Correlation between the High-Temperature Local Mobility of

Aug 19, 2016 - The present study provides insights into the changes of the mechanical properties of heterocyclic polymers which are directly connected...
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Correlation between the High-Temperature Local Mobility of Heterocyclic Polyimides and Their Mechanical Properties Victor M. Nazarychev,† Alexey V. Lyulin,‡ Sergey V. Larin,† Iosif V. Gofman,† José M. Kenny,†,§ and Sergey V. Lyulin*,† †

Institute of Macromolecular Compounds, Russian Academy of Sciences, Bol’shoi pr. 31 (V.O.), 199004 St. Petersburg, Russia Theory of Polymers and Soft Matter Group, Technische Universiteit Eindhoven, PO Box 513, 5600 MB, Eindhoven, The Netherlands § Materials Science and Technology Centre, University of Perugia, Loc. Pentima, 4, 05100 Terni, Italy ‡

S Supporting Information *

ABSTRACT: The present study provides insights into the changes of the mechanical properties of heterocyclic polymers which are directly connected to their local segmental mobility above the glass-transition point. By performing both fully atomistic molecular dynamic simulations and physical experimentation, we, for the first time, focus on the mechanical behavior of the thermoplastic polyimide R-BAPS with the repeating unit consisting of 1,3-bis(3′,4-dicarboxyphenoxy)benzene (dianhydride R) and 4,4′-bis(4″-aminophenoxy)biphenyl sulfone (diamine BAPS). The previous computer simulations of this polyimide established the significant role of the partial charges to interpret the experimental thermal properties of R-BAPS. The present study determines the influence of the electrostatic interactions on the local mobility of R-BAPS, which, in turn, is to a large extent responsible for its mechanical behavior in the glassy state. It is demonstrated that accounting for partial charges increases the average translational and orientational relaxation times by approximately 2 orders of magnitude as compared to the systems without partial charges. We show that this segmental mobility reduction above the glass transition leads to the improved polyimide mechanical properties in the glassy state. With proper accounting for partial charges in the simulations, the R-BAPS yield stress increases, and the Poisson’s ratio is reduced, as compared to the systems without partial charges. At the same time, all the simulated samples show similar dependence of mechanical properties on the cooling and deformation rates. The Eyring theory formalism has been used to assess the plastic deformation-related kinetic properties. The interrelation between the activation energy during the plastic deformation and the thermal history (cooling rate) of the simulated samples is shown.

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INTRODUCTION

does not allow the polymer slowest relaxation processes to reach the final equilibrium stage. Polymer sample at temperatures below the corresponding glass-transition point (Tg) may retain some small-scale movements of individual segments or atomic groups, with no visible cooperative rearranging on scales of few repeating units, or reorientations of polymer chain as a whole.6,7 Therefore, the final mechanical properties of a polymer glass are determined, in particular, by the total interatomic potential energy which is a function of the coordinates of all atoms. The shape and depth of these potential energy maxima and minima (the potential landscape) depend on the polymer thermal history, in particular, on the implemented cooling rate.8 Therefore, the analysis of the polymer relaxation processes at

Polyimides (PIs) are promising polymeric binding materials which have a great potential of being used in space, aircraft, and automotive industry as light and strong coatings, substrates in microelectronic devices, gas-separation membranes, etc.1,2 The changes in the PI chemical structure allow for purposeful modification of their thermal,3 mechanical,4,5 and dielectric properties. For example, synthesizing new thermoplastic PIs with improved performance1,2 is nowadays a common practice to incorporate oxygen or nitrogen heteroatoms or polar sulfone groups into the PI repeating unit. The synthesis of new thermoplastic PIs with improved mechanical properties without simultaneous significant loss of their thermal plasticity and the heat resistance is an important objective of modern polymer physicochemistry.1,2 The transition of a polymer sample into a glassy state usually occurs upon cooling the polymer melt down with some fixed cooling rate. The external time scale imposed by this cooling © XXXX American Chemical Society

