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# Formation of Crystallites with Low Surface Energy

Masatoshi Tosaka, Shohei Oue, Alice Gros, Bertrand Huneau, Erwan Verron, Sirilux Poompradub

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## To cite this version:

Masatoshi Tosaka, Shohei Oue, Alice Gros, Bertrand Huneau, Erwan Verron, et al., Formation of Crystallites with Low Surface Energy. Journal of Fiber Science and Technology, 2018, 74 (6), pp. 133-142. 10.2115/fiberst.2018-0019. hal-04371341

HAL Id: hal-04371341

https://hal.science/hal-04371341v1

Submitted on 11 Sep 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Formation of Crystallites with Low Surface Energy—
A Key to Understand the Crystallization of
Oriented Polymer

Masatoshi TOSAKA*¹,#, Shohei OUE*¹, Alice GROS*², Bertrand HUNEAU*²,
Erwan VERRON*², and Sirilux Poompradub*³

**Abstract:** In order to demonstrate that lower surface energies of bundle-like crystals have a dominant effect for the enhancement of strain-induced crystallization of natural rubber (NR), an estimation of the end-surface energy ($\sigma_e$) was attempted. NR samples were quickly stretched at predetermined crystallization temperature ($T_c$) to a prescribed draw ratio ($\alpha_s$), and the wide angle X-ray diffraction patterns during the subsequent stepwise heating process were investigated. Melting temperature ($T_m$) was found to depend solely on $\alpha_s$ and to be independent of $T_c$. Simultaneously, average crystallite size in the chain direction ($\langle L_c \rangle$) was primarily determined by network-chain density and was also independent of $T_c$. As a result, the plot of $T_m$ versus $\langle L_c \rangle^{-1}$ suggested that $\sigma_e$ is negligibly low and estimated $T_m$ is an approximation of the equilibrium melting temperature.

## 1. Introduction

Crystalline thermoplastic polymers are widely used in modern society. The melt of these materials are subjected to shear or extensional flow during the molding process of products. The flow induces orientation of polymer chains. The resulted crystalline microstructures greatly affect the mechanical properties of the products. Therefore, correct understanding of the crystallization mechanism in the oriented polymers is an important issue. However, with respect to the crystallization of oriented polymer, even a fundamental framework for understanding has not been established. In order to recognize this situation, we first review polymer crystallization in quiescent conditions[1], which is relatively well-understood.

Under quiescent conditions, thin plate-like crystals (lamellar crystals) are usually formed in supercooled polymer melt. Polymer chains in the

lamellar crystals are folded back and forth on the basal surface. The high energy of the fold surface influences melting and formation of the lamellar crystals as stated below.

Melting of lamellar crystals can be explained by the equilibrium thermodynamics. When the influence of the crystal surface can be ignored, the Gibbs energy change between crystal and amorphous, $\Delta F$, is given by,

$$ \Delta F = \Delta H - T\Delta S \quad (1) $$

where $T$ is temperature, $\Delta H$ and $\Delta S$ are the changes of enthalpy and entropy, respectively, accompanied by the phase transition. The temperature at which $\Delta F = 0$ corresponds to the equilibrium melting temperature, $T_m^0$, such that

$$ T_m^0 = \frac{\Delta H}{\Delta S} \quad (2) $$

<footnote>*¹Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan</footnote>

<footnote>*²Ecole Centrale de Nantes, Institut de Recherche en Génie Civil et Mécanique (GeM), UMR CNRS 6183, BP 9210,
Nantes Cedex 3, France</footnote>

<footnote>*³Department of Chemical Technology, Faculty of Science, Chulalongkorn University, Phaya Thai Rd., Patumwan,
Bangkok 10330, Thailand</footnote>

<footnote># corresponding author: Masatoshi Tosaka (E mail: tosaka@scl.kyoto u.ac.jp)</footnote>
Now, melting of a finite size crystal is affected by the surface energy. As a result, melting temperature, T_m, of lamellar polymer crystals is expressed by the Gibbs-Thomson equation[1],

$$T_m = T_m^0 \left( 1 - \frac{2\sigma_c}{l\Delta H} \right) \quad (3)$$

where σ_c is the surface energy per area and l is the lamellar thickness. On the basis of this equation and knowing ΔH, σ_c can be estimated from the slope of the plot of T_m versus the inverse of lamellar thickness, 1/l. This plot is referred to as the Gibbs-Thomson plot.

