ARTICLE pubs.acs.org/IECR
Artificial Neural Network Modeling of Surface Tension for Pure Organic Compounds Aliakbar Roosta, Payam Setoodeh, and Abdolhossein Jahanmiri* School of Chemical and Petroleum Engineering, Shiraz University, Shiraz, Iran ABSTRACT: Surface tension as an important characteristic in much scientific and technological research is a function of liquid materials’ physical properties. Thus, it is desirable to have an accurate correlation between effective parameters and surface tension. This study investigates the applicability of artificial neural networks as an efficient tool for the prediction of pure organic compounds’ surface tensions for a wide range of temperatures. The experimental data gathered for training and verification of the network are related to a wide variety of materials such as alkanes, alkenes, aromatics, and sulfur, chlorine, fluorine, and nitrogen containing compounds. The most accurate network among several constructed configurations has one hidden layer with 20 neurons. The average absolute deviation percentage obtained for 1048 data points related to 82 compounds is 1.57%. The results demonstrate that the multilayer perceptron network could be an appropriate lookup table for the determination of surface tension as a function of physical properties.
’ INTRODUCTION Surface tension is a phenomenon in which the surface of a liquid, where the liquid is in contact with a vapor, i.e., its own vapor, or a gas, acts like a thin elastic sheet. If the surface is between two liquids or between a liquid and a solid, the phenomenon is called “interfacial tension”. Various intermolecular forces, such as van der Waals forces, draw the liquid particles together. Along the surface, the particles are pulled toward the rest of the liquid. There is always a tendency for the surface layer to minimize its area according to the total mass, constraints posed by the container, and external forces. The surface tension, generally employed as a quantitative index of this tension, is defined as the force exerted in the plane of the surface per unit length. Scientific and technological research in many areas needs data on surface tension of the used materials, e.g., thin layer technologies, microelectronics, electronic functional units, solgel technologies for material production, development of compound materials, phase separation techniques, matrix systems for chemical reactions, drug carriers, treatment of raw materials, chemical synthesis catalyzed by micelles, washing processes, and tertiary oil recovery. Many handbooks, e.g., the CRC Handbook of Chemistry and Physics1 and Surface Tension of Pure Liquids and Binary Liquid Mixtures,2 contain data or, at least, constants of fitting functions for the surface tension of liquids. Furthermore, several attempts have been made to predict surface tension based on thermodynamic properties. One of the simplest methods to estimate the surface tension of pure liquids and liquid mixtures was proposed by Macleod.3 It expresses the surface tension of a liquid in equilibrium with its own vapor as a function of the liquid and vapor phase densities according to eq 1. σ ¼ KðFl Fv Þ4
Figure 1. Schematic diagram of the ANN model. Pc (bar), critical pressure; ω, acentric factor; Tr, reduced temperature; Tnbr, reduced normal boiling temperature; s, specific gravity at normal boiling point; σ (mN/m), surface tension.
