# Modeling Temporal Lobe Epilepsy during Music Large-Scale Form Perception
using the Impulse Pattern Formulation (IPF) Brain Model

Rolf Bader  Institute of Musicology
University of Hamburg
Neue Rabenstr. 13, 20354 Hamburg, Germany

###### Abstract

Musical large-scale form is investigated using an Electronic Dance Music (EDM)
piece fed into a Finite-Difference Time Domain (FDTD) physical model of the
cochlear which again inputs into an Impulse-Pattern Formulation (IPF) brain
model. In previous studies, experimental EEG data showed an enhanced
correlation between brain synchronization and the musical piece’s amplitude
and fractal correlation dimension in good agreement with a FitzHugh-Nagumo
oscillator model48. Still, this model cannot display temporal developments of
large-scale forms. The IPF Brain model also shows a high correlation between
cochlear input and brain synchronization at the gamma band range around 50 Hz,
but also a strong negative correlation for low frequencies, associated with
musical rhythm, during time frames of low cochlear input amplitude. Such high
synchronization corresponds to temporal lobe epilepsy, often associated with
creativity or spirituality. Therefore, the IPF Brain model suggests those
conscious states to happen in times of low external input at low frequencies
where isochronous musical rhythms are present.

##  I Introduction

Musical large-scale forms, the temporal organization of a musical piece over
its whole length, e.g., the song structure of verse and refrain, Electronic
Dance Music (EDM) tension build-up and relaxation, or sonata song structures
are poorly researched in terms of brain dynamics3332. It is widely accepted
that music is organized according to Gestalt principles1323, often with
hierarchical structures, psychologically20, or in terms of music theory36, and
subject to long-term memory in music18.

The processing of sound in the brain has historically been taken as a bottom-
up process, starting from the transition of sound in the cochlear into neural
spikes9 further processed in the auditory pathway47, showing coincidence
detection7, tonotopy, interaural level and time difference detection, or pitch
perception1784, next to others. Still, even within the auditory cortex,
multiple bottom-up and top-down connections are present21 and so viewing the
auditory pathway as a complex, self-organizing neural network seems more
appropriate.

Such a self-organizing view is standard in terms of cortical processing and
existing neural models vary in terms of complexity and scaling34. Only a few
brain models try to understand the brain with simple principles like the free-
energy principle26 assuming adaptation of the brain to external surprises, a
global workspace view1 assuming synchronization and de-synchronization of
brain parts over time, or a Synergetic approach of Gestalt perception31.

Machine-learning models also view conscious content as a process of complex,
often nonlinear interactions between single neurons leading to a heuristic and
coherent Gestalt, like connectionist29, 30 music models2814 also when
analyzing large musical databases as Computational Phonogram Archiving512 for
streaming platforms or archives in general11.

The self-organizing idea is also reflected in the idea of 50-millisecond
intervals of organized neural spatiotemporal patterns followed by short,
chaotic disturbances associated with olfactory35 or auditory4544 conscious
content. Enlarging the picture to interactions between subjects results in the
idea of a self-organizing society of brains24 or including cultural artifacts
and nature in a Physical Culture Theory6.

Incorporating brains with cultural objects asks for a general framework which
has been proposed as Impulse Pattern Formulation (IPF) at first developed for
musical instruments. As physically the only force we experience - next to
gravity - is electricity it is straightforward to formulate acoustic as well
as electric nervous spike impulses as being of the same nature. The IPF then
takes a viewpoint neuron or musical instrument part from which an impulse is
sent out to several other neurons, musical instrument parts, or any kind of
object. This impulse is processed, damped, and returns back to the viewpoint
object10404. Such an iterative, nonlinear-dynamical process is scale-free,
capable of modeling sudden phase changes, and includes convergent,
bifurcating, complex, and chaotic states. Although often modeling a system
with only very few nodal points, the IPF has already been shown to be of high
precision in musical instrument applications373941 as well as in rhythm
perception and production38.

The model was also formulated as a brain model3 with neural adaptation and
plasticity non-trivially finding the concentration of inhibitory vs.
excitatory neurons of 10-20% as a maximum of possible system convergence, a
maximum reflection strength around the usual time for Event-Related Potentials
(ERP), as well as a decay of memory in the system corresponding to short-time
memory. These general findings are strongly pointing to the validity of the
model.

Investigating brain dynamics of large-scale musical form has already shown
musical tension to be represented in increasing synchronization of brain
parts333248. Such findings correspond to synchronization caused by
expectancy15 towards a climax. A main finding is that brain synchronization is
strongest around 50 Hz, so in the gamma band of brain dynamics.

An interesting aspect of the IPF Brain model is the presence of convergence,
i.e., the complete synchronization of the neurons in the model. This
corresponds to epilepsy27 which is not the usual and a very dangerous brain
state. Still, partial synchronization in the brain is well known as chimera
states46. Also in terms of the auditory cortex, temporal lobe epilepsy has
often been reported43 and associated with spiritual or meditative states of
mind. Such states are naturally associated with larger time spans than pitch
or rhythm and are subject to large-scale forms.

The present paper applies the IPF Brain model as suggested before3 to the case
of an Electronic Dance Music (EDM) piece _One Mic_ of the Rapper NAS which has
been investigated before in EEG experiments3332 and by using a FitzHugh-Nagumo
dynamical brain model48. Strong correlations between the large-scale form of
the model, associated with the sound amplitude and fractal correlation
dimension as a measure of musical density10 have been found experimentally and
in the model with in both cases a maximum of synchronization in the gamma band
of brain dynamics around 50 Hz. Still, due to the FitzHugh-Nagumo model not
dissolving the temporal development of the brain dynamics, a deeper
understanding of the reason for this frequency-dependency of synchronization
asks for a model capable of such a temporal resolution, which is performed
using the IPF Brain model in this paper.

The paper first introduces the method used. As in the previous paper, the
input to the Brain IPF is the output of a Finite-Difference Time-Domain (FDTM)
physical model of the cochlear to which the musical sound is used as an input.
The cochlear input to the Brain IPF results in the development of both the IPF
system parameter g, the state of the viewpoint neuron, as well as the
adaptation of the reflection points, the neurons. To estimate the influence of
the amount of neurons used in the model, N=50, 100, and 200 neurons are taken,
showing the independence of the model with respect to this parameter. In the
method section, the post-processing of these modeled parameters is discussed,
especially the Kuramoto order parameter to measure the synchronization of
neurons and the correlation of this parameter with the cochlear input. The
result section then mainly concentrates on this correlation and discusses the
reason for the model behavior. In the conclusions, an overview of different
conscious states with respect to the synchronization frequency concludes the
paper.

