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PAPER
Cite this: New J. Chem., 2021,
45, 18584
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Prediction of ionic conductivity of imidazoliumbased ionic liquids at different temperatures using
multiple linear regression and support vector
machine algorithms†
Zi Kang Koi,a Wan Zaireen Nisa Yahya
*ab and Kiki Adi Kurniac
Ionic liquids (ILs) are well recognized as promising and environmentally friendly solvents owing to their
remarkable features, which have captured the imagination of the global research community to expand
ILs’ usage across various industrial processes and applications. Nevertheless, the formulation of ILs with
desired properties requires strenuous effort in discovering the suitable combination of cations and
anions. Hence, it is imperative to develop simple and accurate models to predict the physicochemical
properties of ILs prior to experimentation. Ionic conductivity is one of the most significant intrinsic
properties which affects the transport capabilities of ILs and it has been a limiting factor in the design of
suitable ILs. In the present study, the conductivity of different imidazolium-based ILs has been estimated
and correlated via the Quantitative Structure–Property Relationship (QSPR) approach using two different
algorithms, namely multiple linear regression (MLR) and support vector machine (SVM) regression
coupled with stepwise model-building. A set of descriptors, including interaction energies as well as
dielectric energy of the ILs’ cation–anion pairs generated by the Conductor-like Screening Model for
Real Solvents (COSMO-RS), were employed to derive the best-fit model. The models were developed
using experimental data of imidazolium-based ionic liquids collected from the literature with
conductivity in the range of 0.008–5.1 S m1 at temperatures between 268.15 K and 398.15 K. The
coefficients of determination (R2) for the MLR and SVM model’s entire data set are 0.8556 and 0.9906,
respectively, while the average absolute relative deviations (AARD) are 46.55% and 7.15%, respectively.
Received 15th April 2021,
Accepted 1st September 2021
This suggests that the non-linear model developed using the SVM regression algorithm fits better with
DOI: 10.1039/d1nj01831k
that conductivity is highly affected by van der Waals forces and temperature, followed by electrostatic
the conductivity data set and is more reliable than the MLR algorithm. The stepwise approach reveals
forces and dielectric energy to some extent. The prediction results from this work will aid the screening
rsc.li/njc
process of suitable ILs with desired conductivity for specific applications.
Introduction
Ionic liquids (ILs), which are molten salts at ambient temperature,
represent an important class of materials which are currently
widely investigated for numerous potential applications, such as
extraction and separation processes, lubricants, waste recycling,
catalysis, gas separation, CO2 capture, electrochemical energy
a
Department of Chemical Engineering, Universiti Teknologi PETRONAS,
32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia.
E-mail: zaireen.yahya@utp.edu.my; Tel: +605-368 7584
b
Center of Research in Ionic Liquids, Universiti Teknologi PETRONAS,
32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia
c
Department of Chemical Engineering, Faculty of Industrial Technology, Institut
Teknologi Bandung, Jl. Ganesha No. 10, Bandung 40132, Indonesia
† Electronic supplementary information (ESI) available. See DOI: 10.1039/
d1nj01831k
18584 | New J. Chem., 2021, 45, 18584–18597
storage, and energy devices.1–3 These applications are driven by
ILs’ desirable features of salts such as nonvolatility, superior
electrochemical and thermal stability, high polarity, and high
ionic conductivity.4–7 The common classes of cations encompass
imidazolium, pyridinium, phosphonium, and ammonium,
whereas the anions are usually inorganic in nature such as
halides, sulfates, and phosphates.4 Their tunable nature and
environmentally friendly characteristics offer numerous
opportunities for improving existing processes and exploring
new possibilities, thus making it essential to understand
their behavior in the systems.8 Exemplarily, a study of the ILs’
properties and structure–property relationships for ILs is
significant to provide valuable information on the design of
suitable ILs. However, due to the immense number of ILs, one
can envisage that possibly more than 108 ILs can be formed by
combining different sets of anions and cations. The experimental
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determination of ILs’ properties is time-consuming, expensive,
and unrealistic.9,10 Therefore, methods for a priori prediction of
ILs’ properties are highly desirable as they would provide a
preliminary understanding that aids in the selection and design
of suitable ILs for specific applications. To date, there have been
several attempts at building predictive models via different
approaches such as quantitative structure–property relationship
(QSPR), group contribution method (GCM), ab initio molecular
dynamics (AIMD), and machine learning.11,12
One of the critical properties that has drawn considerable
attention from many researchers is ionic conductivity (s).
Conductivity and its inverse, resistivity, represent the strength
of cation–anion motion in ILs under an electric potential
difference.13 Conductivity is also the measure of a material’s
ability to allow the transport of free ions, which is a piece of
crucial information in devising electrochemical applications.14–16
ILs with high conductivity at low temperatures are of extreme
interest as they can replace organic solvents for charge transport
in energy generators and batteries.17
Previously, Abbott18 successfully developed a model to
predict the conductivity of ambient-temperature molten salts
based on the Stokes–Einstein equation as follows:
s¼
z2 Fer 1
Rþ þ R1
6pZMw
(1)
where s is the conductivity, z is the charge on the ion, F is the
Faraday constant, e is the electronic charge, r is the density, Z is
the viscosity, Mw is the molar mass of the IL, and R+ and R are
the radii of the cation and anion respectively. The model pivots
around the concept of ‘‘hole theory’’ in which the IL system is
assumed to behave like an ideal gas. Its motion is limited by the
availability of sites or holes for the migration of ions during
charge transfer.19 Although the model shows a good correlation,
this method has several drawbacks which limit its application
for predicting the properties of unknown ILs. These include
lengthy computational procedures, the need for some experimental data from the studied ILs and the fact that it is solely
applicable for ambient-temperature ILs.
Slattery et al.20 in a different approach proposed a simple
volume-based method to predict ILs’ conductivity. They
concluded that the conductivity decreases exponentially with
increasing molecular volume and such a relationship can be
interpreted in terms of ion mobility, whereby the conductivity
is proportional to the motion of charge carriers in the ILs.
