- Email: [email protected]
Brownian motion description of activation energies from NMR-relaxation times for rotating molecular-groups
9
Chemical Physics 56 (19Sl) 9-14 North-Holland PubMing Company
BROWNTANMOTION DESCRIF’TION(JF ACTIVATION ENERGIES FROM NMR-RELAXATION TIMES FOR ROTATING MOLECULAR-GROUPS 0. EDHOLM and C. BLOMBERG
Department of Theoretical Received 23 September
P/~yrics,Royal Institute of Technology, S-100 44 Stockholm 70. Sweden
1980
A general forrnuia is derived (containing eigenvalues and eigenfunctions of some diffusion q~.tion) for activation energies from NM%relaation times for rotding molecular groups.Numerical calculations for a trigonometric three well potential in the strong &ping (Smolwhowski) limit show that there may be a considerable differencr: b-ctween activation energy ad barrier height In the low damping limit there is fai11y good agreement.
1. Introduction
Rotating molecular groups, for instance methyl groups in a larger molecule, are subject to hindrances from the rest of the molecule which is described by a potential barrier. The height of this barrier can be derived from different experiments, perhaps most directly by using sptittings in the microwave spectrum. For an extensive discussion of internal rotation in molecules, see e.g. ref. [l]. Information about the barrier height can also be gained through dynamic experiments as e.g. nuclear magnetic resonance (NhlR). The NMR-relaxation times give information about the rate of reorientation, which of course becomes smaller the higher the potential barrier is. Knowing the rate of reorientation of the free rotor, the barrier height could be estimated, see e.g. ref. 121.Such an estimate is, however, rather uncertain, and a more suitable way is to do the measurements at different temperatures and fit the reorientation rates, R, to an Arrhenius type expression [3]: R = A &p(-E_JkB
T).
(1)
tial factor, some of which have been discussed by Brot [4]. Kowalewaki aud Liijefors [S] have shoRn, that the absolute rate theory gives an activation energy, that is very close to the barrier height, and a preexponential factor that is almost constant with temperature. In this wcrk, the starting point wih be taken by the classical equations for brownian rotation in a potential, see e.g. ref. [6]. First, however, we give in section 2 following Woessner [7] the appropriate formula relating N?NRrelaxation time and orientational correlation functions As a second step, general formulae for these relations and for the activation energy, expressed in eigenvalues and eigenfunctions of some diffusion equation, are given. We take the Fokker-Planck equation as fundamental, but since it would be very cumbersome to furnish a general (numerical) solution, two limiting cases are considered. The strong damping (diffusive) limit, in which the diffusive motion dominates over the deterministic oscillations, is described by the Smoluchowski equation. This case is discussed in section 3. The iow damping (inertial) limit, in which one has to wait for several deten&istic oscillaticns ‘before any appreciable effect of the diff&ive motion is seen, is discussed-in section 4. In the summary (section 5) a brief discussion of the applicability of the two limiting cases is given.
Here, the preexponenti factor,d. and the activation energy,EA, are adjustable.Parameten, both being phenomeno~ogical and having no immediate physical interpretation~.One therefore wants to relate these $$rym+s tq ynti&s such as potential barrier height..*and rotational diffusionconstanf A number of : expressrons L have be& proposed for the preexponen-_; . . ~~‘301~&01Q4~81/0000~~oooo/S 6250 @‘North-HoUarrd~PubJishingCompany . : i .. -_-. :.. 1. : .. _.. .... .‘.’ ..’,_.. (.., .-. _-.: ._:_ ~ .,.‘. : : ..-: _ : :
0. Edholm, C. Biomberg / Activation energies for rotating moleculargroup~
10
2. NMR-relaxation
obtains:
times and activation energies
1= $r-r
Tine NMR-relaxation times are determined by the spectral density of correlation functions of spherical harmonics Y,#(t), r(t)):
X {A +B(exp[i(@
exp(-6Do)
- @c)] ) + C
(6) with
X exp(iwr) dr.
A = $(3 cos*A - 1)2,
(2)
B = acti 2Aj2,
C=@~IIA)~.
