PHYSICAL REVIEW MATERIALS
5
, 013604 (2021)
Interface pinning causes the hysteresis of the hydride transformation in binary metal hydrides
Nicholas J. Weadock
,
1
,
*
Peter W. Voorhees,
2
,
3
and Brent Fultz
1
1
Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, California 91125, USA
2
Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA
3
Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
(Received 10 September 2020; accepted 8 January 2021; published 28 January 2021)
Hydriding and dehydriding transitions in bulk and nanocrystalline binary metal hydrides were studied us-
ing the Pd-H model system by measuring pressure-composition isotherms with
in situ
x-ray diffractometry.
Nanocrystalline Pd showed a smaller pressure hysteresis, solvus hysteresis, and hysteresis in lattice parameter,
compared to bulk Pd. The time-dependence of pressure equilibration was measured after dosing with aliquots of
hydrogen, giving equilibration times that were much faster in the single-phase regions than in the two-phase
plateaus. In the broad two-phase plateaus, the pressure relaxations were exponential functions of time. An
explanation of hysteresis is developed that is based on a dissipative potential barrier that impedes the motion
of the interface due to interactions between lattice defects and the two-phase interface. The exponential pressure
relaxations and hysteresis are consistent for this mechanism. For a simple model of the pinning potential, the
potential barrier maximum is an order of magnitude less than typical grain boundary energies. These pinning
effects are substantially different in the nanocrystalline Pd, suggesting differences in the hydriding mechanism.
DOI:
10.1103/PhysRevMaterials.5.013604
I. INTRODUCTION
Hydrogen-based energy systems exist today and show
promise for reducing carbon emissions. Metal hydride sys-
tems are now used in battery, fuel cell, thermal storage, and
compression technologies [
1
–
5
]. One aspect of metal hydrides
that lowers their efficiency is hysteresis—the absorption and
desorption of hydrogen does not occur with thermodynamic
reversibility. This hysteresis has been a research topic for
many years. Hysteresis in metal hydrides manifests as a dif-
ference in the hydrogen absorption and desorption pressures
(pressure hysteresis) and as a difference in the terminal phase
boundary compositions during hydriding and dehydriding
(solvus hysteresis). The origin of hysteresis remains unclear.
We begin by summarizing some important theories of hystere-
sis that are based on different principles.
Schwarz and Khachaturyan (S-K) presented an analysis of
hysteresis that considers how elastic strain energy modifies
the thermodynamics of two-phase hydriding and dehydriding
phase transitions [
6
–
8
]. The S-K theory is based on Eshelby’s
analysis of elastic inclusions in an infinite matrix. The S-K
theory gives the following expression for the pressure hystere-
sis:
ln
(
P
abs
P
des
)
=
4
ρ
H
Y
η
2
(
c
β
−
c
α
)
k
B
T
,
(1)
where
Y
=
E
/
(1
−
ν
),
E
is Young’s modulus,
ν
is Poisson’s
ratio,
η
the fractional change in lattice parameter with respect
*
Present address: SSRL Materials Science Division, SLAC Na-
tional Accelerator Laboratory, Menlo Park, CA 94025; nwead-
ock@slac.stanford.edu
to hydrogen concentration, and
ρ
H
is a molar volume term
which, when fit to molecular dynamics simulations of hydride
formation in palladium, was found to be approximately equal
to the volume expansion induced by the H interstitial [
8
].
The expression also depends on the width of the coherent
two-phase region,
c
β
−
c
α
. The S-K theory assumes coherent
interfaces between the matrix and hydride phase; thus elastic
energy gives a “macroscopic” nucleation barrier that depends
on the hydrogen concentration. In this theory, defects do not
relieve the elastic stress.
In an earlier model, Flanagan and Clewley proposed that
hysteresis originates with plastic deformation during absorp-
tion and desorption [
9
]. The addition of an enthalpic term
corresponding to dislocation formation
H
δ
gives a pressure
hysteresis of 4
H
δ
/
RT
. The addition of
H
δ
also manifests
in solvus hysteresis. This theory requires a balanced annihila-
tion of dislocations generated during dehydriding, however.
Recently, Griessen
et al.
developed a mean-field model
to predict size-dependent spinodal pressures and associated
hysteresis in Pd nanostructures [
10
]. This model assumes a co-
herent core-shell hydriding geometry with different hydrogen
concentrations in each component. Experimental hysteresis
values were reproduced for several Pd nanostructures by
assuming a coherent transition with a modified surface-shell-
core coupling. An increased solubility of hydrogen in the
low-hydrogen-content
α
phase and reduction in total capacity
is attributed to thermodynamically distinct absorption sites
on or near nanostructure surfaces [
10
–
12
]. However, recent
experimental studies demonstrate that spherical cap nuclei
are energetically favorable compared to core-shell geometries
[
13
,
14
].
