Kern and Frenkel patchy model: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
m (partial save)
No edit summary
 
(23 intermediate revisions by 5 users not shown)
Line 1: Line 1:
The '''Kern and Frenkel''' <ref>[http://dx.doi.org/10.1063/1.1569473 Norbert Kern and Daan Frenkel "Fluid–fluid coexistence in colloidal systems with short-ranged strongly directional attraction", Journal of Chemical Physics 118, 9882 (2003)]</ref> [[Patchy particles |patchy model]] is an amalgamation of the [[hard sphere model]] with
The '''Kern and Frenkel''' <ref>[http://dx.doi.org/10.1063/1.1569473 Norbert Kern and Daan Frenkel "Fluid–fluid coexistence in colloidal systems with short-ranged strongly directional attraction", Journal of Chemical Physics 118, 9882 (2003)]</ref> [[Patchy particles |patchy model]] published in 2003 is an amalgamation of the [[hard sphere model]] with
attractive [[Square well model | square well]] patches (HSSW). The potential has an angular aspect, given by (Eq. 1)
attractive [[Square well model | square well]] patches (HSSW). The model was originally developed by Bol (1982),<ref>[http://dx.doi.org/10.1080/00268978200100461 W. Bol "Monte Carlo simulations of fluid systems of waterlike molecules", Molecular Physics '''45''' pp. 605-616 (1982)]</ref> and later Chapman (1988) <ref name="Chapman">[W.G. Chapman, Doctoral Thesis, Cornell University (1988)]</ref> <ref>[G. Jackson, W.G. Chapman, K.E. Gubbins, Molecular Physics 65, 1-31 (1988)]</ref> reinvented the model as the basis for numerous articles describing properties of associating particles from molecular simulation and theory. The computational advantage of Bol's model is that only a simple dot product is required to determine if a particle is in the bonding orientation.
The potential has an angular aspect, given by (Eq. 1)




:<math>\Phi_{ij}({\mathbf r}_{ij}; \tilde{{\mathbf \Omega}}_i, \tilde{{\mathbf \Omega}}_j)  =\Phi_{ij}^{ \mathrm{HSSW}}({\mathbf r}_{ij}) \cdot f(\tilde{{\mathbf \Omega}}_i, \tilde{{\mathbf \Omega}}_j) </math>
:<math>\Phi_{ij}({\mathbf r}_{ij}; \tilde{ {\mathbf \Omega}}_i, \tilde{ {\mathbf \Omega}}_j)  =\Phi_{ij}^{ \mathrm{HSSW}}({\mathbf r}_{ij}) \cdot f(\tilde{ {\mathbf \Omega}}_i, \tilde{ {\mathbf \Omega}}_j) </math>




Line 19: Line 20:


:<math>
:<math>
f_{ij} \left(\hat{ {\mathbf r}}_{ij}; \tilde{{\mathbf \Omega}}_i, \tilde{{\mathbf \Omega}}_j \right) =  
f_{ij} \left(\hat{ {\mathbf r}}_{ij}; \tilde{ {\mathbf \Omega}}_i, \tilde{ {\mathbf \Omega}}_j \right) =  
\left\{ \begin{array}{clc}
\left\{ \begin{array}{clc}
1        & \mathrm{if}        & \left\{ \begin{array}{ccc}    &  (\hat{e}_\alpha\cdot\hat{r}_{ij} \leq \cos \delta) & \mathrm{for~some~patch~\alpha~on~}i  \\  
1        & \mathrm{if}        & \left\{ \begin{array}{ccc}    &  (\hat{e}_\alpha\cdot\hat{r}_{ij} \geq \cos \delta) & \mathrm{for~some~patch~\alpha~on~}i  \\  
                                                             \mathrm{and} & (\hat{e}_\beta\cdot\hat{r}_{ji} \leq \cos \delta)  & \mathrm{for~some~patch~\beta~on~}j  \end{array} \right. \\
                                                             \mathrm{and} & (\hat{e}_\beta\cdot\hat{r}_{ji} \geq \cos \delta)  & \mathrm{for~some~patch~\beta~on~}j  \end{array} \right. \\
0        & \mathrm{otherwise} &  \end{array} \right.
0        & \mathrm{otherwise} &  \end{array} \right.
</math>
</math>


