Chain Network: Key to the Ductile Behavior of Polymer Glasses

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Chain Network: Key to the Ductile Behavior of Polymer Glasses Zhuonan Liu, Xiaoxiao Li, Yexin Zheng, Shi-Qing Wang,* and Mesfin Tsige* Department of Polymer Science, University of Akron, Akron, Ohio 44325, United States ABSTRACT: Polymer glasses can be either ductile, undergoing yielding and necking, or brittle, suffering from crazing and brittle fracture. Until recently, there was no satisfactory molecular-level interpretation of these processes. In this work, carefully designed polystyrene glass systems with new deformation protocol have been modeled using all-atom molecular dynamics simulation. The effect of chain networking in terms of yielding taking place in the presence of long chains embedded in short chains is investigated. The results are compared with a recently proposed molecular model, which suggests that the structural integrity, necessary to prevent brittle fracture, is mainly due to chain networking.

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INTRODUCTION Understanding mechanical behavior of polymer glasses is one of several fundamental challenges in polymer physics.1,2 Since only glassy polymers of sufficiently high molecular weight are ductile, some type of chain networking must play an important role in minimizing craze formation and ensuring ductility. For example, the popular concept of polymer entanglement has been used3 to indicate that somehow intermolecular chain uncrossability is important. Recently, we have proposed4 that a particular kind of chain network is necessary to drive a glassy polymer to yield. The microscopic structure of this molecular network differs from that of the entanglement network described by the packing model.5−7 Our molecular model4 emphasized the specific role played by the chain network, while the past studies indicated that (a) yielding could take place without help from the chain network in a nonlinear Langevin equation (NLE)-based microscopic theory8−10 and (b) polymer glasses of low molecular weight could undergo plastic flow without brittle failure in molecular dynamics simulations.11 Because of the lack of effective experimental tools to directly visualize molecular responses during deformation of amorphous polymeric solids, our latest understanding about yielding and failure has remained difficult to verify. It is experimentally well-known that glassy polymers of low molecular weight cannot undergo plastic deformation and are typically mechanically weak in both tensile extension12 and uniaxial compression.13,14 Actually, nonpolymeric organic glasses are all and always brittle. For example, let us consider a macroscopic specimen (say 10 mm in length) of polystyrene of 30 kg/mol in molecular weight (PS30K) in tensile extension. Well below the glass transition temperature, e.g., room temperature, the polystyrene molecules in the specimen have insufficient mobility even on the monomeric length scales. If only the monomers at the moving end of the macroscopic specimen are displaced as is the case for tensile extension, how can monomers far away from this end keep up with the moving © XXXX American Chemical Society

boundary as apparent strain increases? How could the displacement of one end (while holding the other fixed) cause the entire specimen to deform beyond the linear response regime? It is actually the consensus of the field that the spatially sharp structural/mechanical failure of polystyrene at room temperature, independent of molecular weight, is an indication of the lack of yielding. While polystyrene of low molecular weight (e.g., PS30K) remains brittle even in compression14 as long as the experimental temperature is below the glass transition temperature, high molecular weight polystyrene (e.g., PS150 K) does yield and turn ductile at 70 °C in a certain range of extensional rate.15 Since sufficient pressurization can cause polystyrene to be ductile at room temperature and yield at a substantially higher stress than the breaking stress under atmospheric pressure,16 the standard LDWO hypothesis17−20 is clearly not a valid account of the brittle−ductile transition. Brittle fracture is not due to chain scission.21 A plausible molecular mechanism for the brittle−ductile transition states that the transition temperature is Goldilocks temperature4 where the chain network is marginally effective to activate the glassy state into plasticity. Yielding of the molecular structure due to intersegmental attractions is a prerequisite for plastic flow. This work attempts to accomplish two objectives: (a) to explore how brittle failure may be observed in atomistic molecular dynamics simulation and (b) to gain some insight into how yielding takes place by the formation of activation zones due to the active displacement of network strands. Specifically, all-atom molecular dynamics (MD) simulations were performed to study the mechanical behaviors of Received: August 16, 2017 Revised: January 9, 2018