##  II Method

###  A Cochlear Model

The present model assumes a differential equation of a membrane

$\frac{K(x)}{\mu(x)}\frac{\partial^{2}u}{\partial x^{2}}-d(x)\frac{\partial
u}{\partial t}=\frac{\partial^{2}u}{\partial t^{2}}+f(t)\ ,$ (1)

with BM displacement u along a one-dimensional axis x, BM stiffness
$K(x)=2\times 10^{9}e^{-3.4x}dyn/cm^{3}$ changing along x and linear density
$\mu(x)=m/A(x)$ with mass over area again changing along the BM and
$A(x)=.1cm\times(.1cm+0.02cm\times x/l)$ with BM length $l=3.5cm$ taking the
slight widening of the BM over its length into account.

A 1-D model is sufficient to model a BM as shown by 2 comparing a 1D and 2D
model based on the anisotropy of the BM as discussed by 42. Here it is found
numerically and based on experimental data that the inclusion of a second
dimension contributes less than 1% of the results already obtained by a 1-D
model. This is reasonable as the BM has dimensions of about 3.5 cm in length
but only about 1 mm in width and is therefore more a rod than a membrane. Also
the Young’s modulus in y-direction is only about 10% of that in x-direction 2
and does therefore not add considerably to the overall BM movement.

To confirm this finding, a 2-dimensional model was built

$\frac{K_{x}(x)}{\mu(x,y)}\frac{\partial^{2}u}{\partial
x^{2}}+\frac{K_{y}(x)}{\mu(x,y)}\frac{\partial^{2}u}{\partial
y^{2}}-d(x,y)\frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial
t^{2}}+f(t)\ ,$ (2)

with Young’s modulus in x-direction $K_{x}(x)=2\times
10^{9}e^{-3.4x}dyn/cm^{3}$ as above and in y-direction $K_{y}(x)=.1\times
K_{x}(x)$ according to the literature 2. The linear density $\mu(x,y)=mu(x)$
of the 1D model, which holds also for damping $d(x,y)=d(x)$. The model
consisted again of 48 modal points in x-direction and 6 nodal points in
y-direction. The boundary conditions were again simply supported. Still the
results did not differ considerably as shown below comparing the tuning curves
of the 1D model in Fig. 3 and those of the 2D model shown in Fig. 4. For sake
of simplicity, also taking the computational cost into consideration the 1D
model was then used further.

The Electronic Dance Music (EDM) piece _One Mic_ by the Rapper NAS was used as
a sound input to the cochlear model. The FDTM sample rate was 192 kHz to
ensure model stability. The output is a set of spikes time points $S^{i}_{B}$
at each of the 24 Bark bands B with i = 1,2,3…$N_{B}$, where $N_{B}$ is the
maximum number of spikes at Bark band B. Each $S^{i}_{B}$ has an associated
amplitude $A^{i}_{B}$. Although single spikes have a more or less uniform
amplitude, the cochlear nerve fiber output of the model sums many spikes at
each position $S^{i}_{B}$. Therefore $A^{i}_{B}$ represents the amount of
spikes or the output strength.

As input to the IPF Brain model all output as accumulates with respect to 20
ms time intervals, corresponding to $f_{m}ax=$ 500 Hz which then is the time
constant of the IPF Brain model discussed below. Therefore, the frequency
range of interest in the brain up to about 200 Hz is well represented. The
resulting cochlear output time series then is denoted

$C_{t}=\sum_{B=1}^{24}\sum_{i=1}^{f_{max}\times
T_{max}}G(A^{i}_{B})H(S^{i}_{B})\ ,$ (3)

where $T_{max}$ = 266 seconds, the length of the musical piece, and the
function G and H detect if the respective spike is within the respective time
window and gives $G(A^{i}_{B})=A^{i}_{B}$ and $H(S^{i}_{B})=1$ if so. For
further correlation with the IPF Brain model output, $C_{T}$ is calculated as
the time series averaged over time windows of 1 second. The unitless mean
amplitude of the $\overline{C_{t}}$ = 0.000118.

###  B Brain Impulse Pattern formulation (IPF)

The brain is modeled using N = 50 neurons. Each neuron is a reflection point,
returning impulses from a viewpoint neuron. The system state of the viewpoint
neuron is g, which represents a time period and amplitude strength. Each
reflection neuron i has a damping $\alpha_{i}$. The IPF then is

$g_{t}=\frac{1}{N}\left|\sum_{i=1}^{N}P_{i}\ln\alpha_{t}^{i}\
g_{t-i}\right|+w_{2}\ g^{I}_{t}\ .$ (4)

Here the viewpoint neuron is i=1 the reflections come from neurons i=2,3,4,…N.
Pi are the polarizations of the neuron, where Pi = 1 is an excitatory and Pi =
-1 is an inhibitory neuron. Throughout the paper a relation of 10% of
inhibitory neurons is used. Note that the sum of all reflections is normalized
using the amount of neurons N. The model is discrete with time steps
t=0,1,2,3… Therefore, the earlier states of the viewpoint neuron, which this
neuron has sent out to the other neurons, are returning after a delay in a
damped and polarized form.

For a deeper discussion of the model see3.

###  C Plasticity Model

The plasticity of each neuron is calculated for each time step t. Plasticity
means a change in the damping parameter $\alpha_{i}$, where each time step t
then might have a different damping $\alpha_{t}^{i}$. Note that in the IPF
reasoning, the damping is originally $1/\alpha$, which, for the sake of
convenience, is skipped, now using $\alpha$ instead.

For each time step, the new damping is calculated like

$\alpha_{t}^{i}=\left|\alpha_{t-1}^{i}+w_{1}\ln(1+(g_{t-i}-g_{t}))\right|\ .$
(5)

If the reflection point neuron t-i has the same value $g_{t-i}$ as the
viewpoint neuron value $g_{t}$ the logarithm becomes zero, and no change in
the damping $\alpha_{t^{i}}$ happens. If the reflection point neuron t-i has a
larger value than the viewpoint neuron, the logarithm becomes larger than
zero, and $\alpha_{i}$ increases. Otherwise, the logarithm assures a negative
influence, and $\alpha_{i}$ decreases. The plasticity process is generally
modeled using a constant $w_{1}$. Therefore, plasticity can be switched off in
the model by using $w_{1}=0$. To examine different model behavior, $w_{1}$
will systematically be altered, shown below. Again, the absolute value of
$\alpha$ is used, not allowing negative or complex values. This, again, does
not change the model behavior due to the logarithms used. Still, positive
values are more convenient. Indeed, negative arguments of the logarithm in the
simulations shown below appear very rarely and are additionally suppressed by
using the absolute value.

