# A Predictive Model for Oil Market under Uncertainty: Data-Driven System
Dynamics Approach

Sina Aghaei  Department of Industrial & Systems Engineering, University of
Southern California, Los Angeles, CA 90007, USA<EMAIL_ADDRESS>Amirreza
Safari Langroudi  School of Management, Sabanci University, Istanbul 34956 ,
Turkey  Masoud Fekri  School of Industrial Engineering, Iran University of
Science and Technology, Tehran 16846-13114 , Iran

###### Abstract

In recent years, there have been a lot of sharp changes in the oil price.
These rapid changes cause the traditional models to fail in predicting the
price behavior. The main reason for the failure of the traditional models is
that they consider the actual value of parameters instead of their
expectational ones. In this paper, we propose a system dynamics model that
incorporates expectational variables in determining the oil price. In our
model, the oil price is determined by the expected demand and supply vs. their
actual values. Our core model is based upon regression analysis on several
historic time series and adjusted by adding many casual loops in the oil
market. The proposed model in simulated in different scenarios that have
happened in the past and our results comply with the trends of the oil price
in each of the scenarios.

###### keywords:

Oil Price, Oil Supply , Oil Demand, Economic Model, Expectations on Demand and
Supply , System Dynamics Model , Trend Estimation , Oil Price Shocks

##  Appendix A Introduction

In recent years, there have been multiple efforts to model the oil market to
capture its volatile behavior. In most of the models, both supply and demand
side have different dependencies of the consumers on the oil price. However,
in almost all of the traditional models, the actual value of the supply and
demand is used for modeling the oil market [1, 2, 3]. However, due to the
failure of the traditional models in capturing price trend, [4, 5, 6] started
to use expectational values instead. The accuracy of the model proposed in [7]
shows the validity of the expectational models.

The emerging markets like Russia, India, Brazil, and China (RIBC) are heavily
thirsty of oil and their industries are getting even more dependent on oil
consumption [8, 9, 10]. However, in the European Union and United States (EUS)
there are usually policies on limiting the pollution and air emission which
directly affect the amount of fuel consumption and the standards that are used
in the manufacturers like car producers [11]. Moreover, most of the researches
on oil substitutes are performed and enforced in the EUS countries. Therefore
the variables on different sides of the demand involve different types of
dependencies on the oil price. It should be noted that the two sides of the
demand chain (EUS and RIBC) are actually interrelated, as well [12].

Therefore, the factors enforcing the oil price are more than just the
aggregated demand and supply. For instance, after the OPEC meeting on November
2014, the oil price gained a much higher speed in reduction [13]. This was
because, unlike the expectations, the OPEC producers decided to keep the
production rate unchanged, where as the oil price had already fallen by $20
per barrel since June 2014. Or as another instance, one could think of the new
types of energy resources that have been explored. For instance, as a result
of the Shale boom, the US is now the largest producer of the natural gas [14]
which puts even a greater pressure on the oil price [15].

In this paper, we extend the proposed model in [16] to capture some other
parameters and variables that are specially involved in determining the oil
price and the expectations that shape it other than just the demand and
supply. First, we make a cause and effect model based on the casual
relationships among the variables. Next, we add the formula and build the
stock and flow SD model upon the causal one. Similar to [17, 18], we separate
the total oil demand to three different categories of the demand by EUS, RIBC
and the other countries. Likewise, the supply side is categorized as OPEC, US,
spare oil supply, smuggled oil production, and the production by other
countries. Our model incorporates the new expectational parameters.
Particularly, the two variables “Expectation on Demand Side" and “Expectation
on Supply Side" summarize effects of the unconventional factors [19]. These
variables play a vital role in our model. They are amplifier or shrinker of
the total supply and demand in this model. We build the model, its causal
loops, and the mathematical formula in section B. Similar to [20], we use real
data of oil price, demand, supply and economic growths for building the
mathematical relations of the core model using the regression analysis on key
variables. Then, in section C, we provide our simulation results for five
scenarios and compare the results with the time-wise corresponding real data.
We show that our results successfully comply with the real data’s trends in
each of the scenarios that have happened in the past. Moreover, we provide
several cases that try to predict the oil price in a 60-day time window for
some possible scenarios. Finally, section D concludes the paper.

##  Appendix B The model structure

In this section, we describe details of the proposed model. In the first
section, we will explain the supply side of the model and the second part
describes the demand section. Details of the models is described in the
following. The proposed model could also be modified to capture the volatility
in car market [7].

###  B.1 The Supply Loop

In this subsection, we provide the main causal loops on the supply side of the
chain. Figure 1 depicts this main loop whose parameters and variables are
explained next.

Figure 1: The main supply loop and the sign of its arrows.

