INTRODUCTION

Flow- or strain-induced crystallization (FIC or SIC) of polymers occurs not only in processing like film blowing, injection molding and fiber spinning, but also in service of some polymers like natural rubber and shape memory materials.^[1-5] Though large efforts from both academic and industrial communities have been devoted for understanding FIC, it still remains as a classic and important challenge in polymer physics.^[5,6] In recent two decades, new techniques such as in situ synchrotron radiation X-ray scattering and computer simulations provide more in-depth information about the kinetic pathway of FIC, but we still lack a thermodynamic theory of FIC to incorporate these new observations, such as the mesophase or precursor, shish-kebab, and the emergence of new crystal forms.^[7-15]

At present, Flory’s conformational entropy reduction model (CERM) proposed in 1947 is the most widely recognized model for FIC, which was originated from uniaxial stretching of natural rubber.^[16] CERM depicts a process of flow-induced chain straightening, which results in conformational entropy reduction (CER, $\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}$) of melt. Based on the classic nucleation theory (see Fig. 1a), CERM simply incorporates $\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}$ (see Fig. 1b) and expresses the nucleation barrier under flow $\Delta {G}_{\rm{f}}^{*}$ as

1Full-screen View

Schematic illustration of the free energy barrier of nucleation (a) at quiescent condition and (b) under flow from the conformational entropy reduction (CERM) model. $\Delta {G}_{0}^{*}$ and $\Delta {G}_{\rm{f}}^{*}$ are the nucleation barrier at quiescent condition and under flow; $\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}$ is the conformational entropy reduction of melt induced by flow.

1 $\Delta {G}_{\rm{f}}^{*}=\Delta {G}_{0}^{*}+T\Delta {S}_{\rm{c}\rm{o}\rm{n}}$

where $\Delta {G}_{0}^{*}$ represents the nucleation barrier at quiescent condition. As $\Delta {S}_{\rm{c}\rm{o}\rm{n}}$ is negative, thus $\Delta {G}_{\rm{f}}^{*}< \Delta {G}_{0}^{*}$, namely the nucleation barrier is reduced. Correspondingly, the equilibrium melting temperature ${T}_{\rm{mf}}^{0}$ of the crystal increases under flow, which can be written as

2 $\frac{1}{{T}_{\rm{m}\rm{f}}^{0}}=\frac{1}{{T}_{\rm{m}}^{0}}-\frac{\Delta {S}_{\rm{c}\rm{o}\rm{n}}}{\Delta H}$

where ${{T}}_{\mathrm{m}}^{0}$ and $\Delta H$ are the equilibrium melting temperature and the crystallization enthalpy of the crystal at quiescent condition, respectively. CERM can qualitatively account for flow-induced enhancement of nucleation rate and has been widely employed to explain FIC experiments and simulations.^[12-14] Later on, although various expressions of $\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}$ have been introduced by different groups and chain dynamics of polymer without crosslink is also considered, new models keep the essential physics of CERM and take $\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}$ as the core contributor in understanding FIC.^[17-19]

