An inhomogeneous random recursive lattice is constructed from the multibranched Husimi square lat... more An inhomogeneous random recursive lattice is constructed from the multibranched Husimi square lattice. The number of repeating units connected on one vertex is randomly set to be 2 or 3 with a fixed ratio P 2 or P 3 with P 2 + P 3 = 1. The lattice is designed to describe complex thermodynamic systems with variable coordinating neighbors, e.g. the asymmetric range around the surface of a bulk system. Classical ferromagnetic spin-1 Ising model is solved on the lattice to achieve an annealed solution via the local exact calculation technique. The model exhibits distinct spontaneous magnetization similar to the deterministic system, with however rigorous thermal fluctuations and significant singularities on the entropy behavior around the critical temperature, indicating a complex superheating frustration in the cross-dimensional range induced by the stochasticity. The critical temperature was found to be exponentially correlated to the structural ratio P with the coefficient fitted as 0.53187, while the ground state energy presents linear correlation to P , implying a well-defined average property according to the structural ratio.
A second order cross-link network is set onto the classical Husimi lattice, to investigate the ro... more A second order cross-link network is set onto the classical Husimi lattice, to investigate the role of a "phantom" non-neighboring interactions of mid-and long-range in Bethe-like lattices for the first time. Since antiferromagnetic Ising model on Husimi lattice has been exactly solved and successfully presented the melting and glass transition, the Phantom Cross-link Network (PCN) is introduced here to understand the relationship between glassy defect and long-range interactions in small molecule systems, and the concept is inspired from the classical rubber network theory (Flory, 1985). One random site out of four on the recursive unites with certain distance I (the net size) is selected to be linked onto the PCN. The solutions are still in the fashion of normal antiferromagnetic Ising model, with expected frustrations along with the net size. Beside the regular Curie transition, several interesting thermodynamics are observed in this toy model, and as the main found, PCN clearly introduces glassy portion into the system, identified by the supercooling behavior with lower TC, the metastable entropy curve and the Kauzmann paradox.
A theoretical self-sustainable economic model is established based on the fundamental factors of ... more A theoretical self-sustainable economic model is established based on the fundamental factors of production, consumption, reservation and reinvestment, where currency is set as a unconditional credit symbol serving as transaction equivalent and stock means. Principle properties of currency are explored in this ideal economic system. Physical analysis reveals some facts that were not addressed by traditional monetary theory, and several basic principles of ideal currency are concluded: 1. The saving-replacement is a more primary function of currency than the transaction equivalents; 2. The ideal efficiency of currency corresponds to the least practical value; 3. The contradiction between constant face value of currency and depreciable goods leads to intrinsic inflation.
Two kinds of recursive lattices with the same coordination number but different unit cells (2-D s... more Two kinds of recursive lattices with the same coordination number but different unit cells (2-D square and 3-D cube) are constructed and the antiferromagnetic Ising model is solved exactly on them to study the stable and metastable states. The Ising model with multi-particle interactions is designed to represent a monatomic system or an alloy. Two solutions of the model exhibit the crystallization of liquid, and the ideal glass transition of supercooled liquid respectively. Based on the solutions, the thermodynamics on both lattices was examined. In particular, the free energy, energy, and entropy of the ideal glass, supercooled liquid, crystal, and liquid state of the model on each lattice were calculated and compared with each other. Interactions between particles farther away than the nearest neighbor distance are taken into consideration. The two lattices show comparable properties on the transition temperatures and the thermodynamic behaviors, which proves that both of them are practical to describe the regular 3-D case, while the different effects of the unit types are still obvious.