How about the formation of lamellar crystals, that is, crystallization? Below T_m^0, primary nuclei are sporadically generated by the Brownian motion of molecules. However, since the nucleation leads to the generation of surface energy, nuclei having a large specific surface area (namely, small ones) disappear. Only when the size of a nucleus exceeds a certain critical size, this nucleus can grow to the larger size. Nuclei having the critical size are referred to as critical nuclei. The change in Gibbs energy for the formation of a critical nucleus can be described theoretically[2]. Moreover, the rate of crystal growth after the formation of a primary nucleus can be described by the theory of Lauritzen and Hoffman[3, 4]. Using this theory, for example, σ_c can be estimated from the dependence of the growth rate on crystallization temperature.

As mentioned above, σ_c can be estimated by two different methods; by the measurement of T_m and by the measurement of crystal growth rate. The values of σ_c obtained by these completely different approaches have been shown to coincide well with each other[3]. Conversely, if σ_c and other required parameters are known, one can predict both T_m and growth rate of polymer crystals in quiescent conditions. In this way, the crystallization of polymer under quiescent conditions has been quantitatively understood in details, already in the 1970 s.

Then, to what extent is crystallization of oriented polymer understood? So far, some morphological and qualitative features have been established[5-8]. However, results given by attempts of quantitatively describing crystallization rate or melting point are lacking of versatility so far. As an example and to the authors' knowledge, we still cannot predict crystal nucleation rate in oriented polymer on the basis of standard thermodynamic parameters[8].

Here, conventional theories describing polymer

crystallization under orientation[9-21] should be reviewed. We can notice that, in either theory, only the rise of melting temperature due to the change in conformational entropy of molecular chains has been considered as the factor that promotes crystallization. A simplified representation of this concept, which is has been first proposed to explain the strain-induced crystallization of natural rubber[9, 10], is as follows[11]. When an amorphous rubber sample is stretched to a draw ratio α, end-to-end distances of the network chains increase. As a result, conformational entropy of the network chains changes by the amount ΔS_{dcf}(α). Therefore, the change of entropy due to phase transition becomes ΔS_a. That is,

$$\Delta S_{\alpha} = \Delta S_{1} - \Delta S_{dcf}(\alpha) \quad (4)$$

where subscripts 1 and α designate the draw ratio. Then, the equilibrium melting temperature, T_m^0, of unstretched sample is given by

$$T_{m,1}^0 = \frac{\Delta H_1}{\Delta S_1} \quad (5)$$

and the one of a stretched sample is

$$T_{m,\alpha}^0 = \frac{\Delta H_{\alpha}}{\Delta S_{\alpha}} \quad (6)$$

If the enthalpy change does not depend on stretch (ΔH_1 = ΔH_α), Eqs. (4-6) lead to

$$\frac{1}{T_{m,\alpha}^0} = \frac{1}{T_{m,1}^0} - \frac{\Delta S_{dcf}(\alpha)}{\Delta H_1} \quad (7)$$

This equation shows that T_{m,α}^0 is higher than T_{m,1}^0. Thus, equilibrium melting temperature is elevated, and accordingly, supercooling increases due to stretching. As a result, crystallization is promoted. This explanation sounds reasonable. However, why can't we use such concept to quantitatively describe crystallization and melting? (An example will be shown shortly.) There has been no appropriate answer to this question; in fact, even such a question has been hardly raised so far. Considering this circumstance, the authors stated that even a fundamental framework for understanding has not been established about the crystallization of the polymer under orientation.