Conformational entropy is a global parameter on the individual chains, while nucleation and crystallization are packing processes involving local multi-chain segments, suggesting that $\Delta {S}_{\rm{c}\rm{o}\rm{n}}$ may not be the most important contributor in FIC. Indeed, CERM misses several important physics observed in experiments and simulations.^[20-25] Fig. 2 presents a schematic illustration of the factors modified by flow. Starting from the classic nucleation theory with liquid and crystal states, we first examine what happen in the initial polymer liquid under flow. (i) Chain orientation. Chain straightening or $\Delta {S}_{\rm{c}\rm{o}\rm{n}}$ only considers individual chains and does not take chain orientation or parallel packing involving multi-chain segments into account, while parallel packing of segments is inevitably required in crystal nucleation. In uniaxial flow, the chain straightening and chain orientation occur synchronously, leading to little attention on the effect of orientation on FIC for a long period. However, a recent discovery about the frustrating SIC of natural rubber during biaxial stretching has suggested the crucial role of chain orientation on FIC.^[20] Therefore, in addition to $\Delta {S}_{\rm{c}\rm{o}\rm{n}}$, entropy change related to orientation $\Delta {S}_{\rm{o}\rm{r}\rm{i}}$ should be put into the theory of FIC separately. (ii) Chain interaction. Flow-induced preordering or precursor has been widely reported in FIC experiments, in which non-crystalline shish detected with in situ X-ray scattering is a representative example.^[9,26] These experimental observations reveal that not only entropy but also enthalpy or molecular interactions are changed by flow. Thus flow-induced enthalpy change of polymer liquid $\Delta {H}_{\rm{f}}^{\rm{l}}$ should be considered in FIC theory. After depicting the initial liquid, we then shift our attention to the final crystalline state. (iii) Crystal morphology and structure. Flow modifies crystal morphology (e.g., from lamellae to shish-kebab) and crystal form (e.g., from orthorhombic to hexagonal crystals for polyethylene (PE)) has been largely documented since the earliest FIC study,^[27-33] but historically, these factors had been surprisingly overlooked in FIC theory. If flow induces new crystal forms, enthalpy difference $\Delta {H}_{\rm{f}}^{\rm{c}}$ between crystals at quiescent and strained conditions should be introduced. Meanwhile, transforming crystal morphology from lamellae to shish should change the free energy density of end surface, that is, from folded-chain surface ${\sigma }_{\rm{e}}$ to extended-chain surface ${\sigma }_{\rm{ef}}$. Evidently, in addition to $\Delta {S}_{\rm{c}\rm{o}\rm{n}}$ in CERM, the new FIC theory should consider the above three factors.

2Full-screen View

Flow-induced changes of the physical parameters in FIC. $\Delta {S}_{\rm{c}\rm{o}\rm{n}}$ and $\Delta {S}_{\rm{o}\rm{r}\rm{i}}$ are the entropy change of conformation and orientation of melt induced by flow; $\Delta {H}_{\rm{f}}^{\rm{l}}$ is the enthalpy change of polymer melt due to the interaction of ordered segments; $\Delta {H}_{\rm{f}}^{\rm{c}}$ is the enthalpy difference between two crystal forms like flow-induced orthorhombic and hexagonal phases of PE; ${\sigma }_{\rm{e}}$ and ${\sigma }_{\rm{ef}}$ are the free energy densities of crystal end surface at quiescent condition and under flow.

Based on the experiment and simulation results, FIC models incorporating the effects of orientation, chain interaction, new crystal morphology and structure have been proposed in recent years, which, however, were considered separately. Chen et al.^[20] introduced the orientation entropy and the chain interaction between the ordered segments, while Liu et al.^[24] took the free energy change of the final crystalline state into account and proposed the entropy reduction-energy change (ER-EC) model. Although these efforts make a stepwise progress for understanding FIC, a unified FIC theory incorporating the above three factors has not been developed yet.

In this work, starting from classic nucleation theory, we propose a thermodynamic model for FIC to unify flow-induced changes of conformational entropy $\Delta {S}_{\rm{c}\rm{o}\rm{n}}$, orientation entropy $\Delta {S}_{\rm{o}\rm{r}\rm{i}}$, chain interaction of polymer liquid $\Delta {H}_{\rm{f}}^{\rm{l}}$, and modifications of crystal structure $\Delta {H}_{\rm{f}}^{\rm{c}}$ and morphology ${\sigma }_{\rm{ef}}$. Our model is a thermodynamically phenomenological model but explicitly unifies all structural changes in FIC study. For the convenience of description, we name it as uFIC model hereafter. The possible calculation methods for the changes of conformational entropy $\Delta {S}_{\rm{c}\rm{o}\rm{n}}$, orientation entropy $\Delta {S}_{\rm{o}\rm{r}\rm{i}}$ and chain interaction of polymer liquid $\Delta {H}_{\rm{f}}^{\rm{l}}$ are given in the Appendix, as they have been discussed in early studies.^[16,20]

