statistical motions m. bée institut de biologie structurale cea-cnrs laboratoire de biophysique...
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STATISTICAL MOTIONSSTATISTICAL MOTIONS
M. Bée
Institut de Biologie Structurale CEA-CNRS Laboratoire de Biophysique Moléculaire
41 rue J. Horowitz, 38027 Grenoble Cedex-1
Institut Laue-LangevinBP 156, 6 rue J. Horowitz, 38042 Grenoble Cedex-9
Laboratoire de Spectrométrie Physique Université J. Fourier Grenoble 1BP 87 38402 Saint-Martin d’Hères Cedex
M. Bée
Institut de Biologie Structurale CEA-CNRS Laboratoire de Biophysique Moléculaire
41 rue J. Horowitz, 38027 Grenoble Cedex-1
Institut Laue-LangevinBP 156, 6 rue J. Horowitz, 38042 Grenoble Cedex-9
Laboratoire de Spectrométrie Physique Université J. Fourier Grenoble 1BP 87 38402 Saint-Martin d’Hères Cedex
0kkQ
Neutronsincidents(E0, 0, k0)
Détecteur
EF < E0
kF < k0
Perte d’énergie
du neutron
EF > E0
kF > k0
Gain d’énergie
du neutron
EF = E0
kF = k0
Pas d’échange
d’énergie
Q
Q
Q
k0Echantillon
PRINCIPLE OF A NEUTRON EXPERIMENT
ii) Energy transfer, difference between final, E, and initial, E0, neutron energy,
)(2
20
22
0 kkm
EE
i) Momentum transfert, Q, difference between final, k, and initial, k0 neutron momentum
Two relevant quantities are measured:
dtetQ.ri
eQ.ri
ebbNk
k
dd
d ti )(
)0(
.2
11
0
2
Average over spin states Average over positions
Differential scattering cross sections
Thermal average over the positions of the nuclei and over their spin states.In general, can be taken separately.
dtetQ.ri
eQ.ri
ebbNk
k
dd
d ti )(
)0(
2
11
0
2
One component system
cohb b
22 incb b b
2 2
0
22
0
(0) ( )1 1.
2
(0)1 1 ( ). 2
i t
i t
i i td kb e e e dt
d d k N
ik i tb b e e e dtk N
Q.r Q.r
Q.r Q.r
Incoherent scattering length
Coherent scattering length
SCATTERING FUNCTIONS2 2 2
coh inc
d d d
d d d d d d
Coherent scattering Incoherent scattering
Intermediate scattering function
( )1 (0)( , ) .
i tiF t e e
N
Q.rQ.rQ
Spatial Fourier transform of time-dependent pair correlation function
3
1( , )
(2 )iG t F( ,t) e d
Q.rr Q Q
Time Fourier transform: dynamical scattering function (scattering law, dynamical structure factor)
1( , )
2i tS F( ,t) e dt
Q Q
( , ) iF t G( ,t) e d
Q.rQ r r
1 ( , )
2i tF( ,t) S e d
Q Q
( )1( , )
2i tS G( ,t) e d dt
Q.rQ r r( )
3
1( , )
(2 )i tG t S( , ) e d d
Q.rr Q Q
Intermediate incoherent scattering funtion
(0)1 ( )( , ) . inci i tF t e e
N
Q.r Q.rQ
1( , )
2i t
inc incS F ( ,t) e dt
Q Q
Space-time Fourier transform of time dependent autocorrelation function
( )1( , )
2i t
inc sS G ( ,t) e d dt
Q.rQ r r
( )3
1( , )
(2 )i t
s incG t S ( , ) e d d
Q.rr Q Q
INCOHERENT SCATTERINGSame scatterer interacts with neutron at times 0 and t
Incoherent scattering funtion
Finally 2
0
1( , ) ( , )
4 coh inc inc
d kS S
d d k
Q Q
Coherent scatteringcross section 2
4 cohcoh b 2
4 incinc b
Incoherent scatteringcross section
Lattice vibrations Equilibrium position within molecule with orientation Internal molecular vibrations
( ) ( ) ( , ) ( )t t t t r R uv
SEPARATION OF ATOMIC MOTIONS
Instantaneous position of a scattering nucleusBelonging to a reorienting molecule or chemical group:
TWO CASES
No long-range motions (rotator phases in solids, liquid crystals…)
Existence of long-range displacements(liquid, intercalated species…)
( ) ( ) ( , ) ( )t t t t r T R u
C.O.M translation Equilibrium position within molecule with orientation Internal molecular vibrations
Basic assumption: all the components of the atomic displacement are completely decoupled
Occur on different time scales:Lattice vibrations and internal molecular vibrations: 10-13 – 10-14 sMolecular reorientations: 10-11 – 10-12 sTranslation of molecule C.O.M in liquids: 10-9 - 10-10 s
-4
-2
0
2
4
6
8
10
0 20 40 60 80 100
Temps (picosecondes)
xx
y
z
External (lattice)and internalvibrations
Moleculereorientations
Example: molecular dynamic simulation of hexamethylethane (CH3)3-C-C-(CH3)3
( , ) exp . (0) .exp . ( )
exp . (0) .exp . ( )
exp . ,0) .exp . ( , )
incF t i i t
i i t
i i t
Q Q Q
Q u Q u
Q R( Q R
v v
Ω Ω
Intermediate scattering function:No long-range displacements Long-range displacements
( , ) exp . (0) .exp . ( )
exp . (0) .exp . ( )
exp . ,0) .exp . ( , )
incF t i i t
i i t
i i t
Q Q T Q T
Q u Q u
Q R( Q RΩ Ω
( , ) ( , ). ( , ). ( , )inc V U RF t F t F t F tQ Q Q Q ( , ) ( , ). ( , ). ( , )inc T U RF t F t F t F tQ Q Q Q
Incoherent scatteringfunctions
( , ) ( , ) ( , ) ( , )inc U RS S S S Q Q Q QV ( , ) ( , ) ( , ) ( , )inc T U RS S S S Q Q Q Q
Quantum mechanicClassical mechanic
Quantum mechanicClassical mechanic
LATTICE AND MOLECULAR VIBRATIONS
0
0
(0) ( )( , ) .