Main Components of the Supply: The expected total oil supply consists of the
‘OPEC Oil Supply ($S_{OPEC}$)’, ‘US Oil Supply ($S_{US}$)’, ‘Other Countries
Oil Supply ($S_{Other}$)’, ‘Smuggled Oil Supply ($S_{Smuggled}$)’, and ‘Spare
Oil Supply ($S_{Spare}$)’. The ‘Smuggled Oil Supply’ models the production of
the oil in the regions with political upsets that the governments of the
producing countries do not have the political controls over the country [21,
22]. For instance, in 2014, oil production in Syrian and Iraqi regions that
were out of the control of their governments was approximately 600,000 barrels
a day [23, 24]. After political troubles in that region, the expectations
favored a reduction in the oil production in the corresponding countries for
long-run. However, this only happened for a short period of time just before
the market was overwhelmed by the smuggled oil production which accounts for
less than one mb/d. However, this is a considerable amount of production in
the oil market, where only one mb/d surplus in the production can cause the
prices to plunge sharply. We express the mathematical relations between
$S_{Smuggled}$, $S_{Spare}$ and the parameter ‘OPEC’s Decision’, which models
the effect of OPEC’s decisions in the oil market, in the next subsection where
we talk about Geopolitical Upsets and the Expectation on Supply. Moreover, we
have separated the producing countries’ shares in the oil market. This is
mainly due to the fact that the production resources in different countries
differ from each other. Therefore, the Expected Total Oil Supply can be
written as follows:

$\displaystyle TOS=S_{OPEC}+S_{US}+S_{Other}+S_{Smuggled}+S_{Spare}.$ (1)

###  B.2 The Expectation on Oil Supply Loops

In this subsection we provide factors that constitute the ‘Expectation on the
Oil Supply ($ExS$)’. Figure 2 depicts the main loops and causal relations in
determining the value of this variable.

Figure 2: The main expectation on supply loop and the sign of its arrows.

The Expectation on the Oil Supply: The $ExS$ variable models the factors that
form the expectations in the changes of the oil supply. Particularly, in the
occurrence of some event that affects the expectation on the direction of
future changes of supply, these expectations eventually are transformed
through some variables and parameters into actual changes in the oil price. In
such cases, although the actual change in the oil supply might not have
occurred yet, the oil price changes accordingly as if the supply has actually
changed in the predicted direction. This is why many of the variables that are
involved specifically in the expectation loops are modeled such that they get
influenced through the direction of changes of the main variables. For
instance, some of them change according to the smoothed derivatives of main
variables such as oil revenues or economic growth. Because indeed, in
practice, the expectations are shaped according to the (recent) direction of
changes of the definitive economic factors. Thus, in a novel way, we have
modeled these kinds of parameters to affect the oil price only through the oil
supply. The $ExS$ is a dynamic coefficient that acts as an amplifier or
suppressor on the global oil supply. Thus, the oil price is not changed
directly, and rather, the expectation on supply changes the supply value
artificially, and it is only through the supply to demand ratio that the oil
price changes. In other words, it is as if the oil supply has really changed
and the oil price takes appropriate response to such a change. Finally, the
actual supply is still shown in the model through the variable ‘Actual Total
Supply’, which is the ‘Expected Total Oil Supply ($TOS$)’ divided by the
$ExS$.

###  B.3 The Demand Loop

In this subsection, we provide the causal loops that determine the total
amount of global oil demand. Figure 3 reflects the main loops that are
effective in the demand side of the chain.

Figure 3: The main demand loop and the sign of its arrows.

Main variables on Demand: The ‘Expected Total Oil Demand ($TOD$)’ is
determined by the ‘EU and US Oil Demand ($D_{EUS}$)’, ‘Russia, India, Brazil,
China Oil Demand ($D_{RIBC}$) and the ‘Other Countries Oil Demand
($D_{Other}$)’. The reason to separate RIBC oil demand is that these countries
have a growing economy that is heavily thirsty of oil [25]. Therefore, their
industries are increasingly consuming oil and this makes it slightly different
from the EU and US side, where the oil demand growth rate is much lower.
Particularly, through new polices on oil consumption or even finding
substitute energy resources, they even tend to decrease their oil dependencies
in the future. Therefore, structure of the oil demand and even the economic
growth is different on the two sides of demand. Thus, there is a need to model
their causal loops in slightly different ways with different variables and
functions. Thus we can compute $TOS$ as:

$\displaystyle TOD=D_{EUS}+D_{RIBC}+D_{Other}.$ (2)

Throughout this paper, by total oil demand we refer to Expected Total Oil
Demand, unless explicitly stated.

Energy Resources and Policies on Oil Consumption: The variable ‘Policies on
Oil Consumption’ reflects the effects of policies and regulations on the oil-
dependent industries like car factories. Generally, due to the factors such as
air pollution caused by burning the fossil fuels, limitations on the oil
reserves [26], and vulnerability of the economic growth caused by the oil
price, there is a tendency to reduce economy’s dependency on the oil
consumption [27]. For instance, the newer cars have higher MPGs (Miles Per
Gallon). Therefore, we have modeled the effect of those regulations, which are
themselves a functions of the economic health, with the aforementioned
variable. Moreover, the variable ‘Other Energy Resources’ reflects the effect
of oil substitutes such as natural gas that have been growingly used by the EU
and US countries. Particularly, since the Shale Boom, the US has been the
first producer of the natural gas in the world [28, 14]. This has also put
higher pressures on the oil price which was very volatile in the second half
of 2014, and seems to be onwards (in 2015) [29, 30]. As another instance, the
lithium battery powered cars are being considered as the new trend in the car
industry which also reduces the oil dependency [31], if the electrical energy
that is used to charge those batteries are themselves produced by the
renewable energy resources or other types of fossil fuels. These are some of
the variables that are specially important on reducing the oil dependency of
the EU and US countries. Unfortunately these kinds of policies are sometimes
overlooked or less considered in RIBC countries. This is the reason that we
have not included those in that side of the model.

