New insight into statistics of the autohesion origination in polymers: a symmetric amorphous polyether-polyether interface

New insight into statistics of the autohesion origination in polymers: a symmetric amorphous polyether-polyether interface | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Short Report New insight into statistics of the autohesion origination in polymers: a symmetric amorphous polyether-polyether interface Yuri M. Boiko This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8308321/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract For the first time, a comprehensive analysis of the statistical distribution of the lap-shear strength ( σ ) developed at a symmetric interface of amorphous poly(2,6-dimethyl-1,4-phenylene oxide) (PPO) at a temperature of the autohesion origination ( T a origin ) is carried out. After the contact of the two PPO specimens for 24 h at a temperature well below the PPO bulk glass transition temperature ( T g b ), at T a origin = T g b − 126 o C, the PPO−PPO interface was self-bonded and thereafter fractured at ambient temperature. The σ distribution dataset σ ( n ) including n measurements is investigated on its conformity to the Weibull and normal distributions. It is shown that the Weibull and normal probability plots, the Gaussian probability density function histogram, and a number of the standard tests for normality describe the σ distribution with varying degrees of success. Among those, the most favorable statistical results supporting the null hypothesis of normality are obtained in the standard tests for normality (in 6 out of 8 performed). An attempt to describe the distribution curve σ ( n ) more correctly is undertaken. It is found that the most appropriate distribution forms are the phenomenological logarithmic and exponential relationships log σ ∼ n , σ ∼ e (a + b n ) and σ ∼ (1– e a n ) where a and b are the constants. amorphous polymers interface self-healing adhesion statistics Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction Once two glassy polymer pieces are brought into physical contact at the molecular level, the autohesion or adhesion (joining of two identical or two dissimilar materials, respectively) have little, if any, chance to occur since the interaction between the contacting samples’ surfaces is provided exclusively by weak van der Waals (v-d-W) forces between the surface molecular groups [1−4]. In fact, the v-d-W forces represent a weak electrical attraction between neutral atoms and molecules arising from temporary dipole moments induced due to varying electron positions. For this reason, those forces are not sufficient to give rise to the formation of a mechanically sustainable adhesive joint (AJ). However, if the surface polymer chains may diffuse across the interface, the interdiffusion depth increases with time ( t ) with the corresponding increase in the concentration of the new-created v-d-W intermolecular bonds per unit of the contact area, and, as a consequence, with an increase in the AJ strength ( σ ) [5−13]. Statistical characterization of the self-healing process of weak polymer−polymer interfaces, in particular, at a temperature of the autohesion or adhesion origination ( T a origin ), is an interesting and new developing field in polymer science [14−16]. Actually, T a origin seems to be an important thermal transition characteristic of a polymer material since it divides the range of the self-healing temperatures ( T ) into two intervals, T T a origin , wherein the chain mobility differs drastically. At T < T a origin , the long-range translation-rotation segmental motion in the chain backbone, a key factor contributing to the adhesion between polymers, is not activated at the polymer surface whereas it is at T > T a origin . Therefore, it is of principal significance as whether the polymer system is at T T a origin since no adhesion is feasible at T < T a origin at the expense of the v-d-W interaction between the surface molecular groups only. So, studying the statistical aspects of these phenomena at T = T a origin provides a better understanding of the problems of the activation of the chain interdiffusion and of the self-bonding strength ( σ ) origination. Indeed, using various statistical methods to probe the σ distribution behavior gives a deeper insight into the mechanisms of the interface bonding and fracture. For instance, if the Weibull method is more correct than the Gaussian one, it suggests that the brittle fracture mechanism dominates. If the Gaussian distribution holds, it implies that the fracture mechanism is controlled by many independent and equally weighted factors. However, it should be noted that, within the normal distribution itself, there is an uncertainty in the criteria of the validity of normality estimated in the different normal distribution methods [14−16] which should be clarified. The statistical self-healing behavior of a number of weak polymer−polymer interfaces such as the polystyrene (PS)−PS, poly(methyl methacrylate) (PMMA)−PMMA, PS−PMMA, and PS−poly(2,6-dimethyl-1,4-phenylene oxide) (PPO) interfaces has been investigated at T < T g b in detail (see, e.g., [14−16]. It is noteworthy that the diffusion-controlled adhesion at T < T g b would not be feasible (i) if the surface T g ( T g s ) would not be decreased as compared to T g b [17−22] and (ii) if the interface contact layer would not preserve the viscoelastic physical state of the surface at the early stages of the interface self-healing [ 23 ]. In this respect, studying the molecular mechanisms of the early stages of the interface self-healing process at T < T g b is an interesting and exclusive subject of investigation. Actually, on one hand, the long-range segmental motion is frozen in the bulk of the contacted sample. On the other hand, this mode of the molecular motion can be activated at its interface, giving rise to the development of a certain level of the adhesion strength ( σ ). So, it is important to investigate the molecular mechanisms of the early stages of the interface self-healing process at T < T g b , especially, at T = T a origin . As far as a symmetric polyether−polyether interface is concerned, to our knowledge, it has not yet been investigated in this respect. This situation seems to be surprising because namely such a polyether as PPO has demonstrated the best self-bonding ability reported in the literature so far ( T a origin = T g b − 126 o C) [ 23 ]. So, a comprehensive statistical analysis of the lap-shear strength ( σ ) distribution at the PPO−PPO interface should be initiated, and the most interesting is to investigate the σ distribution at this interface after its self-healing at T a origin , i.e. at the threshold of the autohesion origination. Thus, the objective of this work is to perform, for the first time, a comprehensive statistical analysis of the σ distribution at the PPO−PPO interface after its self-healing at T a origin = T g b − 126 o C by employing the approach proposed and developed earlier [14−16] for other weak polymer−polymer interfaces. It includes the construction and analysis (i) of the Weibull and (ii) normal probability plots, and (iii) a PDF ( σ ) histogram. Besides, (iv) a series of the standard tests for normality such as the Shapiro–Wilk, Lilliefors, Kolmogorov–Smirnov, Anderson–Darling, Chen–Shapiro and three D’Agostino–K squared tests [24−28] will be performed. In those tests, the acceptance of the null hypothesis of normality ( H 0 ) will be checked by a comparison of the so-called normal probability ( p -) parameter calculated using software provided by OriginLab with a generally accepted critical level of significance of p of 0.05. The latter will be served as a measure of the acceptance of H 0 if p ≥ 0.05. Statistical analysis The Weibull model describes the probability of failure of identical samples (in the present work, of the AJs self-bonded at identical time-temperature conditions) at or below a stress σ via the cumulative probability function Р ( σ ) as [29−34] Р ( σ ) = 1 − exp [−( σ / σ 0 ) m ] (1), where m is the shape parameter (the so-called Weibull modulus), which is a measure of the data scatter, and σ 0 is the scale parameter (an equivalent of σ av in normal statistics). The cumulative probability of failure of the j th result in the set of n fractured AJs is given by Р j = ( j − 0.5)/ n (2). By rearranging Eq. (1), taking the natural logarithm of its both sides, and replacing Р ( σ ) by Р j , one obtains Eq. (3): lnln[1/(1 − Р j )] = m ⋅ ln σ − m ⋅ ln σ 0 (3). Eq. (3) can be expressed in a simpler form as y = a + bx (4), where y = lnln[1/(1 − Р j )], b = m , x = ln σ , and a = − m ⋅ln σ 0 . After estimating m as a tangent to the linear curve “lnln[1/(1 − Р j )] − ln σ ”, one can easily calculate the σ 0 value as σ 0 = exp (− a / m ) (5). It should be noted that, prior to construct the Weibull plot giving an opportunity to calculate the two statistical parameters − the data scatter parameter m and the mean σ value σ 0 − several calculation steps should be carried out by employing Eq. (2), Eq. (3), Eq. (4), and Eq. (5), which is a time-consuming procedure. By contrast, the normal probability analysis provided by OriginLab software requires only the input of the σ values into the datasheet. Thereafter, the construction of the normal probability plot generated in the form “normal percentile− σ ”, its linear fit, and the calculation of the arithmetic average σ value ( σ av ) and standard deviation (SD) are performed automatically. So, the latter method seems to be easier than the former if the estimates of the mean value and its scatter are needed only. The σ 0 value calculated in the Weibull analysis using Eq. (5) will be compared with the σ av value calculated in the normal probability analysis because these both parameters characterize the mean adhesion strength. The data scatter parameters estimated in the Weibull and normal probability statistical methods are m and