Experiments on the chain melting thermal transition in simple biological membranes are briefly re... more Experiments on the chain melting thermal transition in simple biological membranes are briefly reviewed and shown to indicate that a microscopic order-disorder model may be appropriate to describe the thermodynamics of the transition. To test this conclusion further a pair of two dimensional lattice models (A and B) are introduced and the statistical mechanics is solved exactly using dimer techniques. The phenomenological parameters required by the models are evaluated from experiments on other systems. The more realistic of the two models (model A) has a second order transition at a temperature of 353°K compared to 315°K for dipalmitoyl-L-α-lecithin membranes. However, the specific heat peaks do not have the same shape and the transition is broader for model A than for the experiment. In comparison, the less realistic model B has a first order transition at 925°K, considerably higher than the experimental transition temperature. From these results it seems likely that the points of disagreement between model A and experiment may be due to the simplified features of model A. It is concluded that the basic order-disorder model is likely to be correct for the main transition. There is also a smaller transition at a lower temperature which is also discussed in terms of an order-disorder model involving the phospholipid head groups. In this case the calculation shows that the simple model is either wrong or that some additional features must be added.
Dimer models with excluded volume interactions and classical anisotropic dispersion interactions ... more Dimer models with excluded volume interactions and classical anisotropic dispersion interactions are solved exactly in the limit of close packing on the two dimensional square, triangular and honeycomb lattices. These models exhibit nematic phase transitions for the aforementioned three lattices at temperatures of 2.55b/k, 1.51b/k and 3.10b/k respectively, where - b is the ground state energy per molecule.
The extensive surface area A and the conjugate surface pressure Π are introduced into a cooperati... more The extensive surface area A and the conjugate surface pressure Π are introduced into a cooperatively interacting hydrocarbon chain model. The exact statistical mechanical method of solution is extended to accommodate the new variables. The previously discussed 3/2 order transition becomes a critical point in phase diagrams involving Π and A. The values of the usual critical exponents are β=1, δ=2, γ=γ′=1, and α=0, and the 3/2 order exponents appear on the critical isobar, Π=Πc. The model shows Π-A phase behavior which is strikingly different from the P–V behavior, in agreement with the difference between the experimental Π-A monolayer behavior and the P–V bilayer behavior. This supports the theory that the bilayer is two weakly coupled monolayers.
Abstract A brief survey is given of a number of exactly solved two-dimensional dimer models with ... more Abstract A brief survey is given of a number of exactly solved two-dimensional dimer models with K-type transitions and with isomorphisms to chain melting and/or domain wall, adsorbed atom models.
The area of critical phenomena is a rather large one, both quantitatively, as measured by the num... more The area of critical phenomena is a rather large one, both quantitatively, as measured by the number of researchers and the number of publications, and qualitatively, as judged by the percentage of “hard” results such as exact calculations and highly precise experiments. This makes it impossible to write an article or even a book which is completely self-contained and which includes all that has been accomplished.
The lattice statistical problem of calculating the residual entropy of ice has been considered in... more The lattice statistical problem of calculating the residual entropy of ice has been considered in some detail for the hexagonal and cubic ice lattices as well as for a two-dimensional icelike lattice. Even for the two-dimensional lattice, this problem appears to be intractable using exact methods, so an approximation method is in order. The series method of DiMarzio and Stillinger has been developed so that the series is completely characterized by the numbers of various kinds of cycles on the lattice. The first five terms of the series have been evaluated and used to extrapolate values of the residual entropy S(0) within rather narrow limits for all practical purposes. The result for hexagonal ice and cubic ice is S(0) = .8145 ± .0002 cal/deg/mole which agrees with experiment even better than Pauling's original approximation. Some other methods are also discussed, and their results tend to confirm the series results.
Page 1. 2 |Dimer Models on Anisotropic Lattices John F. Nagle s Department of Physics, Carnegie-M... more Page 1. 2 |Dimer Models on Anisotropic Lattices John F. Nagle s Department of Physics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA Carlos S. 0. Yokoi Instituto de Fisica, Universidade de S50 Paulo, Caixa ...