###  B.4 The Expectation on Demand Loops

In this subsection, we provide the loops that are important in forming the
Expectation on Oil Demand. The main loops are depicted in Fig. 4. The rest of
this section elaborates the structure of the dependencies in these loops.

Figure 4: The main expectation on demand loop and the sign of its arrows.

The Expectation on Oil Demand: Likewise the supply chain, the $ExD$ variable
models the factors that form the expectations in the changes of demand. To
emphasize the need for this variable, consider a circumstance where the
International Monetary Fund (IMF) declares the expected economic growth of EU,
US or China [32]. Usually, whenever these kinds of declarations happen, the
oil price reacts to them. This is because the expected economic growth, as
discussed in the previous subsection, is interrelated with the global oil
demand. Therefore, an expectation on the economic growth rate is translated
through some variables and parameters into an expectation over the global oil
demand. In such a case, although the actual change in the oil demand might not
have occurred yet, the oil price changes slightly as if the demand has
actually changed in the predicted direction. Therefore, one of the
contributions of this paper is to model those kind of parameters in
determining the oil price. Likewise supply, $ExD$ is a dynamic coefficient
that acts as an amplifier or suppressor on the global oil demand. Therefore,
once again, instead of directly changing the price, the expectation on demand,
changes the demand value artificially and it is only through the demand that
the oil price changes. In other words, it is as if the oil demand has slightly
changed and the oil price responds to that correspondingly. Notice that, the
‘Actual Total Demand’ is the ‘Expected Total Oil Demand’ divided by $ExD$.

Economic Depression: One of the main ongoing economic events since 2008 has
been the economic depression. Although US economy is said to have recovered
from the depression in 2015 [33], some of the economies in the European Union
are still suffering from the effects of this crisis. This is one of the
factors that has a direct impact on the economic growth, for instance, through
unemployment rate, exports, gross domestic product, etc. Moreover, through the
economic growth change rate data, it can be understood that this factor is
indeed important in forming the expectations on the demand change rate, whence
the oil price [34]. In our model, the ‘Economic Depression ($EconDep$)’ is an
initial coefficient that acts on the loops containing the economic growth and
the expectations on demand to reflect the adverse effects of the depression on
the global oil demand. Particularly, in the initiation of Economic Depression,
$ExD$ decreases, which causes the price to reduce. This can help recovering
the growth, however, it is only until a long-term recovery of the economy that
the demand increases back. Therefore, we have used a $PULSE(start,duration)$
to model the $EconDep$ and to trigger the loops that involve $EconDep$ for the
duration of time that we want to model the existence of depression.

Effects of the Policies and Regulations on Oil Consumptions: Although the
policies and regulations have been considered in determining the actual amount
of the oil demand with a great time constant, it is necessary to consider
their effects on forming the expectations on demand with a lower time
constant, as well. Particularly, on one hand, the regulations might take a
much greater period of time to actually change the amount of demand by
changing the oil-dependency [35]. On the other hand, just as the polices get
passed in the parliaments or organizations, they form the expectations on the
demand change rate. However, their effects through this loop might be lower
than the aforementioned side.

Now that we have elaborated the causal loops in the model, we are ready to
simulate the model and bring the results corresponding to several different
scenarios. The next section talks about these scenarios in detail.

##  Appendix C Simulations and Scenarios

In this section, we evaluate the accuracy of the proposed model in different
scenarios. Our simulations involve the cases where the parameters on the
expectation on either demand or the supply side changes and the results are
reflected in the oil price. In our simulations we have considered scenarios
that are consistent with the real world scenarios and have used the West Texas
Intermediate (WTI) index as our resource for the oil price real data. The WTI
crude oil price history is shown in Fig. 5 with the data taken from [36].
Moreover, in Fig. 6 we have shown the parts of the data that we have used in
our simulations with three rectangles. Moreover, as will be explained later,
in scenario A, we have used the data from 30th of Dec. 2011 to 10th of Jan.
2012, that we have not shown in this figure for the sake of cleanness of the
figure. Moreover, in scenario E, we do a simulation for the period of time
starting from Jan. 2012 until July 2015. Finally, in the last scenario, we
provide future predictions of the price changes in several possible cases.

Figure 5: WTI oil price history. Figure 6: Parts of the WTI oil price history
that we have used in our simulations, shown with rectangles.

###  C.1 Scenario A: Sanctions on the Production

In this subsection, we simulate a scenario where the sanctions on oil
producers have caused a reduction in their production.