SD, respectively. A direct comparison of these parameters is meaningless, first, since m is dimensionless while SD has the units of stress. Second, an increase in m and an increase in SD mean a decrease and an increase in in the data scatter, respectively. Hence, a correct quantitative comparison of these two data scatter parameters can be made by reducing them, first, to the same trend of their variations (i.e., whether an increase or a decrease) and, second, to the same units. Recently [14−16], it has been argued that the only way to solve this problem is to compare 1/ m with SD/ σ av , when the two parameters become dimensionless. So, this approach will be employed in this work as well. Besides, reliability each of these methods will be compared via root mean square deviation ( R 2 ), R 2 (W) calculated when performing the linear fitting of the graphs “lnln[1/(1 − Р j )] − ln σ ”, and R 2 (n) calculated when performing an additional linear fitting of the graphs “normal percentile − σ ” generated by OriginLab software. Materials and methods Materials The polymer used in this study was a commercial amorphous PPO (General Electric, USA) with a number-average molecular weight M n = 23 kg/mol and a weight-average molecular weight M w = 44 kg/mol, and T g b = 216 o C. The samples of PPO represented the rectangular strips with a width of 5 mm, a length of 30 mm and a thickness of ∼100 µm which were cut from the central part of the extruded tapes produced by melt extrusion using a Haake-Bühler Rheocord System 40 tween-screw lab extruder. This central tape portion did not touch the receiving rolls of the extruder because it was thinner as compared to the tape edges. This allowed the polymer melt free surface to equilibrate until its solidification and avoid an undesirable surface perturbation by preventing the contact of the central tape portion with the rolls, thus, making it smooth. Interface self-healing procedure In the self-healing experiments, the two PPO samples were brought into contact at T = 90 o C (= T g b – 126 o C) at an overlapped area of 5 × 5 mm 2 under a small contact pressure of 0.8 MPa [ 2 , 9 – 11 ] and held for 24 h in a Carver press. The as-formed mechanically sustainable PPO−PPO lap-shear AJ were submitted to mechanical testing. Fracture tests The PPO−PPO AJs were shear-fractured in tension on an Instron-5565 tensile tester at ambient temperature at a crosshead speed of 5 mm/min. The distance between the tester grips was 50 mm with the joint located in the middle. The interface lap-shear strength σ was calculated as fracture load divided by the contact area of 5×5 mm 2 . The procedures of the interface self-healing and fracture thereof are schematically depictured in Fig. 1 . Experiments were performed according to ASTM D3163−01 (2023). Results and discussion In Fig. 2 , the values of σ developed at the PPO–PPO interface at T h = T g b – 126 o C for 24 h are plotted as a function of the AJ number n . To construct this plot, the σ dataset, first, was arranged in ascending order and, second, each of the σ values of the dataset obtained was assigned to the corresponding n value. To put it differently, e.g., the lowest and highest σ values were coupled with n = 1 and 17, respectively. As is seen, σ monotonically increases with n . Analyze this plot in the frameworks of the Weibull and Gaussian distributions, the two statistical approaches which have been most widely used in the literature [ 29 – 40 ]. Both of them are appropriate for this purpose. In particular, because the Weibull model [ 29 – 34 ] has been proposed initially for brittle materials, and weak polymer–polymer interfaces belong namely to this type of objects [ 14 – 16 ]. The Weibull plot for the dataset of Fig. 2 is constructed in Fig. 3 A. The Gaussian model has been used for a tremendous variety of the properties and phenomena of various natures (see, e.g. [ 35 – 40 ]). This model has several different methods, and three of them will be used in this work: the construction of the normal probability plot (see Fig. 3 B) and the histogram PDF ( σ ) (see Fig. 3 C), and performing a number of the standard tests for normality (see the histogram in Fig. 3 D and Table 1 ). Table 1 Statistical parameters of the lap-shear strength distribution at the PPO–PPO interface self-bonded at T = T g b − 126 o C for 24 h estimated in several tests for normality Test type Statistic p -value Decision at level 5%* Shapiro–Wilk Lilliefors Kolmogorov–Smirnov Anderson–Darling D’Agostino–K squared: Omnibus Skewness Kurtosis 0.86687 0.19298 0.19298 0.66249 2.07714 1.15878 –0.85696 0.04751 0.1876 0.67358 0.06408 0.35396 0.24655 0.39147 – (±) + + + + + + * “+” and “–” in column 5 mean “can’t reject normality” and “reject normality”, respectively. As follows from Figs. 3 A,B, to a first approximation, both the Weibull and normal probability plots can be fitted with a linear function. However, the fitting results which are characterized by rather low values of R 2 (W) = 0.937 and R 2 (n) = 0.942 cannot be treated as satisfactory in view of the observed deviations from linearity at very small strengths. In part, this behavior may be explained by the instability of the self-healing process at the adhesion strength origination threshold, at T a origin . Nevertheless, these two procedures give an opportunity to compare the statistical parameters estimated in different ways and basing on the completely different statistical basis and data treatment in this interesting but poorly studied low-temperature self-healing range T a origin < < T g b (see Table 2 ). Table 2 Results of the normal probability and Weibull analysis of the adhesion strength distribution at the PPO–PPO interface self-bonded at T = T g b − 126 o C for 24 h σ av , MPa SD, MPa SD/ σ av R 2 (n) σ 0 , MPa m R 2 (W) σ 0 / σ av 1/ m (SD/ σ av )/(1/ m ) 0.03338 0.02453 0.7349 0.942 0.0371 1.434 0.937 1.11 0.6974 1.05 The calculation of the two Weibull statistical parameters, the Weibull modulus m (a data scatter parameter) and the scale parameter σ 0 (an equivalent of σ av ), has been performed from the constructed Weibull plot as the curve slope ( m ) and the curve intersection point with the ordinate axis ( σ 0 ). Prior to make a correct comparison between the calculated values of σ 0 and m with the corresponding normal probability statistical parameters SD and σ av , which were calculated automatically by the standard OriginLab software when constructing the normal probability plot, the following points should be addressed. First, the data scatter parameters SD and m should be reduced to one and the same units. Second, they should characterize one and the same trend of variation (an increase or a decrease). Since SD has the units of stress (MPa) whereas m is dimensionless, on one hand, it has been proposed by us recently [14−16] to obtain a dimensionless normal probability parameter by normalizing SD by σ av and introducing a dimensionless parameter SD/ σ av . Second, since an increase in the normal statistical parameter SD means an increase in the data scatter while an increase in the Weibull statistical parameter m means, vice versa, its decrease, the problem of the correct comparison can be solved easily by introducing the reciprocal m value (1/ m ). Now, the two modified data scatter statistical parameters, SD/ σ av and 1/ m , first, are dimensionless and, second, both of them characterize the same trend in the data scatter variation, its increase with an increase in SD/ σ av or1/ m and its decrease with a decrease in SD/ σ av or1/ m . So, they can be compared correctly. Furthermore, as a more convenient way of their comparison, their ratio (SD/ σ av )/(1/ m ) can be estimated and related to a reference point of 1 for clarity. Actually, as follows from Table 2 , (SD/ σ av )/(1/ m ) = 1.05 which is close to unity. Therefore, in spite of the completely different statistical basis and data treatment procedures used for the Weibull and normal distributions, the two methods give similar statistical parameters: average values and data scatter. Second, by analogy, compare the mean strength values σ 0 and σ av estimated using the Weibull and normal probability approaches, respectively. As is seen, the ratio σ 0 / σ av is equal to 1.11 which is also close to unity. It is interesting that these two different approaches, rather simple procedure to calculate σ av and more sophisticated procedure to calculate σ 0 , give similar results which are mutually correlated. In this sense, an easier obtained σ av value may be recommended to estimate the σ 0 value. So, it may be concluded that any of these methods can be used to estimate the mean strength and the dispersion parameter despite the fact that the shape of the Weibull and normal probability plots, especially the latter one, did not represent the perfect linear fitting curves with R 2 → 1. In this respect, the estimate of these parameters via the construction of the normal probability plots may be recommended as a less sophisticated procedure which requires only the input of the experimental data points into a software datasheet to be analyzed. Expand the normal probability analysis by considering the histogram PDF( σ ) (see Fig. 3 C). As is seen, the lap-shear strength values are distributed not uniformly and shifted to low strengths. For this reason, the histogram does not have the shape of a bell curve. Hence, it does not conform to the normal distribution. As far as the tests for normality are concerned [see Table 1 and Fig. 3 D wherein the p -values are proposed to be visualized as a histogram p (test type) for clarity], the majority of them (six out of seven) evidence in favor of the validity of normality, especially the Kolmogorov-Smirnov and D’Agostino–K squared tests ( p > 0.247). The only exception is the Shapiro-Wilk test with p = 0.048. However, that p -value is very close to the critical p -value of 0.05 accepting H 0 . So, it may be concluded that, in total, the results obtained in