The angles fl and y detemine the orientation of the vector between two interacting nuciei and the magnetic field. For dipolar relaxation between two un3ike nuclei, the longitudinal relaxation time, T,, is given by:
(7)
The averages may be calcu!ated knowing the probability distribution for the angle Q as a function of time. This is governed by some stochastic equation, e.g., a Fokker-Planck equation (see e.g. ref. [6]):
where the probability P(&, w, t) is a function of angle @, angular velocity w and time t. U(q) is the potential giving the deterministic part of the motion, fl is the angular friction coefficient and I is the moment of inertia. in the strong damping case this simplifies to the Smoluchowski equation:
See, for instance ref. [S], although his normalization is somewhat different giving different coefficients. yr and ys are the gyromagnetic ratios of the involved nuclei. S is the spin of the interacting nuclei, r the distance and wI and os the I-armor frequencies. Following Woessner [7], the angles A and Cp describing the orientation of tht- internuclear axis with respect to the axis about which it rotates, and the angles 0 and 0’ describing the orientation of this latter axis with respect to the laboratory coordinate system are introduced. The result of Woessner [7] is most simply achieved uring the transformation properties of the spherical harmonics under rotation:
ap
-CD
at
( )I*
a2P I a --pau _+_[aa .2,2-a@ a*
(9)
with the diffusion constant D being kBT/Zfi and the probability P(q, t) corresponding to that of eq. (8) integrated over all angular velocities. Averages as those on the rhs of eq. (6) have earlier been calculated from ?he Smoluchowski equation for a rotor in a two-well potential by Dianoux and Voko 191, aiming at understanding neutron diffraction against liquid crystal molecules. Lassier and Brot [lo] have simulated various correlation functions for such a rotor. Edholm and Blomberg [I l] have done caktiations for a three-well [email protected] aiming at interpreting NMR relaxation in hydrocarbon chains. We wilI follow the procedure of ref.. [Ill. Splitting off a time dependent fact& eyU from eq. (8) or (9). an eigenvahre’ problem _$formed by the: eqtition together wit! G’per&idic boundary cticdi- .‘. tion. W&denote’the ei&$&r& & and the.eigenfimc-. ti~nsfs(&) (i;l-~~e_F‘okker~-~anckcask‘f,(~)-~e ihe: .eige+rnctions titeg+$~o~er+I &g&r v+ities). Since in g&er~;th.% equatio~‘~enot-self-ahjdiiit,.: :
(Y;i(t + 7) Yzi(t))
-where the two rotations have been assumed to be ‘. independent of one another. The subscript 0 denotes time zero and @$9, @‘)are second order rotation matrix elements__The first average i&now easiljr calculated assuming isotropic ;eodent.ation (Da is ,$I&rota-. tion& $fusion co&ant for the:wh&z molecule): 1
.,
the ~igenfunctiom~,(~)~of__. the.adjoint eqtitiotirare:’ .-.. . . ,. ._
-‘.
: .’ ., ‘. .. .: .. : -, .. ;;.. ,:,‘:“,
. . ... . ,. .‘... .: ... .:’ .__.. .__ ~.
.’ :
.:,
.,
‘..
_;
.” _.i.
..
..;
.‘, _.
.‘..
.:.-
~
__.
..:.
:
.:.
.:
,,‘...
:.-
‘..
. ;.:
:.-
‘., :
..
. .
.
_
:, ;__..
0. Edholm, C. Hornberg /Activation
’ a3so needed. In terms of these quantities the averages in eq. (6) can be obtained as an expansion: (exp[ip(@ -@a)]
1= C
where the coefficients;;)
A!) exp(-X,r),
(10)
are:
with s = 0 corresponding
to the zero eigenvaluets). For diffusion in a potential well of height much larger than kBT, Y is a rather general feature of the eigenvalues that one of them (take s = 1) is much smaller than the others and corresponds to passage over the potential barrier out of the well. The other eigenvalues correspond to the motion within the potential well. Thus the spectral densities become:
enegies for rota*
defmed as:
EA=+-
+ h*) (A\% (60,
f npcJ
+ Xz)* + w2
.-.
1 ;
(13
(14)
In this approximation, the activation enew, EA: is in fact the activation energy of the integral of the time correlation function of second order spherical harmonics. This is a quantity that might be of interest also for other spectroscopic techniques than NMR. By formula (13) we then have the activation energy: EA=VO+E,
‘Et
+E3,
05)
with
El=-Vo
1-t [
E,
+ 2(6D*
11
molecular groups
=
V,
bwp
a(v
+cny9,
06b)
0
E3 = V.
