pyPRISM.calculate.chi module

pyPRISM.calculate.chi.chi(PRISM, extrapolate=True)[source]

Calculate the effective interaction parameter, \(\chi\)

Parameters:
  • PRISM (pyPRISM.core.PRISM) – A solved PRISM object.
  • extrapolate (bool, optional) – If True, only return the chi value extrapolated to \(k=0\) rather than returning \(\chi(k)\)
Returns:

chi – PairTable of all \(\chi(k)\) or \(\chi(k=0)\) values
Return type:

pyPRISM.core.PairTable

Mathematical Definition

\[\hat{\chi}_{\alpha,\beta}(k) = \frac{0.5 \rho}{R^{+0.5} \phi_{\alpha} + R^{-0.5} \phi_{\beta}} (R^{-1} \hat{C}_{\alpha,\alpha}(k) + R \hat{C}_{\beta,\beta}(k) - 2 \hat{C}_{\alpha,\beta}(k))\]
\[R = v_{\alpha}/v_{\beta}\]

Variable Definitions

  • \(\hat{\chi}_{\alpha,\beta}(k)\)

    Wavenumber dependent effective interaction parameter between site types \(\alpha\) and \(\beta\)

  • \(\rho\)

    Total system density from the pyPRISM.core.Density instance stored in the system object (which is stored in the PRISM object)

  • \(\phi_{\alpha},\phi_{\beta}\)

    Volume fraction of site types \(\alpha\) and \(\beta\).

    \[\phi_{\alpha} = \frac{\rho_{\alpha}}{\rho_{\alpha} + \rho_{\beta}}\]
  • \(v_{\alpha},v_{\beta}\)

    Volume of site type \(\alpha\) and \(\beta\)

Description

\(\hat{\chi}_{\alpha,\beta}(k)\) describes the overall effective interactions between site types \(\alpha\) and \(\beta\) as a single number. While there are many different definitions of \(\chi\), this is an effective version that takes into account both entropic and enthalpic interactions. In this way, this \(\chi\) is similar to a second virial coefficient. In terms of value, \(\chi<0\) indicates effective attraction and \(\chi>0\) effective repulsion.

As most theories do not take into account the (potentially contentious [2,3]) wavenumber dependence of \(\chi\), the zero-wavenumber extrapolation is often used when reporting PRISM-based \(\chi\) values. For convenience, the full wavenumber dependent curve can be requested, but only the \(k=0\) values are returned by default.

Warning

Using standard atomic closures (e.g, PY, HNC, MSA), PRISM theory may not predict the correct scaling of spinodal temperatures for phase separating systems. While this issue is mitigated by using molecular closures,[3] these closures are not currently implemented in pyPRISM. For more information, this issue is referenced in the pyPRISM paper.[5]. We urge users to do their due diligence in understanding how well these closures and PRISM theory perform for their systems of interest.

Warning

The \(\chi\) calculation is only valid for multicomponent systems i.e. systems with more than one defined type. This method will throw an exception if passed a 1-component PRISM object.

Warning

This calculation is only rigorously defined in the two-component case. With that said, pyPRISM allows this method to be called for multicomponent systems in order to calculate pairwise \(\chi\) values. We urge caution when using this method for multicomponent systems as it is not clear if this approach is fully rigorous.

Warning

Passing an unsolved PRISM object to this function will still produce output based on the default values of the attributes of the PRISM object.

References

  1. Schweizer, Curro, Thermodynamics of Polymer Blends, J. Chem. Phys., 1989 91 (8) 5059, DOI: 10.1063/1.457598 [link]
  2. Zirkel, A., et al., Small-angle neutron scattering investigation of the Q-dependence of the Flory-Huggins interaction parameter in a binary polymer blend. Macromolecules, 2002. 35(19): p. 7375-7386. [link]
  3. Cabral, Higgins, Small Angle Neutron Scattering from the Highly Interacting Polymer Mixture TMPC/PSd: No Evidence of Spatially Dependent chi Parameter. Macromolecules, 2009. 42(24): p. 9528-9536. [link]
  4. Schweizer, K.S. and A. Yethiraj, POLYMER REFERENCE INTERACTION SITE MODEL-THEORY - NEW MOLECULAR CLOSURES FOR PHASE-SEPARATING FLUIDS AND ALLOYS. Journal of Chemical Physics, 1993. 98(11): p. 9053-9079. [link]
  5. Martin, T.B.; Gartner, T.E. III; Jones, R.L.; Snyder, C.R.; Jayaraman, A.; pyPRISM: A Computational Tool for Liquid State Theory Calculations of Macromolecular Materials, Macromolecules, 2018, 51 (8), p2906-2922 [link]

Example

import pyPRISM

sys = pyPRISM.System(['A','B'])

# ** populate system variables **

PRISM = sys.createPRISM()

PRISM.solve()

chi = pyPRISM.calculate.chi(PRISM)

chi_AB = chi['A','B']
chi_AA = chi['A','A'] #returns None because self-chi values are not defined