# Discriminator Synthesis: On reusing the other half of Generative Adversarial
Networks

Diego Porres
Computer Vision Center (CVC), Universitat Autònoma de Barcelona
Barcelona, Spain, 08193
<EMAIL_ADDRESS>

###### Abstract

Generative Adversarial Networks have long since revolutionized the world of
computer vision and, tied to it, the world of art. Arduous efforts have gone
into fully utilizing and stabilizing training so that outputs of the Generator
network have the highest possible fidelity, but little has gone into using the
Discriminator after training is complete. In this work, we propose to use the
latter and show a way to use the features it has learned from the training
dataset to both alter an image and generate one from scratch. We name this
method Discriminator Dreaming, and the full code can be found at
https://github.com/PDillis/stylegan3-fun.

##  1 Introduction

Generative adversarial networks (GAN) [9] have, irrevocably, altered the
course of machine-aided art. Some architectures are better suited for the task
[23, 4, 28], mainly due to the eventual ease of use and evaluation of these
models, as well as the availability of data and compute. However, the main
focus in the literature has been to increase the fidelity and stabilization of
the images being generated [12, 13, 14, 15] and, while the Discriminator has
to also increase its capability in order for these larger networks to be
trained, it is usually left unexplored if not outright discarded.

While previous efforts have gone into using the Discriminator as part of an
optimization process, either to use it in order to generate images it deems to
be fake [3] or to use it alongside another classifier to update the weights of
the Generator [2], more effort can be done here, especially to use these large
networks in an artistic manner. These networks have learned unique features
from both the real and fake dataset, and they are simply being thrown away.

This work has two main sources of inspiration. The first is by Robbie Barrat
who used the Discriminator from a trained GAN to optimize the tile placements
of a nude portrait [1]. The second is the recent global effort to use large
pre-trained models such as CLIP [24], which would not have been able to be
trained by an individual researcher or artist. Two notable examples are the
Big Sleep [21] and Control The Soul [6], who use CLIP alongside BigGAN and
StyleGAN, respectively, to generate images from prompts.

Even though the features learned by a Discriminator may not be used for other
downstream classification tasks (e.g., such as the models trained on ImageNet
[7] which are then fine-tuned for other tasks), it is still worth exploring
the learned features, especially for artistic endeavors. Indeed, what warrants
as a ’fake’ or ’real’ image for the network itself will not translate to
logical human sight features per se, but nevertheless these may still hold
beauty.

We can use the Discriminator in a variety of ways. For example, we could use
it in lieu of a VGG network [26] for style transfer [8], or when projecting an
image using the Generator of the same network (or even using another Generator
altogether). Staying within StyleGAN, we could use it as a sort of feedback
loop with the Generator, as the shape of the fully-connected layer before the
final output (that is, the output shape of D.b4.fc) has the same shape as the
latent space $\mathcal{Z}$.

As a proof of concept, we use StyleGAN2’s Discriminator instead of the
Inception network [27] for Inceptionism/DeepDream [20]. We provide a starter
code on how to extract the intermediate features of the Discriminator (with
residual [10] connections) of StyleGAN2, in particular, its ADA (PyTorch)
version, but is also compatible with StyleGAN3’s [16] repository and models.
Our code is built upon the DeepDream-PyTorch repository [19] and we name our
algorithm Discriminator Dreaming.

##  2 Discriminator Dreaming

The following are some preliminary results we have obtained, though the final
code is subject to change. For this work, we used a vast collection of
available pre-trained models, some trained by third parties or provided by the
authors of the original papers, others trained by ourselves. These official
models were introduced in StyleGAN2 and StyleGAN2-ADA papers, and can be found
in their respective GitHub repositories. These include: at $1024$ resolution,
FFHQ and MetFaces; at $512$ , FFHQ-512, Cars, and AFHQ-Cat; and at $256$,
FFHQ-256, LSUN Horse, and LSUN Church.

Other models include, at $1024$ resolution, MinecraftGAN, a model trained with
images taken from a first-person POV in the videogame Minecraft throughout
different weathers and biomes [11]. At $512$ resolution, the Huipiles model
was trained with images from huipiles from Guatemala and Mexico [22]. At $256$
resolution, Doors is a model trained on art nouveau doors found in Barcelona
[17].

In Figure 1, we can see the result of Discriminator Dreaming using different
layers for various models. We use, as a starting image, 00012.png from the
FFHQ dataset introduced in StyleGAN. This is using a single frame, so for a
video results, refer to Appendix A.1. As per this section, for future work, we
could set the zoom, rotation, and translation to be variable instead of fixed,
by e.g. using an audio-reactive code [5].