( , ) exp .exp
. (0) . ( )
inc
t
t
i i tF t e e
F t U U
U i U i t
Q.u Q.uQ
Q
Q u Q u
Operators U0 and Ut do not commutte, but each of them commutte with their commutator [U0, Ut].
0 0 0
20 0
exp( ).exp( ) exp( )exp([ , ])
( , ) exp exp
t t t
t
U U U U U U
F t U U U
Q
2
22
( , ) exp . (0) .exp . (0) . ( )
1exp . (0) . 1 . (0) . ( ) . (0) . ( ) ...
2
F t t
t t
Q Q u Q u Q u
Q u Q u Q u Q u Q u
Purely inelastic term involving in its expansion
the autocorrelation function of the atom
displacement.
Expansion to second order of the second exponential term
2 2 2exp - (0) exp -
exp -2 ( )
Q u
W Q
Q.u
Debye-Waller term: interference between the neutronic wave scattered by atom at times 0 and t is attenuated
by its own thermal vibrations
( , ) exp 2 ( ) ( ) ( , ) inélS W Q S Q Qv v v
0Energy exchange
Example: Methyl torsion
30 meV
INTERNAL VIBRATIONS
Debye Waller term 2 22 ( )W Q Q v v
( , ) exp 2 ( ) ( ) ( , ) inélu u uS W Q S Q Q
0
Acoustic modesBrillouin zone limits
Optical modes:Brillouin zone limitsCentre of Brillouin zone
5 –10 meV
Debye Waller term
Energy exchange
EXTERNAL VIBRATIONS
2 22 ( )W Q u Q u
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0 5 10 15 20 25 30 35 40
Vib
ratio
nal D
ensi
ty o
f S
tate
s (A
rb. u
nits
)
Energy (meV)
0
0.0001
0.0002
0.0003
0.0004
0.0005
0 5 10 15 20 25 30 35 40
Vib
ratio
nal D
ensi
ty o
f S
tate
s (A
rb. u
nits
)
Energy (meV)
(a)
(b)
100 K
271 K
235 K
185 K
300 K320 K
2 = 95°
2 = 25°
2 = 48°
Example 2: camphor sulfonic acid
0 50 100 150 200 250
Nombre d'onde (cm-1)
0 500 1000 1500 2000
0
Inte
nsité
(un
ité a
rbitr
aire
)
Energie (meV)
Example 1: Cytidine T = 15 KGaigeot, M.-P., Leulliot, N., Ghomi, M., Jobic, H. Coulombeau, C. Bouloussa, O.,Chemical Physics 261 (2000) 217
CASE OF A MOLECULE
3N - 6 harmonic oscillators
N atoms: 3N – 6 frequencies 3N – 6 Dirac peaks
Time of flight(10-11-10-12 s)
Multichoppers
IN5 ILL GrenobleMibemol LLB SaclayNEAT HMI BerlinDCS NIST USA
Crystal monochromator
IN6 ILL GrenobleFOCUS PSI SwitzerlandFCS NIST USA
Inverted TOF geometry
IRIS RAL UK
Backscattering(10-9-10-10 s)
Doppler effect
IN10 ILL GrenobleBSS KFA Jülich IN16 ILL GrenobleHFBS NIST USAFRMII (project) Munich
Thermal expansion
IN13 ILL Grenoble
Spin echo(10-8-10-9 s)
Standard
IN11 ILL GrenobleMESS LLB SaclayIN15 ILL Grenoble
Zero field
HMI BerlinLLB Saclay
OVERVIEW OF SPECTROMETERS FOR QENS
PROPERTIES OF TRANSLATIONAL AND ROTATIONALSCATTERING FUNCTIONS
( , ) (0); ( ) . (0) .exp . (0) .exp . ( ) . ( ). (0)incF t p t p i i t d t d Q R R R Q R Q R R R
)();0( tRRp Probability for a scattering atom, located at R(0) at initial time, to be at R(t) at time t
)0(Rp Distribution of probabilities at initial time
( (0); ( )) ( (0)) if p R R p R t
( , ) ( ) (0) exp . (0) exp . ( ) ( ) (0)
(0) exp . (0) (0). ( ) exp . ( ) ( )
incF p p i i d d
p i d p i d
Q R R Q R Q R R R
R Q R R R Q R R
Return to equilibrium:
System in equilibrium ( ( )) ( (0)) p R p R
2
2
( , ) (0) exp . (0) (0)
( ) exp . ( ) ( )
incF p i d
p i d
Q R Q R R
R Q R R
2( , ) exp( . (0))incF Q iQ R
At any arbitrary time, formal separation
time independing part
( , ) ( , ) ' ( , )inc inc incF t F F t Q Q Q
Time-Fourier transformation
Purely elastic component
1 1( , ) ( , ) exp( ) ' ( , ) exp( )
2 21
( , ). ( ) ' ( , ) exp( )2
inc inc
inc inc
S F i t dt F t i t dt
F F t i t dt
Q Q Q
Q Q
' ( , ) ( ,0) ( , ) .expinc inc inct
F t F F Q Q Q
2 2
( , ) ( , ). ( )
1( ,0) ( , . .