The palladium-hydrogen system is the prototypical metal
hydride and a convenient one for studying hysteresis. At
2475-9953/2021/5(1)/013604(10)
013604-1
©2021 American Physical Society
WEADOCK, VOORHEES, AND FULTZ
PHYSICAL REVIEW MATERIALS
5
, 013604 (2021)
moderate temperatures and pressures, fcc palladium ab-
sorbs up to 0.7 H per Pd while undergoing an isostructural
expansion from the
α
phase to the hydrogen-rich
β
phase.
This transition results in a volume change of approximately
10% [
15
]. Here we characterize the absorption and desorp-
tion of hydrogen by Pd by measuring pressure-composition
isotherms with simultaneous
in situ
x-ray diffraction and by
measuring the equilibration times during the individual steps
of isotherm measurements. The same hysteresis is found even
for incomplete phase transformations—if a partial absorption
is reversed, the hydrogen partial pressure shifts to the plateau
pressure of the opposite branch before significant changes
in hydrogen concentration occur [
16
]. Hysteresis, relaxation
times, and the composition dependencies of lattice parameters
are consistent with an explanation for the hysteresis that is
based on an interface mobility which is impeded by pinning
forces in the microstructure. The pinning potentials are an
order of magnitude less than grain boundary energies and vary
over a length scale of order 10 nm. The nanocrystalline Pd
has crystallite sizes smaller than this, and it has a different
hydriding behavior.
II. INTERFACE MOTION AS THE ORIGIN OF
HYSTERESIS
A. Chemical potential and interface velocity
Palladium hydrides nucleate coherently. Then, due to their
large misfit, they lose coherency as they grow above a crit-
ical size [
13
,
17
]. This loss of coherency creates an array of
defects, such as twin boundaries, slip bands, dislocations, and
possibly, point defects [
17
]. The resulting dislocation densities
can be quite high, so after the earliest stages of growth, the
hydride-matrix interfaces propagate through arrays of defects.
Due to elastic and plastic interactions between the defects and
interface, a driving force is needed for the interface to move.
Such impediments to interface motion have been observed,
for example, during martensitic transformations where the
interfaces are moving through an array of dislocations [
18
,
19
].
We propose that interface pinning causes the hysteresis
between absorption and desorption. Assume a planar interface
with a local normal velocity of the interface
v
is related to the
driving force for interface migration [
20
],
v
=−
M
{
[[
V
]]
−
T
β
ij
n
j
[[
E
ik
]]
n
k
+
∂φ
p
/∂
x
}
,
(2)
where [[
ξ
]]
ξ
β
−
ξ
α
for a quantity
ξ
,
x
is a displacement
of the interface along the normal direction,
M
is the mobility
of the interface,
V
is the grand potential density,
T
ij
is the
stress tensor,
n
j
is the normal to the interface pointing from
the
β
phase to the
α
phase,
E
ij
is the total strain tensor,
φ
p
is the pinning potential on the interface caused by defects in
the matrix, and summation from 1 to 3 over repeated indices
is assumed. The second term on the right-hand side comes
from the work required to keep the interface coherent during
growth, which we keep at this point to illustrate the potential
role of coherency-generated stresses.
If the quantity in the large braces in Eq. (
2
) is negative, then
the
β
phase will grow from
α
. In the absence of stress and
pinning, the jump in grand potential is the chemical driving
force for interface motion; if the grand potential of
β
is less
than that for
α
,the
β
phase will grow. The
∂φ
p
/∂
x
is chosen
so that when it is positive, it impedes the motion of
β
growing
into
α
. We assume that the diffusion of H is very fast, and its
concentration is in local equilibrium at the moving interface.
Thus gradients in H composition within a single phase are
small, as observed experimentally, and there is no jump in the
diffusion potential of H at the interface,
[[
M
HV
]]
=
0
,
(3)
where
M
HV
is the diffusion potential.
Since H is an interstitial atom in the Pd lattice, it is neces-
sary to account for the constraint that
N
I
=
N
H
+
N
V
, where
N
I
is the number of interstitial sites,
N
H
is the number of H
atoms, and
N
V
is the number of vacancies on the H sublattice.
The diffusion potential is [
20
]
M
HV
=
ρ
0
[
μ
H
(
c
)
−
μ
V
(
c
)]
−
η
T
kk
,
(4)
where
μ
H
and
μ
V
are the stress-free chemical potentials of H
and vacancies, respectively,
ρ
0
is the molar density of inter-
stitial lattice sites, and
η
is the solute expansion coefficient,
where
η
=
(1
/
a
)
da
/
dc
(
a
is the lattice parameter and
c
is the
mole fraction of H). We assume Vegard’s law is valid and thus
η
is a constant. The molar grand potential for each phase is
[
20
]
=
F
(
c
)
+
ρ
−
1
0
W
e
−
M
HV
c
,
(5)
where the molar Helmholtz free energy at zero pressure in
the absence of stress is
F
=
μ
H
c
+
μ
V
(1
−
c
), and
W
e
is the
elastic energy density.