The expression is as follows:
s = cedVm
(2)
where c and d are empirical constants of best fit, and Vm is the
molecular volume. It was found that the experimental and the
predicted conductivity are highly correlated with an R2 value of
0.9871. Despite the simplicity, the caveat is that the model was
developed based on pure ILs that are liquid at limited
temperatures spanning from 293.15 to 295.15 K. Besides, the
model does not account for generality as the empirical constant
needs to be derived separately for different cation–anion pairs.
For this equation to be temperature-dependent, Wileńska et al.21
introduced a T0/T term with T0 being set at 298.15 K, as
shown below:
T0 s ¼ cedVm T
(3)
However, it was reported that this equation only performs
relatively well in the temperature range of 298.15–343.15 K,
which could limit its predictive capability.
From the experimental point of view, ionic conductivity data
are commonly described using the Arrhenius model, which is
correlated with temperature as follows:
1
s¼f
(4)
T
Based on such temperature dependency, with 300 data
points for 15 ILs, Coutinho and Gardas22 applied Vogel–
Fulcher–Tammann (VFT) equation, which is also a function
of temperature to predict the conductivity as follows:
ln s ¼ ln As þ
Bs
ðT T0s Þ
(5)
where As, Bs and T0s are adjustable parameters, from which As
and Bs can be obtained with a group contribution method
(GCM). The GCM, also known as an additive method, is often
used to develop a relationship between the chemical substructures and the desired physical property as the summation
of the contributions of certain defined groups of atoms allowing
prediction of the estimated property value.23,24 Their model
indeed shows a good agreement between experimental and
calculated conductivity values (R-squared (R2) = 0.9974, average
absolute relative deviation (AARD) = 4.57%). However, an empirical
fitting parameter T0s, also known as the Vogel temperature, is
required. Estimation of this parameter is done by assuming that
the glass transition temperature (Tg) of the IL is known; otherwise
the VFT equation is not applicable for the prediction of the
IL conductivity. Nevertheless, the VFT equation is widely used
as the initial expression to generate models for the conductivity of
ILs.11,21
With the reported success of previous work on the viscosity
model,25 Chen et al. employed a similar GCM equation to
predict the conductivity as follows:26
2
s
100
100
ln
¼ As þ Bs
(6)
þ Cs
R0s
T
T
where s is the electrical conductivity in S m1, T is the
temperature in K, R0s is an adjustable parameter, also with a
unit of S m1, while As, Bs and Cs are calculated from the group
contribution method. They investigated 1578 experimental data
points of conductivity, including 77 ILs covering 8 cation cores,
34 anions, and 4 side groups in a wide range of temperature
(248.05–468.15 K) and conductivity (0.0017–9.167 S m1).
Although the model has an acceptable AARD of 3.30%, the
GCM heavily relies on the contribution values of the groups.
Thus, its capability to estimate the conductivity for some ILs
may be hindered due to the lack of some group contribution
values.9,27
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Nilsson-Hallén et al.11 offered another thought-provoking
approach by comparing two VFT based models for IL
conductivity which were developed using volume, mass, and
moment of inertia of the constituting ions and the energy of the
interaction between cation and anion. Both equations are as
follows:
b
Eint
s¼
(7)
exp c
T T0
Vtot Mred Iþ
b
E
int
s¼
exp c
Vtot Mred Iþ
T Tg þ d
!
(8)
where Vtot is the sum of the cation and anion volumes, Mred is
the reduced mass, I+ is the moment of inertia of the cation, and
Eint is the interaction energy, while b, c, d and T0 are fitting
parameters and Tg is the glass transition temperature from the
literature. The difference between both models is that one is
with the inclusion of experimental glass transition temperatures
(R2 = 0.9600) while the other has T0 as a fitting parameter (R2 =
0.9508). They investigated a total dataset of 176 data points for
22 ILs and found that the inclusion of experimental glass
transition temperatures did improve the fit but not to a large
extent, and that it will narrow the temperature range that the
model can predict. Despite the high R2 value, the models do not
provide any valuable information on the relationship between
each parameter and conductivity.
The second approach attempted by Coutinho and Gardas22
was to correlate the molar conductivity of ILs with reciprocal
of viscosity based on the Walden rule, which can be written as
LZa = constant
(9)
where L is the molar conductivity and Z is the viscosity.
The expression of molar conductivity is as follows:
L = sM/d
(10)
1
where M is the molecular weight (g mol ) and d is the density
(g cm3) of the IL. The correlation was indicated in a log–log
plot with the Walden line slope obtained as 0.935 0.008.
According to Galiński et al.,28 they investigated and discovered
that the Walden rule is applicable for a wide range of ILs while
confirming the significant influence of viscosity on ionic liquid
conductivity.
Combining both the Walden rule and the Stokes–Einstein
relation, the self-diffusivity can be correlated with conductivity
through a Nernst–Einstein relation as follows:29
sNE ¼
Npair 2
qþ Dþ þ q2 D
Vkb T
(11)
where Npair is the number of ion pairs, q+ and q are the cation
and anion charges, and D+ and D are the self-diffusivities
of the cations and anions, respectively. The difference
between the actual ionic conductivity and the conductivity
predicted by the Nernst–Einstein relation is usually expressed
as follows:30
s = sNE(1 D)
18586 | New J. Chem., 2021, 45, 18584–18597
(12)
where D is a Nernst–Einstein deviation parameter which is
related to differences in cross-correlations of ionic velocities.
Liu and Maginn29 employed the Nernst–Einstein model to
study the conductivity of six different ionic liquids containing
imidazolium, pyrrolidinium, and ammonium cations paired
with bis(trifluoromethylsulfonyl)imide and bis(perfluoroethylsulfonyl)
imide anions using molecular dynamics simulations. Although
the model suggests that these ILs follow the Walden rule over
the entire temperature range closely, the model prediction
performance is subpar and all predicted ionic conductivity
values were underestimated by a factor of 2 to 10. Additionally,
the complex force-field parameters and intramolecular
potential parameters need to be determined while electronic
structure calculations need to be performed using special software
such as Large-scale Atomic/Molecular Massively Parallel Simulator
(LAMMPS).31 Essentially, a robust predictive model should
preferably be user-friendly, non-empirical, and accurate.