1
inc
inc inc
S F
F F
Q Q
Q Q
time depending part
Inelastic component the shape and width of whichdepend on the nature and of the characteristic timesassociated to nucleus motions
Very simple case: exponential decreasing, with a unique characteristic time,
Temps
Iinc
(Q,t)
(Iinc
(Q,0)-Iinc
(Q,)).exp(-t/
Iinc
(Q,0)
Iinc
(Q,
0
5
10
15
20
25
30
-600 -400 -200 0 200 400 600
Echange d'énergie du neutron (eV)
2/
Elastic Incoherent
Structure Factor (EISF)
2( ) ( , ) exp( . ( )
( )
( ) ( )
inc
EL
EL QEL
EISF Q F Q iQ R
Int Q
Int Q Int Q
Immediate information about motion geometry.Vanishing in the case of long-range diffusive motions (liquids).For localised motions, the more the proton distribution is isotropic in space,the weaker is the elastic contribution to scattered intensity at large momentum transfer.
LONG RANGE DIFFUSION
2 2 22
2 2 2
( , ) ( , ) ( , ) ( , )( , )
c t c t c t c tD c t D
t x y z
r r r rr
c(r, t) concentration of diffusing atomsD, diffusion coefficient (macroscopic)
( , )( , )s
c tG t
N
rr
Fick’s equation is also fulfilled by Gs(r, t ). Solution: normalised gaussian function
Phenomenological equation: second law of Fick (1855)
System at equilibrium: van Hove autocorrelation function,
Gs(r, t) = probability for an atom, initially located at origin,to be at r at time t:
N = total number of atoms
2
3/ 2
1( , ) exp
(4 ) 4s
rG t
Dt Dt
r Initial condition: Gs(r, t) = δ(r).
Function isotropic in space (no driving force in any direction)Mean square displacement of an atom proportional to time
2 ( ) 6r t Dt
Corresponds to the microscopic description of Einstein’s random walk modelEinstein (1905), Smoluchowski
a
TkD B
6
Translational diffusion coefficient:Brownian tracer sphere with radius a in a fluid with shear viscosity Brownian motion (1827)
Can be still valid when sizes of tracer and solvent molecules are comparable
Earlier QENS experiments (30 years ago) mainly devoted to studies of liquidsEarlier QENS experiments (30 years ago) mainly devoted to studies of liquids
Second Fick’s law
2 2 22
2 2 2
( , ) ( , ) ( , ) ( , )( , )
c t c t c t c tD c t D
t x y z
r r r rr
Spatial Fourier transform intermediate scattering function
2( , ) ( , ) exp( )F t F Q t DQ t Q
Gaussian function isotropic with Q
Remark:
lim ( , ) 0t
F Q t
Any atom can move away from origin from an arbitrary distance, the EISF is zero
Spatial Fourier transform dynamical structure factor
Lorentzian function, isotropic with Q Halfwidth at half maximum, , increases as a function of momentum transferaccording to a parabolic law
2
22 2
1( , )inc
DQS Q
DQ
SCATTERING LAW; DYNAMICAL STRUCTURE FACTORH
alw
idth
at h
alf
maxi
mu
m
Q2
HWHM = DQ2
0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0 0,05 0,1 0,15 0,2 0,25 0,3
Liquid paraxylene
Hal
f Wid
th a
t H
alf M
axim
um (
meV
)
Q2(A-2)
DQ2 law
Earlier studies:Liquid argon
Dasannacharya and RaoPhys. Rev. 137 (1965) A417
Coherent scattering:de Gennes narrowing
More recently observedfor liquid paraxylene
Jobic et al. (2001)
Study of liquid waterJ. Texeira et al. J. Phys. Coll. 45, C7 (1982) 65J. Texeira et al. Phys. Rev. A 31 (1985) 1913
Description with the modelOf Singwi and Sjölander
DQ2 law actually verified at small Q values
Correspond to large distances (several tens de of molecular diameters).
At large Q values, systematic deviations: over small distances, microscopic mechanisms intervene.
JUMP DIFFUSION
Deviation with respect to Fick’s Discontinuous character of diffusion mechanism.
Different models: application to different systems Hydrogen in metals Molecules adsorbed inside porous media (zeolites) Molecular liquids (water)
The model of Chudley and Elliott Initially developped for liquids with a short-range order.Numerous applications (e.g. diffusion of hydrogen atoms adsorbed in metals)
Hypotheses:
Vibrations about an average position (site) during ,Displacement towards another site by instantaneous jumps,Jump distance, l, >> vibrational displacements, Equilibrium sites on a Bravais lattice.
1
( , ) 1( , ) ( , )
n
ii
P tP t P t
t n
r
r l r
Diffusion equation (master equation) Analogous to Fick’s equation
P(r, t): probability for one atom to occupy a given siteSum over the n nearest neighbour sites located at distances li.
Fourier transformation: intermediate scattering function F(Q, t)
Differential equation
1
( , ) 11 exp( . ) ( , )
n
i
F ti F t
t n
i
QQ l Q
Solution
1
1( , ) exp 1 exp( . )
n
i
F t i tn
iQ Q l
Intermediate scattering function non isotropic: depends on the orientation of scattering vector Q with respect to cristallographic directions.
22
1 ( )( , )
( )incS
Q
Dynamical structure factor
Halfwidth at half-maximum
1
1( ) 1 exp( . )
n
i
in
iQ Q l
DEPENDS ON THE DIRECTION OF THE SCATTERING VECTOR Q
Remarks:
Expressions more complicated in the case of jumps towards sites of different types involving several jump probabilities,Dependence with the direction de Q enables to discriminate between several hypotheses of dynamical models
Case of powders Average over all the orientations of the crystallites with respect to Q
No analytical expression . Numerical average.