Bulletin of the American Physical Society, Mar 16, 2016
Univ-As one of the few exactly solvable thermodynamic models, the Ising model on recursive lattic... more Univ-As one of the few exactly solvable thermodynamic models, the Ising model on recursive lattice is featured by its impressive advantages and successful applications in various thermodynamic and statistical researches. However this model was considered that, since the recursive calculation demands homogeneous structure, it can only describe the bulk and even systems with narrow utilization. In this work we figured out a practical methodology to extend the conventional homogeneous structure of single-unit Husimi lattice to be random inhomogeneous lattices with variable units and structures, while keeping the feature of exact calculation. Three designs of inhomogeneous recursive lattices: the random-angled rhombus lattice, the Husimi lattice of variable units, and the randomly multi-branched Husimi square lattice; and the corresponding exact recursive calculations based on the partial partition function algorithm, which is derived from the Bethe Cavity method, have been investigated and developed. With the "total-symmetry assumption" and the "iterative-replica trick" we were able to exactly solve the classical ferromagnetic spin-1 Ising models on these lattices, to describe the complex systems that can only be solved by approximations or simulations on regular lattices. Our work may enhance the application of the exact calculation on recursive lattices in various fields of materials science and applied physics, especially it may serve as a powerful tool to explore the cross-dimensional thermodynamics and phase transitions.
A many-body Ising lattice model of monatomic systems is solved exactly on a new recursive lattice... more A many-body Ising lattice model of monatomic systems is solved exactly on a new recursive lattice with the aim to study the metastability in supercooled liquids and the ideal glass transition. Interactions between particles farther away than the nearest neighbor distance are taken into consideration. The model has a strong antiferromagnetic interaction to give rise to an ordered phase identified as a crystal. Thermal properties including free energy, energy and entropy of the ideal glass and supercooled liquid state of the model are calculated. The computation results show both the first order melting transition and second order ideal glass transition (entropy crisis). The effects of different energy terms are studied. We also study the defects in the ideal glass, supercooled liquid and the crystal to support the theory that a glass can be treated as a highly defective crystal, since the ideal glass at absolute zero has much higher energy than the crystal.
Since the first paper by Keddie et al. published on 1994 [21], the glass transition of polymer sy... more Since the first paper by Keddie et al. published on 1994 [21], the glass transition of polymer systems on surface/thin film has been an active research field and attracted many groups interests. Numerous works have been done, in both experimental and computation approaches, to investigate this subject. In this paper we reviewed the milestone findings in the last twenty years. Generally with only minor disagreements in the mechanism all the mainstream works are consistent in the conclusions that: 1) Geometric confinement in thin film or on surface reduces the glass transition temperature Tg comparing to the bulk behavior; 2) For supported film the substrate-film interaction is critical and its effect may surpass the geometry effects and rise increase on Tg; 3) Chain mobility and molecular weight are critical but the detailed phenomena vary with systems. Notwithstanding the achievement has been made, due to the controversy of glass transition itself and technology limitation on characterization on glass transitions on thin film, the research in this field is still a long-marching effort and breakthrough findings are expected for the development in materials science and engineering and feedback knowledge to understand the glass transition on the theoretical base.
An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting ... more An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting a regular square lattice, has been developed to investigate the thermodynamics and phase transitions on surface by the methodology of recursive lattice, which we have previously applied to study polymers near a surface. The model retains the advantages of simple formulation and exact calculation of the conventional Bethelike lattices. An antiferromagnetic Ising model is solved on the surface of this lattice to evaluate thermal properties such as free energy, energy density and entropy, from which we have successfully identified a first-order order-disorder transition other than the spontaneous magnetization, and a secondary transition on the supercooled state indicated by the Kauzmann paradox.
A quasi 2-dimensional recursive lattice formed by planar elements have been designed to investiga... more A quasi 2-dimensional recursive lattice formed by planar elements have been designed to investigate the surface thermodynamics of Ising spin glass system with the aim to study the metastability of supercooled liquids and the ideal glass transition. The lattice is constructed as a hybrid of partial Husimi lattice representing the bulk and 1D single bonds representing the surface. The recursive properties of the lattices were adopted to achieve exact calculations. The model has an anti-ferromagnetic interaction to give rise to an ordered phase identified as crystal, and a metastable solution representing the amorphous/metastable phase. Interactions between particles farther away than the nearest neighbor distance are taken into consideration. Free energy and entropy of the ideal crystal and supercooled liquid state of the model on the surface are calculated by the partial partition function. By analyzing the free energies and entropies of the crystal and supercooled liquid state, we are able to identify the melting transition and the second order ideal glass transition on the surface. The results show that due to the coordination number change, the transition temperature on the surface decreases significantly compared to the bulk system. Our calculation agrees with experimental and simulation results on the thermodynamics of surfaces and thin films conducted by others.