###  D External Musical Instrument Input IPF

Like with a previous study48 the Electronic Dance Music (EDM) piece _One Mic_
of the Rapper NAS has been used.

###  E Detection of System Behavior

Each neuron has its own time series

$g^{i}_{t}=P_{i}\ln\alpha_{t}^{i}\ g_{t-i}$ (6)

when reflecting back to the central neuron, as can be seen in Eq. 4, where
$g^{i}$ is the ith neuron at time point t with a certain $\alpha^{i}_{t}$ at
that time point. These time series are Fourier analyzed with time windows of
one second, resulting in $F^{i}_{T}(f)$, where $0\leq T\leq 266$ seconds, the
length of the musical piece. All IPF simulations in this paper are performed
with a maximum of $f_{max}$ = 500 Hz, therefore 1 $\leq$ f $\leq$ 500 Hz. The
time series of the central neuron will be labeled below as simply $g$.

Synchronization is measured, as in a previous paper, using the Karumoto order
parameter

$K_{T}(f)=\frac{1}{N}\left|\sum_{i=1}^{N}e^{\imath\ \theta^{i}_{T}(f)}\right|\
,$ (7)

where $\theta^{i}_{T}(f)$ is the phase of the ith neuron at time interval T of
frequency f taken from $F^{i}_{T}(f)$. N is the amount of neural reflection
points, in this study N=50, N=100, and N=200.

The synchronization order parameter $K_{T}(f)$ is time-dependent. To estimate
the overall synchronization strength, from $K_{T}(f)$ a time-averaged mean
$K(f)$ is calculated.

Also, the correlation of $K_{T}(f)$ with the cochlear input time series
$C_{T}$ is performed, leading to $K^{C}(f)$, an estimation of the frequency-
dependency of the correlation strength of the synchronization with the time
series.

##  III Results

To estimate the influence of the amount of neurons used, analysis was
performed for N=50, like in a previous study, as well as N=100, and N=200.

###  A System behavior

At first, the system parameter g is expected to follow the cochlear input
systematically. Fig. 1 shows the cochlear input $C_{T}$ at the bottom (blue)
and the system parameter g of the central neuron (yellow) at the top. The
simulation was performed for N=50, N=100, and N=200 neural reflection points,
where the figure shows the N=100 case. Visually, the time series correspond
well. The correlations are 0.37 for N=50, 0.38 for N=100, and 0.40 for N=200.
This is the first example of a high independence of the results with respect
to the amount N of neural reflection points.

Figure 1: Time series of cochlear input $C_{t}$ (blue) and system parameter g
(yellow) over the 266 seconds of the musical piece _One Mic_ by NAS, using
N=100 reflection points. Both time series correlate with 0.38.

###  B Gamma band synchronization strength

The IPF Brain model is expected to reproduce experimental data. It was found
that the strongest correlation between brain synchronization and musical sound
input is in the gamma band around 50 - 80 Hz. Below and above the gamma band,
synchronization is still there, but decreases considerably.

Figure 2: Frequency-dependent correlation of cochlear input with Kuramoto
synchronization $K^{C}(f)$. As expected, the gamma band around 50-80 Hz shows
strongest correlation, while correlation decreases to lower and higher
frequencies. Very low frequencies have negative correlations, pointing to a
conversion of the system during low sound input time windows.

Fig. 2 shows the $K^{C}(f)$, the correlation of the Kuramoto order parameter
with cochlear input $C_{T}$ for frequencies up to 200 Hz for the three
reflection point N=50, N=100, and N=200. As expected, there is a peak of
$K^{C}(f)$ around 50 Hz with decreasing correlation both above 50 Hz, up to
about 90-100 Hz, and below 50 Hz. This is consistent through all N=50, 100,
and 200, again pointing to the independence of the choice of N. Still, the
correlation for N=50 is slightly lower than for N=100 or N=200, a tendency
already found above.

As an example of the correspondence between $K^{C}(f)$ and $C_{T}$ in Fig. 3
the time series of $C_{T}$ and $K^{C}(50)$, so for 50 Hz for the N=100 case is
shown. Clearly, synchronization follows the amplitude of the cochlear input,
the musical piece, nicely. The synchronization is not as smooth as found
experimentally, still, the model is only about listening to this musical piece
while the EEG experimental data measure the whole brain performing many other
tasks at the same time.

Correlations decreases up to 90-100 Hz and increase again to higher
frequencies. Oscillations higher than about 100 Hz are not considered within
’classical’ bands like delta (0.5-3.5 Hz), theta (4-7Hz), alpha (8-12 Hz),
beta (13-30 Hz), or the gamma band which is found at frequencies above 30
Hz22, or split into lower gamma range (30 -60 Hz) and higher gamma range
(60-120 Hz)25. Frequencies above 120 Hz are referred to as fast oscillations16
and only briefly associated with conscious content.

Figure 3: Kuramoto order parameter $K_{T}(f)$ of 100 neural reflection points
at a frequency of 50 Hz (yellow) and cochlear input $C_{T}$ (blue). Mainly,
during windows of strong cochlear input corresponds to large Kuramoto
synchronization and vice versa.

Looking for an explanation of the gamma band strength, we could expect some
connection between the frequency-dependency of $K^{C}(f)$ and the cochlear
input $C_{t}$ or the system parameter g. Still, connections in terms of their
spectral amplitudes are not straightforward. In Fig. 4 both spectra are shown.
Both spectra are very similar, pointing to a response of the Brain IPF to its
input, as expected. Indeed, around 50 Hz a peak is present in both time
series, corresponding to the 50 Hz peak of $K^{C}(f)$. Still, the 50 Hz peak
in $C_{t}$ and g are much sharper. Also, $C_{t}$ and g spectra have a wider
peak around 90-100 Hz while the correlation with Kuramoto synchronization has
a gap in this region.

Furthermore, both spectra of $C_{t}$ and g have enhanced energy in the low-
frequency range, pointing to the rhythm content of the musical piece. The
piece has 92 BPM corresponding to 1.53 Hz. During the high amplitude sections
of the piece, semiquaver notes are played, resulting in 6.1 Hz, during the low
amplitude sections the piece uses mainly quaver notes at 3.1 Hz. So much
energy is expected below about 6 Hz which is the case.

Again, connections to $K^{C}(f)$ are not straightforward, as about below 10 Hz
$K^{C}(f)$ is mainly negative. This is not found experimentally and is
surprising at first. Still when examining an example of N=100 at 2 Hz as shown
in Fig. 5 the reason becomes clear. During times of low cochlear input,
amplitude synchronization is very strong, while it is lower when the cochlear
input is stronger. In a previous study on the principle behavior of the IPF
Brain model3 it appeared that at a relation of 10% of inhibitory vs.
excitatory neurons in the brain, an optimum in terms of convergence or
stability of the system is reached. Therefore, we expect the system to
converge when low external input is presented which is the case here. This
leads to a systematic increase of synchronization at times of low external
input and therefore to a negative correlation.

