Miguel A Gonzalez Institut Laue Langevin Grenoble France gonzalezmilleu Outline General remarks and reminders The main equations and their physical meaning QENS models for translational diffusion and localized motions ID: 315111
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Slide1
Quasielastic Neutron Scattering
Miguel A. Gonzalez
Institut
Laue-
Langevin
(Grenoble, France)
gonzalezm@ill.euSlide2
Outline
General remarks and reminders
The main equations and their physical meaning
QENS models for translational diffusion and localized motions
The EISF and its physical interpretation
Instrumentation: A Neutron Backscattering spectrometer (IN16
)
Examples
Complex systems and MD simulations
Conclusions and referencesSlide3
Neutron scattering: What can we see?Slide4
Coherent and incoherent neutron scattering•
Incoherent scattering
appears when there is a random variability in the
scattering lengths of the atoms in the sample, e.g. different isotopes or isotopes with non-zero nuclear spin
so (b
+
= I + ½)
(b
= I
½) .
•
Coherent scattering:
Information on spatial correlations (structure) and/or collective
motion.
– Elastic: Where are the atoms? What are the shape of objects?
– Inelastic: What is the excitation spectrum in crystals – e.g. phonons?
– Quasielastic: Correlated diffusive motions.
•
Incoherent scattering:
Information on single-particle dynamics.
– Elastic: Debye-Waller factor, Elastic Incoherent Structure Factor (EISF)
geometry of diffusive motion (continuous, jump, rotations)
–
Inelastic
:
Molecular
vibrations
– Quasielastic: Diffusive
dynamics
, diffusion coefficients.
Here
focus on
quasielastic
incoherent
neutron
scattering
(
QEINS or QENS)
!Slide5
From
Jobic
&
Theodorou
,
Micropor
.
Mesopor
. Mater.
102, 21-50 (2007)
When will we have incoherent neutron scattering?
Mainly incoherent scatterers: H 49Ti V 53Cr Co Sm
Or if polarized neutrons are used to separate coherent and incoherent scattering! Slide6
From
Heberle
et al.,
Biophys
. Chem.
85
, 229-248 (2000)
EINS and QEINS: Main information
Elastic intensity
Quasielastic intensity
Quasielastic broadening
Debye-Waller factor: Vibrational amplitudesA0 = EISF (ratio elastic/total): Geometry of motion Width: Characteristic time scaleSlide7
PELICAN@ANSTO
A true QEINS spectrum: water
Teixeira et al.,
Phys. Rev. A
31
, 1913 (1985)
Qvist
et al.,
J. Chem. Phys.
134
, 144508 (2011) Neutron exchanges small amount of energy with atoms in the sample: Typically from 0.1 eV (BS) to 5-10 meV
(TOF).
•
Vibrations normally appear just like flat background and treated as
Debye-Waller.
•
Maximum of intensity is at
= 0.
•
Low-Q – typically < 5 Å
1
and often <2-3 Å
1
.
IN6@ILL
IN5@ILLSlide8
The instrumental resolution and the
dynamical
window (maximum
energy transfer) determine
the observable timescales:
IN16:
1
eV
min
0.1
eV
t
max
2/
min
40 ns
max
15
eV
tmin 275 ps IN13: 8 eV min 1 eV tmax 4 ns max 100 eV tmin 40 ps IN5: 50 eV min 5 eV tmax 800 ps max 10 meV tmin 0.4 ps The Q-range determines the spatial properties that are observable. Typical range (IN16, IN5) is 0.2 – 2 Å1 3 – 30 Å. In IN13, Qmax 5 Å1 dmin 1 Å. Instrumental limitations (limited Q-range, resolution and energy range) together with the complexity of the motion(s) can make interpretation difficult.
Instrumental constraintsSlide9
QEINS is associated with relaxation phenomena, such as translational diffusion, molecular reorientations
,
confined
motion within a pore, hopping among sites, etc
But how is related the QEINS signal or broadening with the physical information of interest to us?Slide10
Master equation
intermediate scattering function, I(
Q,t
)
DIRECT RELATION: Measured quantity
Physical
information
d
2
/
dd
S(Q
, )
We can measure the
double differential cross section
, i.e. the number of neutrons scattered into a detector having a solid angle
and with an energy between and +d
and this can be easily related to the
dynamical structure factor
, S(Q
,
), which
is a correlation function related only to the properties of the scattering system.Slide11
Self correlations (incoherent scattering)
self intermediate function
FT
in time
FT
in space
S
inc
(
Q
,
)
I
self
(
Q
,t
)
G
self
(
r
,t
)
[energy]
1
[] [volume]1Slide12
Physical meaning of G
self
(
r,t
)
G
s
cl
(
r
,t)dr is the probability that, given a particle at the origin at time t=0, the same particle is in the volume dr at the position r at time t !
