Prediction of Lubrication Performances of Vegetable Oils by Genetic Functional Approximation Algorithm

Abstract

Vegetable oils, which are considered potential lubricants, are composed of different types and proportions of fatty acids. Because of their diverse types and varying compositions, they exhibit different lubrication performances. The genetic function approximation algorithm was used to model the quantitative structure–property relationship between fatty acid structure and the wear scar diameter and friction coefficients measured by four-ball friction and wear tests. Based on the models with adjusted R² greater than 0.9 and fatty acid compositions of vegetable oils, the wear scar diameter and friction coefficients of Xanthoceras sorbifolia bunge oil and Soybean oil as validation oil samples were predicted. The difference between the predicted and experimental values was small, indicating that the models could accurately predict the lubrication performances of vegetable oils. The lubrication performances of 14 kinds of vegetable oils were predicted by GFA-QSPR models, and the primary factors influencing their lubrication properties were studied by cluster analysis. The results show that the content of C18:1 has a positive effect on the lubrication performances of vegetable oils, while the content of C18:3 has a negative effect, and the length of the carbon chain of fatty acids significantly affects their lubrication properties.

1. Introduction

Lubricating oil is a common chemical product in production and daily life, capable of effectively reducing energy loss in the engine, cutting, and other areas [1,2,3]. Currently, different types of lubricants are widely used across various industries. Mineral oil offers stability and cost-effectiveness, but its refining process has negative environmental impacts. Synthetic oil boasts excellent performance and stability, albeit at a higher cost [4]. Both mineral oil and most synthetic oils are challenging to biodegrade, hindering plant growth. Discharging them into water results in polluted waters covered with an oil film, leading to a significant decrease in water oxygen content and the production of new toxic substances. This, in turn, causes a large number of animals and plants to die, damaging the entire ecological environment [5]. Vegetable-based oil, derived from renewable resources, is environmentally friendly with high biodegradability, providing excellent lubrication and extending the lifespan of machinery and equipment. Therefore, vegetable-based oil holds a distinct advantage in the lubricant field and deserves further research and development [6,7,8,9].

Vegetable oils display varying lubrication performances due to their different types and compositions. Yin et al. investigated the interfacial lubrication characteristics of various vegetable oils like cotton oil, palm oil, castor oil, and peanut oil [10]. Reeves et al. examined the wear resistance and heat stability of avocado, canola oil, corn, olive, peanut, safflower oil, sesame oil, and SBO to understand the impact of the vegetable oil components on their properties [11]. Additionally, Sajeeb et al. evaluated the anti-wear performance of coconut oil, mustard oil, and their blend, demonstrating that the mixed oil had enhanced multiple properties and outperformed mineral oil [12]. However, the fatty acid content of vegetable oils obtained from different sources or extraction methods varies, leading to differences in tribological properties. Zulhanafi et al. investigated the tribological properties of castor oil with varying refinement levels and highlighted the significant influence of different fatty acid compositions on its tribological properties [13]. The tribological process is intricate, with numerous influencing factors like friction heat, contact area, unintentional particles, and noise impacting the test outcomes. Liu et al. conducted multiple tribological experiments on rapeseed oil under identical conditions, yielding slightly varied results [14,15]. Clearly, obtaining the lubrication properties of different vegetable oil sources and types solely through friction tests would be a time-consuming and labor-intensive endeavor.

If an accurate mathematical model can be established correlating test results or data information with the molecular structures of the compounds, predicting their properties precisely would significantly reduce test costs and time. The quantitative structure–property relationship (QSPR) model is a mathematical model established through theoretical calculations and statistical analysis to express the quantitative relationship between the structure and properties of compounds. QSPR typically utilizes experimental data and molecule descriptors to construct a regression model, ensuring model accuracy through statistical indicators and providing guidelines for structure–property relationships. Recently, researchers have applied QSPR techniques to tribology using various statistical methods. Weinebeck et al. developed a QSPR model for the tribological properties and structure of fuels using the elastic net regression method, distinguishing between effective and ineffective lubricants and offering a rapid screening tool for selecting potential biofuel molecules [16]. Liu et al. utilized rapeseed oil as the base oil, employing machine learning (ML) and logistic regression (LR) to analyze the correlation between the wear scar diameter (WSD) and the molecular structure of anti-wear additives, resulting in a well-fitted QSPR model with predictive capabilities [17]. Wan et al. used the least squares support vector regression (LS-SVR) method to establish an efficient QSPR model, selecting key molecular descriptors as inputs to characterize lubricants and identify ester-based compounds suitable for use as lubricants [18]. Despite the valuable results achieved in designing and enhancing lubricating oil additives and ester-based oils through QSPR technology, there is limited literature on studying the lubrication properties of vegetable oils using QSPR.