the standard tests for normality listed in Table 1 and depictured in Fig. 3 D accept H 0 and testify in favor of the validity of the normal distribution for the σ values developed at the PPO–PPO interface at the autohesion threshold conditions, T = T g b – 126 o C and t = 24 h. The results of one more, the Chen–Shapiro test are a specified statistical value ( s ) and a critical value ( c ) at a 5% significance level instead of a p -value considered above in the other tests. In this test, H 0 is rejected if s > c . The values of c and s calculated in this test were 0.06116 and 0.06903, respectively, i.e. s > c . Hence, the Chen–Shapiro test rejects H 0 . Nevertheless, in total, the majority of the normality tests carried out (six out of eight) support the conformity of the σ distribution to normality. The statistical analysis carried out above has revealed a rather sophisticated character of the statistical distribution of the adhesion strength developed at the PPO–PPO interface at the autohesion threshold time-temperature conditions. In particular, a discrepancy between the arguments of the validity of normality has been observed even within one and the same distribution type, the normal distribution. The Weibull model also turned out not to be working properly, in contrast to other weak polymer–polymer interfaces for which it worked properly [ 14 – 16 ], including self-healing at T = T a origin . So, these results motivate searching for more appropriate analytical descriptions of the σ statistical distribution at the PPO–PPO interface at the autohesion origination. For this purpose, first, consider the simplest analytical form σ = a + b n where ‘a’ and ‘b’ are the constants and apply it to analyze the initial dataset σ ( n ) using a linear fitting procedure (see Fig. 4 A). Now, the fitting result with Eq. (6) with R 2 = 0.956 seems to be more correct as compared to the Weibull and normal probability plots with R 2 = 0.937 and 0.942, respectively, though the dataset approximation with two straight lines looks more preferable. Nevertheless, it is quite surprising that the two classical statistical Weibull and Gaussian models describe the σ distribution less satisfactorily than even the simplest relationship σ ∼ n . Further improving the data description can be obtained by plotting them in the coordinates log σ – n , i.e. in the form log σ = a + b n . In this case, a linear fitting result with Eq. (7) with R 2 = 0.985 (see Fig. 4 B) is the best as compared to the others estimated above. It implies that the σ distribution is expected to obey an exponential function. Actually, after considering the dataset as ln σ vs n (see Fig. 4 C), its linear fitting with Eq. (8) with R 2 = 0.985. To express σ directly via n , Eq. (8) is transformed to Eq. (9). Alternatively, the original dataset can be directly described as y = a[1– exp(–b x )] (see Fig. 4 D) with Eq. (10), which seems also to be correct, though the fitting result is less reliable ( R 2 = 0.970) but still satisfactory. σ = − 0.00877 + 0.00602 n (6) log σ = − 2.21638 + 0.08834 n (7) ln σ = − 5.1034 + 0.20342 n (8) σ = e (–5.1034 + 0.20342 n ) (9) σ = − 0.02034 (1– e 0.12271 n ) (10) So, quite surprisingly, it turned out that the fairly unsophisticated functions can more correctly describe the lap-shear strength distribution upon the autohesion origination at the PPO–PPO interface as compared to those used in the classical normal and Weibull distributions. It is interesting to note that, to our knowledge, the statistical laws of such types have not been used earlier. Hence, they can be employed to analyze the statistical distributions of other properties and phenomena of various natures. The results obtained in the present work motivate further investigations of the adhesion strength development at the PPO–PPO and other polymer−polymer interfaces after self-healing at T ≥ T a origin on the conformity to these new phenomenological statistical relationships. At the same time, it should be noted that namely an excellent self-bonding ability of PPO as compared to those of other polymers [ 23 ] made it possible to investigate the adhesion strength distribution at such low T as T a origin = T g b – 126 o C. This temperature reduced to the PPO T g b is notably lower and still not attainable for other polymers. Actually, the following values of T a origin have been reported for the PS−PS, PMMA–PMMA, and PET–PET interfaces, respectively: T a origin = T g b – 53 o C, T a origin = T g b – 65 o C, and T a origin = T g b – 17 o C [ 23 ]. The state-of-the-art of this issue requires its further clarification by investigating the interfaces of other polymers in this respect. Conclusions The comprehensive analysis of the statistical distribution of the lap-shear strength σ developed at the symmetric amorphous PPO−PPO interface at the temperature of the autohesion origination T a origin corresponding to an extremely low T as compared to T g b , T = T g b − 126 o C, has been performed. The distribution dataset σ ( n ) for the self-bonded PPO−PPO interface has been investigated on its conformity to the Weibull and normal distributions. It has been shown that the Weibull and normal probability plots, the Gaussian probability density function histogram, and a number of the standard tests for normality describe the σ distribution with varying degrees of success. Among the methods of the normal distribution analysis, the most favorable statistical results supporting the conformity to normality have been obtained in the standard tests for normality. The disagreement between the three different methods used for checking the validity of the normal distribution points to a rather sophisticated statistical character of the adhesion strength origination at the PPO−PPO interface. It has been shown that the mean strength values σ av and σ 0 and the data scatter parameters SD/ σ av and 1/ m estimated using the normal probability and Weibull approaches, respectively, are close. It suggests their general statistical nature in spite of the completely different statistical basis and data treatment procedures involved. It has been found that the most appropriate analytical forms to describe the distribution curve σ ( n ) at the PPO−PPO interface are log σ ∼ n , σ ∼ e (a + b n ) and σ ∼ (1– e a n ). These findings suggest that there is a deviation from the classical statistical laws which are not working properly whether at such extremely low self-bonding temperatures or for polyethers in particular. In fact, due to the best self-bonding ability of PPO reported so far in the literature, it turned out to be possible to find the new phenomenological statistical relationships between σ and n which have not been used yet. In this respect, the question arises concerning the existence of a ‘statistical transition temperature’ below which the adhesion strength distribution at a polymer–polymer interface does not obey the classical statistical models but above which it does. An answer to this important question requires carrying out the statistical investigations on the lap-shear strength distributions for the PPO−PPO interface at T > T a origin and for other polymer–polymer interfaces, which are characterized by markedly higher T a origin ’s reduced to T g b as compared to that for PPO, at T ≥ T a origin . This work is now in progress, and its results will be reported in the near future. Declarations CRediT authorship contribution statement Yuri M. 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Phys Rev E 85:051106. https://doi.org/10.1103/PhysRevE.85.051106 Iwuoha SE, Seim W, Olaniran SO (2023) Statistical distributions and their influence on the material property values of tropical timber: case study of. Gmelina Arborea Struct 53:205–213. https://doi.org/10.1016/j.istruc.2023.04.059 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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1","display":"","copyAsset":false,"role":"figure","size":65675,"visible":true,"origin":"","legend":"\u003cp\u003eGeneral sketch of the experimental procedures of self-bonding of a PPO-PPO interface followed by fracture of as-formed AJ PPO–PPO\u003c/p\u003e","description":"","filename":"1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-8308321/v1/7a2eb3ff906673175d6de92b.jpeg"},{"id":100692929,"identity":"fb9c1bf1-9acd-4d3c-b325-9229b90fcfee","added_by":"auto","created_at":"2026-01-20 14:23:59","extension":"jpeg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":44828,"visible":true,"origin":"","legend":"\u003cp\u003eAdhesion strength \u003cem\u003es \u003c/em\u003edeveloped at a PPO–PPO interface\u003csup\u003e \u003c/sup\u003eat \u003cem\u003eT\u003c/em\u003e = \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e – 126 \u003csup\u003eo\u003c/sup\u003eC for 24 h plotted in ascending order as a function of AJ number \u003cem\u003en\u003c/em\u003e\u003c/p\u003e","description":"","filename":"2.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-8308321/v1/31d8113303631dd5a8f344d0.jpeg"},{"id":100693042,"identity":"2fc08750-94ab-4019-9157-7a1142477c32","added_by":"auto","created_at":"2026-01-20 14:25:24","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":291896,"visible":true,"origin":"","legend":"\u003cp\u003e(A) Weibull and (B) normal probability plots, and (C) Gaussian and (D) \u003cem\u003ep\u003c/em\u003e-value histograms for adhesion strength \u003cem\u003es \u003c/em\u003eof a PPO–PPO interface developed at \u003cem\u003eT \u003c/em\u003e= \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e - 126 \u003csup\u003eo\u003c/sup\u003eC for 24 h\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-8308321/v1/8c0886f4355fb8ca2a27e893.png"},{"id":100693365,"identity":"3d60115c-64f2-4239-baf1-4493525a61d5","added_by":"auto","created_at":"2026-01-20 14:28:11","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":291830,"visible":true,"origin":"","legend":"\u003cp\u003e(A) Adhesion strength \u003cem\u003es \u003c/em\u003eas a function of AJ number \u003cem\u003en\u003c/em\u003e for a PPO–PPO interface self-healed at \u003cem\u003eT \u003c/em\u003e= \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e - 126 \u003csup\u003eo\u003c/sup\u003eC for 24 h (a linear fit is shown with the solid line; the dashed lines are drawn as a guide to the eye),\u003cem\u003e \u003c/em\u003e(B) its semi-log plot (a linear fit is shown with the solid line), and (C) its exponential fit (shown with the solid line)\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-8308321/v1/729bf60bad7288aec0b92ac4.png"},{"id":104783468,"identity":"d5eaf28e-a3a4-4e2c-87ff-81343ab08764","added_by":"auto","created_at":"2026-03-17 07:59:03","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1322536,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8308321/v1/c005c400-99ad-48d9-86e4-e23ec2f4d12f.