Here we have brought in the barrier height V, since the eigenvalues and eigenfunctions depend upon temperature through V,-Jc~Z’.
with A, B and C given by eq. (7). Here the higher terms decrease rapidly and it is often sufficient to keep the lowest terms. Under the further assumption of extreme narrowing (i.e. As Z+ws, oI and DO S us, wI) the NMR-relaxation time is obtained in a simple form:
3. The diffusion Iimit In the strong damping (diffusion) limit, a simplified equation, the Smoluchowski equation, is valid. A numerical solution for the potential: V(%) = V.&j1
with
(n being the number $f spins interacting with the. relaxing s$n). If Do is ddteaed independently as in ref;, 131, the first term ir~eq. (13) card be subtractedaway from J/T,,.w%iIe D,.oftkn can be &glected ‘. compared to.-& in-the other.t&ms: In an-&hen@ plot of this quantity the activatidn energy is thus ... : : ,:
:
.’ :..
which is suitable for describing, for instance, rotating methyl groups, has been obtained by Edholm and Blomberg [9]. The dependence of the eigenvalues & and the coefficients A$) upon VO/kBT may, be found in ref. ill]. Before calculating the derivatives according to eqs. (16a)-(16c) a Won constant, D = Q&T/@), is taken out of all eigenvalues giving: 4, =E;.+E4,
Wa)
_:_.
...’ .
(17)
.,.
.. ._. .
- cos 3CJ))
_..:,
_I
.,...
‘..
_;,
_.
:...,..,
:
_’
‘.
~
;’
:
:
:
.’
;.’
.,,
0. Edholm, C. Blomberg / Activation energies for rorating mdlecular groups
with
(18b)
and E4 = -
(W
E’,, E2 and E3 are shown in fig. 1 as functions of the barrier height in thermal units. E’, is the correction to transition state behaviour, exp(- V,,/kBT), of the first eigenvalue. According to a formula due to Kramers [12]: A; a (VO/kBT)exd-vO/kB~ f&r large
Vop which
in practice means
0% 5 5keT.
E:
becomes in this limit: E: = -kBT.
E3 comes directly from the fast small&ale motion * within the wells and is important only for. barrier heights below 2-3 kBT Finally, there is the temperature dependence of the rotational diffirsion constant. According to the classiczl theory the rotational diffusion constant for a macroscopic sphere is:
(21)
D = kBTf8m$t3,
with R being the radius of the rotating sphere and n the visccisity of the surrounding fluid_ This can be modified for more realistic particle shape and for more realistic boundary conditions than the “sticky” ones used to arrive at formula (21), but that will not change the temperature dependence (w’hich comes in directly and through the viscosity). The macroscopic viscosity of relevant substances can in a limited temperature interval be roughly approximated by:
03)
The second correction term is an effect from the fast small-scale motion within the wells which reduces the amplitude for the part of the correlation functions corresponding to the barrier crossing. This correction is negligible for higher barriers than 6-7 k&
(22)
V~OT^(: with 7 being in the range 2-10 tion becomes:
[13]. Then the correc-
E4 ~(7 + 1)kuT.
(23)
If this macroscopic theory can be extended to systems as small as this, which is not in any way proven although often done, we see that E., dominates over the other corrections, and might well be larger than the barrier height itself. We conclude that in the strong damping Limit there is no simple reIation between barrier height and activation energy. The differences come from two sources Firstly there are correctionscoming from the solution of the Smoluch&ski equation, which can be. cakulated if the approximate shape. of the potential barrier is known. Secondly, there.are corrections due ,fothe temperature dependence of the rotational diffusion
constant.
.:.I
Knowing the.barrier height~.f& other s&ces,‘the temperature dependence of rotatiorial d&&ion. constants could be.detertniued-from’NMR activation ener-
0. Edholrn, C. Bbmberg
Table
4. The intectial limit Now, the fXl Fokker-Planck equation has to be considered. For barrier heights d&n to a few kiT, Garners 1121 has however furnished an approximate formula for the transition rate between different wells: XI =R =R,(&/D)([l
+ (D/D,)‘]‘R
- 11 (24)
x exP(-v&q-
Here R/R, and D/D, are the transition rate and the diffusion constant respectively, in dimensionless urrits. With the potential given by eq. (17) we have: D, = kBT/(18Vol)1R
13
/ Activation energies for rotatirg molmdmmups
CW
1
Comparison of Bh(ll) + CA?) 3s calculated by a numerical solution of the Smaluchowski eqution and by the approG mate formuia (28) VohT
Bh$‘)
1
0.391
2 3 4
0.533 0.635 0.701 0.744 0.172 0.792 0.806 0.817 0.825
5 6 1 8 9
10
f CA5’)
Bte? 0.420 0.537 0.631 0.698 0.742 0.772
4 + Ctez’? ‘4
.
0.792 0.806 0.817 0.825
and R, = (3/2rr) (V&Y)‘“.