Sum of Lorentzian functions with diferent widths.Low values of Q: Sinc(Q, ) Lorentzian shapeLarge values of Q: clear differences.
Jumps between octahedral sites
Distance 3 Å
-1.0 -.5 0.0 .5 1.0
Inte
nsité
(u
.a.)
0
1
2
3
4
-1.0 -.5 0.0 .5 1.0
0
1
2
3
Energie
-1.0 -.5 0.0 .5 1.0
0
2
4
Q = 1 Å-1
Q = 2 Å-1
Q = 3 Å-1
Evaluated
profile Lorentzian
Q (Å-1)
0 1 2 3 4
HW
HM
0.00
.05
.10
.15
.20
.25
.30
Halfwidth of the average
of theLorentzianfunctions
Chudley and Elliott
0
1 1( ) 1 exp( cos )sin
2
1 sin( )1 )
Q iQl d
Ql
Ql
Noticeably differs from the correct result (from the halfwidth of the average of the lorentzian functions)
Chudley and Elliott suggest an averageof the halfwidth at half maximum over all directions of Q
Q2 (Å-2)0 1 2 3 4
HW
HM
0
1
2
3
4
5
6
Q2 (Å-2)
0.0 .2 .4 .6 .8
HW
HM
0
1
2
3
4
5
6
(a)
(b)
At large Q values, oscillatory term negligeable
1lim ( )Q
Q
2 2
0
2 2
1lim ( ) 1 1 ...
3!
6
Q
Q lQ
Q l
Einstein relation
At small Q values, development of the sine function
At low values of Q, Chudley-Elliott model
DQ2 law
Diffusion constant, D, related to residence time t,
2
6
lD
1 sin( )( ) 1 ( ) )
QlQ g l dl
Ql
Other models
Chudley-Elliott model can be generalised:distribution of jump distances, g(l)
Hall and Ross: 2 2
2300
2( ) exp
22
l lg l
ll
2 2
01( ) 1 exp
2
Q lQ
Singwi and Sjölander:
succession of vibratory and oscillatorystates (water).
Chudley-Elliott delocalisation Singwi-Sjölander
r (Å)
0 5 10 15 20 25 30
(r)
0.00
.02
.04
.06
.08
.10
.12
Jobic: delocalisation of the molecules around their sites.
Chudley Elliott
Hall- Ross
Singwi-Sjolander
Hydrogen diffusion
Dominant role within the past (inc(H) = 80 10-24 cm-2)
Still very active field
Hydrogen in TaV5
A.V. Skripov et al.1996
Proton diffusion in relation with protonic conductivity
Perovskites aliovalently doped or non with stochiometric composition: At high temperature: Oxygen ionic conductors
In moist atmosphere can dissolve several mol% of water: proton conductivity
Grotthuss mechanism: proton exchange along an hydrogen bond + OH rotation
Yb-doped SrCeO3 in the temperature range 400-1000°C
(Hempelmann et al., 1995; Karmonik et al,. 1995) Water-doped Ba[Ca0.39Nb0.61]O2.91 between 460 and 700 K. (Pionke et al., 1997)
Systems with hydrogen bonds in stoichiometric quantityHigh conductivity in high temperature phases Several energetically equivalent sites available for each protonCsHSO4 (Colomban et al. 1987; Belushkin et al. 1992; Belushkin et al. 1994)CsOH.H2O (Lechner et al. 1991; Lechner et al. 1993)
ROTATION
EXCHANGE
Solid ionic conductors
Activity has grown rapidly because of potential applications:batteries, fuel cells, electrochemical and photoelectrochemical devices
REQUIREMENTS FOR QENS Favourable neutron cross sections, Ions with high ionic mobility small ionic radius single charge (H+, Li+, Na+, Ag+ or OH-, F- and Cl-)
Ag+ diffusivity in AgI, Ag2Se, Ag2Se, and Rb4Ag4I5
(Funke 1987, 1989; 1993, Funke et al. 1996, Hamilton et al. 2001)
Na+ motions in the two-dimensional ionic conductor -Al203
(Lucazeau et al. 1987) Na+ motions sodium silicate glass, Na2O.2 SiO2
(Hempelmann et al. 1994)
Li+ cations in antifluorite Li2S: diffusion via interstitial hopping
(Altorfer et al. 1994a) Li+ in Li12C60 fulleride (Cristofolini et al., 2000)
Cl- diffusion in SrCl2 and yttrium doped single crystal (Sr,Y)Cl2.03 (Goff et al. 1992)
Intermetallic alloys
Applications for high-temperature environments (aerospace industry)
Diffusion of the constituents themselves
Knowledge of the kinetics on the atomic scale permits:
to control industrial crucial processesto optimize macroscopic characteristics
CsCl structures: NiGa CoGaM. Kaisermayr et al. Phys. Rev. B 61 12038 (2000)
Mechanism of self diffusion not yet fully understoodStructure can host a large amount of vacanciesJump to vacant sites
1- Nearest Neighbour ? Disturbs local order Shorter jump distance
2- Next Nearest Neighbour ? Same sublattice Longer jump distance
E ( µ e V )
- 4 - 2 0 2 4
S(Q,
) (arb. units)
0,0 0,5 1,0 1,5 2,00,0
0,1
0,2
0,3
Two distinct time-scalesLong residence time on regular latticeSmall width
Short residence time on antistructure sitesLarge width
IN16
Quasielastic width
Zeolite MFI structure (silicalite, ZSM-5)
Unit cell: (a = 20.07 Å, b = 19.92 Å, c = 13.42 Å) 96 tetrahedral units, • Si or Al as central atom• Oxygen as corner atoms•4 straight channels•4 zigzag channels•4 channel intersections•10-membered oxygen rings (Ø ~ 5.5 Å)
LONG RANGE DIFFUSION IN MICROPOROUS MATERIALS
Confined media: many effects can be observed: finite size effects, interface effects, dimensionality, liquid dynamics, thermodynamics
Challenge for the theoreticiansImportant industrial processes:• catalysis,• separation of gases and liquidsWide variety of experimental techniques:
Results can vary by up to six ordersof magnitude• Differences between microscopic and macroscopic • QENS and pulsed-field gradient NMR complementary
Workshop« Dynamics in confinement »
Jan 26-29, 2000 GrenobleB. Brick, R. Zorn, H. Büttner
J.Physique IV, Vol 10 PR7-2000
E (µeV)
-10 -5 0 5 10
Inte
nsity
(a.
u.)