International Sanctions: The international society can decide to establish
economic, political or other types of sanctions on the oil producing countries
due to various reasons. There are many instances of those cases over the past
50 years. For instance, as mentioned before in the 90s, there were heavy
economical sanctions on Iraq which had infected their economy with lots of
problems. Over the past three years (since 2012), there has been many
economical sanctions on Iran (as an OPEC member), which have limited their oil
sales and production. We have used the real data of oil price for the year
2012, where these sanctions were one of the main reasons of the oil market’s
volatility. These kinds of sanctions have many adverse effects on the
economies of the sanctioned countries. We have modeled these effects to
gradually reduce oil production of the sanctioned countries using the real
data, and have reflected the results of introduction of sanctions on one of
the OPEC producers in Fig. LABEL:subfig:Scenario_C_vensim. In the simulation,
we have assumed a placement of sanctions on one of the producers (at time 1)
whose effects leads to a shortage in the supply with a reasonable delay (due
to the stock and flow nature of the model), consequently giving an increasing
rate to the oil price. Moreover, the OPEC Decision is still zero which
simulates a case that the OPEC countries have not yet decided on compensating
the supply shortage, which is consistent with the real data in 2012.
Rationally, it is expected that these sanctions should cause the price to face
a gradual increase. This is mainly due to the fact that the sudden reduction
in oil production of one the countries might not be compensated by the usual
growth of the oil supply. However, this is not what happens in reality. As it
is seen in Fig. LABEL:subfig:Scenario_C_vensim, the oil price keeps its trend
for slightly decreasing. This is mainly due to the fact that the US oil
production has increased with a noticeable rate during the same period in
2012. Therefore, although, the OPEC oil production has reduced due to the
reduction of oil production in Iran, the US oil production has compensated the
same amount and even, by the end of 2012, the total oil production in the
world has increased. This is indeed reflected in the figure and our
simulations predict the oil price pretty well in this scenario, as well. The
only slight increase occurs in the first immediate months that follow the
imposition of the sanctions. However, this volatility is compensated and
eliminated in the following months, and this is reflected in the figure, as
well. The mechanism for these reductions and growths is as explained in the
previous section through the supply and its corresponding expectation loops.

###  C.2 Scenario B: Economic Depression

In this subsection, we simulate effects of an economic depression in oil
consuming countries.

Lower oil demand due to depression: Figure LABEL:subfig:Scenario_D_vensim
reflects the fluctuations of the oil price in a case where an economic
depression has happened, specially in the EU and US. As shown in this figure,
the oil price decreases in this case, due to a reduction in the demand.
However, we assume that the depression starts to get better after 210 days,
which causes the demand and subsequently the oil price to increase. We have
used the year-long real data for the period of time from 30th of May, 2008 to
30th of May 2009. In the first months after the economic depression, the
demand for oil reduces sharply in the EU and US side, which causes the oil
price to fall down, however, it is the increased volatility that makes the
prices dive from about $133 in 30th of May, 2008 to about $40 by the end of
2008. As mentioned before, this volatility is reflected in our model through
the expectation variables, and indeed it is the effect of the better
expectations in the beginning of 2009 (in addition to the increased demand of
the RIBC), that contributes in increasing the oil price in 2009. Note that in
the beginning of 2008, the oil price had seen sharp increases which had
various cause and reasons. Some of these reasons were the strikes and
unexpected events in the oil producing countries like Nigeria, Scotland and
Canada [37, 38]. Therefore, the growth of oil price was mainly due to the
volatility in the supply and expectation on the future of the supply side of
the market. In the later months of the 2008, some of these volatility sources
were still existent which can be seen from the supply data in those days.
Therefore, we have modeled those supply volatilities in the supply loops and
the overall result is shown in Fig. LABEL:subfig:Scenario_D_vensim which shows
that our simulation results can follow the oil price trend in the reflected
period of time.

##  Appendix D Conclusion and Future Work

In this paper we proposed a model for oil market. The novelty of the model is
in modeling the supply and the demand loops in which instead of the actual
values, we used the expected values of supply and demand in determining the
ratio. Our approach was to build a Stock and Flow model based on a cause and
effect model containing the causal relations among various determinants, and
train the raw model with the historic real data of price, demand and supply to
make a core model that explains the price changes in normal situations. Then,
based on various events that included unexpected oil price changes, we
adjusted the model and its related loops that are active in each of those
scenarios. Our model was successful in predicting the past trends of the oil
price in both short and long term examples on the past eight years. The future
models may include more expectational variables . Moreover, there are some
specific reasons for the oil price changes that has influenced the market in
the past. For instance, extensive use of computerized marketing in 2011 caused
a decay in the oil price towards the middle of that year . Similarly, factors
like US dollar strength, stock market indices (that get affected by many other
commodities), and big political decisions are some of the other parameters
that might be considered more extensively in future.

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# Rheological model for the alpha relaxation of glass-forming liquids and its
comparison to data for DC704 and DC705

Tina Hecksher<EMAIL_ADDRESS>Glass and Time, IMFUFA, Department of Science and
Environment, Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark
Niels Boye Olsen  Glass and Time, IMFUFA, Department of Science and
Environment, Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark
Jeppe C. Dyre<EMAIL_ADDRESS>Glass and Time, IMFUFA, Department of Science and
Environment, Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark

###### Abstract

Dynamic shear-modulus data are presented for the two silicone oils DC704 and
DC705 for frequencies between 1 mHz and 10 kHz at temperatures covering more
than five decades of relaxation-time variation. The data are fitted to the
alpha part of a phenomenological model previously shown to describe well the
dynamic shear modulus of squalane, which has a large beta process [Hecksher et
al., J. Chem. Phys. 146, 154504 (2017)]; that model is characterized by
additivity of the alpha and beta shear compliance and by a high-frequency
decay of the alpha process in proportion to $\omega^{-1/2}$ in which $\omega$
is the angular frequency. The fits of the alpha part of this model to the
DC704 and DC705 data are compared to fits by a Havriliak-Negami type model,
the Barlow-Erginsav-Lamb model, and a Cole-Davidson type model. At all
temperatures the best fit is obtained by the alpha part of the squalane model.
This strengthens the conjecture that so-called $\sqrt{t}$-relaxation, leading
to high-frequency decays proportional to $\omega^{-1/2}$, is a general
characteristic of the alpha relaxation of supercooled liquids [Dyre, Phys.
Rev. E 74, 021502 (2006); Nielsen et al., J. Chem. Phys. 130, 154508 (2009);
Pabst et al., J. Phys. Chem. Lett. 12, 3685 (2021)].