(Xb)
In the strong damping limit (D/D, << !) eq. (24) becomes: AI = (9/4n> DV’olkB0 in accordance have:
exp(-v&an
(26)
with eq. (19). In the opposite limit we
(Vo/V)lRexP(-vO/kBT)z
A, = (VW
(27)
in agreement with the absolute rate theory. In this case the pre-exponential factor is independent of D and T and thus El, E’, and Ea are zero. E2 can be calculated in an approximate way valid both for low and high damping but only for large barrier heights. It is then assumed that the motions within the we& have time enough to come to an equilibrium before there is any appreciable passage over the barriers, which is true if X1 4 Xi, i = 2,3, .._ 00. AF’ is then the amplitude of the correlation function when there is equilibrium ovir the potential well but no barrier passage has yet occurred:
A(I)
25
P
(eiP*}eq,
(28)
where the a&age is @en over a BoItzmann distribution over one potential well. In table 1, BAY) + CAY) is compared with the approximate quantity by eq. (20) for yar@ig barrier height, showing that the agreemerit .js good down to abopt 2 kBT_ Tlus, provided Xhk’qactig of the eigenvalues is not entirely different @-the Ibw dainping:case; for high eri&gb barrier, the
:.
.;
1
‘.
.
The discussion has been separated into two limiting by the rotational diffusion constant being much larger (tow damping) and much smaller (strong damping) respectively than a critical diffusion constant D,. The behaviour of the activation energy differs drastically for the two cases. II-Iboth cases there are differences beMeen the activation energy and the barrier height due to contributions to the NhiR-relaxation from other types of motion than barrier passage (motions within the potential wells)_ These terms,&‘? and Ex above, are small. In the low damping case the rate of barrier passage is @ven by eq. (27) giving Arrhenius behaviour, while in the strong damping case there are IoDe corrections [eq. (2611. It is therefore crucial to know in which region we are. D/D, can be given through eqs- (2i) and (25a) inserting the moment of inertia of a homogeneoussphere,I=(8n/15)pRS,withp being the dencases, which are characterized
_.
,:~.,,.
_
5. summary
.‘. .-
.-
correction E2 is the same as in the strong damping case. Finally there is Es, which has the same sign and probably is of about the same size as in the strong damping case, and thus anyhow is small. Summarizing, with E, being zero and E2 and ES small and of opposite sign, we may conclude that the activation energy and the barrier height are close in the Iow damping limit.
.
.I
‘.
. . ..
i,
-’ .
:._
. . .
:
._
.;. :.
. :.
j
‘..
..:
.
..
0. Edholm. C. Blomberg / Activation energies for rotaiing moIecuhrgroups
14
121 K.F. Kuhlmann and D.M. Grant, J. Chem. Phys. 55
sity:
- (1971) 2998.
D/DC = (l/q) (3pV0’o120~R)’ n.
(29)
For a rotating methyl group in an extremly non-vis-
cous medium we may put the viscosity to 0.5 cP, the density to lo3 kg/m3, the barrier height to 10 W/m01 and the radius to IO-” m, giving DjDc = 0.2, which indicates being in the strong damping limit. It should, however, be pointed out that more realistic boundary conditions change eq. (21) and increase D, perhaps encugh to bring it into the neighbourhood of the critical value. (See calculations on rotating clipsoids with “slippy” boundary conditions by Hu and Zwanzig [14].) As a further test we have calculated rotationai diffusion constants from the experimental NR1R-relaxation times of ref. [3], assuming a realistic potential barrier. They are a factor 2-3 sm-Jer than the critical value.
References - [I] W-T.Orville-Thomas, eb, Internal rotation in molecules (Wiley, New York, 1974).
f3] A. Ericsson, J. Kowalewski, T. Liijefors and P. Stilbs, 5.
Mqn. Res. 38 (1980) 9. [4] C. Brat, Chem. Phys. Letters 3 (1969) 319. [S] J. KowaJewski and T. Liljefors, Cheni. Phyr Letters 64 (1979) 170. [6] S. Chandnsekhx, Rev. Mod.Phys. 15 (1943) 1. 17) D.E. Woessner, 1. Chem. Phys. 36 (1972) 1. [81 A Abragam, The principles of nuclear magnetism (Oxford University Press, Oxford, 1961). [9] A.J. Dianous and F. Voliio, Mol. Phys. 34 (1977) 1263. [IO) B. Lzssier and C. Brot, Discussions Faraday Sot. 48 (1969) 39. [ll] 0. Edhohn artd C. Blornberg, Chcm. Phys.42 (1979) 449. 1121 H.A. Kramers, Physica 7 (1940) 284. [ 13 1R.C. Weast, ed., Handbook of chemistry and physics, 58th Ed. (Chemical Rubber Company Press, Cleveland, 1977). !14] C. Hu and R. Zwanzig, J. Chem. Phys. 60 (1974) 4354.