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"New insight into statistics of the autohesion origination in polymers: a symmetric amorphous polyether-polyether interface","fulltext":[{"header":"Introduction","content":"\u003cp\u003eOnce two glassy polymer pieces are brought into physical contact at the molecular level, the autohesion or adhesion (joining of two identical or two dissimilar materials, respectively) have little, if any, chance to occur since the interaction between the contacting samples\u0026rsquo; surfaces is provided exclusively by weak van der Waals (v-d-W) forces between the surface molecular groups [1\u0026minus;4]. In fact, the v-d-W forces represent a weak electrical attraction between neutral atoms and molecules arising from temporary dipole moments induced due to varying electron positions. For this reason, those forces are not sufficient to give rise to the formation of a mechanically sustainable adhesive joint (AJ). However, if the surface polymer chains may diffuse across the interface, the interdiffusion depth increases with time (\u003cem\u003et\u003c/em\u003e) with the corresponding increase in the concentration of the new-created v-d-W intermolecular bonds per unit of the contact area, and, as a consequence, with an increase in the AJ strength (\u003cem\u003eσ\u003c/em\u003e) [5\u0026minus;13].\u003c/p\u003e \u003cp\u003eStatistical characterization of the self-healing process of weak polymer\u0026minus;polymer interfaces, in particular, at a temperature of the autohesion or adhesion origination (\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e), is an interesting and new developing field in polymer science [14\u0026minus;16]. Actually, \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e seems to be an important thermal transition characteristic of a polymer material since it divides the range of the self-healing temperatures (\u003cem\u003eT\u003c/em\u003e) into two intervals, \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e and \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e, wherein the chain mobility differs drastically. At \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e, the long-range translation-rotation segmental motion in the chain backbone, a key factor contributing to the adhesion between polymers, is not activated at the polymer surface whereas it is at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e. Therefore, it is of principal significance as whether the polymer system is at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e or at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e since no adhesion is feasible at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e at the expense of the v-d-W interaction between the surface molecular groups only. So, studying the statistical aspects of these phenomena at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e provides a better understanding of the problems of the activation of the chain interdiffusion and of the self-bonding strength (\u003cem\u003eσ\u003c/em\u003e) origination.\u003c/p\u003e \u003cp\u003eIndeed, using various statistical methods to probe the \u003cem\u003eσ\u003c/em\u003e distribution behavior gives a deeper insight into the mechanisms of the interface bonding and fracture. For instance, if the Weibull method is more correct than the Gaussian one, it suggests that the brittle fracture mechanism dominates. If the Gaussian distribution holds, it implies that the fracture mechanism is controlled by many independent and equally weighted factors. However, it should be noted that, within the normal distribution itself, there is an uncertainty in the criteria of the validity of normality estimated in the different normal distribution methods [14\u0026minus;16] which should be clarified.\u003c/p\u003e \u003cp\u003eThe statistical self-healing behavior of a number of weak polymer\u0026minus;polymer interfaces such as the polystyrene (PS)\u0026minus;PS, poly(methyl methacrylate) (PMMA)\u0026minus;PMMA, PS\u0026minus;PMMA, and PS\u0026minus;poly(2,6-dimethyl-1,4-phenylene oxide) (PPO) interfaces has been investigated at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e in detail (see, e.g., [14\u0026minus;16]. It is noteworthy that the diffusion-controlled adhesion at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e would not be feasible (i) if the surface \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e (\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003es\u003c/sup\u003e) would not be decreased as compared to \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e [17\u0026minus;22] and (ii) if the interface contact layer would not preserve the viscoelastic physical state of the surface at the early stages of the interface self-healing [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. In this respect, studying the molecular mechanisms of the early stages of the interface self-healing process at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e is an interesting and exclusive subject of investigation. Actually, on one hand, the long-range segmental motion is frozen in the bulk of the contacted sample. On the other hand, this mode of the molecular motion can be activated at its interface, giving rise to the development of a certain level of the adhesion strength (\u003cem\u003eσ\u003c/em\u003e). So, it is important to investigate the molecular mechanisms of the early stages of the interface self-healing process at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e, especially, at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e. As far as a symmetric polyether\u0026minus;polyether interface is concerned, to our knowledge, it has not yet been investigated in this respect. This situation seems to be surprising because namely such a polyether as PPO has demonstrated the best self-bonding ability reported in the literature so far (\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e = \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026minus; 126 \u003csup\u003eo\u003c/sup\u003eC) [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. So, a comprehensive statistical analysis of the lap-shear strength (\u003cem\u003eσ\u003c/em\u003e) distribution at the PPO\u0026minus;PPO interface should be initiated, and the most interesting is to investigate the \u003cem\u003eσ\u003c/em\u003e distribution at this interface after its self-healing at \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e, i.e. at the threshold of the autohesion origination.\u003c/p\u003e \u003cp\u003eThus, the objective of this work is to perform, for the first time, a comprehensive statistical analysis of the \u003cem\u003eσ\u003c/em\u003e distribution at the PPO\u0026minus;PPO interface after its self-healing at \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e = \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026minus; 126 \u003csup\u003eo\u003c/sup\u003eC by employing the approach proposed and developed earlier [14\u0026minus;16] for other weak polymer\u0026minus;polymer interfaces. It includes the construction and analysis (i) of the Weibull and (ii) normal probability plots, and (iii) a PDF (\u003cem\u003eσ\u003c/em\u003e) histogram. Besides, (iv) a series of the standard tests for normality such as the Shapiro\u0026ndash;Wilk, Lilliefors, Kolmogorov\u0026ndash;Smirnov, Anderson\u0026ndash;Darling, Chen\u0026ndash;Shapiro and three D\u0026rsquo;Agostino\u0026ndash;K squared tests [24\u0026minus;28] will be performed. In those tests, the acceptance of the null hypothesis of normality (\u003cem\u003eH\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) will be checked by a comparison of the so-called normal probability (\u003cem\u003ep\u003c/em\u003e-) parameter calculated using software provided by OriginLab with a generally accepted critical level of significance of \u003cem\u003ep\u003c/em\u003e of 0.05. The latter will be served as a measure of the acceptance of \u003cem\u003eH\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e if \u003cem\u003ep\u003c/em\u003e\u0026thinsp;\u0026ge;\u0026thinsp;0.05.\u003c/p\u003e \u003cdiv id=\"Sec2\" class=\"Section2\"\u003e \u003ch2\u003eStatistical analysis\u003c/h2\u003e \u003cp\u003eThe Weibull model describes the probability of failure of identical samples (in the present work, of the AJs self-bonded at identical time-temperature conditions) at or below a stress \u003cem\u003eσ\u003c/em\u003e via the cumulative probability function \u003cem\u003eР\u003c/em\u003e (\u003cem\u003eσ\u003c/em\u003e) as [29\u0026minus;34]\u003c/p\u003e \u003cp\u003e \u003cem\u003eР\u003c/em\u003e(\u003cem\u003eσ\u003c/em\u003e)\u0026thinsp;=\u0026thinsp;1 \u0026minus; exp [\u0026minus;(\u003cem\u003eσ\u003c/em\u003e /\u003cem\u003eσ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e)\u003csup\u003em\u003c/sup\u003e] (1),\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003em\u003c/em\u003e is the shape parameter (the so-called Weibull modulus), which is a measure of the data scatter, and \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e is the scale parameter (an equivalent of \u003cem\u003eσ\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e in normal statistics).