##  I Introduction

Dynamic shear properties are central for the description of glass-forming
liquids Barlow  _et al._ (1967); Harrison (1976); Christensen and Olsen
(1995); Deegan  _et al._ (1999); Jakobsen  _et al._ (2005); Dyre (2006a).
Historically, many different expressions have been fitted to the dynamic shear
modulus $G(\omega)=G^{\prime}(\omega)+iG^{\prime\prime}(\omega)$ in which
$\omega$ is the angular frequency. In a paper published in 2017 we proposed an
expression for the dynamic shear compliance $J(\omega)\equiv 1/G(\omega)$,
which was shown to fit well to squalane data over a sizable range of
temperatures and frequencies Hecksher  _et al._ (2017). The model also
describes well data for 1-propanol Weigl  _et al._ (2021).

Squalane, which is a reference liquid for rheological measurements Comunas
_et al._ (2013); Schmidt  _et al._ (2015), has a strong beta process. The
model proposed in Ref. Hecksher  _et al._ , 2017 was based on two assumptions:
1) The dynamic shear compliance is additive in the alpha and beta processes;
2) for $\omega\to\infty$ the alpha process obeys for its imaginary part
$J^{\prime\prime}(\omega)\propto\omega^{-1/2}$ or, equivalently,
$G^{\prime\prime}(\omega)\propto\omega^{-1/2}$. The second property can be
justified by different arguments, many of which date far back in time Glarum
(1960); Isakovich and Chaban (1966); Barlow  _et al._ (1967); Doremus (1970);
Montrose and Litovitz (1970); Majumbar (1971); Kimmich and Voigt (1978); Lamb
(1978); Wyllie (1979); Cunat (1988); Condat (1989); Lishchuk and Malomuzh
(1997); Dyre (2006b, 2007); this so-called $\sqrt{t}$-relaxation recently
received strong support from light-scattering measurements on 16 supercooled
liquids of quite different chemistry Pabst  _et al._ (2021). An
$\omega^{-1/2}$ high-frequency decay of a dynamic response function
corresponds via the fluctuation-dissipation theorem to a short-time behavior
of the relevant time-autocorrelation function that deviates from a constant by
a term proportional to the square root of the time, hence the name
$\sqrt{t}$-relaxation. Dielectric data also indicate that this short-
time/high-frequency behavior of the alpha process may be universal Nielsen
_et al._ (2009), but the experimental evidence is here less convincing than
for light scattering.

The model proposed in Ref. Hecksher  _et al._ , 2017 was based on an
electrical equivalent circuit, which is reproduced in the Appendix where its
translation into a rheological circuit is given. It leads to the following
expression for the dynamic shear compliance in which $\tau_{\alpha}$ and
$\tau_{\beta}$ are the characteristic times for the alpha and beta processes
and $J_{\alpha}$ is the high-frequency limit of the compliance (often denoted
by $J_{\infty}$)

$J(\omega)=J_{\alpha}\left(1+\frac{1}{i\omega\tau_{\alpha}}+\frac{k_{1}}{1+k_{2}\sqrt{i\omega\tau_{\alpha}}}\right)+\frac{J_{\beta}}{1+(i\omega\tau_{\beta})^{b}}\,.$
(1)

The first term is the contribution from the alpha process, which is a sum of a
constant, a dc flow term, and a Cole-Cole type term Cole and Cole (1941) with
exponent 1/2. The second term is the contribution from the beta process with
compliance strength $J_{\beta}$, which is a Cole-Cole type term with exponent
$b$. Equation (1) has a total of seven parameters; the alpha term has four
parameters: two of dimension ($J_{\alpha}$ and $\tau_{\alpha}$) and the two
dimensionless shape parameters $k_{1}$ and $k_{2}$.

Figure 1: Squalane data at 172 K (gray crosses) and the prediction of Eq. (1)
(full curves) in different representations (data and fits are from Ref.
Hecksher  _et al._ , 2017). (a) and (b) give the real and imaginary parts of
the shear modulus, respectively, while (c) is the corresponding Nyquist plot.
Figures (d), (e), and (f) give the same data in the compliance representation.
In all the figures, the dashed curves are the predictions of the alpha part of
Eq. (1), which does not provide a convincing fit to the low-frequency part of
the data. In order to validate the alpha part of Eq. (1) one needs to study
liquids with little or no beta relaxations, which motivates the present paper.