Chemical Physics 56 (19Sl) 9-14 North-Holland PubMing Company
BROWNTANMOTION DESCRIF’TION(JF ACTIVATION ENERGIES FROM NMR-RELAXATION TIMES FOR ROTATING MOLECULAR-GROUPS 0. EDHOLM and C. BLOMBERG
Department of Theoretical Received 23 September
P/~yrics,Royal Institute of Technology, S-100 44 Stockholm 70. Sweden
1980
A general forrnuia is derived (containing eigenvalues and eigenfunctions of some diffusion q~.tion) for activation energies from NM%relaation times for rotding molecular groups.Numerical calculations for a trigonometric three well potential in the strong &ping (Smolwhowski) limit show that there may be a considerable differencr: b-ctween activation energy ad barrier height In the low damping limit there is fai11y good agreement.
1. Introduction
Rotating molecular groups, for instance methyl groups in a larger molecule, are subject to hindrances from the rest of the molecule which is described by a potential barrier. The height of this barrier can be derived from different experiments, perhaps most directly by using sptittings in the microwave spectrum. For an extensive discussion of internal rotation in molecules, see e.g. ref. [l]. Information about the barrier height can also be gained through dynamic experiments as e.g. nuclear magnetic resonance (NhlR). The NMR-relaxation times give information about the rate of reorientation, which of course becomes smaller the higher the potential barrier is. Knowing the rate of reorientation of the free rotor, the barrier height could be estimated, see e.g. ref. 121.Such an estimate is, however, rather uncertain, and a more suitable way is to do the measurements at different temperatures and fit the reorientation rates, R, to an Arrhenius type expression [3]: R = A &p(-E_JkB
T).
(1)
tial factor, some of which have been discussed by Brot [4]. Kowalewaki aud Liijefors [S] have shoRn, that the absolute rate theory gives an activation energy, that is very close to the barrier height, and a preexponential factor that is almost constant with temperature. In this wcrk, the starting point wih be taken by the classical equations for brownian rotation in a potential, see e.g. ref. [6]. First, however, we give in section 2 following Woessner [7] the appropriate formula relating N?NRrelaxation time and orientational correlation functions As a second step, general formulae for these relations and for the activation energy, expressed in eigenvalues and eigenfunctions of some diffusion equation, are given. We take the Fokker-Planck equation as fundamental, but since it would be very cumbersome to furnish a general (numerical) solution, two limiting cases are considered. The strong damping (diffusive) limit, in which the diffusive motion dominates over the deterministic oscillations, is described by the Smoluchowski equation. This case is discussed in section 3. The iow damping (inertial) limit, in which one has to wait for several deten&istic oscillaticns ‘before any appreciable effect of the diff&ive motion is seen, is discussed-in section 4. In the summary (section 5) a brief discussion of the applicability of the two limiting cases is given.
Here, the preexponenti factor,d. and the activation energy,EA, are adjustable.Parameten, both being phenomeno~ogical and having no immediate physical interpretation~.One therefore wants to relate these $$rym+s tq ynti&s such as potential barrier height..*and rotational diffusionconstanf A number of : expressrons L have be& proposed for the preexponen-_; . . ~~‘301~&01Q4~81/0000~~oooo/S 6250 @‘North-HoUarrd~PubJishingCompany . : i .. -_-. :.. 1. : .. _.. .... .‘.’ ..’,_.. (.., .-. _-.: ._:_ ~ .,.‘. : : ..-: _ : :
0. Edholm, C. Biomberg / Activation energies for rotating moleculargroup~
10
2. NMR-relaxation
obtains:
times and activation energies
1= $r-r
Tine NMR-relaxation times are determined by the spectral density of correlation functions of spherical harmonics Y,#(t), r(t)):
X {A +B(exp[i(@
exp(-6Do)
- @c)] ) + C
(6) with
X exp(iwr) dr.
A = $(3 cos*A - 1)2,
(2)
B = acti 2Aj2,
C=@~IIA)~.
