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Prediction of ionic liquids heat capacity at variable temperatures based on the norm indexes
Fluid Phase Equilibria 500 (2019) 112260
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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Prediction of ionic liquids heat capacity at variable temperatures based on the norm indexes Wensi He a, Fangyou Yan a, *, Qingzhu Jia b, Shuqian Xia c, Qiang Wang a a
School of Chemical Engineering and Material Science, Tianjin University of Science and Technology, 13St. 29, TEDA, 300457, Tianjin, PR China School of Marine and Environmental Science, Tianjin University of Science and Technology, 13St. 29, TEDA, 300457, Tianjin, PR China c Key Laboratory for Green Chemical Technology of the State Education Ministry, Collaborative Innovation Center of Chemical Science and Engineering (Tianjin), School of Chemical Engineering and Technology, Tianjin University, 300072, Tianjin, PR China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 May 2019 Received in revised form 20 July 2019 Accepted 1 August 2019 Available online 2 August 2019
The heat capacity of ILs is essential in the chemical engineering and design applications. In this study, the dataset consisting of 5964 heat capacity data points within a wide temperature range (191.13e550.00 K) of 185 ILs was collated. Based on the concept of norm index, a serial of norm indexes were derived and adopted to develop the QSPR model. Here, the temperature parameters were adjusted on the basis of the molecular structure information. The linear model is precise for estimating heat capacity of ILs at variable temperatures with square of correlation coefficient R2 ¼ 0.996, average absolute relative deviation AARD ¼ 2.49% and the root mean square error RMSE ¼ 19.43. Besides, the good results of multiple verification methods further illustrate the stability and reliability of this work. Consequently, this normindex based work could provide a correlation method to describe the heat capacity of different structures ILs at variable temperatures. © 2019 Elsevier B.V. All rights reserved.
Keywords: Heat capacity Ionic liquids Norm indexes QSPR
1. Introduction Recently, the study of ionic liquids (ILs), in both academic research and industry fields, has attracted worldwide attention. Because of the unique properties of ILs, such as easy circulation, high solubility to polar/apolar compounds, good thermal and chemical stability, etc., ILs are widely applied in many areas of science and engineering, especially in the fields of electrochemistry [1,2], catalysis [3,4], gas separations [5,6], biotechnology [7,8] and other engineering technology [9e12]. Heat capacity (CP), one of the elemental thermodynamic properties of ILs, is defined as the partial derivative of the enthalpy with respect to temperature under constant pressure. Heat capacity is often required in the thermodynamics calculation, and is also the indispensable parameter in the chemical process designing. Based on the heat capacity, some other thermodynamic properties, such as entropy, enthalpy, Gibbs free energy, Helmholtz free energy, etc. [13], can be achieved. Similarly, the heat capacity data will bring a profound influence on the unit operation in chemical engineering like heat exchangers, distillation column and so on [14]. Various
* Corresponding author. E-mail address: [email protected] (F. Yan). https://doi.org/10.1016/j.fluid.2019.112260 0378-3812/© 2019 Elsevier B.V. All rights reserved.
measurement methods are available for obtaining the ILs heat capacity, such as high-precision heat capacity drop calorimetry [15], adiabatic calorimetry [16], differential scanning calorimetry [17,18] and others [19]. But in fact, there is still a lack of data on the heat capacity of many ILs. The heat capacity of these ILs have either not been measured yet, or are available just at a few temperatures. Besides, due to the tunable properties of ILs, numerous potential ILs can be yield via the different combinations of cation and anion. Thus, experimental determining the heat capacity of these incredibly high number of ILs over a wide operating conditions range by means of observation technology is unrealistic and unworkable. Therefore, it is essential to explore ways and means of acquiring the ILs heat capacity fastly and reliably. Computational approaches may be alternative methods to acquire the heat capacity of ILs. Recently, methods of calculation, such as simple correlation, group contribution method (GCM), and quantitative structureproperty relationship (QSPR), have been widely applied to estimate the physicochemical properties including the ILs heat capacity. Farahani et al. [14] developed a general correlation based on 2940 experimental data for 56 ILs within a wide temperatures range (188.06e663.10 K). Ahmadi et al. [20] established a GCM model to calculate 128 ILs heat capacities, consisting of 4822 data
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W. He et al. / Fluid Phase Equilibria 500 (2019) 112260
points over a temperature range from 190 to 663 K. Valderrama et al. [21] used artificial neural networks to predict heat capacities of 31 ILs with 477 data points. Sattari et al. [13] developed a QSPR model to predicte heat capacities of 82 ILs including 3726 data points covering the temperatures range of 180.06e663 K. Although the data points are in large quantity and the results of these models are effective and satisfactory, the types of ILs used in these studies are in a limited range. It is much more advisable to propose a model based on various ILs to estimate the heat capacity at different temperatures. To improve the predictive accuracy, the parameters for temperature were adjusted based on molecular structures in this study. In this work, attention is focused to propose a QSPR model to describe the heat capacity of various ILs under variable temperatures based on norm-index. The QSPR approach is one of the widely used computational methods, the fundamental theory of QSPR is to detect the connection between the structure of molecules and their physicochemical or thermodynamic properties [22,23]. Recently, the concept of norm index was proposed and QSPR models have been proposed to describe different properties of diverse substances, such as the toxicity of ILs to Staphylococcus aureus [24] and Candida albicans [25], thermal conductivity [26] and viscosity [27] of ILs, the adsorption data of organic compounds [28], heat capacity of gas [29] and so on [30e32]. In these works above, our group provided accurate and reliable models for different properties of varied compounds. This paper was an impactful extension of our study on norm index. The main purpose of this work was to calculate the heat capacity at variable temperatures for large amounts of ILs with various structures.
MS ¼ aij
aij ¼
8 <
n :0
if the path between atom i and j is n if i ¼ j (1)
8 < 1 rij if isj MD ¼ aij aij ¼ if i ¼ j :0 rij is the Euclidean spatial distance between atom i and j (2)
MP1 ¼ ½AW=Ra
(3)
MP2 ¼ ½EN Ra
(4)
MP3 ¼ ½expðAW=MWÞ
(5)
MP4 ¼ ½AQ =AW
(6)
MP5 ¼ ½1=ð1 þ expðBDÞÞ
(7)
Based on the matrices above, four new matrices are further defined as follows.
2
CM1;m
3 MP m 5 ¼4 MS
(8)
2
2. Methodology 2.1. Data on the heat capacity of ILs This present work was carried out based on 5964 experimental heat capacity data (in the unit of J$K1$mol1) covering a wide range of temperatures (191.13e550.00 K) for 185 ILs. The structures of these ILs included 97 imidazolium (Im), 10 pyrrolidinium (Pyr), 43 pyridinium (Py), 3 piperidinium (Pip), 23 ammonium (Am), 6 phosphonium (Ph), 2 quinolinium (Qui) and 1 morpholinium (Mor) based ILs. The observed values of heat capacity for these ILs are presented in Data Sheet of the supplementary Material (Supplementary Material.xlsx).