At present, common statistical methods for constructing QSPR models include ML, LR, principal component regression (PCR), partial least squares (PLS), genetic function approximation (GFA) algorithm, etc. The GFA algorithm is a promising statistical method for develo** the QSPR model. Rogers et al. developed the GFA algorithm, which combines GA and multivariate adaptive regression spline [19]. It automatically screens features by obtaining descriptors and completing the experimental descriptor screening. Each GFA iteration starts from a group, facilitating the approximation of the global optimal solution. It excels in finding a good solution from multiple initial solutions while mitigating the impact of partial noise in the equation. Abdulfatai et al. designed multiple lubricant antioxidants without sulfated ash, phosphorus, and sulfur based on the QSPR model developed using the GFA method [20]. Nasab et al. established a QSAR model to predict the viscosity index and pour point of ester oils using partial least squares combined with Leardi’s genetic algorithms (GAs) [21]. Currently, in tribology, GFA is primarily utilized in researching the QSPR of additives, with limited application in the lubrication properties of vegetable oils. Hence, the research team opted to employ the GFA algorithm for constructing the QSPR model.

Vegetable oils possess lubrication potential, yet the lubrication performances of numerous vegetable oils remain unknown. Conducting trial-and-error tribological experiments for all vegetable oils is impractical and time-consuming. The paper aims to utilize the GFA algorithm to establish the relationship between the fatty acid structure and tribological properties, predicting the lubrication properties of vegetable oils during friction by leveraging the fatty acid composition in vegetable oils and the established QSPR model. The WSDs and friction coefficients (COFs) of different fatty acids and vegetable oils were evaluated according to predicted outcomes. Based on fatty acid compositions and cluster analysis (CA), the primary factors affecting the lubrication performance of vegetable oils were examined. This study aims to establish a new QSPR-based calculation model for investigating the lubrication performances of vegetable oils and to offer data for the screening and design of bio-based lubricants.

2. Materials and Methods

2.1. Raw Materials

Thirteen common fatty acids present in vegetable oils, such as palmitic acid, palmitoleic acid, stearic acid, oleic acid, linoleic acid, linolenic acid, erucic acid, lauric acid, myristic acid, behenic acid, arachidic acid, and decanoic acid, were selected for friction experiments to determine their WSDs and COFs. Xanthoceras sorbifolia bunge oil (XSBO) and soybean oil (SBO) were chosen as validation samples for the predictive model. All materials were procured in Wuhan, Hubei province, China. Real images of the samples at room temperature are depicted in Figure 1, while the melting points of fatty acids and pour points of vegetable oils are presented in

Table 1. Property, WSD (mm), and COF of fatty acids and vegetable oils.

No	Name		Melting Point /Pour Point *	WSD (mm)		COF
				Experimental	Error	Experimental	Error
1	Palmitic acid	C16:0	62	0.2363	0.0170	0.0282	0.0026
2	Palmitoleic acid	C16:1	0.5	0.2446	0.0102	0.0210	0.0028
3	Stearic acid	C18:0	67–72	0.2665	0.0310	0.0147	0.0015
4	Oleic acid	C18:1	13–14	0.2461	0.0270	0.0116	0.0046
5	Linoleic acid	C18:2	−5	0.2208	0.0059	0.0271	0.0010
6	Linolenic acid	C18:3	−11	0.6000	0.0240	0.0552	0.0186
7	Ricinoleic acid	C18:1-12OH	22.5–24.5	0.2692	0.0023	0.0278	0.0027
8	Erucic acid	C22:1	28–32	0.2779	0.0087	0.0125	0.0039
9	Lauric acid	C12:0	44–46	0.2770	0.0082	0.0039	0.0008
10	Myristic acid	C14:0	52–54	0.3134	0.0110	0.0255	0.0035
11	Behenic acid	C22:0	72	0.2834	0.0118	0.0374	0.0013
12	Arachidic acid	C20:0	74	0.2765	0.0135	0.0173	0.0012
13	Decanoic acid	C10:0	27–32	0.3878	0.0232	0.0190	0.0016
14	XSBO	32%C18:1 47%C18:2	−21	0.2270	0.0355	0.0263	0.0040
15	SBO	16%C18:2 65%C18:3	−9	0.4390	0.0118	0.0437	0.0019

* The melting points of fatty acids and the pour points of vegetable oils.

. The molecular structures of these fatty acids has been previously reported and are not reiterated here [22].