An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting ... more An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting a regular square lattice, has been developed to investigate the thermodynamics and phase transitions on surface by the methodology of recursive lattice, which we have previously applied to study polymers near a surface. The model retains the advantages of simple formulation and exact calculation of the conventional Bethe-like lattices. An antiferromagnetic Ising model is solved on the surface of this lattice to evaluate thermal properties such as free energy, energy density and entropy, from which we have successfully identified a first-order order–disorder transition other than the spontaneous magnetization, and a secondary transition on the supercooled state indicated by the Kauzmann paradox.
Square unit in Husimi lattice (the Bethe square lattice) is generalized to be rhombus with random... more Square unit in Husimi lattice (the Bethe square lattice) is generalized to be rhombus with randomly variable angles. The independence feature of unit cells in recursive lattice makes the random angle conformation possible in the model construction, which is unfeasible in the conventional lattice. Since the randomness of the conformations in real system is naturally introduced into the model, this new lattice methodology can describe the off-crystal metastable states without artificial randomness. With reasonable simplification, a coefficient A(θ) is formulated to present the effect of angle in the rhombus unit. A "visit and count" recursive technique is developed to numerically calculate the thermodynamics. While the computation randomizes a quenched configuration in each iteration, the calculation counts and averages a large number of random units to deal with a system in equilibrium with annealed randomness at particular temperature. The critical temperature Tc of spontaneous magnetization transition is lowered with the presence of angle randomness, which implies a less stable system. Besides consistent results to the regular lattice, the random-angled lattice features a distribution of solutions and the thermal fluctuation with exact calculation. The effects of the variation of energy and ground state parameters on thermodynamics are investigated.
Two kinds of recursive lattices with the same coordination number but different unit cells (2-D s... more Two kinds of recursive lattices with the same coordination number but different unit cells (2-D square and 3-D cube) are constructed and the antiferromagnetic Ising model is solved exactly on them to study the stable and metastable states. The Ising model with multi-particle interactions is designed to represent a monatomic system or an alloy. Two solutions of the model exhibit the crystallization of liquid, and the ideal glass transition of supercooled liquid respectively. Based on the solutions, the thermodynamics on both lattices was examined. In particular, the free energy, energy, and entropy of the ideal glass, supercooled liquid, crystal, and liquid state of the model on each lattice were calculated and compared with each other. Interactions between particles farther away than the nearest neighbor distance are taken into consideration. The two lattices show comparable properties on the transition temperatures and the thermodynamic behaviors, which proves that both of them are practical to describe the regular 3-D case, while the different effects of the unit types are still obvious.
By defining the dimension of natural numbers as the number of prime factors, all natural numbers ... more By defining the dimension of natural numbers as the number of prime factors, all natural numbers smaller than 2^(n+1) (n is a natural number) can be classified by their dimensions, and the count of numbers of each dimension gives a dimensions distribution directly related to the distribution of prime numbers. A triangle similar to Pascals triangle from some aspects can be obtained by the ensemble of dimensions distributions. Some superficial explorations have been done to this interesting triangle.