From “Neutron and X-ray spectroscopy” (Hercules school)Slide13
Properties of G
self
(
r,t
),
I
self
(
r,t
) and
S
inc(Q,)
G
s
(r,0) =
(r)
G
s
(r,t
)
1/VSlide14
FROM THE GENERAL EXPRESSION TO USEFUL MODELSSlide15
Self intermediate scattering functionSlide16
A full analytical evaluation of Is(Q,t) is impossible*
unless we assume that we can separate motions having different time scales and neglect any coupling between them:
Vibrations: internal (molecule), external (lattice vibrations).
Local motions: local diffusion,
molecular
reorientations
.
Translational diffusion.
This is valid to separate vibrations from translations or rotations, as they have very different time scales (typically 10
14
s for vibrations and 1012 -1011 s for diffusive motions, either reorientations or translational diffusion). Separating translational and rotational diffusive motions is less satisfactory, but nevertheless accepted in most cases as the only way to proceed (again the importance of roto-translational coupling in the experimental spectra can only be judged from computer simulations, e.g. work of Liu, Faraone and Chen on water).* Is(
Q,t
) can be computed without approximations from a computer simulation trajectory (as we have
r
(t) for all atoms). This can be compared to experimental results, but there is not yet a direct way to refine it using the experimental S(Q,
).
Approximations or assumptionsSlide17
Self intermediate scattering
function and incoherent dynamical structure factorSlide18
Vibrational termsSlide19
A first (too general) expression to fit to our data
Adding instrument resolution and assuming that vibrations appear as flat background:
S
inc
(Q,
) = B(Q) + e
u
2
Q
2
[ST(Q,) SR(Q,)] R(Q,)Slide20
Brownian motionE.g. liquid argon:
Very weak interactions + small random displacements. Collisions are instantaneous, straight motion between them and random direction after collision.
If Q is low enough to loose the details of the jump mechanism (because we look to a large number of jumps) we can use the same expression used to describe macroscopic diffusion (
Fick’s
law).Slide21
Fick’s 2nd law tells how diffusion causes concentration to change with time:
Translational diffusion (Brownian motion)
We can arrive to an equivalent expression by introducing P(l,
), which is
the probability of a particle travelling a distance l during a time
, after a collision:
And we have the following conditions
:
Slide22
Translational diffusion (Brownian motion)
Neutron spectrum is a
lorentzian
function with a width increasing strongly with Q: HWHM = DQ
2
.
FT in space
FT in timeSlide23
Translational diffusion (Chudley
-Elliott model)
Model for jump diffusion in liquids (1961).
Atoms or molecules ‘caged’ by other atoms and jumping into a
neighbouring
cage from time to time.
Jump length
l
identical for all sites.
Can be applied to atom diffusion in crystalline lattices.
l
= 1 Å
D = 0.1 Å
2
m
eV =
1.519 × 10
5
cm
2
/sSlide24
Jump diffusion in cubic lattices
Lattice constant
a
and coordination number
z
= 6.
Jump vectors
(
a
, 0, 0), (0,
a
, 0), and (0, 0, a). If crystal oriented with x-axis parallel to Q: = 1 meV1 = 0.658 psSlide25
Localized motion
Hopping between 2 or more sites, e.g. CsOH
H
2
O, crystals, …
Intramolecular reorientations, e.g. CH
3
jumps, motion of side groups in polymers and proteins, …
Molecular rotations, e.g. plastic crystals, liquid crystals, …
Confined motion, e.g. in a pore
All such motions are characterized by the existence of a non-null Q-dependent elastic contribution
elastic incoherent structure factor (EISF). Slide26
r
1
r
2
Jump model between two equivalent sites
And assuming that at t = 0, the atom is at r
1
:
Solutions are:
Slide27
r
1
r
2
Jump model between two equivalent sitesSlide28
r
1
r
2
Jump model between two equivalent sites
If powder, average over all possible orientations
d
EISF
QISFSlide29
r
1
r
2
Jump model between two equivalent sites
d
Half
width
~
1/
(independent of Q)
w
0
A
0
d
(
w
)Slide30
r
1
r
2
Jump model between two equivalent sites
d
Qr
A
0
(Q)
= ½[
1+j
0
(
Qd
)/(
Qd
)]
EISF
1
½
Q
HWHMSlide31
EISFs corresponding to different rotation modelsSlide32
EISFs and widths of different rotation modelsSlide33
Physical meaning of the EISF
And if the system is in equilibrium, there are no correlations between positions at t=0 and t=
, so:
And the EISF is easily obtained as the ratio between the elastic intensity and the total (elastic + quasielastic, no DW) intensity:
Direct information about the region of space accessible to the
scatterers
(Bee,
Physica
B
182
, 323 (1992)) Slide34
Physical meaning of the EISF
2
/Q
If the atom moves out of the volume defined by 2
/Q in a time shorter than
t
max
set by the instrument resolution it will give rise to some quasielastic broadening
loss of elastic intensity.