Experiments on the chain melting thermal transition in simple biological membranes are briefly re... more Experiments on the chain melting thermal transition in simple biological membranes are briefly reviewed and shown to indicate that a microscopic order-disorder model may be appropriate to describe the thermodynamics of the transition. To test this conclusion further a pair of two dimensional lattice models (A and B) are introduced and the statistical mechanics is solved exactly using dimer techniques. The phenomenological parameters required by the models are evaluated from experiments on other systems. The more realistic of the two models (model A) has a second order transition at a temperature of 353°K compared to 315°K for dipalmitoyl-L-α-lecithin membranes. However, the specific heat peaks do not have the same shape and the transition is broader for model A than for the experiment. In comparison, the less realistic model B has a first order transition at 925°K, considerably higher than the experimental transition temperature. From these results it seems likely that the points of disagreement between model A and experiment may be due to the simplified features of model A. It is concluded that the basic order-disorder model is likely to be correct for the main transition. There is also a smaller transition at a lower temperature which is also discussed in terms of an order-disorder model involving the phospholipid head groups. In this case the calculation shows that the simple model is either wrong or that some additional features must be added.
Dimer models with excluded volume interactions and classical anisotropic dispersion interactions ... more Dimer models with excluded volume interactions and classical anisotropic dispersion interactions are solved exactly in the limit of close packing on the two dimensional square, triangular and honeycomb lattices. These models exhibit nematic phase transitions for the aforementioned three lattices at temperatures of 2.55b/k, 1.51b/k and 3.10b/k respectively, where - b is the ground state energy per molecule.
The extensive surface area A and the conjugate surface pressure Π are introduced into a cooperati... more The extensive surface area A and the conjugate surface pressure Π are introduced into a cooperatively interacting hydrocarbon chain model. The exact statistical mechanical method of solution is extended to accommodate the new variables. The previously discussed 3/2 order transition becomes a critical point in phase diagrams involving Π and A. The values of the usual critical exponents are β=1, δ=2, γ=γ′=1, and α=0, and the 3/2 order exponents appear on the critical isobar, Π=Πc. The model shows Π-A phase behavior which is strikingly different from the P–V behavior, in agreement with the difference between the experimental Π-A monolayer behavior and the P–V bilayer behavior. This supports the theory that the bilayer is two weakly coupled monolayers.
Abstract A brief survey is given of a number of exactly solved two-dimensional dimer models with ... more Abstract A brief survey is given of a number of exactly solved two-dimensional dimer models with K-type transitions and with isomorphisms to chain melting and/or domain wall, adsorbed atom models.
The area of critical phenomena is a rather large one, both quantitatively, as measured by the num... more The area of critical phenomena is a rather large one, both quantitatively, as measured by the number of researchers and the number of publications, and qualitatively, as judged by the percentage of “hard” results such as exact calculations and highly precise experiments. This makes it impossible to write an article or even a book which is completely self-contained and which includes all that has been accomplished.
The lattice statistical problem of calculating the residual entropy of ice has been considered in... more The lattice statistical problem of calculating the residual entropy of ice has been considered in some detail for the hexagonal and cubic ice lattices as well as for a two-dimensional icelike lattice. Even for the two-dimensional lattice, this problem appears to be intractable using exact methods, so an approximation method is in order. The series method of DiMarzio and Stillinger has been developed so that the series is completely characterized by the numbers of various kinds of cycles on the lattice. The first five terms of the series have been evaluated and used to extrapolate values of the residual entropy S(0) within rather narrow limits for all practical purposes. The result for hexagonal ice and cubic ice is S(0) = .8145 ± .0002 cal/deg/mole which agrees with experiment even better than Pauling's original approximation. Some other methods are also discussed, and their results tend to confirm the series results.
Page 1. 2 |Dimer Models on Anisotropic Lattices John F. Nagle s Department of Physics, Carnegie-M... more Page 1. 2 |Dimer Models on Anisotropic Lattices John F. Nagle s Department of Physics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA Carlos S. 0. Yokoi Instituto de Fisica, Universidade de S50 Paulo, Caixa ...
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