Received: March 23, 2016 Revised: August 12, 2016

A

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Macromolecules temperatures above Tg and the polymer segmental mobility in a viscous-flow state might help to provide insights into the transformation mechanisms governing the mechanical properties of a thermoplastic polymer glass. It is exactly our goal in this study to try to connect the mechanical properties of thermoplastic PIs in the glassy state with their dynamic peculiarities at temperatures above the glass transition. The presence of bulky phthalimide cycles and phenylene rings in the chemical structure of the thermoplastic PI repeating unit suggests some reduced local translational and segmental mobility as compared to those of simpler polymers at temperatures close to the corresponding Tg. Despite the fact that experimental methods (dielectric spectroscopy (DS),9−15 dynamic mechanical analysis (DMA),16−27 and nuclear magnetic resonance (NMR)28) are widely used to evaluate the PI mobility, the experimental approaches do not always allow to investigate the individual contributions of various interand intramolecular interactions into the full system energy and to determine their possible influence on the segmental mobility. One of the most efficient methods to quantitatively assess the local segmental mobility of polymer chains is a computer-aided simulation using fully atomistic models. The molecular dynamics (MD) is widely used to study the local mobility of commodity polymers, such as polyethylene and polystyrene, for example.29−31 In simulations of atactic polystyrene,32,33 the authors considered how the changes of various external conditions (as deformation rate, temperature, pressure, etc.) impact the mechanical characteristics. The results obtained correlate very well with the available experimental data. At the same time, no quantitative estimates of the time scales of the segmental mobility have been performed for polyimides. The main reason is probably due to the fact that rather long and computationally demanding simulations of the cooling procedure are required to correctly determine the corresponding relaxation times.34 In the computer simulation of many commodity polymers the electrostatic interactions are usually not taken into account.32,33 It should not be the case for thermoplastic polyimides with repeating units including nitrogen, sulfur, and oxygen heteroatoms. The electrostatic interactions cannot be neglected in this case.3 Thermoplastic PIs containing polar (such as sulfone) hinge groups may undergo changes in the system energy due to the changes either in the polymer chain flexibility or in the interand intramolecular dipole−dipole interactions between the groups with high values of dipole moments.3 The incorporation of polar groups may lead to increase in Tg and to a decrease of the thermal expansion coefficient (CTE) as compared to samples without polar groups.3,35 For PI R-BAPS,34,36 with the repeating unit consisting of 1,3-bis(3′,4-dicarboxyphenoxy)benzene (dianhydride R) and 4,4′-bis(4″-aminophenoxy)biphenyl sulfone (diamine BAPS) (see Figure 1), a strong

influence of partial charges on polymer thermal properties was found in our previous study.3 This effect may be attributed to rather different energy barriers restricting the segmental mobility of polymer chains at temperatures above and below T g. Our previous simulations34−37 of thermoplastic PIs showed that their thermal properties may depend strongly on the implemented parametrization of the electrostatic interactions (EI).3 The obtained results demonstrated that for the MD simulations using the popular Gromos53a5 force field the best correlation with the available experiments is achieved with partial charges calculated using the Hartree−Fock (HF) approach with the basis set of wave functions 6-31G* and Mulliken method of calculation, HF/6-31G* (Mulliken). This type of PI and the HF/6-31G* (Mulliken) method are also used in the present study. Calculated values of partial charges for PI R-BAPS were published in our previous study.3 The present study is from our point of view the logical continuation of our research on the influence of the electrostatic interactions on the static, thermal, dynamic, and mechanic properties of polar thermoplastics.3,34−39 As far as we know, the segmental relaxation for these classes of industrially important thermoplastics has not been investigated yet. The influence of the deformation and cooling rates on the Poisson’s ratio and yield stress for the samples of polar thermoplastics is analyzed in the present study for the systems with and without partial charges which allows better understanding how the simulation data can be directly compared with corresponding experimental data and the partial charges importance for the atomistically detailed simulation. We also perform a comparison of the computer simulation results for models with and without partial charges with the experimental yield stress values measured upon the uniaxial stretching of PI R-BAPS film samples. The simulated Poisson’s ratio is compared with literature data. The rest of the paper is organized as follows. The next section describes the model used and the corresponding simulation methods. It is followed by the main part, which discusses our findings concerning the PI segmental dynamic and the mechanical properties, with and without EI accounted for. The final section offers major conclusions.