Figure 4: Spectra of the cochlear input $C_{t}$ and the system parameter g for
N=100. Peaks are present at about 50 Hz and 100 Hz. Strong amplitudes at very
low frequencies pointing to rhythm content of the musical piece.

For comparison, an IPF Brain model fed with white noise was performed to see
if the low-frequency behavior might be a systematic bias of the model. Still,
over the whole frequency range low correlations around zero were found with
slightly positive and negative values and no frequency-dependency, as
expected.

Such convergence would correspond to epilepsy, as all neurons are
synchronized. Indeed, epilepsy is reported in the auditory cortex, associated
with creativity or spirituality43. Such non-everyday experiences are known to
occur as trance, often associated with either long periods of isochronous
rhythm or during silence. Both is present in the musical piece during times of
low cochlear input. Of course, as discussed above, the model is only about
listening to a musical piece without performing other tasks like a real brain
does. Still, temporal lobe epilepsy is only present at the auditory cortex,
and therefore, the negative correlations might model such a case.

Indeed, in the EEG data, there is enhanced synchronization at such low
frequencies again, not found with the FitzHugh-Nagumo model48. This
synchronization is positive and not negative. This might be due to the EEG
data taken from all regions of the human skull, while the IPF Brain model is
not specifying brain regions and, therefore, is also suitable to represent the
auditory cortex only. Zhe enhanced synchronization at lower frequencies might
also be caused by motor action taken in the brain at low frequencies in the
body movement dancing range.

Figure 5: Cochlear input $C_{T}$ and Kuramoto order parameter $K_{T}(2Hz)$ for
the N=100 neuron reflection point case. During times of low cochlear input
synchronization is very strong, up to 0.8, pointing to a convergence behavior
of the system during times of low input at low frequencies, similar to a
temporal lobe epilepsy.

###  C Scaling law

EEG data of the brain show a scaling law in the brain spectrum. Plotting this
spectrum in a log-log plot exhibits a constant slope over a wide range of
frequencies35. This corresponds to a so-called 1/f-noise, as the spectrum
follows a $1/f^{\alpha}$ rule, where a fractal dimension of $\alpha\sim 2$ is
found. The scaling law points to a close and systematic connection between the
different frequency regions of brain activity and the self-organizing nature
of the brain.

The IPF Brain model also shows such a scaling law as shown in Fig. 6.
Experimental EEG data also show a plateau or even a positive slope up to about
5 Hz, consistent with the IPF Brain findings. The rippling of the spectrum
indicates enhanced activity in certain regions like with the gamma band found
already above. The only aspect not corresponding to EEG data is the fractal
dimension of IPF at $\sim 0.43$ rather than the expected $\sim 2$. The reason
here is unclear and subject to future analysis.

##  IV Conclusions and Discussion

The association of temporal lobe epilepsy with spiritual or meditative
experiences corresponds well with the intention of the musical piece _One Mic_
used in this investigation. The lyrics report about the hard struggle in
criminal gangs with police interaction and shootings. Several references are
made to spiritual, especially Christian symbols and ways of similar suffering.
Such lyrics are rapped during the large amplitude time frames of the piece,
the verses, where also a police siren can be heard. These sections are
followed by the low-amplitude parts where the refrain _All I need is one mic_
is repeated, pointing to music and lyrics production as an alternative or a
weapon against such a hard struggle. These sections contrasts the verses as
they are presented in a contemplative or meditative way.

This compositional tool of presenting a meditative alternative by reducing the
volume, reducing the beat from semi-quavers to quavers and omitting most of
the sound effects and keyboard pads of the verse is shown in this paper to
produce strongly enhanced synchrony and convergence of the neural network.
This synchrony is found in the low frequency range where the musical rhythm is
represented.

Such musical structures are very simple and present in many musical pieces,
where regions of low volume and isochronous rhythm are present. Further
investigations are needed to model and measure the neural reaction to such
musical content in more detail. Still, due to the simplicity of this
compositional tool, one can expect composers and musicians to use it in many
musical scenarios.

Figure 6: Scaling law of brain synchronization measured as Kuramoto order
parameter $K^{C}(f)$. In a log-log plot over a wide range of frequencies a
constant slope is pointing to the scaling law of the IPF Brain model
consistent with EEG data.

In a previous similar EEG study of a musical piece _Classical Symphony_ by
Shemian again, increased brain synchronization is found towards an expectancy
point, after which synchronization decreases again32. This is a typical EDM
piece such that a tension is built up by compositional tools like increased
amplitude and event density only to climax at a point where the dense
structure ends and a four-to-the-floor bass drum starts. Again, this is a
typical compositional tool in EDM, a tension build-up and decay repeated
dozens of times during a dance night. There again this compositional tool is
clearly present in the EEG data in its large-scale musical form.

In this study brain synchronization followed the reasoning of a coincidence
detection mechanism of cortical oscillators modeling neural activity in the
striatum15. After a start of neural oscillation increased synchronization
peaks at a point a subject expects an event to happen, like waiting at a
traffic light to turn green. The oscillation is then expected to include motor
regions, making us nervously shaking towards the expected time point.

This might be considered another compositional tool to make people dance. A
tension built-up leading to a neural oscillation and including the motor
region enhances the will of subjects to move and, in the case of music, to
dance. Although there is strong evidence that this is the case, there is no
final proof, as measurements in the motor region are still missing in the
musical case.

This line of reasoning seems fundamentally different than that followed in the
present paper of neural synchronization leading to a meditative mood. Still,
both cases are confirmed experimentally, also that of the temporal lobe
epilepsy43. The difference might indeed be found in the different frequency
ranges, where increased synchronization at higher frequencies, around 50 Hz,
might cause a perception of increased tension and synchronization at low
frequencies contrary that of a meditative state. Taking into account that
synchronization and de-synchronization are fundamental activities in the
brain, and brain activity represents all possible states of mind, perception,
and consciousness, such differentiation seems plausible.

