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Machine learning models for solvent effects on electric double layer capacitance
Accepted Manuscript Machine Learning Models for Solvent Effects on Electric Double Layer Capacitance Haiping Su, Cheng Lian, Jichuan Liu, Honglai Liu PII: DOI: Reference:
S0009-2509(19)30252-0 https://doi.org/10.1016/j.ces.2019.03.037 CES 14860
To appear in:
Chemical Engineering Science
Received Date: Revised Date: Accepted Date:
27 September 2018 10 March 2019 15 March 2019
Please cite this article as: H. Su, C. Lian, J. Liu, H. Liu, Machine Learning Models for Solvent Effects on Electric Double Layer Capacitance, Chemical Engineering Science (2019), doi: https://doi.org/10.1016/j.ces.2019.03.037
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Machine Learning Models for Solvent Effects on Electric Double Layer Capacitance
Haiping Su1, Cheng Lian1*, Jichuan Liu2, Honglai Liu1* 1
State Key Laboratory of Chemical Engineering, and Department of Chemistry, East China University of Science and Technology, Shanghai, 200237, P. R. China 2
*
Department of Chemical Engineering, University College London, London WC1E 7JE
Correspondence concerning this article should be addressed: [email protected] , or [email protected] .
1
Highlights The solvent effects on electrochemical capacitances were investigated by combining experimental observation and machine learning (ML) models. The classical density functional theory (CDFT) could be used for better understanding of the solvent effects derived from the ML models. The experimental-CDFT-ML combined method provides a useful benchmark for computational and experimental screening of new materials for electrochemical applications.
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Abstract The role of solvent molecules in electrolytes for supercapacitors, representing a fertile ground for improving the capacitive performance of supercapacitors, is complicated and has not been well understood. Here, a combined method is applied to study the solvent effects on capacitive performance. To identify the relative importance of each solvent variable to the capacitance, five machine learning (ML) models were tested for a set of collected experimental data, including support vector regression (SVR), multilayer perceptions (MLP), M5 model tree (M5P), M5 rule (M5R) and linear regression (LR). The performances of these ML models are ranked as follows: M5P > M5R > MLP > SVR > LR. Moreover, the classical density functional theory (CDFT) is introduced to yield more microscopic insights into the conclusion derived from ML models. This method, by combining machine learning, experimental and molecular modeling, could potentially be useful for predicting and enhancing the performance of electric double layer capacitors (EDLCs). Keywords: Solvent effects, Electric double layer capacitance, Machine learning, Classical density functional theory
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Graphical Abstract
4
1 Introduction Electric double layer capacitors (EDLCs), also known as supercapacitors, present remarkable advantages including rapid charging/discharging kinetics, high power densities, and long cycling lifespans (Dou et al., 2017; Lian et al., 2018a; Simon and Gogotsi, 2012; Su et al., 2017; Zhan et al., 2017). EDLCs store electrical energy by physical adsorption of ions onto the surfaces of porous electrodes, and the capacitance of supercapacitors is strongly reliant on the nonredox ionic behavior inside the microspores of porous electrodes (Burt et al., 2014; Di et al., 2018; Fedorov and Kornyshev, 2014; Frackowiak et al., 2013; Jiang and Wu, 2013; Shao et al., 2015; Vatamanu and Bedrov, 2015; Zhong et al., 2015). The energy density per surface area of an EDLC device is proportional to its capacity and the square of its operating potential window (OPW), E CV 2 2 , where C is the capacitance per surface area and V the maximum value of the OPW (Fedorov and Kornyshev, 2008; Feng et al., 2015; Kornyshev, 2007; Pilon et al., 2015). Accordingly, the energy density of an EDLC can be enhanced by improving the capacitance of the electric double layer (EDL) and expanding the OPW through various methods. Among them, changing the solvent addition within the electrolyte is treated as an effective strategy. Many experimental efforts have been devoted to study the solvent effects on the performance of EDLC. Chimiola and Largeot (Chmiola et al., 2006; Largeot et al., 2008) reported that an “anomalous” increase of the carbon capacitance occurred when the ionic liquid with proper size was added into the organic electrolyte. Morita et al. (Morita et al., 1992) found that improvements in the capacitance and the electrolytic conductivity were observed by mixing organic solvents. Intensive studies conducted by Janes et al. (Arulepp et al., 2004; Jänes et al., 2004; Lust et al., 2004) also demonstrated the influences of organic solvent on solution resistance, EDL charging dynamics, and EDL capacitance. The investigation by Kim and
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coworkers (Kim et al., 2010) revealed that small solvent molecules could enter into the nanopores, resulting in enhanced capacitance by shifting the potential of zero charge (PZC). Liu et al. (Liu et al., 2012) presented improved capacitive performances by optimizing the ratio of EMIM-BF4 and acetonitrile in the electrolyte. Decaux et al. (Decaux et al., 2014) demonstrated that the polarizing salt ions in solvents do not completely lose the solvation shell and that the performance of EDLC depends greatly on the interaction between the solvents and ions. Apart from the experimental works, molecular theory and molecular simulation method have been applied to study the solvent effects on the capacitive behavior of EDLCs. Cummings et al. (Li et al., 2014; Uysal et al., 2014) discovered that the solvent addition (dicationic ionic liquid) contribute greatly to the performance enhancement by the molecular dynamics (MD) simulation method. Furthermore, they found that the solvent polarity of room temperature ionic liquid (RTIL) plays an important role on the capacitive behavior (Osti et al., 2016) and that the humidity of the solvent also affects the performance to a great extent (Osti et al., 2017). A capacitance maximum was found in RTIL through dilution with organic solvents from the MD simulation (Bozym et al., 2015; Uralcan et al., 2016). Bedrov et al. (Li et al., 2015) used MD to study the effect of organic solvents on the solvation and dynamics of Li+ ions in RTIL electrolytes, and the results showed that the addition of solvent could accelerate the self-diffusion coefficient of Li+ ions and the conductivity of electrolytes, ultimately leading to an enhanced performance. Theoretically, the molecular-scale solvent effects are not well considered in the traditional EDL models such as the Helmholtz model and the Gouy-Chapman-Stern theory (GCS). Kondrat and Kornyshev (Kondrat and Kornyshev, 2016; Rochester et al., 2016) developed a statistical theory to describe the charging behavior of cations, anions and solvent
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molecules in narrow pores. They found that the solvent addition such as “ionophobic agents” in the electrolyte could change the charging mechanisms in EDLC, ultimately leading to the improvement of energy density. Wu and coworkers applied CDFT to study the solvent polarity effect on the EDL capacitance (Jiang and Wu, 2014), and they also found that the organic solvent effect differs when the electrode varies (Jiang et al., 2012; Jiang and Wu, 2013; Lian et al., 2018b) and that the solvent-like additions (neutral molecular, or dipole) could also affect the capacitive behavior (Lian et al., 2017; Liu et al., 2017; Liu and Wu, 2016). Although it has been widely known that the EDL capacitance is significantly affected by solvent properties, our understanding of the interactions between solvent properties and capacitive performance is still limited in the following capacities: (i) experimentally, it is difficult to elucidate the solvent effects in practical electrochemical tests, and the nanoscale electrode-electrolyte interface behaviors could only be speculated from the results of electrochemical experiments; (ii) theoretically, traditional EDL models only provide a mathematical equation or a statistical regression, and it is difficult to describe solvent effects by providing a dielectric constant (εr); and (iii) molecular theory and simulation tools are considered as effective tools to investigate these nanoscale problems, but they are very time-consuming for large systems. It is also a significant challenge for molecular simulation to bridge the gap between microscopic details and macroscopic properties. Although different solvent parameters have been studied by different researchers, they often consider one factor for each experiment or calculation, which may ignore the coupled solvent effect on the EDL capacitance. Hence, a more comprehensive method is needed for better understanding the role of the solvent on supercapacitors.
