Multicomponent Thermodynamics of Strain-Induced Polymer

Article pubs.acs.org/JPCB

Multicomponent Thermodynamics of Strain-Induced Polymer Crystallization Liyun Zha,† Yixian Wu,‡ and Wenbing Hu*,† †

Department of Polymer Science and Engineering, State Key Laboratory of Coordination Chemistry, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210093, China ‡ State Key Laboratory of Chemical Resource Engineering, Beijing University of Chemical Technology, Beijing 100029, China ABSTRACT: We developed a linear combination of two Flory’s meltingpoint theories, one for stretched and the other for solution polymers, to predict the melting point of stretched solution polymers. The dependences of the melting strains on varying temperatures, polymer volume fractions, and solvent qualities were verified by the onset strains of crystallization in our dynamic Monte Carlo simulations of stretched solution polymers under a constant strain rate. In addition, owing to phase separation before crystallization in a poor solvent, calibration of polymer concentration to the polymer-rich phase appears necessary for the verification. Our results set up a preliminary thermodynamic background for the investigation of the multicomponent effect on strain-induced crystallization of polymers in rubbers and gels as well as on shear-induced crystallization of polymers in solutions and blends.

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INTRODUCTION

the corresponding predictions of the new equation. The article ends with a summary of our conclusions.

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Strain-induced crystallization of network polymers is an important issue in the physics of polymer networks, which, for instance, results in a significant strain-hardening phenomenon and reversible heat release from natural rubber.1,2 The issue associated with polymer relaxation is also important for polymer crystallization in a flow,3−13 which determines the oriented semicrystalline texture for a high mechanical performance of polymeric products via industrial polymer processing, such as plastic molding, thin-film stretching, and fiber spinning.14,15 The strain in the thermoreversible polymer crystalline gel also changes the course of crystallization as well as the crystallite morphology.16 In practice, the stretched polymers are often mixed with other components of either small molecular solvents/additives or other blending polymers. However, so far, the multicomponent effects on strain-induced crystallization behaviors of polymers, although being broadly studied as a fundamental issue,17−19 have not yet reached a quantitative level. In this article, we set up a preliminary thermodynamic approach for a further quantitative discussion. First, we develop a linear combination of two Flory’s thermodynamic equations of polymer melting points, one on the solution polymers and another on the stretched polymers (either in a stretched network or in a shear flow). The new equation can predict the melting strains of stretched solution polymers at various temperatures, polymer concentrations, and solvent qualities. To validate this new equation, we perform dynamic Monte Carlo (MC) simulations of stretched solution polymers and fit the onset strains for crystallization under a constant strain rate with © XXXX American Chemical Society

THERMODYNAMIC PREDICTION With regard to the thermodynamic aspects, stretching of network polymers lowers the conformational entropy and hence increases the melting point, according to the first equation derived by Flory in 194720 1/2 1 1 k ⎡⎛⎜ 6 ⎞⎟ ⎢ (ε + 1) − 0 =− Tms Δh ⎢⎣⎝ πN ⎠ Tms 1 ⎛ ε 2 + 2ε + 1 1 ⎞⎤ − ⎜ + ⎟⎥ 2 N⎝ ε + 1 ⎠⎥⎦

=−

k f ( ε) Δh

(1)

T0ms

Here, Tms and denote the melting points of stretched and unstretched polymers, respectively, k is Boltzmann’s constant or the ideal gas constant when applied, Δh represents the heat of fusion, N is the number of monomers in each polymer chain, and ε represents the strain defined by (l − l0)/l0 × 100% (l and l0 are the lengths after and before stretching along the stretching direction of the sample, respectively). The whole content in brackets is taken as a variable for briefness. On the other hand, mixing of other components increases the conformational entropy of polymers and hence lowers the Received: May 29, 2016 Revised: June 23, 2016

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DOI: 10.1021/acs.jpcb.6b05404 J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

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melting point, according to the first expression derived again by Flory in 194921 1 1 k − 0 = (ϕ1 − χϕ12) Ts Δh Tm