Figure 1: Discriminator Dreaming with StyleGAN2 models of different
resolutions. From left to right and top to bottom: ($1024$ resolution)
MetFaces and MinecraftGAN; ($512$ resolution) Huipiles, FFHQ-512, Cars, and
AFHQ-Cat; ($256$ resolution) Doors, FFHQ-256, LSUN Horse, and LSUN Church.
Results shown to scale. Best viewed in color.

###  2.1 Limitations

Due to the iterative nature of this algorithm, it is slow to synthesize
images, so perhaps more efficient algorithms could be used. Resizing the
octaves (see Appendix A) usually makes the code run twice as slow on a RTX
2070, making it less energy-efficient as a trained Generator. However, if we
do not resize the octaves, then the image loses its sharpness, resulting in
fuzzy images, as well as sometimes giving us vastly different results. Thus, a
lot of tweaking and exploration will be needed in order to get the desired
result. See Figures 2 and 3 for a difference when resizing or not the octaves.

## Acknowledgments and Disclosure of Funding

Diego Porres acknowledges the financial support to perform his Ph.D. given by
the grant PRE2018-083417.

## Ethical implications

This work is meant for purely artistic use. We have limited our work to using
StyleGAN2 and StyleGAN3’s Discriminator, for which usually the available pre-
trained models are based on publicly-available datasets. However, the trained
Discriminator may have learned the distribution of images it was trained on
for which we might not have the rights to. Fully utilizing the Machine
Learning models we train is what this work strives towards, as the
environmental implication of training each from scratch and then discarding
half of the network should not be so easily dismissed.

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##  Appendix A Appendix

#### Discriminator architecture

A full breakdown of StyleGAN2’s Discriminator architecture is beyond the scope
of this work, though we refer the interested reader to its respective paper.
In short, we will be mostly interested in the two main convolutional layers
per block (e.g., D.b1024.conv0 and D.b1024.conv1 for the block at $1024\times
1024$ scale). There will be $\log_{2}{(\texttt{D.img\\_resolution}/4)}$ blocks
in total, and the last one at $4\times 4$ scale will have a minibatch
standard-deviation layer, a convolutional layer, and two fully-connected
layers, the latter one giving the actual output of the Discriminator (i.e.,
the same as if passing D(input_image, None) to an unconditional model).

#### Discriminator Dreaming with intermediate layers

We show the preliminary results of Discriminator Dreaming with a StyleGAN2
Discriminator. In Figures 2 and 3 we show results using the intermediate
layers: from blocks at scales from $256\times 256$ down to $4\times 4$. In the
selected models, the blocks at higher scale generally did not produce
meaningful results, and the last fully-connected layers added little of
interest, compared to the convolutional layer in the block at scale $4\times
4$. However, all of the available layers can be accessed and used by the user
if so is desired.

We show results with two additional models at $1024\times 1024$ resolution.
Earthview is a model trained with images from Earth View from Google, which
are pre-selected satellite images of the most beautiful landscapes in Google
Earth [18]. FreaGAN is a model by Derrick Schultz which was trained on Frea
Buckler’s art [25].

For both of these networks, we can see in Figures 2 and 3 that they were
trained via transfer learning from FFHQ, as the results when using the
convolutional layers of the larger blocks are quite similar. This makes sense,
as the training of these models was limited compared to that of FFHQ, mainly
due to a difference in the size of each dataset, so the gradients were mostly
concentrated in the lower resolution blocks.

#### Resizing octaves

We can apply the algorithm to a single image, but that would rarely produce
significant amount of alteration. In order to affect different layers of
granularity of the image, we can apply Discriminator Dreaming at different
scales by reducing the size of the image and adding this loss to the overall
image. Each scale of the image is referred to as an octave.

However, given that our Discriminator has pretty aggressive downscaling per
block (each block lowers the resolution by a factor of 2), we might eventually
encounter errors of size mismatch. To solve this, we can simply resize each
octave back to the original resolution. This will make the code run slower,
but we also obtain different results, as can be seen in Figures 2 and 3 (using
the same parameters, except whether or not we resize the octaves).

Figure 2: Discriminator Dreaming with different models of $1024\times 1024$
resolution, without resizing the octaves. Columns denote the individual layer
we are using to do the dreaming (starting from block $256\times 256$ down to
$4\times 4$) and rows the models used. We use image 00012.png as a starting
image. Zoom in for details. Best viewed in color. Figure 3: Discriminator
Dreaming with different models of $1024\times 1024$ resolution, plus resizing
the octaves. Columns denote the individual layer we are using to do the
dreaming (starting from block $256\times 256$ down to $4\times 4$) and rows
the models used. We use image 00012.png as a starting image. Zoom in for
details. Best viewed in color.