Submitted for the MAR09 Meeting of The American Physical Society Exact thermodynamic calculation ... more Submitted for the MAR09 Meeting of The American Physical Society Exact thermodynamic calculation of a monatomic system and its ideal glass transition on a new recursive lattice formed by cubic units RAN HUANG, PURU GUJRATI, The Univeristy of Akron-A many-body Ising lattice model is used to represent monatomic systems and is solved exactly on a new recursive lattice with the aim to study the metastability in supercooled liquids and the ideal glass transition. Interactions between particles farther away than the nearest neighbor distance are taken into consideration. The Ising model is antiferromagnetic in nature so that its ordered phase represents an alloy-type crystal of alternating species (A-B or particle-void). The new recursive lattice appears quite reliable to represent a cubic lattice. Thermal properties including free energy, energy and entropy of the ideal crystal and supercooled liquid state of the model are calculated. The computation results show a first order melting and second order ideal glass transition (entropy crisis) in the supercooled liquid phase. The effects of different energy terms on the two transitions are studied. We also study the defects in the ideal glass, supercooled liquid and the crystal to support the theory that a glass can be treated as a highly defective crystal.
The multi-branched Husimi recursive lattice has been extended to a virtual structure with fractio... more The multi-branched Husimi recursive lattice has been extended to a virtual structure with fractional numbers of branches joined on one site. Although the lattice is undrawable in real space, the concept is consistent with regular Husimi lattice. The Ising spins of antiferromagnetic interaction on such a sets of lattices were calculated to check the critical temperatures (T c) and ideal glass transition temperatures (T k) variation with fractional branch numbers. Besides the similar results of two solutions representing the stable state (crystal) and metastable state (supercooled liquid) and indicating the phase transition temperatures, the phase transitions show a well-defined shift with branch number variation. Therefore the fractional branch number as a parameter can be used as an adjusting tool in constructing a recursive lattice model to describe real systems.
An inhomogeneous random recursive lattice is constructed from the multibranched Husimi square lat... more An inhomogeneous random recursive lattice is constructed from the multibranched Husimi square lattice. The number of repeating units connected on one vertex is randomly set to be 2 or 3 with a fixed ratio P 2 or P 3 with P 2 + P 3 = 1. The lattice is designed to describe complex thermodynamic systems with variable coordinating neighbors, e.g. the asymmetric range around the surface of a bulk system. Classical ferromagnetic spin-1 Ising model is solved on the lattice to achieve an annealed solution via the local exact calculation technique. The model exhibits distinct spontaneous magnetization similar to the deterministic system, with however rigorous thermal fluctuations and significant singularities on the entropy behavior around the critical temperature, indicating a complex superheating frustration in the cross-dimensional range induced by the stochasticity. The critical temperature was found to be exponentially correlated to the structural ratio P with the coefficient fitted as 0.53187, while the ground state energy presents linear correlation to P , implying a well-defined average property according to the structural ratio.
The porous poly(lactic acid) (PLA) foams potential for tissue engineering usage are prepared by a... more The porous poly(lactic acid) (PLA) foams potential for tissue engineering usage are prepared by a modified solvent casting/particulate leaching method with different crystallinity. Since in typical method the porogens are dispersed in the solution and flow with the polymers during the casting and the crystallinity behavior of PLA chains in the limited space cannot be tracked, in this work the processing is modified by diffusing the PLA solution into a steady salt stack. With a thermal treatment before leaching while maintaining the stable structure of the porogens stack, the crystallinity of porous foams is made possible to control. The characterizations indicate that the porous PLA foams have a lower crystallizability than the bulk materials. Pores and caves of around 250 μm size are obtained in samples with different crystallinity. The macro-structures are not much impaired by the crystallization nevertheless the morphological effect of the heating process is still obvious.
An inhomogeneous random recursive lattice is constructed from the multibranched Husimi square lat... more An inhomogeneous random recursive lattice is constructed from the multibranched Husimi square lattice. The number of repeating units connected on one vertex is randomly set to be 2 or 3 with a fixed ratio P 2 or P 3 with P 2 + P 3 = 1. The lattice is designed to describe complex thermodynamic systems with variable coordinating neighbors, e.g. the asymmetric range around the surface of a bulk system. Classical ferromagnetic spin-1 Ising model is solved on the lattice to achieve an annealed solution via the local exact calculation technique. The model exhibits distinct spontaneous magnetization similar to the deterministic system, with however rigorous thermal fluctuations and significant singularities on the entropy behavior around the critical temperature, indicating a complex superheating frustration in the cross-dimensional range induced by the stochasticity. The critical temperature was found to be exponentially correlated to the structural ratio P with the coefficient fitted as 0.53187, while the ground state energy presents linear correlation to P , implying a well-defined average property according to the structural ratio.