THEORY AND METHOD

To deduce the uFIC model, we first incorporate flow-induced variations of entropy and enthalpy into the chemical potential $\mu =H-TS$. The chemical potential difference per unit volume between polymer crystal and melt (liquid) phases under flow can be written as

3 $\Delta {\mu }_{\rm{f}}=\Delta {H}_{\rm{f}}-T\Delta {S}_{\rm{f}}=\Delta H+\Delta {H}_{\rm{f}}^{\rm{c}}-\Delta {H}_{\rm{f}}^{\rm{l}}-T\left(\Delta S-\left(\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}+\Delta {S}_{\mathrm{o}\mathrm{r}\mathrm{i}}\right)\right)$

where $\Delta H$ and $\Delta {H}_{\rm{f}}=\Delta H+\Delta {H}_{\rm{f}}^{\rm{c}}-\Delta {H}_{\rm{f}}^{\rm{l}}$ are the crystallization enthalpies of polymer melt at quiescent condition and under flow, respectively. While $\Delta S$ and $\Delta {S}_{\rm{f}}=\Delta S-\left(\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}+\Delta {S}_{\mathrm{o}\mathrm{r}\mathrm{i}}\right)$ are the crystallization entropies of polymer melt at quiescent condition and under flow, respectively. Here, the contribution of conformation and orientation to the entropy change of the melt are considered. Note in current work, the symbols with subscript “f” represent the physical parameters under flow. In Eq. (3), we assume that the difference in the entropies of crystal phase between quiescent and strained conditions is negligible. At the equilibrium melting temperature under flow ${T}_{\rm{mf}}^{0}$, the chemical potential of these two phases is equal, $\Delta {\mu }_{\rm{f}}=0$, which results in

4 ${T}_{\rm{mf}}^{0}=\frac{\Delta {H}_{\rm{f}}}{\Delta {S}_{\rm{f}}}=\frac{\Delta H+\Delta {H}_{\rm{f}}^{\rm{c}}-\Delta {H}_{\rm{f}}^{\rm{l}}}{\Delta S-\left(\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}+\Delta {S}_{\mathrm{o}\mathrm{r}\mathrm{i}}\right)}$

Then, Eq. (3) can be written as

5 $\Delta {\mu }_{\rm{f}}=\Delta {H}_{\rm{f}}\left(1-\frac{T}{{T}_{\rm{mf}}^{0}}\right)=\frac{\Delta {H}_{\rm{f}}\Delta {T}_{\rm{f}}}{{T}_{\rm{mf}}^{0}}$

where $\Delta {T}_{\rm{f}}$ is the supercooling under flow. Inserting ${T}_{\rm{m}}^{0}=\dfrac{\Delta H}{\Delta S}$ into Eq. (4) and taking the reciprocal lead to the following equation

6 $\frac{1}{{T}_{\rm{mf}}^{0}}=\frac{\Delta H}{\Delta {H}_{\rm{f}}{T}_{\rm{m}}^{0}}-\frac{\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}+\Delta {S}_{\mathrm{o}\mathrm{r}\mathrm{i}}}{\Delta {H}_{\rm{f}}}$

The CERM and other FIC models generally assume that the free energies of the critical nucleus and the final state under flow keep the same as those at quiescent condition. However, at the same crystallization temperature ${T}_{\rm{c}}$, the size of the critical crystal nucleus should decrease as the equivalent supercooling under flow is expected to increase. Obviously, the free energy landscape changes under flow, which will be considered in our uFIC model (see Fig. 3a). Following the classical nucleation theory,^[34] the free energy change for the formation of a cylindrical crystal nucleus from the liquid phase under flow is given by

