Soft sphere potential

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The soft sphere potential is defined as

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_{12}\left( r \right) = \left\{ \begin{array}{lll} \epsilon \left( \frac{\sigma}{r}\right) ^n & ; & r \le \sigma \\ 0 & ; & r > \sigma \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 \Phi_{12}\left(r \right) } is the intermolecular pair potential between two soft spheres separated by a distance 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 r := |\mathbf{r}_1 - \mathbf{r}_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 \epsilon } is the interaction strength 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 \sigma } is the diameter of the sphere. Frequently the value of 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 n} is taken to be 12, thus the model effectively becomes the high temperature limit of the Lennard-Jones model [1]. If 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 n\rightarrow \infty} one has the hard sphere model. For 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 n \le 3} no thermodynamically stable phases are found.

Equation of state

The soft-sphere equation of state[2] has recently been studied by Tan, Schultz and Kofke[3] [4] and expressed in terms of Padé approximants. For 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 k_BT/\epsilon=1.0} 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 n=6} one has (Eq. 8):


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 Z_{n=6} = \frac{1 + 7.432255 \rho + 23.854807 \rho^2 + 40.330195 \rho^3 + 34.393896 \rho^4 + 10.723480 \rho^5}{1+ 3.720037 \rho + 4.493218 \rho^2 + 1.554135 \rho^3}}


and for 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 n=9} one has (Eq. 9):


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 Z_{n=9} = \frac{1 + 3.098829 \rho + 5.188915 \rho^2 + 5.019851 \rho^3 + 2.673385 \rho^4 + 0.601529 \rho^5}{1+ 0.262771 \rho + 0.168052 \rho^2 - 0.010554 \rho^3}}

Virial coefficients

Tan, Schultz and Kofke[3] have calculated the virial coefficients at 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 k_BT/\epsilon=1.0} (Table 1):

n=12 n=9 n=6
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 B_3} 3.79106644 4.27563423 5.55199919
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 B_4} 3.52761(6) 3.43029(7) 1.44261(4)
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 B_5} 2.1149(2) 1.08341(7) -1.68834(9)
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 B_6} 0.7695(2) -0.21449(11) 1.8935(5)
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 B_7} 0.0908(5) -0.0895(7) -1.700(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 B_8} -0.074(2) 0.071(4) 0.44(2)

Melting point

For 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 n=12}

pressure 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 \rho_{\mathrm {melting}}} 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 \rho_{\mathrm {freezing}}} Reference
22.66(1) 1.195(6) 1.152(6) Table 1 [5]
23.24(4) 1.2035(6) 1.1602(7) Table 2 [3]

For 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 n=9}

pressure 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 \rho_{\mathrm {melting}}} 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 \rho_{\mathrm {freezing}}} Reference
36.36(10) 1.4406(12) 1.4053(14) Table 3 [3]

For 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 n=6}

pressure 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 \rho_{\mathrm {melting}}} 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 \rho_{\mathrm {freezing}}} Reference
100.1(3) 2.320(2) 2.295(2) Table 4 [3]

Glass transition

[6][7]

Transport coefficients

[8]

Radial distribution function

[9]

References

  1. ↑ Jean-Pierre Hansen "Phase Transition of the Lennard-Jones System. II. High-Temperature Limit", Physical Review A 2 pp. 221-230 (1970)
  2. ↑ William G. Hoover, Marvin Ross, Keith W. Johnson, Douglas Henderson, John A. Barker and Bryan C. Brown "Soft-Sphere Equation of State", Journal of Chemical Physics 52 pp. 4931-4941 (1970)
  3. ↑ 3.0 3.1 3.2 3.3 3.4 Tai Boon Tan, Andrew J. Schultz and David A. Kofke "Virial coefficients, equation of state, and solid-fluid coexistence for the soft sphere model", Molecular Physics 109 pp. 123-132 (2011)
  4. ↑ N. S. Barlow, A. J. Schultz, S. J. Weinstein, and D. A. Kofke "An asymptotically consistent approximant method with application to soft- and hard-sphere fluids", Journal of Chemical Physics 137 204102 (2012)
  5. ↑ Nigel B. Wilding "Freezing parameters of soft spheres", Molecular Physics 107 pp. 295-299 (2009)
  6. ↑ D. M. Heyes, S. M. Clarke, and A. C. Brańka "Soft-sphere soft glasses", Journal of Chemical Physics 131 204506 (2009)
  7. ↑ Junko Habasaki and Akira Ueda "Several routes to the glassy states in the one component soft core system: Revisited by molecular dynamics", Journal of Chemical Physics 134 084505 (2011)
  8. ↑ D. M. Heyes and A. C. Branka "Density and pressure dependence of the equation of state and transport coefficients of soft-sphere fluids", Molecular Physics 107 pp. 309-319 (2009)
  9. ↑ A. C. Brańka and D. M. Heyes "Pair correlation function of soft-sphere fluids", Journal of Chemical Physics 134 064115 (2011)
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