\u003c/p\u003e \u003cp\u003eThe cumulative probability of failure of the \u003cem\u003ej\u003c/em\u003eth result in the set of \u003cem\u003en\u003c/em\u003e fractured AJs is given by\u003c/p\u003e \u003cp\u003e \u003cem\u003eР\u003c/em\u003e \u003csub\u003ej\u003c/sub\u003e = (\u003cem\u003ej\u003c/em\u003e \u0026minus; 0.5)/\u003cem\u003en\u003c/em\u003e (2).\u003c/p\u003e \u003cp\u003eBy rearranging Eq.\u0026nbsp;(1), taking the natural logarithm of its both sides, and replacing \u003cem\u003eР\u003c/em\u003e(\u003cem\u003eσ\u003c/em\u003e) by \u003cem\u003eР\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e, one obtains Eq.\u0026nbsp;(3):\u003c/p\u003e \u003cp\u003elnln[1/(1 \u0026minus; \u003cem\u003eР\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e)]\u0026thinsp;=\u0026thinsp;\u003cem\u003em\u003c/em\u003e \u0026sdot; ln\u003cem\u003eσ\u003c/em\u003e \u0026minus; \u003cem\u003em\u003c/em\u003e \u0026sdot; ln\u003cem\u003eσ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e (3).\u003c/p\u003e \u003cp\u003eEq.\u0026nbsp;(3) can be expressed in a simpler form as\u003c/p\u003e \u003cp\u003e \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003ea\u003c/em\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003ebx\u003c/em\u003e (4),\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;lnln[1/(1 \u0026minus; \u003cem\u003eР\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e)], \u003cem\u003eb\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003em\u003c/em\u003e, \u003cem\u003ex\u003c/em\u003e\u0026thinsp;=\u0026thinsp;ln\u003cem\u003eσ\u003c/em\u003e, and \u003cem\u003ea\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u0026minus;\u003cem\u003em\u003c/em\u003e\u0026sdot;ln\u003cem\u003eσ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e. After estimating \u003cem\u003em\u003c/em\u003e as a tangent to the linear curve \u0026ldquo;lnln[1/(1 \u0026minus; \u003cem\u003eР\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e)] \u0026minus; ln\u003cem\u003eσ\u003c/em\u003e\u0026rdquo;, one can easily calculate the \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e value as\u003c/p\u003e \u003cp\u003e \u003cem\u003eσ\u003c/em\u003e \u003csub\u003e0\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;exp (\u0026minus;\u003cem\u003ea\u003c/em\u003e/\u003cem\u003em\u003c/em\u003e) (5).\u003c/p\u003e \u003cp\u003eIt should be noted that, prior to construct the Weibull plot giving an opportunity to calculate the two statistical parameters \u0026minus; the data scatter parameter \u003cem\u003em\u003c/em\u003e and the mean \u003cem\u003eσ\u003c/em\u003e value \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e \u0026minus; several calculation steps should be carried out by employing Eq.\u0026nbsp;(2), Eq.\u0026nbsp;(3), Eq.\u0026nbsp;(4), and Eq.\u0026nbsp;(5), which is a time-consuming procedure. By contrast, the normal probability analysis provided by OriginLab software requires only the input of the \u003cem\u003eσ\u003c/em\u003e values into the datasheet. Thereafter, the construction of the normal probability plot generated in the form \u0026ldquo;normal percentile\u0026minus;\u003cem\u003eσ\u003c/em\u003e\u0026rdquo;, its linear fit, and the calculation of the arithmetic average \u003cem\u003eσ\u003c/em\u003e value (\u003cem\u003eσ\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e) and standard deviation (SD) are performed automatically. So, the latter method seems to be easier than the former if the estimates of the mean value and its scatter are needed only.\u003c/p\u003e \u003cp\u003eThe \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e value calculated in the Weibull analysis using Eq.\u0026nbsp;(5) will be compared with the \u003cem\u003eσ\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e value calculated in the normal probability analysis because these both parameters characterize the mean adhesion strength. The data scatter parameters estimated in the Weibull and normal probability statistical methods are \u003cem\u003em\u003c/em\u003e and SD, respectively. A direct comparison of these parameters is meaningless, first, since \u003cem\u003em\u003c/em\u003e is dimensionless while SD has the units of stress. Second, an increase in \u003cem\u003em\u003c/em\u003e and an increase in SD mean a decrease and an increase in in the data scatter, respectively. Hence, a correct quantitative comparison of these two data scatter parameters can be made by reducing them, first, to the same trend of their variations (i.e., whether an increase or a decrease) and, second, to the same units. Recently [14\u0026minus;16], it has been argued that the only way to solve this problem is to compare 1/\u003cem\u003em\u003c/em\u003e with SD/\u003cem\u003eσ\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e, when the two parameters become dimensionless. So, this approach will be employed in this work as well. Besides, reliability each of these methods will be compared via root mean square deviation (\u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e), \u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e (W) calculated when performing the linear fitting of the graphs \u0026ldquo;lnln[1/(1 \u0026minus; \u003cem\u003eР\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e)] \u0026minus; ln\u003cem\u003eσ\u003c/em\u003e\u0026rdquo;, and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e (n) calculated when performing an additional linear fitting of the graphs \u0026ldquo;normal percentile \u0026minus; \u003cem\u003eσ\u003c/em\u003e\u0026rdquo; generated by OriginLab software.\u003c/p\u003e \u003c/div\u003e"},{"header":"Materials and methods","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003eMaterials\u003c/h2\u003e \u003cp\u003eThe polymer used in this study was a commercial amorphous PPO (General Electric, USA) with a number-average molecular weight \u003cem\u003eM\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e = 23 kg/mol and a weight-average molecular weight \u003cem\u003eM\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e = 44 kg/mol, and \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e = 216 \u003csup\u003eo\u003c/sup\u003eC. The samples of PPO represented the rectangular strips with a width of 5 mm, a length of 30 mm and a thickness of \u0026sim;100 \u0026micro;m which were cut from the central part of the extruded tapes produced by melt extrusion using a Haake-B\u0026uuml;hler Rheocord System 40 tween-screw lab extruder. This central tape portion did not touch the receiving rolls of the extruder because it was thinner as compared to the tape edges. This allowed the polymer melt free surface to equilibrate until its solidification and avoid an undesirable surface perturbation by preventing the contact of the central tape portion with the rolls, thus, making it smooth.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eInterface self-healing procedure\u003c/h3\u003e\n\u003cp\u003eIn the self-healing experiments, the two PPO samples were brought into contact at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;90 \u003csup\u003eo\u003c/sup\u003eC (=\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026ndash; 126 \u003csup\u003eo\u003c/sup\u003eC) at an overlapped area of 5 \u0026times; 5 mm\u003csup\u003e2\u003c/sup\u003e under a small contact pressure of 0.8 MPa [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e, \u003cspan additionalcitationids=\"CR10\" citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] and held for 24 h in a \u003cem\u003eCarver\u003c/em\u003e press. The as-formed mechanically sustainable PPO\u0026minus;PPO lap-shear AJ were submitted to mechanical testing.\u003c/p\u003e\n\u003ch3\u003eFracture tests\u003c/h3\u003e\n\u003cp\u003eThe PPO\u0026minus;PPO AJs were shear-fractured in tension on an Instron-5565 tensile tester at ambient temperature at a crosshead speed of 5 mm/min. The distance between the tester grips was 50 mm with the joint located in the middle. The interface lap-shear strength \u003cem\u003eσ\u003c/em\u003e was calculated as fracture load divided by the contact area of 5\u0026times;5 mm\u003csup\u003e2\u003c/sup\u003e. The procedures of the interface self-healing and fracture thereof are schematically depictured in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Experiments were performed according to ASTM D3163\u0026minus;01 (2023).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"Results and discussion","content":"\u003cp\u003eIn Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, the values of \u003cem\u003e\u0026sigma;\u003c/em\u003e developed at the PPO\u0026ndash;PPO interface at \u003cem\u003eT\u003c/em\u003e\u003csub\u003eh\u003c/sub\u003e = \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026ndash; 126 \u003csup\u003eo\u003c/sup\u003eC for 24 h are plotted as a function of the AJ number \u003cem\u003en\u003c/em\u003e.