The EISF is essentially the probability that a particle can be found in the same volume of space after the time
t
max
.Slide35
The EISF can be obtained without any ‘a priori’ assumption and compared to any of the many physical models available in the literature (see M. Bee: “Quasielastic Neutron Scattering”, 1988).
In this way we can determine the geometry of the motion that we observe and then apply the correct model to obtain the characteristic times.
Caveat: In complex systems this is not a trivial task and can be even impossible. In such cases it is useful to recourse to computer simulations.Slide36
A NEUTRON BACKSCATTERING SPECTROMETER:IN16Slide37
Backscattering is a special kind of TASSlide38
Best resolution when 2
= 180 (backscattering)Slide39
BS instruments in the practiceSlide40
IN16 at ILL
Si(111)
Si(111)Slide41
Performing an energy scan
- Move monochromator with velocity v
D
parallel to reciprocal lattice vector
.
- Energy of reflected neutrons modified
by a longitudinal Doppler effect (the
neutrons see a different lattice
constant in case of a moving lattice).
- Register scattered neutrons as a
function of Doppler velocity vD
.
- Maximum achievable speed determines
max energy transfer (~10-40 eV)
- Or change the lattice distance of the monochromator by heating/cooling.
- Need crystals having a large thermal
expansion coefficient, good energy
resolution and giving enough intensity.
- Possible energy transfers > 100 eV Slide42
IN16: Resolution
better than 1
eVSlide43
Fixed window scan: Measure S(Q,
~0)
Obtain an effective
mean square displacement!
Dynamical transition in proteins (
Doster
et al., Nature 1989)Slide44
Tunnelling spectrum of NH
4
ClO
4
and with different levels of partial deuteration
Probe potential energy barriers and rotational potentials (test for simulations)
Nuclear hyperfine splitting of
Nd
Low-frequency excitationsSlide45
Quasielastic scattering: motions in a polymerSlide46
EXAMPLESSlide47
- Dislocation pipe diffusion
enhanced atomic migration along dislocations due to a reduced activation barrier.
- Can improve diffusivity by orders of magnitude.Slide48
Hydrogen diffusion in
Pd
QENS spectra (BASIS, SNS) & fits
Line widths (
Chudley
-Elliot model)
l &
- D is lower by 2-3 orders of magnitude compared to regular bulk diffusion.
- Diffusivities for hydrogen DPD characterized by much lower E
a.Heuser
et al.,
PRL 2014Slide49
Hydrogen diffusion in Pd
Heuser
et al.,
PRL 2014
- Suggest existence of a continuum of lattice sites associated with dislocations.
- Reduced site blocking.
- H de-population of dislocation trapping sites
goes as
e
kT
bulk regular diffusion above 300 K.- DFT shows metastable sites characterized by a lower activation energy for diffusion.
- DPD expected to depend on H concentration and dislocation density.
QENS represent a unique experimental scenario that allows the diffusivity associated to dislocation pipe diffusion to be directly quantified!
(QENS ~ 230
meV
)
(QENS ~ 40-80
meV
)Slide50
- Fe(
pyrazine
) [Pt(CN)
4
] spin crossover (SCO) compound.
-
Neutron diffraction points to free rotations of the ligand in the HS, which are blocked in the LS.Slide51
Rodriguez-
Velamazan
et
al.,
JACS 2012
HS
LS
Bz
295 K
(IN5)- Switching of rotation associated with change of spin state.- In HS,
pz
rings perform 4-fold jump motion about the coordinating N axis.