As mentioned above, the concept of the rise of melting temperature due to conformational entropy change was first proposed to explain strain-induced crystallization of rubber. According to the rubber
elasticity theory, we can relate the conformational entropy change $\Delta S_{de}f(\alpha)$ and the tensile stress $f$ as follows[22, 23],

$$ \Delta S_{de}(\alpha) = -\frac{1}{T} \int_{J_1}^{\alpha} f(x) dx \qquad (8) $$

where $x$ is a variable representing the draw ratio. The right-hand-side of this equation corresponds to the area below the uniaxial stress-strain curve. Combination of Eqs. (7-8) indicates that, when we compare unfilled rubbers with different tensile modulus, the one with the higher modulus should show the more pronounced increase in $T_m^0$ during stretching. As a consequence, strain-induced crystallization would occur at the smaller draw ratio for the rubber with the higher modulus. However, it is experimentally observed that for different natural rubbers (NR) with different moduli, crystallization starts at the same draw ratio[23-26]. This example highlights that crystallization of oriented polymers cannot be explained solely by the conformational entropy of molecular chains.

In the way of considering this problem, we examined the condition of nucleation in stretched NR by the combination of nucleation theory and the rubber elasticity theory. As a result, it has been concluded that, in the case of strain-induced crystallization of NR, the activation energy of nucleation is reduced by two or three orders of magnitude because bundle-like nuclei with lower surface energies form[27]. For the reduction of the activation energy, the effect of surface energies overwhelms the effect of the change in conformational entropy. That is to say, the formation of bundle-like nuclei with a lower surface energy is essential in the case of strain-induced crystallization of NR. Such result is quite important, considering the current understanding about the crystallization of oriented polymer.

In that previous work[27], we did not estimate $\sigma_c$. In the present study, we attempt to estimate $\sigma_c$ of the bundle-like nuclei of strain-induced crystals of NR.

## Experimental

### Samples

Sheets (2 mm thick) of sulfur cured NR with different network-chain densities were prepared. NR and other ingredients were mixed on a two-roll mill. The corresponding recipes and curing times are listed

<table><caption>**Table 1** Recipes for NR compounding in the unit of part by weight per hundred part of rubber (phr)</caption><thead><tr><th>Sample code</th><th>NR-S1.125</th><th>NR-S2.25</th><th>NR-S4.5</th></tr></thead><tbody><tr><td>NR<sup>a</sup> (phr)</td><td>100</td><td>100</td><td>100</td></tr><tr><td>Stearic acid (phr)</td><td>2</td><td>2</td><td>2</td></tr><tr><td>ZnO (phr)</td><td>1</td><td>1</td><td>1</td></tr><tr><td>CBS<sup>b</sup> (phr)</td><td>0.75</td><td>1.5</td><td>3</td></tr><tr><td>Sulfur (phr)</td><td>1.125</td><td>2.25</td><td>4.5</td></tr><tr><td>Curing time<sup>c</sup> (min)</td><td>35</td><td>25</td><td>20</td></tr></tbody></table>

<sup>a</sup> RSS #1

<sup>b</sup> N-cyclohexyl-2-benzothiazole sulfenamide

<sup>c</sup> Curing temperature of 140°C

in Table 1. The NR compounds were then cured at 140 °C by compression molding. Stress-strain curves of these samples at various temperatures were obtained using a conventional tensile tester (AGS-1 kNG, Shimadzu Co., Ltd.). From the initial slope of the stress-strain curve, network-chain density, $v$, of each specimen at different temperature was estimated on the basis of the rubber elasticity theory[22]. It is noted that the apparent value of $v$ estimated from the stress-strain curve changes with temperature because effect of entanglement points which play the role of temporal cross-linking points also change with chain mobility. The values of $v$ are listed in Table 2. Ring-shaped specimens (circumference: 50 mm) were punched out from the sample sheets. The initial length corresponds to the half of the circumference, i. e. 25 mm.

### X-ray diffraction measurements

The two-dimensional wide-angle X-ray diffraction (WAXD) patterns were recorded by a conventional X-ray instrument (Nano-Viewer operated at 40 kV, 30 mA, Rigaku Co., Ltd.), in which a custom-made tensile tester was placed[28]. This tensile tester can control specimen temperature between ca. 10 °C and 100 °C. The stretching direction of the specimen was tilted so that the 002 reflection satisfies the Bragg reflection condition. The wavelength of X-ray, $\lambda$, was 0.154 nm and the camera length was ca. 100 mm.

