How to determine diffusion rates
experimentally ?
Patrice Theulé
Aix-Marseille Université
Physics of Ionic and Molecular Interactions laboratory
Grain-surface networks and data for astrochemistry workshop, Leiden 2014
Outline
 generalities on diffusion
 surface diffusion
 volume diffusion
why is diffusion important ?
 for solid-state reactivity
• diffusion-limited reactivity
• cage effect
 for gas-solid interaction
• diffusion-limited desorption
• trapping
why is diffusion important ?
 for solid-state reactivity
• diffusion-limited reactivity
• cage effect
 for gas-solid interaction
• diffusion-limited desorption
• trapping
why is diffusion important ?
 for solid-state reactivity
• diffusion-limited reactivity
• cage effect
 for gas-solid interaction
• diffusion-limited desorption
• trapping
how do we consider diffusion ?
our representation of the diffusion phenomenon (the model) underlies:
 the experiments carried out
 the way we analyze the experimental data
 the measured quantities
Diffusion: the macroscopic approach
The diffusion equation
particle flux
2D diffusion
[particles.cm-1.s-1]
n [particles.cm-2] [cm2.s-1]
3D diffusion
n [particles.cm-3]
[particles.cm-2.s-1]
[cm2.s-1]
The diffusion equation
3D diffusion
[particles.cm-2.s-1]
particle flux
the diffusion equation
[cm-2.s-1] [cm-3.s-1]
2D diffusion
[particles.cm-1.s-1]
experiments usually designed to have diffusion
along a single dimension
[cm-2.s-1] [cm-2.s-1]
[cm2.s-1]
The diffusion coefficient
The Stockes-Einstein equation
(for spherical particles diffusion and low Reynolds numbers)
D=
k BT
6π η r
the size of the diffusing particles
the viscosity (Pa.s-1 or kg.m-2.s-1)
Diffusion: the microscopic approach
Site to site diffusion
the Einstein-Smoluchowski random walk (Brownian motion)
•
thermal hopping from one binding site to another

Eb
mπ 2

k
exp
=
−
thermal hopping rate: hop
 k T
2n s E D
 B grain





-1
[s ]
ν0
•
models : 1 D view




tunneling diffusion through the binding energy barrier
tunneling rate:
ktun = v0 exp[-(4πa/h)(2mEb)1/2]
[s-1]
Macroscopic-microscopic approaches relation
The Einstein equation relates macroscopic transport coefficient D with microscopic information
on the mean square distance <r2(t)> of molecular migration over the time t
2D diffusion
2
r (t ) = 4 × D × t
r2=x2+y2
3D diffusion
2
r (t ) = 6 × D × t
r2=x2+y2+z2
Outline
 generalities on diffusion
 surface diffusion
 volume diffusion
Outline
 generalities on diffusion
 surface diffusion
•
•
•
•
secondary ion mass spectroscopy (SIMS)
atomic force microscopy (AFM)
thermal desorption spectroscopy (TDS) experiments
laser resonant desorption (LRD)
 volume diffusion
Secondary ion mass spectroscopy
1-2 nm
a very sensitive surface analysis technique, but not quantitative
Secondary ion mass spectroscopy