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Machine learning (ML) technique, a data-based method, has been treated as an alternative tool (supplementing the theory, experiment, and simulation methods shown in Figure 1) to solve many practical problems. Recently, several chemical engineering issues have been efficiently studied by ML methods (Chakraborty et al., 2014; Sui et al., 2016; Turan et al., 2011; Wu et al., 2013), which inspired us to investigate the solvent effects on the EDL capacitance thorough the ML method.
Figure 1. Scientific understanding involves the formulation of theory, experimental observation, modeling/simulation, and machine learning methods. These four methods are closely related and supplement each other. In this work, the experimental data of different solvents were collected with the purpose of linking the physicochemical properties of solvents and the EDL capacitive behavior. Using these empirical data as inputs for different ML models, the performance of each ML model was evaluated to produce the best model for this issue. Here, ML models were used as black boxes to elucidate the connections and rules between the capacitance and several solvent properties. Then, the CDFT was applied to yield microscopic insights into the connections and rules built by ML
8
models. Hence, the capacitive performance of EDLCs with ionic liquids in different solvents would be accurately predicted by the mentioned method.
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2. Methods and models To predict the EDL capacitance without further experimental measurements, a sequential procedure is used, as shown in Figure 2. In the first step, the experimental or calculated capacitance values with different solvents were collected as data source. Then, the obtained empirical data were applied to assess the performance of five different machine learning models. Finally, the best one was chosen to predict the EDL capacitance.
Figure 2. Illustration of the sequential approach used for predicting EDL capacitance by machine learning models. 2.1 Data collection Several solvents including acetonitrile, aniline, benzyl alcohol, dichloromethane (DCM), dimethyl sulfoxide (DMSO), ethanol, ethylene glycol, formamide, glycerol, n-methylformamide (NMF), propylene carbonate (PC), tetrahydrofuran (THF), and water were gathered from Hou’s work (Hou et al., 2014), in which experimentally controllable solvent variables for EDL capacitance were obtained by a dual parallel platinum electrodes system in the polarized potential domain. The authors carefully selected these solvents with a wide range of diameters (dor and dmv), dipole moments (µ), viscosities (η), boiling temperatures (Tbp), and dielectric 10
constants (εr). Frequency-dependent capacitance (C)1Hz was measured in the experiments, and the solvent variable effects on the capacitance (C)1Hz are shown in Table 1. The authors found that the values of capacitance show linear relations with the logarithmic frequency lnf, regardless of the solvent type, i.e., C C 1Hz k ln f , where k is a constant depending on the solvents and (C)1Hz is the capacitance at f = 1 Hz with voltages between the two platinum electrodes. Therefore, the characteristic variable of the capacitance at 1 Hz was selected for further comparison of solvent effects on the capacitance. This observed capacitance should be the differential capacitance, which is potential dependent. As we know, the capacitance could be divided into Helmholtz layer capacitance (CH) and the diffusion layer capacitance (CD), given by 1/C = 1/CH + 1/CD. From Gouy-Chapman theory or classical DFT, the CH is constant at different potentials, but CD always depends on applied potentials. From their experimental results, it was found that 1/CH > 1/CD at platinum electrodes near the potential of zero charge, and the Helmholtz layer capacitance (CH) dominates the observed capacitance. The values of dmv could be obtained from the molar volume as: dmv = (M/ρNA) 1/3, where M is the molecular weight, ρ the density of solvent, and NA is the Avogadro constant. Furthermore, dor is the longest lengths of the oriented molecules, which could be estimated from ChemOffice. For an ideal dipole solvent, the dor is the distance between the positive center and negative center.
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Table 1. Solvent variables and EDL capacitance for different solvents(Hou et al., 2014)
acetonitrile
0.400
0.444
3.405
0.3436 128.308 37.5
11
2
aniline
0.640
0.534
1.541
3.8060 202.154
7.3
12
3
benzyl alcohol
0.536
0.557
1.622
5.4562 224.308
13
10
4
0.263
0.475
1.581
0.4448
8.93
15
0.369
0.491
3.811
1.9799 180.000 46.7
14
6
dichloromethane (DCM) dimethyl sulfoxide (DMSO) ethanol
0.316
0.46
1.784
1.1012 209.538 24.3
11
7
ethylene glycol
0.415
0.453
2.189
16
216.923
37
20
8
formamide
0.304
0.405
3.365
945
229.846
84
24
9
glycerol
0.543
0.495
1.743
9.3293 292.615 42.5
17
10
n-methylformamide (NMF) propylene carbonate (PC) tetrahydrofuran (THF) water
0.544
0.461
3.770
1.6811
16
0.614
0.521
4.865
2.4562 253.846
0.431
0.513
1.703
0.4544
0.158
0.312
1.865
0.8303 139.385
12 13
12
Tbp/℃
C/µFcm-2
1
11
µ/D
εr
Solvents
5
dor/nm dmv/nm
η/mPas
No.