0
100
200
300
400
-10 -5 0 5 10
Inte
nsity
(a.
u.)
100
200
300
400
500
n-octaneT = 400 K
isobutaneT = 570 K
Q = 0.87 Å-1
J. Phys. Chem. B 103 (1999) 1096
IN16
Q = 0.87 Å-1
n – octane
T = 400K
Isobutane
T = 570K
H. Jobic et al.J. Phys. Chem. B 103 (1999) 1096
Number of carbon atoms
0 2 4 6 8 10 12 14 16 18
D (
cm2 s-1
)
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
QENS
hierarchicalsimulation
MD
single-crystal
ZLC
linear alkanes / MFI (T = 300 K)
J. Mol. Catal. A 158 (2000) 135
Deviations increase with the number of carbon atoms along the chain:
MD: infinite dilution, rigid framework, no thermalisationBetter agreement at higher loading
Kinetic separation of linear/branched alkanes in Zeolite MFI structure (silicalite, ZSM-5)
time (ns)
0.1 1 10
I(Q,t)
/I(Q
,0)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
C6D6 / ZSM-5 (Q = 0.4 Å-1)
655 K
300 K
465 K555 K
605K
1000/T (K)
1.5 2.0 2.5 3.0 3.5D
(m
2 s-1)
10-16
10-15
10-14
10-13
10-12
10-11
Shah et al.Doelle et al.Zikanova et al. Förste et al.Beschmann et al.Ruthven et al. Rees et al. Karge & NieenSnurr et al. Forester & SmithNSE (this work)
Benzene / MFI
silicalite (˜ , ¿ , q ) ZSM-5 (™, ¯ , s )
J. Phys. Chem. B 104 (2000) 7130H. Jobic et al.
J. Phys. Chem. B 104 (2000) 2878
Benzene / ZSM5 (NSE)
Time-scale extended to longer timesFlux: high loadingAnalysis at low Q
Deuterated samples required
-0.2 -0.1 0.0 0.1 0.2
0
10
20
30
40
50
-0.2 -0.1 0.0 0.1 0.2
0
1
2
3
CH4 /
AlPO4-5
CH4 /
ZSM-48
E (meV)
Inte
nsity
(a.
u.)
Diffusion in 1D channel systems
Resolution: FWHM: 18 µeV ; Q = 0.3 Å-1
D = 10-8
m2/s
F = 10-11
m2/s1/2
CH4 molecules can cross each other
CH4 molecules cannotcross each other
tDx 2 2
tFx 2 2
LOCALISED MOTIONS Jump over two inequivalent sites
11
1 2
11 2
( , ) 1 1( , ) ( , )
( , ) 1 1 ( , ) ( , )
dp tp t p t
dt
dp tp t p t
dt
rr r
rr r
2
22
Equilibrium solution
11 1
1 2
22 2
1 2
( ,0) ( , )
( ,0) ( , )
p p
p p
r r
r r
Solutions:
2
1
( , ) exp( / )
( , ) exp( / )
p t A B t
p t A B t
1
2
r
r
Characteristic time, defined as
1 2
1 1 1
1 2
1
1
2
1
Constant values A and B to be determined from initial conditions
Particle initially on site 1 Particle initiallly on site 2
1 1
22 1
1
( ,0; ,0) 1
( ,0; ,0) 0
p A B
p A B
r r
r r
1 2
22 1
1
( ,0; ,0) 0
( ,0; ,0) 1
p A B
p A B
r r
r r
Conditional probabilities
1 21 1
1 2 1 2
2 22 1
1 2 1 2
( , ; ,0) exp( / )
( , ; ,0) exp( / )
p t r t
p t r t
r
r
1 11 2
1 2 1 2
2 12 2
1 2 1 2
( , ; ,0) exp( / )
( , ; ,0) exp( / )
p t r t
p t r t
r
r
Normalisation1 1 1
1 2 2
( , ; ,0) ( , ; ,0) 1
( , ; ,0) ( , ; ,0) 1
p t p t
p t p t
2
2
r r r r
r r r r
At infinite time, these solutions yield to equilibrium expressions:
11 2 1 1 1
1 2
22 1 2 2 2
1 2
lim ( , ; ,0) lim ( , ; ,0) ( , )
lim ( , ; ,0) lim ( , ; ,0) ( , )
t t
t t
p t r p t r p
p t r p t r p
r r r
r r r
Particle has forgottenits history.
At any arbitrary time
1 1 1 1 1 2 2
2 2 1 1 2 2 2
( , ) ( , ; ,0) ( ,0) ( , ; ,0) ( ,0)
( , ) ( , ; ,0) ( ,0) ( , ; ,0) ( ,0)
p t p t p p t p
p t p t p p t p
r r r r r r r
r r r r r r r
1 1 2 2 1 11
1 2 1 2 1 2 1 2 1 2 1 2
1 2 2 2 2 1
1 2 1 2 1 2 1 2 1 2 1 2
( , ) exp( / ) exp( / )
( , ) exp( / ) exp( / )
p t t t
p t t t
2
r
r
11
1 2
22
1 2
( , )
( , )
p t
p t
r
r
1( , ) ( , ) 1p t p t 2r r
The system is always in equilibrium.