The experimental procedure was as follows. First, temperature of the specimen was adjusted at the predetermined crystallization temperature, $T_c$. The range of $T_c$ was between 20 °C and 60 °C, which was

<table><caption>**Table 2** Network-chain density (mol/cm³) of the sample at various temperature</caption><thead><tr><th>Temperature (°C)</th><th>NR-S 4.5</th><th>NR-S 2.25</th><th>NR-S 1.125</th></tr></thead><tbody><tr><td>20</td><td>2.08 × 10<sup>-4</sup></td><td>1.45 × 10<sup>-4</sup></td><td>0.94 × 10<sup>-4</sup></td></tr><tr><td>40</td><td>1.72 × 10<sup>-4</sup></td><td>1.28 × 10<sup>-4</sup></td><td>0.89 × 10<sup>-4</sup></td></tr><tr><td>60</td><td>1.57 × 10<sup>-4</sup></td><td>1.25 × 10<sup>-4</sup></td><td>0.94 × 10<sup>-4</sup></td></tr><tr><td>80</td><td>1.55 × 10<sup>-4</sup></td><td>1.26 × 10<sup>-4</sup></td><td>1.02 × 10<sup>-4</sup></td></tr><tr><td>100</td><td>1.55 × 10<sup>-4</sup></td><td>1.28 × 10<sup>-4</sup></td><td>0.91 × 10<sup>-4</sup></td></tr></tbody></table>
wider than the previous research in which $T_m^0$ of NR in quiescent condition was determined by the Gibbs-Thomson plot[29]. Then, the specimen was quickly stretched to a prescribed draw ratio, $\alpha_s$, at 1000 mm/s (40 s⁻¹), minimizing crystallization during deformation. After allowing crystallization for more than 5 min, the WAXD pattern was recorded. Next, the specimen temperature was elevated and another WAXD pattern was recorded. The stepwise temperature rising and recording of WAXD patterns were repeated until the crystalline reflections disappear or up to the upper limit temperature of the instrument (100 °C). Figure 1 schematically shows the temperature profile and corresponding WAXD patterns. As shown in this figure, the intensity of crystalline reflections gradually decreases with the increase in temperature.

Each WAXD pattern was converted into the equatorial intensity profile by integrating the intensity in the range of ±10 ° from the equator. Also the meridional intensity profile was obtained from each of the WAXD pattern in the similar way. Each intensity profile was separated into peaks of crystalline reflections and amorphous halo.

<figure><img src="image_4.png" /><figcaption>**Fig. 1** Schematic illustration of the temperature profile</figcaption></figure>

## Results and discussion

First, we have to relate $T_m$ with surface energies. Here, we assume a parallelepiped crystal with side surface energy $\sigma$ and end surface energy $\sigma_e$; these values are defined for a unit area. The sizes of the crystal are $L_a$, $L_b$ and $L_c$; $L_c$ is the size in the molecular-chain direction. By redefining $\Delta F$ to be the value per unit volume, the condition for the stable existence of this crystal is written as

$$2(L_a + L_b)L_c\sigma + 2L_aL_b\sigma_e - L_aL_bL_c\Delta F < 0 \quad (9)$$

(It is noted that this condition is applicable to existing crystals, and accordingly, crystallization process is irrelevant. For example, though instantaneous temperature rise at the moment of rapid extension of the specimen may have certain effect to the crystallization process, we can neglected such an effect in the following analysis.) In the case of lamellar crystal, $L_a$ and $L_b$ are considerably larger than $L_c$. Therefore, Eq. (9) can be simplified to erase $L_a$ and $L_b$ if we divide both sides by $L_aL_b$ and assume $L_c/L_a \approx L_c/L_b \approx 0$. Substituting $\Delta F$ with

$$\Delta F = \Delta H - \frac{T}{T_m}\Delta H \quad (10)$$

which is derived from Eqs. (1) and (2), we can obtain Eq. (3), namely the Gibbs-Thomson equation. In the current case, according to previous studies[24, 25], $L_a$, $L_b$ and $L_c$ are not so different. Strictly speaking, therefore, we have to keep Eq. (9) without approximation, which gives