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MODEL AND SIMULATION METHOD Model and Simulations Details. The details of the MD simulations related to the generation of the polymer initial configurations, the equilibration criteria, and the parametrization of EI can be found in our previous publications on the subject.3,34−37,39,40 We explain here only the major details. The initial configuration of the simulated system was made up of 27 partially coiled polymer chains with a polymerization degree Np = 8 (i.e., about 15 Kuhn segments per chain). The initial lowdensity “polymer gas” state was isotropically compressed at the initial high temperature, T = 800 K, without partial charges taken into account. To achieve the equilibrium state, a sequence of compression, annealing, and microsecond-scale long-time equilibration runs has been carried out.36 After such a thoroughly performed equilibration, a main production was carried out for 1 μs, which was accompanied by saving the instantaneous polymer configurations every 100 ns; in total, 11 configurations have been saved. To explore the influence of the dipole−dipole interactions, each of the samples under study was additionally equilibrated for 100 ns accounting for partial charges. They, in turn, were calculated using the quantum-

Figure 1. Chemical structure of the repeating unit of the thermoplastic PI R-BAPS, used in the present MD simulations and experiments. Unit vectors normal to the phenylene rings PH1, PH2, PH3, PH4, and PH5 investigated in the present simulation are shown by arrows. The considered last ring of the 8th monomer unit is shown by the dotted circle. B

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Figure 2. Flowchart of the simulations performed to investigate the dynamical and mechanical properties of the thermoplastic PI R-BAPS.

chemical HF/6-31G* (Mulliken) method.3,36,37 These long equilibration runs helped to ensure the completion of the local chains reorganization. The polymer samples obtained with and without partial charges were further cooled to the room temperature. In order to investigate the influence of the cooling rate (thermal history) on the PI mechanical properties, the MD simulations have been performed using the stepwise cooling from 800 to 290 K with various cooling rates γc, spanning 4 decades, from 1.5 × 1010 to 1.5 × 1014 K/min. The local segmental mobility above Tg was investigated by the analysis of trajectories obtained during the slow cooling with γc = 1.5 × 1011 K/min. To study the influence of the deformation rate on the mechanical properties of PI R-BAPS, the polymer samples in a glassy state, cooled with γc = 1.5 × 1011 K/min, were further uniaxially deformed along the X-, Y-, or Z-axis, with different deformation rates γda, from 10−1 to 10−5 nm/ps. For all samples the initial dimension of the simulated periodic cubic box was L0 ∼ 5.8 nm; the corresponding deformation rate γd = γda/L0 is varied from 1.8 × 106 to 1.8 × 1010 s−1. Figure 2 shows the general flowchart of the performed simulations. The MD simulations were performed using Gromacs 4.5 package,41,42 with Gromos53a5 force field.43 To account for the EIs, the particle-mesh Ewald method (PME) was used.44 Simulation of the Uniaxial Deformation. Prior to the deformation, the P-LINCS45 algorithm used to fix the length of chemical bonds was deactivated, followed by a 400 ps simulation, in order to avoid the observed instabilities during the deformation at rates exceeding 1010 s−1. In all further calculations, the chemical bond stretching energy was described using the harmonic potential of the Gromos53a5 force field.43 As before,32 the uniaxial deformation of the samples under study was carried out by changing the size of the simulation periodic box with a constant deformation rate along each of the positive directions. An anisotropic Berendsen barostat with the time constant τp = 1 ps has been used. Along the deformation direction, the compressibility of the samples was equal to zero, whereas in the direction perpendicular to the deformation, the compressibility of the sample was set equal to 4.5 × 10−10 Pa−1, which is a default compressibility value in the Gromacs package. At a constant external pressure of 1 atm the periodic size

dimensions were allowed to change in the direction perpendicular to the deformation (see Figure 3). Notably, the

Figure 3. Initial (left) and final (right) configuration of the PI R-BAPS sample during the typical uniaxial deformation along the X-axis. Black lines mark the periodic box before and after the deformation.

suggested deformation algorithm differs from the simulations of stretching of polymer films by applying the force to the confining crystalline substrates,46 where no periodic box dimensional changes in the transversal direction are allowed. Such a deformation, for example, makes it impossible to calculate the Poisson’s ratio. During the deformation, the stress σ and the strain ε have been calculated every 1 ps: σ = −Pii ε=

Li − L0i L 0i

(1)

where i = {x, y, z} is the deformation direction, Pii are the diagonal components of the stress tensor, Li is the current dimension of the simulation box, and L0i is the edge size of the simulation cubic box at starting time t = 0. The initial (up to ∼2% of the strain) stress−strain dependence σ = f(ε) is well described by the linear Hooke’s law.47,48 An important mechanical characteristic of a polymer sample is its Poisson’s ratio μ, which indicates the relative decrease of the transversal dimension γW of the deformed body to the increase of its longitudinal dimension48 γ μ = − lim W (2) ε→ 0 ε C

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γW =

W − W0 W0

(3)

where W and W0 are current and effective widths (the square root of the cross-section area A of the periodic box in the direction perpendicular to the deformation) of the box, respectively.32,49 These measurements have been performed in the reported simulations.

