Exact Finite Method of Lattice Statistics. III. Dimers on the Square Lattice

1970 ◽  
Vol 11 (3) ◽  
pp. 842-850 ◽  
Author(s):  
L. K. Runnels
Keyword(s):  
Square Lattice ◽  
Lattice Statistics ◽  
Finite Method

Exact Finite Method of Lattice Statistics. II. Honeycomb‐Lattice Gas of Hard Molecules

1967 ◽  
Vol 47 (10) ◽  
pp. 4015-4020 ◽  
Author(s):  
L. K. Runnels ◽  
L. L. Combs ◽  
James P. Salvant
Keyword(s):  
Lattice Gas ◽  
Honeycomb Lattice ◽  
Lattice Statistics ◽  
Finite Method

A LATTICE STATISTICS MODEL DERIVED FROM THE TWO-LAYER ADSORPTION PROBLEM

1965 ◽  
Vol 43 (6) ◽  
pp. 980-985
Author(s):  
D. D. Betts ◽  
D. L. Hunter
Keyword(s):  
Nearest Neighbor ◽  
Chemical Potential ◽  
Physical Adsorption ◽  
Square Lattice ◽  
Lateral Interaction ◽  
Lattice Statistics ◽  
Regular Lattice ◽  
Adsorption Sites ◽  
Layer Adsorption ◽  
Gas Molecules

A model is proposed for the physical adsorption of two layers of gas molecules at the sites of a regular lattice with lateral interaction between nearest-neighbor molecules. The model is more complicated than the two-dimensional Ising model. However, for a particular relation among the three energy parameters and at a particular value of the chemical potential the model simplifies considerably. For the simplified model and a square lattice of adsorption sites, high- and low-temperature series expansions for the specific heat have been obtained and the transition temperature estimated.

Lattice statistics in a magnetic field. II. Order and correlations of a two-dimensional super-exchange antiferromagnet

1960 ◽  
Vol 256 (1287) ◽  
pp. 502-513 ◽  
Keyword(s):  
Magnetic Field ◽  
Long Range ◽  
Pair Correlation ◽  
Configurational Entropy ◽  
Square Lattice ◽  
Lattice Statistics ◽  
Zero Field ◽  
Two Dimensional ◽  
Ising Lattice ◽  
Pair Correlations

The long-range order and pair correlation functions of a two-dimensional super-exchange antiferromagnet in an arbitrary magnetic field are derived rigorously from properties of the standard square Ising lattice in zero field. (The model investigated was described in part I: it is a decorated square lattice with magnetic spins on the bonds coupled antiferromagnetically via non-magnetic spins on the vertices.) The behaviour near the transition temperature in a finite field is similar to that of the normal plane lattice, i. e. the long-range orders or spontaneous magnetizations of the sublattices vanish as ( T t – T ) ⅛ and the pair correlations behave as ω c + W ( T – T t ) ln | T – T t |. The configurational entropy is discussed and the anomalous entropy in the critical field at zero temperature is calculated exactly.

Exact Finite Method of Lattice Statistics. IV. Square Well Lattice Gas

1970 ◽  
Vol 52 (5) ◽  
pp. 2352-2358 ◽  
Author(s):  
L. K. Runnels ◽  
J. P. Salvant ◽  
H. R. Streiffer
Keyword(s):  
Lattice Gas ◽  
Lattice Statistics ◽  
Square Well ◽  
Finite Method

Lattice statistics in a magnetic field, I. A two-dimensional super-exchange antiferromagnet

1960 ◽  
Vol 254 (1276) ◽  
pp. 66-85 ◽  
Keyword(s):  
Magnetic Field ◽  
Critical Point ◽  
Continuous Functions ◽  
Square Lattice ◽  
Lattice Statistics ◽  
Zero Field ◽  
Two Dimensional ◽  
Small Fields ◽  
Arbitrary Magnetic Field ◽  
Ising Spins

The partition function of a two-dimensional ` super-exchange ’ antiferromagnet in an arbitrary magnetic field is derived rigorously. The model is a decorated square lattice in which magnetic Ising spins on the bonds are coupled together via non-magnetic Ising spins on the vertices. By use of the decoration transformation all the thermodynamic and magnetic properties of the model are derived from Onsager’s solution for the standard square lattice in zero field. The transition temperature T t (H) is a single-valued, decreasing function of the field H . The energy and the magnetization are continuous functions of T for all magnetic fields; but the specific heat and the temperature gradient of the magnetization become infinite as — In | T — T t |. The initial ( H = 0) susceptibility is a continuous and smoothly varying function of T with a maximum 40 % above the critical point; but ∂x/∂ T becomes infinite at T = T c . In a non-vanishing field the susceptibility has a logarithmic infinity at T = T t . For small fields the behaviour near the critical point is given by X ≈ ( N μ/ kT ) {2—√2— D ( T—T c ) ln ∣ T — T c ∣ — D´H 2 ln ∣ T — T c ∣}, where D and D' are constants.

Influence of Flexible Side-Chains on the Breathing Phase Transition of Pillared Layer MOFs: A Force Field Investigation

2020 ◽  
Author(s):  
Julian Keupp ◽  
Johannes P. Dürholt ◽  
Rochus Schmid
Keyword(s):  
First Principles ◽  
Good Accuracy ◽  
Field Investigation ◽  
Square Lattice ◽  
Side Chain ◽  
Second Step ◽  
Side Chains ◽  
Dynamics Simulations ◽  
Complex Phenomena ◽  
Closed Pore

The prototypical pillared layer MOFs, formed by a square lattice of paddle-<br>wheel units and connected by dinitrogen pillars, can undergo a breathing phase<br>transition by a “wine-rack” type motion of the square lattice. We studied this not<br>yet fully understood behavior using an accurate first principles parameterized force<br>field (MOF-FF) for larger nanocrystallites on the example of Zn 2 (bdc) 2 (dabco) [bdc:<br>benzenedicarboxylate, dabco: (1,4-diazabicyclo[2.2.2]octane)] and found clear indi-<br>cations for an interface between a closed and an open pore phase traveling through<br>the system during the phase transformation [Adv. Theory Simul. 2019, 2, 11]. In<br>conventional simulations in small supercells this mechanism is prevented by periodic<br>boundary conditions (PBC), enforcing a synchronous transformation of the entire<br>crystal. Here, we extend this investigation to pillared layer MOFs with flexible<br>side-chains, attached to the linker. Such functionalized (fu-)MOFs are experimen-<br>tally known to have different properties with the side-chains acting as fixed guest<br>molecules. First, in order to extend the parameterization for such flexible groups,<br>1a new parametrization strategy for MOF-FF had to be developed, using a multi-<br>structure force based fit method. The resulting parametrization for a library of<br>fu-MOFs is then validated with respect to a set of reference systems and shows very<br>good accuracy. In the second step, a series of fu-MOFs with increasing side-chain<br>length is studied with respect to the influence of the side-chains on the breathing<br>behavior. For small supercells in PBC a systematic trend of the closed pore volume<br>with the chain length is observed. However, for a nanocrystallite model a distinct<br>interface between a closed and an open pore phase is visible only for the short chain<br>length, whereas for longer chains the interface broadens and a nearly concerted trans-<br>formation is observed. Only by molecular dynamics simulations using accurate force<br>fields such complex phenomena can be studied on a molecular level.

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