Kern and Frenkel patchy model: Difference between revisions

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:<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>




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:<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>
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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.
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.
==Two patches==
==Two patches==
The "two-patch" Kern and Frenkel model has been extensively studied by  Giacometti et al. <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>.
The "two-patch" Kern and Frenkel model has been extensively studied by Sciortino and co-workers <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==
==Four patches==
:''Main article: [[Anisotropic particles with tetrahedral symmetry]]''
:''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="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
;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/10.1063/1.3689308 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]]

Revision as of 22:03, 25 May 2021

The Kern and Frenkel [1] patchy model is an amalgamation of the hard sphere model with attractive square well patches (HSSW). The potential has an angular aspect, given by (Eq. 1)



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

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

where is the solid angle of a patch () whose axis is (see Fig. 1 of Ref. 1), forming a conical segment.

Two patches

The "two-patch" Kern and Frenkel model has been extensively studied by Sciortino and co-workers [2][3][4].

Four patches

Main article: Anisotropic particles with tetrahedral symmetry

Single-bond-per-patch-condition

If the two parameters and fullfil the condition

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 [2][4]

Hard ellipsoid model

The hard ellipsoid model has also been used as the 'nucleus' of the Kern and Frenkel patchy model [5].

References

Related reading