#### Random image

In lack of a starting image, as per the original DeepDream results, we can
start from a random noise image (initialized with a seed), to see what the
network generates. Figure 4 shows the result of Discriminator Dreaming with
two different layers, starting from a random image with seed $0$.

#### Using more than one layer

The user may also use more than one layer to play with. One of the layers may
overtake the other, so we can normalize each by either the number or square
root of elements of elements per layer. There is no limit on the number of
layers to use, though it is recommended to start with a lower amount for
better control of the final result.

(a) Random noise image (seed 0)

(b) b16_conv1

(c) b8_conv1

(d) b16_conv1 and b8_conv1, no normalization

(e) b16_conv1 and b8_conv1, with normalization

(f) b16_conv1 and b8_conv1, with square-root normalization

Figure 4: Results of Discriminator Dreaming starting from a random noise
image, using the Earthview model and two different layers. We use $10$ octaves
(and resize them), with an octave scale of $1.4$, learning rate of $0.01$, for
a total of $20$ iterations. When using multiple layers, we can normalize each
by either its number of elements or square-root of the number of elements, so
as to let each layer have equal effect on the final image.

###  A.1 Video

Applying the algorithm iteratively on the outputs, as per the original
DeepDream, we can create some interesting video outputs. In each case, the
user can set the frames per second (FPS) and length of the video, learning
rate (so as to better control the effect per frame), the number of octaves,
the octave scale, which layers to use from the Discriminator, should each
layer be normalized, etc. Each frame will be saved in the resulting directory,
allowing for the code to save as a video in the end or for the users to use
each individual frame as they please. The full code and list of available
settings can be found at https://github.com/PDillis/stylegan3-fun.

We showcase some examples of zoom, rotation, and translation. For each, if the
image is zoomed or rotated, the background will be black by default, but this
can be changed by the user. Note that each of the following can also be $0$,
so the generated video will be iteratively applying the effect to a static
image.

#### Zoom

We zoom the dreamed image after each iteration by a fixed amount of pixels and
then we resize it back again to the Discriminator’s expected image size,
forming a loop. A sample video of zooming in (by one pixel) can be found at
https://youtu.be/HAO3Kj2Y2Ok. In it, we use the layer b16_conv1 of the
MetFaces model, starting from image 00012.png of FFHQ. We use 5 octaves whilst
resizing them, a learning rate of $5\times 10^{-3}$, for a total of $30$
seconds at $30$ FPS, with $10$ iterations per frame. Note that the number of
pixels to zoom by can be of arbitrary magnitude and sign, so a zoom-out result
can also be obtained if it is set to e.g. $-1$.

#### Rotation

We rotate each dreamed image by a fixed angle counterclockwise, usually low,
before feeding it back to the Discriminator (without expanding the image). A
sample video of rotation (rotating by $0.2\deg$ counter-clockwise) can be
found at https://youtu.be/0QD1O_g-VcE. In it, we use layer b8_conv1 of the
MinecraftGAN model, using $5$ octaves without resizing, a learning rate of
$5\times 10^{-3}$, for a total of $60$ seconds at $30$ FPS, with $20$
iterations per frame.

The rotation angle can be higher (or negative), it can be set so that the user
gets the desired result. Another sample video using both rotation (by
$0.1\deg$ counter-clockwise) and zoom (by $-1$ pixel) can be found at
https://youtu.be/hEJKWL2VQTE. We use the same settings as the previous video,
but an FPS of $25$ as well as reversing the final video, resulting in a fake
zoom-in.

#### Translation

We can translate the image in both horizontal and vertical axes in an
independent manner, giving a panning result. The image is translated from left
to right and from top to bottom when using positive values of translation. As
an example, a sample video using both horizontal and vertical translation can
be found at https://youtu.be/_WCyhrymo-0. We use the b16_conv1 layer of the
MinecraftGAN model and translate both axes by $1$ pixel per frame, a learning
rate of $5\times 10^{-3}$ with $20$ iterations, using 5 octaves without
resizing.


# Is hydrogen diffusion in amorphous metals non-Arrhenian?

Chunguang Tang1,2<EMAIL_ADDRESS>Gang Sun3 Yun Liu1
<EMAIL_ADDRESS>1 Research School of Chemistry, The Australian National
University, Canberra, Australia;
2 Institute of Climate, Energy and Disaster Solutions, The Australian National
University, Canberra, Australia;
3 Department of Fundamental Engineering, Institute of Industrial Science,
University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan.

