Editing Rotational relaxation
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'''Rotational relaxation''' refers to the decay of certain [[autocorrelation]] | '''Rotational relaxation''' refers to the decay of certain [[autocorrelation]] | ||
magnitudes related to the orientation of molecules. | magnitudes related to the orientation of molecules. | ||
If a molecule has an orientation along a unit vector | |||
If a molecule has an orientation along a unit vector '''n''', its autocorrelation | |||
will be given by | will be given by | ||
:<math>c_1(t)=\langle \mathbf{n}(0)\cdot\mathbf{n}(t) \rangle.</math> | :<math>c_1(t)=\langle \mathbf{n}(0)\cdot\mathbf{n}(t) \rangle.</math> | ||
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a characteristic relaxation time (either from the long-time exponential decay, or | a characteristic relaxation time (either from the long-time exponential decay, or | ||
from its total integral, see [[autocorrelation]]). This magnitude, which | from its total integral, see [[autocorrelation]]). This magnitude, which | ||
is readily computed in a [[ | is readily computed in a [[simulation]] is not directly accessible experimentally, | ||
however. Rather, relaxation times of the second | however. Rather, relaxation times of the second | ||
[[spherical harmonics|spherical harmonic]] are obtained: | [[spherical harmonics|spherical harmonic]] are obtained: | ||
:<math> | :<math>c_1(t)=\langle P_2( \mathbf{n}(0)\cdot\mathbf{n}(t) ) \rangle,</math> | ||
where <math>P_2(x)</math> is the second [[Legendre polynomials|Legendre polynomial]]. | where <math>P_2(x)</math> is the second [[Legendre polynomials|Legendre polynomial]]. | ||
According to simple [[rotational diffusion]] theory, the relaxation time | According to simple [[rotational diffusion]] theory, the relaxation time | ||
for <math>c_1(t)</math> would be given by | for <math>c_1(t)</math> would be given by | ||
<math>\tau_1 = | <math>\tau_1 = 1/2D_\mathrm{rot}</math>, and the relaxation time for | ||
<math>c_2(t)</math> would be <math>\tau_2 = | <math>c_2(t)</math> would be <math>\tau_2 = 1/6D_\mathrm{rot}</math>. | ||
Therefore, <math>\tau_1= 3 \tau_2</math>. This ratio is actually lower in simulations, | Therefore, <math>\tau_1= 3 \tau_2</math>. This ratio is actually lower in simulations, | ||
and closer to <math>2</math>; the departure from a value of 3 signals rotation | and closer to <math>2</math>; the departure from a value of 3 signals rotation | ||
processes "rougher" than what is assumed in simple [[rotational diffusion]] (Ref 1). | processes "rougher" than what is assumed in simple [[rotational diffusion]] (Ref 1). | ||
==Water== | ==Water== | ||
Often, molecules are more complex geometrically and can not be described by a single | Often, molecules are more complex geometrically and can not be described by a single | ||
orientation. In this case, several vectors should be considered, each with its own | orientation. In this case, several vectors should be considered, each with its own | ||
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| HH || H-H axis || <math>\tau_2=2.0</math>ps (H-H dipolar relaxation NMR) | | HH || H-H axis || <math>\tau_2=2.0</math>ps (H-H dipolar relaxation NMR) | ||
|- | |- | ||
| OH || O-H axis || <math>\tau_2=1.95</math>ps (< | | OH || O-H axis || <math>\tau_2=1.95</math>ps (<math>^{17}</math>O-H dipolar relaxation NMR) | ||
|- | |- | ||
| <math>\mu</math> || dipolar axis || not measurable, but related to bulk dielectric relaxation | | <math>\mu</math> || dipolar axis || not measurable, but related to bulk dielectric relaxation | ||
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|- | |- | ||
|} | |} | ||
==References== | |||
*[http://dx.doi.org/10.1063/1.476482 David van der Spoel, Paul J. van Maaren, and Herman J. C. Berendsen "A systematic study of water models for molecular simulation: Derivation of water models optimized for use with a reaction field", J. Chem. Phys. '''108''' 10220 (1998)] | |||
==See also== | ==See also== | ||
*[[ | |||
*[[ | *[[rotational diffusion]] | ||
*[[ | *[[diffusion]] | ||
*[[autocorrelation]] | |||
[[Category: Non-equilibrium thermodynamics]] | [[Category: Non-equilibrium thermodynamics]] |