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Figure 1. Comparison of two deformation protocols on low molecular weight PS glass (system I). (a) Conventional affine deformation of a periodic system consisting of short PS chains. The system has been stretched to 20%. The carbon atoms on the rings are blue, the hydrogen atoms are white, and the backbone carbon atoms are green. (b) 20% uniaxial extension of the same system by removing the periodicity in the z-direction and pulling the top and bottom layers colored in gray referred as the new deformation protocol. (c) The stress−strain curves from the two deformation protocols. The strain rate for both systems is 8.6%/ns. (d, e) The average mean-square-displacement of atoms as a function of their z coordinates at different times, after subtracting the affine deformation background (S), for the system deformed with the conventional affine protocol and the new deformation protocol, respectively. The extension for a period of 0.8 ns corresponds to 7% elongation. (f, g) The density along z-direction (extension direction) at different time of the two systems as shown in (a) and (b), respectively.

make it continuous at the cutoff. In our previous studies of polystyrene glasses, the same set of force field parameters have been used and the simulation results have shown that these parameters can provide structural and dynamics properties of polystyrene that match well with experimental measurements.25,26 The Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)27 package was used for performing all the molecular dynamics (MD) simulations in this work. Integration of the equations of motion was performed using the velocityVerlet algorithm. The long-range Coulombic interactions were calculated by the particle−particle/particle-mesh (PPPM) Ewald algorithm.28 The time step of the simulations was 1 fs in order to capture the vibrational movements of hydrogen atoms. Low molecular weight polystyrene was prepared by randomly adding 250 polystyrene chains of 20-mers each in the simulation box. This system was initially equilibrated at 500 K for 10 ns in order to accelerate the equilibration process in the melt state and will be hereafter referred to as system I. Two additional statistically independent systems were also generated by equilibrating the original system for an additional 10 ns or a total of 20 ns at 500 K (hereafter referred to as system II) and for an additional 20 ns or a total of 30 ns at 500 K (hereafter

polystyrene in the glassy state in order to gain information at the molecular scale.

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MODEL AND METHODS Polystyrene was chosen due to its broad industrial application and has been well-studied experimentally and theoretically and has also received a lot of attention in the modeling community. In order to achieve the two objectives laid out above, we have performed all-atom molecular dynamics simulations of polystyrene using the Optimized Potentials for Liquid Simulations All-Atom (OPLS-AA) force field22−24 based on its previous success in correctly predicting the structural properties of polystyrene.25,26 The total potential energy of the system in OPLS-AA force field is represented as a sum of four types of potentials: nonbonded, bond stretching, angle bending, and torsion. The nonbonded potential was modeled as a combination of Lennard-Jones 12−6 potential and Coulomb potential. The bond stretching and angle bending terms were both modeled as harmonic potentials while the torsion is represented by a Fourier series of up to order four of the dihedral angle ϕ. The force field parameters used in this study were directly taken from the OPLS-AA force field.22−24 The cutoff distance for Lennard-Jones potential used here was set to 12 Å, and the potential was shifted by a constant to B

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Figure 2. Demonstrating the reproducilility of our new deformation protocol. In the figure, the stress−strain curves using the conventional affine deformation (open symbols) and the new deformation protocol (closed symbols) for the three different short chain polystyrene systems are shown: triangles (system I), circles (system II), and squares (system III). Open and closed symbols of the same type of symbol represent the stress−strain behavior of a system in the conventional and the new deformation protocol, respectively. The green × symbols overlapped at the end of the closed symbols represent the failure strain values. N