An obvious question is the following: Is the excellent fit of Eq. (1) to
squalane data Hecksher  _et al._ (2017) due to the many fit parameters or does
it reflect physical reality? If the latter is the case, the model provides a
good starting point for arriving at a better understanding of the physics of
supercooled liquids, for instance described in terms of independent and
additive single-particle mean-square displacements deriving from the alpha and
beta processes Hecksher  _et al._ (2017). That would imply additivity of the
alpha and beta response to an external field coupling to the single-particle
motion, which by a Stokes-Einstein type argument translates into additivity of
the shear compliance contributions from the alpha and the beta processes.
Whether or not such an additivity of compliance reflects an element of
physical reality cannot be judged from the squalane data of Ref. Hecksher  _et
al._ , 2017, however, because of the sizable beta process.

To illustrate this we plot in Fig. 1 squalane data for the dynamic shear
modulus at the temperature $T=172$K. (a) and (b) show the real and imaginary
parts as functions of the frequency ($\omega/2\pi$) while (c) shows a Nyquist
plot, i.e., with the real part along the x-axis and the imaginary part along
the y-axis. The data, which in (d), (e), and (f) are given in the compliance
representation, are marked as gray crosses while the full curves are the best-
fit model predictions (with the same parameters for both representations). The
dashed curves are the predictions of the alpha part of Eq. (1). This analysis
does not confirm that the alpha process of Eq. (1) has independent physical
reality. In order to properly investigate whether the first term of Eq. (1)
corresponds to an element of physical reality, one should compare this term to
data for liquids with little or no mechanical beta relaxations. Although such
liquids are rare, they do exist. This paper presents new data for the two
silicone diffusion pump oils DC704 (tetramethyltetraphenyl trisiloxane) and
DC705 (trimethylpentaphenyl trisiloxane). These liquids have no observable
beta relaxation in the shear mechanics when probed with our PSG shear
transducer covering more than six decades of frequency Christensen and Olsen
(1995), and they are therefore ideal for testing the alpha part of Eq. (1). We
show below that this “alpha model” fits well to the data and conclude that the
alpha-beta additivity of the dynamic compliance of Eq. (1) is a physically
meaningful assumption.

Figure 2: Dynamic shear modulus data for DC704 and DC705 over a range of
temperatures. (a), (b), (d), and (e) give the real and imaginary parts of the
frequency-dependent shear modulus $G(\omega)$ of the two liquids, while (c)
and (f) present the data in normalized Nyquist plots. The dashed curves are
best fits to Eq. (2).

##  II Data and model fits

Data for the dynamic shear-mechanical properties of DC704 and DC705 were
obtained by the three-disc piezo-ceramic shear transducer described in Refs.
Christensen and Olsen, 1995 and Hecksher  _et al._ , 2013; references Igarashi
_et al._ , 2008a and Igarashi  _et al._ , 2008b give details about the
cryostat and the impedance-measuring setup. Data were obtained with 2 K
intervals using a home-built cryostat that keeps the temperature stable within
10 mK. The data are from previously unpublished measurements but consistent
with published data Hecksher  _et al._ (2013).

Figure 2 shows the dynamic shear modulus data as colored symbols in which the
color reflects the temperature. The four upper figures show the real and
negative imaginary parts of
$G(\omega)=G^{\prime}(\omega)+iG^{\prime\prime}(\omega)$, while (c) and (f)
show the corresponding normalized Nyquist plots. The latter are sensitive to
changes in the relaxation-time distribution, i.e., to violations of time-
temperature superposition (TTS). While there is not exact TTS for either
liquid, the deviations are minor. We observe a slight narrowing of the loss as
the temperature is lowered, but it should be emphasized that this happens
while at the same time the average relaxation time changes by more than five
decades. In Fig. 2 the dashed curves are fits by the alpha part of Eq. (1),
i.e., by $G(\omega)=1/J(\omega)$ in which

$J(\omega)=J_{\alpha}\left(1+\frac{1}{i\omega\tau_{\alpha}}+\frac{k_{1}}{1+k_{2}\sqrt{i\omega\tau_{\alpha}}}\right)\,.$
(2)

The rheological circuit corresponding to Eq. (2) is given in the Appendix.

Figure 3: Best-fit parameters as functions of the temperature for DC704 and
DC705 in which the vertical lines are error bars. (a) and (d) show the
relaxation time $\tau_{\alpha}$, (b) and (e) show $J_{\alpha}$, while (c) and
(f) show the two shape parameters $k_{1}$ and $k_{2}$. The fact that the
latter are not temperature independent shows that time-temperature
superposition (TTS) is not rigorously obeyed.

In the low-frequency limit of Eq. (2) one has $J(\omega)\cong
J_{\alpha}/({i\omega\tau_{\alpha}})$. Since the dynamic shear viscosity
$\eta(\omega)$ is calculated from the shear compliance by means of
$\eta(\omega)=1/[i\omega J(\omega)]$ Harrison (1976), the dc viscosity
$\eta_{0}$ is given by

$\eta_{0}\,=\,\frac{\tau_{\alpha}}{J_{\alpha}}\,.$ (3)

In the high-frequency limit, Eq. (2) implies $J(\omega)\cong J_{\alpha}$ which
means that the high-frequency plateau modulus $G_{\infty}$ is given by

$G_{\infty}\,=\,\frac{1}{J_{\alpha}}\,.$ (4)

Combining Eq. (3) with Eq. (4) leads to the well-known Maxwell relation
$\tau_{\alpha}=\eta_{0}/G_{\infty}$ Harrison (1976).