2.2. Tests on Lubrication Performances

The anti-wear properties of the lubricant are typically evaluated through the ASTM D4172 four-ball test, conducted at various durations and loads [23]. The tribological behavior of different fatty acids and vegetable oils was investigated using an MRS-10A friction wear tester (**an YiHua Tribology Testing Technology Co., Ltd., Shandong, China) (Figure 2) following the ASTM D4172 standard. The test parameters included 1200 rpm, 100 N, 75 ± 2 °C, and 30 min, with GCr15 bearing steel balls of 12.7 mm diameter and 64 HRC hardness. It should be noted that liquid fatty acids at room temperature were directly subjected to friction tests, while some solid fatty acids, initially in a solid state at room temperature, were melted into a liquid state by heating to 75 °C before conducting the four-ball test. The experimental results were averaged after three repetitions. The reduced friction and anti-wear properties of fatty acids and vegetable oils were assessed based on the average COF and average WSD, respectively.

2.3. Fatty Acid Composition Analysis Method

The distribution analysis of fatty acids in vegetable oils in experimental studies is crucial as it directly impacts the lubricating properties of vegetable oils. The fatty acid composition of XSBO and SBO was analyzed using a gas chromatography (GC) analyzer (Agilent 7890A, Santa Clara, CA, USA) with the SP-2560 column. The experimental conditions included a constant-flow mode and a shunt ratio of 30:1. The initial temperature was 160 °C for 5 min, followed by temperature increments to 240 °C at rates of 5 °C/min, 8 °C/min, and 5 °C/min until the end of the experiment. The fatty acid compositions of 12 commonly used vegetable oils in industrial applications were obtained through a literature search [24,25,26,27,28]. The tribological properties of vegetable oils were predicted based on the tribological properties and the compositions of fatty acids.

2.4. Modeling and Calculation Methods

The fatty acid structure was optimized geometrically using the DFT/B3LYP algorithm in the ORCA 5.0 software and then inputted into E-DRAGON 1.0 for calculations. A total of 1600 molecule descriptors were automatically derived from the calculations, with over 120 selected after excluding invalid descriptors, including electronic structure, topological, and electrical descriptors. These key variables were utilized to establish a relationship model between fatty acids and tribological properties using the GFA method in the gplearn library in Python 3.11.0. The implementation process of this study began by randomly selecting from over 120 molecular descriptors to create the initial population. Subsequently, a randomly chosen equation from the initial set was designated as the in-variant equation and intersected with each of the other equations to produce descendant equations. Each offspring equation underwent evaluation, and this process continued until the most suitable QSPR model with the highest predictive power was achieved. Due to the small size of the original dataset, a 10:1 ratio was maintained between the training and validation sets during model development. The data were divided using the random division method. The flowchart illustrating the model establishment is depicted in Figure 3.

Friedman’s lack-of-fit (LOF) score was utilized as the criterion for assessing the quality of the GFA model. This LOF score was employed to estimate the fitness of potential QSPR models [29]. The GFA algorithm integrated Friedman’s LOF error measure to determine the optimal number of features, prevent overfitting, and regulate the fitted smoothness. Additionally, R-squared, Adjusted R-squared, and cross-validated R-squared (Q_CV²) were employed to assess the model’s suitability and consistency. Cross-validation was conducted using the leave-one-out method, which evaluates the model by treating each data point in the training set as a separate test set [30]. The formula for calculating the cross-validation regression coefficient (Q_CV²) is as follows:

$$Q_{CV}^2=1-\frac{\sum {(Y_i-Y_P)}^2}{\sum {(Y_i-\overline{Y_{tr}})}^2}$$

(1)

where Y_i and Y_P represent the experimental and predicted values of the samples, respectively; $\overline{Y_{tr}}$ denotes the average values of the experimental values in the training set. To evaluate the predictive performance of the developed model, C10:0 and C20:0 fatty acids were utilized for external validation. The model’s external validation was assessed based on the R-squared of external test set (R_test²). The R_test² is defined as

$$R_{test}^2=1-\frac{\sum {(Y_{P_{test}}-Y_{i_{test}})}^2}{\sum {(Y_{P_{test}}-\overline{Y_{tr}})}^2}$$

(2)

where $Y_{P_{test}}$ and $Y_{i_{test}}$ are the predicted and experimental values of the test set, while $\overline{Y_{tr}}$ represents the mean values of experimental values of the training set. Subsequently, the tribological characteristics of the untested fatty acids were forecasted by the GFA model. The findings, combined with the obtained fatty acid compositions in the reported vegetable oils, were used to predict their tribological properties. Meanwhile, CA analysis of the fatty acid composition database of vegetable oils was performed by PSPP 1.6.2 software to cluster different vegetable oils. Finally, a clustering map of vegetable oil tribological properties was obtained to assist in the screening of vegetable base oils with good tribological properties.

