-edge photoabsorption spectra of Kr, Br2 and GeCU comparison of theory and experiment. Phys Rev Lett 34 1361-1364 Koch EE (1983) Handbook on Synchrotron Radiation (Vol. lb). North-Holland, New York, 506 p Koningsberger DC, Prins R (1988) X-ray absorption Principles, Applications, Techniques of EXAFS, SEXAFS, and XANES. Wiley, New York, 673 p Kossel W (1920) Zum Bau der Rontgenspektren. Z Phys 1 119-134... 

Lee PA, Beni G (1977) New method for the calculation of atomic phase shifts apphcations to extended X-ray absorption fine stracture (EXAFS) in molecules and crystals. Phys Rev B 15 2862-2883 Lee PA, Citrin PH, Eisenberger P, Kincaid BM (1981) Extended X-ray fine stracture Its strength and limitations as a stractnral tool. Rev Modem Phys 53 769-806 Lee PA, Pendiy JB (1975) Theory of extended X-ray absorption fine stractrrre. Phys Rev B11 2795-2811 Li D, Bancroft, GM Fleet, ME Feng, XH (1995a) Silicon X-edges XANES spectra of silicate minerals. Phys Chem Minerals 22 115-122... 

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