##  V Acknowledgement

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# Macroscopic analysis of time dependent plasticity in Ti alloys

Yi Xiong  Department of Materials, University of Oxford, Parks Road, Oxford,
OX1 3PH, United Kingdom<EMAIL_ADDRESS>Phani S. Karamched  Chi-
Toan Nguyen  Safran SA, Safran Tech, Department of Materials and Processes,
78772 Many-les-Hameaux, France  David M. Collins  School of Metallurgy and
Materials, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United
Kingdom  Department of Engineering Science, University of Oxford, Parks Road,
Oxford, OX1 3PJ, United Kingdom  Christopher M. Magazzeni  Edmund Tarleton
Angus J. Wilkinson

###### Abstract

Component failure due to cold dwell fatigue of titanium and its alloys is a
long-standing problem which has significant safety and economic implications
to the aviation industry. This can be addressed by understanding the governing
mechanisms of time dependent plasticity behaviour of Ti at low temperatures.
Here, stress relaxation tests were performed at four different temperatures on
three major alloy systems: commercially pure titanium (two alloys with
different oxygen content), Ti-6Al-4V (two microstructures with differing
$\beta$ phase fractions) and Ti-6Al-2Sn-4Zr-$x$Mo (two alloys with different
Mo content $x$=2 or 6, and portion of $\beta$ phase). Key parameters
controlling the time dependent plasticity were determined as a function of
temperature. Both activation volume and energy were found to increase with
temperature in all six alloys. It was found that the dwell fatigue effect is
more significant by oxygen alloying but is suppressed by the addition of Mo.
The presence of the $\beta$ phase did not strongly affect the dwell fatigue,
however, it was suppressed at high temperature due to the low strain rate and
strain rate sensitivity.

###### keywords:

Cold dwell fatigue , Ti alloys , Stress relaxation , Time dependent plasticity

††journal: Journal of Materials Science & Technology

[table]capposition=top

Cold dwell fatigue is the critical technological issue in Ti alloys used in
the cooler front sections of aerospace jet engines. When subjected to a
sustained load (stress dwell), the lifetime of these Ti components can be
drastically reduced by as much as an order of magnitude [1, 2, 3, 4]. Without
a clear understanding of the effect, safety concerns are raised. Aeroengine
manufacturers currently manage this problem through heavily conservative
component design and operation. Cold dwell fatigue is mainly manifested in the
$\alpha$ phase (HCP) Ti [5, 6], due to its anisotropic property [7, 8].
Computational models indicate that crack nucleation during cold dwell fatigue
is dominated by so-called rogue hard-soft grain pairs. During a stress dwell
period, localised time dependent plasticity in the ‘soft’ grain (grain
oriented favourably for the dominate slip systems) leads to load shedding and
an increase in elastic stress within the neighbouring ‘hard’ grain (grain
oriented badly for the dominant slip systems) [9, 10]. With repeated load
cycling, the stress in the ‘hard’ grain may become sufficiently high to
initiate cracking, which ultimately leads to catostraphic failure.

Commonly used alloys in aeroengine compressor disks, such as
Ti-6Al-2Sn-4Zr-$x$Mo alloys show accelerated failure via the accumulation of
plastic strains when subjected to a dwell stress [11]. The role of bulk
composition is known to influence dwell damage; the propensity for dwell debit
is minor in high-Mo containing Ti6246 alloys, whereas it is significant in
low-Mo containing Ti6242 alloys which have a lower $\beta$ phase fraction, and
is relatively minor in Ti6246 with a greater $\beta$ phase fraction. The role
of chemistry and microstructure, both known to affect the severity of dwell
fatigue, is not well established. Although this problem has drawn attention
for many decades and great efforts have been made to understand this problem,
the majority of the work has been performed at room temperature. In practice,
it is found that cold dwell fatigue is suppressed at elevated temperatures,
and is no longer present for operating temperatures exceeding 200 ∘C [12, 13].
To establish a better understanding of the cold dwell fatigue problem, a
broader study that explicitly explores the sensitivity of temperature to time
dependent plasticity is necessary.

This proposed experiment seeks to gain mechanistic understanding and
quantification of key parameters across a range of industrially relevant Ti
alloys and microstructures by using simple stress relaxation loading at
different temperatures up to 250 ∘C. The effect of chemistry, microstructure
and temperature on time dependent plasticity has been explored here, which
aims to benefit the future design of alloys and provide better lifetime
prediction under conditions observed in service. Although not capturing the
local extreme conditions at the grain level, the macroscopic stress-strain-
time response gives readily accessible route to assess the propensity for
different alloy compositions, microstructures and textures to influence the
extent of time dependent plasticity in the temperature regime of interest for
cold dwell behaviour. Macroscopic assessment of time dependent plasticity is
the aim of this paper.

Table 1: Materials composition (in wt.%) and microstructure information. | Al | V | Sn | Zr | Mo | Fe | O | N |  | Others
---
total
Microstructure |  |  $\beta$ phase
---
content
| Grain size
---
($\mu$m)
CP-Ti grade 1 |  $<$0.05 | 0.11 |  $<$0.05 |  $<$0.05 |  $<$0.05 |  $<$0.05 | 0.077 | 0.02 | 0.2 |  Pure $\alpha$ | 0% | 30
CP-Ti grade 4 |  $<$0.05 | 0.15 |  $<$0.05 |  $<$0.05 |  $<$0.05 | 0.055 | 0.32 | 0.006 | 0.4 |  Pure $\alpha$ | 0% | 17
Ti64 $\alpha$ | 6.74 | 3.98 |  $<$0.05 |  $<$0.05 |  $<$0.05 |  $<$0.05 | 0.218 | 0.001 | 0.2 |  Primary $\alpha$ | 6% | 15
Ti64 Bimodal | 6.63 | 3.87 |  $<$0.05 |  $<$0.05 |  $<$0.05 |  $<$0.05 | 0.191 | 0.002 | 0.3 | $\alpha+\beta$ | 18% | 8
Ti6242 | 6.21 | 0.28 | 1.27 | 4.21 | 1.86 |  $<$0.05 | 0.079 |  $<$0.001 | 0.4 |  Primary $\alpha$ | 10% | 12
Ti6246 | 5.98 | 0.15 | 0.87 | 4.24 | 5.35 |  $<$0.05 | 0.157 |  $<$0.001 | 0.3 | $\alpha+\beta$ | 60% | 6