3. Results and Discussion

3.1. Friction Test Results and Fatty Acid Compositions

The WSD and COF are important indicators of the tribological properties of lubricants [31]. The WSDs and COFs for fatty acids and vegetable oils [21], as well as their corresponding experimental errors, are presented in Table 1. It was observed that the WSDs of fatty acids ranged from 0.2208 to 0.6000 mm and the COFs ranged from 0.0116 to 0.0552. Among these fatty acids, C18:3 had the worst friction properties, with a WSD of 0.6 mm and a COF of 0.0552. On the other hand, C18:2 had the best WSD of 0.2208 mm and an excellent COF of 0.0271. It can be seen that the tribological properties of fatty acids fluctuated with changes in carbon chain length and unsaturation. However, the changes in their tribological properties cannot be directly summarized from their chain length and unsaturation [32]. The error value was the difference between the maximum and minimum values obtained from multiple experiments. The error values for WSD and COF ranged 0.0059–0.0310 mm and 0.0008–0.0186, respectively. The minor errors observed in multiple experiments demonstrated the strong repeatability of the experiments, confirming the reliability of the data source and providing data support for subsequent prediction experiments.

Twelve vegetable oils reported in the literature and two tested oils, XSBO and SBO, were selected for the study. The fatty acid compositions from the literature and GC tests for XSBO and SBO are depicted in Figure 4. The chosen oils exhibited varying ratios of fatty acids, each possessing distinct characteristics that covered a broad spectrum of fatty acid compositions found in vegetable oils. The fatty acid profiles of the 12 selected vegetable oils, excluding jojoba oil, mainly consist of C18:0, C18:1, C18:2, and C18:3 fatty acids. However, there were significant differences in composition, such as rapeseed oil, containing 63% C18:1; grapeseed oil, with 70% C18:2; and flaxseed oil, with 56% C18:3. This highlighted the substantial diversity in the fatty acid composition among vegetable oils, which may result in varying tribological properties. By combining Table 1 and Figure 4, it is evident that SBO exhibited poor tribological properties with a WSD of 0.439 mm and a COF of 0.0437, possibly due to the high presence of 65% linolenic acid in its fatty acid composition. In contrast, XSBO demonstrated good tribological properties with a WSD of 0.227 mm and a COF of 0.0263, which could be attributed to its significant composition of 47% C18:2. Both oils fell within the experimentally validated range of fatty acid tribological properties, indicating the potential to predict specific tribological properties of vegetable oils based on their fatty acid composition.

3.2. Prediction Model for Lubrication Properties of Fatty Acids

The QSPR model was constructed with fatty acids 1–11 as the data set. Selecting 10 fatty acid samples as the training set, 1 sample as the test set at random, and the 12th and 13th samples as the external validation set. Prior to GFA, the WSDs and COFs of fatty acids were analyzed univariately, as detailed in

Table 2. Univariate analysis of the fatty acid data.

	WSD	COF
Number of sample points	11	11
Range	0.3790	0.0514
Maximum	0.6000	0.0552
Minimum	0.2210	0.0039
Mean	0.2941	0.0241
Median	0.2690	0.0255
Variance	9.95 × 10−3	1.804 × 10−4
Standard deviation	0.1046	0.0141
Mean absolute deviation	0.0591	0.0103
Skewness	2.2059	0.6517
Kurtosis	3.6898	−0.2580

, to evaluate the data’s suitability for further statistical analysis. The maximum and minimum values represented the extremes of the data distribution, with the range between them indicating the spread of values. The difference in WSD was 0.379 mm and COF was 0.0513. The mean values of the input data were WSD 0.2941 mm and COF 0.0241, while the median values were WSD 0.2941 mm and COF 0.0255, respectively. Standard deviation, mean absolute deviation, and variance were utilized to characterize the dispersion of the data set. These two data sets exhibited standard deviations of 0.1046 and 0.0141, suggesting a low dispersion and a concentration around the mean. Skewness and kurtosis indicated the symmetry of the data distribution. From Table 2, both data sets showed skewness values exceeding 0, with 2.2059 and 0.6517 separately, suggesting a right-skewed distribution for WSD and COF. This implied that most of the data were on the left, with extreme values isolated to the right. The kurtosis value indicated a normal distribution [33]. The kurtosis value of 3.6898 for the WSD dataset was above 3, indicating a fat-tailed distribution, while the kurtosis value of −0.258 for the COF dataset was significantly below 3, indicating a fine-tailed distribution for the COF data.