Using Eqs. (
4
) and (
5
)inEqs.(
2
) and (
3
), the velocity of
the interface is
v
=−
M
{
ρ
0
[[
μ
V
(
c
)]]
+
[[
W
e
]]
+
η
[[
cT
kk
]]
−
T
β
ij
n
β
j
[[
E
ik
]]
n
β
k
+
∂φ
p
/∂
x
}
(6)
and the diffusion potential is
M
HV
=
ρ
0
[[
μ
H
(
c
)
−
μ
V
(
c
)]]
−
η
[[
T
kk
]]
=
0
.
(7)
To define the compositions at the interface, it is necessary
to determine the stress in the system. We employ a parallel-
plate geometry of
α
and
β
phases in a linearly elastic isotropic
solid in which the elastic constants of the two phases are
identical and constant. The stress is generated because the
compositions of the two phases are different and thus so are
their lattice parameters. For a coherent system, the elastic
energy and trace of the stress can be determined [
21
],
[[
T
kk
]]
=−
2
Y
η
(
e
e
+
η
[[
c
−
c
e
]] )
,
(8)
[[
W
e
]]
+
η
[[
cT
kk
]]
+
T
β
ij
n
β
j
[[
E
ik
]]
n
β
k
=
Ye
2
e
−
2
η
e
e
Yc
β
e
−
2
η
2
Y
[[
c
e
(
c
−
c
e
)]]
,
(9)
where
c
α
e
,
c
β
e
are the equilibrium, stress-free compositions
without a pinning potential. The misfit between the two phases
is
e
e
=
η
[[
c
]], and the lattice parameter of the
α
phase at
c
α
e
is
used as the reference state for strain. Using Eqs. (
8
) and (
9
)in
Eqs. (
6
) and (
7
) yields the velocity
v
=−
M
{
ρ
0
[[
μ
V
(
c
)]]
+
Ye
2
e
−
2
η
e
e
Yc
β
e
−
2
η
2
Y
[[
c
e
(
c
−
c
e
)]]
+
∂φ
p
/∂
x
}
(10)
013604-2
INTERFACE PINNING CAUSES THE HYSTERESIS OF ...
PHYSICAL REVIEW MATERIALS
5
, 013604 (2021)
and the diffusion potential condition
ρ
0
[[
μ
H
(
c
)
−
μ
V
(
c
)]]
−
2
Y
η
2
[[
c
]]
=
0
.
(11)
These two equations, along with the stress-free chemical po-
tentials (for example, those giving the miscibility gap in the
Pd-H system), yield two nonlinear equations for the two in-
terfacial compositions for a given velocity and mobility of
the interface. Since the elastic state of one phase is indepen-
dent of the other, the elastic energies do not depend on the
volume fraction of the phases and thus there is no elastic-
stress-generated metastability and nucleation barrier [
21
–
24
].
B. Phase compositions, gas pressures
When the interface is pinned,
v
=
0, and thus the terms in
the brackets on the RHS of Eq. (
10
), along with Eq. (
11
)give
the concentrations at the interface. These concentrations then
define the diffusion potential in the solid, which is equal to
the chemical potential of H in the gas. For an approximation
that captures the essential physics of the pinning process,
the two equations defining the concentrations are solved in
the limit that the concentrations of each phase are close
to the equilibrium, stress-free compositions without pinning,
c
α
e
,
c
β
e
. These compositions give the equilibrium incoherent
(stress-free, pinning-free) phase diagram. In this limit, the
concentration dependence of the terms involving the chemical
potentials are given by Taylor expansions about the stress-free
equilibrium state of each phase. It is also necessary to linearize
the terms involving the stress to first order in
c
−
c
e
.Using
linearized chemical potentials and the linearized forms of the
elastic terms yields two linear equations for the two unknown
compositions of the phases that give the phase compositions
at which the interface is pinned, see Supplemental Materials
[
25
]:
c
α
p
−
c
α
e
=
∂φ
p
/∂
x
+
Y
η
2
[[
c
e
]]
2
ρ
0
[[
c
e
]]
G
α
m
(1
+
B
α
)
,
c
β
p
−
c
β
e
=
∂φ
p
/∂
x
−
Y
η
2
[[
c
e
]]
2
ρ
0
[[
c
e
]]
G
β
m
(1
+
B
β
)
,
(12)
where
B
i
=
2
η
2
Y
/ρ
0
G
i
m
and
G
i
m
=
∂
2
G
i
m
/∂
c
2
,
i
=
α, β
, eval-
uated at the equilibrium incoherent compositions of the
phases. Since we have assumed that the diffusion potentials
are constant, these different compositions give the same diffu-
sion potential in the two-phase mixture.