Another intriguing approach that has been widely used to
correlate the conductivity of ILs is the quantity structure–property
relationship (QSPR) approach. Tochigi and Yamamoto6 compared
two different models that were generated via the polynomial
expansion (PE) method and the multiple regression (MR) method,
respectively, in predicting conductivity. The parameters used
include dipole moment, ionization potential, lowest unoccupied
molecular orbital (LUMO) energy, the charge on the nitrogen
atom, area, volume, and ovality. Interestingly, they found that the
PE method results in a higher R2 value (R2 = 0.9745, absolute
average error = 0.457) as compared to the MR method (R2 =
0.9089, absolute average error = 0.928). Despite the high goodness
of fit, the model is considerably complicated, with 25 input
variables in the equation. Some variables are attributed to specific
anions, which may confine its application to predicting the
conductivity of other ILs with different anions. Additionally, the
conductivity term in both PE and MR equations was not
transformed into a logarithmic function.
Recently, machine learning models such as the Artificial
Neural Network (ANN), Fuzzy Logic System (FLS), Adaptive
Network-based Fuzzy Inference System (ANFIS), and Support
Vector Machine (SVM) have been recognized as the next frontier
of statistical learning approaches.4,32–34 Kianfar et al.12
proposed an ANN model for predicting the conductivity of
pyridinium-based hydrophobic ionic liquids using 36 data
points for the network training and 12 data points for testing
the neural network (R2 = 0.9999). It is undeniable that their
model generates high goodness of fit. Nevertheless, the number
of data points used to train the model is comparatively
insufficient, and the predictive performance of this method is
not fully established. Although the ANN has been successfully
applied, there are some underlying concerns regarding its
model architecture, for instance, over-fitting training, local
minima, network optimization, low generalizability, as well
as reproducibility of results, which may be attributed to the
networks’ random initialization and variation of stopping
criteria.35
SVM is a new popular supervised learning approach to
address regression problems. It has attracted significant
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attention due to its extraordinary aptitude and generalization
performance in interpreting the non-linear relationships
between molecular structure and properties.35–37 One of the
key advantages of the SVM algorithm is that the data overfitting can be effectively minimized as compared with the ANN
algorithm.35 LIBSVM, a library for SVMs provided by Chang and
Lin,38 has been successfully used to solve regression problems.
Gharagheizi et al.39 established a least square support vector
machine-group contribution (LSSVM-GC) model, which
included an extensive set of 1077 experimental conductivity
data for 54 ILs with temperature ranging from 238 to 480 K and
conductivity ranging from 0.00009 to 20 S m1. Given that they
have used an extensive database, an R2 of 0.997 and an overall
AARD of 3.3% can be perceived as highly satisfactory.
Nevertheless, as mentioned earlier, the data available for group
contribution are limited, which may lead to an inability to
estimate the conductivity of other ILs.
Fayyaz et al.40 introduced an LSSVM model optimized by the
coupled simulated annealing (CSA) algorithm for an accurate
estimation of the conductivity of ILs in propylene carbonate as
a function of molecular weight of the IL, temperature, and IL
concentration in solution (R2 = 0.996). A total number of 828
data points covering 7 ILs with the temperature ranging from
258.15 to 363.15 K and the conductivity ranging from 0.142 to
10.48 S m1 were used for the model development. Their model
has indeed proven its reliability and robustness; however, it
does not provide molecular insights and understanding on the
contribution of each parameter to conductivity. It is essential to
develop a model that offers a valuable theoretical basis justifying
each factor’s significance in affecting the conductivity.
Following the drawbacks of these models, we find it meaningful
to build a comprehensive QSPR model which possesses high
accuracy of predictive capability yet offers simplicity and provides
valuable insights in terms of the impact of each predictor, namely
the molecular interaction energy of the ILs on the conductivity.
In our previous work, we have developed a correlation model
between viscosity and interaction energies via a stepwise multiple
linear regression (MLR).41 In this work, we propose 2 different
models, which are stepwise MLR and stepwise SVM regression,
to compare both models’ predictive capability. The performance
and accuracy of both models are evaluated by undertaking
statistical error analyses of the results.
Methodology
Dataset
A total of 239 experimental conductivity data points with a wide
range of temperatures (268.15–398.15 K) for 25 ILs were used to
build the predictive models.26,42–44 The cations include 1-ethyl3-methylimidazolium [C2C1im]+, 1-butyl-3-methylimidazolium
[C4C1im]+, 1-propyl-3-methylimidazolium [C3C1im]+, 1-hexyl-3methylimidazolium [C6C1im]+ and 1-octyl-3-methylimidazolium
[C8C1im]+, whereas the anions include bis(trifluoromethylsulfonyl)
imide
[N(CF3SO2)2],
bis(pentafluoroethylsulfonyl)imide
[N(C2F5SO2)2], methanesulfonate [CH3SO3], tetrafluoroborate
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[BF4], acetate [CH3CO2], ethylsulfate [C2H5SO4], octylsulfate
[C8H17SO4], hexafluorophosphate [PF6], methylsulfate
[CH3SO4], tetracyanoborate [B(CN)4], thiocyanate [SCN],
tricyanomethanide [C(CN)3], trifluoroacetate [CF3CO2],
tris(pentafluoroethyl)trifluorophosphate [(C2F5)3PF3] and trifluoromethanesulfonate [CF3SO3]. Note that ILs with halide
anions are not included in the data set as most of these ILs (for
short alkyl chains) exist in the solid phase at room temperature
due to higher melting points.45,46 The chemical structures of the
cations and anions used are depicted in Fig. 1. By the rule of
thumb, the data were randomly divided into training and test
sets, with 190 data points (80%) chosen as the training set and
the remaining 49 data points (20%) used as the test set, the latter
of which consists of ILs at different temperatures as well as the
ILs that were not used in the training set.47 The training set is
used to develop the model, while the test set is used to validate
and fine-tune the model. Both the training and test set data are
provided in Table S1 in the ESI.†
COSMO-RS
Following our previous work on the correlation between
viscosity and interaction energies,41 we aim to use similar
parameters for the development of a predictive model that can
estimate a priori conductivity and gather more in-depth insight
into the mechanisms that impact the conductivity; thus the
Conductor-like Screening Model for Real Solvents (COSMO-RS)
was used herein. COSMO-RS is a thermodynamic model coined
by Klamt48 with a quantum chemistry-based statistical framework that is used to predict the thermodynamic properties of
fluid and liquid mixtures. In the molecular interaction approach,
there are three intermolecular interaction energies, namely
electrostatic or misfit (EMF), hydrogen-bonding (EHB) and van
der Waals forces (EvdW), that can be estimated by COSMO-RS.49
These energies are described by eqn (13)–(15), respectively:
a0
2
EMF ¼ aeff ðs þ s0 Þ
2
(13)
EHB = aeffcHBmin(0;min(0;sdonor + sHB) max(0;sacceptor sHB))
(14)
0
EvdW ¼ aeff tvdW þ tvdW
(15)
where s and s 0 are the screening charge densities of two
different segments; aeff, a 0 , cHB and sHB are the effective contact
surface area, the misfit energy constant, the hydrogen bond
coefficient, and the cut-off of the hydrogen bond, respectively;
sdonor and sacceptor represent the screening charge densities of
hydrogen bond donor and acceptor segments, respectively;
and tvdW is the element-specific vdW interaction parameter.