###### Abstract

Hydrogen diffusion is critical to the performance of metals for hydrogen
storage as well as other important applications. As compared to its
crystalline counterpart which follows the Arrhenius relation, hydrogen
diffusion in amorphous metals sometimes are experimentally found to be non-
Arrhenian. In this work we studied the diffusion of hydrogen in amorphous Pd-H
and Zr-Cu-H alloys based on molecular dynamics simulations. Our simulations
confirm Arrhenian diffusion behaviour for hydrogen in amorphous alloys, in
contrast to previous theoretical studies which predict non-Arrhenian
behaviour. We show that the simulated non-Arrhenian diffusion based on
molecular dynamics could result from a systematic error related to too short
simulation time. We also discussed the experimental non-Arrhenian behaviour of
hydrogen diffusion within the framework of quantum tunneling and amorphous-
amorphous phase transformations.

Published at International Journal of Hydrogen Energy, 47, 9627 (2022).

##  I Introduction

Hydrogen, as the smallest element, can easily diffuse into and interact with
materials. Specifically, the interactions between hydrogen and metals have
remained as an important research topic for more than a centurary. For
example, the old phenomenon of hydrogen embrittlement johnson_remarkable_1875
; zhong_computer-simulation_1993 ; song_atomic_2013  is a factor to be
considered for the design of internal combustion engines for hydrogen-rich
fuels saborio_gonzalez_review_2018  and transport of hydrogen through
pipelines takayama_hydrogen_2011 ; thomas_hydrogen_2020 . Recently metal
hydrides have attracted extensive research activities for their broad
applications, such as hydrogen storage rusman_review_2016 , smart window and
sensors yoshimura_metal_2013 . Specifically, the past decades have seen
intensive research interests in amorphous metals for hydrogen-related
applications yamaura_hydrogen_2005 ; dolan_composition_2006 ;
lin_hydrogenation_2019 . Compared to their crystalline counterparts, amorphous
metal hydrides exhibit some favorable properties, such as higher hydrogen
storage capacity maeland_hydrides_1980 ; lin_room_2012 , easy tuning of
dehydrogenation conditions lin_towards_2016 , and enhanced (de)hydrogenation
kinetics of metal hydrides zhang_remarkably_2015 ; el-
eskandarany_metallic_2019 .

In the above examples and also a wider range of applications
gapontsev_hydrogen_2003 , hydrogen diffusion is a critical factor that impacts
on material properties and performance. The diffusion of hydrogen in
crystalline metals is relatively simple and the temperature ($T$) dependence
of diffusivity $D$ follows the Arrhenius equation

$D=D_{0}{\rm{exp}}(\frac{-E_{a}}{RT})$ (1)

where $E_{a}$ is the activation enthalpy for the diffusion, $R$ is the gas
constant, and $D_{0}$ is a constant. On the other hand, hydrogen diffusion in
amorphous metals exhibits complicated features. For example, it is sensitive
to hydrogen concentrations, which can be explained by the assumption of
Gaussian energy distribution for hydrogen in various interstitial sites
kirchheim_hydrogen_1988 . At higher hydrogen concentrations, hydrogen atoms
tend to occupy those higher-energy sites, resulting in increased diffusivity.
Hydrogen diffusion in amorphous metals intriguingly deviate from the Arrhenius
equation. For example, in a series of amorphous metals bowman_nmr-studies_1981
; bowman_proton_1982 ; bowman_hydrogen_1983 , such as TiCuHx ($x$ from 1.3 to
1.4) and Zr2PdH2.88, the Arrhenius plot of log$D$ versus $1/T$ breaks into
several segments over the temperature range between 115 K and 420 K, with the
activation enthalpy being higher at higher temperatures. Although the normal
Arrhenius relation is also observed in many amorphous systems
lee_hydrogen_1985 ; markert_hydrogen_1988 , the temperature ranges for these
measurements are small compared to those of the non-Arrhenian datasets, and it
is not clear how the diffusion behaviour changes in a larger temperature
range.

The non-Arrhenian hydrogen diffusion has attracted a number of theoretical
investigations. Monte Carlo simulations kirchheim_monte-carlo_1987  have
reported negative curvatures of the Arrhenius curve (i.e., $E_{a}$ decreases
as $T$ increases) or linear Arrhenius lines, depending on the assumption of a
Gaussian distribution or a single value of hydrogen interstitial site energy.
A structure model that relates the negative curvature to the temperature
dependence of the short-range order in amorphous metals has also been proposed
eliaz_new_1999 . On the other hand, a molecular dynamics simulation work
lee_comparative_2014  has reported a positive curvature for H diffusion in Zr-
Cu alloys. Overall, however, these theoretical studies (for the Monte Carlo
simulations kirchheim_monte-carlo_1987 , if Gaussian distribution is assumed)
predict rather smoothly curved Arrhenius plots, instead of linear segments as
observed in experiments. A recent study based on Monte Carlo simulation
hung_estimation_2010  reported a kinked Arrhenius relation for hydrogen
diffusion in amorphous iron, but without a detailed discussion of it.