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EXPERIMENTAL METHODS

Film samples of PI R-BAPS for experimental measurements of tensile mechanical properties were prepared as described in ref 35. The yield stress values σy of the PI R-BAPS 30 μm thick films were determined by the uniaxial extension at T = 293 K. The samples with the working part size of 1 mm × 20 mm were deformed in the AG100kNX (Shimadzu, Japan) machine at the deformation velocity of γda = 10 mm/min, which corresponds to the relative stretching rate of γd = 8.4 × 10−3 s−1. The yield stress value σy was determined as the maximum on the stress−strain curve σ(ε). The resulting data were obtained by averaging dependences for at least five different deformed samples.

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RESULTS AND DISCUSSION Segmental Dynamic Properties above Tg. Translational Mobility of Atoms. To study the local mobility above Tg, we investigated the translational mobility of the end diamine fragment atoms of the thermoplastic PI R-BAPS polymer chains. These atoms are located at the end of the eighth repeating unit (see Figure S1 in the Supporting Information), i.e., in the phenylene ring on the edge of each polymer chain, where the mobility is higher as compared to that of other polymer fragments (see Figures S2 and S3).50 For these atoms the mean-square displacement at different temperatures has been calculated ⟨Δr 2(t )⟩ = ⟨( r (⃗ t1) − r (⃗ t 2))2 ⟩

Figure 4. Time dependence of the mean-square displacement ⟨Δr2(t)⟩ of carbon atoms located at the end of the eighth repeating unit of each PI R-BAPS polymer chain in the phenylene ring, at different temperatures (a) with partial charges and (b) without partial charges.

(4)

where r(⃗ ti) is a vector determining the current position of each atom. The calculation of ⟨Δr2(t)⟩ was performed both for samples with and without partial charges accounted for (Figure 4). In both cases, the Rouse-like regime was observed,50 where ⟨Δr2(t)⟩ ∼ tn. The slope in dependence ⟨Δr2(t)⟩ was around ∼0.56−0.58 (more than 0.5 which may be related with the finite length of the macromolecules). The temperature analysis of the mean-square displacements (see Figure 4) predictably shows the deceleration of the translational mobility at lower temperatures. The characteristic relaxation times τα = Dα−1 have been extracted by the powerlaw fitting of the ⟨Δr2(t)⟩ dependences in the Rouse-like regime:50 ⟨Δr 2(t )⟩ ∼ (Dα t )n

Figure 5. Temperature dependence of the characteristic time τα of the translational relaxation for samples with and without partial charges accounted for. Solid lines are fits of the mode coupling theory (MCT) (see eq 6), and symbols refer to the MD results.

magnitude higher than those for the samples without partial charges. The critical temperature Tc and the MCT parameter γ for the samples with partial charges are 701 ± 4 K and 3.0 ± 0.4, respectively, and for the samples without partial charges they are 492 ± 4 K and 2.6 ± 0.14, respectively. Tc for the samples without partial charges exceeds the simulated34 glasstransition temperature Tg = 459 ± 15 K by approximately 40 deg. For the samples with partial charges the value of Tc is higher than the corresponding glass-transition temperature, Tg = 640 ± 15 K.34 This result correlates very well with other simulation findings; i.e., the MCT critical temperature value only exceeds the volumetric Tg52 and shows once more the importance of the proper accounting for the electrostatic contributions.

(5)

The mode coupling theory (MCT)6,51 states that the temperature dependence of the characteristic relaxation time τα (see Figure 5) above some critical temperature Tc > Tg algebraically diverges τα ∼ (T − Tc)−γ

(6)

Figure 5 shows these dependences for simulated PIs, with and without partial charges accounted for. The analysis of the results demonstrates that at the fixed cooling rate the characteristic relaxation times τα for the samples with partial charges are approximately 2 orders of D

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Macromolecules Orientational Segmental Dynamics. The local orientational mobility of PI R-BAPS fragments in the high-frequency interval of the relaxation spectrum has been simulated by measuring the autocorrelation functions P1(t) of the unit vector b normal to the surface of the different phenylene rings both in the diamine and in the dianhydride fragments of the PI R-BAPS repeating unit:9,53 P1(t ) = ⟨b(0)b(t )⟩

The time dependences P1(t) for the unit vectors normal to the phenylene rings PH1, PH2, PH3, PH4, and PH5 are presented in Figures S3 and S4. The time dependences P1(t) were approximated using the Kohlrausch−Williams−Watts (KWW) stretched exponentials54 ⎛ ⎛ t ⎞β⎞ P1(t ) = A exp⎜ −⎜ ⎟ ⎟ ⎝ ⎝τ⎠ ⎠

(7)

Here the brackets shows the averaging over all 27 polymer chains and 11 samples. Figure 6 represents the corresponding time dependences for the samples with and without partial charges, for different temperatures.