referred to as system III). After these long equilibration runs, all the three systems were quenched down to room temperature with a cooling rate of 20 K/2 ns at atmospheric pressure to obtain a glassy state. Similarly, two entangled long chains were first prepared by manually hooking them at the middle point of each chain. Then the two long chains were mixed with the pre-equilibrated short chain systems by turning off the LJ interactions and adding a soft potential to mildly push the close contact atoms apart in order to avoid any overlap. Then the normal LJ potential was turned on, followed by heating the mixture to 500 K for fast equilibration, similar to the treatment of the short chain systems. The systems were then cooled down to 300 K at a rate of 20 K/2 ns. The mechanical behavior of all the systems was investigated. Two different deformation protocols of glassy polymers were used in this work. The first one is the widely used approach by changing the volume or shape of the simulation box during a dynamics run affinely. The second approach proposed in this work is completely different where the periodicity in one direction of the simulation box is removed by chopping the chains crossing the original periodic boundaries. Then two layers on the top and bottom of the new simulation box are grabbed and pulled in opposite directions mimicking the real process of performing the extensional deformation in experiments. In this study, for the first approach, the deformation speed of the glassy polymer systems was about 2.5 nm/ns in the z-direction, and for the second approach the pulling speed of the top and bottom layers was also about 2.5 nm/ns. The experimental deformation speed4 is in the order of 0.1 mm/s, which is about 4 orders of magnitude slower than the speed used in our simulations. The deviation of the mean-squared displacement (MSD) from affine deformation per layer was calculated using the following equation:

S=

1 z ∑ ∑ [(dik)2 − (rik0λk − rik0)2 ] Nz i = 1 k = x , y , z

where Nz is the number of particles in a layer of interest, dik is kcomponent of the actual displacement of particle i: dik = (rik − rik0) where rik is the k-component of the coordinate vector of particle i, and the subscript zero denotes the initial value. The global deformation is characterized by λk = Lk/Lk0, with k = x, y, or z and Lk and Lk0 being the dimensions of the simulation box along the k-axis at time t and initially, respectively.

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RESULTS AND DISCUSSION Case for a New Deformation Protocol. At 300 K, the mechanical behaviors of three systems containing only short polystyrene chains were investigated using the conventional uniaxial deformation protocol with atmospheric pressure controlling the stress in the lateral direction. This conventional deformation protocol resulted in ductile extension of the short chain PS system, showing no sign of failure/breaking up to a draw ratio of L/L0 = 1.5 as shown in Figure 1a for system I. The corresponding stress−strain relationship is shown in Figure 1c (upper curve). It is worth mentioning that previous MD simulations have also found low molecular weight polymer glasses to behave in a ductile manner during uniaxial extension.11 According to experiment, low molecular weight PS cannot undergo yielding well below Tg. Such PS has little mechanical strength and breaks up well before 5% extension. Thus, the apparent ductility in the present and past MD simulations reveals a serious and perhaps inherent shortcoming of the standard simulation approach. This protocol moves all chain segments uniformly at every time step according to the affine deformation scheme; i.e., all segments are made to displace in spite of the caging constraint causing the system to yield and draw significantly. To overcome the limitation of the conventional deformation protocol, we have explored a new simulation protocol C

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system I* is shown in Figure 3a, where short chains are not shown.

mimicking an actual uniaxial extension experiment. Specifically, in our simulation the periodicity of the simulation box in the zdirection was removed by cleaving chains crossing the periodic boundary at the top and bottom of the box. The initial height of the three short chain systems is about 29 ± 1 nm. The systems were then deformed by pulling two layers of each 1.5 nm in thickness at the top and bottom of the system in opposite directions. In this protocol the movements of segments occur via intersegmental attractions in response to the moving top and bottom layers. Figure 1b shows the state, for the chopped system I, at 20% extension where the sample already fractured, and Figure 1c shows the corresponding stress−strain relationship (lower curve). By comparison, the new, more realistic protocol produces a rather different result. All the three different systems show similar behavior under the two deformation protocols as shown in Figure 2, validating our new deformation protocol. To further understand why different outcomes are observed depending on the deformation protocol, we examine how polymer chains displace during deformation by calculating their mean-squared displacement (MSD). Figures 1d and 1e respectively show the average deviation of the MSD from the affine deformation (denoted by S) as a function of their z coordinates for the two different deformation protocols. For the case of the standard deformation protocol, Figure 1d confirms that the system uniformly follows the affine deformation since the deviation is fluctuating around zero. Whereas, through the new deformation protocol, the dynamics of PS chains are found to significantly deviate from uniform affine deformation (Figure 1e). The short chains that are close to the moving top and bottom layers follow the movement of these layers closely, while chains further away from these layers, such as those in the middle of the film, are left behind. This gradient in the motion of chains in the film eventually leads to a brittle fracture as shown in Figure 1b. This is further confirmed through the huge fluctuation observed in the film density during deformation as a function of distance from the layers (Figure 1g), whereas almost uniform density distribution is observed for the affine deformation case (Figure 1f). The same behavior is also observed for the other two systems (not shown). Thus, the unrealistic uniformity created by the conventional affine deformation protocol with a Poisson’s ratio of almost 0.5, as can be seen in Figure 1f, with little change in density during deformation, seems to be the reason for low molecular weight polystyrene to exhibit the unexpected ductile behavior. Chain Network and Emergence of Activation Zone. Having shown through our new deformation protocol that low molecular weight polystyrene glass is prone to brittle failure, we can study the effect of chain network on polymer ductility. To mimic chain network, two long PS chains were introduced into the previous systems of short PS chains (will correspondingly refer them herafter as system I*, system II*, and system III*). Specifically, two long PS chains of each 120 repeating units in a hairpin shape were hooked to form an “entanglement”, mimicking a network junction, and were immersed into a matrix of 250 short PS chains of each 20 repeating units. Each of the mixtures was initially equilibrated at 500 K and at atmospheric pressure for 10 ns to make sure all chains were relaxed. The temperature of each system was then lowered down to 300 K with a cooling rate of 20 K/2 ns at atmospheric pressure, in order to obtain a glassy state. A representative final configuration of the entangled two long chains at 300 K for