SIMULATION VERIFICATION Recently, we employed dynamic MC simulations of a lattice polymer model to investigate the microscopic mechanism of strain-induced crystallization.30 The results verified the straininduced increase in the melting point and revealed a transition of crystal nucleation from intramolecular to intermolecular modes upon further application of stretching. 30 At a quantitative level, the onset strains in our simulation approach have verified the melting-point equation of stretched random copolymers derived by a linear combination of two Flory’s thermodynamic equations of melting points,31 separately for stretched polymers21 and random copolymers.32 Here, we employed the same approach to verify the melting-point equation of stretched solution polymers, that is, eq 3. In our dynamic MC simulations of strain-induced polymer crystallization,30 polymer chains dissolved in solvent were realized in a lattice box of 80 × 128 × 128 (X × Y × Z) cubic cells surrounded by hard walls. We used hard walls to simplify the situation because as soon as the polymers were stretched, they detached from the lateral boundaries due to normal contraction (see, e.g., Figure 1). Figure 1a demonstrates the

(2)

Here, Ts and T0m denote the melting points of solution and bulk (meaning pure) polymers, respectively, ϕ1 is the volume fraction of the solvent, and χ is the Flory−Huggins interaction parameter. Later, this equation was verified by experiments22,23 as well as molecular simulations.24 It is important to note that in a poor solvent, liquid−liquid phase separation potentially competes with crystallization.25,26 The simulation approach for this competition was previously applied to investigate the structure formation during the fiber-spinning process of polymer solutions,27 when the subtle interplay between liquid−liquid phase separation and crystallization greatly affected the inner structure of the fiber. For solution polymers in a stretched state, the equilibrium melting point, T0ms, refers to the Ts of unstretched solution polymers. Thus, we can add eqs 1 and 2, and let the terms T0ms and Ts cancel each other. Then, we derive the melting-point equation for stretched solution polymers as follows 1 1 k − 0 =− (f (ε) − ϕ1 + χϕ12) Tms Δh Tm

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(3)

The new equation takes account of both the entropy loss on stretching and the entropy gain on dilution, applicable for the stretched solution polymers. As a matter of fact, Flory’s estimation for entropy elasticity in eq 1 has raised arguments,28 whereas the Flory−Huggins interaction parameter in eq 2 appears to be concentrationdependent.29 In this sense, the thermodynamic prediction of eq 3 just provides a preliminary thermodynamic description of the effects of multicomponent factors on the practical observations of strain-/shear-induced polymer crystallization. From an experimental point of view, there exists a practical difficulty in approaching the equilibrium melting point. In principle, the melting point (Tms) of solution polymers under a certain extent of strain is the thermodynamic equilibrium temperature between melting and crystallization at which phase transition appears as infinitely slow and the crystal appears to be infinitely large, both formidable for a direct approach. One may extrapolate the melting points on heating or releasing strain from the variable crystallization conditions under which small crystallites are formed. Because the crystallites are sensitive to crystallization conditions, their melting temperatures/strains would be diverse in small systems of simulations, poorly reproducible for long-range extrapolation. The situation could be better if we observe the onset temperature/strain for crystallization upon cooling/stretching of polymers from a homogeneous phase, as previously adopted by Flory,21 although an overshooting is always required for the initiation of primary crystal nucleation. As will be confirmed below by our simulation approach, although both thermodynamic factors such as mixing interaction parameters and kinetic factors such as the overshooting on stretching bring some shifts of onset strains away from the expected equilibrium melting points, the dependence of onset strains on these thermodynamic factors follows well with the basic rules of melting strains predicted by eq 3.

Figure 1. Snapshots before (a) and after (b) stepwise stretching of solution polymers from strain 0 to 400% in an athermal solvent (B/Ec = 0) with the solvent volume fraction of 0.25 at kT/Ec = 4.0 in the lattice box of 80 × 128 × 128. All of the amorphous bonds are drawn as blue cylinders, and after stretching, the crystalline bonds containing more than four parallel neighbors are shown in yellow.