Sugden4 modified this expression as follows: σ ¼ ½PðFl Fv Þ4
Sugden called the temperature-independent parameter P the parachor and described a way to estimate it based on molecular structure. According to Escobedo and Mansoori’s study,5 there are various shortcomings associated with the use of eq 2. To overcome the shortcomings of the above equations, Escobedo and Mansoori5 proposed an expression for surface tension based on statistical thermodynamics. Their parameters are molar densities of liquid and vapor, reduced temperature, critical temperature, critical
ð1Þ
Received: August 6, 2011 Accepted: December 9, 2011 Revised: November 21, 2011 Published: December 09, 2011
where K is a constant which is independent of temperature and is a characteristic of the liquid and Fl and Fv are the molar densities of liquid and vapor, respectively. r 2011 American Chemical Society
ð2Þ
4
561
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pressure, and normal boiling temperature. The relation is similar to Sugden’s,4 where P is given by the following equations: 0:30066 9 þ 0:86442Τr P ¼ P0 ð1 Τr Þð0:37ÞTr exp Tr
deviation (AAD/%) was 1.05 for 94 organic compounds, mostly comprised of saturated and unsaturated paraffinic and aromatic compounds. Kavun et al.6 developed a quantitative structure activity relationship model for calculation of surface tension ranging from 9.49 to 67 mN/m. this model had eight parameters and was fitted to data for 72 organic compounds. The predictions for 22 test compounds using this model were 30% of the measured surface tension. Kumar et al.7 used artificial neural networks (ANNs) as an approach to predict surface tensions of 166 organic compounds at 20 °C, mostly comprised of saturated and unsaturated paraffinic and aromatic ones. Their model parameters were the liquid density, the refractive index at 20 °C, and the parachor. AAD/% of 38 test compounds was 3.24. Lin et al.8 used the gradient theory for prediction of the surface tensions of pure fluids and binary mixtures. In their study, the volume-translated Peng Robinson (VTPR) and SoaveRedlichKwong (VTSRK) equations of state were applied to determine the Helmholtz free energy density and the bulk properties. AAD values of the surface tensions for 83 pure components were less than 0.24 (mN/m). Fu et al.9 proposed a method for prediction of surface tensions of nonpolar fluids based on the density functional theory. They used the BarkerHenderson perturbation theory and statistical associating fluid theory to establish an equation of state. The
ð3Þ 13=12 R Tc P0 ¼ 39:6431 0:22217 0:00291042 Tbr 2 Pc 5=6 ð4Þ where P0 is a temperature-independent parameter and Tbr is the reduced normal boiling point. R* is equal to Rm/Rm,ref, where Rm is the molar refraction and Rm,ref is the molar refraction of the reference fluid (methane). The overall average absolute percent
Table 2. AAD/% Values, MSE Values, and R2 Values for Training Data and Test Data training data
Figure 2. Absolute average percent deviation between experimental data and ANN results vs number of neurons in hidden layer.
test data
AAD/%
MSE
R2
AAD/%
MSE
R2
1.41
0.09
0.999
1.95
0.11
0.998
Table 1. Parameters of Hidden and Output Layers hidden layer weight matrix
output layer bias vector
0.548 0.365
76.163 7.311
7.629 14.297
406.356 1.230
transposed weight vector
0.195 12.529
265.503 33.733
1.383 9.784 224.382
0.212
33.137
0.242
163.930
5.066
111.260
2.505
17.123
88.321
15.393
39.863
3.964
3.071
0.656
30.748
22.138
1.235
6.071
24.195
17.831
0.022
13.219
29.971
46.770
15.046
33.661
66.361
0.002
4.659
0.407
21.264
0.042
12.354
292.126
0.743
4.004
0.372
15.750
14.425
20.971
26.916
0.987 0.247
33.927 39.903
2.983 0.055
135.835 215.847
49.319 4.905
23.020 140.694
2.036 132.437
0.149
20.287
10.083
151.032
9.785
101.916
0.866
0.534
101.300
6.605
64.561
2.743
8.173
1.354
22.570
465.544
86.398
183.407
746.415
157.602
0.525
3.960
842.332
8.515
1.873
789.895
172.631
2.615 294.382
0.002
3.728
0.605
16.251
0.021
9.373
0.319
52.871
1.214
150.093
0.904
100.652
8.906
8.147 0.214
270.342 33.650
1.891 0.179
211.231 166.542
77.755 5.105
82.378 113.021
1.724 223.785
0.073
14.951
0.123
81.994
0.895
51.610
22.836
0.614
37.916
8.162
30.288
22.470
51.449
34.284
562
bias 0.685
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Table 3. Comparison between Correlated and Experimental Data of Surface Tension no.