Figure 3. Uniaxial extension of two entangled long chains surrounded by short PS chains (system I*). (a) Snapshot of the two entangled long chains after quenched into glassy state; the short chains are not shown for clarity. (b) The two long chains after pulled taut; the pulling speed was 2.5 nm/ns. The aromatic rings are colored blue, while the backbones of the two long chains are colored in red and green, respectively. (c) The change of conformations for one long chain and representative two short chains before and after pulling the chain ends of the long chains for 5 ns. Short chain 1 is about 3 nm away from the closest long chain, while short chain 2 is very close to the long chains. Here the backbones of short chains and long chains are colored in green and red-yellow, respectively. (d) The average mean-squared displacement of all the short chain atoms with respect to their distance to the nearest long chain. The same calculation has been performed at different time during pulling the chain ends of the two long chains. The results indicate two types of dynamics of short chains, activated and nonactivated, depending on the distance between the short chains and the long chains.

At 300 K, the four ends of the two long chains in the mixture were pulled in opposite directions while keeping the simulation box dimensions fixed. The pulling changes the conformations of the two long chains. Toward the end of the simulation when the chain ends were very close to the boundaries of the simulation box, the two long chains were nearly straightened as shown in Figure 3b. During the pulling phase, the short chains close to the long chains were also influenced by the deformation of the long chains. Nearby short chains (for example, short chain 2 in Figure 3c) moved considerably. The short chains that are far away from the two long chains (for example, short chain 1 in Figure 3c) were not directly affected by the stretching of the long chains. These observations resemble a phenomenological molecular model proposed recently.4 In this molecular model, the structure of a polymer glass material is envisioned in terms of D

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Figure 4. Uniaxial deformation of a polystyrene glass in the presence of a single chain network surrounded by short chains (system I*). (a) All the four chain ends of the two long chains were pulled with the top and bottom layers; (b) only one chain end on each long chain was pulled with the top and bottom layers. The backbones of the long chains are colored in yellow and red. The short chains are rendered transparent; (c) the stress− strain behaviors corresponding to the two cases and (d, e) mobility of short chains around the long chains characterized by MSD calculations as a function of the distance from the long chains (similar to Figure 2d), for (a) and (b), respectively. The MSD were calculated from a 40 Å thick layer in the middle of the simulation box, 290 Å long, where the entanglement point of the two long chains was located.