initial state of polymers in a space of 16 × 128 × 128 from X = 33−48. At the beginning, a fixed amount of polymer chains, with a preset concentration, was put in this limited space, with each chain occupying 128 consecutive lattice sites and folded 15 times along the X axis in a fold length of eight monomers extending along the Y axis and spreading one-by-one over each XY plane. Two chain ends were mobile but remained separate at two surface layers of X = 33 and 48. The rest vacancy sites were occupied by the solvent. In our simulations, chain motion was realized by microrelaxation with single-site jumping and local sliding diffusion along the chain.33,34 Double occupation of monomers on one site or bond crossing was forbidden similar to the effect of the excluded volume of polymer chains. During relaxation, two ends of the polymer chain were mobile but remained constrained on their YZ planes. The preset folded chains were relaxed to the equilibrium homogeneous solution under athermal conditions for 106 MC cycles, and each MC cycle as our simulation time unit was defined as the total trial moves for all of the monomers sampled once on average. Such a group of polymer chains stretching in parallel at both ends B


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Figure 2. (a) Strain-evolution curves of crystallinity for solution polymers stretched in an athermal solvent (B/Ec = 0) with solvent volume fraction of 0.25 at various temperatures as labeled. (b) Onset strains of crystallization as a function of temperature for polymers in an athermal solvent with four solvent volume fractions as labeled. Each onset strain with its error bar is an average result from three independent simulations under the same conditions.

where Ep is the energy benefit for each pair of neighboring bonds packing in parallel, B is the energy change during the formation of each solvent−monomer pair, same as that in the Flory−Huggins interaction parameter, Ec is the energy penalty of noncollinear connection between two sequential bonds along the polymer chains, Ea is the normal stress to expel the new vacancy site introduced by each step of stretching away from the central X axis, p is the net change of the parallelpacking pairs, b is the net change of solvent−monomer pairs, a is the net change of noncollinear connection pairs of sequential bonds, and Δr2 is the net square distance of the new vacancy site from the central X axis. Ep/Ec was set at 1 as the reduced driving force for polymer crystallization. B/Ec was set at 0, 0.3, or −0.3 to reflect the different qualities of a solvent. Ea/Ec was set at 1, which was large enough to expel the new vacancy sites from the sample body during stretching. kT/Ec was selected as relatively high for spontaneous crystallization in the stretched rather than in the relaxed states because of the reduced system temperature in our MC simulations. We start with the stretching of solution polymers in an athermal solvent, that is, B/Ec = 0, under constant strain rates. Four different volume fractions of 0.625, 0.53125, 0.4375, and 0.25 of the solvent were considered. Each polymer solution with a specific volume fraction of solvent was stretched under various temperatures. At each temperature, crystallinity was monitored with a stepwise increase of strain, defined as the fraction of crystalline bonds in the total amount of polymer bonds, whereas a crystalline bond was defined as the bond containing more than four parallel neighbors. Here, each bond in the lattice can have 24 possible parallel-neighboring positions, as a result of subtracting two connected bonds from the coordination number of the cubic lattice 26 (including 6 neighbors along 3 axes, 12 along face diagonals, and 8 along body diagonals). We chose the criteria of four parallel neighbors as the margin between the extremely amorphous bond (0 neighbors) and the fully ordered bond (24 neighbors) to take into account the less-perfectly packed bonds at the edges of the crystallites. According to this definition of crystallinity, the small horizontal level is observed at the

and sharing the same solution conditions represents an affine deformation of a network, an assumption also adopted in Flory’s network theory. After athermal relaxation, polymer chains in homogeneous solutions were stepwise stretched under a certain temperature and a fixed strain rate, as demonstrated in Figure 1b.30 In practice, first we chose a random X site inside the sample space, whose YZ plane splits the solution into two parts. We moved the +X part toward one more lattice site in the +X direction and then reconnected all of the broken chain segments on the selected YZ plane by local sliding of the rest of the segments in the −X part. After all of the broken chain segments were reconnected, the sample was allowed to relax for a longer period (5000 MC cycles, whereas the Rouse time of bulk 128mers is around 3000 MC cycles30) to expel the newly inserted athermal vacancy sites similar to that on normal stress of stretched polymers. So the strain rate was fixed at 6.25% strain/ 5000 MC cycles. To increase the efficiency, we allowed any chain ends to leave its YZ plane temporarily during chain sliding diffusion described above, but they would be immediately attracted back by that plane during the subsequent relaxation period. Next, we randomly chose another X site of the XY plane, moved the −X part one lattice site toward the −X direction, and then reconnected all of the broken chain segments by local sliding of the rest of the segments in the +X part. The relaxation of 5000 MC cycles was given again. Alternative stretching of the sample in the two opposite directions was stopped when the two YZ planes of chain ends arrived at X = 1 and 80, respectively, as demonstrated in Figure 1b. During each microrelaxation step in the relaxation process above, the well-known Metropolis sampling algorithm was employed, and whether chain motion was accepted could be judged by the total energy change, as given by30 aEc + pEp + bB + Δr 2·Ea ΔE = kT kT ⎛ Ep E ⎞ E B = ⎜a + p · + b· + Δr 2· a ⎟ · c Ec Ec Ec ⎠ kT ⎝