In this work, we have studied the stress relaxation behaviour of 6 different
Ti alloys. An overview of the chemistry and microstructures of the alloys is
summarised in Table 1. Samples were cut from the raw materials then ground and
polished to a colloidal silica finish. Microstructural analyses were conducted
on these polished samples using a Zeiss Merlin scanning electron microscope
(SEM) equipped with a Bruker e-flash detector, operating with a beam current
of 20 nA and an accelerating voltage of 20 kV. Exemplar microstructures of
these alloys are shown in Figure 1(a)-(f). The two grades of commercially pure
Ti (CP-Ti) were used to study the effect of the common interstitial, oxygen,
on cold dwell fatigue. Two Ti64 alloys with different microstructures were
studied to establish the effect of the $\beta$ phase content. In the two Ti64
alloys, the $\beta$ phase was found to appear at the $\alpha$ grains
boundaries and mainly at the triple junctions. The Ti6242 and Ti6246 alloys
were used to study the effect of both the $\beta$ phase and the alloying
element Mo on time dependent plasticity. In the Ti-6Al-2Sn-4Zr-$x$Mo alloys,
the $\beta$ phase had a lamellar structure. Texture is an important factor
that affects the deformation behaviour of $\alpha$-Ti due to the anisotropic
properties of the HCP crystal structure. The CP-Ti grade 4, CP-Ti grade 1,
Ti64 $\alpha$ and Ti6242, each supplied as rolled sheets, all have a volume
fraction of the $\alpha$ phase exceeding 90%. Their prior processing gives a
strong texture in these four materials (as shown in Figure 1(g)-(j)). The pole
figures, calculated from EBSD measurements, were obtained from larger areas
that contains over 3000 grains. To ensure the major slip systems are
activated, there should be a considerable number of ‘soft’ grains with respect
to the loading direction. The test samples were cut from the raw materials
with the loading direction along the X0 direction in these pole figures.

Figure 1: Microstructures of the raw materials: (a) EBSD map of the CP-Ti
grade 1 alloy (IPF map in normal direction of the raw material); EBSD map of
the CP-Ti grade 4 alloy (IPF map in normal direction of the raw material); SE
images of the (c) Ti64 $\alpha$ alloy; (d) Ti64 Bimodal alloy; (e) Ti6242
alloy and (f) Ti6246 alloy; Pole figures of the raw material with a 10∘ width
used for contours plotting calculation: (g) CP-Ti grade 1 alloy; (h) CP-Ti
grade 4 alloy; (i) Ti64 $\alpha$ alloy and (j) Ti6242 alloy.

A series of samples in each alloy derivative were subjected to stress
relaxation loading (load-up and strain hold) using an Instron ETMT (Electro-
Thermal-Mechanical Tester), where test samples were subjected to user defined
mechanical and thermal profiles. Sample temperature control was achieved
through a DC electric current that was fixed between water cooling grips.
R-type thermocouples were spot-welded to the centre of each sample gauge to
provide accurate temperature surveillance during the tests. Sample deformation
was applied through an in-line drive, displacement controlled 5 kN mechanical
loading assembly [14, 15, 16]. Figure 2(a) shows the geometry of the ETMT
dogbone sample, having a 52 mm length and the gauge length of 16 mm with a 2
mm width and a 1 mm thickness. Each alloy was tested at four different
temperatures: room temperature, 75∘C, 145∘C and 250∘C (CP-Ti grade 1 at 250∘C
test was abandoned due to its extremely low strength at this temperature). A
set of trial tests were performed to determine the target loads, where the
macroscopic yield points were reached and were exceeded by a small plastic
strain. This was assessed via changes in the gradient of the load-stroke
(stress-strain) curves. Samples were then loaded up to the target loads at
constant stress rates (different for different samples) in around 100 to 150s,
corresponding to macroscopic strain rates of approximately $3\times
10^{-5}~{}\textrm{s}^{-1}$ (as measured from the ETMT stroke rate) within the
elastic region. A typical stress relaxation loading curve is shown in Figure
2(b).

Figure 2: (a) Geometry of the ETMT tensile test sample; (b) An example of the
stress relaxation loading (Ti64 $\alpha$ at room temperature).

The stress relaxation curves were numerically analysed using a constitutive
law [9, 17, 18, 19, 20], which links the macroscopic plastic strain rate,
$\dot{\varepsilon}_{p}$, when $\sigma\geqslant\sigma_{c}$, to the macroscopic
applied stress, $\sigma$:

$\dot{\varepsilon}_{p}=\rho
b^{2}\nu\exp\left(-\frac{\Delta{F}}{k_{B}T}\right)\sinh\left(\frac{(\sigma-\sigma_{c}){\Delta}V}{k_{B}T}\right)$
(1)
$\displaystyle\Delta{\varepsilon}_{e}=-\Delta{\varepsilon}_{p}=-\dot{\varepsilon}_{p}\Delta{t}$
$\displaystyle\rightarrow\Delta{\sigma}=E\Delta{\varepsilon_{e}}$ (2)
$\displaystyle\rightarrow\sigma_{i+1}=\sigma_{i}+\Delta{\sigma}$

where $\rho$ is the density of gliding dislocations, $b$ is the magnitude of
the Burgers vector, $\nu$ is the jump frequency, $k_{B}$ is the Boltzmann
constant. These parameters are constant and are summarised in Table 2. $T$ is
the absolute temperature. The remaining three terms are material dependent and
are the key parameters controlling plasticity: $\Delta{F}$ is the thermal
activation energy, $\sigma_{c}$ is the critical stress and $\Delta{V}$ is the
activation volume [9]. The three key parameters were determined via this
constitutive law and the experimental stress relaxation curves.

The mobile dislocation density, $\rho$, in Equation (1) was considered as an
averaged value and was held constant in this equation (the value of $\rho$ was
taken from other studies on CP-Ti, Ti6Al, Ti6242 and Ti6246 alloys on cold
dwell fatigue) [21, 22, 23, 24]. The validation of this assumption can be
found in prior work [24]. The sensitivity of the three fitted parameters
($\Delta{V}$, $\Delta{F}$ and $\sigma_{c}$) to change of $\rho$ can be found
in A.

$\rho$ | 5 $\mu\text{m}^{-2}$
---|---
$b$ | 0.295 nm
$\nu$ |  $10^{11}$ Hz
$k_{B}$ | $1.38\times 10^{-23}~{}\text{JK}^{-1}$
Table 2: Values of the fixed parameters in the slip law [25, 21, 23, 22, 24]

To obtain the slip law parameters, the following steps were used, as shown in
Equation (2), the initial applied stress, $\sigma_{i}$, was substituted into
the Equation (1) to generate the plastic strain rate, $\dot{\varepsilon}_{p}$.
The change in plastic strain, $\Delta{\varepsilon}_{p}$, over a time step,
$\Delta{t}$ (here taken to be 0.2 s) was then be calculated by the product of
$\dot{\varepsilon}_{p}$ and $\Delta{t}$. The change in elastic strain,
$\Delta{\varepsilon}_{e}$, was matched by the negative change in plastic
strain, $\Delta{\varepsilon}_{p}$ during the stress relaxation (as the total
strain was held constant). The change in stress, $\Delta{\sigma}$, was
obtained by multiplying the Young’s modulus, $E$ for Ti at each temperatures
[13, 26] to the elastic strain, $\Delta{\varepsilon}_{e}$. The stress at the
next time step, $\sigma_{i+1}$,was then calculated by adding the
$\Delta{\sigma}$ to the $\sigma_{i}$. This process was repeated so that a
modelled stress relaxation curve could be generated. This method is widely
used for creep stress analysis [27]. Three key parameters were optimised using
a fitting tool in MATLAB so that the modelled relaxation curves and the
experimental relaxation curves show good agreement with errors of less than
1%, as shown in Figure 3(a). The confidence intervals for the three parameters
are $\pm 0.2\times 10^{-20}$J for $\Delta{F}$, $\pm 1b^{3}$ for $\Delta{V}$
and $\pm 5$MPa for $\sigma_{c}$. This numerically fitted parameters are
broadly similar to those used by other studies of titanium alloys [3, 9, 21].