The angles fl and y detemine the orientation of the vector between two interacting nuciei and the magnetic field. For dipolar relaxation between two un3ike nuclei, the longitudinal relaxation time, T,, is given by:
(7)
The averages may be calcu!ated knowing the probability distribution for the angle Q as a function of time. This is governed by some stochastic equation, e.g., a Fokker-Planck equation (see e.g. ref. [6]):
where the probability P(&, w, t) is a function of angle @, angular velocity w and time t. U(q) is the potential giving the deterministic part of the motion, fl is the angular friction coefficient and I is the moment of inertia. in the strong damping case this simplifies to the Smoluchowski equation:
See, for instance ref. [S], although his normalization is somewhat different giving different coefficients. yr and ys are the gyromagnetic ratios of the involved nuclei. S is the spin of the interacting nuclei, r the distance and wI and os the I-armor frequencies. Following Woessner [7], the angles A and Cp describing the orientation of tht- internuclear axis with respect to the axis about which it rotates, and the angles 0 and 0’ describing the orientation of this latter axis with respect to the laboratory coordinate system are introduced. The result of Woessner [7] is most simply achieved uring the transformation properties of the spherical harmonics under rotation:
ap
-CD
at
( )I*
a2P I a --pau _+_[aa .2,2-a@ a*
(9)
with the diffusion constant D being kBT/Zfi and the probability P(q, t) corresponding to that of eq. (8) integrated over all angular velocities. Averages as those on the rhs of eq. (6) have earlier been calculated from ?he Smoluchowski equation for a rotor in a two-well potential by Dianoux and Voko 191, aiming at understanding neutron diffraction against liquid crystal molecules. Lassier and Brot [lo] have simulated various correlation functions for such a rotor. Edholm and Blomberg [I l] have done caktiations for a three-well [email protected] aiming at interpreting NMR relaxation in hydrocarbon chains. We wilI follow the procedure of ref.. [Ill. Splitting off a time dependent fact& eyU from eq. (8) or (9). an eigenvahre’ problem _$formed by the: eqtition together wit! G’per&idic boundary cticdi- .‘. tion. W&denote’the ei&$&r& & and the.eigenfimc-. ti~nsfs(&) (i;l-~~e_F‘okker~-~anckcask‘f,(~)-~e ihe: .eige+rnctions titeg+$~o~er+I &g&r v+ities). Since in g&er~;th.% equatio~‘~enot-self-ahjdiiit,.: :
(Y;i(t + 7) Yzi(t))
-where the two rotations have been assumed to be ‘. independent of one another. The subscript 0 denotes time zero and @$9, @‘)are second order rotation matrix elements__The first average i&now easiljr calculated assuming isotropic ;eodent.ation (Da is ,$I&rota-. tion& $fusion co&ant for the:wh&z molecule): 1
.,
the ~igenfunctiom~,(~)~of__. the.adjoint eqtitiotirare:’ .-.. . . ,. ._
-‘.
: .’ ., ‘. .. .: .. : -, .. ;;.. ,:,‘:“,
. . ... . ,. .‘... .: ... .:’ .__.. .__ ~.
.’ :
.:,
.,
‘..
_;
.” _.i.
..
..;
.‘, _.
.‘..
.:.-
~
__.
..:.
:
.:.
.:
,,‘...
:.-
‘..
. ;.:
:.-
‘., :
..
. .
.
_
:, ;__..
0. Edholm, C. Hornberg /Activation
’ a3so needed. In terms of these quantities the averages in eq. (6) can be obtained as an expansion: (exp[ip(@ -@a)]
1= C
where the coefficients;;)
A!) exp(-X,r),
(10)
are:
with s = 0 corresponding
to the zero eigenvaluets). For diffusion in a potential well of height much larger than kBT, Y is a rather general feature of the eigenvalues that one of them (take s = 1) is much smaller than the others and corresponds to passage over the potential barrier out of the well. The other eigenvalues correspond to the motion within the potential well. Thus the spectral densities become:
enegies for rota*
defmed as:
EA=+-
+ h*) (A\% (60,
f npcJ
+ Xz)* + w2
.-.
1 ;
(13
(14)
In this approximation, the activation enew, EA: is in fact the activation energy of the integral of the time correlation function of second order spherical harmonics. This is a quantity that might be of interest also for other spectroscopic techniques than NMR. By formula (13) we then have the activation energy: EA=VO+E,
‘Et
+E3,
05)
with
El=-Vo
1-t [
E,
+ 2(6D*
11
molecular groups
=
V,
bwp
a(v
+cny9,
06b)
0
E3 = V.