\u003c/p\u003e\n\u003cp\u003eTo construct this plot, the \u003cem\u003e\u0026sigma;\u003c/em\u003e dataset, first, was arranged in ascending order and, second, each of the \u003cem\u003e\u0026sigma;\u003c/em\u003e values of the dataset obtained was assigned to the corresponding \u003cem\u003en\u003c/em\u003e value. To put it differently, e.g., the lowest and highest \u003cem\u003e\u0026sigma;\u003c/em\u003e values were coupled with \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1 and 17, respectively. As is seen, \u003cem\u003e\u0026sigma;\u003c/em\u003e monotonically increases with \u003cem\u003en\u003c/em\u003e. Analyze this plot in the frameworks of the Weibull and Gaussian distributions, the two statistical approaches which have been most widely used in the literature [\u003cspan class=\"CitationRef\"\u003e29\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e40\u003c/span\u003e]. Both of them are appropriate for this purpose. In particular, because the Weibull model [\u003cspan class=\"CitationRef\"\u003e29\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e34\u003c/span\u003e] has been proposed initially for brittle materials, and weak polymer\u0026ndash;polymer interfaces belong namely to this type of objects [\u003cspan class=\"CitationRef\"\u003e14\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e]. The Weibull plot for the dataset of Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e is constructed in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eA. The Gaussian model has been used for a tremendous variety of the properties and phenomena of various natures (see, e.g. [\u003cspan class=\"CitationRef\"\u003e35\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e40\u003c/span\u003e]). This model has several different methods, and three of them will be used in this work: the construction of the normal probability plot (see Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eB) and the histogram PDF (\u003cem\u003e\u0026sigma;\u003c/em\u003e) (see Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eC), and performing a number of the standard tests for normality (see the histogram in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eD and Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e).\u003c/p\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n\u003ch2\u003e\u0026nbsp;\u003c/h2\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab1\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eStatistical parameters of the lap-shear strength distribution at the PPO\u0026ndash;PPO interface self-bonded at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026minus; 126 \u003csup\u003eo\u003c/sup\u003eC for 24 h estimated in several tests for normality\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eTest type\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eStatistic\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cem\u003ep\u003c/em\u003e-value\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eDecision at level 5%*\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eShapiro\u0026ndash;Wilk\u003c/p\u003e\n\u003cp\u003eLilliefors\u003c/p\u003e\n\u003cp\u003eKolmogorov\u0026ndash;Smirnov\u003c/p\u003e\n\u003cp\u003eAnderson\u0026ndash;Darling\u003c/p\u003e\n\u003cp\u003eD\u0026rsquo;Agostino\u0026ndash;K squared:\u003c/p\u003e\n\u003cp\u003eOmnibus\u003c/p\u003e\n\u003cp\u003eSkewness\u003c/p\u003e\n\u003cp\u003eKurtosis\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.86687\u003c/p\u003e\n\u003cp\u003e0.19298\u003c/p\u003e\n\u003cp\u003e0.19298\u003c/p\u003e\n\u003cp\u003e0.66249\u003c/p\u003e\n\u003cp\u003e2.07714\u003c/p\u003e\n\u003cp\u003e1.15878\u003c/p\u003e\n\u003cp\u003e\u0026ndash;0.85696\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.04751\u003c/p\u003e\n\u003cp\u003e0.1876\u003c/p\u003e\n\u003cp\u003e0.67358\u003c/p\u003e\n\u003cp\u003e0.06408\u003c/p\u003e\n\u003cp\u003e0.35396\u003c/p\u003e\n\u003cp\u003e0.24655\u003c/p\u003e\n\u003cp\u003e0.39147\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026ndash; (\u0026plusmn;)\u003c/p\u003e\n\u003cp\u003e+\u003c/p\u003e\n\u003cp\u003e+\u003c/p\u003e\n\u003cp\u003e+\u003c/p\u003e\n\u003cp\u003e+\u003c/p\u003e\n\u003cp\u003e+\u003c/p\u003e\n\u003cp\u003e+\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003e* \u0026ldquo;+\u0026rdquo; and \u0026ldquo;\u0026ndash;\u0026rdquo; in column 5 mean \u0026ldquo;can\u0026rsquo;t reject normality\u0026rdquo; and \u0026ldquo;reject normality\u0026rdquo;, respectively.\u003c/p\u003e\n\u003cp\u003eAs follows from Figs.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eA,B, to a first approximation, both the Weibull and normal probability plots can be fitted with a linear function. However, the fitting results which are characterized by rather low values of \u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e (W)\u0026thinsp;=\u0026thinsp;0.937 and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e (n)\u0026thinsp;=\u0026thinsp;0.942 cannot be treated as satisfactory in view of the observed deviations from linearity at very small strengths. In part, this behavior may be explained by the instability of the self-healing process at the adhesion strength origination threshold, at \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e. Nevertheless, these two procedures give an opportunity to compare the statistical parameters estimated in different ways and basing on the completely different statistical basis and data treatment in this interesting but poorly studied low-temperature self-healing range \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e (see Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e).\u0026nbsp;\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tab2\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eResults of the normal probability and Weibull analysis of the adhesion strength distribution at the PPO\u0026ndash;PPO interface self-bonded at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026minus; 126 \u003csup\u003eo\u003c/sup\u003eC for 24 h\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e, MPa\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSD, MPa\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSD/\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e (n)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e, MPa\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cem\u003em\u003c/em\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e (W)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e/\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e1/\u003cem\u003em\u003c/em\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e(SD/\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e)/(1/\u003cem\u003em\u003c/em\u003e)\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.03338\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.02453\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.7349\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.942\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.0371\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.434\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.937\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.11\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.6974\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.05\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eThe calculation of the two Weibull statistical parameters, the Weibull modulus \u003cem\u003em\u003c/em\u003e (a data scatter parameter) and the scale parameter \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e (an equivalent of \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e), has been performed from the constructed Weibull plot as the curve slope (\u003cem\u003em\u003c/em\u003e) and the curve intersection point with the ordinate axis (\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e). Prior to make a correct comparison between the calculated values of \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e and \u003cem\u003em\u003c/em\u003e with the corresponding normal probability statistical parameters SD and \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e, which were calculated automatically by the standard OriginLab software when constructing the normal probability plot, the following points should be addressed.\u003c/p\u003e\n\u003cp\u003eFirst, the data scatter parameters SD and \u003cem\u003em\u003c/em\u003e should be reduced to one and the same units. Second, they should characterize one and the same trend of variation (an increase or a decrease). Since SD has the units of stress (MPa) whereas \u003cem\u003em\u003c/em\u003e is dimensionless, on one hand, it has been proposed by us recently [14\u0026minus;16] to obtain a dimensionless normal probability parameter by normalizing SD by \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e and introducing a dimensionless parameter SD/\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e. Second, since an increase in the normal statistical parameter SD means an increase in the data scatter while an increase in the Weibull statistical parameter \u003cem\u003em\u003c/em\u003e means, vice versa, its decrease, the problem of the correct comparison can be solved easily by introducing the reciprocal \u003cem\u003em\u003c/em\u003e value (1/\u003cem\u003em\u003c/em\u003e). Now, the two modified data scatter statistical parameters, SD/\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e and 1/\u003cem\u003em\u003c/em\u003e, first, are dimensionless and, second, both of them characterize the same trend in the data scatter variation, its increase with an increase in SD/\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e or1/\u003cem\u003em\u003c/em\u003e and its decrease with a decrease in SD/\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e or1/\u003cem\u003em\u003c/em\u003e. So, they can be compared correctly. Furthermore, as a more convenient way of their comparison, their ratio (SD/\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e)/(1/\u003cem\u003em\u003c/em\u003e) can be estimated and related to a reference point of 1 for clarity. Actually, as follows from Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, (SD/\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e)/(1/\u003cem\u003em\u003c/em\u003e)\u0026thinsp;=\u0026thinsp;1.05 which is close to unity. Therefore, in spite of the completely different statistical basis and data treatment procedures used for the Weibull and normal distributions, the two methods give similar statistical parameters: average values and data scatter.