$$T_m = T_m^0 \left[ 1 - 2 \left( \frac{1}{L_a} + \frac{1}{L_b} \right) \frac{\sigma}{\Delta H} - \frac{2\sigma_e}{L_c\Delta H} \right] \quad (11)$$

and this equation should be used for the estimation of $\sigma$ and $\sigma_e$. In our preliminary examination, such an analysis has been done, assuming $L_a = L_b$ and using the crystallite sizes estimated from the 200 and 002 reflections (Examples are presented in Appendix). However, the conclusion was the same as the one presented below. For the simplicity, we adopt the Gibbs-Thomson plot using the values of $T_m$ and $L_c$ in the following discussion.

For the estimation of $T_m$, a series of equatorial intensity profiles was used. The integrated intensity of the 200-reflection normalized by the total intensity of the equatorial intensity profile was defined as $I_{200}$ and regarded as a measure of crystallinity. (In the case of stretched NR, the amorphous halo is almost isotropic even at very high draw ratio[25, 30-32], and accordingly, localization of amorphous halo on the equator, which could affect the value of $I_{200}$, can be ignored.) As shown in Figure 2a, $I_{200}$ decreased almost linearly with the increase in temperature. Then $T_m$ is defined as the temperature at the intercept of the extrapolation of the linear plot with $I_{200} = 0$. Strictly speaking, the $T_m$ determined in this way reflects the melting of the most thermally stable part of the crystal; melting of other parts occurs at the lower temperature. As shown in Figure 2b, $T_m$ increases with the increase in $\alpha$. In this figure, data points for
<figure><img src="image_5.png" /><figcaption>Fig. 2 Example of the estimation of T_m for NR-S1.125.
(a) Dependence of I<sub>200</sub> on temperature.
(b) Dependence of T<sub>m</sub> on α<sub>s</sub>.</figcaption></figure>

various T_c are mostly overlapping with each other which indicates that T_m is independent of T_c. As shown in Appendix, T_m is also independent of network-chain density of the material[33]. Consequently, in the case of constant-length condition, T_m is determined solely by α_s in the case of strain-induced crystals of NR. This is a totally different situation from the crystallization of linear polymers under quiescent condition.

As for Figure 2a, Trabelsi et al. have reported similar plots for samples drawn at 90 °C and crystallized by keeping at -25°C[24]. Probably due to the different experimental procedure, our results are different from theirs. In their experiment, for example, crystallization took tens of hours to attain saturated crystallinity, while in the current experiment, crystallization is almost finished in ten seconds according to previous studies[27, 28, 34-36]. Such a big difference in crystallization kinetics should affect the nature of resulted crystals.

L_c was estimated by using the Scherrer equation,

$$L_c = \frac{K\lambda}{\beta\cos\theta} \qquad (12)$$

where β is the half width of the 002 reflection in each meridional intensity profile and θ is the Bragg angle of the 002 reflection. For the Scherrer constant, K = 0.89 was adopted[37].

Figure 3a presents an example of temperature dependence of L_c. According to this figure, L_c is also independent of T_c. Most of the plots can be divided into a low temperature region exhibiting a moderate

<figure><img src="image_6.png" /><figcaption>Fig. 3 Example of estimation of <L_c>. (a) Variation of L_c during heating process. Symbols designate the difference of T_c. The sample is NR-S 1.125. (b) Relationship between <L_c> and network-chain density at T_c.</figcaption></figure>

slope and a high temperature region defined by a steeper slope. Though the heating process was very slow, no annealing effect leading to thickening of the crystallite can be recognized. For the Gibbs-Thomson plot, we used the average value of L_c in the low temperature region for each plot. This average value is denoted as <L_c>. Figure 3b shows the plot of <L_c> versus network-chain density, v, at T_c. Similarly than in Figure 3a, the plots can be divided into two regions; the low v region in which <L_c> decreases as v increases, and the high v region in which <L_c> is almost constant. According to this figure, <L_c> is found to be determined primarily by v and secondarily by α_s. That is to say, growth of strain-induced crystals in NR is limited by the size between cross-linking points or entanglement points[24].