91.385
204.000 36.7 69
113.538 7.58 78
9 11 35
2.2 Machine learning methods Many machine learning methods, such as artificial neural network (ANN) based models, rule-learning algorithms, tree-learning techniques, and traditional modeling approaches, such as linear regression, have been developed and widely used. Following are the five methods used in this work: Linear regression (LR) is a statistical tool for the regression of the variables in a subsample. The subsample is selected by iteratively removing the one with the smallest standardized coefficient until no improvement is observed in the error estimation. Support vector regression (SVR) is a regression technique with excellent performances in regression and time series prediction application, allowing categorization of the input data using separating lines or planes (Smola and Schölkopf, 2004). Nevertheless, it contains all the main features that characterize the maximum margin algorithm: a nonlinear function is learned by linear learning machine mapping into high dimensional kernel induced feature space. To determine the optimal separating line or set of separating lines in a high- or infinite-dimensional space, SVR employs an iterative training algorithm, such as the sequential minimal optimization (SMO) algorithm used in this study, to minimize an error function (Flake and Lawrence, 2002). Multilayer perceptron (MLP) is a widely used ANN model which generally includes an input layer, hidden layers and an output layer, shown in Figure 3 (Bishop, 1995). Each layer consists of nodes that are connected with a certain weight to all nodes in the next layer. Except for the input nodes, each node is a processing element with a nonlinear activation function, such as a sigmoid function that enables the network to compute complex nonlinear problems (Haykin, 1999). MLP utilizes the 'back propagation' technique to train the network, which means to change connection weights after each data point is processed while passing through the nodes in
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the network in order to minimize the error in the output compared to the expected result. The training process halts automatically when the generalization stops improving, which means that no further decrease occurs in the errors of cross-validation samples (Inal, 2006). The hiddeninput and hidden-output connection weights are saved and used to quantify the relative contribution (RC) of input variables to the predictive output of the employed MLP models (Garson, 1991).
Figure 3. Architecture of the MLP model used in this study. M5 Model Trees (M5P) is a model generalizing the concept of regression trees (Breiman et al., 1984) in which each branch represents a choice between a number of alternatives, and each node represents a decision. The basic idea is to split the input space into areas (subspaces) and to build a local specialized linear regression model into each of them. M5 Model Rules (M5R) is one of the rule-based learning models which could generate a decision list for regression problems by applying the separate-and-conquer algorithms. It finds a rule that explains the training examples, separates these examples and recursively conquers the other remaining examples by searching for more rules, and the searching process ends when there are
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no more examples remaining. This process could ensure that each variable in the original training dataset is covered by at least one rule.
2.3 Parameters optimization and performance measurement The algorithms of the five ML models were conducted in the machine learning open source package WEKA. Each method includes its own parameters that must be specified, and parameters for the SVR and MLP were obtained by using the CVParameterSelection module in WEKA, as both methods are sensitive to parameters. For M5P, M5R and LR, the WEKA could generate parameters itself. The correlation coefficient (R), mean absolute error (MAE), and root mean square error (RMSE) were used to evaluate the performance of different ML models in predicting EDL capacitance. n
R
y y y yˆ i 1
i
i
i
(1)
n y yˆ n
RMSE
y yˆ i
i 1
i
(2)
n n
MAE
y yˆ i 1
i
n
i
(3)
To determine which ML model provided the best approximate EDL capacitance, the calculated outputs from each model were compared with the experimental results. The 13-fold cross-validation methodology was used for the generalization of results. Following this process, the datasets were evenly split into 13 folds. The instances from 12 folds were used for training while the remaining fold was used for testing. The calculation process was repeated 13 times
15
using a different fold for testing at each cycle. The performance of each model was given by the average of the 169 (13x13) calculated RMSE.
2.4 Classical density functional theory To understand the molecular details of solvent effects on the EDL structure and capacitance, the CDFT (Amin et al., 2016; Evans, 1979; Lian et al., 2014; Robert et al., 2016; Wu and Li, 2007) was used to obtain the EDL capacitance of a charged electrode. When a room temperature ionic liquid (RTIL) was added into the electrolyte solvent, the ionic liquid (THAClO4) was represented as charged hard spheres, while the solvent was represented as a dipole. The details of the CDFT calculations could be found in previous papers. (Henderson et al., 2012; Jiang et al., 2012; Jiang and Wu, 2013; Lian et al., 2016a; Lian et al., 2016b; Lian et al., 2016c) Briefly, we obtained the surface charge densities at various electrical potentials. Given the number densities of ions and solvent molecules in the bulk, the system temperature (298 K), the pore size and the surface electrical potential, we solve for the one-dimensional density profiles of cations and anions
(z),
, as well as the solvent segments
(z),
,
across the slit pore by minimization of the grand potential
M R , a r F M R , a r
R M
where
1
M
M R dR a r a a r dr
(4)
a
k BT , R (r ,r ) represents two coordinates specifying the positions of two
segments in each solvent molecule, is the chemical potential of an ionic species, M is the chemical potential of the solvent, a r stands for the external potential for ions, M R is
16
the summation of the external potential for a solvent molecule, i.e., M R
i ,
i ri , and F
is the total intrinsic Helmholtz energy. The number densities of the positive and negative segments of the solvent are calculated from
r dR r r M R
(5)
r dR r r M R
(6)
The intrinsic Helmholtz energy F includes an ideal-gas contribution and an excess contribution due to intermolecular interactions F ex
F
òéëln R å òéëln
1ù û
M
a
r
a
R dR
M
1ù û
a
r dr
òV
b
R
F ex
M
R dR (7)
where Vb stands for the bonding potential of the solvent molecule. The detailed expression for each contribution and the numerical details can be retrieved from (Henderson et al., 2012; Jiang et al., 2012; Lian et al., 2018c). By evaluation of the Coulomb energy, we calculate the mean electrostatic potential (MEP) from the density distributions of the ions by using the Poisson equation
2 (r)
e
0
[ (r) (r)]
(8)
Equation (8) can be integrated with the boundary conditions that defined by the operation potential. The surface charge density Q is obtained from the condition of overall charge neutrality.
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3 Results and discussions 3.1 Overall performance of each model From the above discussion, capacitance is highly related to the molecular size of the solvent and the dielectric constant. According to the conventional Helmholtz model (Newman and Tiedemann, 1975; Yates et al., 1974),
C
1 d
(9)
where d is the size of counterions, the cations and anions used in this study are the same and the counterion effect could be ignored. It is easy to determine whether the EDL capacitance is more reliable with respect to the inverse of solvent size. Next, we use 1/dor, 1/dmv, εr, µ, η, and Tbp as inputs to ML models. An overview of the independent solvent variables used for each ML model is shown in Table 1. Each method has its own parameters that must be specified. The learning rate (L) and the number of nodes in the hidden layer (H) for MLP are 0.13 and 12, respectively. The complexity parameter (C) that determines the trade-off between the complexity of the SMO model and the tolerance of errors in SVR is optimized as 200. For M5P, M5R and LR, these ML models could generate parameters themselves. Several evaluating parameters, such as root mean square error (RMSE), were used to evaluate the overall performance of 5 ML models in predicting EDL capacitance with different solvents.
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Table 2. Correlation coefficient (R), mean absolute error (MAE), root mean squared error (RMSA) of each model.
MLP
SVR
M5R
M5P
LR
R
0.729
0.661
0.875
0.890
0.425
MAE
3.490
4.053
2.645
2.510
4.558
RMSE
5.056
5.990
3.232
3.158
7.362
To evaluate the performance of each ML model, the EDL capacitance of each solvent predicted by the ML models is compared with experimental data, as shown in Figure 4. The estimated R, MAE, and RMSE for each model are presented in Table 2. It is easy to determine that the ML methods are superior to the traditional LR method. Deep learning model, such as MLP, M5R, and M5P, exhibiting better performances than the SVR model, which has the highest RMSE. According to the RMSA, MAE, and R values, the overall performance of the algorithms for predicting solvent the effect on EDL capacitance can be ranked as follows: M5P > M5R > MLP > SVR > LR.