\u003c/p\u003e\n\u003cp\u003eSecond, by analogy, compare the mean strength values \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e and\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e estimated using the Weibull and normal probability approaches, respectively. As is seen, the ratio\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e/\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e is equal to 1.11 which is also close to unity. It is interesting that these two different approaches, rather simple procedure to calculate\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e and more sophisticated procedure to calculate \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e, give similar results which are mutually correlated. In this sense, an easier obtained \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e value may be recommended to estimate the \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e value.\u003c/p\u003e\n\u003cp\u003eSo, it may be concluded that any of these methods can be used to estimate the mean strength and the dispersion parameter despite the fact that the shape of the Weibull and normal probability plots, especially the latter one, did not represent the perfect linear fitting curves with \u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e \u0026rarr; 1. In this respect, the estimate of these parameters via the construction of the normal probability plots may be recommended as a less sophisticated procedure which requires only the input of the experimental data points into a software datasheet to be analyzed.\u003c/p\u003e\n\u003cp\u003eExpand the normal probability analysis by considering the histogram PDF(\u003cem\u003e\u0026sigma;\u003c/em\u003e) (see Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eC). As is seen, the lap-shear strength values are distributed not uniformly and shifted to low strengths. For this reason, the histogram does not have the shape of a bell curve. Hence, it does not conform to the normal distribution. As far as the tests for normality are concerned [see Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eD wherein the \u003cem\u003ep\u003c/em\u003e-values are proposed to be visualized as a histogram \u003cem\u003ep\u003c/em\u003e (test type) for clarity], the majority of them (six out of seven) evidence in favor of the validity of normality, especially the Kolmogorov-Smirnov and D\u0026rsquo;Agostino\u0026ndash;K squared tests (\u003cem\u003ep\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;0.247). The only exception is the Shapiro-Wilk test with \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.048. However, that \u003cem\u003ep\u003c/em\u003e-value is very close to the critical \u003cem\u003ep\u003c/em\u003e-value of 0.05 accepting \u003cem\u003eH\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e. So, it may be concluded that, in total, the results obtained in the standard tests for normality listed in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e and depictured in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eD accept \u003cem\u003eH\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e and testify in favor of the validity of the normal distribution for the \u003cem\u003e\u0026sigma;\u003c/em\u003e values developed at the PPO\u0026ndash;PPO interface at the autohesion threshold conditions, \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026ndash; 126 \u003csup\u003eo\u003c/sup\u003eC and \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;24 h. The results of one more, the Chen\u0026ndash;Shapiro test are a specified statistical value (\u003cem\u003es\u003c/em\u003e) and a critical value (\u003cem\u003ec\u003c/em\u003e) at a 5% significance level instead of a \u003cem\u003ep\u003c/em\u003e-value considered above in the other tests. In this test, \u003cem\u003eH\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e is rejected if \u003cem\u003es\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;\u003cem\u003ec\u003c/em\u003e. The values of \u003cem\u003ec\u003c/em\u003e and \u003cem\u003es\u003c/em\u003e calculated in this test were 0.06116 and 0.06903, respectively, i.e. \u003cem\u003es\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;\u003cem\u003ec\u003c/em\u003e. Hence, the Chen\u0026ndash;Shapiro test rejects \u003cem\u003eH\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e. Nevertheless, in total, the majority of the normality tests carried out (six out of eight) support the conformity of the \u003cem\u003e\u0026sigma;\u003c/em\u003e distribution to normality.\u003c/p\u003e\n\u003cp\u003eThe statistical analysis carried out above has revealed a rather sophisticated character of the statistical distribution of the adhesion strength developed at the PPO\u0026ndash;PPO interface at the autohesion threshold time-temperature conditions. In particular, a discrepancy between the arguments of the validity of normality has been observed even within one and the same distribution type, the normal distribution. The Weibull model also turned out not to be working properly, in contrast to other weak polymer\u0026ndash;polymer interfaces for which it worked properly [\u003cspan class=\"CitationRef\"\u003e14\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e], including self-healing at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e. So, these results motivate searching for more appropriate analytical descriptions of the \u003cem\u003e\u0026sigma;\u003c/em\u003e statistical distribution at the PPO\u0026ndash;PPO interface at the autohesion origination.\u003c/p\u003e\n\u003cp\u003eFor this purpose, first, consider the simplest analytical form \u003cem\u003e\u0026sigma;\u003c/em\u003e\u0026thinsp;=\u0026thinsp;a\u0026thinsp;+\u0026thinsp;b\u003cem\u003en\u003c/em\u003e where \u0026lsquo;a\u0026rsquo; and \u0026lsquo;b\u0026rsquo; are the constants and apply it to analyze the initial dataset \u003cem\u003e\u0026sigma;\u003c/em\u003e (\u003cem\u003en\u003c/em\u003e) using a linear fitting procedure (see Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003eA). Now, the fitting result with Eq.\u0026nbsp;(6) with \u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.956 seems to be more correct as compared to the Weibull and normal probability plots with \u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.937 and 0.942, respectively, though the dataset approximation with two straight lines looks more preferable. Nevertheless, it is quite surprising that the two classical statistical Weibull and Gaussian models describe the \u003cem\u003e\u0026sigma;\u003c/em\u003e distribution less satisfactorily than even the simplest relationship \u003cem\u003e\u0026sigma;\u003c/em\u003e \u0026sim; \u003cem\u003en\u003c/em\u003e. Further improving the data description can be obtained by plotting them in the coordinates log\u003cem\u003e\u0026sigma;\u003c/em\u003e \u0026ndash; \u003cem\u003en\u003c/em\u003e, i.e. in the form log\u003cem\u003e\u0026sigma;\u003c/em\u003e\u0026thinsp;=\u0026thinsp;a\u0026thinsp;+\u0026thinsp;b\u003cem\u003en\u003c/em\u003e. In this case, a linear fitting result with Eq.\u0026nbsp;(7) with \u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.985 (see Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003eB) is the best as compared to the others estimated above. It implies that the \u003cem\u003e\u0026sigma;\u003c/em\u003e distribution is expected to obey an exponential function. Actually, after considering the dataset as ln\u003cem\u003e\u0026sigma;\u003c/em\u003e vs \u003cem\u003en\u003c/em\u003e (see Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003eC), its linear fitting with Eq.\u0026nbsp;(8) with \u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.985. To express \u003cem\u003e\u0026sigma;\u003c/em\u003e directly via \u003cem\u003en\u003c/em\u003e, Eq.\u0026nbsp;(8) is transformed to Eq.\u0026nbsp;(9). Alternatively, the original dataset can be directly described as \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;a[1\u0026ndash; exp(\u0026ndash;b\u003cem\u003ex\u003c/em\u003e)] (see Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003eD) with Eq.\u0026nbsp;(10), which seems also to be correct, though the fitting result is less reliable (\u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.970) but still satisfactory.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u0026sigma;\u003c/em\u003e = \u0026minus;\u0026thinsp;0.00877\u0026thinsp;+\u0026thinsp;0.00602\u003cem\u003en\u003c/em\u003e (6)\u003c/p\u003e\n\u003cp\u003elog\u003cem\u003e\u0026sigma;\u003c/em\u003e = \u0026minus;\u0026thinsp;2.21638\u0026thinsp;+\u0026thinsp;0.08834\u003cem\u003en\u003c/em\u003e (7)\u003c/p\u003e\n\u003cp\u003eln\u003cem\u003e\u0026sigma;\u003c/em\u003e = \u0026minus;\u0026thinsp;5.1034\u0026thinsp;+\u0026thinsp;0.20342\u003cem\u003en\u003c/em\u003e (8)\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u0026sigma;\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003ee\u003c/em\u003e \u003csup\u003e(\u0026ndash;5.1034 + 0.20342\u003cem\u003en\u003c/em\u003e)\u003c/sup\u003e (9)\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u0026sigma;\u003c/em\u003e = \u0026minus;\u0026thinsp;0.02034 (1\u0026ndash; \u003cem\u003ee\u003c/em\u003e \u003csup\u003e0.12271\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e) (10)\u003c/p\u003e\n\u003cp\u003eSo, quite surprisingly, it turned out that the fairly unsophisticated functions can more correctly describe the lap-shear strength distribution upon the autohesion origination at the PPO\u0026ndash;PPO interface as compared to those used in the classical normal and Weibull distributions. It is interesting to note that, to our knowledge, the statistical laws of such types have not been used earlier. Hence, they can be employed to analyze the statistical distributions of other properties and phenomena of various natures.