- Correlation between rotation of
pz
and change of spin state practical element for creating artificial molecular machines.Slide52
- Benzene and (
PyH
)I (at high-T) show a 6-fold potential with equivalent minima.
- At low-T, (
PyH
)I has a different crystalline phase and NMR indicates that reorientations in this phase take place in an asymmetric potential.
- MD in good agreement with QENS/NMR data and indicates that asymmetric potential is due to the formation of weak H-bonds N-H
I
.
Bz
(PyH)I EISF (6 equivalent sites)Slide53
Pajzderska
et
al.,
JCP 2013
(
PyH
)NO
3
@ IN10 (ILL)
MD snapshot for (
PyH
)NO3 @ 290 K- In (PyH)NO3 only two orientations are significantly populated.
- Two-well asymmetric potential related to the two orientations where
N-H
O
hydrogen bonds can be formed.
- Picture confirmed by MD simulations.
QENS
MDSlide54
Dealing with complex systems …
How does the microscopic structure and dynamics change with varying alkyl chain length?
- In most cases, need some kind of computational model to understand and interpret the QENS spectra.
- The most useful tool is MD (either using empirical potentials or using ab initio DFT to compute interatomic forces)
Solve Newton’s equation for a molecular system:
m(d
2
r
i
/dt
2
) =
f
i
=
u(
r
)
- From the MD simulation we will get the trajectory of all the atoms in our model (typically 10
2
for DFT, 10
5
for classical MD) during the simulation time (typically several
ps
for DFT, hundreds of ns for classical MD)
Compute all kind of properties and, in particular, I(
Q,t
).
- Today there are many available tools that can help us doing this.Slide55
Molecular dynamics in metallic and highly plastic compounds of polyaniline
A study using quasi-elastic neutron scattering measurements and molecular dynamics simulations
(
Maciek
Sniechowski
, David
Djurado
, Marc Bee, Miguel Gonzalez …)
reduced
oxidized Polyaniline –
Emeraldine base
(insulating form)
H
+
Emeraldine salt
(conducting form)
H
N
N
N
N
H
H
H
+
+
H
N
N
N
N
HSlide56
Structural
Analysis
3.5
Å
~25-38
Å
Structural
modelSlide57
Quasi-elastic neutron scattering (QENS) studies of
polyaniline
/DB3EPSA
TOF Spectrometers:
IN6
ILL (50-100
m
eV)
MIBEMOL
LLB Saclay (85
meV)IRIS at ISIS GB (15
m
eV)
DB3EPSA
- Classical
QENS data analysis
in
terms of EISF
- MD
Simulations (Compass in
Cerius
/MS)
->
S(
Q,w
), I(
Q,t
):
comparison
with experiment
confirmation of theoretical modelSlide58
The radii of spheres are distributed in size along the
alkoxy
tails according to a
gamma function: 3
parameters to adjust for varying the curve
shape!
R1
R3
R4
R5
R6
R7
R8
R9
R10
Model of local diffusion of protons in spheres
Volino
and
Dianoux
,
Mol. Phys.41, 271,(1980)Slide59
R(
q,t
) :
resolution function of IN6
Dynamic structure factor
S
inc
(
q,w
) IN6 spectrometerSlide60
R1
R2
R6
R10
R0
DB3EPSA
PANI chain
R1
R0
R2
R6
R10
T=340 K
Analysis of the individual atom trajectories
Mean square displacement:
MD confirms the model employed to fit the QENS spectra!Slide61
A complex example containing several contributions …
How does the microscopic structure and dynamics change with varying alkyl chain length?
EmimBr
or C
2
mimBr
BmimBr
or C
4
mimBr
HmimBr
or C
6
mimBr
Room temperature ionic liquids based on the
imidazolium
cationSlide62
F(
Q,t
)
DW x T(
Q,t
) R(
Q,t
) L(
Q,t
)
S(Q,
) exp(Q
2
u
2
) × [T(Q, ) R(Q, ) L(Q, )]
T(Q,
) L(
T
(Q)DQ
2
)
R(Q,
) A
0
R
+ (1-A
0
R
) L(
R
)
L(Q, ) A0L + (1-A0L) L(L)S(Q,) A0RA0L L(T) + (1A0R) A0L L(T +R) + A0R (1A0L) L(T +L) + (1A0R) (1A0L) L(T +R +L) If R 0 (MD, NMR Imanari 2010) then: S(Q,) A0L L (T) + (1A0L) L (T +L) QENS analysisSlide63
T
(K)
D
(10
10
m
2
s
1
)
t
0
(
ps
)
353
2.7 ± 0.2
3.9 ± 0.6
373
3.4 ± 0.5
2.6 ± 0.4
392
5.1 ± 0.7
3.2 ± 0.2
412
6.6 ± 0.9
2.5 ± 0.2
D follows Arrhenius
law:
D = D
0
exp
(
E
a
/ RT
)
with
D
0
= (1.7±0.8).