Here, we comment on the implication of the estimated values of L_c. Considering the length of network chains and crystal fraction of the sample, only one crystallite should exist between cross-linking points. For example, in the case of NR-S 1.125 which has the longest network-chain length, the number of monomers per chain is about 145 on average, while the number of monomers corresponding to the value of L_c is calculated to be 19~36. Based on these values, fraction of one crystallite per one network chain ranges between 0.13 and 0.24. Because total fraction of
strain-induced NR crystal (namely crystallinity) is up to 0.2[32], formation of two or more crystallites along one network chain is ruled out. Chain folding is also considered to be a miner event that may occur only for relatively long chains. This consideration strongly suggests that strain-induced NR crystals are of the bundle type.

Figure 4 shows the Gibbs-Thomson plot for the estimation of $\sigma_s$, namely the plot of $T_m$ versus $\langle L_c \rangle^{-1}$. In this figure, fitting lines are drawn for the data points corresponding to the same $\alpha_s$ because $T_m^0$ changes with $\alpha_s$ according to Eqs. (7–8). The dependence of $T_m$ on $\langle L_c \rangle^{-1}$ is weak and no clear tendency stands out, though the range of $T_c$ was sufficiently wide, as stated in the Experimental section. Considering that both $T_m$ and $L_c$ are independent of $T_c$, this result suggests that there is no correlation between $T_m$ and $L_c$ in the case of strain-induced crystals of NR.

Let us finally discuss the implication of this result for Eq. (3). According to this equation, the correlation between $T_m$ and $L_c$ disappears when the $\sigma_c$ becomes infinitely low ($\sigma_c \approx 0$). In such a case, Eq. (3) reduces to

$$T_m \approx T_m^0 \qquad (13)$$

That is to say, $T_m$ of strain-induced crystals plotted in Figure 2b corresponds to equilibrium melting temperature, $T_m^0$. This result is verified by extrapolating the plot in Figure 2b to $\alpha_s = 1$ (namely, unstretched state): it gives $T_m = \text{ca. } 35 \,^\circ\text{C}$ at $\alpha_s = 1$ and it is very close to the values reported for $T_m^0$ of NR (35–36 $^\circ\text{C}$)[29, 38]. Thus, it confirms that $\sigma_c$ for strain-induced crystal in cross-linked NR is below the measurement limit. The same logic holds even if we consider Eq. (9), because $T_m$ was independent of $T_c$ and

there was no correlation between $T_c$ and crystallite sizes.

Because $\sigma_c$ is very low, the activation energy of formation for such a crystallite is drastically reduced which leads to the acceleration of the nucleation rate [27]. This effect greatly exceeds the effect due to the conformational entropy of the molecular chain. This is the factor that dominates crystallization in oriented polymers. Furthermore, crystallization kinetics will change from the usual nucleation-controlled type to the diffusion-controlled one. This expectation has been verified by the recent kinetic study of Brüning et al. who concluded that strain-induced crystallization of NR is diffusion-controlled[36].

The structure of the most thermally stable part of the crystal (probably a nucleus) before melting may be illustrated as in Figure 5. The end surfaces are of bundle-like ones with density gradient regions. Side surfaces are in contact with oriented amorphous

<figure><img src="image_7.png" /><figcaption>**Fig. 5** Illustration of the most thermally stable part of a strain-induced NR crystal before melting.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 4** Gibbs-Thomson plot for strain-induced crystals of NR-S1.125.</figcaption></figure>
chains aligned parallel to the crystallized chains. In both cases, the interfaces should be ambiguous, and accordingly, surface energy may be quite low. Though the bundle-like end surface of macroscopic crystal is demonstrated to have higher surface energy than chain-folded one, such a discussion does not apply to the case of nuclei with small number of chain stems[39].