compound name
formula
Tmin (K)
Tmax (K)
AAD/%
1
bromochlorodifluoromethane
CBrClF2
200
400
1.13
2
dichlorodifluoromethane
CCl2F2
130
370
2.72
3
bromotrifluoromethane
CBrF3
110
340
4.45
4
trifluoromethane
CHF3
210
270
2.02
5
methane
CH4
6
methanol
CH4O
7
methylamine
CH5N
204.15
308.15
0.45
8 9
ethene ethyne
C2H4 C2H2
116 183.15
180.65 223.15
0.52 1.73
10
ethane
C2H6
89.87
274.49
1.24
11
ethanol
C2H6O
158.5
513.92
1.37
12
acetic acid
C2H4O2
293.15
523.15
1.26
13
1,2-dichloropropane
C3H6Cl2
287.25
335.45
4.62
14
propylamine
C3H9N
292.35
314.75
0.98
15
propene
C3H6
193.15
250.25
2.15
16 17
2-propanone propanoic acid
C3H6O C3H6O2
200.15 238.05
353.15 413.45
1.76 2.68
18
propane
C3H8
273.15
360.63
5.40
19
1-propanol
C3H8O
243.16
513.15
1.55
20
2-methylpropene
C4H8
187.15
257.15
0.77
21
butanoic acid
C4H8O2
281.75
404.45
1.31
22
2-methylpropane
C4H10
113.74
370
2.79
23
1-butene
C4H8
218.15
293.15
0.73
24 25
2-methylpropanoic acid acetic acid ethyl ester
C4H8O2 C4 H8O2
273.15 293.15
405.35 513.15
0.46 4.90
26
2-methyl-2-propanol
C4H10O
298.15
338.15
1.04
27
diethyl sulfide
C4H10S
286.65
350.55
1.57
28
1-pentene
C5H10
243.15
298.15
1.45
29
2-pentanone
C5H10O
198.95
372.75
1.93
30
pentane
C5H12
144.18
407.25
0.96
31
pyridine
C5H5N
273.15
381.35
0.83
32 33
2-methyl-2-butene 1-pentanol
C5H10 C5H12O
203.15 243.52
343.15 553.15
0.84 1.50 5.83
96.84
183.68
1.69
175.59
512.64
1.22
34
tetradecafluorohexane
C6F14
293.15
432.25
35
chlorobenzene
C6H5Cl
227.95
580
1.90
36
benzene
C6H6
293.15
553.15
1.87
37
phenol
C6H6O
273.15
423.15
1.13
38
aniline
C6H7N
273.15
453.15
1.46
39
methylcyclopentane
C6H12
283.15
333.15
1.15
40 41
1-hexene 2-hexanone
C6H12 C6H12O
283.15 298.15
333.15 323.15
1.79 0.24
42
hexane
C6H14
175.12
447.13
1.38
43
2-methylpentane
C6H14
273.15
333.15
0.39
44
1-hexanol
C6H14O
242.95
593.15
2.08
45
hexadecafluoroheptane
C7F16
303.15
434.35
2.69
46
toluene
C7H8
178.15
270
0.79
47
2-methylphenol
C7H8O
285.65
457.35
0.97
48 49
2,4-dimethylpyridine methylcyclohexane
C7H9N C7H14
293.15 146.58
358.15 540
1.30 2.86
50
1-heptene
C7H14
273.15
348.15
0.62
51
2-methylpropanoic acid propyl ester
C7H14O2
288.45
360.05
0.91
52
heptane
C7H16
183.21
507.94
4.08
53
3-methylhexane
C7H16
253.15
363.15
0.79
54
1-heptanol
C7H16O
243.67
613.15
1.83
563
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Table 3. Continued no.