As can be seen, to date the diffusion of hydrogen in amorphous metals is not
well understood. The theoretical interpretations of hydrogen diffusion in
amorphous metals are inconsistent with each other as well as inconsistent with
the experiments. Moreover, the mechanisms of the non-Arrhenian (or segmented
Arrhenian) diffusion behaviour are not clear. There exists evidence that
hydrogen also impacts on the structural/thermal stability and mechanical
properties of amorphous alloys jayalakshmi_effect_2006 . Hence, for amorphous
metals to reach their full potential for hydrogen-related applications, it is
necessary for us to fundamentally understand the abnormal diffusion of
hydrogen. In an attempt to shed further light on this issue, we here study the
diffusion of hydrogen in amorphous palladium (Pd) and ZrCu alloy based on
molecular dynamics simulations. We choose these two systems partially because
palladium, as a single-element metal, has very low glass forming ability while
ZrCu is a good glass former and represents the prototype of many important
amorphous alloys. As a first discovered metal with hydrogen absorption
capacity graham_absorption_1866 , Pd still finds active applications in the
hydrogen energy sector dekura_hydrogen_2019 ; lamb_ammonia_2019 .

##  II Methods

To obtain the amorphous structures, the Pd-H alloy of composition H/Pd=5%
(containing 4000 Pd and 200 H atoms) was well liquefied at 2100 K and quenched
to 300 K at 1014 K/s, and the Zr-Cu-H alloy with H/metal=10% (with 2197 Zr,
2197 Cu, and 439 H atoms) was well liquefied at 2000 K and quenched to 300 K
at 1.25$\times$1012 K/s. The amorphous states of the resulting structures were
confirmed by their pair distribution functions (Fig. 2). We annealed the
amorphous structures at various temperatures to obtain the equilibrium system
volume. These computations were carried out using the NPT (constant atom
number, pressure, and temperature) ensemble by setting the normal pressures to
zero. Then for each temperature we built a structure by scaling the amorphous
structure to the corresponding equilibrium volume and the mean squared
displacements (MSDs) of H atoms were calculated using this structure based on
the NVT (constant atom number, volume, and temperature) ensemble, following an
initial relaxation (for 5 to 50 ps) that equilibrates the structure. For each
temperature we performed 10 sets of MSD computations by setting different
initial velocities of the atoms. At low temperatures the initial relaxation
upon different initial velocities may not result in very different structures
of the metallic hosts because of the slow dynamics, and hence it remains
unclear whether the resulting MSD data are representative. We note that, due
to the enough large size of the systems, the single structure obtained from
the melting-quenching is representative of the amorphous structure
characteristics. To confirm this, we built 10 more independent Pd-H amorphous
structures by further relaxing the system at 2100 K for 10 to 100 ps before
quenching, and performed one MSD computation for each structure at 300 K
following the above method. It was found that the MSD data based on these
independent structures are essentially identical to those based on the single
amorphous structure with different initial velocities.

The obtained MSDs can be used to compute the self diffusion coefficient of H
atoms following Einstein relation

$D=\frac{1}{6}\lim_{t\to\infty}\frac{\partial\langle r^{2}(t)\rangle}{\partial
t}$ (2)

where $t$ is time and $\langle r^{2}(t)\rangle$ is the ensemble average of the
MSDs of diffusing atoms under investigation. The atomic interactions of the
Pd-H system were described by an embedded atom method (EAM) potential
zhou_embedded-atom_2008 , and for the Zr-Cu-H system, a modified EAM potential
lee_comparative_2014  was used. Tests zhou_embedded-atom_2008 ;
lee_comparative_2014  of thermal properties, hydrogen migration energies, and
crystal lattice constants show comparable results with experiments and/or
first principles computations, indicating the reliability of the potentials.
All the simulations were performed using the code LAMMPS plimpton_fast_1995
with Nos$\rm{\acute{e}}$-Hoover thermostat and barostat, and the timestep for
all computations was set as 1 fs for Pd-H and 0.5 fs for Zr-Cu-H,
respectively. We chose 0.5 fs for Zr-Cu-H following ref lee_comparative_2014
for comparison. Ten independent MSD calculations for each temperature were
carried out for statistics unless otherwise specified.

Figure 1: Experimental hydrogen diffusion data replotted from Fig. 3 of
reference [34] for Pd80Si20 and Fig. 6 of [18] for Ni35Ti65, with CH being the
ratio of hydrogen to metal. The inverse diffusion correlation time
($\tau_{\rm{C}}^{-1}$), based on proton rotating-frame relaxation and
indicative of hydrogen diffusion coefficientbowman_nmr-studies_1981 , is
replotted from reference [21]. The diffusion is Arrhenian in small temperature
ranges but the Arrhenian behaviour becomes segmented in a larger temperature
range.