Additionally, these three interaction energies form the total
interaction energy, EINT, as described in eqn (16):
EINT = EMF + EHB + EvdW
(16)
In addition to these descriptors, here we introduced an
additional descriptor, the dielectric energy, EDiel, which can
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Fig. 1 Chemical structures of cations and anions composing the studied ILs. Cations: (a)
methylimidazolium, (c) 1-butyl-3-methylimidazolium, (d) 1-hexyl-3-methylimidazolium, and
bis(trifluoromethylsulfonyl)imide, (b) bis(pentafluoroethylsulfonyl)imide, (c) methanesulfonate, (d)
(g) octylsulfate, (h) hexafluorophosphate, (i) methylsulfate, (j) tetracyanoborate, (k) thiocyanate,
tris(pentafluoroethyl)trifluorophosphate and (o) trifluoromethanesulfonate.
be described as half of the electrostatic interaction energies of
the ideally screened and self-consistently polarized solutes with
their screening charges based on eqn (17):50
EDiel ¼
1X
1X
Fv qv ¼
Fv sv sv
2 v
2 v
(17)
where Fv is the electrostatic potential, qv is the ideal screening
charge, sv is the area, and sv is the ideal screening charge
density on the v segment, respectively.
Several researchers have made successful attempts at developing
models based on these energy descriptors generated by COSMO-RS
so as to predict the hydrogen-bond basicity of ILs,51 the ability of ILs
as a thermodynamic hydrate inhibitor for methane hydrate,52 as
well as the contact angles and wettability of ILs on polar and nonpolar surfaces.53 In this work, the COSMOthermX program with the
parameter file BP_TZVP_C30_1201 was used in all the calculations.
All the COSMO files required are available in the COSMOthermX
database. The EMF, EHB, EvdW and EDiel values for the studied ILs are
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1-ethyl-3-methylimidazolium, (b) 1-propyl-3(e) 1-octyl-3-methylimidazolium. Anions: (a)
tetrafluoroborate, (e) acetate, (f) ethylsulfate,
(l) tricyanomethanide, (m) trifluoroacetate, (n)
provided in Table S1 in the ESI.† Another descriptor is also
considered in this model, which is 1/T as the conductivity is
temperature dependent.
QSPR models
To select proper and suitable descriptors capable of forming a
relationship with conductivity, a stepwise regression was
used.53 The approach starts with a single input variable as an
initial establishment of the correlation model. The model is
then recurrently improved by adding a new variable or removing
less significant variables until a satisfactory and accurate model
is obtained.
The MLR and SVM algorithms were employed to develop
new models for the prediction of conductivity of ILs. MLR is a
popular QSPR method because of its simplicity, transparency,
reproducibility, and easy interpretability.47,54 This method
correlates a set of data points of the desired property (y) with
one or more descriptors (xn) in a linear form as described in
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Results and discussion
eqn (18):55
y ¼ a0 þ
n
X
an xn
(18)
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1
where y represents the property value, xn is any molecular
descriptor, and an is the regression coefficient.
SVM is an intelligent and supervised algorithm initially
developed by Cortes and Vapnik. It has been used for
regression, clustering, and pattern recognition problems in
various fields of science and engineering.4,56–59 SVM has been
proven to be very effective in solving non-linear regression
problems.35,60 Generally, the SVM function can be defined as
follows:
y = wTF(x) +b
(19)
where x represents the input variables with an N n matrix
(N and n are the total number of data points and number of
input variables, respectively), F(x) is the kernel function, wT is
the output layer transposed vector, and b denotes the bias.
Although SVM’s general equation is very similar to that of MLR,
SVM allows the input variables to be mapped to a highdimensional feature space via a selected kernel function, and
then the linear regression occurs in the feature space.61 Such
robust non-linear feature mapping enables a non-linear
problem to be solved in a linear space.
The selection of the kernel function is very crucial as each
type of function distributes the data points in the highdimensional feature space differently, thus generating different
correlation coefficients. One of the most common kernel
functions, which is the radial basis function (RBF), is used in
this study because it is effective and fast in the training
process.35 The RBF is described in eqn (20):
K(x, xk) = exp(g8xk x82)
(20)
where K(x, xk) is the Kernel function calculated from the inner
product of the two vectors x and xk in the feasible region and g
is the tuning parameter. For a RBF kernel, there are 2 main
tuning parameters, namely g and a cost function, C. C can be
defined as a regularization constant that identifies the trade-off
between minimization of the training error and maximization
of the margin. The optimization of C and g is required to
improve the accuracy of the model in predicting unknown data
in the test set.62 For instance, if the C value is too small,
insufficient stress will be placed on fitting the training data,
which results in underfitting of the model. If the C value is too
large, the model gets overfitted. The g parameter determines
the distance of a single data sample that exerts influence. In
our work, the best values of C and g were identified via a ‘‘gridsearch’’ method.