\u003c/p\u003e\n\u003cp\u003eThe results obtained in the present work motivate further investigations of the adhesion strength development at the PPO\u0026ndash;PPO and other polymer\u0026minus;polymer interfaces after self-healing at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026ge;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e on the conformity to these new phenomenological statistical relationships. At the same time, it should be noted that namely an excellent self-bonding ability of PPO as compared to those of other polymers [\u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e] made it possible to investigate the adhesion strength distribution at such low \u003cem\u003eT\u003c/em\u003e as \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e = \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026ndash; 126 \u003csup\u003eo\u003c/sup\u003eC. This temperature reduced to the PPO \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e is notably lower and still not attainable for other polymers. Actually, the following values of \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e have been reported for the PS\u0026minus;PS, PMMA\u0026ndash;PMMA, and PET\u0026ndash;PET interfaces, respectively: \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e = \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026ndash; 53 \u003csup\u003eo\u003c/sup\u003eC, \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e = \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026ndash; 65 \u003csup\u003eo\u003c/sup\u003eC, and \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e = \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026ndash; 17 \u003csup\u003eo\u003c/sup\u003eC [\u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e]. The state-of-the-art of this issue requires its further clarification by investigating the interfaces of other polymers in this respect.\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003e\u0026nbsp;\u003c/h3\u003e"},{"header":"Conclusions","content":"\u003cp\u003eThe comprehensive analysis of the statistical distribution of the lap-shear strength \u003cem\u003eσ\u003c/em\u003e developed at the symmetric amorphous PPO\u0026minus;PPO interface at the temperature of the autohesion origination \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e corresponding to an extremely low \u003cem\u003eT\u003c/em\u003e as compared to \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e, \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026minus; 126 \u003csup\u003eo\u003c/sup\u003eC, has been performed. The distribution dataset \u003cem\u003eσ\u003c/em\u003e (\u003cem\u003en\u003c/em\u003e) for the self-bonded PPO\u0026minus;PPO interface has been investigated on its conformity to the Weibull and normal distributions. It has been shown that the Weibull and normal probability plots, the Gaussian probability density function histogram, and a number of the standard tests for normality describe the \u003cem\u003eσ\u003c/em\u003e distribution with varying degrees of success. Among the methods of the normal distribution analysis, the most favorable statistical results supporting the conformity to normality have been obtained in the standard tests for normality. The disagreement between the three different methods used for checking the validity of the normal distribution points to a rather sophisticated statistical character of the adhesion strength origination at the PPO\u0026minus;PPO interface.\u003c/p\u003e \u003cp\u003eIt has been shown that the mean strength values \u003cem\u003eσ\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e and \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e and the data scatter parameters SD/\u003cem\u003eσ\u003c/em\u003e\u003csub\u003eav\u003c/sub\u003e and 1/\u003cem\u003em\u003c/em\u003e estimated using the normal probability and Weibull approaches, respectively, are close. It suggests their general statistical nature in spite of the completely different statistical basis and data treatment procedures involved.\u003c/p\u003e \u003cp\u003eIt has been found that the most appropriate analytical forms to describe the distribution curve \u003cem\u003eσ\u003c/em\u003e(\u003cem\u003en\u003c/em\u003e) at the PPO\u0026minus;PPO interface are log\u003cem\u003eσ\u003c/em\u003e \u0026sim; \u003cem\u003en\u003c/em\u003e, \u003cem\u003eσ\u003c/em\u003e \u0026sim; \u003cem\u003ee\u003c/em\u003e\u003csup\u003e(a + b\u003cem\u003en\u003c/em\u003e)\u003c/sup\u003e and \u003cem\u003eσ\u003c/em\u003e \u0026sim; (1\u0026ndash; \u003cem\u003ee\u003c/em\u003e\u003csup\u003ea\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e). These findings suggest that there is a deviation from the classical statistical laws which are not working properly whether at such extremely low self-bonding temperatures or for polyethers in particular. In fact, due to the best self-bonding ability of PPO reported so far in the literature, it turned out to be possible to find the new phenomenological statistical relationships between\u003cem\u003eσ\u003c/em\u003e and \u003cem\u003en\u003c/em\u003e which have not been used yet. In this respect, the question arises concerning the existence of a \u0026lsquo;statistical transition temperature\u0026rsquo; below which the adhesion strength distribution at a polymer\u0026ndash;polymer interface does not obey the classical statistical models but above which it does. An answer to this important question requires carrying out the statistical investigations on the lap-shear strength distributions for the PPO\u0026minus;PPO interface at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e and for other polymer\u0026ndash;polymer interfaces, which are characterized by markedly higher \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e\u0026rsquo;s reduced to \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e as compared to that for PPO, at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026ge;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e. This work is now in progress, and its results will be reported in the near future.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eCRediT authorship contribution statement\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eYuri M. Boiko:\u0026nbsp;\u003c/strong\u003econceptualization (lead), data curation (lead), formal analysis (lead), investigation (lead), methodology (lead), writing – original, draft (lead), writing – review and editing (lead).\u003c/p\u003e\n\u003ch3\u003eDisclosure statement\u003c/h3\u003e\n\u003cp\u003eNo potential conflicts of interest were reported by the author.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData availability statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable.\u003c/p\u003e\n\u003ch3\u003eFunding\u003c/h3\u003e\n\u003cp\u003eThis research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eVoyutskii SS (1963) The diffusion theory of autohesion and adhesion of high polymers. 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Gmelina Arborea Struct 53:205\u0026ndash;213. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1016/j.istruc.2023.04.059\u003c/span\u003e\u003cspan address=\"10.1016/j.istruc.2023.04.059\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"amorphous polymers, interface, self-healing, adhesion, statistics","lastPublishedDoi":"10.21203/rs.3.rs-8308321/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8308321/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFor the first time, a comprehensive analysis of the statistical distribution of the lap-shear strength (\u003cem\u003eσ\u003c/em\u003e) developed at a symmetric interface of amorphous poly(2,6-dimethyl-1,4-phenylene oxide) (PPO) at a temperature of the autohesion origination (\u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e) is carried out. After the contact of the two PPO specimens for 24 h at a temperature well below the PPO bulk glass transition temperature (\u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e), at \u003cem\u003eT\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003csup\u003eorigin\u003c/sup\u003e = \u003cem\u003eT\u003c/em\u003e\u003csub\u003eg\u003c/sub\u003e\u003csup\u003eb\u003c/sup\u003e \u0026minus; 126 \u003csup\u003eo\u003c/sup\u003eC, the PPO\u0026minus;PPO interface was self-bonded and thereafter fractured at ambient temperature. The \u003cem\u003eσ\u003c/em\u003e distribution dataset \u003cem\u003eσ\u003c/em\u003e(\u003cem\u003en\u003c/em\u003e) including \u003cem\u003en\u003c/em\u003e measurements is investigated on its conformity to the Weibull and normal distributions. It is shown that the Weibull and normal probability plots, the Gaussian probability density function histogram, and a number of the standard tests for normality describe the \u003cem\u003eσ\u003c/em\u003e distribution with varying degrees of success. Among those, the most favorable statistical results supporting the null hypothesis of normality are obtained in the standard tests for normality (in 6 out of 8 performed). An attempt to describe the distribution curve \u003cem\u003eσ\u003c/em\u003e(\u003cem\u003en\u003c/em\u003e) more correctly is undertaken. It is found that the most appropriate distribution forms are the phenomenological logarithmic and exponential relationships log\u003cem\u003eσ\u003c/em\u003e \u0026sim; \u003cem\u003en\u003c/em\u003e, \u003cem\u003eσ\u003c/em\u003e \u0026sim; \u003cem\u003ee\u003c/em\u003e\u003csup\u003e(a + b\u003cem\u003en\u003c/em\u003e)\u003c/sup\u003e and \u003cem\u003eσ\u003c/em\u003e \u0026sim; (1\u0026ndash; \u003cem\u003ee\u003c/em\u003e\u003csup\u003ea\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e) where a and b are the constants.\u003c/p\u003e","manuscriptTitle":"New insight into statistics of the autohesion origination in polymers: a symmetric amorphous polyether-polyether interface","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-20 11:58:42","doi":"10.21203/rs.3.rs-8308321/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"b9db309d-cb98-4835-b854-827777ca09ab","owner":[],"postedDate":"January 20th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-04-28T12:08:57+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-20 11:58:42","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8308321","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8308321","identity":"rs-8308321","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}