The data obtained from the univariate analysis can be utilized for QSPR analysis. Subsequently, over 120 molecule descriptors were acquired through E-DROGON computational screening. QSPR models between fatty acid structures and tribological properties (WSD and COF) were established by GFA. Descriptors served as independent variables, while two tribological indexes were the response variables. During GFA regression computation, a population size of 50 and a mutation rate of 0.1 were utilized, with a maximum evolutionary generation set at 1000. If convergence was not achieved, the number of maximum evolutionary generations would be increased. Each equation began with an initial term of 5 and had a maximum length of 10.

The predicted equations of the WSD and COF, along with their analysis, are detailed in

Table 3. The genetic function approximation validation parameters.

Parameters	$$\begin{gathered} \mathrm{W}\mathrm{S}\mathrm{D} \\ =0.160675\times {\mathrm{X}}_1 \\ -4.822002\times {\mathrm{X}}_2 \\ -0.008440\times {\mathrm{X}}_3 \\ +10.882637 \end{gathered}$$	$$\begin{gathered} \mathrm{C}\mathrm{O}\mathrm{F} \\ =0.007718\times {\mathrm{X}}_4 \\ -0.005111\times {\mathrm{X}}_5 \\ +0.003419\times {\mathrm{X}}_6 \\ -0.139028\times {\mathrm{X}}_7 \\ -0.023146\times {\mathrm{X}}_8 \\ +0.100953 \end{gathered}$$
Friedman LOF	1.54 × 10−3	3.3 × 10−5
R-squared	0.9501	0.9856
Adjusted R-squared	0.9289	0.9712
Cross-validated R-squared (QCV2)	0.7145	0.7466
Significant Regression	Yes	Yes
Significance-of-regression F-value	44.4719	68.4366
Critical SOR F-value (95%)	4.5239	5.1301
Replicate points	0	0
Computed experimental error	0.0000	0.0000
Lack-of-fit points	7	5
Min expt. error for non-significant LOF (95%)	0.0196	0.0016
R-squared external test set (Rtest2)	0.8682	0.8525

. The descriptors and their corresponding values are presented in

Table 4. Descriptors used in the QSPR model and their corresponding values.

No	Name	X1	X2	X3	X4	X5	X6	X7	X8
		AlogP98	E-State Keys (Sums): S_sCH3	Shadow Area: XY Plane	Total Dipole	Atomic Composition (Total)	Shadow Area: ZX Plane	Shadow Area Fraction: YZ Plane	Propyl
1	Palmitic acid	6.3930	2.2604	89.5046	4.9780	56.1928	85.1321	0.6687	1.0000
2	Palmitoleic acid	5.9484	2.2492	86.6844	2.9080	54.8715	85.7617	0.6567	1.0000
3	Stearic acid	7.3054	2.2700	96.2423	2.1470	62.0359	98.1686	0.6891	1.0000
4	Oleic acid	6.8608	2.2588	103.6651	2.7960	60.7315	95.1076	0.7387	1.0000
5	Linoleic acid	6.4162	2.2347	106.5797	2.8110	59.3642	94.3883	0.6897	1.0000
6	Linolenic acid	5.9716	2.1479	105.7513	2.8200	57.9274	94.8382	0.6877	0.0000
7	Ricinoleic acid	5.6239	2.1997	105.5990	4.8030	65.1875	101.9273	0.7271	1.0000
8	Erucic acid	8.6856	2.2753	125.8767	2.0470	72.3452	113.9757	0.7312	1.0000
9	Lauric acid	4.5682	2.2281	72.1495	2.1760	44.3626	66.6541	0.6586	1.0000
10	Myristic acid	5.4806	2.2471	76.3903	4.4560	50.3061	78.2255	0.6920	1.0000
11	Behenic acid	9.1302	2.2829	124.1960	2.1550	73.6250	120.8127	0.6712	1.0000
12	Arachidic acid	8.2178	2.2773	111.6795	1.5190	67.8444	109.2919	0.6962	1.0000
13	Decanoic acid	3.6558	2.1988	60.8896	4.9050	38.3422	56.8992	0.7017	1.0000

. Evaluation of the QSPR model in GFA involved measuring LOF, F, and R² scores. Higher LOF and F values indicate better model-fitting performance. The F values of 44.4719 and 68.4366 for WSD and COF models, respectively, were deemed acceptable for data fitting. An R-squared (R²) value exceeding 0.8 signifies a good model fit. The R² values of 0.9501 and 0.9856 for the two models indicate strong fit performance. Additionally, the adjusted R-squared (R²_adj) values of 0.9289 and 0.9712 demonstrate the models’ ability to explain over 90% of the data in the training set. This suggests that the models fit the data well and have strong generalization capabilities. The cross-validated R-squared (Q_CV²) serves as a measure of the model’s predictive power. It is generally believed that Q_CV² is greater than 0.6, and the difference between R² and Q_CV² is less than 0.3, indicating that the model has reliable prediction ability [34]. The Q_CV² values for these two models were 0.7145 and 0.7466, respectively, confirming the robustness of the models. From Table 4, the QSPR model indicators of tribological properties of fatty acids show that both WSD and COF prediction models have good performance and can predict their individual tribological properties well.