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Figure 4. Statistical performance of each ML model by comparing the EDL capacitance from ML models (y-axis) and the experimental capacitance (x-axis). The dots for certain solvents which lie on the dashed line indicate that the predicted capacitance and input (experimental) capacitance are the same. Here, we discuss different models. The RMSE of MLP is slightly higher than that of M5R and M5P. The relative contribution of solvent variables to the EDL capacitance from MLP is shown in Figure 5. The size of solvent molecule (both dor and dmv) and the dielectric constant (εr) are the most important variables, with weights of 37%, 22%, and 16%, respectively. The dipole moments (µ), viscosities (η), and boiling temperatures (Tbp) with weights of 13%, 10%, and 2%, respectively, show smaller impacts on the capacitive performances than the sizes and dielectric constants of solvents.
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Figure 5. The relative contribution of solvent variables to the predictive EDL capacitance of the MLP model. The performances of M5P and M5R differed slightly: the differences in the R, RMSE and MAE values between them were less than 1.7%, 5% and 2%, respectively. However, M5P was more preferable than M5R because M5P does not find the optimal parameters by trial and error, and it generates simpler rules which we could understand more easily. The rules of the M5P models are shown as:
C
11.99 2.481 0.128 r 8.7165 d mv
(10)
From the above rules obtained by the M5P model, it was noted that the solvent molecule dmv, the dielectric constant (εr), and the dipole moments (µ) are the three most important variables, since they were highly correlated with the EDL capacitance. Other variables such as viscosity and boiling temperatures have less important effects on the EDL capacitance. However, questions remain. What is the reason for the solvent effects on the EDL capacitance? Why does the counterion have little effect on the EDL capacitance for the solvent
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system? It is very interesting to compare the differences between the Helmholtz model and the solvent effect rules from ML. As shown in Figure 6, the results from the ML models show that the EDL capacitance is primarily related to the size, dipole moment, and the dielectric constant of the solvent, which is totally different from the conventional Helmholtz model (in which case the capacitance is only related to the size of counter-ion and dielectric constant). What are the reasons for and physical insights into these differences between the rules obtained by the ML and conventional model? To find the physical pictures of the solvent effects, CDFT was used to study the structures of solvent and ions near charged electrodes.
Figure 6. Comparison between the traditional Helmholtz model and the model from the ML model. 3.2 Molecular insights into the solvent effects by CDFT We could obtain the relationship between the EDL capacitance and the solvent variables, which will help us to optimize the performance of EDL capacitor devices. However, the molecular details and understanding for the solvent effects are not well understood. To better understand the microscopic insights of the ionic liquid and solvent near the charged electrode surface, the CDFT calculation is used to study the EDL structure. In CDFT calculation, we use a dipole model to represent solvents, which consists of two charged segments of equal size, and charged hard spheres to represent cations and anions. The charges for the cation and anion are +1
22
and -1, while the δ charges for the positive and negative segments are +0.3 and -0.3. The diameters for dipole segments and cations/anions are 0.3 nm and 0.5 nm, respectively. The surface potential for the electrode is fixed as -0.1 V, the reduced density for anion and cation are set as 0.01, and the reduced density for the solvent is set as 0.50. Figure 7 shows the reduced density profiles of the cation (red sphere), anion (blue sphere), the positive segment (pink) of the dipolar solvent, and the negative segment (green) of the dipolar solvent near the negative-charged electrode surface. It is easily observed that the electric double layers (EDL) include not only a layer of the counterions (cations) but also a layer of the aligned solvent molecules. As far as we know, there is only a counterion layer near the charged surface for traditional EDL, and the first counterion dominates the EDL structure and properties. Interestingly, the first layer includes a positive segments layer of solvent, and the density of this layer is significantly greater than the density of the counterion layer.
Figure 7. Calculated EDL structure of the ionic liquid electrolytes with solvents from CDFT.
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From the density profile above, we could find that the first solvent layer dominates the EDL, since it acts in the role of the Helmholtz layer (first counterion layer) in traditional EDL, and it is reasonable that the EDL capacitance is directly related to the inverse of solvent size rather than that of counterions. Similarly, for the EDL structures shown in Figure 6 at low potential, the solvent will still align near the electrode surfaces within the potential window of 3.0 V from the classical DFT in a previous report (Jiang and Wu, 2014). The aligned solvents near the surfaces will dominate the Helmholtz layer. 3.3 Prediction of Solvent effects on the Capacitance With the relationship between the EDL capacitance and the solvent variables above, we could enhance the performance of EDL capacitor devices by screening solvents. Figure 8 shows the capacitance of different solvents calculated by equation (10) with the same conditions in the mentioned experiments (Hou et al., 2014).
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Figure 8. Prediction of the solvent effect on the EDL capacitances.
25
Interestingly, our predicted results indicate that there exists the optimal solvent that yields improved capacitive performance. Our work illustrates the rich behavior of the organic electrolytes and suggests new experiments for selecting organic electrolytes for EDLCs. However, the predicted capacitances with different solvents under a specific condition cannot capture the entire picture of solvent effects, and further studies with different conditions (salt types, concentration, applied potential, and electrodes) should be addressed by ML.
4 Conclusion In summary, we investigate the influence of solvent effects on the electric double layer (EDL) capacitance by using machine learning (ML) models. The experimental data for different solvent variables are collected as the inputs for different ML models. The performance of the different ML models is ranked as follows: M5P > M5R > MLP > SVR > LR. Among these models, the M5P model shows the best performance because it is simple and provides wellunderstood rules. ML model results could successfully identify the relative importance of each solvent variable on the EDL capacitance: the solvent molecule and the dielectric constant are the most important variables, since they were highly correlated with the EDL capacitance, while the dipole moments, viscosity, and boiling temperature of the solvent have little effect on the EDL capacitance. The rules provided by the ML models could also be applied to predict the EDL capacitance. Furthermore, the CDFT calculation is also used to enable a better understanding of the rules obtained from ML models. This sequential procedure consisting of machine learning, experimentations and molecular theory modeling could potentially be useful for predicting and enhancing the performance of EDLCs. This present method draws conclusions from experimental and calculated data by using machine learning. We hope that this method could
26
inspire further efforts toward the application of machine learning in electrochemistry and chemical engineering.
Acknowledgements This work was sponsored by the National Natural Science Foundation of China (No. 91534202, 91834301, 21808055), National Natural Science Foundation of China for Innovative Research Groups (No. 51621002), the 111 Project of China (No. B08021), the China Postdoctoral Science Foundation (2017M620137), Shanghai Sailing Program (18YF1405400), Fundamental Research Funds for the Central Universities (WJ1814016), and the National Postdoctoral Program for Innovative Talents (BX201700076).