The best MLR and SVM equations were selected based on
the lowest Average Absolute Relative Deviation (AARD) and the
highest R-squared (R2) value as screening criteria. The QSPR
models’ accuracy was ascertained by further statistical analyses,
including Standard Deviation Error (STDE) and Root Mean
Square Error (RMSE).
The electrical conductivity of ILs is one of the intrinsic properties that is critical, especially for electrochemical applications.
Therefore, systematic knowledge of the effect of interaction
energies on conductivity is of high importance. As all systems
studied in this work are based on the imidazolium cation, it
allows us to evaluate the impact of the alkyl chain length
and anion on conductivity. The experimental and predicted
conductivity values are given in Table S2 in the ESI.†
Results of the stepwise MLR algorithm
The correlations between selected descriptors and conductivity
upon stepwise MLR regression are summarized in Table 1 and
Fig. 2. A total of 190 experimental ionic conductivity data points
of imidazolium-based ionic liquids, gathered from Chen
et al.,26 were used as the training set data to build the predictive
model. The first three equations (cf. eqn (21)–(23)) consist of (i)
individual interaction energy that arises from each cation and
anion and (ii) temperature.
Table 1 shows that the hydrogen-bonding interaction has
less influence on conductivity than the electrostatic interaction
and van der Waals force, which is a similar outcome for
viscosity property as presented in our previous work.41 This is
likely to be attributed to the complexity and asymmetry of such
polar and non-coordinating liquids.46,63 It was discovered that
the contribution of dispersion force, which is part of the van
der Waals forces, can outbalance the hydrogen-bonding energy
with increasing temperature.64 The equation with EvdW
produced the highest R2 value and the lowest AARD value when
compared to other single interaction energy correlations. This
elucidates that the van der Waals interaction energies between
cation and anion of the ionic liquid have a pivotal impact on
the electric potential of these fluids. Despite being simple, the
yielded R2 value (0.5193) and AARD (72.81%) of the 3-descriptor
equation are far below satisfactory. Therefore, a multidescriptor approach was then employed.
Since the EvdW was identified as the interaction energy with
the dominant contribution, extra descriptors were further
added to eqn (23), resulting in a series of eqn (24)–(26) as listed
in Table 1. The addition of EMF to eqn (23) results in a higher R2
value and a lower AARD as compared to the addition of EHB due
to the smaller influence of hydrogen bonding. Interestingly, the
combination of all three types of interaction energies in
eqn (26) yields a nearly similar R2 value and AARD compared
to eqn (25). This is in good agreement with the research outcome
by Shi et al.65 in which the intermolecular attraction force in
ionic liquids is mainly from the sum of electrostatic force and
van der Waals force. According to Delhorbe et al.,66 both glass
transition temperature and charge carrier density, which are
associated with the polymer backbone’s segmental motion, are
not the only factors that influence the charge transport in 1-alkyl3-vinylimidazolium bis-(trifluoromethane)sulfonimide-derived
homopolymers. The charge mobility is also attributed to its
self-assembling nanostructure, which is determined by the
sum of Coulombic repulsion and van der Waals interactions.
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Table 1 Stepwise MLR to evaluate the relationship between the experimental conductivity values and the interaction energies (in kcal mol1) together
with dielectric energy (in kcal mol1) estimated using COSMO-RS
R2
AARD (%)
Eqn
lnðsÞ ¼ 6:5868 EHB;Cation þ ð6:5336Þ EHB;Anion
0.3427
99.13
(21)
1
þ 11:2667
T
lnðsÞ ¼ ð0:2413Þ EMF; Cation þ ð0:0510Þ EMF;Anion
0.3931
92.81
(22)
1
þ 13:1135
T
lnðsÞ ¼ 0:3495 EvdW;Cation þ 0:0192 EvdW;Anion
0.5193
72.81
(23)
1
þ 13:9012
T
lnðsÞ ¼ ð11:0498Þ EHB;Cation þ 11:3014 EHB;Anion
0.5009
66.13
(24)
0.6783
64.56
(25)
0.7032
64.53
(26)
0.8653
45.86
(27)
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þ ð3839:5942Þ þ ð4027:5787Þ þ ð3630:7134Þ þ 0:4244 EvdW;Cation þ 0:1003 EvdW;Anion
1
þ 14:1212
T
lnðsÞ ¼ 0:0713 EMF;Cation þ ð0:6712Þ EMF; Anion
þ ð3214:1704Þ þ 0:4112 EvdW; Cation þ ð0:2034Þ EvdW; Anion
1
þ 16:3842
T
lnðsÞ ¼ 12:7664 EHB;Cation þ ð13:1097Þ EHB;Anion
þ ð4121:3225Þ þ ð0:2606Þ EMF;Cation þ ð0:6731Þ EMF; Anion
þ 0:3229 EvdW; Cation þ ð0:1776Þ EvdW;Anion
1
þ 18:5985
T
lnðsÞ ¼ 0:9284 EMF; Cation þ ð0:9506Þ EMF; Anion
þ ð4661:9406Þ þ 0:5266 EvdW;Cation þ ð0:3081Þ EvdW; Anion
þ ð0:7667Þ EDiel; Cation þ 0:0983 EDiel; Anion
þ ð3513:9528Þ 1
þ ð21:1878Þ
T
The linear model based on eqn (25) was then further improved
with the addition of EDiel, which represents the energy needed
for dielectric polarization, which explains its good correlation
with ionic conductivity.67
Next, the test set’s conductivity was predicted using eqn (27)
to evaluate its reliability and goodness of fit. The cross-plot of
experimental data versus calculated/predicted data and the
relative deviations for both training and test sets are presented
in Fig. 4. The calculated statistical error parameters for s are
shown in Table 3. As can be seen from Fig. 4(a), the calculated
conductivity values were not in good agreement with
the experimental conductivity values. As shown in Fig. 4(b),
the AARD of the whole set is 46.55%, and almost 77% of the
predicted ILs’ conductivity values have above 15% deviation.