a chain network surrounded by a primary structure. The chain network are made of load-bearing strands, originating from the uncrossability of long polymer chains. The primary structure is composed of non-load-bearing strands. When a glassy polymer is under deformation, strands in the chain network deform. Their displacements create activation zones (AZs) in the surrounding primary structure (i.e., non-load-bearing strands).4 This model suggests that a polymer glass is brittle if the AZs are very small in size or low in density thus unable to spread across the macroscopic sample and overlap before the chain network fails through chain pullout. The present simulation permits us to characterize any emergence of activation zones surrounding load-bearing strands, i.e., the two long chains. Specifically, we detect the appearance of activation zones by analyzing the MSD of short chains during deformation at different distances from the two long chains as shown in Figure 3d. The activation zone defines the region where the atoms of the PS chains have moved at least more than their van der Waals radius, that is, beyond the simple vibrational motions observed in a glassy state. In the MSD figure shown in Figure 3d, the activated and nonactivated zones show different MSD slope. Short chains close to the long chains are more “activated” and have higher mobility than those chains that are far from the long chains. There are clearly two different regions, depending on the distance to the backbones of the long chains: an activated region closely surrounding the long chains and a nonactivated region in the rest of the system. Note that the activation zone around the two long chains increases with deformation as can be seen in Figure 3d and reveals the AZs to be on the length scale of 1−1.5 nm. Effect of Chain Network on Polymer Ductility. Having demonstrated the role of chain network to mobilize the vitreous surroundings, we wonder whether this mixture would gain ductility relative to the pure short-chain PS system. To answer this question, we deformed the systems that were created in the

preceding simulation where the two looped long PS chains were stretched relative to the glassy surrounding and the periodicity in the stretching direction was removed. Specifically, the top and bottom layers of the systems were displaced in opposite directions along with the ends of the long chains. The end pulling of the long chains mimics the effect of chain network deformation under external extension. There are several posibilities regarding chain end pulling and in the present investigation either all four ends or one chain end from each of the two long chains were pulled with the top and bottom layers. As shown in Figure 3b, the two loops were already in their taut states before extension. Thus, when all ends of the long chains were pulled in the Z-direction along with the top and bottoms layers, a “locking” behavior was observed, meaning the long chains get more and more straigtened with no observable movement leading to a high stress along the whole chain (see higher initial stress in Figure 4c). In contrast, when only one end of each chain was pulled, because of the remaining free chain ends the long chains are able to slide around the entanglement point (Figure 4b), and as a result, the two chains can move more freely. This huge variation in the mobility of the long chains in the two cases has an effect on the dynamics of the surrounding short chains. Figures 4d and 4e show the difference in the mobility of the short PS chains around the two long entangled chains for the two different cases. When all the chain ends of the long chains in the taut state are pulled, they activate the surrounding area slightly (Figure 4d), but their locking position prohibits the spreading of the activation, and as a result failure occurs near the middle of the sample. Whereas, when the long chains move more freely as only one chain end of each long chain was pulled, they activate the surrounding short chains much more effectively than the other case (Figure 4e). Consequently, the system is much more E

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Macromolecules ductile as indicated by the absence of failure until L/L0 = 1.25 in Figure 4b. Our observation is that when the two long chains are sufficiently taut, they mimic the condition of being load-bearing and their affine-like displacement can effectively activate the vitreous surroundings and ensure ductility, as shown in Figures 4b and 4e, and result in ductile behavior of the glass system, whereas the scenario of holding all four ends during pulling mimics the condition of dense cross-linking, which does not improve ductility effectively. Figure 5 shows the MSD of short

pulling all ends of the two long chains in the system is also observed in the other two systems. The slight variation observed in the stress−strain behavior of the three different systems for a given deformation approach is due to a slight difference in the initial conformation of the two long chains in their taut state before extension. The other two systems also display a similar dynamic behavior of the short chains around the long chains. Lastly, we examine the case where the long chains were initially in a fairly coiled state, not mimicking the condition of being load-bearing. Along with the displacement of the top and bottom layers, one end of each long chain was also pulled. In spite of the existence of the entangled long chains, the extension resulted in premature failure at L/L0 = 1.15 (Figure 7a) similar to the purely short chain system case shown in Figure 1. In order to understand why coiled strands are ineffective to activate the glassy state, we evaluated the dynamics of the short chains surrounding the long chains. Figure 7b shows the state of activation of short chains in a 20 Å thick layer in the middle of the system. No short chain was activated, as the long chains remain coiled. However, when the same characterization was performed in a 20 Å thick layer close to the boundary of the box, where the long chains are pulled taut, i.e., undergoing significant displacement, there is considerable movement of short chains surrounding the displacing long chains, as shown in Figure 7c. Figure 7d summarizes the comparison between Figures 7b and 7c and also the remaining two layers at L/L0 = 1.07.