10
7
m
2
s
-1
E
a
= 19 ± 2 kJ
mol
-1
Reasonable agreement with NMR (Every, PCCP 2004), although D values 3-4 times larger.
Data fitted with two
lorentzians
: 1 translational-like + 1 local-like
QENS: C
2
mimBr translational dynamicsSlide64
(solid)
(liquid)
QENS: C
2
mimBr local dynamicsSlide65
QENS analysis using MD input (C
2
mimBr)
Aoun
et al. ,
J. Phys. Chem. Letters
1
, 2503 (2010)Slide66
Quasielastic widths: Simulation
vs
experiment
Liquid 360K
Liquid 360K
Crystal 300K
Simulation:
D (from width of narrow line) = 4.9 x 10
10
m
2
/s
vs
3.2
x 10
10
m
2
/s obtained directly from
m.s.d
.!Slide67
EISF: Simulation
vs
experiment
Qualitative or even
semiquantitative
agreement between experimental (fitted S(Q,
)) and simulated (fitted F(
Q,t
) with equivalent model)
widths and EISF’s.Slide68
Simulated spectra and components
COM trajectory
Ring rotation (with fixed methyl + alkyl)
Local motions (no COM or global rotation)Slide69
Center of mass trajectory
Self-diffusion coefficient consistent with value of D extracted directly from the mean square displacements. Slide70
Global rotation
EISF by groups
When looking to individual groups, reasonable agreement with model of diffusion on the surface of a sphere.Slide71
Local motions: Dihedral torsionSlide72
Simulated spectra: EISF for chain motions
Possible to fit to model of rotation in a circle. But meaningful? Slide73
Local motions: Spatial distribution (in the crystal)
C6 & C7
Methyl
CH
2
in ethyl chain
CH
3
in ethyl chain
MD can give a much clearer picture of how the molecules really move,
but they can also be misleading, so they should be validated using experimental data!Slide74
• Applicable to wide range of scientific areas: – Biology: dynamic transition in proteins, hydration water, ...
–
Chemistry:
complex fluids, ionic liquids, porous media, surface
interactions, water at interfaces, clays, ...
–
Materials science:
hydrogen storage, fuel cells, polymers, ...
•
Probes true “diffusive” motions.• Range of analytic function models
systematic comparisons.• Close ties to theory – particularly Molecular Dynamics simulations. Complementary to techniques such as light spectroscopy, NMR,dielectric relaxation, etc.• Unique – Can answer questions you cannot address otherwise: – (Q, ) information: provides information about the dynamics on length scales given by Q. – Very sensitive to H –
Able to test microscopic models of motion and MD simulations
–
Large range of time scales:
From sub-
picosecond
to several ns
CONCLUSIONS
Or why should I use Quasi-elastic Neutron Scattering? Slide75
REFERENCES
•
Quasielastic Neutron Scattering,
M. Bee (Bristol, Adam
Hilger
1988)
•
Quasielastic Neutron Scattering and Solid State Diffusion,
R.
Hempelmann
(Oxford University Press 2000).
• Neutron and X-ray Spectroscopy, F. Hippert et al. (eds) (Springer 2006): Focused more on instrumentation.• Collection of articles from JDN8 school (Diffusion Quasiélastique des Neutrons): In french, but free access from SFN web page (www.neutron-sciences.org Écoles thématiques) .•
Quasielastic Neutron Scattering,
G. R. Kneller (Lecture for Hercules course, available at http://dirac.cnrs-orleans.fr/~kneller/HERCULES/hercules2004.pdf)
•
Quasi-elastic neutron scattering and molecular dynamics simulation as complementary techniques for studying diffusion in
zeolites
,
H.
Jobic
and D. N.
Theodorou
,
Micropor
.
Mesopor
. Mater.
102
, 21-50 (2007).