Here we have brought in the barrier height V, since the eigenvalues and eigenfunctions depend upon temperature through V,-Jc~Z’.
with A, B and C given by eq. (7). Here the higher terms decrease rapidly and it is often sufficient to keep the lowest terms. Under the further assumption of extreme narrowing (i.e. As Z+ws, oI and DO S us, wI) the NMR-relaxation time is obtained in a simple form:
3. The diffusion Iimit In the strong damping (diffusion) limit, a simplified equation, the Smoluchowski equation, is valid. A numerical solution for the potential: V(%) = V.&j1
with
(n being the number $f spins interacting with the. relaxing s$n). If Do is ddteaed independently as in ref;, 131, the first term ir~eq. (13) card be subtractedaway from J/T,,.w%iIe D,.oftkn can be &glected ‘. compared to.-& in-the other.t&ms: In an-&hen@ plot of this quantity the activatidn energy is thus ... : : ,:
:
.’ :..
which is suitable for describing, for instance, rotating methyl groups, has been obtained by Edholm and Blomberg [9]. The dependence of the eigenvalues & and the coefficients A$) upon VO/kBT may, be found in ref. ill]. Before calculating the derivatives according to eqs. (16a)-(16c) a Won constant, D = Q&T/@), is taken out of all eigenvalues giving: 4, =E;.+E4,
Wa)
_:_.
...’ .
(17)
.,.
.. ._. .
- cos 3CJ))
_..:,
_I
.,...
‘..
_;,
_.
:...,..,
:
_’
‘.
~
;’
:
:
:
.’
;.’
.,,
0. Edholm, C. Blomberg / Activation energies for rorating mdlecular groups
with
(18b)
and E4 = -
(W
E’,, E2 and E3 are shown in fig. 1 as functions of the barrier height in thermal units. E’, is the correction to transition state behaviour, exp(- V,,/kBT), of the first eigenvalue. According to a formula due to Kramers [12]: A; a (VO/kBT)exd-vO/kB~ f&r large
Vop which
in practice means
0% 5 5keT.
E:
becomes in this limit: E: = -kBT.
E3 comes directly from the fast small&ale motion * within the wells and is important only for. barrier heights below 2-3 kBT Finally, there is the temperature dependence of the rotational diffirsion constant. According to the classiczl theory the rotational diffusion constant for a macroscopic sphere is:
(21)
D = kBTf8m$t3,
with R being the radius of the rotating sphere and n the visccisity of the surrounding fluid_ This can be modified for more realistic particle shape and for more realistic boundary conditions than the “sticky” ones used to arrive at formula (21), but that will not change the temperature dependence (w’hich comes in directly and through the viscosity). The macroscopic viscosity of relevant substances can in a limited temperature interval be roughly approximated by:
03)
The second correction term is an effect from the fast small-scale motion within the wells which reduces the amplitude for the part of the correlation functions corresponding to the barrier crossing. This correction is negligible for higher barriers than 6-7 k&
(22)
V~OT^(: with 7 being in the range 2-10 tion becomes:
[13]. Then the correc-
E4 ~(7 + 1)kuT.
(23)
If this macroscopic theory can be extended to systems as small as this, which is not in any way proven although often done, we see that E., dominates over the other corrections, and might well be larger than the barrier height itself. We conclude that in the strong damping Limit there is no simple reIation between barrier height and activation energy. The differences come from two sources Firstly there are correctionscoming from the solution of the Smoluch&ski equation, which can be. cakulated if the approximate shape. of the potential barrier is known. Secondly, there.are corrections due ,fothe temperature dependence of the rotational diffusion
constant.
.:.I
Knowing the.barrier height~.f& other s&ces,‘the temperature dependence of rotatiorial d&&ion. constants could be.detertniued-from’NMR activation ener-
0. Edholrn, C. Bbmberg
Table
4. The intectial limit Now, the fXl Fokker-Planck equation has to be considered. For barrier heights d&n to a few kiT, Garners 1121 has however furnished an approximate formula for the transition rate between different wells: XI =R =R,(&/D)([l
+ (D/D,)‘]‘R
- 11 (24)
x exP(-v&q-
Here R/R, and D/D, are the transition rate and the diffusion constant respectively, in dimensionless urrits. With the potential given by eq. (17) we have: D, = kBT/(18Vol)1R
13
/ Activation energies for rotatirg molmdmmups
CW
1
Comparison of Bh(ll) + CA?) 3s calculated by a numerical solution of the Smaluchowski eqution and by the approG mate formuia (28) VohT
Bh$‘)
1
0.391
2 3 4
0.533 0.635 0.701 0.744 0.172 0.792 0.806 0.817 0.825
5 6 1 8 9
10
f CA5’)
Bte? 0.420 0.537 0.631 0.698 0.742 0.772
4 + Ctez’? ‘4
.
0.792 0.806 0.817 0.825
and R, = (3/2rr) (V&Y)‘“.