The solid is in equilibrium with the gas,
μ
g
H
=
M
HV
(
c
,
T
kk
)
,
(13)
where
μ
g
H
is the chemical potential of H in the gas. For small
deviations in the gas pressure about the equilibrium pressure
in the absence of stress and pinning, and using Eq. (
12
)to
determine the diffusion potential
M
HV
, the deviation of the
equilibrium pressure
P
at which
v
=
0 from the equilibrium
value
P
e
is
ln
(
P
P
e
)
=
2
∂φ
p
/∂
x
+
2
Y
η
2
[[
c
e
]]
2
ρ
0
RT
[[
c
e
]]
,
(14)
where
R
is the gas constant and the chemical potential of
the gas is
μ
g
H
=
μ
o
+
(
RT
/
2) ln
P
,
μ
o
is the standard state
chemical potential, and
P
is the partial pressure of H
2
gas. The
equilibrium pressure is shifted by the presence of stress but
does not change on charging and discharging, and coherency-
induced stress does not cause hysteresis.
C. Thermodynamics with interface pinning
The pinning potential causes a hysteresis of absorption
and desorption when the pinning force changes sign with the
direction of
v
. To illustrate, consider a pinning potential with
a periodic dependence,
φ
p
=
A
2
[
1
−
cos(
kx
)
]
,
(15)
where
A
sets the strength of pinning, and
k
=
2
π/
L
is the
inverse length scale of the interaction. With a greater driving
force, the interface is displaced from its equilibrium value
at the bottom of the well located at
x
=
0. The direction of
displacement changes with absorption (
β
phase grows) or
desorption (
α
phase grows). The maximum force acting on
the interface is given by the interface displacement at which
∂
2
φ
p
∂
x
2
=
0
,
(16)
which is
x
c
=±
L
2
(
n
+
1
/
2),
n
=
0
,
±
1
,
±
2
, ...
.Using
Eq. (
15
) and
x
c
gives the maximum pinning force
F
p
,
m
as
F
p
,
m
=−
∂φ
p
∂
x
∣
∣
∣
∣
max
=±
A
π
L
(17)
The plus is the maximum force when charging (
β
grows) and
the minus is the maximum force for discharging (
α
grows).
The pressure hysteresis is evaluated by subtracting desorption
from absorption in Eq. (
14
):
ln
(
P
abs
P
des
)
=
2
∂φ
p
/∂
x
)
abs
−
∂φ
p
/∂
x
)
des
ρ
0
RT
[[
c
e
]]
.
(18)
Thus there is no effect of coherency stress on the hystere-
sis, since the equilibrium pressure due to coherency stress
does not depend on whether hydrogen is being absorbed or
desorbed. Using the maximum pinning force derived from
Eq. (
15
),
ln
(
P
abs
P
des
)
=
4
A
π
L
ρ
0
RT
[[
c
e
]]
.
(19)
The pinning also causes different phase compositions for ab-
sorption and desorption. Using Eq. (
17
)inEq.(
12
) yields
the pinning compositions of the
α
phase on absorption and
desorption,
c
α
abs
−
c
α
e
=
A
π/
L
+
Y
η
2
[[
c
e
]]
2
[[
c
e
]]
ρ
0
G
α
m
(1
+
B
α
)
,
c
α
des
−
c
α
e
=
−
A
π/
L
+
Y
η
2
[[
c
e
]]
2
[[
c
e
]]
ρ
0
G
α
m
(1
+
B
α
)
,
(20)
expressions that are valid away from the critical temperature,
and coherent spinodal.
We expect that the pinning potential
φ
p
(
x
) will be more
complicated than Eq. (
15
), so in this case we can use a general
form as a Fourier transform,
φ
p
(
x
)
=
∫
∞
−∞
ψ
p
(
k
)
e
ikx
dk
.
(21)
013604-3
WEADOCK, VOORHEES, AND FULTZ
PHYSICAL REVIEW MATERIALS
5
, 013604 (2021)
The pinning force
F
p
(
x
)isthus
F
p
(
x
)
=−
∂φ
p
∂
x
=−
∫
∞
−∞
ik
ψ
p
(
k
)
e
ikx
dk
.