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Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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S0009-2509(19)30252-0 https://doi.org/10.1016/j.ces.2019.03.037 CES 14860
To appear in:
Chemical Engineering Science
Received Date: Revised Date: Accepted Date:
27 September 2018 10 March 2019 15 March 2019
Please cite this article as: H. Su, C. Lian, J. Liu, H. Liu, Machine Learning Models for Solvent Effects on Electric Double Layer Capacitance, Chemical Engineering Science (2019), doi: https://doi.org/10.1016/j.ces.2019.03.037
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Machine Learning Models for Solvent Effects on Electric Double Layer Capacitance
Haiping Su1, Cheng Lian1*, Jichuan Liu2, Honglai Liu1* 1
State Key Laboratory of Chemical Engineering, and Department of Chemistry, East China University of Science and Technology, Shanghai, 200237, P. R. China 2
*
Department of Chemical Engineering, University College London, London WC1E 7JE
Correspondence concerning this article should be addressed: [email protected] , or [email protected] .
1
Highlights The solvent effects on electrochemical capacitances were investigated by combining experimental observation and machine learning (ML) models. The classical density functional theory (CDFT) could be used for better understanding of the solvent effects derived from the ML models. The experimental-CDFT-ML combined method provides a useful benchmark for computational and experimental screening of new materials for electrochemical applications.
2
Abstract The role of solvent molecules in electrolytes for supercapacitors, representing a fertile ground for improving the capacitive performance of supercapacitors, is complicated and has not been well understood. Here, a combined method is applied to study the solvent effects on capacitive performance. To identify the relative importance of each solvent variable to the capacitance, five machine learning (ML) models were tested for a set of collected experimental data, including support vector regression (SVR), multilayer perceptions (MLP), M5 model tree (M5P), M5 rule (M5R) and linear regression (LR). The performances of these ML models are ranked as follows: M5P > M5R > MLP > SVR > LR. Moreover, the classical density functional theory (CDFT) is introduced to yield more microscopic insights into the conclusion derived from ML models. This method, by combining machine learning, experimental and molecular modeling, could potentially be useful for predicting and enhancing the performance of electric double layer capacitors (EDLCs). Keywords: Solvent effects, Electric double layer capacitance, Machine learning, Classical density functional theory
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Graphical Abstract
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1 Introduction Electric double layer capacitors (EDLCs), also known as supercapacitors, present remarkable advantages including rapid charging/discharging kinetics, high power densities, and long cycling lifespans (Dou et al., 2017; Lian et al., 2018a; Simon and Gogotsi, 2012; Su et al., 2017; Zhan et al., 2017). EDLCs store electrical energy by physical adsorption of ions onto the surfaces of porous electrodes, and the capacitance of supercapacitors is strongly reliant on the nonredox ionic behavior inside the microspores of porous electrodes (Burt et al., 2014; Di et al., 2018; Fedorov and Kornyshev, 2014; Frackowiak et al., 2013; Jiang and Wu, 2013; Shao et al., 2015; Vatamanu and Bedrov, 2015; Zhong et al., 2015). The energy density per surface area of an EDLC device is proportional to its capacity and the square of its operating potential window (OPW), E CV 2 2 , where C is the capacitance per surface area and V the maximum value of the OPW (Fedorov and Kornyshev, 2008; Feng et al., 2015; Kornyshev, 2007; Pilon et al., 2015). Accordingly, the energy density of an EDLC can be enhanced by improving the capacitance of the electric double layer (EDL) and expanding the OPW through various methods. Among them, changing the solvent addition within the electrolyte is treated as an effective strategy. Many experimental efforts have been devoted to study the solvent effects on the performance of EDLC. Chimiola and Largeot (Chmiola et al., 2006; Largeot et al., 2008) reported that an “anomalous” increase of the carbon capacitance occurred when the ionic liquid with proper size was added into the organic electrolyte. Morita et al. (Morita et al., 1992) found that improvements in the capacitance and the electrolytic conductivity were observed by mixing organic solvents. Intensive studies conducted by Janes et al. (Arulepp et al., 2004; Jänes et al., 2004; Lust et al., 2004) also demonstrated the influences of organic solvent on solution resistance, EDL charging dynamics, and EDL capacitance. The investigation by Kim and
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coworkers (Kim et al., 2010) revealed that small solvent molecules could enter into the nanopores, resulting in enhanced capacitance by shifting the potential of zero charge (PZC). Liu et al. (Liu et al., 2012) presented improved capacitive performances by optimizing the ratio of EMIM-BF4 and acetonitrile in the electrolyte. Decaux et al. (Decaux et al., 2014) demonstrated that the polarizing salt ions in solvents do not completely lose the solvation shell and that the performance of EDLC depends greatly on the interaction between the solvents and ions. Apart from the experimental works, molecular theory and molecular simulation method have been applied to study the solvent effects on the capacitive behavior of EDLCs. Cummings et al. (Li et al., 2014; Uysal et al., 2014) discovered that the solvent addition (dicationic ionic liquid) contribute greatly to the performance enhancement by the molecular dynamics (MD) simulation method. Furthermore, they found that the solvent polarity of room temperature ionic liquid (RTIL) plays an important role on the capacitive behavior (Osti et al., 2016) and that the humidity of the solvent also affects the performance to a great extent (Osti et al., 2017). A capacitance maximum was found in RTIL through dilution with organic solvents from the MD simulation (Bozym et al., 2015; Uralcan et al., 2016). Bedrov et al. (Li et al., 2015) used MD to study the effect of organic solvents on the solvation and dynamics of Li+ ions in RTIL electrolytes, and the results showed that the addition of solvent could accelerate the self-diffusion coefficient of Li+ ions and the conductivity of electrolytes, ultimately leading to an enhanced performance. Theoretically, the molecular-scale solvent effects are not well considered in the traditional EDL models such as the Helmholtz model and the Gouy-Chapman-Stern theory (GCS). Kondrat and Kornyshev (Kondrat and Kornyshev, 2016; Rochester et al., 2016) developed a statistical theory to describe the charging behavior of cations, anions and solvent
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molecules in narrow pores. They found that the solvent addition such as “ionophobic agents” in the electrolyte could change the charging mechanisms in EDLC, ultimately leading to the improvement of energy density. Wu and coworkers applied CDFT to study the solvent polarity effect on the EDL capacitance (Jiang and Wu, 2014), and they also found that the organic solvent effect differs when the electrode varies (Jiang et al., 2012; Jiang and Wu, 2013; Lian et al., 2018b) and that the solvent-like additions (neutral molecular, or dipole) could also affect the capacitive behavior (Lian et al., 2017; Liu et al., 2017; Liu and Wu, 2016). Although it has been widely known that the EDL capacitance is significantly affected by solvent properties, our understanding of the interactions between solvent properties and capacitive performance is still limited in the following capacities: (i) experimentally, it is difficult to elucidate the solvent effects in practical electrochemical tests, and the nanoscale electrode-electrolyte interface behaviors could only be speculated from the results of electrochemical experiments; (ii) theoretically, traditional EDL models only provide a mathematical equation or a statistical regression, and it is difficult to describe solvent effects by providing a dielectric constant (εr); and (iii) molecular theory and simulation tools are considered as effective tools to investigate these nanoscale problems, but they are very time-consuming for large systems. It is also a significant challenge for molecular simulation to bridge the gap between microscopic details and macroscopic properties. Although different solvent parameters have been studied by different researchers, they often consider one factor for each experiment or calculation, which may ignore the coupled solvent effect on the EDL capacitance. Hence, a more comprehensive method is needed for better understanding the role of the solvent on supercapacitors.