where <math>\delta</math> is the solid angle of a patch (<math>\alpha, \beta, ...</math>) whose axis is <math>\hat{e}</math> (see Fig. 1 of Ref. 1), forming a conical segment.
==Multiple patches==
The "two-patch" and "four-patch" Bol (Chapman or Kern and Frenkel) model was extensively studied by Chapman and co-workers for bulk and interfacial systems using hard sphere and Lennard-Jones reference systems.  Later other groups, including Sciortino and co-workers, considered stronger association energies for the "two-patch" hard sphere reference <ref name="bianchi">[http://dx.doi.org/10.1063/1.2730797  F. Sciortino, E. Bianchi, J. Douglas and P. Tartaglia "Self-assembly of patchy particles into polymer chains: A parameter-free comparison between Wertheim theory and Monte Carlo simulation", Journal of Chemical Physics '''126''' 194903 (2007)]</ref><ref>[http://dx.doi.org/10.1063/1.3415490 Achille Giacometti, Fred Lado, Julio Largo, Giorgio Pastore, and Francesco Sciortino "Effects of patch size and number within a simple model of patchy colloids", Journal of Chemical Physics 132, 174110 (2010)]</ref><ref name="rovigatti">[http://dx.doi.org/10.1063/1.4737930  José Maria Tavares, Lorenzo Rovigatti, and Francesco Sciortino "Quantitative description of the self-assembly of patchy particles into chains and rings", Journal of Chemical Physics '''137''' 044901 (2012)]</ref>.
==Four patches==
:''Main article: [[Anisotropic particles with tetrahedral symmetry]]''
==Single-bond-per-patch-condition==
If the two parameters <math>\delta</math> and <math>\lambda</math> fullfil the condition
:<math>
\sin{\delta} \leq \dfrac{1}{2(1+\lambda\sigma)}
</math>
then the patch cannot be involved in more than one bond. Enforcing this condition makes it possible to compare the simulations results with [[Wertheim's first order thermodynamic perturbation theory (TPT1)| Wertheim theory]] <ref name="Chapman"/><ref name="bianchi"/><ref name="rovigatti"/>
==Hard ellipsoid model==
The [[hard ellipsoid model]] has also been used as the 'nucleus' of the Kern and Frenkel patchy model <ref>[http://dx.doi.org/10.1063/1.4969074  T. N. Carpency, J. D. Gunton and J. M. Rickman "Phase behavior of patchy spheroidal fluids", Journal of Chemical Physics '''145''' 214904 (2016)]</ref>.
==References==
==References==
<references/>
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.3689308 Christoph Gögelein, Flavio Romano, Francesco Sciortino, and Achille Giacometti "Fluid-fluid and fluid-solid transitions in the Kern-Frenkel model from Barker-Henderson thermodynamic perturbation theory", Journal of Chemical Physics '''136''' 094512 (2012)]
*[http://dx.doi.org/10.1063/1.4722477 Emanuela Bianchi, Günther Doppelbauer, Laura Filion, Marjolein Dijkstra, and Gerhard Kahl "Predicting patchy particle crystals: Variable box shape simulations and evolutionary algorithms", Journal of Chemical Physics '''136''' 214102 (2012)]
*[http://dx.doi.org/10.1063/1.4960423  Z. Preisler, T. Vissers, F. Smallenburg and F. Sciortino "Crystals of Janus colloids at various interaction ranges", Journal of Chemical Physics '''145''' 064513 (2016)]


[[category: models]]
[[category: models]]

Latest revision as of 01:41, 21 September 2023

The Kern and Frenkel [1] patchy model published in 2003 is an amalgamation of the hard sphere model with attractive square well patches (HSSW). The model was originally developed by Bol (1982),[2] and later Chapman (1988) [3] [4] reinvented the model as the basis for numerous articles describing properties of associating particles from molecular simulation and theory. The computational advantage of Bol's model is that only a simple dot product is required to determine if a particle is in the bonding orientation. 

