pyPRISM.closure.HyperNettedChain module¶
-
class
pyPRISM.closure.HyperNettedChain.HNC(apply_hard_core=False)[source]¶ Bases:
pyPRISM.closure.HyperNettedChain.HyperNettedChainAlias of HyperNettedChain
-
class
pyPRISM.closure.HyperNettedChain.HyperNettedChain(apply_hard_core=False)[source]¶ Bases:
pyPRISM.closure.AtomicClosure.AtomicClosureHyperNettedChain closure
Mathematial Definition
\[c_{\alpha,\beta}(r) = \exp\left(\gamma_{\alpha,\beta}(r)-U_{\alpha,\beta}(r)\right) - 1.0 - \gamma_{\alpha,\beta}(r)\]\[\gamma_{\alpha,\beta}(r) = h_{\alpha,\beta}(r) - c_{\alpha,\beta}(r)\]Variables Definitions
- \(h_{\alpha,\beta}(r)\)
- Total correlation function value at distance \(r\) between sites \(\alpha\) and \(\beta\).
- \(c_{\alpha,\beta}(r)\)
- Direct correlation function value at distance \(r\) between sites \(\alpha\) and \(\beta\).
- \(U_{\alpha,\beta}(r)\)
- Interaction potential value at distance \(r\) between sites \(\alpha\) and \(\beta\).
Description
The Hypernetted Chain Closure (HNC) is derived by expanding the direct correlation function, \(c(r)\), in powers of density shift from a reference state. See Reference [1] for a full derivation and discussion of this closure.
The change of variables is necessary in order to use potentials with hard cores in the computational setting. Written in the standard form, this closure diverges with divergent potentials, which makes it impossible to numerically solve.
Compared to the PercusYevick closure, the HNC closure is a more accurate approximation of the full expression for the direct correlation function. Depsite this, it can produce inaccurate, long-range fluctuations that make it difficult to employ in phase-separating systems. The HNC closure performs well for systems where there is a disparity in site diameters and is typically used for the larger site.
References
- Hansen, J.P.; McDonald, I.R.; Theory of Simple Liquids; Chapter 4, Section 4; 4th Edition (2013), Elsevier [link]
Example
import pyPRISM sys = pyPRISM.System(['A','B']) sys.closure['A','A'] = pyPRISM.closure.PercusYevick() sys.closure['A','B'] = pyPRISM.closure.PercusYevick() sys.closure['B','B'] = pyPRISM.closure.HypernettedChain() # ** finish populating system object ** PRISM = sys.createPRISM() PRISM.solve()