Intermediate scattering function
1 1 2 1 2 1 1
1 2 1 2 2 2 2
( , ) ( , ; ,0) ( , ; ,0)exp .( ) ( ,0)
( , ; ,0)exp .( ) ( , ; ,0) ( ,0)
F t p t p t iQ p
p t iQ p t p
Q r r r r r r r
r r r r r r r
1 2 2 2 12 1
1 2 1 2 1 2 1 2 1 2
1 1 2 1 21 2
1 2 1 2 1 2 1 2 1 2
( , ) exp( / ) exp( / ) exp .( )
exp( / ) exp .( ) exp( / )
F t t t i
t i t
Q Q r r
Q r r
2 2 1 2
1 2 1 22 2
1 2 1 2
21( , ) 2 cos( ) 1 cos( ) exp( / )F t t
Q Q.r Q.r
Depends on the orientation of Q = Q() with respect to the jump vector r = r1 – r2
( , ) ( , )sin F Q t F t d d QPolycrystalline sample
At contrast with the case of long range jump diffusion over absorption sites of a lattice , the exponential term does not depend on the scattering vector Q The powder average yields an analytical expression with a unique exponential function
Powder average
2 2 1 2
1 2 1 22 2
1 2 1 2
21 sin( . ) sin( . )( , ) 2 1 exp( / )
. .
Q r Q rF Q t t
Q r Q r
Term independent on time:localisation of the particle in space
2 2
1 2 1 22
1 2
lim ( , ) ( , )
1 sin( . )2
.
tF Q t F Q
Q r
Q r
Limit at infinite time, of theintermediate scattering function
ELASTIC INCOHERENTSTRUCTURE FACTOR (EISF)
Term dependent on time:Exponential decay of fluctuations
2 21 2
2
1 2
lim ( , )Q
F Q
Oscillatory, with pseudo-période Q.rLimit at high Q
2 21 2
2
1 2
Two sites energetically equivalents
min
min min
min
1 sin( . ) 1 sin( . )( , ) 1 1 exp( / )
2 . 2 .
1 sin( . )lim ( , ) ( , ) 1
2 .
1lim ( , )
2
t
Q
Q r Q rF Q t t
Q r Q r
Q rF Q t F Q
Q r
F Q
If the particle stays indefinitely ober one of the sites whatever it is
2 2 2 21 2 1 2
2 2
1 2 1 21 2lim lim 1
Fourier transform dynamical scattering function
2 21 2 1 22
1 2
1 22 2 2
1 2
1 sin( . )( , ) 2 ( )
.
2 sin( . ) 11
. 1
Q rS Q
Q r
Q r
Q r
32 2
1 sin( 3 2 sin( 3 1( , ) 1 2 ( ) 1
3 33 3 1sites Qr Qr
S QQr Qr
JUMPS OVER 3 EQUIVALENT SITESPolycristalline sample
Example: methyl groups of monohydrated camphor sulfonic acid, C10H15O4S
- - H3O+ (CSA)
3
2 2
12 6( , ) ( ) ( , )
18 18
1 sin( 37 2 ( )
9 3
2 sin( 3 11
9 3 1
sitesS Q S Q
Qr
Qr
Qr
Qr
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0 1 2 3 4 5
Sauts de 120° T = 200 KT = 225 KT = 250 KEchange de protonE
last
ic I
nco
her
ent S
tru
ctu
re F
acto
r
Q (A-1)
Limit of IN6 and IN10
Analysis from 15 to 340 K, with backscattering and time of flight
spectrometries. Uncertainties over experimental
points especially high. Low neutron flux
Ratio mobile/total hydrogen atoms 6:18
0
0.01
0.02
0.03
0.04
0.05
1.5 2 2.5 3 3.5 4 4.5 5
T = 200 K
Q (Å-1)
IN13
0
0.0002
0.0004
0.0006
0.0008
0.001
1 1.2 1.4 1.6 1.8 2
T = 150 K
Jum
p-ra
te (
meV
)Q (Å-1)
IN100.02
0.04
0.06
0.08
0.1
1 1.5 2
Q (Å-1)
IN6 T = 300 K
IN10 T = 150 K
IN13 T = 200 K
Halfwidth at half maximumof quasielastic part of spectra:
HWHM is constant with Q. 1 single lorentzian (2 or 3 sites)
CONTINUOUS ROTATIONAL DIFFUSION ON A CIRCLE
),,(.),,( 02
2
0 tPDtPt R
) exp( )(exp2
1),,( 2
00 tnDtintP R
n
0000 )(),( )0,(),(exp),( ddPtPtitI Rinc RRQ.Q
m
RmRinc tmDQRJtI ) exp()sin(),( 22Q
221
01
1 )( )( )(),(
m
m
m
mAAS
QQQ
with the EISF )sin()( 200 QRJA Q
and quasielastic incoherent structure factors
)sin(2)( 2 QRJA mm Q
Rm Dm21 correlation times tm
rotational diffusion coefficient DR
ISOTROPIC ROTATIONAL DIFFUSION OF AN ATOM ON A SPHERE
),,(.),,( 00 tPDtPt R
1
220 )1(exp )( )12( )(),(
l
RlRinc tDllQRjlQRjtQI
221
01
1 )( )( )(),(
l
l
l
l QAQAQS
)()( 200 QRjQA
Coefficients Al(Q) with l 0 are the quasielastic incoherent structure factors
)()12()( 2 QRjlQA ll
The hwhm of the successive Lorentzian functions in the expansion are directly related to the rotational diffusion coefficient DR and increase with l according to
Rl Dll )1(1
The norbornane molecule
The OPCTS molecule
ELASTIC SCANS; THE FIXED-WINDOW METHOD
Quasielastic broadening
Rapid inspection of dynamicsas a function of temperature.