Figure 3 shows the best-fit parameters as functions of the temperature. The
increase of $J_{\alpha}$ with temperature is consistent with the well-known
fact that $G_{\infty}$ usually decreases as the temperature is increased Dyre
_et al._ (1996); Dyre (2006a). The shape parameters are not entirely constant,
confirming the already mentioned fact that TTS is slightly violated.

Figure 4: Real and imaginary parts of the shear compliance as functions of the
frequency. The colored symbols are data, the black dashed lines are model
fits. The data do not confirm the alpha-model predction of a low-frequency
plateau of the real part ((a) and (c)), but there are indications of the
corresponding peak in the imaginary part of the recoverable shear compliance
defined in Eq. (5) (lower curves of (b) and (d)).

The dynamic shear modulus has a different emphasis than the dynamic shear
compliance $J(\omega)=J^{\prime}(\omega)-iJ^{\prime\prime}(\omega)$, and it is
important to investigate whether the model also fits data well in the latter
representation. Figure 4 shows the real and imaginary parts of $J(\omega)$
where the colored symbols are the data and the black dashed lines are the
model fits (with the same parameters as in Fig. 2, compare Fig. 3). The model
predicts a low-frequency plateau for $J^{\prime}(\omega)$, compare Eq. (1),
which is not possible to measure with our setup that is optimized for moduli
above $10^{7}$ Pa. The data do not show evidence for the plateau prediction,
but they also do not contradict it. If the purely imaginary flow term
$1/(\eta_{0}i\omega)$ is subtracted from $J(\omega)$ to form the “recoverable
shear compliance” Plazek (1965); Buchenau (2016, 2017), which except for a
constant is identical to the so-called retardational compliance Harrison
(1976), i.e.,

$J_{R}(\omega)\,\equiv\,J(\omega)-J_{\alpha}/(i\omega\tau_{\alpha})=J(\omega)-1/(\eta_{0}i\omega)\,,$
(5)

there are hints of a peak in the imaginary part as predicted by the model
(Fig. 4(b) and (d)).

Figure 5: Nyquist plots of the $G(\omega)$ data at selected temperatures
compared to the best fits of different models. The fits of the alpha and the
HN models are all good, whereas the fits by the BEL and CD models deviate
notably from the data.

##  III Comparing to fits by three other models

This section compares to fits by other models. Recalling that
$G(\omega)=1/J(\omega)$, it is sometimes convenient to formulate a model in
terms of one or the other representation, e.g., writing the dynamic shear
modulus as $G_{\infty}$ minus a function commonly used for fitting dielectric
relaxation data, as done in two cases below. For all three functions, the
dynamic shear-modulus goes to zero at low frequencies and reaches a plateau at
high frequencies as required for any realistic model of an alpha process.

  * •

The Havriliak-Negami (HN) type model is defined Hecksher  _et al._ (2017) by

$G(\omega)\,=\,\frac{1}{J_{\alpha}}\,\left(1-\frac{1}{(1+(i\omega\tau_{\alpha})^{c})^{a}}\right)\,.$
(6)

  * •

The Barlow-Erginsav-Lamb (BEL) model is defined Barlow  _et al._ (1967) by

$J(\omega)\,=\,\left(\frac{1}{\sqrt{G_{\infty}}}+\frac{1}{\sqrt{i\omega\eta_{0}}}\right)^{2}\,=\,\frac{1}{G_{\infty}}+\frac{1}{i\omega\eta_{0}}+\frac{2}{\sqrt{G_{\infty}i\omega\eta_{0}}}\,.$
(7)

This can be obtained from the alpha model Eq. (2) by substituting $k_{1}=2$
and $k_{2}=1$ and replacing unity by zero in the last denominator. The latter
corresponds to removing a spring in the rheological circuit, compare the
Appendix, thus arriving at

$J(\omega)=J_{\alpha}\left(1+\frac{1}{i\omega\tau_{\alpha}}+\frac{2}{\sqrt{i\omega\tau_{\alpha}}}\right)\,.$
(8)

From this perspective, Eq. (2) may be regarded as an extension of the BEL
model by introducing two shape parameters.

  * •

The Cole-Davidson (CD) type model is defined by

$G(\omega)\,=\,\frac{1}{J_{\alpha}}\,\left(1-\frac{1}{({1+i\omega\tau_{\alpha}})^{a}}\right)\,.$
(9)

In all cases $J_{\alpha}$ is the high-frequency limit of $J(\omega)$ and
$J_{\alpha}=1/G_{\infty}$.

Figure 6: Logarithmic Nyquist plots of compliance data and fits at selected
temperatures. The approach to vertical tangents for the alpha and HN models
reflects the existence of a low-frequency plateau in the real part of the
shear compliance in these two models. The approach to a plateau is fitted best
by the alpha model.

All models have two parameters of dimension, one determining the scale of the
time and one determining the scale of the compliance. In regard to the number
of shape parameters, the alpha and HN models have two such parameters, the BEL
model has none, and the CD model has one. This, of course, influences the
quality of the fits obtained, compare Fig. 5 that shows all data and fits in
Nyquist plots of the dynamic shear modulus. We see generally good fits of the
alpha and HN models, while the BEL and and CD models do not give good fits.