A second order cross-link network is set onto the classical Husimi lattice, to investigate the ro... more A second order cross-link network is set onto the classical Husimi lattice, to investigate the role of a "phantom" non-neighboring interactions of mid-and long-range in Bethe-like lattices for the first time. Since antiferromagnetic Ising model on Husimi lattice has been exactly solved and successfully presented the melting and glass transition, the Phantom Cross-link Network (PCN) is introduced here to understand the relationship between glassy defect and long-range interactions in small molecule systems, and the concept is inspired from the classical rubber network theory (Flory, 1985). One random site out of four on the recursive unites with certain distance I (the net size) is selected to be linked onto the PCN. The solutions are still in the fashion of normal antiferromagnetic Ising model, with expected frustrations along with the net size. Beside the regular Curie transition, several interesting thermodynamics are observed in this toy model, and as the main found, PCN clearly introduces glassy portion into the system, identified by the supercooling behavior with lower TC, the metastable entropy curve and the Kauzmann paradox.
A theoretical self-sustainable economic model is established based on the fundamental factors of ... more A theoretical self-sustainable economic model is established based on the fundamental factors of production, consumption, reservation and reinvestment, where currency is set as a unconditional credit symbol serving as transaction equivalent and stock means. Principle properties of currency are explored in this ideal economic system. Physical analysis reveals some facts that were not addressed by traditional monetary theory, and several basic principles of ideal currency are concluded: 1. The saving-replacement is a more primary function of currency than the transaction equivalents; 2. The ideal efficiency of currency corresponds to the least practical value; 3. The contradiction between constant face value of currency and depreciable goods leads to intrinsic inflation.
Two kinds of recursive lattices with the same coordination number but different unit cells (2-D s... more Two kinds of recursive lattices with the same coordination number but different unit cells (2-D square and 3-D cube) are constructed and the antiferromagnetic Ising model is solved exactly on them to study the stable and metastable states. The Ising model with multi-particle interactions is designed to represent a monatomic system or an alloy. Two solutions of the model exhibit the crystallization of liquid, and the ideal glass transition of supercooled liquid respectively. Based on the solutions, the thermodynamics on both lattices was examined. In particular, the free energy, energy, and entropy of the ideal glass, supercooled liquid, crystal, and liquid state of the model on each lattice were calculated and compared with each other. Interactions between particles farther away than the nearest neighbor distance are taken into consideration. The two lattices show comparable properties on the transition temperatures and the thermodynamic behaviors, which proves that both of them are practical to describe the regular 3-D case, while the different effects of the unit types are still obvious.
Bulletin of the American Physical Society, Mar 16, 2016
Univ-As one of the few exactly solvable thermodynamic models, the Ising model on recursive lattic... more Univ-As one of the few exactly solvable thermodynamic models, the Ising model on recursive lattice is featured by its impressive advantages and successful applications in various thermodynamic and statistical researches. However this model was considered that, since the recursive calculation demands homogeneous structure, it can only describe the bulk and even systems with narrow utilization. In this work we figured out a practical methodology to extend the conventional homogeneous structure of single-unit Husimi lattice to be random inhomogeneous lattices with variable units and structures, while keeping the feature of exact calculation. Three designs of inhomogeneous recursive lattices: the random-angled rhombus lattice, the Husimi lattice of variable units, and the randomly multi-branched Husimi square lattice; and the corresponding exact recursive calculations based on the partial partition function algorithm, which is derived from the Bethe Cavity method, have been investigated and developed. With the "total-symmetry assumption" and the "iterative-replica trick" we were able to exactly solve the classical ferromagnetic spin-1 Ising models on these lattices, to describe the complex systems that can only be solved by approximations or simulations on regular lattices. Our work may enhance the application of the exact calculation on recursive lattices in various fields of materials science and applied physics, especially it may serve as a powerful tool to explore the cross-dimensional thermodynamics and phase transitions.