Thus, these results suggest that eqn (27) established by the
MLR algorithm is not a suitable model. Hence, it is necessary to
develop a non-linear regression model to fit the data.
Results of the stepwise SVM regression algorithm
The same descriptors selected for the MLR algorithm were
applied as the input parameters to establish a non-linear model
18590 | New J. Chem., 2021, 45, 18584–18597
using the SVM algorithm. The correlations between selected
descriptors and conductivity upon stepwise SVM regression are
summarized in Table 2 and Fig. 3. Similarly, a stepwise
approach was applied to evaluate the significance of each
interaction energy parameter. Although the R2 value of the
EvdW-based model (0.7596) is slightly lower than that of the
EMF-based model (0.8435), van der Waals force was again
identified as the principal contributor as its AARD of 28.17%
is significantly lower than those of eqn (28) and (29), which are
72.12% and 53.54%, respectively. Also, the addition of EMF to
eqn (30) results in a higher R2 value and a lower AARD as
compared to the addition of EHB. The statistical result of
eqn (32) is on par with the model which combines all three
interaction energies, and thus the EDiel descriptor was added to
eqn (32). When C = 1024, g = 0.0156 and epsilon = 0.0800 with
37 support vectors, the best model was obtained.
The cross-plot of experimental data versus calculated/
predicted data and the relative deviations for both training
and test sets are portrayed in Fig. 5. The calculated statistical
error parameters for s are shown in Table 3. Based on Fig. 5(a),
the results are concentrated on the unit slope line (line Y = X),
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Fig. 2 Stepwise MLR models: (a) eqn (21), (b) eqn (22), (c) eqn (23), (d) eqn (24), (e) eqn (25), (f) eqn (26), and (g) eqn (27). R2 values: (a) 0.3427, (b) 0.3931,
(c) 0.5193, (d) 0.5009, (e) 0.6783, (f) 0.7032, and (g) 0.8653.
Table 2 Stepwise SVM to evaluate the relationship between the experimental conductivity values and the interaction energies (in kcal mol1) together
with dielectric energy (in kcal mol1) estimated using COSMO-RS
Parameter (x)
EHB
lnðsÞ ¼
n
P
2
ai exp gkxi xnew k þ b
EMF
EvdW
EDiel
—
—
i
—
—
—
—
—
—
—
—
—
—
—
—
—
—
1
T
—
—
—
—
—
—
—
—
g
b
C
Epsilon
nSVs
R2
AARD (%)
Eqn
0.3333
0.3333
0.3333
0.2000
0.2000
0.1429
0.1429
0.0156
0.3681
0.0066
0.3023
0.5790
0.1328
0.2066
0.3796
5.1408
1
1
1
1
1
1
1
1024
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.0800
138
127
108
87
79
75
67
37
0.4488
0.8435
0.7596
0.7826
0.9657
0.9588
0.9788
0.9915
72.12
53.54
28.17
15.78
13.50
14.06
12.95
6.99
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
which elucidates that the SVM regression algorithm’s new model
is robust and has a more superior prediction capability than the
MLR algorithm. Moreover, the resulting AARD of the whole set is
7.15%, and more than 95% of the data points show deviations
within 15%, which indicates a good accuracy for the imidazolium-based ILs. Some of the observed deviations may be due to
impurities or water as it may improve the ionic conductivity.68
The mathematical expression of the SVM model is in the
form of a Gaussian radial basis function kernel (RBF), which is
expressed as eqn (36):
lnðsÞ ¼
n
X
2
ai exp 0:0156kxi xnew k 5:1408
(36)
i
where ai, xi, and xnew are the coefficients of the support vector,
support vectors (SVs), and new input variables, respectively. ai
and xi are provided in Table S6 in the ESI.† It is of high
importance that normalization processing should be first
carried out before using the model. Please note that to easily
calculate the conductivity values of ILs in the MATLAB software,
the LIBSVM toolbox should be initially installed (which can be
downloaded online for free).38
Effect of the alkyl chain length on the ionic conductivity
Understanding how the alkyl chain length might affect the
transport properties of ILs is fundamentally important. Fig. 6
portrays the effect of the alkyl chain length on the conductivity
of ILs. As shown in Fig. 6, the conductivity of imidazoliumbased ILs paired with the bis(trifluoromethylsulfonyl)imide
anion decreases with increasing alkyl chain length, which can
be explained by the increase in the viscosity of ILs due to the
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Fig. 3 Stepwise SVM regression models: (a) eqn (28), (b) eqn (29), (c) eqn (30), (d) eqn (31), (e) eqn (32), (f) eqn (33), (g) eqn (34), and (h) eqn (35). R2 values:
(a) 0.4488, (b) 0.8435, (c) 0.7596, (d) 0.7826, (e) 0.9657, (f) 0.9588, (g) 0.9788, and (h) 0.9915.
Fig. 4 MLR plots using eqn (27): (a) experimental versus calculated conductivity with an R2 value of 0.8556 and (b) relative deviations between the
experimental and calculated conductivity. Symbols: ( ), training set and ( ), test set.
increase of the side chain length.26 In other words, an increase
in the cationic size or –CH2– units will increase the van der
Waals interactions in terms of the alkyl chain-ion inductive
forces and the CH–CH bond interaction.70,71 This is consistent
with the observations by Sun et al.72 in which the conductivity
increases from [P2224]+ to [P2228]+-based ILs even at different
temperatures due to the elongation. The ion-pairing interaction
and ion mobility are slightly reduced for ILs with longer alkyl
chain length which results in a lower threshold positive
potential to oxidize the anion, thus leading to lower ionic
conductivity. Hanabusa et al.73 deduced that the ionic
18592 | New J. Chem., 2021, 45, 18584–18597
conductivity is highly correlated with the viscosity upon
discovering that the ionic conductivity of [DBUH]+-based protic
ILs increases as the carbon number in the carboxylate ion
increases to 8. In contrast, the viscosity decreases for the same
carbon number range.