In order to better be able to compare the two best models according to Fig. 5
– the alpha and HN models – we introduce in Fig. 6 a “log-compliance” Nyquist
plot. This way of showing data has a different focus than the standard Nyquist
plot of Fig. 5. In particular, one can here investigate the low-frequency real
part of the compliance, which has a plateau for the alpha and CD models, but
not for the BEL and HN models. Figure 6 shows that the approach to a plateau
is best represented by the alpha model.

Figure 7: Quality of all fits at all temperatures measured as indicated by the
squared difference between data and model prediction of the dynamic shear
modulus averaged over all $N$ measured frequencies in which $n$ is the number
of fit parameters that is 4 for the alpha and HN models, 3 for the CD model,
and 2 for the BEL model. For both liquids, the alpha model is superior at all
temperatures.

Figure 7 compares the quality of all fits by giving at each temperature the
average deviation between data and best-fit predictions. For both liquids the
alpha model is best at all temperatures.

##  IV Summary

Using shear-dynamical data for DC704 and DC705 this paper has investigated the
question to which extent the separation of the compliance function proposed in
Ref. Hecksher  _et al._ , 2017 into additive alpha and a beta parts makes
sense physically. If this were the case, then the alpha part of the model
should fit well to data for liquids with no mechanical beta relaxations. The
two silicone oils studied here fall into this category.

The alpha model fits data well for both DC704 and DC705. We take this as a
confirmation that the simple $\omega^{-1/2}$ high-frequency behavior of the
imaginary parts of the modulus and compliance reflects an important
characteristic of the two liquids studied, confirming previous and recent
works Dyre (2007); Nielsen  _et al._ (2009); Pabst  _et al._ (2021). An
important further test of the alpha model would be to check whether the
predicted low-frequency plateau in the real part of the shear compliance
exists and is predicted correctly, because this feature distinguishes the
alpha model from some of the other models studied here. Our shear transducer
Christensen and Olsen (1995) is optimized for high-modulus samples and cannot
be used for reliable measurements at the relevant low frequencies,
unfortunately. Such measurements are important also because they would allow
for testing the prediction of the alpha model that the Nyquist plot of
$J_{R}(\omega)$ (Eq. (5)) is symmetrical and identical to the standard
dielectric relaxation Nyquist plot of the Cole-Cole function with exponent 1/2
Cole and Cole (1941).

###### Acknowledgements.

This work was supported by the VILLUM Foundation’s Matter grant (16515).

## Data availability

The data that support the findings of this study are available from the
corresponding authors upon request.

## Appendix: Diagrams

This Appendix presents the rheological model behind Eq. (1) in two equivalent
formulations: the electrical-equivalent circuit used in Ref. Hecksher  _et
al._ , 2017 and the spring-dashpot formulation that is standard in rheology.
Even though not visually obvious, these two formulations are mathematically
equivalent. In the present case of highly viscous glass-forming liquids where
intertia plays no role, a parallel combination of electrical circuit elements
corresponds to a mechanical series combination and vice versa. This is because
currents are additive for an electrical parallel combination and velocities
(currents) are additive in a mechanical series combination. Likewise, the
potential drop across an electrical series combination is additive and
forces/stresses (potential drops) are additive for a mechanical parallel
combination.

For translating between the two types of circuits, besides the above
parallel/series interchange one reasons as follows. Electrical charge
corresponds to (shear) displacement and electrical current corresponds to
shear rate. A resistor in an electrical circuit is an element for which the
current is proportional to the potential drop. This corresponds to a dashpot
in a rheological circuit, the element characterized by a constant frequency-
independent viscosity for which the shear rate is proportional to the force
(stress). A capacitor in an electrical circuit is an element for which the
charge is proportional to the potential drop, which translates into a spring
in a rheological circuit for which the (shear) displacement is proportional to
the force (stress).

With this in mind, Fig. 8 shows the relevant diagrams. Figure 8(a) reproduces
the electrical equivalent circuits of Ref. Hecksher  _et al._ , 2017 and (b)
is the alpha part of this circuit. Note that these electrical diagrams are
equivalent diagrams of the mechanical response; they are not diagrams for the
dielectric response. In particular, no additivity is implied for the alpha and
beta dielectric responses. The rheological diagram of the alpha model is given
in (d), corresponding to the electrical equivalent circuit (b). To arrive at
Eq. (1) for the (shear) compliance one uses the fact that compliances are
additive for a mechanical series connection (the force is the same for each
element while displacements are added).

Figure 8: Diagrammatic representations of shear rheological models. (a)
Electrical-equivalent diagram of the alpha-beta model of Ref. Hecksher  _et
al._ , 2017. A triangle is a constant-phase element characterized by an
admittance that varies with frequency as $(i\omega)^{-n}$, where $n=1/2$ for
the alpha element while $n=b$ is a free parameter for the beta element. (b)
The alpha part of the alpha-beta model, i.e., the electrical equivalent
circuit of Eq. (1). (c) Rheological model corresponding to the alpha-beta
electrical-equivalent diagrams. Besides the standard spring and dashpot
symbols, a box is a constant-phase element with the indicated power-law
frequency dependence of the shear compliance. (d) Rheological model
corresponding to the alpha model Eq. (1).

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