The potential has an angular aspect, given by (Eq. 1)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{ij}({\mathbf r}_{ij}; \tilde{ {\mathbf \Omega}}_i, \tilde{ {\mathbf \Omega}}_j) =\Phi_{ij}^{ \mathrm{HSSW}}({\mathbf r}_{ij}) \cdot f(\tilde{ {\mathbf \Omega}}_i, \tilde{ {\mathbf \Omega}}_j) }


where the radial component is given by the square well model (Eq. 2)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{ij}^{ \mathrm{HSSW}} \left({\mathbf r}_{ij} \right) = \left\{ \begin{array}{ccc} \infty & ; & r < \sigma \\ - \epsilon & ; &\sigma \le r < \lambda \sigma \\ 0 & ; & r \ge \lambda \sigma \end{array} \right. }

and the orientational component is given by (Eq. 3)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{ij} \left(\hat{ {\mathbf r}}_{ij}; \tilde{ {\mathbf \Omega}}_i, \tilde{ {\mathbf \Omega}}_j \right) = \left\{ \begin{array}{clc} 1 & \mathrm{if} & \left\{ \begin{array}{ccc} & (\hat{e}_\alpha\cdot\hat{r}_{ij} \geq \cos \delta) & \mathrm{for~some~patch~\alpha~on~}i \\ \mathrm{and} & (\hat{e}_\beta\cdot\hat{r}_{ji} \geq \cos \delta) & \mathrm{for~some~patch~\beta~on~}j \end{array} \right. \\ 0 & \mathrm{otherwise} & \end{array} \right. }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} is the solid angle of a patch (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha, \beta, ...} ) whose axis is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{e}} (see Fig. 1 of Ref. 1), forming a conical segment.

Multiple patches[edit]

The "two-patch" and "four-patch" Bol (Chapman or Kern and Frenkel) model was extensively studied by Chapman and co-workers for bulk and interfacial systems using hard sphere and Lennard-Jones reference systems. Later other groups, including Sciortino and co-workers, considered stronger association energies for the "two-patch" hard sphere reference [5][6][7].

Four patches[edit]

Main article: Anisotropic particles with tetrahedral symmetry

Single-bond-per-patch-condition[edit]

If the two parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} fullfil the condition

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{\delta} \leq \dfrac{1}{2(1+\lambda\sigma)} }

then the patch cannot be involved in more than one bond. Enforcing this condition makes it possible to compare the simulations results with Wertheim theory [3][5][7]

Hard ellipsoid model[edit]

The hard ellipsoid model has also been used as the 'nucleus' of the Kern and Frenkel patchy model [8].

References[edit]

  1. ↑ Norbert Kern and Daan Frenkel "Fluid–fluid coexistence in colloidal systems with short-ranged strongly directional attraction", Journal of Chemical Physics 118, 9882 (2003)
  2. ↑ W. Bol "Monte Carlo simulations of fluid systems of waterlike molecules", Molecular Physics 45 pp. 605-616 (1982)
  3. ↑ 3.0 3.1 [W.G. Chapman, Doctoral Thesis, Cornell University (1988)]
  4. ↑ [G. Jackson, W.G. Chapman, K.E. Gubbins, Molecular Physics 65, 1-31 (1988)]
  5. ↑ 5.0 5.1 F. Sciortino, E. Bianchi, J. Douglas and P. Tartaglia "Self-assembly of patchy particles into polymer chains: A parameter-free comparison between Wertheim theory and Monte Carlo simulation", Journal of Chemical Physics 126 194903 (2007)
  6. ↑ Achille Giacometti, Fred Lado, Julio Largo, Giorgio Pastore, and Francesco Sciortino "Effects of patch size and number within a simple model of patchy colloids", Journal of Chemical Physics 132, 174110 (2010)
  7. ↑ 7.0 7.1 José Maria Tavares, Lorenzo Rovigatti, and Francesco Sciortino "Quantitative description of the self-assembly of patchy particles into chains and rings", Journal of Chemical Physics 137 044901 (2012)
  8. ↑ T. N. Carpency, J. D. Gunton and J. M. Rickman "Phase behavior of patchy spheroidal fluids", Journal of Chemical Physics 145 214904 (2016)
Related reading
Retrieved from "http://www.sklogwiki.org/SklogWiki/index.php?title=Kern_and_Frenkel_patchy_model&oldid=20617"