CM2;m
3 MP m 5 ¼4 MD
(9)
h i CM3;m ¼ MP Tm MPm þ MS
(10)
h i CM4;m ¼ MP Tm MPm þ MD
(11)
where m ¼ 1, 2, 3, 4 and 5, which represent the property matrices MP1 to MP5. Three norm indexes are specially defined as follows.
# " p X norm CMn;m ; 1 ¼ max CMn;m i;j j
j ¼ 1; /; q
i¼1
(12) 2.2. Method proposed The property of a substance is inextricably linked to its structure, and the same is true for IL heat capacity. At the present work, by using the HyperChem 7.0 software (http://www.hyper.com/), the stable spatial ionic structures were achieved. The ab initio method in quantum chemistry at the ST0-3G level was adopted during the optimizing procedure. ILs are constituted by a pair of ions, i.e. a cation and an anion, here, the cation and anion of each IL were optimized separately. On the basis of the optimized molecular structure, the step matrix (MS) and the Euclidean spatial distance matrix (MD) of each ion were first obtained. Five property matrices (MP1-5) were used to further distinguish the constituent of an ion. Here, the basic atomic properties constituted the MP1-5, and these properties contained atom weight (AW), atomic radius (Ra), electronegativity (EN), atom charge (AQ), branching degree (BD) and molecular weight (MW). The matrices mentioned above were described in detail as follows.
norm CMn;m ; 2 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
max l1 CMH n;m CMn;m
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u0 q p u 2 u@ X X CMn;m ij A norm CMn;m ; 3 ¼ t
(13)
(14)
j¼1 i¼1
where n ¼ 1, 2, 3 and 4, representing the combined matrix; p and q are the numbers of rows and columns of matrix CMn;m , respectively. The l1 refers to the matrix eigenvalue. The CM H n;m is the Hermite matrix of matrix CMn;m . ILs are made up solely of the cations and anions, and their thermodynamic property are thus closely related to the character of cation and anion. Undoubtedly, the IL heat capacity is not just the simply superposition of the cation contribution and the anion contribution, and the interaction between cations and anions
W. He et al. / Fluid Phase Equilibria 500 (2019) 112260
should also not be ignored. Accordingly, another set of norm indexes is specially defined to depict the cation-anion interaction, which is defined as follow.
3
test (F), the R2 (shown as Eq. (16)), AARD (shown as Eq. (17)) and the RMSE (shown as Eq. (18)) were employed to demonstrate the quality of the developed model. Also, the internal and external
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi normIL CMn;m ; w ¼ CF normCa CMn;m ; w þ AF normAn CMn;m ; w w ¼ 1; 2; 3
In this equation, CF and AF represent the mass fraction of the cation and anion, respectively. The normCa ðCMn;m ; wÞ and normAn ðCMn;m ; wÞ are the norm indexes of the matrices generated from the cations and anions, respectively.
validations [33,34] were used in this study. The former indicated the robust performance of the model, while the latter reflected the predictive performance of the model. Moreover, Y-randomization [35,36], and applicability domain analysis [37] methods were further applied to demonstrate the robust and predictive performance of this work.
2.3. CP(T) model 2
Temperatures greatly affect the heat capacity of ILs. The CP(T) model shown as Eq. (19) is used at present study. Here, a is related to the structures of ILs. As for the parameters b and g, there are two ways to be acquired: (1) b and g are set to a constant for all ILs, (2) b and g are adjusted on account of the molecular structure. For comparing these two methods, the data points of IL heat capacity used in the work are fitted using Eq. (19) via the aforesaid two ways. The fitting results are shown in Fig. 1a and b. It can be seen that the data points in Fig. 1b are closer to the bisection than these in Fig. 1a, so the second method is much better than the first method. Comparing the b and g being adjusted on the basis of the molecular structure parameters before and after, the square of the correlation coefficient (R2) increases from 0.9942 to 0.9996 and the average absolute relative deviation (AARD) decreases from 1.87% to 0.63%. Therefore, according to the statistical parameters results, it is advisable to adjust the temperature parameters with the molecular structure for calculating the heat capacity at variable temperatures of ILs. 2.4. Model evaluation To strengthen the scientific validity of the QSPR model and to assist its acceptance for application purpose, validation is necessary before putting the QSPR model into extensive use. Here, the performance parameters of the model including the results of the F-
(15)
R ¼1
n X
CPðcalc:;iÞ CPðobs:;iÞ
i¼1
2
,
n X
CPðcalc:;iÞ CP
2
(16)
i¼1
n
. X CPðobs;iÞ AARD ¼ CPðcalc:;iÞ CPðobs:;iÞ
, n
(17)
i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , u n
2 uX CPðcalc:;iÞ CPðobs:;iÞ RMSE ¼ t n
(18)
i¼1
where the CPðcalc:Þ is the calculated heat capacity value, the CPðobs:Þ is the experimental heat capacity value, the CP is the average value of the whole set experimental values, and the n is the number of experimental values of ILs heat capacity used herein. 3. Results and discussion 3.1. Calculation results of the CP(T) model On the basis of Eqs. (1)e(15), 60 matrix norm-indexes were come into being for each cation, each anion, and for the cationanion interaction, respectively. Besides, 3 other parameters were applied to convey the effect of the mass fraction of cation, mass fraction of anion, and atom number of hydrogen-suppressed molecular structure on the ILs heat capacity. However, not every
Fig. 1. Correlation between calculated and experimental heat capacity values for the two methods of obtaining parameters b and g in Eq. (19). a: b and g being as constants for all ILs, b: b and g being related with the molecular structure.
4
W. He et al. / Fluid Phase Equilibria 500 (2019) 112260
descriptor has a prominent contribution to the heat capacity of the ILs, herein, the most decisive descriptors were screened out to build the model. The model was described as following Eq. (19):
CP ¼ a þ b T þ g T 2
a ¼ 523:575887 þ
4 X
(19) bi norm CMn;m ; w i
Table 2 Summary of regression results for this work. States
Sample
D.P.