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Machine learning (ML) technique, a data-based method, has been treated as an alternative tool (supplementing the theory, experiment, and simulation methods shown in Figure 1) to solve many practical problems. Recently, several chemical engineering issues have been efficiently studied by ML methods (Chakraborty et al., 2014; Sui et al., 2016; Turan et al., 2011; Wu et al., 2013), which inspired us to investigate the solvent effects on the EDL capacitance thorough the ML method.
Figure 1. Scientific understanding involves the formulation of theory, experimental observation, modeling/simulation, and machine learning methods. These four methods are closely related and supplement each other. In this work, the experimental data of different solvents were collected with the purpose of linking the physicochemical properties of solvents and the EDL capacitive behavior. Using these empirical data as inputs for different ML models, the performance of each ML model was evaluated to produce the best model for this issue. Here, ML models were used as black boxes to elucidate the connections and rules between the capacitance and several solvent properties. Then, the CDFT was applied to yield microscopic insights into the connections and rules built by ML
8
models. Hence, the capacitive performance of EDLCs with ionic liquids in different solvents would be accurately predicted by the mentioned method.
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2. Methods and models To predict the EDL capacitance without further experimental measurements, a sequential procedure is used, as shown in Figure 2. In the first step, the experimental or calculated capacitance values with different solvents were collected as data source. Then, the obtained empirical data were applied to assess the performance of five different machine learning models. Finally, the best one was chosen to predict the EDL capacitance.
Figure 2. Illustration of the sequential approach used for predicting EDL capacitance by machine learning models. 2.1 Data collection Several solvents including acetonitrile, aniline, benzyl alcohol, dichloromethane (DCM), dimethyl sulfoxide (DMSO), ethanol, ethylene glycol, formamide, glycerol, n-methylformamide (NMF), propylene carbonate (PC), tetrahydrofuran (THF), and water were gathered from Hou’s work (Hou et al., 2014), in which experimentally controllable solvent variables for EDL capacitance were obtained by a dual parallel platinum electrodes system in the polarized potential domain. The authors carefully selected these solvents with a wide range of diameters (dor and dmv), dipole moments (µ), viscosities (η), boiling temperatures (Tbp), and dielectric 10
constants (εr). Frequency-dependent capacitance (C)1Hz was measured in the experiments, and the solvent variable effects on the capacitance (C)1Hz are shown in Table 1. The authors found that the values of capacitance show linear relations with the logarithmic frequency lnf, regardless of the solvent type, i.e., C C 1Hz k ln f , where k is a constant depending on the solvents and (C)1Hz is the capacitance at f = 1 Hz with voltages between the two platinum electrodes. Therefore, the characteristic variable of the capacitance at 1 Hz was selected for further comparison of solvent effects on the capacitance. This observed capacitance should be the differential capacitance, which is potential dependent. As we know, the capacitance could be divided into Helmholtz layer capacitance (CH) and the diffusion layer capacitance (CD), given by 1/C = 1/CH + 1/CD. From Gouy-Chapman theory or classical DFT, the CH is constant at different potentials, but CD always depends on applied potentials. From their experimental results, it was found that 1/CH > 1/CD at platinum electrodes near the potential of zero charge, and the Helmholtz layer capacitance (CH) dominates the observed capacitance. The values of dmv could be obtained from the molar volume as: dmv = (M/ρNA) 1/3, where M is the molecular weight, ρ the density of solvent, and NA is the Avogadro constant. Furthermore, dor is the longest lengths of the oriented molecules, which could be estimated from ChemOffice. For an ideal dipole solvent, the dor is the distance between the positive center and negative center.
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Table 1. Solvent variables and EDL capacitance for different solvents(Hou et al., 2014)
acetonitrile
0.400
0.444
3.405
0.3436 128.308 37.5
11
2
aniline
0.640
0.534
1.541
3.8060 202.154
7.3
12
3
benzyl alcohol
0.536
0.557
1.622
5.4562 224.308
13
10
4
0.263
0.475
1.581
0.4448
8.93
15
0.369
0.491
3.811
1.9799 180.000 46.7
14
6
dichloromethane (DCM) dimethyl sulfoxide (DMSO) ethanol
0.316
0.46
1.784
1.1012 209.538 24.3
11
7
ethylene glycol
0.415
0.453
2.189
16
216.923
37
20
8
formamide
0.304
0.405
3.365
945
229.846
84
24
9
glycerol
0.543
0.495
1.743
9.3293 292.615 42.5
17
10
n-methylformamide (NMF) propylene carbonate (PC) tetrahydrofuran (THF) water
0.544
0.461
3.770
1.6811
16
0.614
0.521
4.865
2.4562 253.846
0.431
0.513
1.703
0.4544
0.158
0.312
1.865
0.8303 139.385
12 13
12
Tbp/℃
C/µFcm-2
1
11
µ/D
εr
Solvents
5
dor/nm dmv/nm
η/mPas
No.
91.385
204.000 36.7 69
113.538 7.58 78
9 11 35
2.2 Machine learning methods Many machine learning methods, such as artificial neural network (ANN) based models, rule-learning algorithms, tree-learning techniques, and traditional modeling approaches, such as linear regression, have been developed and widely used. Following are the five methods used in this work: Linear regression (LR) is a statistical tool for the regression of the variables in a subsample. The subsample is selected by iteratively removing the one with the smallest standardized coefficient until no improvement is observed in the error estimation. Support vector regression (SVR) is a regression technique with excellent performances in regression and time series prediction application, allowing categorization of the input data using separating lines or planes (Smola and Schölkopf, 2004). Nevertheless, it contains all the main features that characterize the maximum margin algorithm: a nonlinear function is learned by linear learning machine mapping into high dimensional kernel induced feature space. To determine the optimal separating line or set of separating lines in a high- or infinite-dimensional space, SVR employs an iterative training algorithm, such as the sequential minimal optimization (SMO) algorithm used in this study, to minimize an error function (Flake and Lawrence, 2002). Multilayer perceptron (MLP) is a widely used ANN model which generally includes an input layer, hidden layers and an output layer, shown in Figure 3 (Bishop, 1995). Each layer consists of nodes that are connected with a certain weight to all nodes in the next layer. Except for the input nodes, each node is a processing element with a nonlinear activation function, such as a sigmoid function that enables the network to compute complex nonlinear problems (Haykin, 1999). MLP utilizes the 'back propagation' technique to train the network, which means to change connection weights after each data point is processed while passing through the nodes in
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the network in order to minimize the error in the output compared to the expected result. The training process halts automatically when the generalization stops improving, which means that no further decrease occurs in the errors of cross-validation samples (Inal, 2006). The hiddeninput and hidden-output connection weights are saved and used to quantify the relative contribution (RC) of input variables to the predictive output of the employed MLP models (Garson, 1991).