(4) C


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Figure 3. Inverse melting points (crystallization temperatures in simulations) of solution polymers as a function of strains (onset strains in simulations) with four solvent volume fractions as labeled, as the linear relationship derived from eq 3, separately in (a) an athermal solvent with B/ Ec = 0, (b) a good solvent with B/Ec = −0.3, and (c) a poor solvent with B/Ec = 0.3.

Figure 4. Strain-evolution curves of (a) demixing parameters and (b) crystallinity of solution polymers for three different solvent qualities as labeled, with the solvent volume fraction of 0.25 at kT/Ec = 4.0. The demixing parameter obtained at the onset strain of crystallization is demonstrated in the red curve in (a).

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Figure 5. (a) Standard calibration curve of the demixing parameter vs solvent volume fraction obtained from the simulations of homogeneous 128mer solutions under athermal conditions. (b) Inverse crystallization temperatures of solution polymers as a function of the onset strains and calibrated solvent volume fractions for the poor solvent with B/Ec = 0.3 according to eq 3. The solvent volume fractions at the onset strains of crystallization were calibrated in (a) according to the demixing parameter of polymers in the concentrated phase.

beginning of stretching, owing to the thermal fluctuations in the system. Figure 2a shows a group of strain-evolution curves of crystallinity for solution polymers stretched in the athermal solvent with the solvent volume fraction of 0.25 at various high temperatures. The sudden increase in crystallinity demonstrates the initiation of crystallization induced by stepwise stretching. As usual, higher temperatures require larger strains to initiate polymer crystallization, and the saturated crystallinities appear as insensitive to the temperatures. One can determine the onset strain of crystallization at the cross-over point between the two extrapolation lines separately from the initial horizontal and transition zones, as also demonstrated by the highesttemperature curve in Figure 2a. The onset strains for solution polymers in the athermal solvent with four solvent volume fractions at various temperatures are summarized in Figure 2b. Next, we transformed the parameters applied in our simulations and fitted the obtained onset strains into the predictable format of eq 3. The results for three solvent qualities are summarized in Figure 3. One can see from Figure 3a that the results for the athermal solvent with four solvent volume fractions align well with a linear curve as predicted by eq 3. This linear correlation implies that the fixed strain rates seem to make parallel overshooting of the onset strains away from the equilibrium values under various thermodynamic conditions. The results for the good solvent (B/Ec = −0.3, Figure 3b) with four solvent volume fractions also align well with the linear prediction but apparently not for the poor solvent (B/Ec = 0.3, Figure 3c). What happened in the poor solvent? In the polymers stretched in a poor solvent, liquid−liquid demixing may take place before crystallization and thus changes the course of the latter. To identify liquid−liquid demixing during the stretching of solution polymers in the poor solvent, we traced the demixing parameters when the polymers were stretched in the solvents of different qualities, as shown in Figure 4a. Here, the demixing parameter was defined as the average fraction of monomers occupying the neighboring positions of each monomer. In Figure 4a, in comparison to the crystallinity curves under