R2/Q2
AARD
RMSE
The model Five-fold CV Training set Testing set
185 185 148 37
5964 5964 4610 1354
0.996 0.996 0.995 0.994
2.49% 2.48% 2.36% 4.03%
19.43 19.48 18.01 33.71
(19a)
i¼1
b ¼ 25:620578 þ d1 þ
19 X
Table 3 Overall of the classified results for AARD of Eq. (19)
bj norm CMn;m ; w j
(19b)
j¼1
g ¼ d2 þ
10 X
bk norm CMn;m ; w k
Chemical Family
Im
Pyr
Py
Pip
Am
Ph
Qui
Mor
All
No. of ILs D.P. AARD%
97 3911 2.31
10 241 3.28
43 604 4.21
3 56 2.84
23 520 2.60
6 522 1.14
2 90 3.65
1 20 2.86
185 5964 2.49
(19c)
k¼1
d1 ¼ 1:830741 expðMWAn =MWÞ d2 ¼ 0:000768
(19d)
qffiffiffiffiffiffiffiffiffiffiffiffiffi NAn_H þ 0:004854 expðMWCa =MWÞ (19e)
N ¼ 185 D.P. ¼ 5964 F ¼ 35665 R2 ¼ 0.996 AARD ¼ 2.49% RMSE ¼ 19.43 Here, MWAn is the molecular weight of the anion; MWCa is the molecular weight of the cation; NAn_H is the nonhydrogen atom number for the anion. N is the number of ILs covered by this work; D.P. is the number of data points for the observed heat capacity values. The b parameters are listed in Table 1. All the observed and calculated values of heat capacity used in the CP(T) model are given in Data Sheet of the supplementary material (Supplementary Material.xlsx). The performance summary of Eq. (19) is shown in Table 2. The F is 35665, the R2 is 0.996, the AARD is 4.62% and the RMSE is 19.43. All these statistical parameters show accurate and reliable performance of this work. A summary about the performance parameter AARD of the classified results are shown in Table 3, from which it can be seen that the AARD values of each category are approximal to the overall AARD but the Pyr and Qui based ILs. The scatter plot of the
Table 1 Coefficients and the types of norm indexes for Eq. (19) i
norm(CMn,m, w)
bi
j
norm(CMn,m, w)
bj
1 2 3 4
normCa(CM2,1, 1) normCa(CM3,1, 1) normCa(CM4,1, 1) normIL(CM2,4, 2)
12.134896 0.525900 0.507940 184.509915
1 2 3 4 5 6 7 8
normCa(CM1,1, normCa(CM2,1, normCa(CM2,2, normCa(CM2,2, normCa(CM2,4, normCa(CM2,5, normCa(CM3,2, normCa(CM3,2,
0.188190 0.070882 2.384047 3.594319 30.648439 33.585766 0.008309 0.127771
k
norm(CMn,m, w)
bk
9
normCa(CM3,2, 3)
0.054665
1 2 3 4 5 6 7 8 9 10
normAn(CM1,2, normAn(CM1,4, normAn(CM2,4, normAn(CM2,4, normAn(CM3,5, normAn(CM4,3, normAn(CM4,4, normAn(CM4,4, normAn(CM4,5, normAn(CM4,5,
0.002206 0.000107 0.014436 0.091299 0.002179 0.000314 0.013594 0.095681 0.001976 0.005165
10 11 12 13 14 15 16 17 18 19
normCa(CM3,3, 1) normCa(CM4,4, 1) normAn(CM3,2, 1) normAn(CM3,2, 3) normAn(CM3,3, 1) normIL(CM1,1, 2) normIL(CM1,1, 3) normIL(CM2,3, 2) normIL(CM4,2, 1) normIL(CM4,5, 2)
0.219337 2.459231 0.020263 0.026593 0.065904 0.672496 0.832386 6.955550 0.907004 2.623174
3) 3) 1) 2) 3) 1) 1) 2) 1) 3)
2) 3) 1) 2) 2) 2) 1) 2)
Fig. 2. Correlation between the calculated and experimental values of heat capacity.
calculated heat capacity values by Eq. (19) vs. experimental values is shown in Fig. 2. As shown in Fig. 2, for this linear model, the data points are distributed in the vicinity of the bisector, which indicates that the calculated values agree well with experimental data. Additionally, the residual error for the heat capacity of every IL at different temperatures calculated with Eq. (19) is schematically displayed in Fig. 3, which showed that the residual of this work greater than 50 J K1.mol1 is only a small fraction of the total. Subsequently, Eq. (19) is acceptable and reliable for calculating the ILs heat capacity at different temperatures.
Fig. 3. Residual vs. experimental heat capacity values as determined via Eq. (19).
W. He et al. / Fluid Phase Equilibria 500 (2019) 112260
5
Table 4 Comparison of this work with four prior studies. Ref.
NO. of ILs
D.P.
T/K
R2
AARD%
RMSE
1 [14] 2 [38] 3 [39] 4 [40] This work
56 82 46 56 185
2940 3726 2416 2940 5964
188.06e663.10 189.66e663 223.1e663 188.06e663.1 191.13e550.00
0.975 0.993 0.985 0.992 0.996
2.48 1.70 2.72 1.04 2.49
19.261 15.110 11.050 19.432
3.2. Comparison with prior studies According to Figs. 2 and 3, the performance of this linear model is almost consistent with the trend of experimental data, which demonstrates that the established model is remarkably accurate. In addition, the predictive performance of this work was compared with four recent built linear models for estimation of ILs heat capacity. The comparison statistical parameters of these studies are listed in Table 4. It is evident that the statistical results of these models are quite good. Meanwhile, although the AARD and RMSE of this work are slightly worse than those of prior studies, the R2 (0.996) is comparable to those of the previous works. For these references works, the small number of ILs in modeling work might confine the predictive performance and application scope of their models. While, the lareger dataset of ILs with various structures was used for our modelling work, and satisfactory prediction results further confirmed that this proposed model is robust and reliable.
3.3. Internal validation Here, five-fold cross-validation (five-fold CV) is applied to validate the stability and robustness performance of this work [33,41]. The results of five-fold CV are given in Table 2. Table 2 shows that the Q2 (consistent with the meaning of R2), AARD and RMSE are agreeable, and are almost consistent with the results from Eq. (19). As illustrated in Fig. 4, the predicted values by five-fold CV are compared with experimental values of heat capacity. Additionally, the residual distributions (RDs) of the predicted heat capacity values by five-fold CV in comparison with that of Eq. (19) are shown in Fig. 5. As seen from Fig. 5, the RDs for the five-fold CV predictions are similar with those of the model Eq. (19), with most residuals range from 25 to 25 J K1 mol1. Thus, this further confirms that the norm-index based model for calculating heat capacity of various ILs at different temperatures is stable and reliable.
Fig. 4. Correlation between the predicted values (Five-fold CV) and experimental values of heat capacity.
Fig. 5. Distribution of the residuals by this work and five-fold CV for calculating heat capacity.
3.4. External validation In this work, 148 ILs (4610 data points) are used for training phase and the remaining 37 ILs (1354 data points) are utilized for testing phase. The performance parameters for external validation results are listed in Table 2. It is interesting to note from Table 2 that the R2, AARD and RMSE for the training set are almost the same as those for the whole set. In case of testing set, the AARD and RMSE are a little worse than the whole set and the training set. The calculated heat capacity values versus the experimental heat capacity values for the training set and the testing set are presented in Fig. 6. Fig. 6 shows that the calculated heat capacity values by Eq. (19) have similar trends with the corresponding experimental values, which indicates that the model has a good predictive performance on the heat capacity of ILs. Accordingly, these results well demonstrated that the proposed model is accurate to predict heat capacity of various ILs at different temperature. 3.5. Mean absolute error (MAE) validation The developed model is further validated using the MAE validation techniques to demonstrate the performance. Based on the reference work [42], the objective function must be translated into logarithmic form. And an acceptable error is 10% of the training set range, while a relatively great errors is the error value more than 20% of the training set range. In other words, a good prediction
Fig. 6. Comparison of the experimental and predicted data for heat capacity for the training set and testing set.