(8)

where A ≤ 1, τ is the characteristic relaxation time, and β is a parameter taking into account the nonexponentiality of the relaxation process. We should emphasize that for the approximation by stretched exponent (KWW function) we chosen the parts of P1(t) autocorrelation functions without ballistic regime (1 ps−2 ns time scale for the samples without partial charges and 10 ps−50 ns for the samples with partial charges). Simulated β values in KWW approximation only very weakly depend on temperature in the simulated temperature range for the phenylene rings PH2−PH5 in diamine fragment; their values for samples with partial charges were close to 0.55. The values of β for the samples without partial charges are slightly larger, β ∼ 0.7. This reflects the narrowing of the relaxation spectrum for the samples without partial charges. Upon temperature decrease the values of β decrease as well, for both samples, and approach to β ∼ 0.3, which reflects the broadening of the relaxation spectrum and more heterogeneous segmental dynamics upon approaching the glass transition. However, for the PH1 phenylene rings the values of β parameter are the same for the samples both with and without partial charges as for rather high temperatures (β ∼ 0.30−0.35) as with temperature decrease until the glass transition (β ∼ 0.15). Rather strong heterogeneity of the dynamics has been observed in the relaxation of the phenylene rings in both samples. The temperature dependence of the relaxation times can be fit by the Vogel−Fulcher−Tammann (VFT) dependence for samples without partial charges. The activation energy E′α and the values of the critical temperature T0 are different for different phenylene rings; this could reflect the cooperative nature of the vitrification. For the samples without partial charges the critical temperatures are very low, about 120−200 K below the corresponding glass transition (see Table S1). For the samples with partial charges the much smaller deviations from the Arrhenius temperature dependence of the PH2−PH5 rings relaxation could reflect the important result of the strong dipole−dipole interactions between the sulfone groups. Most likely, the slowing down of the local mobility does not relate only to the increase of the average sample density due to the presence of nonzero partial charges. To test this suggestion, we have performed an additional simulation of samples with and without partial charges but having the same density (we increased the density of the sample without partial charges by applying additional external pressure; see Supporting Information for the details). Obtained results demonstrate that the local orientational mobility of the samples with partial charges is much smaller as compared to those without partial charges, not only because of the difference in average density but also because of the significant influence of the dipole−dipole interactions (see Figure S6).

Figure 6. Time dependence of the autocorrelation function P1(t) of the unit vector normal to the surface of the selected phenylene ring for the samples (a) with partial charges and (b) without partial charges.

In the present study the size of the kinetic segment was evaluated under the plastic deformation. It was shown that it is comparable to the size of the phenylene or phthalimide ring of the PI R-BAPS repeating unit. It is rather natural to assume that it is the phenylene-rings relaxation which starts first upon onset of the plastic deformation. We also have simulated the local orientational mobility of all the phenylene rings, both in the diamine and in the dianhydride fragments of PI R-BAPS (see Figures S1 and S2). First of all, the temperature dependence of the phenylene-rings relaxation times is very similar qualitatively for all the simulated rings (see Figure S5). It should be noted that very long simulations are required to get a good statistics for the analysis of this kind of the relaxation for all the phenylene rings. That is why only the most mobile groups, at the ends of the repeating units, were analyzed in detail, as their relaxation occurs within the simulation time window. In this case the relaxation times can be calculated with sufficient precision. E

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Figure 8. Stress−strain dependence for PI samples deformed with the deformation rate γd = 1.8 × 108 s−1 and cooled down with different cooling rates γc (a) with and (b) without partial charges.