(22)
Equation (
15
) is symmetrical in
±
x
, so its pinning force
F
p
(
x
)
is real, and
F
p
(
x
)
=−
F
p
(
−
x
), a reasonable but not clearly
necessary condition. Equation (
22
) nevertheless shows that
the higher Fourier components of
ψ
p
(
k
) are weighted by
k
and make larger contributions to the pinning force. The spatial
range of the pinning force is therefore
L
2
π/
k
max
, where
k
max
is the largest
k
for which
ψ
p
(
k
) is substantial. Equiva-
lently, with a distribution of defects that contribute to
φ
p
(
x
),
the effective pinning range
L
is that of the smallest features
that offer substantial pinning.
D. Kinetics with interface pinning
We assume that H diffusion in the Pd is fast on the
timescale of interface motion. When a small amount of H is
added to the surrounding gas, it rapidly enters the Pd, leading
to the motion of the interfaces in the two-phase mixture until
the diffusion potential in the material is again equal to the
chemical potential of the H gas. The decay of gas pressure
can be measured experimentally and used as a probe of the
dynamics of interface motion on the two-phase mixture. The
velocity of the interface is given by Eq. (
6
). Assuming small
departures in equilibrium from the pinned state, see Supple-
mental Materials [
25
],
v
=
ρ
0
M
(1
+
B
α
)
G
α
m
[[
c
p
]] (
c
α
−
c
p
)
.
(23)
Defining
f
V
β
/
V
s
as the volume fraction of
β
phase (
V
s
is
the total volume of the material system), the rate of change of
f
is given by
dV
β
dt
=
V
s
df
dt
=
∫
A
v
ds
,
(24)
where
A
is the area of the interface. Using Eq. (
23
)inEq.(
24
),
V
s
df
dt
=
ρ
0
M
(1
+
B
α
)
G
α
m
[[
c
p
]]
∫
A
(
c
α
−
c
α
p
)
ds
.
(25)
Since the composition of the
α
phase is constant along the
interface,
df
dt
=
ρ
0
M
(1
+
B
α
)
G
α
m
[[
c
p
]]
A
(
c
α
−
c
α
p
)
V
s
.
(26)
Defining the interfacial area per volume
S
V
A
/
V
s
,
df
dt
=
ρ
0
M
(1
+
B
α
)
G
α
m
[[
c
p
]]
S
V
(
c
α
−
c
α
p
)
.
(27)
The
S
V
should not vary much after an aliquot of H is added,
but it might change along the isotherm, in the early or late
stages of absorption, for example.
After inserting a small aliquot of H, the composition of the
material system is
c
s
, with H distributed between the
α
and
β
phases:
c
s
=
(1
−
f
)
c
α
+
fc
β
.
(28)
Since
c
s
is constant after the aliquot of H is added, for small
supersaturations (see Supplemental Materials [
25
]),
dc
α
dt
=−
G
β
m
(1
+
B
β
)[[
c
p
]]
G
β
m
(1
+
B
β
)(1
−
f
o
)
+
G
α
m
(1
+
B
α
)
f
o
df
dt
,
(29)
where
f
o
is the volume fraction before the aliquot H is added.
Using Eq. (
27
)inEq.(
29
) yields
dc
α
dt
=−
MS
V
(
t
)
(
c
α
−
c
α
p
)
,
(30)
where
involves a volume fraction weighted average of
thermodynamic terms of each phase,
=
ρ
0
[[
c
p
]]
2
G
α
m
(1
+
B
α
)
G
β
m
(1
+
B
β
)
G
α
m
(1
+
B
α
)
f
o
+
G
β
m
(1
+
B
β
)(1
−
f
o
)
.
(31)
Assuming that
S
V
is little changed when a small aliquot of H
is added,
c
α
(
t
)
=
c
α
p
+
(
c
α
pi
−
c
α
p
)
exp
{
−
MS
o
V
t
}
,
(32)
where
c
α
pi
is the initial pinned composition, which does not
have to be the same after the first absorption because the
defects responsible for the pinning may be different, and
S
o
V
is the surface area per volume before the aliquot is added.
Since the change in composition in the
α
phase fixes the
diffusion potential, which equals the chemical potential in the
gas, Eq. (
32
) yields
P
(
t
)
=
P
p
+
P
δ
exp
{
−
MS
o
V
t
}
,
(33)
where
P
p
is the pinning pressure and the transient pressure
excursion is
P
δ
=
[
ρ
0
(
c
α
pi
−
c
α
p
)(1
+
B
α
)
G
α
m
P
p
]
/
[
k
B
T
] and as-
suming that
|
P
(
t
)
−
P
p
|
is small. We have allowed for
S
V
to
depend on the initial volume fraction
f
o
, so there may be
different relaxation times across an isotherm.
III. METHODS
Palladium powder (200 mesh, 99.95% metals basis) was
purchased from Alfa Aesar (Ward Hill, MA, USA). The pow-
der was annealed at 1273 K for one hour under N
2
flow in
a horizontal tube furnace before any hydriding experiments.
