Soft sphere potential: Difference between revisions

Revision as of 12:12, 19 January 2011

The soft sphere potential is defined as

\Phi _{12}\left(r\right)=\left\{{\begin{array}{lll}\epsilon \left({\frac {\sigma }{r}}\right)^{n}&;&r\leq \sigma \\0&;&r>\sigma \end{array}}\right.

where $\Phi _{12}\left(r\right)$ is the intermolecular pair potential between two soft spheres separated by a distance $r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|$ , $\epsilon$ is the interaction strength and $\sigma$ is the diameter of the sphere. Frequently the value of $n$ 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] 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):

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
$B_{4}$	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 $n=12$

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 ^[4]
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}}}	$\rho _{\mathrm {freezing} }$	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

^[5]

Transport coefficients

^[6]

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)

[Tan-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] Nigel B. Wilding "Freezing parameters of soft spheres", Molecular Physics 107 pp. 295-299 (2009)

[5] D. M. Heyes, S. M. Clarke, and A. C. Brańka "Soft-sphere soft glasses", Journal of Chemical Physics 131 204506 (2009)

[6] 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)

[1]

[2]

[3]

[4]

[5]

[6]

@@ Line 20: / Line 20: @@
 :<math>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}</math>
 ==Virial coefficients==
-Tan, Schultz and Kofke<ref name="Tan">[ </ref> have calculated the [[Virial equation of state | virial coefficients]] at <math>k_BT/\epsilon=1.0</math> (Table 1):
+Tan, Schultz and Kofke<ref name="Tan"> </ref> have calculated the [[Virial equation of state | virial coefficients]] at <math>k_BT/\epsilon=1.0</math> (Table 1):
 :{| border="1"
@@ Line 38: / Line 38: @@
 | <math>B_8</math>  || -0.074(2)  || 0.071(4) || 0.44(2)
 |}
-==Solid phase==
+==Melting point==
-<ref>[http://dx.doi.org/10.1080/00268970802603507 Nigel B. Wilding "Freezing parameters of soft spheres", Molecular Physics '''107''' pp. 295-299 (2009)]</ref>
+For <math>n=12</math>
+:{| border="1"
+|-
+| pressure || <math>\rho_{\mathrm {melting}}</math> || <math>\rho_{\mathrm {freezing}}</math>  || Reference
+|-
+| 22.66(1) || 1.195(6) || 1.152(6) || Table 1 <ref>[http://dx.doi.org/10.1080/00268970802603507 Nigel B. Wilding "Freezing parameters of soft spheres", Molecular Physics '''107''' pp. 295-299 (2009)]</ref>
+|-
+| 23.24(4) || 1.2035(6) || 1.1602(7) || Table 2 <ref name="Tan"> </ref>
+|}
+For <math>n=9</math>
+:{| border="1"
+|-
+| pressure || <math>\rho_{\mathrm {melting}}</math> || <math>\rho_{\mathrm {freezing}}</math>  || Reference
+|-
+| 36.36(10) || 1.4406(12) || 1.4053(14) || Table 3 <ref name="Tan"> </ref>
+|}
+For <math>n=6</math>
+:{| border="1"
+|-
+| pressure || <math>\rho_{\mathrm {melting}}</math> || <math>\rho_{\mathrm {freezing}}</math>  || Reference
+|-
+| 100.1(3) || 2.320(2) || 2.295(2) || Table 4 <ref name="Tan"> </ref>
+|}
 ==Glass transition==
 <ref>[http://dx.doi.org/10.1063/1.3266845 D. M. Heyes, S. M. Clarke, and A. C. Brańka "Soft-sphere soft glasses", Journal of Chemical Physics '''131''' 204506 (2009)]</ref>

Soft sphere potential: Difference between revisions

Revision as of 12:12, 19 January 2011

Contents

Equation of state

Virial coefficients

Melting point

Glass transition

Transport coefficients

References

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Soft sphere potential: Difference between revisions

Revision as of 12:12, 19 January 2011

Equation of state

Virial coefficients

Melting point

Glass transition

Transport coefficients

References

Navigation menu

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