Figure 7. Temperature dependence of the relaxation time τ for P1(t) autocorrelation functions for different phenylene rings (PH1−PH5) for the samples with (a) and without (b) partial charges. The dashed lines in (a) and (b) show Arrhenius τ ≈ τ0 exp[Ea/RT] and Vogel− Fulcher−Tammann (VFT) τ ≈ τ0 exp[E′a/R(T − T0)] approximation, respectively.

both sample types is very well visible at a relative deformation of 10−12%. The samples without partial charges show more distinct yield stress and strain softening regime as compared to the samples with partial charges. In the case of the further plastic deformation, the thermal history influence is negligible (for deformations exceeding 30%); the universal strain hardening regime at strains above 30% is observed for all samples, independently of the simulated values of cooling rate, similar to what was concluded earlier for other polymer glasses, as polystyrene for example.32 The calculated strain-hardening modulus is almost independent of γc and depends only slightly on the presence of partial charges. The yield stress values σy for the samples with partial charges exceed those values for the samples without partial charges, obtained by cooling with the same cooling rate. Obviously, the packing of polymer chains in a glassy state and the average monomer density are also affected by the sample thermal history. The density-strain dependences ρ(ε) for samples with and without partial charges are shown in Figure 9. For all the cooling rates γc, the density of the nondeformed samples with partial charges at T = 290 K are slightly higher, whereas for the samples without partial charges they are slightly lower as compared to the experimental value, ρexp = 1365 ± 0.002 kg/m3.36 For all PI samples, the faster cooling produces samples with lower density. For samples with and without partial charges, the faster cooling rate result in smaller value of the density in the glassy state. The strong dipole−dipole interactions between the polar sulfone groups may lead to some additional ordering,3,40 which helps to pack the PI chains more effectively. This leads to the increase of the sample density (by ∼5% as compared to the samples without partial charges) and can, in turn, influence the final mechanical properties.

The results obtained for orientational segmental relaxation correlate well with the observed deceleration of the translational mobility of individual PI R-BAPS polymer chains. Mechanical Properties of Polyimide R-BAPS in Glassy State. To investigate the interrelation between the observed segmental-dynamic properties in the melt and the mechanical properties in the glassy state, the PI R-BAPS samples were deformed below Tg. The observed changes in the internal energy and in the spatial structure of the polymer chains depend on the simulated values of cooling and deformation rates. These two parameters have rather significant influence on the mechanical properties of PI polymers in the glassy state.29 Influence of the Cooling Rate on the Mechanical Properties. In order to explore the influence of the cooling rate γc on the mechanical properties, we investigated the PI RBAPS samples cooled down to 290 K.34 For the samples without partial charges the cooling rate varied from 1.5 × 1014 to 1.5 × 1010 K/min; for the samples with partial charges, the cooling rate γc varied from 1.5 × 1013 to 1.5 × 1011 K/min. The uniaxial deformation has been performed with the deformation rate of 1.8 × 108 s−1, and the stress−strain dependence σ(ε) for different cooling rates has been calculated (see Figure 8). The results obtained showed that for both samples with (Figure 8a) and without partial charges (Figure 8b) the yield stress σy decreases with increase of γc.33 At high cooling rates γc exceeding 1.5 × 1012 K/min, the yield stress is hardly to be seen. As γc decreases, the curves σ(ε) start to display the yield stress and the strain-softening area. In fact, the yield stress for F

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Figure 9. Density-strain dependence of PI R-BAPS for systems with and without partial charges cooled down with different cooling rates γc. Uniaxial deformation was performed with the deformation rate γd = 1.8 × 108 s−1. Black horizontal dotted line marks the experimental density of the nondeformed sample at a room temperature cooled at the cooling rate γc = 5 K/min.36

Figure 10. Stress−strain dependences of the PI R-BAPS deformed with different deformation rates γd for the samples (a) with and (b) without partial charges. Nondeformed samples were obtained by cooling at cooling rate γc = 1.5 × 1011 K/min.

Upon stretching, the PI density decreases, reaching the saturation above 30% of the deformation, where the strain hardening is observed (see Figure 8). On the other hand, the temperature dependence of the density for samples with partial charges continues to decrease above 30% of elongation, in the strain hardening area (see Figure 9). This behavior can be explained by the effect of some strong dipole−dipole electrostatic interactions between polar groups that counteract the propagation of the plastic deformation in the PI R-BAPS samples with partial charges. Influence of the Deformation Rate on the PI Mechanical Properties. The deformation rate values affect substantially the mechanical properties of bulk polymers in the glassy state. To study this influence, the uniaxial deformation of the PI R-BAPS samples cooled down with the rate γc = 1.5 × 1011 K/min has been performed with different deformation rates from 1.8 × 106 to 1.8 × 1010 s−1 (Figure 10). The simulated values of the Poisson’s ratio μ decrease with the increase of the deformation rate γd for both samples with and without partial charges (Table 1). For the fixed deformation rate, the value of the Poisson’s ratio is slightly higher for samples with partial charges (for example, μ = 0.36 ± 0.01 for PI with partial charges as compared to μ = 0.42 ± 0.01 for PI without partial charges, at γd = 1.8 × 108 s−1). The samples with partial charges are less compressible (lower Poisson’s ratio), which also reflects their better initial chain packing. For all considered values of γd, the increased yield stress value σy of the samples with partial charges correlates with the decreasing Poisson’s ratio μ. In simulations, the Poisson’s ratio remains practically unchanged upon changing the values of the deformation or cooling rates. This very weak dependence justifies the direct comparison of the simulated and experimental data; the conclusion is that the simulated Poisson’s ratio of both samples with and without partial charges correlate with the experimental result, μ = 0.34− 0.40.55−58 It is well-known that the temperature and the deformation rate dependence of the yield stress σy(T,γd) could be well described by the Eyring theory32,59−61