6
W. He et al. / Fluid Phase Equilibria 500 (2019) 112260
should meet the following criterias:
MAE 0:1 training set range
(20)
MAE þ 3 s 0:2 training set range
(21)
In Eq. (21), the s value is the standard deviation of the absolute error values for the testing set data. At present work, the MAE, training set range and s of the model are 0.0108, 0.8985 and 0.0148, respectively. The 0:1 training set range ¼ 0:08985 , MAE þ3 s ¼ 0:0552 , and 0:2 training set range ¼ 0:1797 can be calculated. Accordingly, the MAE validation in this work conforms to the above criterias: MAE 0:1 training set range and MAE þ 3 s 0:2 training set range. Thus, on the basis of these above results, the MAE validation metrics of the model are reasonable, which further demonstrates that this work has a good performance in describing the heat capacity of various ILs at different temperatures. 3.6. Y-Randomization test The randomization test [35] executed on the data matrix provides a valuable technique for evaluating the existence of any chance correlation in the QSPR model. In this work, some randomized models are developed, and the square of the correlation coefficients (R2r and Q2r ) are achieved. A satisfactory QSPR model should commonly have both low R2r and five-fold cross-validation Q2r values. Besides, the degree of difference between R2 and R2r is described by the cR2p parameter (defined as Eq. (22)). The criticality value of cR2p is 0.5, and the model is considered to be accidently achieved if the cR2p is lower than the stated limit [36]. c 2 Rp
¼R
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 R2r
(22)
Under the randomization test, the randomized model is repeatedly built and eight of these models’ performance parameters are listed in Table 5. As shown in Table 5, the R2r and Q2r of the randomized models are all quite close to zero and the cR2p value far exceeds the threshold value (0.5) [36]. This indicates that the original model Eq. (19) is not achieved by chance. 3.7. Application domain analysis
4. Conclusions A CP ðTÞ model was proposed to describe the 185 ILs heat capacity of 5964 experimental data points covering a wide range of temperature (191.13e550.00 K). The model performed well with excellent results (F ¼ 35665; R2 ¼ 0.996; AARD ¼ 2.49%; RMSE ¼ 19.43). The CP ðTÞ model was built by three sets of norm indexes, which were derived from cations, anions, and the interaction between them. These norm indexes were achieved on the basis of the atomic characters and positional distribution. And the main advantage of the present model is that the temperature parameters are adjusted based on molecular structure information, which greatly improves the effectiveness of the model. The validation metrics and the comparison of prior studies are both performed with good results, which indicate that this proposed method based on norm-indexes can be applied to describe the heat capacity of various ILs at variable temperatures. This proposed model could provide a possible alternative approach in the design of ILs for specific industrial applications at the determined temperature. Acknowledgements This research was supported by the National Natural Science Foundation of China [21306137, 21676203, 21808167]; and the Tianjin University of Science and Technology Youth Innovation Fund [2016LG14]. Conflicts of interest Declarations of interest: none. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.fluid.2019.112260.
The application domain [43] of a QSPR model plays a critical role in model establishment and application, and it is a theoretical region in the chemical space surrounding both the model descriptors and modeled response. The standardization approach is applied to depict this work's application domain. The standardization approach theory is based on the Roy et al.’s study [37]. And the algorithm for the standardization approach is described as follows.
Ski ¼
The meaning of each parameter is the same as the Roy et al.’s study [37]. In this work, the ½Si maxðkÞ values of the training set and testing set are 0.4979 and 0.2153, respectively. Both the ½Si maxðkÞ values for the training set and the testing set are lower than 3, which suggests that every training set IL is not an X-outlier, and all testing set ILs are located within the application domain.
X X i ki
(23)
sXi
SnewðkÞ ¼ Sk þ 1:28 sSk
(24)
Table 5 Results of the Y-randomization test in this work. States
1
2
3
4
5
6
7
8
R2r Q2r c 2 Rp
0.0030 0.0033 0.9941
0.0031 0.0026 0.9940
0.0033 0.0021 0.9939
0.0037 0.0048 0.9938
0.0037 0.0017 0.9937
0.0039 0.0004 0.9936
0.0041 0.0006 0.9935
0.0045 0.0002 0.9934
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W. He et al. / Fluid Phase Equilibria 500 (2019) 112260 [11] C.G. Yoo, et al., Ionic liquids: promising green solvents for lignocellulosic biomass utilization, Curr. Opin. Green Sustain. Chem. 5 (2017) 5e11. [12] M. Tuzen, et al., A simple, rapid and green ultrasound assisted and ionic liquid dispersive microextraction procedure for the determination of tin in foods employing ETAAS, Food Chem. 245 (2018) 380e384. [13] M. Sattari, et al., Estimation of the heat capacity of ionic liquids: a quantitative structureeproperty relationship approach, Ind. Eng. Chem. Res. 52 (2013) 13217e13221. [14] N. Farahani, et al., A simple correlation for prediction of heat capacities of ionic liquids, Fluid Phase Equilib. 337 (2013) 73e82. [15] P.B.P. Serra, et al., Phase behavior and heat capacities of the 1-benzyl-3methylimidazolium ionic liquids, J. Chem. Thermodyn. 100 (2016) 124e130. [16] E. Paulechka, et al., Thermodynamics of long-chain 1-alkyl-3methylimidazolium bis(trifluoromethanesulfonyl)imide ionic liquids, J. Chem. Thermodyn. 97 (2016) 331e340. [17] C.M. Tenney, et al., A computational and experimental study of the heat transfer properties of nine different ionic liquids, J. Chem. Eng. Data 59 (2014) 391e399. [18] A. Diedrichs, J. Gmehling, Measurement of heat capacities of ionic liquids by differential scanning calorimetry, Fluid Phase Equilib. 244 (2006) 68e77. [19] C. Su, et al., Isobaric molar heat capacities of 1-ethyl-3-methylimidazolium acetate and 1-hexyl-3-methylimidazolium acetate up to 16 MPa, Fluid Phase Equilib. 427 (2016) 187e193. [20] A. Ahmadi, et al., A simple group contribution correlation for the prediction of ionic liquid heat capacities at different temperatures, Fluid Phase Equilib. 403 (2015) 95e103. [21] J.O. Valderrama, et al., Heat capacity of ionic liquids using artificial neural networks and the concept of mass connectivity, Int. J. Thermophys. 32 (2011) 942e956. [22] K. Roy, et al., Understanding the Basics of QSAR for Applications in Pharmaceutical Sciences and Risk Assessment, Academic press, 2015. [23] J.C. Dearden, The History and Development of Quantitative Structure-Activity Relationships (QSARs), Oncology: Breakthroughs In Research and Practice: Breakthroughs In Research and Practice, 2016, p. 67. [24] W. He, et al., QSAR models for describing the toxicological effects of ILs against Staphylococcus aureus based on norm indexes, Chemosphere 195 (2018) 831e838. [25] F. Yan, et al., QSAR models for describing the toxicological effects of ILs against Candida albicans based on norm indexes, Chemosphere 201 (2018) 417e424. [26] W. He, et al., Description of the thermal conductivity l(T, P) of ionic liquids using the structureeproperty relationship method, J. Chem. Eng. Data 62 (2017) 2466e2472.