Palladium nanopowder (99.95%) was purchased from US
Research Nanomaterials, Inc., (Houston, TX, USA) and de-
gassed at 353 K for at least 8 hours prior to any hydriding
experiments.
Electron micrographs of annealed bulk Pd powder were
acquired with a high-resolution Zeiss 1550VP field emis-
sion scanning electron microscope (SEM). Nanocrystalline Pd
powder was analyzed with bright- and dark-field transmission
electron microscopy (TEM) using an FEI Tecnai F-30UT
STEM. The powder was dispersed in isopropanol and soni-
cated one hour to reduce agglomeration before loading on an
amorphous carbon grid for TEM.
In situ
x-ray diffraction (XRD) experiments were per-
formed using an Inel CPS 120 powder diffractometer utilizing
Mo K
α
radiation. A Si
110
single crystal oriented in the
220
direction was used as the incident beam monochro-
mator. Two-theta calibration of the CPS 120 detector was
performed with a NIST SRM 660a (LaB
6
). At least 200 mg of
sample was first loaded into a temperature-controlled vertical
013604-4
INTERFACE PINNING CAUSES THE HYSTERESIS OF ...
PHYSICAL REVIEW MATERIALS
5
, 013604 (2021)
sample holder and then placed into a stainless-steel chamber
(980 mL) with a Be window. This reactor is connected to a gas
manifold with VCR fittings and MKS Baratron pressure trans-
ducers. Prior to hydrogen uptake, the sample was evacuated at
353 K for 72 h. Each sample was cycled at least once prior
to collection of data presented here. Full diffraction measure-
ments from at least two complete isotherms were obtained for
each temperature.
Raw diffraction data were initially processed with a de-
convolution algorithm to remove instrument effects. Rietveld
refinement with the GSAS-II software package was sub-
sequently used to extract lattice parameter data from the
deconvoluted
in situ
XRD results [
26
]. The Pd-H data were
fit with a two-phase model consisting of the solid-solution
α
phase and the hydride
β
phase, with the sum of the phase
fractions constrained to be unity [
27
].
Pressure-composition isotherms of bulk and nanocrys-
talline Pd powder were also measured on an independent
volumetric Sieverts-type apparatus. At least 1.0 g of sample
were loaded into an AISI 316L stainless-steel reactor (5 mL)
and evacuated (baseline 10
−
5
Pa) at 473 K (bulk Pd) or 353 K
(nanocrystalline Pd) for 8 h. Each sample was cycled at least
once prior to collection of data presented here, and at least
two complete isotherms were measured for each sample and
temperature.
Hydrogen concentrations in the sample were calculated
volumetrically with the NIST
REFPROP
database [
28
,
29
]. Ab-
sorption and desorption were performed at 333 K for both
the
in situ
XRD and Sieverts apparatus experiments. An ad-
ditional pressure-composition isotherm was measured on the
Sieverts apparatus for bulk Pd at 435 K. At each absorption
(desorption) step, equilibration was reached when pressure in
the reactor did not change for a period of 15 min. After equi-
libration, the next quantity of hydrogen was added (removed
for desorption) to the reactor.
IV. RESULTS
The microstructures of the annealed bulk Pd and as-
received nanocrystalline Pd powders were investigated with
scanning electron microscopy (SEM) and transmission elec-
tron microscopy (TEM). A high-resolution TEM image of
nanocrystalline Pd is provided in Fig.
1
, and SEM and ad-
ditional TEM images are provided in Supplemental Figs.
1
and
2
[
25
]. Annealed Pd particles are 43
.
5
±
13
μ
minsize
with distinct internal grains of 3
.
0
±
1
.
3
μ
m in diameter.
The nanocrystalline Pd consists of agglomerated crystallites
7
.
5
±
2
.
3 nm in diameter, as measured from dark-field im-
ages.
Pressure-composition isotherms for bulk and nanocrys-
talline Pd, measured on a Sieverts apparatus, are shown in
Fig.
2(a)
. Uptake values of approximately 0.6 and 0.7 H
/
M
are found for the nanocrystalline and bulk Pd, respectively,
consistent with other reports [
9
,
11
,
12
,
30
–
33
]. Defining the
dimensionless pressure hysteresis, with
h
as the left-hand side
of Eq. (
1
),
h
=
ln
(
P
abs
P
des
)
,
(34)
FIG. 1. High-resolution TEM image of the as-received nanocrys-
talline Pd.
the 333 K pressure hysteresis measured from the isotherms
is 0.37 for nanocrystalline Pd and 0.88 for bulk Pd. The
high-temperature hysteresis of the bulk Pd at 435 K is 0.37. A
significant reduction in the absorption plateau pressure for the
nanocrystalline Pd (from 10 to 6.9 kPa) contributes primarily
to the reduction in hysteresis at 333 K. Only a small increase
in the desorption plateau is observed for the nanocrystalline
Pd. Another important difference is that the transition is more
gradual for the nanocrystalline Pd; a sharp initial transition is
observed for the bulk Pd. We also define solvus hysteresis
c
at the
α
-or
β
-phase boundary as
c
α
=
∣
∣
c
α
abs
−
c
α
des
∣
∣
,
(35)
and similarly for
c
β
for the
β
phase.