Table 1. Poisson’s Ratio for the PI R-BAPS with and without Partial Chargesa Poisson’s ratio μ deformation rate γd (s−1) 1.8 1.8 1.8 1.8 1.8

× × × × ×

without partial charges

10

10 109 108 107 106

0.27 0.37 0.42 0.41 0.40

± ± ± ± ±

0.01 0.01 0.01 0.01 0.01

with partial charges 0.28 0.36 0.36 0.39 0.37

± ± ± ± ±

0.01 0.01 0.01 0.01 0.01

a Samples were deformed with different deformation rates from 1.8 × 106 to 1.8 × 1010 s−1 after cooling down at the cooling rate γc = 1.5 × 1011 K/min.

σy =

2γ ΔH RT + ln d v v γ0

(9)

where ΔH is the activation energy, v is the volume of the element (the activation volume) which starts to move to initiate the plastic yield, and R is the universal gas constant. Initially, the dependence of the yield stress σy on the deformation rate γd is clearly logarithmic, Figure 11. A similar dependence has been obtained in the coarse-grained modeling of a polyethylene melt.62 At very high deformation rates, γd > 1.8 × 1010 s−1, strong deviations from the logarithmic dependence are clearly seen. Such deviations might be related to the contribution of several relaxation processes into the total relaxation spectrum, these processes being activated at high deformation rates.63,64 To determine the values of ΔH and v, the dependence σy(γd) was approximated using eq 9 only for γd < 1.8 × 1010 s−1 (see Table 2). For the samples without partial charges, the activation energy for the plastic flow decreases with the growing cooling rate γc. For the samples with partial charges, this dependence is much weaker; the activation volume is also slightly lower. Again, this G

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In the present study, this extrapolation has been performed (see Figure 11). For both samples, with and without partial charges, such an extrapolation produces negative values of the charges charges yield stress, σw/o ∼ −170 MPa and σwith ∼ −70 MPa. y y The additional extrapolation to the experimental cooling rates (which are much slower as compared to those used in the typical MD simulation) leads to the increase of the calculated values of the yield stress. The best agreement with the experimental value, σexp y = 112 MPa, is obtained for the sample with partial charges. Note, nevertheless, that such a conclusion should be taken with some caution; it has only some qualitative character. Much better and more precise results are necessary to obtain using simulations with much slower cooling rates. These cooling rates require computationally demanding simulations of the order of about 100 μs. We plan to perform these simulations in the near future.

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CONCLUSIONS We investigated the segmental relaxation and macroscopic mechanical properties of the amorphous thermoplastic polyimide R-BAPS containing strongly charged polar groups, using atomistic molecular dynamics simulations and physical experiments. For these polymers we examined the influence of the electrostatic interactions on the dynamic performance in the viscous flow state at temperatures significantly exceeding the glass-transition temperature Tg and on their mechanical properties in the glassy state. To this end, the average relaxation times of the local translational and orientational mobility for PI R-BAPS samples with and without partial charges have been simulated. For the samples with partial charges the much smaller deviations from the Arrhenius temperature dependence of the PH2−PH5 rings relaxation could reflect the important role of the strong dipole− dipole interactions between the sulfone groups. Such a difference in the activation energies might imply a crucial role of dipole−dipole interactions influencing the dynamical and mechanical properties of the PI R-BAPS in the glassy state. A substantial correlation has been established between the changes in the local translational and orientational mobility above Tg and Poisson’s ratio and the yield stress. It has been demonstrated that the reduced local mobility above the glasstransition point leads to an increase in the yield stress σy as well as to the reduction of the Poisson’s ratio μ in a glassy state. The deceleration of the segmental relaxation in PI chains with partial charges correlates with the enhancement of mechanical properties in a glassy state. It has been shown that the samples with reduced segmental mobility may not reach the yield stress during the uniaxial deformation. The density-cooling rate dependence disappears when the sample is stretched above the yield stress, which might be due to the mechanical rejuvenation effect of the polymer material. Our study also included the experimental investigation of mechanical properties of the thermoplastic PI R-BAPS. The results obtained showed that correct simulation of the experimental values requires the account of the difference in cooling and deformation rates. In order to have a better agreement between simulational and experimental data, it is necessary to simultaneously consider the influence of both cooling and deformation rates on the mechanical properties of thermoplastic polyimides, which, however, requires incredibly large computational resources.