Figure 3. Architecture of the MLP model used in this study. M5 Model Trees (M5P) is a model generalizing the concept of regression trees (Breiman et al., 1984) in which each branch represents a choice between a number of alternatives, and each node represents a decision. The basic idea is to split the input space into areas (subspaces) and to build a local specialized linear regression model into each of them. M5 Model Rules (M5R) is one of the rule-based learning models which could generate a decision list for regression problems by applying the separate-and-conquer algorithms. It finds a rule that explains the training examples, separates these examples and recursively conquers the other remaining examples by searching for more rules, and the searching process ends when there are
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no more examples remaining. This process could ensure that each variable in the original training dataset is covered by at least one rule.
2.3 Parameters optimization and performance measurement The algorithms of the five ML models were conducted in the machine learning open source package WEKA. Each method includes its own parameters that must be specified, and parameters for the SVR and MLP were obtained by using the CVParameterSelection module in WEKA, as both methods are sensitive to parameters. For M5P, M5R and LR, the WEKA could generate parameters itself. The correlation coefficient (R), mean absolute error (MAE), and root mean square error (RMSE) were used to evaluate the performance of different ML models in predicting EDL capacitance. n
R
y y y yˆ i 1
i
i
i
(1)
n y yˆ n
RMSE
y yˆ i
i 1
i
(2)
n n
MAE
y yˆ i 1
i
n
i
(3)
To determine which ML model provided the best approximate EDL capacitance, the calculated outputs from each model were compared with the experimental results. The 13-fold cross-validation methodology was used for the generalization of results. Following this process, the datasets were evenly split into 13 folds. The instances from 12 folds were used for training while the remaining fold was used for testing. The calculation process was repeated 13 times
15
using a different fold for testing at each cycle. The performance of each model was given by the average of the 169 (13x13) calculated RMSE.
2.4 Classical density functional theory To understand the molecular details of solvent effects on the EDL structure and capacitance, the CDFT (Amin et al., 2016; Evans, 1979; Lian et al., 2014; Robert et al., 2016; Wu and Li, 2007) was used to obtain the EDL capacitance of a charged electrode. When a room temperature ionic liquid (RTIL) was added into the electrolyte solvent, the ionic liquid (THAClO4) was represented as charged hard spheres, while the solvent was represented as a dipole. The details of the CDFT calculations could be found in previous papers. (Henderson et al., 2012; Jiang et al., 2012; Jiang and Wu, 2013; Lian et al., 2016a; Lian et al., 2016b; Lian et al., 2016c) Briefly, we obtained the surface charge densities at various electrical potentials. Given the number densities of ions and solvent molecules in the bulk, the system temperature (298 K), the pore size and the surface electrical potential, we solve for the one-dimensional density profiles of cations and anions
(z),
, as well as the solvent segments
(z),
,
across the slit pore by minimization of the grand potential
M R , a r F M R , a r
R M
where
1
M
M R dR a r a a r dr
(4)
a
k BT , R (r ,r ) represents two coordinates specifying the positions of two
segments in each solvent molecule, is the chemical potential of an ionic species, M is the chemical potential of the solvent, a r stands for the external potential for ions, M R is
16
the summation of the external potential for a solvent molecule, i.e., M R
i ,
i ri , and F
is the total intrinsic Helmholtz energy. The number densities of the positive and negative segments of the solvent are calculated from
r dR r r M R
(5)
r dR r r M R
(6)
The intrinsic Helmholtz energy F includes an ideal-gas contribution and an excess contribution due to intermolecular interactions F ex
F
òéëln R å òéëln
1ù û
M
a
r
a
R dR
M
1ù û
a
r dr
òV
b
R
F ex
M
R dR (7)
where Vb stands for the bonding potential of the solvent molecule. The detailed expression for each contribution and the numerical details can be retrieved from (Henderson et al., 2012; Jiang et al., 2012; Lian et al., 2018c). By evaluation of the Coulomb energy, we calculate the mean electrostatic potential (MEP) from the density distributions of the ions by using the Poisson equation
2 (r)
e
0
[ (r) (r)]
(8)
Equation (8) can be integrated with the boundary conditions that defined by the operation potential. The surface charge density Q is obtained from the condition of overall charge neutrality.
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3 Results and discussions 3.1 Overall performance of each model From the above discussion, capacitance is highly related to the molecular size of the solvent and the dielectric constant. According to the conventional Helmholtz model (Newman and Tiedemann, 1975; Yates et al., 1974),
C
1 d
(9)
where d is the size of counterions, the cations and anions used in this study are the same and the counterion effect could be ignored. It is easy to determine whether the EDL capacitance is more reliable with respect to the inverse of solvent size. Next, we use 1/dor, 1/dmv, εr, µ, η, and Tbp as inputs to ML models. An overview of the independent solvent variables used for each ML model is shown in Table 1. Each method has its own parameters that must be specified. The learning rate (L) and the number of nodes in the hidden layer (H) for MLP are 0.13 and 12, respectively. The complexity parameter (C) that determines the trade-off between the complexity of the SMO model and the tolerance of errors in SVR is optimized as 200. For M5P, M5R and LR, these ML models could generate parameters themselves. Several evaluating parameters, such as root mean square error (RMSE), were used to evaluate the overall performance of 5 ML models in predicting EDL capacitance with different solvents.
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Table 2. Correlation coefficient (R), mean absolute error (MAE), root mean squared error (RMSA) of each model.
MLP
SVR
M5R
M5P
LR
R
0.729
0.661
0.875
0.890
0.425
MAE
3.490
4.053
2.645
2.510
4.558
RMSE
5.056
5.990
3.232
3.158
7.362
To evaluate the performance of each ML model, the EDL capacitance of each solvent predicted by the ML models is compared with experimental data, as shown in Figure 4. The estimated R, MAE, and RMSE for each model are presented in Table 2. It is easy to determine that the ML methods are superior to the traditional LR method. Deep learning model, such as MLP, M5R, and M5P, exhibiting better performances than the SVR model, which has the highest RMSE. According to the RMSA, MAE, and R values, the overall performance of the algorithms for predicting solvent the effect on EDL capacitance can be ranked as follows: M5P > M5R > MLP > SVR > LR.
