Water molecule bond length - Revision history

Mol: /* Temperature/ Initial distribution */

2020-10-12T20:12:03Z

Temperature/ Initial distribution

← Older revision		Revision as of 23:12, 12 October 2020
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	\exp\left[- \frac12 \frac{mv^2}{k_B T} \right]		\exp\left[- \frac12 \frac{mv^2}{k_B T} \right]
	</math>		</math>

			The mean speed (for 3d?) is

			<math>
			v_\text{mean} = \sqrt{\frac{8k_B T}{\pi m}}
			</math>

			where <math>k_B</math> is Boltzmann constant, here <math>k_B = 1</math>.

	== Results ==		== Results ==

2020-10-12T20:10:33Z

Temperature/ Initial distribution

← Older revision		Revision as of 23:10, 12 October 2020
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	p(v) = \left( \frac{m}{2\pi k_B T} \right)^{1/2}		p(v) = \left( \frac{m}{2\pi k_B T} \right)^{1/2}
	\exp\left[- \frac12 \frac{mv^2}{k_B T} \right]		\exp\left[- \frac12 \frac{mv^2}{k_B T} \right]
	~~</math>~~

	~~The mean speed is (for 3D?)~~

	~~<math>~~
	~~v_\text{mean} = sqrt{ \frac{8k_B T}{\pi m}}~~
	</math>		</math>

2020-10-12T20:10:24Z

Temperature/ Initial distribution

← Older revision		Revision as of 23:10, 12 October 2020
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	p(v) = \left( \frac{m}{2\pi k_B T} \right)^{1/2}		p(v) = \left( \frac{m}{2\pi k_B T} \right)^{1/2}
	\exp\left[- \frac12 \frac{mv^2}{k_B T} \right]		\exp\left[- \frac12 \frac{mv^2}{k_B T} \right]
			</math>

			The mean speed is (for 3D?)

			<math>
			v_\text{mean} = sqrt{ \frac{8k_B T}{\pi m}}
	</math>		</math>

2020-10-12T20:06:21Z

← Older revision		Revision as of 23:06, 12 October 2020
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	==== Velocity Verlet Algorithm ====		=== Velocity Verlet Algorithm ===

	A very good and easy to implement integration method is velocity Verlet:		A very good and easy to implement integration method is velocity Verlet:

2020-10-12T20:05:36Z

← Older revision		Revision as of 23:05, 12 October 2020
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	1D statement		1D statement


			== References ==

			D.T.W. Lin and C.-K. Chen: A molecular dynamics simulation of TIP4P and Lennard-Jones water in nanochannel, acta Mechanica 173, 181.194 (2004).

2020-10-12T20:04:31Z

Velocity Verlet Algorithm

← Older revision		Revision as of 23:04, 12 October 2020
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	</math>		</math>

	where <math>a(t) = - \frac1m \frac{d}{dx}V(x)</math> is given at Section . . .		where <math>a(t) = a(x(t)) = - \frac1m \frac{d}{dx}V(x(t))</math> is given at Section . . .

	=== Temperature/ Initial distribution ===		=== Temperature/ Initial distribution ===

2020-10-12T20:03:34Z

Velocity Verlet Algorithm

← Older revision		Revision as of 23:03, 12 October 2020
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	\end{align}		\end{align}
	</math>		</math>

			where <math>a(t) = - \frac1m \frac{d}{dx}V(x)</math> is given at Section . . .

	=== Temperature/ Initial distribution ===		=== Temperature/ Initial distribution ===

2020-10-12T20:01:48Z

Newton Equations

← Older revision		Revision as of 23:01, 12 October 2020
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	=== Newton Equations ===		=== Newton Equations ===
	<math>F = ma</math> where <math>F = - \nabla V</math>. The system is 1D, thus gradient will be differential, and <math>F= - \frac{d V}{dx}</math>.		<math>F = ma</math> where <math>F = - \nabla V</math>. The system is 1D, thus gradient will be differential, and <math>F= - \frac{d V}{dx}</math>.

			The mass <math>m</math> should be the [https://en.wikipedia.org/wiki/Reduced_mass reduced mass] of the oxygen--hydrogen system, but we use mass <math>m=1</math> here.

	=== Potential Function ===		=== Potential Function ===

2020-10-12T20:00:41Z

Potential Function

← Older revision		Revision as of 23:00, 12 October 2020
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	=== Potential Function ===		=== Potential Function ===

	[https://en.wikipedia.org/wiki/Lennard-Jones_potential Lennard--Jones potential] with parameters for [https://en.wikipedia.org/wiki/Water_model TIPS] model:		[https://en.wikipedia.org/wiki/Lennard-Jones_potential Lennard--Jones potential] with dimensionless parameters for [https://en.wikipedia.org/wiki/Water_model TIPS] model:

	<math>		<math>
	V(x) = 4~~\epsilon~~ \left[ \left( \~~frac\sigma x~~\right)^{12} - \left( \~~frac \sigma~~ x\right)^6 \right]		V(x) = 4 \left[ \left( \frac1x\right)^{12} - \left( \frac1 x\right)^6 \right]
	</math>		</math>

			where the distance <math>x</math> is given in Ångstroms.
	~~<math>~~
	~~\begin{align}~~
	~~A &= 580.0 \times 10^3 kcal A^{12}/mol \\~~
	~~B &= 525.0 kcal A^6/mol~~
	~~\end{align}~~
	~~</math>~~
	where <math>A</math> is ~~Ångströms~~.

	=== Integration ===		=== Integration ===

2020-10-12T19:54:17Z

Potential Function

← Older revision		Revision as of 22:54, 12 October 2020
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	<math>		<math>
	V(x) = 4\epsilon \left[ \left( \frac\sigma r\right)^{12} - \left( \frac \sigma r\right)^6 \right]		V(x) = 4\epsilon \left[ \left( \frac\sigma x\right)^{12} - \left( \frac \sigma x\right)^6 \right]
	</math>		</math>

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	</math>		</math>
	where <math>A</math> is Ångströms.		where <math>A</math> is Ångströms.



	=== Integration ===		=== Integration ===