Figure 11. Simulated dependence of the yield stress σy on the deformation rate γd at T = 290 K for the PI R-BAPS samples (a) with and (b) without partial charges. Dashed line illustrates extrapolation results to the experimental values of the deformation rate γdexp = 8.3 × 10−3 s−1. The dotted lines mark the quantitative extrapolation to the experimental values of the cooling rate.

Table 2. Activation Energy ΔH and Activation Volume v for the PI R-BAPS Samples with and without Partial Charges and Cooled at Different Cooling Rates without partial charges cooling rate γc (K/min) 1.5 1.5 1.5 1.5 1.5

× × × × ×

10

10 1011 1012 1013 1014

ΔH (kJ/mol) 49 46 41 39 34

± ± ± ± ±

4 1 1 2 2

v (nm3) 0.16 0.16 0.15 0.16 0.14

± ± ± ± ±

0.02 0.01 0.01 0.01 0.02

with partial charges ΔH (kJ/mol)

v (nm3)

65 ± 6 62 ± 5 63 ± 6

0.12 ± 0.01 0.12 ± 0.01 0.13 ± 0.02

might be related to the additional packing of the PI R-BAPS polymer chains due to the dipole−dipole interactions between sulfone groups,36 which facilitates plastic flow. From the activation volume data the kinetic segment can be evaluated as about 5 Å for both samples with and without partial charges. This length is comparable to the size of one phenylene or phtalimide group in the PI R-BAPS repeating unit. The experiment has also been performed (see the Experimental Methods section) to define the PI yield stress value, 112 ± 3 MPa, which occurs to be much lower as compared to the simulation results. The yield stress of the glassy PI samples depends logarithmically on both the deformation and cooling rates; see refs 65−68 (however, for some filled polymers this dependence may be rather weak69). Usually, the extrapolation to the experimental values is not carried out, and accurate comparison with experimental data is absent. H

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Macromolecules

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The Eyring phenomenological theory was used to calculate the activation energy ΔH and the activation volume v of the thermoplastic PI R-BAPS with and without partial charges. It has been demonstrated that the value of ΔH has stronger dependence of the thermal history as compared to the corresponding dependence of the activation volume. The calculated length of the polymer chain kinetic segment which starts to displace in the plastic yield was equal to 5 Å; this value is comparable to the size of the phenylene or phtalimide group in the PI R-BAPS repeating unit. Such a small value of the kinetic segment during the plastic deformation might be due to the use of rather large deformation rates in the simulations as compared to those used in the experiment, which obviously leads to the plastic yield on a spatial scale substantially smaller than the size of the PI repeating unit. While computer simulations of PI R-BAPS samples with partial charges require much larger computational resources, the consideration of these charges leads to a much better agreement with the experimental yield stress values, upon extrapolation to the experimental values of the deformation and cooling rates. Correct extrapolation to the experimental cooling rates requires computationally demanding simulations of the order of about 100 μs. We plan to perform these simulations in the near future. Our study has demonstrated the relevant interrelation between changes in the local segmental mobility of thermoplastic PI above the glass-transition point and changes in their mechanical properties in the glassy state. The method applied will be further utilized to simulate the dielectric properties of this important class of polyimides using fully atomistic dynamic simulations.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b00602. Figures S1−S6 and Table S1 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (S.V.L.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The fruitful discussions with Prof. K. Karatasos are gratefully acknowledged. This study has been supported by the Russian Ministry of Education and Science within State Contract 14.Z50.31.0002 (megagrant). All simulations have been performed using the computational resources of the Institute of Macromolecular Compounds, Russian Academy of Sciences, Chebyshev and Lomonosov supercomputers at Moscow State University, and computational resources of MCC NRC “Kurchatov Institute”.

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REFERENCES

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