parallel conditions in Figure 4b, one can see that the demixing parameter curve for the poor solvent (B/Ec = 0.3) exhibits phase separation at the beginning of stretching, whereas crystallization conventionally brings another forced phase separation in the middle region of strains. It is obvious that in the poor solvent (B/Ec = 0.3), liquid− liquid demixing takes place before crystallization, so crystallization is dominated by those polymers in the polymer-rich phase. As a result, at the onset strain of crystallization in the poor solvent, the actual solvent volume fraction in the polymerrich phase would be much lower than its preset value for a homogeneous solution, so it should be calibrated when applied to eq 3. To this end, first we determine the standard correlation between the demixing parameter and the solvent volume fraction for polymers homogeneously distributed in the athermal solvent, as shown in Figure 5a. Because the demixing parameter at the onset strain can be read from the strainevolution curve of the demixing parameter (see, e.g., in Figure 4a), the corresponding solvent volume fraction is obtained from the calibration curve in Figure 5a. Applying the calibrated solvent volume fraction to eq 3, the simulation results for onset strains of crystallization in the poor solvent with four solvent volume fractions become aligned with the linear prediction of eq 3, as shown in Figure 5b. In summary, in Figure 6, we put together the three inverse temperature curves of polymers shown above as a unified function of onset strains, solvent volume fractions, and solvent qualities. In addition, we also added the corresponding results of bulk polymers obtained in the previous simulations.22 One can clearly see that all of the data points align well with the linear relationship predicted by eq 3. Although all of the data points align well with a global line, Figure 6 still shows a slight deviation of slopes and intercepts for the different solvent qualities and for pure polymers. The equilibrium melting points and the heat of fusion for bulk polymers derived separately from the slopes and the intercepts of the linear groups according to eq 3 are listed in Table 1. One can see that the results in the poor solvent are close to those in the bulk state due to prior phase separation, whereas the results in the good solvent deviate the most from those in the bulk E



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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: 0086-25-89686667. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the helpful early-stage investigation by Yang Zhou as a previous undergraduate student and Yijing Nie as a previous Ph.D. student at Nanjing University. Financial support from the National Natural Science Foundation of China (Nos. 21274061 and 21474050), the Program for Changjiang Scholars and Innovative Research Team in University, the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Creativity Plan for Ph.D. Candidates in the Common Universities is acknowledged.

Figure 6. Inverse crystallization temperatures of solution polymers as a function of onset strains and solvent volume fractions for the athermal solvent with B/Ec = 0 (red squares), the good solvent with B/Ec = −0.3 (green triangles), and the poor solvent with B/Ec = 0.3 (blue spheres). The black downward triangles depict the data for pure polymers.16 The straight line is drawn as a guide to the eye, together with the expression of eq 3.

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Table 1. Bulk Equilibrium Melting Points and the Heats of Fusion Derived According to Equation 3 from the Slopes and Intercepts of the Linear Fitting Curves in Figure 6 for Various Solvent Qualities and for Bulk Polymers T0m/Ec/k Δh/Ec

bulk

B/Ec = 0.3

B/Ec = 0

B/Ec = −0.3

3.3 4.00

3.5 4.83

3.8 6.84

3.9 10.02

state. Because the derived bulk melting points and heats of fusion exhibit a weak dependence on the solvent quality, one may believe that such a dependence can be attributed to the profound theoretical bugs in the thermodynamic equations rather than to the kinetic factors in primary crystal nucleation. The complicated concentration-dependent factors could include the mixing interactions, entanglement, and others.

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REFERENCES

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SUMMARY

We developed a linear combination of two Flory’s thermodynamic equations of melting points, one for solution polymers and another for stretched polymers, to derive the melting-point equation of stretched solution polymers. We carried out dynamic MC simulations of strain-induced crystallization of solution polymers for observing the onset strains of crystallization under varying temperatures, solvent volume fractions, and solvent qualities. Under parallel thermodynamic conditions, the simulation results can be fitted well by the linear relationship derived from the new equation, which thus validates the latter. In the poor solvent, the solvent volume fraction has to be calibrated by the concentrated polymer phase owing to phase separation before strain-induced crystallization. Such a theoretical achievement provides a concrete thermodynamic background for a further investigation of the straininduced crystallization behaviors of network polymers mixed with other small or large molecules, as well as for the rheological experiments on liquid−solid transition in multicomponent polymer systems. F


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Multicomponent Thermodynamics of Strain-Induced Polymer

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