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[27] F. Yan, et al., Prediction of ionic liquids viscosity at variable temperatures and pressures, Chem. Eng. Sci. 184 (2018) 134e140. [28] Y. Wang, et al., Assessment for multi-endpoint values of carbon nanotubes: quantitative nanostructure-property relationship modeling with norm indexes, J. Mol. Liq. 248 (2017) 399e405. [29] J. Yin, et al., Predicting heat capacity of gas for diverse organic compounds at different temperatures, Fluid Phase Equilib. 446 (2017) 1e8. [30] Y. Wang, et al., Quantitative structure-property relationship for critical micelles concentration of sugar-based surfactants using norm indexes, J. Mol. Liq. 248 (2017) 399e405. [31] Q. Jia, et al., Quantitative structureeactivity relationship for high affinity 5HT1A receptor ligands based on norm indexes, J. Phys. Chem. B 119 (2015) 15561e15567. [32] X. Xu, et al., Predicting solubility of fullerene C60 in diverse organic solvents using norm indexes, J. Mol. Liq. 223 (2016) 603e610. [33] A. Tropsha, et al., The importance of being earnest: validation is the absolute essential for successful application and interpretation of QSPR models, QSAR Comb. Sci. 22 (2003) 69e77. [34] K. Roy, et al., Comparative studies on some metrics for external validation of QSPR models, J. Chem. Inf. Model. 52 (2012) 396e408. [35] C. Rücker, et al., y-Randomization and its Variants in QSPR/QSAR, J. Chem. Inf. Model. 47 (2007) 2345e2357. [36] I. Mitra, et al., Exploring quantitative structureeactivity relationship studies of antioxidant phenolic compounds obtained from traditional Chinese medicinal plants, Mol. Simul. 36 (2010) 1067e1079. [37] K. Roy, et al., On a simple approach for determining applicability domain of QSAR models, Chemometr. Intell. Lab. 145 (2015) 22e29. [38] M. Sattari, et al., Development of a group contribution method for the estimation of heat capacities of ionic liquids, J. Therm. Anal. Calorim. 115 (2014) 1863e1882. [39] Y. Zhao, et al., Estimation of heat capacity of ionic liquids using ss-profile molecular descriptors, Ind. Eng. Chem. Res. 54 (2015) 12987e12992. [40] A. Barati-Harooni, et al., Prediction of heat capacities of ionic liquids using chemical structure based networks, J. Mol. Liq. 227 (2017) 324e332. [41] K. Roy, et al., Chapter 7 - validation of QSAR models, in: Understanding the Basics of QSAR for Applications in Pharmaceutical Sciences and Risk Assessment, Academic Press, Boston, 2015, pp. 231e289. [42] K. Roy, et al., Be aware of error measures. Further studies on validation of predictive QSAR models, Chemometr. Intell. Lab. 152 (2016) 18e33. [43] D. Gadaleta, et al., Applicability domain for QSAR models: where theory meets reality, Int. J. Quant. Struct. Property Relatsh. (IJQSPR) 1 (2016) 45e63.
Contents lists available at ScienceDirect
Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Prediction of ionic liquids heat capacity at variable temperatures based on the norm indexes Wensi He a, Fangyou Yan a, *, Qingzhu Jia b, Shuqian Xia c, Qiang Wang a a
School of Chemical Engineering and Material Science, Tianjin University of Science and Technology, 13St. 29, TEDA, 300457, Tianjin, PR China School of Marine and Environmental Science, Tianjin University of Science and Technology, 13St. 29, TEDA, 300457, Tianjin, PR China c Key Laboratory for Green Chemical Technology of the State Education Ministry, Collaborative Innovation Center of Chemical Science and Engineering (Tianjin), School of Chemical Engineering and Technology, Tianjin University, 300072, Tianjin, PR China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 May 2019 Received in revised form 20 July 2019 Accepted 1 August 2019 Available online 2 August 2019
The heat capacity of ILs is essential in the chemical engineering and design applications. In this study, the dataset consisting of 5964 heat capacity data points within a wide temperature range (191.13e550.00 K) of 185 ILs was collated. Based on the concept of norm index, a serial of norm indexes were derived and adopted to develop the QSPR model. Here, the temperature parameters were adjusted on the basis of the molecular structure information. The linear model is precise for estimating heat capacity of ILs at variable temperatures with square of correlation coefficient R2 ¼ 0.996, average absolute relative deviation AARD ¼ 2.49% and the root mean square error RMSE ¼ 19.43. Besides, the good results of multiple verification methods further illustrate the stability and reliability of this work. Consequently, this normindex based work could provide a correlation method to describe the heat capacity of different structures ILs at variable temperatures. © 2019 Elsevier B.V. All rights reserved.
Keywords: Heat capacity Ionic liquids Norm indexes QSPR
1. Introduction Recently, the study of ionic liquids (ILs), in both academic research and industry fields, has attracted worldwide attention. Because of the unique properties of ILs, such as easy circulation, high solubility to polar/apolar compounds, good thermal and chemical stability, etc., ILs are widely applied in many areas of science and engineering, especially in the fields of electrochemistry [1,2], catalysis [3,4], gas separations [5,6], biotechnology [7,8] and other engineering technology [9e12]. Heat capacity (CP), one of the elemental thermodynamic properties of ILs, is defined as the partial derivative of the enthalpy with respect to temperature under constant pressure. Heat capacity is often required in the thermodynamics calculation, and is also the indispensable parameter in the chemical process designing. Based on the heat capacity, some other thermodynamic properties, such as entropy, enthalpy, Gibbs free energy, Helmholtz free energy, etc. [13], can be achieved. Similarly, the heat capacity data will bring a profound influence on the unit operation in chemical engineering like heat exchangers, distillation column and so on [14]. Various
* Corresponding author. E-mail address: [email protected] (F. Yan). https://doi.org/10.1016/j.fluid.2019.112260 0378-3812/© 2019 Elsevier B.V. All rights reserved.