Now, one may wonder; if $T_m$= ca. 35 °C at $\alpha$ = 1, then, once strain-induced crystallites are formed, the sample is expected to remain partially crystallized even at the end of recovery. Such samples do exist as shape-memory NR[40]. In the shape-memory NR, fraction of the most thermally stable part may be high. Detailed study related to this topic is currently in progress.

We should like to emphasize that the present theory is applicable not only to NR but also to crystallization in other polymers under orientation. For example, in order to explain the crystallization of oriented polypropylene, Janeschitz-Kriegl and his coworkers have proposed the formation of bundle-like nuclei having the lower surface energy[8]. Indeed, Yamazaki et al. reported the lower surface energies for nuclei in oriented polyethylene and polypropylene [41, 42]. Even in the case of crystallization in quiescent condition, Okada et al. demonstrated that nano-sized nuclei has lower end-surface energy than macroscopic crystal, assuming the formation of bundle-like nuclei [43]. Our study clarified that formation of crystallites (that may be regarded as nuclei) with such low surface energy is the key to understand the

crystallization of oriented polymer[27].

## Conclusion

Through the investigation of melting behavior of strain-induced crystallites in vulcanized NR which was quickly stretched and held at the draw ratio, $T_m$ was found to be independent of $T_c$. As the sizes of the crystallites are also independent of $T_c$, there is no correlation between $T_m$ and $L_c$ in the case of strain-induced crystals of NR. These results indicate that surface energies of strain-induced crystals are negligibly low. The drastic acceleration of crystallization by the application of strain to vulcanized NR is due to the formation of crystallites having such low surface energies. This new concept will contribute for the derivation of a general theory on crystallization of oriented polymer, which has not been accomplished for over 70 years since the 1940's.

## Acknowledgment

The custom-made tensile tester was provided by Sumitomo Bakelite Co., Ltd. This work was supported by Grant-in-Aid for Scientific Research (C) (16K05913).

## Appendix

### 1. Variation of $I_{200}$

Figures A1-A3 show the variation of $I_{200}$ during heating process for NR samples crystallized at different $T_c$.

<figure><img src="image_9.png" /><figcaption>Fig. A1 Dependence of $I_{200}$ on temperature for NR-S1.125</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. A2 Dependence of $I_{200}$ on temperature for NR-S2.25</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. A3 Dependence of $I_{200}$ on temperature for NR-S4.5</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. A4 Variation of $L_c$ during heating process. Symbols designate the difference of $T_c$.</figcaption></figure>

## 2. Variation of $L_c$

Figure A4 shows the variation of $L_c$ during heating process for NR-S 2.25 and NR-S 4.5.

## 3. Relationship between network-chain density and melting temperature

Figure A5 shows the dependence of melting temperature, $T_m$, on draw ratio, $\alpha$, for samples with different network-chain densities. As this figure shows, the data for the different samples are mostly overlapped. This result indicates that $T_m$ is independent of network-chain density of the sample.

<figure><img src="image_13.png" /><figcaption>Fig. A5 Dependence of melting temperature on draw ratio for samples with different network-chain densities.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. A6 Variation of L_a during heating process.</figcaption></figure>

4. Preliminary examination for the estimation of surface energies on the basis of Eq. (11).

NR-S 2.25 was quickly stretched at various T_c. Then, T_m, L_a and L_c were estimated for the strain-induced crystals. Each L_a value was estimated using the half breadth of 200 reflection in the same way as

<figure><img src="image_15.png" /><figcaption>Fig. A7 Results of the fitting the data of T_m, L_a and L_c using Eq. (A1).</figcaption></figure>

L_c. Figure A6 shows the variation of L_a during heating process.

Here, we assumed L_a = L_b and simplified Eq. (11) as,

$$T_m = T_m^0 \left( 1 - \frac{4\sigma}{L_a\Delta H} - \frac{2\sigma_e}{L_c\Delta H} \right) \quad (A1)$$

By fitting the data of T_m, L_a and L_c, we obtained the results shown in Figure A7. As these results show, the sign of the terms of 1/L_a and 1/L_c varies positive to negative. Considering that T_m, L_a and L_c are all independent of T_c, σ and σ_e are thought to be negligibly small, as discussed in the main text.

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