To be completed by quasielastic measurements
Decrease of theobserved elastic intensity.
Backscattering spectrometersIncident neutrons with exactly the same energy as selected by analyseurs.Recording of elastic scatteringVariation of an external parameter (temperature, sometime pressure.)Apparition of a motion in the experimental time-range
Ne
utro
n co
unt
s
Energy transfer
2 2( , 0) exp( )steS Q C Q u
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100 150 200 250 300 350
T(K)
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100 150 200 250 300 350
Cou
nts
(No
rma
lise
d U
nits
)
2 = 29°
2 = 93°
2 = 156°
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100 150 200 250 300 350
T(K)
Diffusive motions too slowHarmonic vibrations<u2(T)> linear with T
Quasielastic broadening too largeHarmonic vibrations<u2(T)> linear with T
<u2(T)> accounts for bothdiffusive and et vibratory motions
2 2( , 0) exp( )steS Q C Q u
2 2( , 0) exp( )steS Q C Q u
Possible extraction of EISF
Example: camphor sulfonic acid
2 2( , 0) exp( )S Q Q v
Example of application: protein dynamicsDynamical transition of myoglobin hydrated with D2O.From 4 K to 180 K elastic intensity varieswith Q according to a Gaussian law For an harmonic solid: Debye-Waller factor
Mean square displacement <v2> proportional to temperature.Near 200 K apparition of new degrees of freedom. Deviation from Gaussian behaviour which increases with temperature
2 22 2
1 2 1 22
1 2
exp( ) sin( . )( , 0) 2
.
Q Q rS Q
Q r
v
Initial slope of the curve.
22 1 2
2 21 20
ln ( ,0)
3( ) ( )Q
d S Q r
d Q
v
From refinement to experimental data,<v2>, r2 et 1/2.
Free energy G1
2exp( / )G RT
Myoglobin (Doster et al.):asymmetry of well DH = 12 kJ.mol-1
Entropy DS/R = 3,0,Jump distance r = 1.5 Å.
EXAMPLES OF ROTATIONAL DIFFUSION OF MOLECULES AND CHEMICAL GROUPS
Mostly based on jump models
Earlier studies: SH reorientations in NaSH, CsSH
Rowe et al., J. Chem. Phys. 58, 5463 (1973)120° jumps of methyl groups in para-azoxy-anisole
Hervet et al. (1976)Rotations of aromatic rings
Chhor et al. (1982)Several axes: use of group theory
Rigny Physica 59 707 (1972)Thibaudier and Volino Mol. Phys. 26 1281 (1973); ibid. 30 1159
(1975)
Some cases of rotational diffusion
OctaphenylcyclotetrasiloxaneBée et al. J. de Chimie Physique 83 10 (1984)
NorbornaneBée et al. Mol. Phys. 51 2 221(1986)
NH NH NH NHn
NH NH Nn
N
DopingAH
Semi-oxydized, non conductive form: emeraldine base
Insoluble powder, non conductive10-10 S/cm
Insoluble powder,1-20 S/cm
Conductive salt of emeraldine
SO3HO
Doping by HCl, H2SO4…
Protonation with a functionalized organicacid: camphor sulfonicacid (CSA)Polar solvent: meta-cresol
Brittle films100-400 S/cm
+
A- A-+
Dynamics of counter-ions in conducting polyaniline.
D. Djurado et al. Phys. Rev. B 65 (2002) 184202
Polyaniline chain
Camphor sulfonic acid
150
200
250
300
350
0 50 100 150 200 250 300
Co
nd
uctivity
(S
/cm
)
Temperature (K)