compound name
formula
Tmin (K)
Tmax (K)
AAD/%
55 56
1,2-dimethylbenzene ethylbenzene
C8H10 C8H10
273.15 178.16
373.15 620
0.51 1.45
57
1-octene
C8H16
283.15
373.15
1.46
58
octane
C8H18
218.51
522.48
2.27
59
2,2,3-trimethylpentane
C8H18
283.15
373.15
1.30
60
1-octanol
C8H18O
257.97
613.15
2.83
61
quinoline
C9H7N
303.15
673.15
2.57
62
propylbenzene
C9H12
283.15
373.15
0.83
63 64
1,2,4-trimethylbenzene 1-nonene
C9H12 C9H18
283.15 283.15
373.15 373.15
1.48 0.68
65
nonane
C9H20
283.15
393.15
0.85
66
2,2,3,4-tetramethylpentane
C9H20
283.15
333.15
0.71
67
1-nonanol
C9H20O
273.15
633.15
1.62
68
naphthalene
C10H8
363.15
673.15
3.03
69
butylbenzene
C10H14
283.15
373.15
0.96
70
decane
C10H22
283.15
393.15
0.82
71 72
1-decanol 1-methylnaphthalene
C10H22O C11H10
278.15 242.27
653.15 710
1.64 1.35
73
undecane
C11H24
283.15
393.15
0.60
74
1,1-biphenyl
C12H10
343.15
673.15
2.13
75
dodecane
C12H26
283.15
393.15
0.19
76
diphenylmethane
C13H12
293.15
373.15
1.18
77
tridecane
C13H28
283.15
393.15
0.33
78
tetradecane
C14H30
283.15
393.15
0.14
79 80
pentadecane hexadecane
C15 H32 C16H34
293.15 293.15
393.15 393.15
0.16 0.50
81
heptadecane
C17H36
273.15
393.15
0.95
82
1-ethoxyhexadecane
C18H38O
273.15
393.15
overall AAD/%
surface tensions for 18 pure nonpolar fluids were predicted with an AAD/% of 3.3. Romero-Martínez et al.10 developed a method to estimate and predict the surface tensions for isomers of pure hydrocarbons by using the surface tension value of the linear or normal member of a given hydrocarbon homologous series. This method was used to calculate surface tension values for 60 isomers with an AAD/% of 1.5. Artificial neural networks (ANNs) with different structures have been proven to be universal function approximators. This ability has been exploited for the approximation of chemical and physical properties.11 The major advantage of the ANN model is construction without detailed information of the underlying process. ANNs as black box modeling tools have already been used for many applications in industry, business, and science.12 Since in white box modeling approaches the model of development is based on the information of relevant equations and detailed knowledge for a specific system is usually not directly available, most efforts in the white box modeling approach are devoted to revealing all relevant mechanisms and quantifying these mechanisms correctly. This usually requires an extensive research program (including experiments, which can also be very time- and money-consuming). Here a compromise must be made in order to save time and money. Therefore, white box models often have limited accuracy because in developing the models minor mechanisms are neglected and only the major mechanisms are taken into account. The major advantage of the artificial neural networks is that they can be constructed
0.69 1.57
without the need for detailed knowledge of the underlying system. One of the applications of neural network models is to map an input space to an output space and function as a lookup table. Thus, in recent years, artificial neural networks have been applied to formulation of chemical and physical properties.1319
’ METHODS: ARTIFICIAL NEURAL NETWORK APPROACH In this study, applicability of a multilayer perceptron (MLP) neural network is investigated to develop an appropriate model for the prediction of surface tensions of organic compounds. The needed experimental data on surface tension were gathered from Surface Tension of Pure Liquids and Binary Liquid Mixtures,2 and the physical properties were gathered from the CRC Handbook of Chemistry and Physics.1 The data set prepared consists of the surface tension values ranging from 0 to 45 mN/m at various temperatures for 82 organic compounds. The total data set contains 1048 data points. About 70% of data is employed as training data, and the remaining 30% is applied for network verification. The MATLAB artificial neural network toolbox is employed in this study for model development and MLP training. A schematic diagram of the ANN model is illustrated in Figure 1. Several physical properties were considered and checked as input parameters, and after many attempts the most effective ones were selected as the inputs of the proposed network, which are critical 564
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(AAD) and mean square error (MSE) values: AAD=% ¼
MSE ¼
1 N jσexp σ cal j 100 N i¼1 σ exp
∑
1 N ðσ exp σcal Þ2 N i¼1
∑
ð5Þ
ð6Þ
where N is the number of data points and σexp and σcal are the available experimental data within the literature and calculated values for surface tension, respectively.
Figure 3. Surface tension as a function of temperature for different compounds. Solid lines represent model results, and points are experimental data.