##  III Results and discussions

Figure 2: Structure of Zr-Cu-H system. (top) Amorphous structure model at 300 K after melting-quenching. Cu: brown, Zr: green, H: white. (bottom) Cu-Cu pair distribution function of the Zr-Cu-H alloy at 2000 K (liquid state) and 300 K (glass state). The splitting of the second g(r) peak is characteristic of the amorphous structure. The functions for Zr-Zr pair and Pd-Pd pair for the Pd-H alloy are similar and not shown. Figure 3: (left) A fixed amorphous Pd host (4000 atoms) corresponding to 300 K, with each dot representing a Pd atom. (right) Relative potential energy of the system (4001 atoms) that contains a single test H atom at one of the $\sim$22300 Pd Voronoi vertices corresponding to the fixed Pd host. Each dot represents a vertex and its color represents the potential energy. Figure 4: H motion, at 300 K if not specified, in Pd-H alloys with Pd atoms fixed (see Fig. 3(a)). (a) The distribution of hydrogen atoms. (b) Example mean squared displacement curves of H with high (red) and low (blue) initial potential energies (PEs). Inset: Relative system PE with only a test H atom at different vertices of Pd Voronoi polyhedra. (c) PE of an example atom as a function of time. The PE at 600 K is shifted upwards by 0.4 eV for clarity. (d) Temperature effect on PE distribution of H atoms. On average, the minimum PE of an H atom is -1.15$\pm$0.016 eV and -1.15$\pm$0.017 eV at 300 and 600 K, respectively. Table 1: Hydrogen diffusion data in amorphous Pd-H and Zr-Cu-H alloys. $t_{0}$ is the time span for computation, $\langle r^{2}\rangle$ is the mean squared displacement value corresponding to time $t_{0}$, and $D$ value was obtained from the slope of linearly fitting $\langle r^{2}(t)\rangle$ versus $t$ using the top 50% $\langle r^{2}(t)\rangle$ values. STD is for standard deviation. $T$ (K) |  $t_{0}$ (ns) |  $\langle r^{2}\rangle$ (Å2) |  $D$ (cm2/s) | STD (cm2/s)
---|---|---|---|---
Pd-H
125 | 0.1 |  $\sim$11 | 1.48e-6 | 3.27e-7
125 | 10 |  $\sim$250 | 3.83e-7 | 8.00e-8
150 | 0.1 |  $\sim$20 | 2.80e-6 | 5.02e-7
150 | 10 |  $\sim$810 | 1.25e-6 | 1.59e-7
200 | 0.1 |  $\sim$60 | 9.80e-6 | 1.34e-6
200 | 1 |  $\sim$490 | 7.87e-6 | 8.70e-7
250 | 0.1 |  $\sim$140 | 2.17e-5 | 3.12e-6
300 | 0.1 |  $\sim$280 | 4.65e-5 | 4.80e-6
350 | 0.1 |  $\sim$440 | 7.14e-5 | 8.12e-6
400 | 0.1 |  $\sim$640 | 1.06e-4 | 1.19e-5
450 | 0.1 |  $\sim$820 | 1.39e-4 | 1.53e-5
500 | 0.1 |  $\sim$1110 | 1.82e-4 | 1.66e-5
Zr-Cu-H
400 | 0.1 |  $\sim$1.3 | 1.30e-7 | 3.80e-8
400 | 10 |  $\sim$22 | 2.88e-8 | 3.67e-9
500 | 0.1 |  $\sim$3.1 | 3.84e-7 | 9.20e-8
600 | 0.1 |  $\sim$7.2 | 9.62e-7 | 1.85e-7
700 | 0.1 |  $\sim$15 | 2.26e-6 | 3.80e-7
800 | 0.05 |  $\sim$17 | 5.13e-6 | 9.14e-7
1000 | 0.05 |  $\sim$60 | 1.81e-5 | 1.63e-6
1100 | 0.05 |  $\sim$90 | 2.92e-5 | 2.55e-6