A many-body Ising lattice model of monatomic systems is solved exactly on a new recursive lattice... more A many-body Ising lattice model of monatomic systems is solved exactly on a new recursive lattice with the aim to study the metastability in supercooled liquids and the ideal glass transition. Interactions between particles farther away than the nearest neighbor distance are taken into consideration. The model has a strong antiferromagnetic interaction to give rise to an ordered phase identified as a crystal. Thermal properties including free energy, energy and entropy of the ideal glass and supercooled liquid state of the model are calculated. The computation results show both the first order melting transition and second order ideal glass transition (entropy crisis). The effects of different energy terms are studied. We also study the defects in the ideal glass, supercooled liquid and the crystal to support the theory that a glass can be treated as a highly defective crystal, since the ideal glass at absolute zero has much higher energy than the crystal.
Since the first paper by Keddie et al. published on 1994 [21], the glass transition of polymer sy... more Since the first paper by Keddie et al. published on 1994 [21], the glass transition of polymer systems on surface/thin film has been an active research field and attracted many groups interests. Numerous works have been done, in both experimental and computation approaches, to investigate this subject. In this paper we reviewed the milestone findings in the last twenty years. Generally with only minor disagreements in the mechanism all the mainstream works are consistent in the conclusions that: 1) Geometric confinement in thin film or on surface reduces the glass transition temperature Tg comparing to the bulk behavior; 2) For supported film the substrate-film interaction is critical and its effect may surpass the geometry effects and rise increase on Tg; 3) Chain mobility and molecular weight are critical but the detailed phenomena vary with systems. Notwithstanding the achievement has been made, due to the controversy of glass transition itself and technology limitation on characterization on glass transitions on thin film, the research in this field is still a long-marching effort and breakthrough findings are expected for the development in materials science and engineering and feedback knowledge to understand the glass transition on the theoretical base.
An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting ... more An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting a regular square lattice, has been developed to investigate the thermodynamics and phase transitions on surface by the methodology of recursive lattice, which we have previously applied to study polymers near a surface. The model retains the advantages of simple formulation and exact calculation of the conventional Bethelike lattices. An antiferromagnetic Ising model is solved on the surface of this lattice to evaluate thermal properties such as free energy, energy density and entropy, from which we have successfully identified a first-order order-disorder transition other than the spontaneous magnetization, and a secondary transition on the supercooled state indicated by the Kauzmann paradox.
A quasi 2-dimensional recursive lattice formed by planar elements have been designed to investiga... more A quasi 2-dimensional recursive lattice formed by planar elements have been designed to investigate the surface thermodynamics of Ising spin glass system with the aim to study the metastability of supercooled liquids and the ideal glass transition. The lattice is constructed as a hybrid of partial Husimi lattice representing the bulk and 1D single bonds representing the surface. The recursive properties of the lattices were adopted to achieve exact calculations. The model has an anti-ferromagnetic interaction to give rise to an ordered phase identified as crystal, and a metastable solution representing the amorphous/metastable phase. Interactions between particles farther away than the nearest neighbor distance are taken into consideration. Free energy and entropy of the ideal crystal and supercooled liquid state of the model on the surface are calculated by the partial partition function. By analyzing the free energies and entropies of the crystal and supercooled liquid state, we are able to identify the melting transition and the second order ideal glass transition on the surface. The results show that due to the coordination number change, the transition temperature on the surface decreases significantly compared to the bulk system. Our calculation agrees with experimental and simulation results on the thermodynamics of surfaces and thin films conducted by others.