The equations for calculating the predicted values of WSD and COF are as follows:

$$\mathrm{W}\mathrm{S}\mathrm{D}=0.160675\times {\mathrm{X}}_1-4.822002\times {\mathrm{X}}_2-0.008440\times {\mathrm{X}}_3+10.882637$$

(3)

$$\mathrm{C}\mathrm{O}\mathrm{F}=0.007718\times {\mathrm{X}}_4-0.005111\times {\mathrm{X}}_5+0.003419\times {\mathrm{X}}_6-0.139028\times {\mathrm{X}}_7-0.023146\times {\mathrm{X}}_8+0.100953$$

(4)

where X₁ is the AlogP98, X₂ is the E-state keys (sums): S_sCH3, X₃ is the shadow area: XY plane, X₄ is the total dipole, X₅ is the atomic composition (total), X₆ is the shadow area: ZX plane, X₇ is the shadow area fraction: YZ plane, and X₈ is the propyl.

The WSDs and COFs of these 11 fatty acids were predicted using the QSPR model, and the residuals were calculated between the predicted and experimental values. The results are shown in Figure 5. It showed the regression equation between the predicted and experimental values obtained from the GFA model with slopes of 0.950 and 0.986, indicating good predictive power for fatty acid friction performance [35]. Also, there was only a small difference between the predicted and experimental values of WSD and COF. Combining these features, it can be concluded that the values predicted by the models match well with the experimental values, and there is no overfitting, so these two GFA models can be used to predict the friction properties of fatty acids.

Due to the limited data in the validation set, the validation set’s LOF and other validation information were not presented. However, the predicted WSD and COF of behenic acid, which was used for validation, were 0.2933 mm and 0.0379, respectively, with minimal error from the actual value. The long-chain and short-chain fatty acids, C20:0 and C10:0, not in the dataset, served as external data points to confirm the accuracy and reliability of the QSPR models. The QSPR models predicted WSDs of C20:0 and C10:0 as 0.2795 mm and 0.3533 mm, respectively, with corresponding COFs of 0.0196 and 0.0167. The predicted values of the validation set, and the external test set showed minor differences from the experimental values. The R-squared calculated for the WSD and COF of the external test set are 0.8682 and 0.8525, respectively, proving the usability of the two models. This outcome demonstrates the high stability and strong predictivity of the two QSPR models developed, allowing for further exploration.

Figure 6 displays the distribution of the residual values concerning the measured WSD and COF values. The residuals are the differences between the measured values and the predicted values by the GFA model. Figure 6a,c depicts the residual values of WSD and COF, while Figure 6b,d shows the residual values of WSD and COF with row numbers. Outlier analysis evaluated the predicted QSAR model based on two standard deviations of the residual mean. Figure 6 reveals that the critical thresholds for four standard deviations within the dashed lines range from −2.5 to 2.5. Only the predicted COF values for linoleic acid slightly deviated, but they ultimately fell within an acceptable range. Therefore, the QSPR model is deemed acceptable.

3.3. The Descriptors in QSPR Models

The above descriptors highlight their significance in describing the anti-wear and friction reduction properties of fatty acid molecules in QSPR models.

The AlogP98 (X₁) is a molecular descriptor that indicates the octanol/water partition coefficient, linking the chemical structure to observed chemical behavior. AlogP98 is associated with the molecule’s hydrophobicity and is directly related to the molecular chain length. The AlogP98 index shows a positive correlation with WSD values, suggesting that fatty acid molecules should not be excessively long as this can lead to an increase in WSD.

The Estate bonds are utilized to compute the sum of electronic state values and to categorize each atomic type [36]. The E-state keys (sums): S_sCH3 (X₂) descriptor represents the total count of carbon atoms with a specific electro-topological state index. The “sCH3” state exists when a carbon atom is bonded to a single non-hydrogen atom and three hydrogen atoms. The descriptor indicates the number of methyl groups in the molecule and plays a significant role in predicting WSD, showing a notable negative correlation with WSD. The electric topological state index characterizes the electronic structure, topological structure, and bonding valence information of various atoms in the compound. It not only includes molecular details like size and branching chain length but also relates to molecular polarity [37]. This value is directly linked to the fatty acid chain length. Additionally, the presence of carbon–carbon double bonds decreases the value, indicating that their presence in fatty acids increases the WSD. Conversely, increasing the chain length of fatty acids helps reduce the WSD [38].