Effect of different anions on the ionic conductivity
Given the established SVM model’s reliable performance,
eqn (35) was used to predict the conductivity values of other
ILs not studied in this work (or ILs with no available experimental data). A database of conductivities comprising 7
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Fig. 5 SVM regression plots using eqn (35): (a) experimental versus calculated conductivity with an R2 value of 0.9906 and (b) relative deviations
between the experimental and calculated conductivities. Symbols: ( ), training set and ( ), test set.
Table 3 The statistical error parameters for s via the MLR and SVM
approach
Statistical parameter
MLR
SVM
Training set
R2
Average absolute relative deviation
Standard deviation error
Root mean square error
No. of data points
0.8653
45.86%
0.4852
0.4868
190
0.9915
6.99%
0.0831
0.0851
190
Test set
R2
Average absolute relative deviation
Standard deviation error
Root mean square error
No. of data points
0.7717
49.21%
0.4034
0.3993
49
0.9900
7.76%
0.0575
0.0570
49
Total
R2
Average absolute relative deviation
Standard deviation error
Root mean square error
No. of data points
0.8556
46.55%
0.4695
0.4702
239
0.9906
7.15%
0.0787
0.0801
239
different 1-alkyl-3 methylimidazolium-based cations combined
with 41 anions resulting in a total of 287 ILs was built. The full
name, abbreviation, and cation or anion numbering of the ILs
are given in Table S3 in the ESI.† The predicted conductivity
values are provided in Table S4 in the ESI† and illustrated in
Fig. 7. ILs paired with halide ions are not presented in this
chart due to their high melting point properties, due to which
they exist in crystalline form at room temperature.46 ILs that are
paired with a larger anion such as [N(CN)3] (volume =
113.8148 Å3) display higher conductivity regardless of the alkyl
chain length as compared to the smaller [BF4]-based (volume =
72.8506 Å3) ILs. The increase in anion size may decrease the
interaction between anion and cation, mainly affected by the
reduced hydrogen bond strength and Coulomb interaction.74
The decrease in interionic interaction results in reduced
hindrance to the migration of charge, resulting in increased
Fig. 6 Predicted vs. experimental conductivity value for [CNC1im][N(CF3SO2)2], where N = 2–9, at T = 298.15 K and P = 0.1 MPa. Symbols:
(m), experimental value collated by Chen et al.;26 (K), experimental value
by Papović et al.;7 (K), experimental value by Widegren et al.;69 and (m),
predicted value using the SVM model.
ionic conductivity. As the weakly coordinating or less symmetric
anions such as [N(CF3SO2)2] and [N(C2F5SO2)2] have lower
viscosities, the observed higher conductivities of these ILs is well
justified.25 In general, the extent of this database can be as large
as desired as the model established only requires two types of
cation–anion interaction energies and dielectric energy that
can be estimated using COSMO-RS. Thus, Fig. 7 can be used
as an a priori tool to screen suitable ILs before running the
experimental study.
Comparison with existing models
In this section, the performance of the developed SVM regression
model in predicting the ionic conductivity of imidazolium-based
ILs is evaluated graphically and statistically. Moreover, the results
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Fig. 7 Predicted ionic conductivity values of 287 imidazolium-based ILs
at T = 298.15 K and P = 0.1 MPa.
of our model are compared with the existing models found in the
literature for verification. Fig. 8 presents the experimental
conductivity values and the predicted conductivity values from
COSMOthermX, Chen et al.,26 Gharagheizi et al.,39 and our
present work for [C4C1im]+ paired with [CH3CO2], while Fig. 9
presents the values for [C4C1im]+ paired with [CF3SO3]. The
conductivity values are provided in Table S5 in the ESI.†
As can be seen from Fig. 8 and 9, our model shows a far
more superior prediction capability with less deviation from
the experimental value as compared to the COSMO model,
which was built with more than 50% of its training data being
comprised of [N(CF3SO2)2]-based ILs.75 The GC-based model
and the LSSVM-GC model proposed by Chen et al.26 and
Gharagheizi et al.,39 respectively, show similar reliability in
comparison to our model but the GCM is limited to the number
of group functions used in the model, and thus cannot be
Fig. 8 Comparison between experimental and predicted conductivity
values from the COSMO model, Chen et al.,26 Gharagheizi et al.,39 and
present work for 1-butyl-3-methylimidazolium acetate at P = 0.1 MPa.
Symbols: (m), experimental value; (K), COSMO model; (K), Chen et al.;
(K), Gharagheizi et al.; and (m), predicted value using the SVM model.
18594 | New J. Chem., 2021, 45, 18584–18597
Fig. 9 Comparison between experimental and predicted conductivity
values from the COSMO model, Chen et al.,26 Gharagheizi et al.,39 and
present work for 1-butyl-3-methylimidazolium trifluoromethanesulfonate
at P = 0.1 MPa. Symbols: (m), experimental value; (K), COSMO model; (K),
Chen et al.; (K), Gharagheizi et al.; and (m), predicted value using the SVM
model.
applied when a new group is introduced. Our model also
provides a much simpler calculation with only 3 energy
parameters that can be easily generated using the COSMO-RS
software. There are only three simple steps required for the
generation of data input. Firstly, in the COSMOthermX
program, the user will need to select the cation and anion in
the study. Next, under the properties tab, the chemical
potential of the mixture is to be selected and then followed
by inserting the corresponding temperature and equimolar
mole fraction before running the simulation.
In terms of the number of data points for the model
development, Chen et al.26 and Gharagheizi et al.39 used a
larger database as compared to our model. One can envisage
that an extensive databank may contribute to the better understanding of a wider range of ILs and offer a higher degree of
generality for the established model. However, in this case,
there are always discrepancies in experimental values reported
by two or more research groups due to sample impurities and
difference in terms of instrument accuracy limit or calibration.