An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting ... more An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting a regular square lattice, has been developed to investigate the thermodynamics and phase transitions on surface by the methodology of recursive lattice, which we have previously applied to study polymers near a surface. The model retains the advantages of simple formulation and exact calculation of the conventional Bethe-like lattices. An antiferromagnetic Ising model is solved on the surface of this lattice to evaluate thermal properties such as free energy, energy density and entropy, from which we have successfully identified a first-order order–disorder transition other than the spontaneous magnetization, and a secondary transition on the supercooled state indicated by the Kauzmann paradox.
Square unit in Husimi lattice (the Bethe square lattice) is generalized to be rhombus with random... more Square unit in Husimi lattice (the Bethe square lattice) is generalized to be rhombus with randomly variable angles. The independence feature of unit cells in recursive lattice makes the random angle conformation possible in the model construction, which is unfeasible in the conventional lattice. Since the randomness of the conformations in real system is naturally introduced into the model, this new lattice methodology can describe the off-crystal metastable states without artificial randomness. With reasonable simplification, a coefficient A(θ) is formulated to present the effect of angle in the rhombus unit. A "visit and count" recursive technique is developed to numerically calculate the thermodynamics. While the computation randomizes a quenched configuration in each iteration, the calculation counts and averages a large number of random units to deal with a system in equilibrium with annealed randomness at particular temperature. The critical temperature Tc of spontaneous magnetization transition is lowered with the presence of angle randomness, which implies a less stable system. Besides consistent results to the regular lattice, the random-angled lattice features a distribution of solutions and the thermal fluctuation with exact calculation. The effects of the variation of energy and ground state parameters on thermodynamics are investigated.
Two kinds of recursive lattices with the same coordination number but different unit cells (2-D s... more Two kinds of recursive lattices with the same coordination number but different unit cells (2-D square and 3-D cube) are constructed and the antiferromagnetic Ising model is solved exactly on them to study the stable and metastable states. The Ising model with multi-particle interactions is designed to represent a monatomic system or an alloy. Two solutions of the model exhibit the crystallization of liquid, and the ideal glass transition of supercooled liquid respectively. Based on the solutions, the thermodynamics on both lattices was examined. In particular, the free energy, energy, and entropy of the ideal glass, supercooled liquid, crystal, and liquid state of the model on each lattice were calculated and compared with each other. Interactions between particles farther away than the nearest neighbor distance are taken into consideration. The two lattices show comparable properties on the transition temperatures and the thermodynamic behaviors, which proves that both of them are practical to describe the regular 3-D case, while the different effects of the unit types are still obvious.
By defining the dimension of natural numbers as the number of prime factors, all natural numbers ... more By defining the dimension of natural numbers as the number of prime factors, all natural numbers smaller than 2^(n+1) (n is a natural number) can be classified by their dimensions, and the count of numbers of each dimension gives a dimensions distribution directly related to the distribution of prime numbers. A triangle similar to Pascals triangle from some aspects can be obtained by the ensemble of dimensions distributions. Some superficial explorations have been done to this interesting triangle.
Submitted for the MAR09 Meeting of The American Physical Society Exact thermodynamic calculation ... more Submitted for the MAR09 Meeting of The American Physical Society Exact thermodynamic calculation of a monatomic system and its ideal glass transition on a new recursive lattice formed by cubic units RAN HUANG, PURU GUJRATI, The Univeristy of Akron-A many-body Ising lattice model is used to represent monatomic systems and is solved exactly on a new recursive lattice with the aim to study the metastability in supercooled liquids and the ideal glass transition. Interactions between particles farther away than the nearest neighbor distance are taken into consideration. The Ising model is antiferromagnetic in nature so that its ordered phase represents an alloy-type crystal of alternating species (A-B or particle-void). The new recursive lattice appears quite reliable to represent a cubic lattice. Thermal properties including free energy, energy and entropy of the ideal crystal and supercooled liquid state of the model are calculated. The computation results show a first order melting and second order ideal glass transition (entropy crisis) in the supercooled liquid phase. The effects of different energy terms on the two transitions are studied. We also study the defects in the ideal glass, supercooled liquid and the crystal to support the theory that a glass can be treated as a highly defective crystal.