(Xb)
In the strong damping limit (D/D, << !) eq. (24) becomes: AI = (9/4n> DV’olkB0 in accordance have:
exp(-v&an
(26)
with eq. (19). In the opposite limit we
(Vo/V)lRexP(-vO/kBT)z
A, = (VW
(27)
in agreement with the absolute rate theory. In this case the pre-exponential factor is independent of D and T and thus El, E’, and Ea are zero. E2 can be calculated in an approximate way valid both for low and high damping but only for large barrier heights. It is then assumed that the motions within the we& have time enough to come to an equilibrium before there is any appreciable passage over the barriers, which is true if X1 4 Xi, i = 2,3, .._ 00. AF’ is then the amplitude of the correlation function when there is equilibrium ovir the potential well but no barrier passage has yet occurred:
A(I)
25
P
(eiP*}eq,
(28)
where the a&age is @en over a BoItzmann distribution over one potential well. In table 1, BAY) + CAY) is compared with the approximate quantity by eq. (20) for yar@ig barrier height, showing that the agreemerit .js good down to abopt 2 kBT_ Tlus, provided Xhk’qactig of the eigenvalues is not entirely different @-the Ibw dainping:case; for high eri&gb barrier, the
:.
.;
1
‘.
.
The discussion has been separated into two limiting by the rotational diffusion constant being much larger (tow damping) and much smaller (strong damping) respectively than a critical diffusion constant D,. The behaviour of the activation energy differs drastically for the two cases. II-Iboth cases there are differences beMeen the activation energy and the barrier height due to contributions to the NhiR-relaxation from other types of motion than barrier passage (motions within the potential wells)_ These terms,&‘? and Ex above, are small. In the low damping case the rate of barrier passage is @ven by eq. (27) giving Arrhenius behaviour, while in the strong damping case there are IoDe corrections [eq. (2611. It is therefore crucial to know in which region we are. D/D, can be given through eqs- (2i) and (25a) inserting the moment of inertia of a homogeneoussphere,I=(8n/15)pRS,withp being the dencases, which are characterized
_.
,:~.,,.
_
5. summary
.‘. .-
.-
correction E2 is the same as in the strong damping case. Finally there is Es, which has the same sign and probably is of about the same size as in the strong damping case, and thus anyhow is small. Summarizing, with E, being zero and E2 and ES small and of opposite sign, we may conclude that the activation energy and the barrier height are close in the Iow damping limit.
.
.I
‘.
. . ..
i,
-’ .
:._
. . .
:
._
.;. :.
. :.
j
‘..
..:
.
..
0. Edholm. C. Blomberg / Activation energies for rotaiing moIecuhrgroups
14
121 K.F. Kuhlmann and D.M. Grant, J. Chem. Phys. 55
sity:
- (1971) 2998.
D/DC = (l/q) (3pV0’o120~R)’ n.
(29)
For a rotating methyl group in an extremly non-vis-
cous medium we may put the viscosity to 0.5 cP, the density to lo3 kg/m3, the barrier height to 10 W/m01 and the radius to IO-” m, giving DjDc = 0.2, which indicates being in the strong damping limit. It should, however, be pointed out that more realistic boundary conditions change eq. (21) and increase D, perhaps encugh to bring it into the neighbourhood of the critical value. (See calculations on rotating clipsoids with “slippy” boundary conditions by Hu and Zwanzig [14].) As a further test we have calculated rotationai diffusion constants from the experimental NR1R-relaxation times of ref. [3], assuming a realistic potential barrier. They are a factor 2-3 sm-Jer than the critical value.
References - [I] W-T.Orville-Thomas, eb, Internal rotation in molecules (Wiley, New York, 1974).
f3] A. Ericsson, J. Kowalewski, T. Liijefors and P. Stilbs, 5.
Mqn. Res. 38 (1980) 9. [4] C. Brat, Chem. Phys. Letters 3 (1969) 319. [S] J. KowaJewski and T. Liljefors, Cheni. Phyr Letters 64 (1979) 170. [6] S. Chandnsekhx, Rev. Mod.Phys. 15 (1943) 1. 17) D.E. Woessner, 1. Chem. Phys. 36 (1972) 1. [81 A Abragam, The principles of nuclear magnetism (Oxford University Press, Oxford, 1961). [9] A.J. Dianous and F. Voliio, Mol. Phys. 34 (1977) 1263. [IO) B. Lzssier and C. Brot, Discussions Faraday Sot. 48 (1969) 39. [ll] 0. Edhohn artd C. Blornberg, Chcm. Phys.42 (1979) 449. 1121 H.A. Kramers, Physica 7 (1940) 284. [ 13 1R.C. Weast, ed., Handbook of chemistry and physics, 58th Ed. (Chemical Rubber Company Press, Cleveland, 1977). !14] C. Hu and R. Zwanzig, J. Chem. Phys. 60 (1974) 4354.