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Schematic illustration of the free energy barrier of nucleation under flow. (a) The uFIC model incorporates the entropy change of polymer melt originated from conformation and orientation $\Delta {S}_{\rm{f}}^{\rm{l}}=\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}+\Delta {S}_{\mathrm{o}\mathrm{r}\mathrm{i}}$, enthalpy change of polymer melt $\Delta {H}_{\rm{f}}^{\rm{l}}$ due to oriented segments and the free energy change of final nucleus with different crystal morphologies and structures induced by flow $\Delta {H}_{\rm{f}}^{\rm{c}}$. (b) The condition for the existence of a thermodynamically metastable intermediate phase under flow.

7 $\Delta {G}_{\rm{f}}=\text{π} {R}^{2}L\Delta {\mu }_{\rm{f}}+2\text{π} RL{\sigma }_{\rm{sf}}+2\text{π} {R}^{2}{\sigma }_{\rm{ef}}$

where ${\sigma }_{\rm{ef}}$ and ${\sigma }_{\rm{sf}}$ are the free energy densities for the end and side surfaces of a cylindrical nucleus of length $L$ and radius $R$ under flow, $\Delta {\mu }_{\rm{f}}$ is the chemical potential difference per unit volume between the crystal and melt phases under flow. Combining Eq. (5), the critical nucleus thickness ${L}_{\rm{cf}}$ and the critical nucleus radius ${R}_{\rm{cf}}$ are obtained by finding the maximum in $\Delta {G}_{\rm{f}}$ with respect to nucleus length $L$ and nucleus radius $R$ in Eq. (7),

8

where we assume that the ratios of these two values to the length ${L}_{\rm{c}}$ and radius ${R}_{\rm{c}}$ of the critical nucleus at quiescent condition are ${K}_{\rm{L}}$ and ${K}_{\rm{R}}$, respectively. Then the free energy densities of the end and side surfaces of crystals under flow can be obtained as

9

Substituting these expressions of ${L}_{\mathrm{c}\mathrm{f}}$ and ${R}_{\mathrm{c}\mathrm{f}}$ into Eq. (7) leads to the free energy barrier of critical crystal nucleus under flow as

10 $\Delta {G}_{\rm{f}}^{*}=\frac{8\text{π} {\sigma }_{\rm{ef}}{\left({\sigma }_{\mathrm{s}\mathrm{f}}\right)}^{2}{\left({T}_{\rm{mf}}^{0}\right)}^{2}}{{\left(\Delta {H}_{\rm{f}}\right)}^{2}{\left(\Delta {T}_{\rm{f}}\right)}^{2}}$

Inserting ${\sigma }_{\rm{ef}}$ and ${\sigma }_{\rm{sf}}$ of Eq. (9) into Eq. (10), we have the nucleation barrier of critical nucleus under flow as

11 $\Delta {G}_{\rm{f}}^{*}=K\left(1+\frac{1}{\Delta {\mu }_{0}}\left({\Delta H}_{\rm{f}}^{\rm{c}}-{\Delta H}_{\rm{f}}^{\rm{l}}\right)+\frac{1}{\Delta {\mu }_{0}}T(\Delta {S}_{\rm{con}}+\Delta {S}_{\rm{ori}})\right)\Delta {G}_{0}^{*}$

where $\Delta {G}_{0}^{*}$ and $\Delta {\mu }_{0}$ are the nucleation barrier of critical nucleus and the chemical potential difference per unit volume between polymer crystal and melt (liquid) phases at quiescent condition, respectively. We also inform a shape factor K as

12 $K={K}_{\rm{L}}{K}_{\rm{R}}^{2}=\frac{{V}_{\rm{cf}}}{{V}_{\rm{c}}}$

where ${V}_{\rm{c}}$ and ${V}_{\rm{cf}}$ are the volume of critical cylindrical nucleus at quiescent condition and under flow, respectively. According to Eq. (3) and Eq. (9), Eq. (11) can be further written as