The multi-branched Husimi recursive lattice has been extended to a virtual structure with fractio... more The multi-branched Husimi recursive lattice has been extended to a virtual structure with fractional numbers of branches joined on one site. Although the lattice is undrawable in real space, the concept is consistent with regular Husimi lattice. The Ising spins of antiferromagnetic interaction on such a sets of lattices were calculated to check the critical temperatures (T c) and ideal glass transition temperatures (T k) variation with fractional branch numbers. Besides the similar results of two solutions representing the stable state (crystal) and metastable state (supercooled liquid) and indicating the phase transition temperatures, the phase transitions show a well-defined shift with branch number variation. Therefore the fractional branch number as a parameter can be used as an adjusting tool in constructing a recursive lattice model to describe real systems.
An inhomogeneous random recursive lattice is constructed from the multibranched Husimi square lat... more An inhomogeneous random recursive lattice is constructed from the multibranched Husimi square lattice. The number of repeating units connected on one vertex is randomly set to be 2 or 3 with a fixed ratio P 2 or P 3 with P 2 + P 3 = 1. The lattice is designed to describe complex thermodynamic systems with variable coordinating neighbors, e.g. the asymmetric range around the surface of a bulk system. Classical ferromagnetic spin-1 Ising model is solved on the lattice to achieve an annealed solution via the local exact calculation technique. The model exhibits distinct spontaneous magnetization similar to the deterministic system, with however rigorous thermal fluctuations and significant singularities on the entropy behavior around the critical temperature, indicating a complex superheating frustration in the cross-dimensional range induced by the stochasticity. The critical temperature was found to be exponentially correlated to the structural ratio P with the coefficient fitted as 0.53187, while the ground state energy presents linear correlation to P , implying a well-defined average property according to the structural ratio.
The porous poly(lactic acid) (PLA) foams potential for tissue engineering usage are prepared by a... more The porous poly(lactic acid) (PLA) foams potential for tissue engineering usage are prepared by a modified solvent casting/particulate leaching method with different crystallinity. Since in typical method the porogens are dispersed in the solution and flow with the polymers during the casting and the crystallinity behavior of PLA chains in the limited space cannot be tracked, in this work the processing is modified by diffusing the PLA solution into a steady salt stack. With a thermal treatment before leaching while maintaining the stable structure of the porogens stack, the crystallinity of porous foams is made possible to control. The characterizations indicate that the porous PLA foams have a lower crystallizability than the bulk materials. Pores and caves of around 250 μm size are obtained in samples with different crystallinity. The macro-structures are not much impaired by the crystallization nevertheless the morphological effect of the heating process is still obvious.
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Papers by Ran Huang
modified solvent casting/particulate leaching method with different crystallinity. Since in typical method
the porogens are dispersed in the solution and flow with the polymers during the casting and the
crystallinity behavior of PLA chains in the limited space cannot be tracked, in this work the processing is
modified by diffusing the PLA solution into a steady salt stack. With a thermal treatment before leaching
while maintaining the stable structure of the porogens stack, the crystallinity of porous foams is made
possible to control. The characterizations indicate that the porous PLA foams have a lower crystallizability than the bulk materials. Pores and caves of around 250 μm size are obtained in samples with
different crystallinity. The macro-structures are not much impaired by the crystallization nevertheless
the morphological effect of the heating process is still obvious.
modified solvent casting/particulate leaching method with different crystallinity. Since in typical method
the porogens are dispersed in the solution and flow with the polymers during the casting and the
crystallinity behavior of PLA chains in the limited space cannot be tracked, in this work the processing is
modified by diffusing the PLA solution into a steady salt stack. With a thermal treatment before leaching
while maintaining the stable structure of the porogens stack, the crystallinity of porous foams is made
possible to control. The characterizations indicate that the porous PLA foams have a lower crystallizability than the bulk materials. Pores and caves of around 250 μm size are obtained in samples with
different crystallinity. The macro-structures are not much impaired by the crystallization nevertheless
the morphological effect of the heating process is still obvious.