We start our study by investigating the local dynamics of hydrogen since,
based on previous theoretical studies, the diverse local environment of
hydrogen interstitials impacts on its diffusion behaviour. Understanding the
detailed local dynamics in amorphous materials represents a nontrivial task
and here we only study the site energetics of H atoms with Pd atoms fixed for
simplification. The fixed Pd atoms are shown in Fig. 3(a). At 300 K, we
collected H positions at the end of MSD calculations and geometrically
optimized their positions. Fig. 4(a) shows the distribution of their distance
to the nearest vertex of Pd Voronoi polyhedra (i.e., Pd Wigner–Seitz cells),
with $\sim$90% of the atoms coincide with the vertices in a tolerance of 0.3
Å. This indicates that the Voronoi vertices are preferred interstitial sites,
although there may be a small fraction of unfavorable vertices corresponding
to small open volume lancon_simulation_1985 . We then put a test H atom at
various vertices and computed the potential energy of the system without
geometrical optimization. Fig. 3(b) shows the space distribution of the
relative potential energy of the single H atom. The statistics of the relative
potential energy features a bell shape, as shown in the inset of Fig. 4(b). We
performed five sets of MSD calculations to test the effects of high and low,
respectively, site energy on diffusion. For each calculation, we placed 200 H
atoms on randomly selected low (or high) energy vertices, with at least 2.5 Å
apart from each other, and computed their MSDs after optimizing their initial
positions. As indicated in Fig. 4(b), initially the atoms of high energies
move much faster than those of low energies, but after $\sim$100 ps the two
groups of MSD curves exhibit similar slopes. This implies that although local
environment impacts on the movement of H atoms, the diffusion coefficient,
which is a long range property, reflects the features of the global potential
energy surface of the system.

To address the temperature effect, we relaxed H atoms at 300 and 600 K,
respectively, with Pd atoms fixed. Fig. 4(c) shows the potential energy of an
example atom as a function of the relaxation time, and Fig. 4(d) shows the
potential energy distribution of all H atoms over the span of relaxation. As
can be seen, as the temperature increases, the probability of H atoms in high
energy states increases, but their preference for low-energy sites does not
change. Indeed, we calculated the average lowest potential energy of the atoms
to be the same for 300 and 600 K. For the effect of temperature on diffusion
in amorphous materials, some Monte Carlo simulationskirchheim_monte-carlo_1987
; limoge_temperature_1990  assume the increasing probability of interstitials
at lower energy states would increase $E_{a}$ at low temperatures, while some
molecular dynamics simulationslee_comparative_2014  argue that those atoms
trapped at low energy sites hardly contribute to diffusion and hence $E_{a}$
would be lower at low temperatures. However, we believe that the measured
$E_{a}$, which is a parameter averaged over all diffusing interstitials, is a
characteristic of the potential energy surface mainly determined by the host
structure (at least for low H concentrations where H-H interactions can be
neglected). At low temperatures, the trapping of H interstitials at those low
energy sites just implies more time is needed for them to sample the potential
energy surface, which results in a decreased $D$ but not necessarily the
change in $E_{a}$. In other words, temperature change only changes $D$ of H
interstitials, but not $E_{a}$, as long as the host structure (and hence the
potential energy surface) is not changed.

Figure 5: (a) Computed diffusion coefficient. The triangular symbols are the
values of $D$ replotted from reference 26. The data difference between this
work and reference 26 is probably due to different time spans used. The
straight lines are for eye guide. The blue solid symbols at relatively low
temperatures indicate the improved accuracy from extended simulations. (b)
Mean squared displacements of H in Zr-Cu-H alloy. The top 50% MSD (highlighted
in blue) are linearly fitted, and the slopes are indicative of the convergence
of $D$ calculations. For the calculation at 400 K for $t$=10 ns (not shown),
the slope increases to 0.64.

To test our above arguments, we computed MSDs of hydrogen in amorphous Pd-H
and Zr-Cu-H alloys (without fixing the host atoms) at various temperatures.
The time span for MSD calculations was 50 ps for temperatures $T\geq$800 K and
100 ps for lower temperatures. The time span chosen here is comparable to that
(30-90 ps) used for previous simulations of Zr-Cu-H alloyslee_comparative_2014
. The obtained data for diffusion coefficient based on Equation 2 are shown in
Table 1 and plotted in Fig. 5(a). As can be seen, the Arrhenius plots for both
alloys exhibit a positive curvature, similar to the previous molecular
dynamics resultslee_comparative_2014 .

Figure 6: Mean squared displacement of H in Pd-H at 300 K. Three regimes can
be identified. The log(MSD)/log($t$) slope of the top 50% MSD is 0.86.