13 $\Delta {G}_{\rm{f}}^{*}=\left(\frac{{\sigma }_{\rm{ef}}}{{\sigma }_{\rm{e}}}\right){\left(\frac{{\sigma }_{\rm{sf}}}{{\sigma }_{\rm{s}}}\right)}^{2}{\left(\frac{\Delta {\mu }_{0}}{\Delta {\mu }_{\rm{f}}}\right)}^{2}\Delta {G}_{0}^{*}$

DISCUSSION

As the uFIC model takes both flow-induced modifications of liquid and crystal into account, it is expected that this model can explain all the experimental observations of FIC. To analyze how flow affects nucleation process, we compare the nucleation barriers at flow and quiescent conditions. Here we assume the free energy density of the crystal side surface under flow is unchanged, $\dfrac{{\sigma }_{\rm{sf}}}{{\sigma }_{\rm{s}}}=1$, then the ratio of the nucleation barriers between these two conditions is

14 $\frac{\Delta {G}_{\rm{f}}^{*}}{\Delta {G}_{0}^{*}}=\left(\frac{{\sigma }_{\rm{ef}}}{{\sigma }_{\rm{e}}}\right){\left(\frac{\Delta {\mu }_{0}}{\Delta {\mu }_{\rm{f}}}\right)}^{2}$

Taking the nucleation barrier expressed in Eq. (14) as the basic starting point, we are going to discuss the 4 characteristic observations in FIC experiments: (i) accelerating nucleation; (ii) changing the nucleus from lamellae to shish; (iii) preordering; and (iv) inducing new crystal form.

(iv) Inducing New Crystal Form

Finally we consider the situation with the competition between two crystal forms like flow-induced orthorhombic and hexagonal phases of PE, where ${\Delta H}_{\rm f}^{\rm c} > 0$. According to Eq. (13), the ratio between their nucleation barriers is written as

18 $\frac{\Delta {G}_{\rm{hf}}^{*}}{\Delta {G}_{\rm {of}}^{*}}=\left(\frac{{\sigma }_{\rm {ef}}^{\rm h}}{{\sigma }_{\rm {ef}}^{\rm o}}\right){\left(\frac{\Delta {\mu }_{\rm {of}}}{\Delta {\mu }_{\rm {hf}}}\right)}^{2}=\left(\frac{{\sigma }_{\rm {ef}}^{\rm h}}{{\sigma }_{\rm {ef}}^{\rm o}}\right){\left(1-\frac{\Delta {H}_{\rm f}^{\rm c}}{\Delta {\mu }_{\rm {hf}}}\right)}^{2}$