The shadow index is employed to describe the molecule’s shape and is a geometric descriptor based on the molecule’s conformation and orientation. It is calculated by projecting the model surface onto three perpendicular planes: XY (X₃), YZ (X₇), and ZX (X₆) [39]. According to the GFA model, the projection area of fatty acid molecules in the XY plane is inversely related to their WSD. It indicates that maintaining a large rectangle shape of fatty acid molecules in the XY plane, with the cross-section along the longest axis of the molecule, can enhance their anti-wear properties. Additionally, the projected area in the YZ plane shows a negative correlation with COF, indicating that maintaining a large cross-section of the longest molecule axis has a positive impact on the friction-reducing properties of fatty acid molecules. However, the projected area in the ZX plane shows a positive correlation with COF, suggesting that the presence of branched chains in fatty acid molecules reduces their friction-reducing properties.

In its turn, the dipole (X₄) descriptor is used to calculate the total dipole moment’s magnitude and the Cartesian component (x, y, z), representing the strength and direction of the molecules in the electrostatic field. The total dipole signifies the overall dipole moment’s magnitude, making a positive contribution to the model and significantly influencing friction reduction performance. Therefore, the presence of the dipole moment in the fatty acid molecule hinders its ability to reduce friction. Furthermore, the atomic composition (total) (X₅) serves as an information–content descriptor, derived by structurally analyzing the molecule and converting it into an average descriptor, then multiplying it by the molecule’s atom count. This descriptor contains all molecular information of the fatty acid molecule, showing a positive correlation with molecule atom count while having a negative correlation with COF. This illustrates that the longer the chain length of the fatty acid molecule, the more effective its friction reduction performance. Additionally, the presence of unsaturated bonds leads to weaker friction reduction performance [40]. It is evident that the descriptors derived from the GFA model associated with the tribological properties of fatty acids are related to the chain length and unsaturated bonds of fatty acids, which are consistent with those obtained experimentally.

3.4. Predicting Lubrication Performance

Utilizing the GFA model, the tribological properties of certain untested fatty acids commonly present in vegetable oils were forecasted. The results for WSD and COF are depicted in Figure 5a,b correspondingly. The maximum, minimum, and median values of WSD and COF for the examined fatty acids were computed and illustrated with dotted lines. The majority of the projected fatty acid WSDs fall within the spectrum of the input fatty acid WSDs. Nevertheless, the highly unsaturated fatty acids C18:4, C20:3, and C20:5 all surpass the least-performing C18:3 with values of 0.61, 0.648, and 0.612 mm, respectively. The anticipated WSD value for C18:1 is also below 0.218 mm but only deviates by 0.0281 mm from the experimental value, falling within an acceptable margin of error. Likewise, some COF forecasts extend beyond the input range—C18:3, C18:4, C20:3, and C21:1 exhibit COF values of 0.055, 0.079, 0.057, and 0.063, respectively. The predictions indicate that utilizing the selected fatty acid tribological properties as input data can better encompass the spectrum of fatty acid tribological properties and enhance prediction accuracy. The outcomes of the projected data mirror those indicated by the descriptors. The anticipated WSDs and COFs of fatty acids were scaled by the percentage of fatty acid content in each vegetable oil and then aggregated to derive the projected WSDs and COFs of the vegetable oils. All vegetable oils exhibited a WSD ranging from 0.2383 to 0.4711 mm and a COF ranging from 0.013 to 0.0447, with none falling below the minimum or exceeding the maximum values of WSD and COF. This suggests the viability of employing this approach to compute the tribological properties of vegetable oils and make more precise predictions. Finally, it is evident that the predicted WSD and COF of two external validation oil samples, XSBO and SBO, exhibited minimal variance compared to those determined using a four-ball friction tester under identical experimental conditions. The predicted values for XSBO were WSD 0.2383 mm and COF 0.0235, with discrepancies of 0.0113 mm and 0.0028 compared to the experimentally obtained values, respectively. The predicted values for SBO were WSD 0.4711 mm and COF 0.0447, with variances of 0.0321 mm and 0.001 compared to the experimentally obtained values. These discrepancies fell within the acceptable experimental error range, showcasing the feasibility of utilizing the WSD and COF of fatty acids to forecast the WSD and COF of vegetable oils based on fatty acid percentage calculations. This also validates the reliability and validity of the constructed GFA model.