measurement methods are available for obtaining the ILs heat capacity, such as high-precision heat capacity drop calorimetry [15], adiabatic calorimetry [16], differential scanning calorimetry [17,18] and others [19]. But in fact, there is still a lack of data on the heat capacity of many ILs. The heat capacity of these ILs have either not been measured yet, or are available just at a few temperatures. Besides, due to the tunable properties of ILs, numerous potential ILs can be yield via the different combinations of cation and anion. Thus, experimental determining the heat capacity of these incredibly high number of ILs over a wide operating conditions range by means of observation technology is unrealistic and unworkable. Therefore, it is essential to explore ways and means of acquiring the ILs heat capacity fastly and reliably. Computational approaches may be alternative methods to acquire the heat capacity of ILs. Recently, methods of calculation, such as simple correlation, group contribution method (GCM), and quantitative structureproperty relationship (QSPR), have been widely applied to estimate the physicochemical properties including the ILs heat capacity. Farahani et al. [14] developed a general correlation based on 2940 experimental data for 56 ILs within a wide temperatures range (188.06e663.10 K). Ahmadi et al. [20] established a GCM model to calculate 128 ILs heat capacities, consisting of 4822 data
2
W. He et al. / Fluid Phase Equilibria 500 (2019) 112260
points over a temperature range from 190 to 663 K. Valderrama et al. [21] used artificial neural networks to predict heat capacities of 31 ILs with 477 data points. Sattari et al. [13] developed a QSPR model to predicte heat capacities of 82 ILs including 3726 data points covering the temperatures range of 180.06e663 K. Although the data points are in large quantity and the results of these models are effective and satisfactory, the types of ILs used in these studies are in a limited range. It is much more advisable to propose a model based on various ILs to estimate the heat capacity at different temperatures. To improve the predictive accuracy, the parameters for temperature were adjusted based on molecular structures in this study. In this work, attention is focused to propose a QSPR model to describe the heat capacity of various ILs under variable temperatures based on norm-index. The QSPR approach is one of the widely used computational methods, the fundamental theory of QSPR is to detect the connection between the structure of molecules and their physicochemical or thermodynamic properties [22,23]. Recently, the concept of norm index was proposed and QSPR models have been proposed to describe different properties of diverse substances, such as the toxicity of ILs to Staphylococcus aureus [24] and Candida albicans [25], thermal conductivity [26] and viscosity [27] of ILs, the adsorption data of organic compounds [28], heat capacity of gas [29] and so on [30e32]. In these works above, our group provided accurate and reliable models for different properties of varied compounds. This paper was an impactful extension of our study on norm index. The main purpose of this work was to calculate the heat capacity at variable temperatures for large amounts of ILs with various structures.
MS ¼ aij
aij ¼
8 <
n :0
if the path between atom i and j is n if i ¼ j (1)
8 < 1 rij if isj MD ¼ aij aij ¼ if i ¼ j :0 rij is the Euclidean spatial distance between atom i and j (2)
MP1 ¼ ½AW=Ra
(3)
MP2 ¼ ½EN Ra
(4)
MP3 ¼ ½expðAW=MWÞ
(5)
MP4 ¼ ½AQ =AW
(6)
MP5 ¼ ½1=ð1 þ expðBDÞÞ
(7)
Based on the matrices above, four new matrices are further defined as follows.
2
CM1;m
3 MP m 5 ¼4 MS
(8)
2
2. Methodology 2.1. Data on the heat capacity of ILs This present work was carried out based on 5964 experimental heat capacity data (in the unit of J$K1$mol1) covering a wide range of temperatures (191.13e550.00 K) for 185 ILs. The structures of these ILs included 97 imidazolium (Im), 10 pyrrolidinium (Pyr), 43 pyridinium (Py), 3 piperidinium (Pip), 23 ammonium (Am), 6 phosphonium (Ph), 2 quinolinium (Qui) and 1 morpholinium (Mor) based ILs. The observed values of heat capacity for these ILs are presented in Data Sheet of the supplementary Material (Supplementary Material.xlsx).
CM2;m
3 MP m 5 ¼4 MD
(9)
h i CM3;m ¼ MP Tm MPm þ MS
(10)
h i CM4;m ¼ MP Tm MPm þ MD
(11)
where m ¼ 1, 2, 3, 4 and 5, which represent the property matrices MP1 to MP5. Three norm indexes are specially defined as follows.
# " p X norm CMn;m ; 1 ¼ max CMn;m i;j j
j ¼ 1; /; q
i¼1
(12) 2.2. Method proposed The property of a substance is inextricably linked to its structure, and the same is true for IL heat capacity. At the present work, by using the HyperChem 7.0 software (http://www.hyper.com/), the stable spatial ionic structures were achieved. The ab initio method in quantum chemistry at the ST0-3G level was adopted during the optimizing procedure. ILs are constituted by a pair of ions, i.e. a cation and an anion, here, the cation and anion of each IL were optimized separately. On the basis of the optimized molecular structure, the step matrix (MS) and the Euclidean spatial distance matrix (MD) of each ion were first obtained. Five property matrices (MP1-5) were used to further distinguish the constituent of an ion. Here, the basic atomic properties constituted the MP1-5, and these properties contained atom weight (AW), atomic radius (Ra), electronegativity (EN), atom charge (AQ), branching degree (BD) and molecular weight (MW). The matrices mentioned above were described in detail as follows.
norm CMn;m ; 2 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
max l1 CMH n;m CMn;m
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u0 q p u 2 u@ X X CMn;m ij A norm CMn;m ; 3 ¼ t
(13)
(14)
j¼1 i¼1
where n ¼ 1, 2, 3 and 4, representing the combined matrix; p and q are the numbers of rows and columns of matrix CMn;m , respectively. The l1 refers to the matrix eigenvalue. The CM H n;m is the Hermite matrix of matrix CMn;m . ILs are made up solely of the cations and anions, and their thermodynamic property are thus closely related to the character of cation and anion. Undoubtedly, the IL heat capacity is not just the simply superposition of the cation contribution and the anion contribution, and the interaction between cations and anions
W. He et al. / Fluid Phase Equilibria 500 (2019) 112260
should also not be ignored. Accordingly, another set of norm indexes is specially defined to depict the cation-anion interaction, which is defined as follow.
3
test (F), the R2 (shown as Eq. (16)), AARD (shown as Eq. (17)) and the RMSE (shown as Eq. (18)) were employed to demonstrate the quality of the developed model. Also, the internal and external
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi normIL CMn;m ; w ¼ CF normCa CMn;m ; w þ AF normAn CMn;m ; w w ¼ 1; 2; 3
In this equation, CF and AF represent the mass fraction of the cation and anion, respectively. The normCa ðCMn;m ; wÞ and normAn ðCMn;m ; wÞ are the norm indexes of the matrices generated from the cations and anions, respectively.
validations [33,34] were used in this study. The former indicated the robust performance of the model, while the latter reflected the predictive performance of the model. Moreover, Y-randomization [35,36], and applicability domain analysis [37] methods were further applied to demonstrate the robust and predictive performance of this work.