Pressure-composition isotherms are also measured inde-
pendently at 333 K with a gas manifold and x-ray transparent
sample chamber designed for
in situ
XRD experiments. No
in situ
data was measured at 435 K because the hydrogen
environment chamber cannot accommodate the required pres-
sures. Diffraction patterns are acquired at several steps along
the isotherm, and full sets of diffraction patterns for the
hydriding and dehydriding transitions in bulk and nanocrys-
talline Pd are provided in Supplemental Figs.
4
and
5
[
25
].
Isotherms measured during the
in situ
hydriding experi-
ments are consistent with those plotted in Fig.
2(a)
.They
are compared in Supplemental Fig. 3 and discussed in the
Supplemental Materials [
25
]. The hysteresis measured in both
apparatus is the same.
Figure
2(b)
plots full and minor loop isotherms for Pd-
H measured on the
in situ
system in which absorption
was stopped at 75% completion, then reversed. Rather than
traversing back along the absorption branch, the pressure de-
creases to the desorption branch before significant quantities
of hydrogen are removed. Similar results have been reported
previously [
16
].
Lattice parameters obtained from refinement of the
in situ
diffraction data are plotted in Fig.
3
. Variation of lattice pa-
013604-5
WEADOCK, VOORHEES, AND FULTZ
PHYSICAL REVIEW MATERIALS
5
, 013604 (2021)
(a)
(b)
FIG. 2. (a) Pressure-composition isotherms for bulk (blue, red
circles) and nanocrystalline (green trianges) Pd measured on a
Sieverts apparatus. The inset shows the plateau region on a linear
pressure scale. Closed symbols denote absorption, and open symbols
correspond to the subsequent desorption. (b) Full and minor loop
isotherms for bulk Pd measured
in situ
. After absorption to approxi-
mately 75% capacity, the isotherm was reversed and the sample was
dehydrided.
rameter with hydrogen concentration in single-phase regions
was fit to a linear function (Vegard’s law); the results are
summarized in Table
I
. Within the two-phase region, lattice
parameters are nearly constant but there is variation for the
nanocrystalline Pd. This is apparent in the continual increase
TABLE I. Vegard’s law relationship and phase boundary com-
positions in bulk and nanocrystalline Pd-H from refined lattice
parameter data. The variation of lattice parameter with hydrogen
concentration (1
/
a
)
da
/
dc
is determined in the single-phase regions.
Region
Bulk
a
Nanocrystalline
α
abs
0
.
045
±
0
.
015
0
.
025
±
0
.
004
α
des
–0
.
018
±
0
.
011
β
abs
0
.
047
±
0
.
013
0
.
036
±
0
.
003
β
des
0
.
039
±
0
.
003
0
.
039
±
0
.
008
a
Empty entries correspond to regions with insufficient data points to
fit.
FIG. 3. Refined lattice parameters of the (a) bulk and
(b) nanocrystalline Pd-H powders during hydrogen absorption and
desorption. The red and blue curves correspond to the solid solution
α
phase and hydride
β
phase, respectively. Error bars are indicated
by the shaded region.
in
α
-phase lattice parameter during absorption. Lattice param-
eters for both phases are larger for absorption than desorption,
with a significant difference for bulk Pd [Fig.
3(a)
].
In principle, x-ray lattice parameters are sensitive to both
hydrogen concentration (through Vegard’s law) and normal
strains from coherency stresses (if present) at
α/β
interfaces.
Averaged over the volume of a polycrystalline solid without
external forces, the distribution of internal strains generally
does not have a substantial effect on the average lattice param-
eter. In what follows we attribute changes in lattice parameter
to changes in hydrogen concentration.
Phase boundary compositions are determined from Fig.
3
by identifying the composition at which the lattice parame-
ters deviate from the values in the two-phase region. These
values are reported in Table
II
. For the nanocrystalline Pd,
the
α
-phase boundaries are instead evaluated from the phase
fraction results in Supplemental Fig. 6 [
25
]. In this case the
phase boundaries are taken as the concentration in which the
phase fraction deviates from the single-phase regime. High-
013604-6
INTERFACE PINNING CAUSES THE HYSTERESIS OF ...