Nevertheless, it is important to highlight that Figure 7b reveals anomalies in the COF prediction of short-chain fatty acids, possibly due to their limited representation in the input data. Consequently, this results in irregularities in predicting the tribological properties of short-carbon chain length fatty acids. However, it is noteworthy that short-chain fatty acids make up only a small fraction of vegetable oils, as depicted in Figure 4. Hence, this has minimal impact on the ultimate prediction outcomes. Overall, the QSPR-GFA model for predicting the tribological properties of vegetable oils is plausible.

3.5. Cluster Analysis of Vegetable Oils

Following a literature review, the research team compiled the fatty acid compositions of 12 commonly used vegetable oils, as displayed in Figure 4. It is evident that the majority of vegetable oils comprise fatty acids with an 18-carbon chain length. Nonetheless, there are variations in the fatty acid composition of each oil. Consequently, CA analysis of the fatty acid composition of different vegetable oils was conducted to achieve a similar classification of fatty acid composition.

Figure 8 depicts a tree diagram illustrating the CA results. The tree extends horizontally from left to right, displaying clustering categories on the left side resembling a root system. It consolidates root categories and extends into the right, gradually forming multiple branch categories before converging into two main categories. Each horizontal line represents a category originating from the root system, continuously merging with other horizontal lines to create highly concentrated categories. Jojoba oil was assigned to Category 1 due to its high content of C20:1 and unique fatty acid C20:5, at 56.01% and 32.3%, respectively. Date seed oil, pongamia pinnata seed oil, moringa oil, and rapeseed oil were grouped under Category 2 as they contained over 40% C18:1. Conifer oil and flaxseed oil were placed in Category 3 since both had more than 40% C18:3. Lastly, coffee oil, hemp seed oil, grapeseed oil, pumpkin seed oil, and corn oil were categorized in Category 4 due to their content of over 40% C18:2.

The WSD and COF results of vegetable oil predicted by the GFA model are presented in Figure 7. It is evident that both Category 2 and Category 4 vegetable oils exhibit superior lubrication performances. The WSD range for Category 2 is 0.2394–0.2635 mm and for Category 4 is 0.2407–0.3119 mm. However, Category 4 demonstrates a higher COF than Category 2, with a COF range of 0.0203–0.0287 compared to 0.013–0.019 for Category 2. Vegetable oils in Category 3 exhibit poor lubrication performances, with both WSD and COF being the least favorable among the selected vegetable oils. The WSD range for Category 3 is 0.4367–0.438 mm and the COF range is 0.0393–0.0419. Finally, the vegetable oil in Category 1 shows average anti-wear performance but lacks friction reduction ability, with a WSD of 0.3614 mm and a COF of 0.0382. In conclusion, vegetable oils in Category 2 demonstrate excellent tribological properties. In summary, the vegetable oils in Category 2 have excellent tribological properties. The content of C18:1 has a positive effect on the tribological properties of vegetable oils, while the content of C18:3 has a negative effect on their tribological properties.

4. Conclusions

The tribological process is intricate, with numerous influencing factors. To obtain the lubrication properties of different vegetable oil sources and types solely through friction tests would be a time-consuming and labor-intensive endeavor. Two reliable QSPR models for predicting WSD and COF of fatty acids are established using the GFA algorithm. Based on the fatty acid composition and QSPR model, the anti-wear and friction reduction properties of vegetable oils can be predicted accurately.

Upon examining the descriptors in the QSPR model, it was found that the length of the carbon chain has a notable impact on the lubrication properties of fatty acids. Additionally, these properties show a positive correlation with most fatty acids. On the other hand, the dipole moment and atomic number of fatty acid molecules exhibit a negative correlation with friction coefficients. It is essential to take this information into account when choosing vegetable oils as environmentally friendly lubricants. The CA analysis indicates that vegetable oils with higher levels of C18:1 fatty acid offer superior lubrication performance. C18:1 positively impacts the lubricating properties of vegetable oil. In future selections, vegetable oils with higher C18:1 fatty acid content can be prioritized as lubricating base oils.

In conclusion, this study illustrates the potential benefits of using the QSPR model to develop and analyze environmentally friendly lubricants derived from vegetable oils. These lubricants are formulated to achieve optimal anti-wear and friction reduction performance while also decreasing the use of mineral oil resources and protecting the environment. Additionally, the established structure–property relationship can offer valuable data support and theoretical insights for identifying vegetable oils with potential industrial applications in the future. Furthermore, this method can be extended to other physicochemical properties of vegetable oils and serve as a reference for their advancement in other fields.