However, it turns out that these curved Arrhenius plots are due to the
systematic errors resulting from the too short time span used for the
calculations. As shown in Fig. 6, the MSD of atoms as a function of time can
be divided into three regimes donati_spatial_1999 . During the ‘ballistic’
regime, atoms move without colliding with other atoms, and MSD is proportional
to $t^{2}$. In the ‘caging’ regime, atoms collide with their neighbours and
move forth and back, and MSD is roughly a constant. The timescale for the
caging regime is temperature- and material-dependent. When the system enters
the deep ‘diffusive’ regime, MSD is proportional to $t$ and the Einstein
relation applies. In the log scales, the ballistic regime has a slope of 2 and
the deep diffusive regime has a slope of 1. Corresponding to these three
stages, the parameter MSD/6$t$=$\langle r^{2}(t)\rangle/6t$ increases to a
maximum at the end of the ballistic regime and then converges towards the
diffusion coefficient defined by Equation 2. Shown as an example in Fig. 5(b),
the slopes of log(MSD)/log($t$) for the Zr-Cu-H alloy are far from 1 at low
temperatures, implying the values of $D$ are far from convergence. For a
remedy, we extended the simulation time for Pd-H for temperatures below 250 K,
and the new Arrhenius plot is well linearly fitted. As shown in the figure,
extension of simulation time for the Zr-Cu-H alloy also improves the accuracy.
We finally note that, since amorphous phases lack the structural periodicity
as in crystals, the convergence of $D$ with respect to MSD for amorphous
materials is generally slower than that for crystals and hence larger MSDs or
longer simulation time is often necessary for accurate computation of $D$.

The above results indicate that hydrogen diffusion in amorphous metals follows
the Arrhenius relation, which is consistent with the experiments as mentioned
earlier. The remaining question is why non-Arrhenian diffusion is sometimes
observed. It is key to note that in this scenario the Arrhenius curves break
into several linear segments bowman_nmr-studies_1981 ; bowman_proton_1982 ;
bowman_hydrogen_1983 , exhibiting abrupt changes in $E_{a}$, instead of a
smooth $E_{a}$ variation over a large temperature range as predicted by
previous simulationskirchheim_modeling_1985 ; lee_comparative_2014 . Such
abrupt changes often imply new diffusion mechanisms and/or structural changes
of the alloys. For example, the change in $E_{a}$ for amorphous TiCuH1.4
bowman_proton_1982  and Zr2PdH2.88 bowman_hydrogen_1983  at $\sim$210 K is
possibly due to quantum tunneling in view that similar transitions in
hydrogenated ZrCr2 crystals were observed between 150 and 200 K and attributed
to quantum tunnelingrenz_pulsed-field-gradient_1994 . The decomposition of
amorphous TiCuHx ($x\sim$1.3) at $\sim$420 K was found to cause an abrupt drop
of hydrogen diffusion coefficient bowman_nmr-studies_1981 . For other $E_{a}$
changes in amorphous alloys bowman_nmr-studies_1981 ; bowman_proton_1982 ;
bowman_hydrogen_1983 , no associated phase transformations, such as
crystallization, were reported. However, these studies were carried out in
decades ago when our knowledge of amorphous structures was limited. Recent
studies reveal amorphous-amorphous phase transitions lan_hidden_2017 ;
liu_evidence_2007  in amorphous alloys, and such transitions may change the
medium-range packing of atomic clusters and hence impact on the diffusion of
hydrogen. From the perspective of potential energy landscape, upon phase
transition the system moves from one subset of the potential energy landscape
to another subset and hence exhibits different energy characteristics or
diffusion activation energy. This is not the case for the systems studied
here.

##  IV Summary

In this work we studied the diffusion of hydrogen in amorphous PdH0.05 and
(ZrCu)H0.1 alloys based on molecular dynamics simulations. Our simulations
confirm the distribution of location-dependent hydrogen interstitial energies
and the influence of temperature on the distribution of hydrogen
interstitials. It was found that hydrogen interstitials prefer the Voronoi
vertices of the host atoms. For the Pd-H system, 90% H atoms were found to be
within 0.3 Å from the vertices and, as temperature increases, the potential
energy of H atoms fluctuates in a wider range with the most probable potential
energy also increasing. In a local scale, those hydrogen interstitials of
higher potential energy are more mobile than those of lower energy. However,
in a global scale the activation energy of hydrogen diffusion reflects the
features of the potential energy landscape that depends on the material
structure. Consistently, we found Arrhenian diffusion behaviour for hydrogen
in amorphous alloys, in contrast to previous theoretical studies which predict
non-Arrhenian behaviour. While only a single composition for each alloy system
was studied in this work, we believe the conclusion for Arrhenian diffusion of
hydrogen is general since the potential energy landscape (which determines the
diffusion activation energy) is fixed for a given composition. A lesson from
this work is that we should be cautious about the convergence when computing
diffusion coefficient of amorphous structures based on molecular dynamics.
Because the potential energy landscape of amorphous structures is generally
more complicated than that of crystalline structures, usually a longer time is
needed for sampling the landscape and obtaining the correct activation
energies. We also discussed the experimental segmented Arrhenian behaviour of
hydrogen diffusion within the framework of quantum tunneling and amorphous-
amorphous phase transformations.

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