Rotational relaxation: Difference between revisions

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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>c_1(t)=\langle P_2(  \mathbf{n}(0)\cdot\mathbf{n}(t) ) \rangle,</math>
:<math>c_2(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 = 1/2D_\mathrm{rot}</math>, and the relaxation time for
<math>\tau_1 = \frac{1}{2D_\mathrm{rot}}</math>, and the relaxation time for
<math>c_2(t)</math> would be <math>\tau_2 = 1/6D_\mathrm{rot}</math>.
<math>c_2(t)</math> would be <math>\tau_2 = \frac{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
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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 (<math>^{17}</math>O-H dipolar relaxation NMR)
| OH  || O-H axis  ||  <math>\tau_2=1.95</math>ps (<sup>17</sup>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

Latest revision as of 07:23, 21 October 2016

Rotational relaxation refers to the decay of certain autocorrelation magnitudes related to the orientation of molecules. If a molecule has an orientation along a unit vector , its autocorrelation will be given by

From the time decay, or relaxation, of this function, one may extract a characteristic relaxation time (either from the long-time exponential decay, or from its total integral, see autocorrelation). This magnitude, which is readily computed in a simulation is not directly accessible experimentally, however. Rather, relaxation times of the second spherical harmonic are obtained:

where is the second Legendre polynomial.

According to simple rotational diffusion theory, the relaxation time for would be given by , and the relaxation time for would be . Therefore, . This ratio is actually lower in simulations, and closer to ; the departure from a value of 3 signals rotation processes "rougher" than what is assumed in simple rotational diffusion (Ref 1).

Water[edit]

Main article Rotational relaxation of water

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 autocorrelation. E.g., typical choices for water molecules would be:

symbol explanation experimental value, and method
HH H-H axis ps (H-H dipolar relaxation NMR)
OH O-H axis ps (17O-H dipolar relaxation NMR)
dipolar axis not measurable, but related to bulk dielectric relaxation
normal to the molecule plane not measurable

See also[edit]

References[edit]

  1. 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)