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Figure 4. Statistical performance of each ML model by comparing the EDL capacitance from ML models (y-axis) and the experimental capacitance (x-axis). The dots for certain solvents which lie on the dashed line indicate that the predicted capacitance and input (experimental) capacitance are the same. Here, we discuss different models. The RMSE of MLP is slightly higher than that of M5R and M5P. The relative contribution of solvent variables to the EDL capacitance from MLP is shown in Figure 5. The size of solvent molecule (both dor and dmv) and the dielectric constant (εr) are the most important variables, with weights of 37%, 22%, and 16%, respectively. The dipole moments (µ), viscosities (η), and boiling temperatures (Tbp) with weights of 13%, 10%, and 2%, respectively, show smaller impacts on the capacitive performances than the sizes and dielectric constants of solvents.
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Figure 5. The relative contribution of solvent variables to the predictive EDL capacitance of the MLP model. The performances of M5P and M5R differed slightly: the differences in the R, RMSE and MAE values between them were less than 1.7%, 5% and 2%, respectively. However, M5P was more preferable than M5R because M5P does not find the optimal parameters by trial and error, and it generates simpler rules which we could understand more easily. The rules of the M5P models are shown as:
C
11.99 2.481 0.128 r 8.7165 d mv
(10)
From the above rules obtained by the M5P model, it was noted that the solvent molecule dmv, the dielectric constant (εr), and the dipole moments (µ) are the three most important variables, since they were highly correlated with the EDL capacitance. Other variables such as viscosity and boiling temperatures have less important effects on the EDL capacitance. However, questions remain. What is the reason for the solvent effects on the EDL capacitance? Why does the counterion have little effect on the EDL capacitance for the solvent
21
system? It is very interesting to compare the differences between the Helmholtz model and the solvent effect rules from ML. As shown in Figure 6, the results from the ML models show that the EDL capacitance is primarily related to the size, dipole moment, and the dielectric constant of the solvent, which is totally different from the conventional Helmholtz model (in which case the capacitance is only related to the size of counter-ion and dielectric constant). What are the reasons for and physical insights into these differences between the rules obtained by the ML and conventional model? To find the physical pictures of the solvent effects, CDFT was used to study the structures of solvent and ions near charged electrodes.
Figure 6. Comparison between the traditional Helmholtz model and the model from the ML model. 3.2 Molecular insights into the solvent effects by CDFT We could obtain the relationship between the EDL capacitance and the solvent variables, which will help us to optimize the performance of EDL capacitor devices. However, the molecular details and understanding for the solvent effects are not well understood. To better understand the microscopic insights of the ionic liquid and solvent near the charged electrode surface, the CDFT calculation is used to study the EDL structure. In CDFT calculation, we use a dipole model to represent solvents, which consists of two charged segments of equal size, and charged hard spheres to represent cations and anions. The charges for the cation and anion are +1
22
and -1, while the δ charges for the positive and negative segments are +0.3 and -0.3. The diameters for dipole segments and cations/anions are 0.3 nm and 0.5 nm, respectively. The surface potential for the electrode is fixed as -0.1 V, the reduced density for anion and cation are set as 0.01, and the reduced density for the solvent is set as 0.50. Figure 7 shows the reduced density profiles of the cation (red sphere), anion (blue sphere), the positive segment (pink) of the dipolar solvent, and the negative segment (green) of the dipolar solvent near the negative-charged electrode surface. It is easily observed that the electric double layers (EDL) include not only a layer of the counterions (cations) but also a layer of the aligned solvent molecules. As far as we know, there is only a counterion layer near the charged surface for traditional EDL, and the first counterion dominates the EDL structure and properties. Interestingly, the first layer includes a positive segments layer of solvent, and the density of this layer is significantly greater than the density of the counterion layer.
Figure 7. Calculated EDL structure of the ionic liquid electrolytes with solvents from CDFT.
23
From the density profile above, we could find that the first solvent layer dominates the EDL, since it acts in the role of the Helmholtz layer (first counterion layer) in traditional EDL, and it is reasonable that the EDL capacitance is directly related to the inverse of solvent size rather than that of counterions. Similarly, for the EDL structures shown in Figure 6 at low potential, the solvent will still align near the electrode surfaces within the potential window of 3.0 V from the classical DFT in a previous report (Jiang and Wu, 2014). The aligned solvents near the surfaces will dominate the Helmholtz layer. 3.3 Prediction of Solvent effects on the Capacitance With the relationship between the EDL capacitance and the solvent variables above, we could enhance the performance of EDL capacitor devices by screening solvents. Figure 8 shows the capacitance of different solvents calculated by equation (10) with the same conditions in the mentioned experiments (Hou et al., 2014).
24
Figure 8. Prediction of the solvent effect on the EDL capacitances.
25
Interestingly, our predicted results indicate that there exists the optimal solvent that yields improved capacitive performance. Our work illustrates the rich behavior of the organic electrolytes and suggests new experiments for selecting organic electrolytes for EDLCs. However, the predicted capacitances with different solvents under a specific condition cannot capture the entire picture of solvent effects, and further studies with different conditions (salt types, concentration, applied potential, and electrodes) should be addressed by ML.
4 Conclusion In summary, we investigate the influence of solvent effects on the electric double layer (EDL) capacitance by using machine learning (ML) models. The experimental data for different solvent variables are collected as the inputs for different ML models. The performance of the different ML models is ranked as follows: M5P > M5R > MLP > SVR > LR. Among these models, the M5P model shows the best performance because it is simple and provides wellunderstood rules. ML model results could successfully identify the relative importance of each solvent variable on the EDL capacitance: the solvent molecule and the dielectric constant are the most important variables, since they were highly correlated with the EDL capacitance, while the dipole moments, viscosity, and boiling temperature of the solvent have little effect on the EDL capacitance. The rules provided by the ML models could also be applied to predict the EDL capacitance. Furthermore, the CDFT calculation is also used to enable a better understanding of the rules obtained from ML models. This sequential procedure consisting of machine learning, experimentations and molecular theory modeling could potentially be useful for predicting and enhancing the performance of EDLCs. This present method draws conclusions from experimental and calculated data by using machine learning. We hope that this method could
26
inspire further efforts toward the application of machine learning in electrochemistry and chemical engineering.
Acknowledgements This work was sponsored by the National Natural Science Foundation of China (No. 91534202, 91834301, 21808055), National Natural Science Foundation of China for Innovative Research Groups (No. 51621002), the 111 Project of China (No. B08021), the China Postdoctoral Science Foundation (2017M620137), Shanghai Sailing Program (18YF1405400), Fundamental Research Funds for the Central Universities (WJ1814016), and the National Postdoctoral Program for Innovative Talents (BX201700076).
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Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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