Figure 3(b)-(d) show the three key parameters vs. temperature for the six
different alloys. Both $\Delta{F}$ and $\Delta{V}$ increased with temperature,
where the relationship between $\Delta{F}$ and temperature was almost linear,
which is consistent with the observations in CP-Ti and BCC steel [28, 24]. The
effect of temperature on $\Delta{F}$ was seen to be markedly stronger than the
alloy-to-alloy variation. CP-Ti had the lowest critical stress and showed a
comparably lower $\Delta{V}$ among the three different alloy systems.
Comparing the two CP-Ti alloys, the grade 4 with a higher oxygen content had a
higher critical stress over the low oxygen content grade 1, corroborating the
known potent strengthening effect of oxygen in Ti [29, 30]. A higher
activation volume was found in the low oxygen content grade 1 samples over the
grade 4. The Ti-6Al-4V alloys system had both moderate critical stress and
activation volume magnitudes among the three alloy systems. With the addition
of Al and V into the pure Ti, the strength was largely increased (i.e. the
$\sigma_{c}$ of Ti64 $\alpha$ had approximately double the value for the CP-Ti
grade 4 and triple the value for the grade 1). With the presence of $\beta$
phase in Ti64, $\Delta{V}$ was slightly reduced, especially at higher
temperatures (145 ∘C and 250 ∘C), but $\sigma_{c}$ was not significantly
affected. With further alloying additions, the most complex alloys system
Ti-6Al-2Sn-4Zr-$x$Mo system had a relatively higher $\Delta{V}$ and the
highest $\sigma_{c}$. There is no significant difference in $\sigma_{c}$
between the Ti6242 and the two Ti64 alloys, but $\sigma_{c}$ for the Ti6246 is
evidently higher than them. This shows the excellent strengthening effect of
Mo [31], where its presence acts as a grain refiner (recall Table 1 the Ti6246
has the finest grain size) and further improves the mechanical strength by the
Hall–Petch effect. The higher Mo content Ti6246 had a higher $\sigma_{c}$ over
the Ti6242, but there was no obvious trend between the $\Delta{V}$ of Ti6242
and Ti6246.

The activation volume, $\Delta{V}$, expresses the volume which is physically
swept by a dislocation from a ground equilibrium state to an activated state
after the deformation, and it is proportional to the dislocation pinning
distance (or the mean free path of dislocation) [9, 32]. In CP-Ti,
interstitial oxygen atoms provide a strengthening effect by acting as
obstacles for dislocations. The pinning distance reduces when there are more
oxygen atoms, resulting in a smaller activation volume in the higher oxygen
content CP-Ti grade 4 compared to the CP-Ti grade 1. Alloying is likely to
increase the pinning distance, because the Ti-6Al-4V alloys have a slightly
higher $\Delta{V}$ over the CP-Ti alloys, and with further alloying, the
Ti-6Al-2Sn-4Zr-$x$Mo system has the highest $\Delta{V}$ among the three alloy
systems. Mo, as a substitutional elements in Ti, was found to increase the
lattice spacing [33], which is likely to lead to a higher dislocation pinning
distance. The $\Delta{V}$ magnitudes increase with temperature for all six
alloys; alloying has no effect on this trend. Therefore, this trend could
solely depend on thermal vibrations. Stronger thermal vibrations at higher
temperatures could lead to a larger dislocation pinning distance and thus a
higher activation volume. Grain boundaries are also likely to affect the
dislocation process, and thus the $\Delta{V}$ could be affected by the grain
size of the samples. However, this grain size effect becomes important only
when the grain size is reduced to below 30-50 nm [34, 35]. As all the Ti
samples examined in this work have grain sizes at the micrometer level, the
effect of grain size on a dislocation process or the activation volume was not
considered. The value of $\Delta{V}$ is an effective reflection of rate-
dependent deformation mechanisms [32]. The intersection of forest dislocations
usually gives rise to an extremely large activation volume (several thousand
$b^{3}$). A dislocation climb process, however, yields a lower activation
volume of 1$b^{3}$ [32]. Overcoming the Peierls barrier, point defect drag and
point defect interactions typically give an activation volume of 10$b^{3}$ to
100$b^{3}$ according to Evans and Rowlings [36]. In this work, the values of
$\Delta{V}$ of all six alloys at all four temperatures were found to lie
within this range, indicating the rate-dependent deformation mechanism remains
unchanged and the mechanism is not affected by alloying or temperatures.

Figure 3: (a) A fitted stress relaxation curve (Ti64 $\alpha$ at room
temperature, as an example). (b) Thermal activation energy, $\Delta{F}$ and
(c) Activation volume, $\Delta{V}$ (d) Critical stress, $\sigma_{c}$, as a
function of temperature for the 6 alloys; (e) Contribution of the three key
parameters to the form of the plastic strain rate when the critical stress is
exceeded by 1 MPa (low stress condition) and (f) 50 MPa (high stress
condition);

Figure 3(e) and (f) show the contribution of the three key parameters into the
plastic strain rate and thus time dependent plasticity. The prefactor
$\rho{b}^{2}\nu=4.35\times 10^{4}s^{-1}$ is constant in Equation (1) and was
ignored in this calculation. The contribution of the three key parameters to
the plastic strain rate was calculated when $(\sigma-\sigma_{c})$=1 MPa (low
stress condition, see Figure 3(e)) and when $(\sigma-\sigma_{c})$=50 MPa (high
stress condition, see Figure 3(f)). When the critical stress was just
exceeded, similar to the dwell fatigue condition, the contribution decreased
as the temperature increased from room temperature to 250∘C and the values
were still of the same order of magnitude. Generally, CP-Ti grade 4 had the
highest strain rate, Ti-6Al-4V systems had a moderate strain rate and Ti6246
had the lowest strain rate. All six alloys had a very low strain rate at
250∘C, and in most of the alloys, there was a sharp drop in strain rate when
the temperature reached 145∘C. For the high stress condition, the trend is
reversed, where the contribution of key parameters to the plastic strain rate
increased with temperature and the difference could be up to 4 orders of
magnitude. The CP-Ti grade 4 had the lowest strain rate and Ti6246 had the
highest. In both stress conditions, the two Ti64 alloys show relatively
similar strain rates. The deformation behaviour was not strongly affected by
the $\beta$ phase fraction.