r 2 (t )
PDMS: polydimethylsiloxane
Secondary ion mass spectroscopy
circular area (radius r)
D= 10-7-10-6 cm2.s-1
Outline
 diffusion
 surface diffusion
•
•
•
•
secondary ion mass spectroscopy (SIMS)
atomic force microscopy (AFM)
thermal desorption spectroscopy (TDS) experiments
laser resonant desorption (LRD)
 volume diffusion
Atomic Force Microscopy
Atomic Force Microscopy
IMC= indomethacin= 1-(p-chlorobenzoyl)-5-methoxy-2-methyl-indole-3-acetic acid
h(t)
2
r (t )
Atomic Force Microscopy
AFM
h(t) grating amplitude
K grating decay constant [s-1]
h(t ) = h0 × exp(− K t )
optical diffraction
Zhu et al., Phys. Rev. Lett., 106, 256103, 2011
Atomic Force Microscopy
• at high temperature (T > 327 K) the viscous flows dominates
• at low temperature (T < 327 K) the surface diffusion dominates
• surface diffusion is at leat 106 times faster than bulk diffusion
Zhu et al., Phys. Rev. Lett., 106, 256103, 2011
Outline
 diffusion
 surface diffusion
•
•
•
•
secondary ion mass spectroscopy (SIMS)
atomic force microscopy (AFM)
thermal desorption spectroscopy (TDS) experiments
laser resonant desorption (LRD)
 volume diffusion
Surface TDS experiments
indirect method, with several processes at work
need a model, assumptions and
several parameters to interpret the data
3. detection: temperature ramp, desorption
and mass spectroscopy detection of the product
increased mobility and reactivity, Edes
2. reactants diffusion + reaction at a fixed temperature
[particles.cm-1.s-1]
D [cm2.s-1]
1. reactants (atoms, simple molecules) deposition (at room temperature)
Flux, sticking coefficient
Ereact, reaction order
Surface TDS experiments
+ other groups
Surface TDS experiments
1. bombardment at fixed temperature (10-15K) for several durations
2. no waiting time at a fixed temperature,
diffusion and reactivity assumed very fast
3. Detection of H2 production by TPD on a ramped temperature
Katz et al., ApJ,522, 305, 1999
Surface TDS experiments
assumptions: Langmuir-Hinshelwood mechanims only, no Eley-Rideal mechanims, thermalized
atoms, single monolayer coverage, no barrier for reaction, orders of diffusion, distributions of sites,
roughness of the surface, microscopic diffusion, limiting rate limit, ν
model:
incoming flux (calibration)
desorption
chemistry: 1 (no barrier)
diffusion: single Arrhenius law
H
H2
chemical desorption
µ= fraction of H2 remaining on the surface
diffusion energy E0
uncertainties on several parameters, correlation, more parameters do not necessarily help
Katz et al., ApJ,522, 305, 1999
Outline
 diffusion
 surface diffusion
•
•
•
•
secondary ion mass spectroscopy (SIMS)
atomic force microscopy (AFM)
thermal desorption spectroscopy (TDS) experiments
laser resonant desorption (LRD)
 volume diffusion
Laser Resonant Desorption
diffusional refilling
LRD surface diffusion measurement
pb: adsorption from the background must be subtracted
from the diffusion refilling signal
upper limit (laser beam spot size) of D < 4 10 -11 cm2.s-1
surface diffusion not measurable for most surface molecules
-> polar molecules : absorption (solvation) into the ice bulk ? -> bulk behavior ?
-> non-polar molecules: formation of small islands ? breaking from the island harder than desorbing ?
Outline
 generalities on diffusion
 surface diffusion
 volume diffusion
Outline
 generalities on diffusion
 surface diffusion
 volume diffusion
•
•
•
•
•
laser resonant desorption
neutron scattering
secondary ion mass spectroscopy
isothermal desorption kinetics
isothermal reactivity kinetics
Laser Resonant Desorption
LRD bulk diffusion measurement: depth-profiling experiments I
laser assisted- desorption (OH stretching vibration)
+ mass detection (QMS)
t=0
t
Livingston F.E., Smith J.A. and George S.M., J.Phys.Chem. A, 106, 6309, 2002
Livingston F.E. and George S.M., J.Phys.Chem. A, 105, 5155, 2001
Livingston F.E., Smith J.A. and George S.M., Anal. Chem., 72, 5590, 2000
Laser Resonant Desorption
LRD bulk diffusion measurement: depth-profiling experiments II
Laser Resonant Desorption
fit against an Arrhenius law