Figure 5. Activation behavior of the system shown in Figure 4b. The mobility of short chains around the long chains characterized by MSD calculations as a function of the distance from the long chains. The MSD is calculated in different layers located at different positions in the box, from the top boundary (0 to 20 Å) to the middle (120 to 140 Å), at 0.8 ns (around 5% of strain, before breaking).

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CONCLUSIONS

In summary, we have applied a more realistic deformation protocol to simulate extension of polystyrene glasses based on all-atom molecular dynamics simulation. We showed that as a reference, low molecular weight PS, i.e., a system made of short PS chains, is brittle, consistent with experimental facts. We then examined several cases where the effect of long PS chains on the overall polymer ductility was demonstrated. For example,

chains at different layers of the film when one chain end on each long chain was pulled. It clearly shows that the short chains close to the entanglement point are more mobile than the rest part of the film. Figure 6 demonstrates that the huge difference observed in the mechanical properties of the system by pulling one end per chain of the two long chains vs by

Figure 6. Demonstrating the reproducibility of the mechanical behavior observed in Figure 4 when pulling one end vs two ends of a chain of the two long chains in three different systems: triangles (system I), circles (system II), and squares (system III). Open and closed symbols of the same type of symbol represent the stress−strain behavior of a system during pulling one end of a long chain and during pulling two ends of a long chain, respectively. The green × symbols represent the failure strain values. F

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported, in part, by the National Science Foundation (DMR-1444859 and DMR-1609977). ABBREVIATIONS LBS, load-bearing strand; AZ, activation zone; MSD, meansquared displacement; MD, molecular dynamics; VDW, van der Waals; LAMMPS, Large-scale Atomic/Molecular Massively Parallel Simulator; PPPM, particle−particle/particle-mesh.

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Figure 7. Effect of the presenence of coiled long PS chains (system I*). (a) Before and after 25% deformation snapshots of a polymer glass system containing two long entangled but coiled PS chains and 85 short PS chains surrounding them. The height of the system is about 95 Å. (b, c, d) Mobility of short chains around the long chains characterized by MSD calculations as a function of the distance from the long chains after subtracting the affine deformation background (S). (b) The MSD of short chains (S) in a 20 Å layer in the middle of the box calculated at different times. (c) Similar measurement made in a 20 Å thick layer close to the top boundary of the box. (d) Comparison of MSD from four different layers, starting from the top layer and going to the middle of the box at 0.4 ns (around 7% of strain, before breaking).

we showed that the displacement of long chains relative to an immobile glassy primary structure (made of short chains) can produce activation zones of high molecular mobility in their proximity. We identified two cases where the presence of long chains did not produce sufficient activation of the glassy primary structure made of short chains. First, the mixture of long and short chains is still prone to premature failure when long chains are in the taut state and unable to make sufficient displacement as shown in Figure 4. Second, the long chains in a highly coiled state are ineffective to mobilize their surroundings because the vitreous matrix resists the transmission of the chain-end pulling into the interior of the coil, as shown in Figure 7. In our simulations, an effective network that is required to ensure ductility is mimicked by (a) introducing a pair of hooked and already extended long chains into a matrix of short chains and (b) pulling only one chain end of each long chain to cause significant movement of the long chains through slippage at the network junction. After detailed analyses we have reached the conclusion that uniform activation by an adequate chain network appears to be necessary for the system to avoid structural failure. Future studies should investigate and explain why significant ductility can be reached after uniaxial melt extension29 and how mechanical rejuvenation makes it easier for yielding to take place.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (M.T.). *E-mail: [email protected] (S.Q.W.). ORCID

Xiaoxiao Li: 0000-0003-0495-3602 Shi-Qing Wang: 0000-0002-0572-7108 Mesfin Tsige: 0000-0002-7540-2050 G

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