where $\Delta {G}_{\rm {of}}^{*}$ ($\Delta {G}_{\rm {hf}}^{*}$) is nucleation barrier of orthorhombic (hexagonal) phase under flow, $\Delta {\mu }_{\rm {of}}$ ($\Delta {\mu }_{\rm {hf}}$) is the chemical potential difference per unit volume between orthorhombic (hexagonal) crystal and polymer melt under flow, and ${\sigma }_{\rm {ef}}^{\rm o}$ $\left({\sigma }_{\rm {ef}}^{\rm h}\right)$ is the free energy density of orthorhombic (hexagonal) crystal end surface under flow. Obviously, $\Delta {\mu }_{\rm {hf}} < 0$ in FIC, as ${\Delta H}_{\rm f}^{\rm c} > 0$, and then $1-\dfrac{\Delta {H}_{\rm f}^{\rm c}}{\Delta {\mu }_{\rm {hf}}} > 1$. According to the relationship of ratio $\dfrac{{\sigma }_{\rm e}}{\Delta H}$ and the crystallization enthalpy ratio between orthorhombic phase and hexagonal phase, ${\left(\dfrac{{\sigma }_{\rm e}}{\Delta H}\right)}_{\rm o}=3.5{\left(\dfrac{{\sigma }_{\rm e}}{\Delta H}\right)}_{\rm h}$ and $\dfrac{\Delta {H}_{\rm h}}{\Delta {H}_{\rm o}}=\dfrac{1}{3}$,^[41-43] then $\dfrac{{\sigma }_{\rm {ef}}^{\rm h}}{{\sigma }_{\rm {ef}}^{\rm o}}=\dfrac{1}{10.5}$. Therefore, the condition for favorable formation of the hexagonal phase $\left(\dfrac{\Delta {G}_{\rm {hf}}^{*}}{\Delta {G}_{\rm {of}}^{*}} < 1 \right)$ requires $\Delta {\mu }_{\rm {hf}} < -0.45{\Delta H}_{\rm f}^{\rm c}$. Combining Eq. (16), the relationship between $\Delta {\mu }_{\mathrm{f}}$ and the critical nucleus size, it proves that for a significantly smaller critical nucleus size, the hexagonal phase is a thermodynamically stable phase, which is in good agreement with early reports.^[41] In addition, the driving force $\left|\Delta {\mu }_{\mathrm{h}\mathrm{f}}\right|$ required for nucleation increases with temperature, indicating that there is a critical temperature ${T}_{\mathrm{h}}$, above which the hexagonal phase is more stable than the orthorhombic phase. Based on the above discussion, the crystal morphologies and structures in the flow-temperature space can be constructed, as shown in Fig. 4. As the temperature increases, the required deformation is increased for FIC to occur. And, taking PE as an example, the crystal morphology and structure gradually changes from orthorhombic lamellar crystal to orthorhombic shish crystal, and finally to hexagonal shish crystal at $T>{T}_{\mathrm{h}}$. This result agrees reasonably well with FIC experiments of PE.^[24]

4Full-screen View

Schematic illustration of the crystal morphologies and structures of PE in the flow-temperature space.

The above discussion is qualitative and based on some reasonable assumptions, which, however, demonstrates the capability of the uFIC model on understanding experimental observations. The CERM may be numerically applied to fit the flow accelerated nucleation rate, but experimental result suggests $\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}$ may make a weak contribution on this effect.^[20] Moreover, other observations of FIC, such as precursor, new crystal forms and morphologies have never been considered in the CERM. Through introducing the changes of orientation entropy $\Delta {S}_{\mathrm{o}\mathrm{r}\mathrm{i}}$, chain interaction of polymer liquid $\Delta {H}_{\rm f}^{\rm l}$, the modifications of crystal structure $\Delta {H}_{\rm f}^{\rm c}$ and morphology ${\sigma }_{\mathrm{e}\mathrm{f}}$, uFIC can not only specify the physical origins for accelerating nucleation rate, but also account for almost all experimental observations in FIC. The above discussion only refers to some specific cases, and the uFIC can be employed to explain other observations of FIC experiments and simulations.

CONCLUSIONS

Taking all the changes of chain conformation, chain orientation, interaction of ordered segments in the initial polymer melt, and the final crystal morphology and structure into account, we propose a uFIC model and rewrite the expression of nucleation free energy barrier under flow, as presented in Eq. (13). Our model can not only effectively explain the flow-accelerated polymer nucleation, but also offer understandings of other FIC experimental observations. It theoretically verifies that the intermediate phase or precursor is a metastable phase only when the free energy density of the interface between the preordering and the melt is smaller than that of the crystal-melt interface. Based on the nucleation barrier under flow, the competition of different crystal forms in the flow can be clarified. Taking PE as an example, the critical transition temperature between orthorhombic and hexagonal crystal forms can be determined by comparing the crystallization enthalpy and the end surface free energy density. In conclusion, our model has the potential to be applied to explain all the observations of FIC experiments which have been done before and can provide a theoretical guidance for further FIC study.