T > 223 K interstitial-mediated (Frenkel) diffusion
T < 223 K vacancy-mediated (Schottky) diffusion
H2O self-diffusion
≈ 65 kJ.mol-1 (in crystalline ice)
1-3 orders of magnitude
Outline
 generalities on diffusion
 surface diffusion
 volume diffusion
•
•
•
•
•
laser resonant desorption
neutron scattering
secondary ion mass spectroscopy
isothermal desorption kinetics
isothermal reactivity kinetics
Neutron scattering
• neutrons can probe the bulk of a material because they are electrically neutral, and as such penetrate
matter deeper than electrically charged particles.
• neutrons can probe translational motion on the nanometer length scale
λ ~ 1 nm vs λ ~ 500 nm for optical radiation, which is more relevant to molecular motion
• neutron time-of-flight scattering
• Bragg reflection (neutron triple-axis spectroscopy,
neutron backscattering)
q = scattering vector
+ quasi-elastic neutron scattering, neutron spin echo
Neutron scattering
TNB= tris(naphtylbenzene)
=1,3-bis-(1-naphtyl)-5-(2-naphtly)benzene
h-TNB and d-TNB (protio and deuteri-TNB))
Swallen et al., J. Chem. Phys., 124, 184501, 2006
Neutron scattering
1
T=342 K
3
5
bulk molecular diffusion on nanometric scale
decay of the structure factor
measured= characteristic time
D = 1x1017 cm2.s-1 at 342 K
Swallen et al. Science, 315, 353, 2007
Neutron scattering
high sensitivity to the deposition temperature
super-exponential decays with an apparent induction time
or onset time (plateau) before any translational motion
occurs
spatially heterogeneous process or highly complex relaxation mechanisms
non-Fickian diffusion ?
Swallen et al., J. Chem.Phys., 128, 214514, 2008
Outline
 generalities on diffusion
 surface diffusion
 volume diffusion
•
•
•
•
•
laser resonant desorption
neutron scattering
secondary ion mass spectroscopy
isothermal desorption kinetics
isothermal reactivity kinetics
Secondary ion mass spectroscopy
IMC= indomethacin= 1-(p-chlorobenzoyl)-5-methoxy-2-methyl-indole-3-acetic acid
d-IMC= indomethacin-d4
stable glass
T= 318 K
850 nm
Sepulveda, Swallen and Edinger, J. Chem. Phys., 138, 12A517-1, 2013
Secondary ion mass spectroscopy
TNB= tris(naphtylbenzene)
=1,3-bis-(1-naphtyl)-5-(2-naphtly)benzene
h-TNB and d-TNB (protio and deuteri-TNB))
T=345 K
Swallen et al., Phys. Rev. Lett., 102, 065503, 2009
Outline
 generalities on diffusion
 surface diffusion
 volume diffusion
•
•
•
•
•
laser resonant desorption
neutron scattering
secondary ion mass spectroscopy
isothermal desorption kinetics
isothermal reactivity kinetics
Isothermal desorption kinetics experiments
[particles.cm-2.s-1]
D [cm2.s-1]
Isothermal desorption kinetic experiments
X = CO, HNCO, H2CO
or NH3
bulk ASW ice
d
Mispelaer A&A 2013
IR spectroscopy monitoring
real isothermal experiments, no need to ramp the temperature for detection
Isothermal desorption kinetic experiments
NH3
115 K
no plateau at t = 0
not the best fit ever
 D(115K)
Isothermal desorption kinetic experiments
absence of plateau
diffusion
diffusion
bulk diffusion or surface diffusion ?
Isothermal desorption kinetic experiments
IK on CO2
IK on NH3
 the water desorption limits the highest temperature
 decay curves cannot be fitted with a diffusion equation only
(desorption + diffusion): limits on the lowest temperature
 upper limits on the diffusion coefficients
Isothermal desorption kinetic experiments
•
lowest temperature limit set by the fast desorption assumption (boundary conditions)
+ experimental conditions (background pressure, long-term stability of the FTIR spectrometer,…)
•
highest temperature limit set by the co-desorption with the ice mantle
 D(T) measurement on a limited range of temperature
 uncertainty of D0 and Ediff
 needs few tens of ML to observe diffusion
 limitation on timescales
Isothermal desorption kinetic experiments
IK at T = 100 K
 sensitivity on the deposition temperature
k= 1.3 +/- 0.7 10-4 s-1
k= 2.5 +/- 0.4 10-4 s-1
Isothermal desorption kinetic experiments
the reproducibility dominates the uncertainty
Isothermal desorption kinetic experiments