The challenge would be to fit all data points and obtain a high
goodness of fit or a low relative deviation. Therefore, in the
model development process, training and validation steps play
a pivotal role in identifying the outliers and anomalies. It is
noteworthy that although our work has a smaller sampling size,
every data point is unique and the test set consists of ILs at
different temperatures as well as the ILs that were not used in
the training set.
In addition, the stepwise approach and the comparison
analysis between linear and non-linear models in the present
study have provided deeper insights and understanding of the
contribution of interaction energies to conductivity and the
suitable correlation between them. As compared to other
machine learning models, SVM offers a significant advantage
in that it is relatively simple to use as there are only a few userdefined parameters. In this work, since the RBF kernel is
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Table 4
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Comparisons of this work with models reported in the literature
Ref.
Method
NIL
D.P.
T (K)
P (MPa)
R2
Coutinho and Gardas22
Chen et al.26
Gharagheizi et al.39
Tochigi and Yamamoto6
Tochigi and Yamamoto6
Eiden et al.75 (COSMO)
This work
This work
GCM
GCM
GCM (LSSVM)
QSPR (PE)
QSPR (MLR)
QSPR
QSPR (MLR)
QSPR (SVM)
15
77
54
79
79
142
25
25
300
1578
1077
150
150
713
239
239
258.15–433.15
248.05–468.15
238–480
243.15–338.15
243.15–338.15
238.15–468.15
268.15–398.15
268.15–398.15
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.9974
0.9973
0.9970
0.9676
0.8697
0.9100
0.8556
0.9906
selected, the user will only need to fine-tune the parameters for
C and g. Moreover, the final results of SVM are reproducible and
stable, unlike other methods such as neural networks, which may
change in each iterative run due to the random initialization of
weights.76 SVM model development can be performed across
various computing platforms and programming languages, for
instance, MATLAB, R, and Python. The optimization process of
the SVM algorithm via grid-search is considerably fast during
execution, and it works adequately with average computer system
requirements.
The statistical comparison in terms of R2 values between our
models and other models reported in the literature is listed in
Table 4. The result shows that the models established by the
MLR algorithm do not provide good accuracy in predicting the
conductivity of ILs in comparison with non-linear models. Our
SVM model has a comparable statistical result as compared to
the GCM models developed by Chen et al.26 and Gharagheizi
et al.,39 which proves the prediction capability of our model.
The R2 value of the COSMO model reported by Eiden et al.75 is
0.91, which is lower than that of our model (R2 = 0.9906).
The PE model established by Tochigi and Yamamoto6
demonstrated a high R2 of 0.9676, but their model requires
numerous quantum data inputs (more than 15 parameters),
which might be excessive and cumbersome for users to obtain.
In contrast, our predictive model applies a user-centric
approach by offering simplicity with only 7 descriptors,
and the algorithm can be easily computed using common
spreadsheet software such as Microsoft Excel.
Furthermore, although several molecular descriptors were
used in Tochigi and Yamamoto’s model, the underlying insights
were not provided. Our model not only works well for a variety of
IL cation and anion types but also explains the theoretical
deductions on the impact of interaction energy descriptors as
the correlations are considerably built using a non-empirical
approach. Based on the results, the proven capability of our
model in predicting conductivity at different temperatures has
overcome the limitation of the VFT model proposed by Coutinho
and Gardas,22 which requires the determination of the Vogel
temperature via trial and error. All these results confirmed the
reliability and robustness of our QSPR-SVM model in describing
the ionic conductivity of ILs at various temperatures.
Following the success of the QSPR model development for
the prediction of ionic conductivity of imidazolium-based ILs,
the model can be further explored to predict the ionic
conductivity of other non-imidazolium-based ILs. Moreover, a
generalized model which can be applied to ILs with any type of
cation can potentially be developed. Nonetheless, more descriptors
may need to be considered in such context, for instance, ring
energy, to improve the correlativity. Besides, the established models
are built based on atmospheric pressure, and thus the models can
be optimized with the addition of a pressure descriptor to allow the
prediction of ILs at any pressure. This study has opened the
research window to discover the correlation between interaction
energies and other physicochemical properties of ILs, which may
be useful for more applications.
Conclusions
In this study, 2 different QSPR models, MLR and SVM regression
algorithms, were built and compared in terms of prediction
capability for the ionic conductivity of imidazolium-based ionic
liquids at different temperatures. A comprehensive data set of
239 ionic conductivity data points belonging to 25 ILs composed
of 15 anions and 5 cations was employed to develop this model.
Upon stepwise approach, eventually, both models were
constructed with 4 parameters, including EMF, EvdW, EDiel and
1/T, in which the first three descriptors were calculated using
COSMO-RS. We discovered that the MLR algorithm performed
poorly for the entire data set, with the R2 and AARD values being
0.8556 and 46.55%, respectively. Comparatively, the model
established by the SVM regression algorithm exhibited accuracy
and correlativity for the whole data set, with the R2 and AARD
values being 0.9906 and 7.15%, respectively. Herein the SVM
model developed in this work is reliable and can be used to
predict conductivity. In contrast to the general predictive
models, we carried out stepwise regression to provide a valuable
and exciting understanding of the impact of different interaction
energies on conductivity. The van der Waals forces were identified
to have the most significant impact on the conductivity of ILs
followed by electrostatic interactions and hydrogen bonding
interactions. Furthermore, dielectric energy is found to have a
reasonable correlation with conductivity as well. It is noteworthy
that this work provides valuable knowledge to the ionic liquid
research community encompassing the impact of interaction energies on ILs’ conductivity for the screening and rational design of ILs.
Conflicts of interest
There are no conflicts to declare.
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Acknowledgements
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This work is funded under YUTP-FRG (015LC0-080) and
received support from the Chemical Engineering Department
and Centre of Research in Ionic Liquids of Universiti Teknologi
PETRONAS.
23
24
25
26
Notes and references
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