2.3. CP(T) model 2
Temperatures greatly affect the heat capacity of ILs. The CP(T) model shown as Eq. (19) is used at present study. Here, a is related to the structures of ILs. As for the parameters b and g, there are two ways to be acquired: (1) b and g are set to a constant for all ILs, (2) b and g are adjusted on account of the molecular structure. For comparing these two methods, the data points of IL heat capacity used in the work are fitted using Eq. (19) via the aforesaid two ways. The fitting results are shown in Fig. 1a and b. It can be seen that the data points in Fig. 1b are closer to the bisection than these in Fig. 1a, so the second method is much better than the first method. Comparing the b and g being adjusted on the basis of the molecular structure parameters before and after, the square of the correlation coefficient (R2) increases from 0.9942 to 0.9996 and the average absolute relative deviation (AARD) decreases from 1.87% to 0.63%. Therefore, according to the statistical parameters results, it is advisable to adjust the temperature parameters with the molecular structure for calculating the heat capacity at variable temperatures of ILs. 2.4. Model evaluation To strengthen the scientific validity of the QSPR model and to assist its acceptance for application purpose, validation is necessary before putting the QSPR model into extensive use. Here, the performance parameters of the model including the results of the F-
(15)
R ¼1
n X
CPðcalc:;iÞ CPðobs:;iÞ
i¼1
2
,
n X
CPðcalc:;iÞ CP
2
(16)
i¼1
n
. X CPðobs;iÞ AARD ¼ CPðcalc:;iÞ CPðobs:;iÞ
, n
(17)
i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , u n
2 uX CPðcalc:;iÞ CPðobs:;iÞ RMSE ¼ t n
(18)
i¼1
where the CPðcalc:Þ is the calculated heat capacity value, the CPðobs:Þ is the experimental heat capacity value, the CP is the average value of the whole set experimental values, and the n is the number of experimental values of ILs heat capacity used herein. 3. Results and discussion 3.1. Calculation results of the CP(T) model On the basis of Eqs. (1)e(15), 60 matrix norm-indexes were come into being for each cation, each anion, and for the cationanion interaction, respectively. Besides, 3 other parameters were applied to convey the effect of the mass fraction of cation, mass fraction of anion, and atom number of hydrogen-suppressed molecular structure on the ILs heat capacity. However, not every
Fig. 1. Correlation between calculated and experimental heat capacity values for the two methods of obtaining parameters b and g in Eq. (19). a: b and g being as constants for all ILs, b: b and g being related with the molecular structure.
4
W. He et al. / Fluid Phase Equilibria 500 (2019) 112260
descriptor has a prominent contribution to the heat capacity of the ILs, herein, the most decisive descriptors were screened out to build the model. The model was described as following Eq. (19):
CP ¼ a þ b T þ g T 2
a ¼ 523:575887 þ
4 X
(19) bi norm CMn;m ; w i
Table 2 Summary of regression results for this work. States
Sample
D.P.
R2/Q2
AARD
RMSE
The model Five-fold CV Training set Testing set
185 185 148 37
5964 5964 4610 1354
0.996 0.996 0.995 0.994
2.49% 2.48% 2.36% 4.03%
19.43 19.48 18.01 33.71
(19a)
i¼1
b ¼ 25:620578 þ d1 þ
19 X
Table 3 Overall of the classified results for AARD of Eq. (19)
bj norm CMn;m ; w j
(19b)
j¼1
g ¼ d2 þ
10 X
bk norm CMn;m ; w k
Chemical Family
Im
Pyr
Py
Pip
Am
Ph
Qui
Mor
All
No. of ILs D.P. AARD%
97 3911 2.31
10 241 3.28
43 604 4.21
3 56 2.84
23 520 2.60
6 522 1.14
2 90 3.65
1 20 2.86
185 5964 2.49
(19c)
k¼1
d1 ¼ 1:830741 expðMWAn =MWÞ d2 ¼ 0:000768
(19d)
qffiffiffiffiffiffiffiffiffiffiffiffiffi NAn_H þ 0:004854 expðMWCa =MWÞ (19e)
N ¼ 185 D.P. ¼ 5964 F ¼ 35665 R2 ¼ 0.996 AARD ¼ 2.49% RMSE ¼ 19.43 Here, MWAn is the molecular weight of the anion; MWCa is the molecular weight of the cation; NAn_H is the nonhydrogen atom number for the anion. N is the number of ILs covered by this work; D.P. is the number of data points for the observed heat capacity values. The b parameters are listed in Table 1. All the observed and calculated values of heat capacity used in the CP(T) model are given in Data Sheet of the supplementary material (Supplementary Material.xlsx). The performance summary of Eq. (19) is shown in Table 2. The F is 35665, the R2 is 0.996, the AARD is 4.62% and the RMSE is 19.43. All these statistical parameters show accurate and reliable performance of this work. A summary about the performance parameter AARD of the classified results are shown in Table 3, from which it can be seen that the AARD values of each category are approximal to the overall AARD but the Pyr and Qui based ILs. The scatter plot of the
Table 1 Coefficients and the types of norm indexes for Eq. (19) i
norm(CMn,m, w)
bi
j
norm(CMn,m, w)
bj
1 2 3 4
normCa(CM2,1, 1) normCa(CM3,1, 1) normCa(CM4,1, 1) normIL(CM2,4, 2)
12.134896 0.525900 0.507940 184.509915
1 2 3 4 5 6 7 8
normCa(CM1,1, normCa(CM2,1, normCa(CM2,2, normCa(CM2,2, normCa(CM2,4, normCa(CM2,5, normCa(CM3,2, normCa(CM3,2,
0.188190 0.070882 2.384047 3.594319 30.648439 33.585766 0.008309 0.127771
k
norm(CMn,m, w)
bk
9
normCa(CM3,2, 3)
0.054665
1 2 3 4 5 6 7 8 9 10
normAn(CM1,2, normAn(CM1,4, normAn(CM2,4, normAn(CM2,4, normAn(CM3,5, normAn(CM4,3, normAn(CM4,4, normAn(CM4,4, normAn(CM4,5, normAn(CM4,5,
0.002206 0.000107 0.014436 0.091299 0.002179 0.000314 0.013594 0.095681 0.001976 0.005165
10 11 12 13 14 15 16 17 18 19
normCa(CM3,3, 1) normCa(CM4,4, 1) normAn(CM3,2, 1) normAn(CM3,2, 3) normAn(CM3,3, 1) normIL(CM1,1, 2) normIL(CM1,1, 3) normIL(CM2,3, 2) normIL(CM4,2, 1) normIL(CM4,5, 2)
0.219337 2.459231 0.020263 0.026593 0.065904 0.672496 0.832386 6.955550 0.907004 2.623174
3) 3) 1) 2) 3) 1) 1) 2) 1) 3)
2) 3) 1) 2) 2) 2) 1) 2)
Fig. 2. Correlation between the calculated and experimental values of heat capacity.
calculated heat capacity values by Eq. (19) vs. experimental values is shown in Fig. 2. As shown in Fig. 2, for this linear model, the data points are distributed in the vicinity of the bisector, which indicates that the calculated values agree well with experimental data. Additionally, the residual error for the heat capacity of every IL at different temperatures calculated with Eq. (19) is schematically displayed in Fig. 3, which showed that the residual of this work greater than 50 J K1.mol1 is only a small fraction of the total. Subsequently, Eq. (19) is acceptable and reliable for calculating the ILs heat capacity at different temperatures.
Fig. 3. Residual vs. experimental heat capacity values as determined via Eq. (19).
W. He et al. / Fluid Phase Equilibria 500 (2019) 112260
5
Table 4 Comparison of this work with four prior studies. Ref.
NO. of ILs
D.P.