r 2 (t ) ≈ D × t
cristallization
water self-diffusion (vacancy diffusion) ?
reorganization
1/200 K
1/100 K
1/67 K
CO
Isothermal desorption kinetic experiments
kinetics of the pore collapse
activation energy 0.9 kJ.mol-1
krelaxation= 1.15 10-3 x exp(0.9 kJ.mol-1 /RT)
10 K trelaxation ≈ 1.4 year
40 K trelaxation ≈ 4.6 hours
Isothermal desorption kinetic experiments

r 2 (t ) ≈ D × t
cristallization
water self-diffusion (vacancy diffusion) ?
reorganization
1/200 K
1/100 K
1/67 K
CO
Isothermal desorption kinetic experiments
molecular translational diffusion appears to occur concomitantly with crystallization (around 155 K)
apparent diffusivity 10 -12+/-1 cm2.s-1 (diffusivity <10-18 cm2.s-1 in crystalline ice at 160 K)
ASW behaves like a very viscous liquid, 107+/-1 times greater than water at 300 K.
Isothermal desorption kinetic experiments
Smith R.S., Matthiesen J., Kay B.D., J.Chem.Phys. 133, 174504, 2010
a rare gas-gas permeation through amorphous overlayers (supercooled liquids)
because of compensation between the parameters, there is an infinite number
of prefactors and activation energies pair that yields the same desorption
peak temperature
only one pair of parameters reproduces the overall TPD lineshape
Isothermal desorption kinetic experiments
universal scaling relationships:
scaling factor
 for an isothermal diffusion kinetics:
 for a temperature programmed permeation experiment:
Paper I : Smith R. S., Matthiesen J. and Kay, B.D., J.Chem.Phys. 133, 174504, 2010
Paper II : Matthiesen J. , Smith R. S. and Kay, B.D., J.Chem.Phys. 133, 174505, 2010
D(T) Fig 7
Arrhenius temperature dependence characteristics of « strong » liquids: the diffusion is related to the
crossing of a single barrier
D(T ) = D0 × exp(− E / RT )
unusually large prefactor (D0=1.85 1019 cm2.s-1) and activation energies (E=64.9 kJ.mol-1)
are characteristics of a fragile liquid (non following a simple Arhenius behavior)
close to the crystallization energy
The metastable phase diagram of water
temperature dependence given by Williams-Landel-Ferry (WLF) and Vogel Fulcher Tamman (VFT)
equations and not by the Arrhenius equation
E=7.2 +/- 0.8 kJ.mol-1
T0=119 +/- 3 K
Smith et al. Chem.Phys. 2000
glass transition of water Tg (H2O) = 136 K
VFT (Vogel-Fulcher-Tammann ) temperature dependence characteristics of « fragile » liquids: the diffusion
is related to collective complex motion, where diffusion is occurring on a rugged energy landscape.
D(T ) = D0 × exp(− B /(T − Ts )
D diverges at Ts
The VFT equation is used for the description of temperature dependence of glassy films viscosity
for Ts < T < Tglass
Ts: threshold of infinitely long relaxation time (disappearance of the configurational entropy)
D0 ≈ 101-2cm2.s-1 , B≈ 1400K-1500 ≈ 11.5-12.5 kJ.mol-1,
Ts ≈ 65 K
very different results, closer to Edes/3
for a thin film (<20 ML) the diffusivity is not representative of bulk diffusivity :
« softened » near surface layer of ≈ 8 ML in which there is a faster diffusion
The metastable phase diagram of water
glass transition of water Tg (H2O) ≈ 136 K
The 'phases' may be stable for hours or days.
metastable 'phase lines' may move with the
direction and time taken of the process .
importance of the time dependence
Molecular-dynamics simulation of amorphous-alloys., 2. self-diffusion, Brandt, E H, J. Phys.: Condens. Matter, 50,
1003-1014, 1989
Anomalous Fickian diffusion in water ice
65 kJ.mol-1
Mispelaer et al. A&A 2013
diffusion probably driven by the crystallization of the ASW ice mantle
k BT
Should we use the Stokes-Einstein equation for diffusion particles through liquid D =
with low Reynold number (very highly viscosity) instead ?
6π × η × r
r radius of the particle
ηliq.water� ηAWS=2.2 x 10-19 m2 s-1 at 150 K � ηcryst.ice
η viscosity
Outline
 generalities on diffusion
 surface diffusion
 volume diffusion
•
•
•
•
•
laser resonant desorption
neutron scattering
secondary ion mass spectroscopy
isothermal desorption kinetics
isothermal reactivity kinetics
Isothermal reactivity kinetic experiments
to monitor the product formation
to use chemical reactions to access smaller time intervals
no need to ramp the temperature: detection in the solid
Isothermal reactivity kinetic experiments
A
+
B
A
A
A
B
H2O molecules :
B
B
the dilution is slowing down the reaction
and the reactivity can be used to measure diffusion rates on small lengthscales
Isothermal reactivity kinetic experiments
A=
A
A
+
B=
B
B
H2O molecules :
A
B
A
B
HNCO
NH3
Reactants decay curve best fitted by a bi-exponential function
(reactivity + reorganization) or by an Avrami equation
HNCO: NH3: H2O
the Johnson-Mehl-Avrami-Kolmogorov (JMAK) or Avrami
equation describes the kinetics of crytallization
HNCO(t)= HNCO(t=0) . exp(-ktn)
time (s)
Mispelaer et al. A&A 2012
Isothermal reactivity kinetic experiments
A=
A
A
+
B=
B
B
A
B
A
B
kreaction
NH3
Noble et al., submitted to PCCP
H2O molecules :
CO2
Isothermal reactivity kinetic experiments
k (T ) = A e
−
Ea
k BT
real isothermal experiments, no need to ramp the temperature for detection
Noble et al., submitted to PCCP
Isothermal reactivity kinetic experiments
ab-initio calculations
Noble et al., submitted to PCCP
Isothermal reactivity kinetic experiments
A=
A
A
A
A
+
B=
B
B
H2O molecules :
B
B
in progress
Conclusion
 diffusion is a fundamental process as it impacts desorption, reactivity and trapping
 different techniques, each one with its pros and cons
 importance of the water ice substrate (morphology, phase change kinetics) since amorphous ice exhibits
liquid-like translational diffusion prior to crystallization
 difference in surface diffusion on « stable » substrates (silicate, graphite, crystalline ice )
and surface/volume diffusion on metastable amorphous water ice
 ice analogues are not totally interstellar ices: to be careful about experimental biases
 lack of knowledge on interstellar ice morphology and its evolution (amorphisation, compactification)
under VUV photons, exothermic reactions, …)
Thank you for your attention