APPENDIX

The estimation of $\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}$ has been proposed by different authors with different models.^[44-46] Here we take the simple one proposed by Flory.^[16] At quiescent condition, the relative coordination of chain ends of the equilibrium polymer melt is approximately considered to obey the Gaussian distribution. Under flow, the amorphous molecular chains are straightened and deviate from the random configurations, resulting in the decrease of conformational entropy. Based on Flory’s CERM model, the conformational entropy reduction $\Delta {S}_{\mathrm{c}\mathrm{o}\mathrm{n}}$ can be expressed as

19

where $x,y$ and $z$ represent the coordinates of one end of the chain with respect to the other at quiescent condition, ${x}',{y}'$ and ${z}'$ are the relative coordination after stretching, ${k}_{{\rm B}}$ is the Boltzmann constant, $\beta ={\left(3/2n\right)}^{0.5}/l$ represents the reciprocal chain displacement value that is the most probable, $l$ is the Kuhn length, $n$ is the segment number per chain, $\xi$ is the number of segments in the flow-induced crystalline phase, ${N}_{\rm{c}}$ is the total number of chains.

Considering the influence of orientation on the free energy of the initial melt, the parallel arrangement of the local stretched segments leads to the reduction in the number of possible conformations of the molecular chain, which is considered as the entropy effect. At the same time, there is an interaction between the parallel oriented segments, which manifests as an influence on enthalpy.^[20] The entropy change of the melt caused by the oriented segments is defined as the orientation entropy, based on statistical mechanics and Maier-Saupe theory,^[47] which can be expressed as

20 $\;\Delta {S_{{\rm{ori}}}} = - {N_{\rm s}}{k_{\rm B}}\left\langle {{\rm{ln}}f\left( \phi \right)} \right\rangle = - \frac{{{a_{\rm N}}}}{{{V^2}}}\frac{{{N_{\rm s}}}}{T}{f^2} + {N_{\rm s}}{k_{\rm B}}{\rm{ln}}\left\{ {\int _0^1{\rm{exp}}\left[ { - \frac{{u\left( {\nu ,f} \right)}}{{{k_{\rm B}}T}}} \right]{\rm{d}}x} \right\}$

where ${N}_{\rm{s}}$ is the number of the ordered segments, $f\left(\phi \right)$ is the distribution function of the angle between segments and the stretching direction, ${a}_{\mathrm{N}}$ is a constant related to the mean field, $V$ is the molar volume of the polymer, $f$ is the Hermans’ orientation parameter, $u\left(\nu ,f\right)$ is the internal energy of a segment, and $\nu$ is the segment vector. Considering that the change of volume of the polymer melt under deformation can be ignored, the enthalpy change of the melt induced by flow ${\Delta H}_{\rm f}^{\rm l}$ equals the internal energy change due to ${N}_{\rm s}$ oriented segments,^[20,47] which can be expressed as

21 ${\Delta H}_{\rm f}^{\rm l}=\Delta {U}_{\mathrm{o}\mathrm{r}\mathrm{i}}=\frac{1}{2}{N}_{\rm s}\frac{{a}_{\rm N}}{{V}^{2}}{f}^{2}$

Through Taylor expansion, ignoring higher-order terms, according to the Eqs. (20) and (21), the contribution of orientation to the free energy of the melt can be written as follows

22 $\Delta {G}_{\mathrm{o}\mathrm{r}\mathrm{i}}^{\rm l}=\Delta {U}_{\mathrm{o}\mathrm{r}\mathrm{i}}-T\Delta {S}_{\mathrm{o}\mathrm{r}\mathrm{i}}=a{f}^{2}+b{f}^{3}+c{f}^{4}$

where a, b, and c are constants related to pressure and temperature.

The existence of crystals with different morphologies and structures under flow indicates that the existence of the enthalpy change of the crystal ${\Delta H}_{\rm f}^{\rm c}$ induced by flow, which can be obtained from the melting enthalpy of the different crystals.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.