Figure 4: (a) Fitting a straight line to the stress vs. strain rate on a log
scale, the gradient represents the strain rate sensitivity exponent $m$ (Ti64
$\alpha$ at room temperature as an example); (b) Strain rate sensitivity vs.
Temperatures for the six different alloys; (c) Percentage of stress relaxed
vs. Temperatures for the six different alloys.

The strain rate sensitivity (SRS) exponent, $m$, can be calculated follow the
equation [37, 38]:

$m=\rm{d}(\log({\sigma}))/\rm{d}(log({\dot{\varepsilon}}))$ (3)

In this stress relaxation experiment, the $m$ values are determined by fitting
a straight line to the stress-strain rate plots on a log scale. The gradient
of the straight line is the strain rate sensitivity exponent; more details of
this method can be found in previous work [17, 24]. As shown in Figure 4(b),
the commercially pure Ti had the highest $m$ values, Ti-6Al-4V had a medium
SRS and the values for the Ti-6Al-2Sn-4Zr-$x$Mo system was the lowest. The SRS
for CP-Ti grade 4 was the highest and Ti6246 was the lowest among the six
alloys. The percentage of stress relaxation was calculated to represent the
accumulation of plastic strain during a stress relaxation cycle, as the total
strain was held constant. The relaxed stress (elastic strain) was matched by
the increased plastic strain. A higher strain rate sensitivity will lead to
more plastic deformation and thus higher plastic strain during stress
relaxation [17, 24], as a result, the trend found in Figure 4(c) is very
similar to the trend in Figure 4(b). It is also noted that, there were sharp
increases in SRS and relaxed stress as temperature increases from room
temperature to 75∘C in the two CP-Ti alloys. This indicates the strong
dislocation activity at this intermediate temperature. This temperature agrees
well with the predicted worst-case scenario temperature (90 to 120 ∘C) for
dwell debit according to Zhang et al. [13].

To summarise, stress relaxation tests were performed on six different alloys
at 4 different temperatures up to 250∘C. Key parameters (thermal activation
energy $\Delta{F}$, activation volume $\Delta{V}$ and critical stress
$\sigma_{c}$) governing the time dependent plasticity were obtained by fitting
the macroscopic stress relaxation cycles to the slip law. Commercially pure Ti
has the lowest strength but the highest strain rate sensitivity, as a result,
the dwell fatigue effect is the strongest among the three alloy systems.
Ti-6Al-4V has both moderate strength and strain rate sensitivity, the dwell
fatigue effect was not significantly influenced by the fraction of $\beta$
phase in Ti64. The Ti-6Al-2Sn-4Zr-$x$Mo system had the highest strength but
the lowest strain rate sensitivity, dwell fatigue effect was highly suppressed
in these alloys. Oxygen has an excellent strengthening effect but does enhance
the dwell fatigue effect. Mo evidently offers both a strengthening effect and
significant resistant to dwell fatigue. At dwell fatigue stress conditions
(when critical stress is just exceeded), both the contribution of the key
parameters to strain rate and strain rate sensitivity sharply decreased in all
six alloys when temperature is above 145∘C. As a result, plastic strain
accumulation at 250∘C is at a very low level, and dwell fatigue is suppressed
at high temperature. Furthermore, the authors believe that the cold dwell
effect could be affected by the material texture. The key parameters of two
Ti64 $\alpha$ alloys with a loading direction 45∘ to each other were compared,
however, there was no difference between them at a macroscopic level. The
ratio of ‘hard’ to ‘soft’ grains in different macroscopic directions could
lead to the activation of different slip systems during deformation. The
effect of texture on time dependent plasticity should be studied in the
context of activation of slip systems.

## Acknowledgements

The authors acknowledge funding from the EPSRC through the HexMat programme
grant (EP/K034332/1) and the Diamond Light Source for beam time under
experiment EE17222. We are grateful for use of characterisation facilities
within the David Cockayne Centre for Electron Microscopy, Department of
Materials, University of Oxford, which has benefitted from financial support
provided by the Henry Royce Institute (Grant ref EP/R010145/1). YX expresses
gratitude to the financial support of China Scholarship Council (CSC) and ET
acknowledges EPSRC for support through Fellowship grant (EP/N007239/1).

##  Appendix A Sensitivity of the fitted parameters

The sensitivity of the three fitted parameters ($\Delta{V}$, $\Delta{F}$ and
$\sigma_{c}$) to the change of $\rho$ is studied here. The $\rho$ was changed
by $\pm$40%, following previous work [24]. Same fitting method was utilised to
find the best fitted $\Delta{V}$, $\Delta{F}$ and $\sigma_{c}$ of the Ti64
$\alpha$ samples. The results are shown in Table A1. It can be found that the
three key parameters were not very sensitive to the changing $\rho$, best
fitted parameters were still within the confidence intervals for the three
parameters ($\pm 0.2\times 10^{-20}$J for $\Delta{F}$, $\pm 1b^{3}$ for
$\Delta{V}$ and $\pm 5$MPa for $\sigma_{c}$).

Table A1: Sensitivity of the three fitted parameters ($\Delta{V}$, $\Delta{F}$ and $\sigma_{c}$) to the change of $\rho$ by $\pm$40%. Using the Ti64 $\alpha$ samples as examples. | Change in $\rho$ | $\Delta{V}(b^{3})$ | $\Delta{F}(10^{-20}\textrm{J})$ | $\sigma_{c}(\textrm{MPa})$
---|---|---|---|---
RT | $\rho$ | 11.71 | 10.28 | 871
$\rho+40\%$ | 11.86 | 10.42 | 873
$\rho-40\%$ | 11.82 | 10.11 | 868
75∘C | $\rho$ | 18.33 | 12.42 | 658
$\rho+40\%$ | 18.02 | 12.58 | 656
$\rho-40\%$ | 17.71 | 12.27 | 655
145∘C | $\rho$ | 34.30 | 15.62 | 562
$\rho+40\%$ | 34.87 | 15.74 | 564
$\rho-40\%$ | 34.50 | 15.46 | 558
250∘C | $\rho$ | 85.05 | 21.02 | 560
$\rho+40\%$ | 84.99 | 21.17 | 567
$\rho-40\%$ | 86.27 | 20.90 | 560

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