T/K
R2
AARD%
RMSE
1 [14] 2 [38] 3 [39] 4 [40] This work
56 82 46 56 185
2940 3726 2416 2940 5964
188.06e663.10 189.66e663 223.1e663 188.06e663.1 191.13e550.00
0.975 0.993 0.985 0.992 0.996
2.48 1.70 2.72 1.04 2.49
19.261 15.110 11.050 19.432
3.2. Comparison with prior studies According to Figs. 2 and 3, the performance of this linear model is almost consistent with the trend of experimental data, which demonstrates that the established model is remarkably accurate. In addition, the predictive performance of this work was compared with four recent built linear models for estimation of ILs heat capacity. The comparison statistical parameters of these studies are listed in Table 4. It is evident that the statistical results of these models are quite good. Meanwhile, although the AARD and RMSE of this work are slightly worse than those of prior studies, the R2 (0.996) is comparable to those of the previous works. For these references works, the small number of ILs in modeling work might confine the predictive performance and application scope of their models. While, the lareger dataset of ILs with various structures was used for our modelling work, and satisfactory prediction results further confirmed that this proposed model is robust and reliable.
3.3. Internal validation Here, five-fold cross-validation (five-fold CV) is applied to validate the stability and robustness performance of this work [33,41]. The results of five-fold CV are given in Table 2. Table 2 shows that the Q2 (consistent with the meaning of R2), AARD and RMSE are agreeable, and are almost consistent with the results from Eq. (19). As illustrated in Fig. 4, the predicted values by five-fold CV are compared with experimental values of heat capacity. Additionally, the residual distributions (RDs) of the predicted heat capacity values by five-fold CV in comparison with that of Eq. (19) are shown in Fig. 5. As seen from Fig. 5, the RDs for the five-fold CV predictions are similar with those of the model Eq. (19), with most residuals range from 25 to 25 J K1 mol1. Thus, this further confirms that the norm-index based model for calculating heat capacity of various ILs at different temperatures is stable and reliable.
Fig. 4. Correlation between the predicted values (Five-fold CV) and experimental values of heat capacity.
Fig. 5. Distribution of the residuals by this work and five-fold CV for calculating heat capacity.
3.4. External validation In this work, 148 ILs (4610 data points) are used for training phase and the remaining 37 ILs (1354 data points) are utilized for testing phase. The performance parameters for external validation results are listed in Table 2. It is interesting to note from Table 2 that the R2, AARD and RMSE for the training set are almost the same as those for the whole set. In case of testing set, the AARD and RMSE are a little worse than the whole set and the training set. The calculated heat capacity values versus the experimental heat capacity values for the training set and the testing set are presented in Fig. 6. Fig. 6 shows that the calculated heat capacity values by Eq. (19) have similar trends with the corresponding experimental values, which indicates that the model has a good predictive performance on the heat capacity of ILs. Accordingly, these results well demonstrated that the proposed model is accurate to predict heat capacity of various ILs at different temperature. 3.5. Mean absolute error (MAE) validation The developed model is further validated using the MAE validation techniques to demonstrate the performance. Based on the reference work [42], the objective function must be translated into logarithmic form. And an acceptable error is 10% of the training set range, while a relatively great errors is the error value more than 20% of the training set range. In other words, a good prediction
Fig. 6. Comparison of the experimental and predicted data for heat capacity for the training set and testing set.
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W. He et al. / Fluid Phase Equilibria 500 (2019) 112260
should meet the following criterias:
MAE 0:1 training set range
(20)
MAE þ 3 s 0:2 training set range
(21)
In Eq. (21), the s value is the standard deviation of the absolute error values for the testing set data. At present work, the MAE, training set range and s of the model are 0.0108, 0.8985 and 0.0148, respectively. The 0:1 training set range ¼ 0:08985 , MAE þ3 s ¼ 0:0552 , and 0:2 training set range ¼ 0:1797 can be calculated. Accordingly, the MAE validation in this work conforms to the above criterias: MAE 0:1 training set range and MAE þ 3 s 0:2 training set range. Thus, on the basis of these above results, the MAE validation metrics of the model are reasonable, which further demonstrates that this work has a good performance in describing the heat capacity of various ILs at different temperatures. 3.6. Y-Randomization test The randomization test [35] executed on the data matrix provides a valuable technique for evaluating the existence of any chance correlation in the QSPR model. In this work, some randomized models are developed, and the square of the correlation coefficients (R2r and Q2r ) are achieved. A satisfactory QSPR model should commonly have both low R2r and five-fold cross-validation Q2r values. Besides, the degree of difference between R2 and R2r is described by the cR2p parameter (defined as Eq. (22)). The criticality value of cR2p is 0.5, and the model is considered to be accidently achieved if the cR2p is lower than the stated limit [36]. c 2 Rp
¼R
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 R2r
(22)
Under the randomization test, the randomized model is repeatedly built and eight of these models’ performance parameters are listed in Table 5. As shown in Table 5, the R2r and Q2r of the randomized models are all quite close to zero and the cR2p value far exceeds the threshold value (0.5) [36]. This indicates that the original model Eq. (19) is not achieved by chance. 3.7. Application domain analysis
4. Conclusions A CP ðTÞ model was proposed to describe the 185 ILs heat capacity of 5964 experimental data points covering a wide range of temperature (191.13e550.00 K). The model performed well with excellent results (F ¼ 35665; R2 ¼ 0.996; AARD ¼ 2.49%; RMSE ¼ 19.43). The CP ðTÞ model was built by three sets of norm indexes, which were derived from cations, anions, and the interaction between them. These norm indexes were achieved on the basis of the atomic characters and positional distribution. And the main advantage of the present model is that the temperature parameters are adjusted based on molecular structure information, which greatly improves the effectiveness of the model. The validation metrics and the comparison of prior studies are both performed with good results, which indicate that this proposed method based on norm-indexes can be applied to describe the heat capacity of various ILs at variable temperatures. This proposed model could provide a possible alternative approach in the design of ILs for specific industrial applications at the determined temperature. Acknowledgements This research was supported by the National Natural Science Foundation of China [21306137, 21676203, 21808167]; and the Tianjin University of Science and Technology Youth Innovation Fund [2016LG14]. Conflicts of interest Declarations of interest: none. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.fluid.2019.112260.
The application domain [43] of a QSPR model plays a critical role in model establishment and application, and it is a theoretical region in the chemical space surrounding both the model descriptors and modeled response. The standardization approach is applied to depict this work's application domain. The standardization approach theory is based on the Roy et al.’s study [37]. And the algorithm for the standardization approach is described as follows.
Ski ¼
The meaning of each parameter is the same as the Roy et al.’s study [37]. In this work, the ½Si maxðkÞ values of the training set and testing set are 0.4979 and 0.2153, respectively. Both the ½Si maxðkÞ values for the training set and the testing set are lower than 3, which suggests that every training set IL is not an X-outlier, and all testing set ILs are located within the application domain.
X X i ki
(23)
sXi
SnewðkÞ ¼ Sk þ 1:28 sSk
(24)
Table 5 Results of the Y-randomization test in this work. States
1
2
3
4
5
6
7
8
R2r Q2r c 2 Rp
0.0030 0.0033 0.9941
0.0031 0.0026 0.9940
0.0033 0.0021 0.9939
0.0037 0.0048 0.9938
0.0037 0.0017 0.9937
0.0039 0.0004 0.9936
0.0041 0.0006 0.9935
0.0045 0.0002 0.9934
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