Figure 4. Absolute average percent deviation as function of reduced temperature.
pressure (Pc, bar), acentric factor (ω), reduced temperature (Tr), reduced normal boiling temperature (Tnbr = Tnbp/Tc), and specific gravity at normal boiling point (s), and also the output is surface tension (σ, mN/m). The required number of training data points and hidden layer neurons are the two challenges that must be tackled appropriately. According to Cybenko,20 a network that has only one hidden layer is able to approximate almost any type of nonlinear mapping. However, the determination of the approximate number of neurons for the hidden layers is difficult and is often done by trial and error. The LevenbergMarquardt algorithm was used in the training procedure.2023 Different neural network topologies were compared using their average absolute deviation
’ RESULTS After many attempts, the best ANN obtained is a one-hiddenlayer MLP with 20 neurons in the hidden layer, as illustrated in Figure 2. The transfer function of the first layers is hyperbolic tangent sigmoid, and that of the second layer is a positive linear function. The parameters of the ANN structure are shown in Table 1. In addition, the AAD/%, MSE, and R2 values for training and test data are listed in Table 2. The results of the ANN model are reported in Table 3. It is noticed from this table that the overall AAD/% of 1.57 in surface tension estimation is obtained for all 82 compounds regarding 1048 data points. In order to show the predicting capabilities of the ANN model, the results relevant to eight different compounds are plotted in Figure 3 in order to demonstrate the model's accuracy. Figure 4 illustrates the effect of reduced temperature as an effective parameter on the network accuracy. As seen, when Tr approaches 1 (near critical temperature), the AAD/% values mostly increase. This is due to the fact that near critical temperature the exact experimental surface tension values which appear in the denominator for calculation of average absolute deviation are too low. ’ DISCUSSION One of the most significant advantages of the proposed approach is fewer input variables and more available parameters employed in comparison to the previous studies. For instance, specific gravity at the normal boiling point (a temperatureindependent term) is used as an input variable instead of (Fl Fv), which is a function of temperature. Moreover, the current model is more general than the previous ones due to the wider range of temperature. To be noted, the average difference between the upper and the lower temperature bounds regarded in this study is 170 K, while that of Escobedo and Mansoori, as the most global and accurate study, is 97 K. In addition, this work considers greater variety for the components than the other studies. Besides the alkanes, alkenes, and aromatics, the model is trained by data sets for sulfur compounds as well as fluorine, nitrogen, and chlorine compounds. At first, it seems that the model proposed by Escobedo and Mansoori is more accurate than this one, regarding the values for AAD/% reported, 1.05 versus 1.57. However, this fact stems from the difference between the corresponding temperature ranges considered. In order to make crystal clear the ability of the current model, Table 4 comprises the AAD/% values related to the models for a number of compounds with similar temperature ranges regarded in both studies. 565
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Table 4. Comparison between Present Study and Escobedo and Mansoori’s work AAD/% formula
name
temperature range (K)
present study
Escobedo and Mansoori5
CH4
methane
90170
1.69
3.39
C2H6
ethane
90275
1.24
3.18
C3H8
propane
273360
5.40
4.38
C5H12
pentane
156440
0.96
2.30
C6H12
methylcyclopentane
283333
1.15
0.31
C6H14
hexane
175450
1.38
1.55
C7H16
heptane
183508
4.08
3.87
C8H10 C9H12
1,2-dimethylbenzene propylbenzene
273343 273373
0.51 0.83
0.62 1.04
C9H12
1,2,4-trimethylbenzene
283373
1.48
0.58
C9H18
1-octene
283373
1.46
0.25
C10H14
butylbenzene
273373
0.96
1.74
C10H22
decane
273393
0.82
1.28
C11H24
undecane
273393
0.60
1.31
C13H28
tridecane
283393
0.33
1.32
1.53
1.81
average
’ CONCLUSION An MLP network was constructed based on experimental data available for organic compounds to correlate the critical pressure, acentric factor, reduced temperature, reduced normal boiling temperature, and specific gravity at the normal boiling point with surface tension. The results demonstrate that the proposed model is appropriate to correlate the surface tensions of a wide variety of organic compounds such as alkanes, alkenes, aromatics, and sulfur compounds as well as fluorine, chlorine, and nitrogen compounds in a wide range of temperatures.