Electrical transition
Semiconductingregime
Metallicregime
Chains are immobileMotions of CSA
Methyl groups
WholeCSA
DEHEPSA1,2 di(2-ethylhexyl)esterof 4-sulphophtalic acid
DPEPSAdi-n-pentyl ester of 4-sulphopthalic acid
O
S
CH
New families of plastdopants: high conductivity, improved flexibility
B. Dufour et al. Chem. Mat. 13, 4032 (2001).
0.01
0 50 100 150 200 250 300 350
Ela
sti
c In
ten
sit
y
T (K)
0 50 100 150 200 250 300 350
0.01
Ela
sti
c In
ten
sit
y
T (K)
0.001
0.01
0 50 100 150 200 250 300 350
Ela
sti
c In
ten
sit
y
T (K)
0 50 100 150 200 250 300 350
0.01
Ela
sti
c In
ten
sit
y
T (K)
2 = 24.5°
Q = 1.2 A-1
2 = 45.6°
Q = 2.8 A-1
2 = 66.8°
Q = 3.1 A-1
2 = 88.0°
Q = 3.9 A-1
PANI / DEHEPSA : IN13 10-10 s time-scale
0.7
0.75
0.8
0.85
0.9
0.95
1
0 0.5 1 1.5 2
T = 330 KT = 302 KT = 281 KT = 251 KT = 331 KT = 301 KT = 281 KT = 251 K
Ela
stic
Inco
here
nt S
truc
ture
Fa
ctor
Q (A-1)
DPEPSADEHEPSA
EISF is temperature dependent:Variation with T- Of the region of space accessible to scatterers- Of the fraction of mobile scatterers
Given T, DPEPSA more ordered than DEHEPSA - Smaller chain- Linear chain: packing favored
High values of EISFFraction of
non mobile scatterers
20 10
( )( ) 3
j QRA Q
QR
Model: Diffusion inside a sphereF. Volino and A.J. Dianoux; Molecular Physics 41 (1980) 271
00 2 2
, 0,0
1( , ) ( ) (2 1) ( )
( )
ll nn l
l m n
DS Q A Q l A Q
D
22
212 2 2
2 22 2
2 2
6 . ( ) . ( )( ) if
( 1)
3 ( 1)( ) ( ). if
2
nl nl l ln ln n
l l
l nn l l
R QR j QR l j QRA Q Q
R l l R Q
Q R l lA Q j QR Q
Q R
EISF
Structure factors
Parameter # 4
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
Incomplete Gamma function
(a,x)
x
a = 7
a = 1
a = 4. ( , )mR R a mp
1 2( , ) . ( ) . ( ) ( , ; )m mm
S Q S Q R Polymer chain Counter-ion head Counter-ion tail
Distribution of radii alongcounter-ion tail
3 ParameterspR1
R2
R3
Biological studiesProteins exhibits a complex structure:
several thousand atomsfolds into a unique 3-D structure
Unfolding of biomolecules Loose of biological activity
Biological function Flexibility
Rich dynamical spectrum: 10-14 to 101 s: manyinstruments concerned
Nanosecond - picosecond regime: major role for intermolecular recognitionand for enzymatic reactions: all QENS instruments are concerned
RigidityPrerequisite of
specificitycompromise
FlexibilityProper function
of protein
Non exchangeable H atoms, distributed homogeneously within the samplegive information on the overall dynamics and on those of the larger groups to which they are bound. (J. C. Smith Quarterly Review of Biophys. J. 76, 1043 (1999)
Flexibility of DNA as a function of hydrationA.P. Sokolov et al. J. Biol. Physics 26 S1 (2000)
Dynamic transition at T ~ 200 - 230 KNot clearly understoodImportant for protein function
Slow relaxationprocess
Fast relaxationprocess
At low temperaturelarger flexibility forlow water content
At high temperature a slow relaxation process appears:Cooperative motion of many DNA’s monomers involving all parts of the molecule (backbone and base pairs)
Strongly depends on the level of hydration
Influence of the solvent in biopolymer dynamics
DEVELOPMENT OF THEORETICAL MODELS
Long range diffusionLattice model with gaussian statistics F. Volino et al. Europhys. J. B 16, 25 (2000)Single-file diffusion K. Hahn et al. Phys. Rev. E 59 6662 (1999) K. Hahn et al. J. Phys. A 28 3061 (1995)Extended Hall and Ross model H. Jobic J. de Physique IV 10 (2000) 77
PolymersMode Coupling TheoryRotation-Rate Distribution Model A.Chahid et al. Macromolecules 17 3282 (1994)
BiologyForce Constant Analysis D. Bicout and G. Zaccaï Biophys. J. 80, 1115 (2001)Combined dynamics: jump and diffusion inside a sphere D. Bicout, Proceedings ILL Millenium Symposium (2001)Diffusion inside two concentric spheres D. Bicout Physical Review E, 62, 1, 261 (2000)Influence of environment fluctuations D. Bicout Physical Review E, 64, (2000)Damped collective vibrations, diffusion in anharmonic potential G. Kneller Chemical Physics 261 1 (2000) K. Hinsen et al. Chemical Physics 261 25 (2000)
MOLECULAR DYNAMIC SIMULATIONS
Seem very promissing for futureStill require long computing time.Up to now application to QENS restricted to small moleculesMethods have been developed for larger molecules K. Hinsen et al. Chemical Physics 261 25 (2000) K. Hinsen and G. Kneller J. Chem. Phys. 111, 24 (1999) 10766
Require a reliable interaction potential Can be a serious difficulty (delocalised charges) Can require long preliminary calculations calculations Need to be checked carefully by comparison with experimental results Necessary to evaluate QENS spectra (nMOLDYN) G. Kneller et al. Comp. Phys. Comm. 91 (1995) 191
Under that conditions, analysis of atoms trajectories can be of considerable help in understanding neutron resultsDistribution of Potential barriers for methyl groups in glassy polyisopreneF. Alvarez et al. Macromolecules 33 (2000) 8077
J. Combet; Journées de la diffusion neutronique 2001
Orientational distribution
FCC PhaseFCC Phase
Dynamics more complexThan an isotropic rotation(preferential orientations).
HCP PhaseHCP Phase
MOLECULAR DYNAMIC SIMULATION OF NORBORNANE
Good agreement with earlier experimental neutron results
CONCLUSIONS AND PERSPECTIVES
Now
Most of samples are very complicated, involving-) many types of scatterers,-) many types of motions, no symmetry-) many time-scales
Models often offer a rather abstract form They can not easily be connected with experimental data.
Molecular dynamic simulations are very long
Pioneer time is finished
“Simple” samples, geometrically well described, stimulated: -) numerous scattering models especially for rotation -) original theoretical approaches based on symmetry
They evidenced the existence and the importance of EISF.
To go further
Collaborations are required:between experimentalists and theoreticiansbetween theoretical calculations and simulations
Multiply environment conditions:temperature, pressure, concentration, selective deuteration, illumination, …
Multiply experimental conditions:analyse sample with different instruments and/or resolutionsanalyse sample with different energy rangesR.E. Lechner Physica B 301 (2001) 83
use polarised neutrons to separate coherent from incoherent scatteringB. Gabrys et al. Physica B 301 (2001) 69
Improve software for data treatment:signal processing (intermediate scattering function)develop new analyses of data (memory function)
Collect maximum of information about the system:from other techniques from other scientific communities (chemistry, physics, biology,
biophysics, ….)