PHYSICAL REVIEW MATERIALS
5
, 013604 (2021)
4
5
6
7
8
9
0.001
Moles H (gas)
1000
500
0
Time (sec)
H/M = 0.410
Data
Fit
9
10
-4
2
Moles H (gas)
1000
500
0
Time (sec)
H/M = 0.013
Data
Fit
80
60
40
20
0
[sec]
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
H in Pd [H/M]
Absorption Cycle 2
Desorption Cycle 1
(a)
(b)
(c)
FIG. 4. Kinetics of equilibration during absorption and desorp-
tion. (a) Equilibration time constants
τ
as a function of hydrogen
concentration during cycling. Raw data from absorption cycle 2
and corresponding fits used to extract
τ
are plotted in (b) for the
single-phase region and in (c) for the two-phase region. Reported
error bars may be smaller than the associated data marker.
600
550
500
450
400
350
300
250
Temperature [K]
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
H in Pd [H/M]
FIG. 5. Absorption and desorption phase boundaries determined
from terminal compositions of bulk and nanocrystalline Pd-H. Bulk
and nanocrystalline Pd-H data from this study are plotted as red
circles and green triangles, respectively. Additional bulk data from
Wicke
et al.
is plotted as upside down triangles [
32
]. 6- and 4-nm
nanoparticle Pd data from Vogel
et al.
is plotted as squares and
diamonds, respectively [
27
]. Filled markers denote absorption, open
markers denote desorption.
TABLE II. Terminal compositions and solvus hysteresis of the
hydriding transition for bulk and nanocrystalline Pd obtained from
in situ
XRD results at 333 K. High-temperature (435 K) composi-
tions for the bulk Pd are evaluated from the pressure-composition
isotherm. The solvus hysteresis
c
α/β
are evaluated using Eq. (
35
).
Sample
Bulk
Nanocrystalline
333 K
435 K
333 K
c
abs
α
0.035
±
0.001
0.13
0.12
±
0.01
c
abs
β
0.62
±
0.01
0.59
0.46
±
0.015
c
des
α
0.012
±
0.0005
0.07
0.115
±
0.005
c
des
β
0.564
±
0.01
0.51
0.45
±
0.01
c
α
0.023
±
0
.
0011
0.06
0.005
±
0
.
011
c
β
0.056
±
0
.
014
0.08
0.01
±
0
.
018
temperature phase boundary compositions were determined
by the graphical method proposed by Frieske and Wicke,
in which the plateaus are extended to the opposite isotherm
branch and the intersection point is taken as the phase bound-
ary [
32
]. The solvus hysteresis is evaluated with Eq. (
35
) and
also reported in Table
I
.
The variation of hydrogen pressure was monitored as a
function of time at each absorption or desorption step during
pressure-composition isotherm measurements and fit to the
exponential function in Eqs. (
32
) and (
33
). A characteristic
equilibration time
τ
, corresponding to [
MS
o
V
]
−
1
, is plotted
in Fig.
4(a)
as a function of hydrogen content for the first
desorption and second absorption cycles. Representative fits
to hydrogen absorption in the single- and two-phase regions
are plotted in Figs.
4(b)
and
4(c)
, respectively. These fits were
also performed for data in the single-phase regions.
Equilibration times increase as the two-phase region is
approached from either side. After the transformation enters
the two-phase region, however, there is little variation in
τ
.
Hydrogen diffusivity in palladium at 333 K is 10
−
6
cm
2
s
−
1
,
corresponding to diffusion times of 5 s into the center of bulk
Pd particles [
34
], and the characteristic time for heating the
reactor vessel was measured to be 5.4 s. Therefore
τ
is not
dominated by the diffusion of hydrogen or heat within the
two-phase region. It is possible that the diffusion of hydrogen
and heat account for much of the relaxation time in the single-
phase regions.
V. DISCUSSION
A. Hysteresis and hysteresis energies
The magnitude of the pinning potential
A
is evaluated
from Eq. (
19
) using the experimentally determined pressure
hysteresis,
ρ
0
, and approximate value of [[
c
e
]]. We assume
a characteristic
L
=
10–35 nm, obtained as the inverse of
the reported dislocation densities in cycled Pd-H [
35
–
37
].
The pinning potentials for bulk Pd-H are 0.1–0.34 and 0.04–
0.15 J m
−
2
at 333 and 435 K, respectively. These energies are
about an order of magnitude less than grain boundary energies
calculated for fcc metals [
38
,
39
]. The solvus hysteresis is
different at the
α
and
β
phase boundaries, as seen in Table
II
.
This behavior is consistent with the predictions of Eqs. (
12
)
and (
20
), as
G
α
m
and
G
β
m
are not equivalent.
013604-7