(11) Bernazzani, L.; Duce, C.; Micheli, A.; Mollica, V.; Sperduti, A.; Starita, A.; Tine, M. R. Predicting Physical-Chemical Properties of Compounds from Molecular Structures by Recursive Neural Networks. J. Chem. Inf. Model. 2006, 46, 2030. (12) Widrow, B; Rumelhart, D. E.; Lehr, M. A. Neural Networks: Applications in Industry, Business and Science. Commun. ACM 1994, 37, 93. (13) Alabi, S. B.; Williamson, C. J. Centrifugal Pump-Based Predictive Models for Kraft Black Liquor Viscosity: An Artificial Neural Network Approach. Ind. Eng. Chem. Res. 2011, 50, 10320. (14) Ghanadzadeh, H.; Fallahi, S.; Ganji, M. LiquidLiquid Equilibrium Calculation for Ternary Aqueous Mixtures of Ethanol and Acetic Acid with 2-Ethyl-1-hexanol Using the GMDH-Type Neural Network. Ind. Eng. Chem. Res. 2011, 50, 10158. (15) Gharagheizi, F.; Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Representation and Prediction of Molecular Diffusivity of Nonelectrolyte Organic Compounds in Water at Infinite Dilution Using the Artificial Neural Network-Group Contribution Method. J. Chem. Eng. Data 2011, 56, 1741. (16) Gharagheizi, F.; Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Determination of Critical Properties and Acentric Factors of Pure Compounds Using the Artificial Neural Network Group Contribution Algorithm. J. Chem. Eng. Data 2011, 56, 2460. (17) Sedighi, M.; Keyvanloo, K.; Towfighi, J. Modeling of Thermal Cracking of Heavy Liquid Hydrocarbon: Application of Kinetic Modeling, Artificial Neural Network, and Neuro-Fuzzy Models. Ind. Eng. Chem. Res. 2011, 50, 1536. (18) Gharagheizi, F.; Abbasi, R. A New Neural Network Group Contribution Method for Estimation of Upper Flash Point of Pure Chemicals. Ind. Eng. Chem. Res. 2010, 49, 12685. (19) Kumar, K. V. Neural Network Prediction of Interfacial Tension at Crystal/Solution Interface. Ind. Eng. Chem. Res. 2009, 48, 4160. (20) Cybenko, G. V. Approximation by superpositions of a sigmoidal function. Math. Control, Signals, Syst. (MCSS) 1989, 2, 303. (21) Marquardt, D. An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 1963, 11, 431. (22) Levenberg, K. A method for the solution of certain problems in least squares. SIAM J. Numer. Anal. 1944, 16, 588. (23) Hagan, M.; Menhaj, M. Training feedforward networks with the Marquardt algorithm. IEEE Trans. Neural Networks 1994, 5, 989.
’ AUTHOR INFORMATION Corresponding Author
*Tel.: +98-711-2303071. Fax: +98-711-6474619. E-mail: jahanmir@ shirazu.ac.ir.
’ REFERENCES (1) Lide, D. R. CRC Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL, 2004. (2) Wohlfarth, C.; Wohlfarth, B. Surface Tension of Pure Liquids and Binary Liquid Mixtures; Springer: New York, 1997. (3) Macleod, D. B. Relation between surface tension and density. Trans. Faraday Soc. 1923, 19, 38. (4) Sugden, S. The Variation of surface tension with temperature and some related functions. J. Chem. Soc. 1924, 125, 32. (5) Escobedo, J.; Mansoori, G. A. Surface tension prediction for pure liquids. AIChE J. 1996, 42, 1425. (6) Kavun, S. M.; Chalykh, A. E.; Palyulin, V. A. Prediction of surface tension of organic liquids. Colloid J. 1995, 57, 767. (7) Kumar, D.; Gupta, S.; Basu, S. Prediction of surface tension of organic liquids using artificial neural networks. Indian Chem. Eng. 2005, 47, 219. (8) Lin, H.; Duan, Y. Y.; Min, Q. Gradient theory modeling of surface tension for pure fluids and binary mixtures. Fluid Phase Equilib. 2007, 254, 75. (9) Fu, D.; Lu, J. F.; Liu, J. C.; Li, Y. G. Prediction of surface tension for pure non-polar fluids based on density functional theory. Chem. Eng. Sci. 2001, 56, 6989. (10) Romero-Martínez, A.; Trejo, A.; Murrieta-Guevara, F. Surface tension of isomers of pure hydrocarbons: a method for estimation and prediction. Fluid Phase Equilib. 2000, 171, 1. 566
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