Nematic Coupling in Polybutadiene from MD Simulations

Jan 3, 2019 - Classic experiments show that polybutadiene oligomers align in a network of stretched chains. Furthermore, the oligomers orient almost a...
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Nematic Coupling in Polybutadiene from MD Simulations Shreya Shetty,† Enrique D. Gomez,†,‡,§ and Scott T. Milner*,†,‡ Department of Chemical Engineering, ‡Department of Materials Science and Engineering, and §Materials Research Institute, The Pennsylvania State University, University Park, Pennsylvania 16802, United States

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ABSTRACT: Classic experiments show that polybutadiene oligomers align in a network of stretched chains. Furthermore, the oligomers orient almost as strongly as the network, which suggests a large nematic coupling, despite polybutadiene being a flexible polymer. Here, we combine self-consistent field theory (SCFT) and atomistic molecular dynamics (MD) simulations of polymer chains under tension to obtain the nematic coupling constant α in polybutadiene. Using α, we compute the ratio of orientation of free chains and stretched chains of polybutadiene in a melt of stretched chains. We show that nematic coupling in polybutadiene, though not quite enough to induce a nematic phase, is surprisingly strong. When extrapolated to the experimental temperature, we find an orientation ratio of 0.8, consistent with the experimental value of 0.9.

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molecules.1−5 Thus, more elaborate models of rubber elasticity often include orientational coupling.4,16,17 Measurements of chain alignment in polybutadiene melts indicate that nematic interactions are important in flexible polymers.2 The orientation of probe molecules within a strained host has been examined using SANS, NMR, and birefringence.1−5 Ylitalo and co-workers studied the orientational order of deuterated polybutadiene oligomers within a strained polybutadiene melt by simultaneously measuring birefringence and dichroism.2 They determined the ratio of probe to host alignment, ϵ:

INTRODUCTION

Oriented polymer systems often have enhanced properties in comparison to their isotropic counterparts.1−5 For thermoplastics like polyethylene and polypropylene, stretching of films and fibers leads to aligned chains. These impart a high modulus in the direction of stretching, leading to applications such as packaging, protection equipment, and high strength fibers.6−8 Polymers with stiff backbones promote local alignment, leading to liquid crystalline phases, which can align under applied fields. Polymers quenched from nematic phases exhibit larger crystalline domains within an amorphous matrix, resulting in higher charge mobilities, increased moduli, and useful optical properties.9−14 Stiffer chains have a stronger tendency to form liquid crystalline phases. As the concentration of stiff chains or rods increases, the reduction in excluded volume overcomes the entropic penalty for aligning chains in a common direction. As first explained by Onsager, the nematic phase occurs for rodlike molecules in solution when random placement of rods results in about one collision per rod, at a critical volume fraction of order d/L (where d is the rod diameter and L is the rod length).15 In this way, excluded volume interactions suffice to couple the orientation of neighboring segments. Anisotropic enthalpic interactions from hydrogen bonds or dipoles, if present, can also contribute to the orientational coupling. Nematic couplings in flexible polymers are often not strong enough to induce a nematic phase. Hence, these interactions are ignored in simple models, such as classical rubber theory, which is based on the assumption of phantom Gaussian chains in an affinely deformed network.4 Simple models successfully predict the stress−strain relationship for small strains and highly swollen networks. But many experiments on flexible polymers indicate that these models are inaccurate for high extension ratios and gels swollen with long slender solvent © XXXX American Chemical Society

ϵ=

qprobe qmelt

(1)

where qprobe and qmelt are orientational order parameters of probe and melt, respectively. Without nematic interactions, the probe molecules would not be influenced by the alignment of strained host chains and would remain isotropic even in a stretched melt, so that ϵ = 0. Instead, Ylitalo et al. found that ϵ was ∼0.9 when the probe molecules were short, indicating a high degree of correlation between the orientation of probe and host molecules, arising from nematic interactions. The nematic coupling α quantifies the interactions that couple the orientation of neighboring polymer segments. In this paper, we use a recently developed approach in our group, which combines self-consistent field theory (SCFT) and atomistic molecular dynamics (MD) simulations of chains under an applied, uniaxial tension, to determine the nematic coupling α for polybutadiene.18 Briefly, we compare the alignment observed in simulations with the alignment Received: October 15, 2018 Revised: December 10, 2018

A

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Macromolecules computed in SCFT at fixed force and different α values to determine the nematic coupling. Using the computed α, we determine the orientational order parameters of probe and stretched host molecules again using SCFT, now for free chains within a melt under tension. We also directly simulate a melt under tension with free probe molecules to validate our results. We compute ϵ to be 0.8 at the experimental temperature (300 K), in good agreement with the experimental value of 0.9. This supports the finding that surprisingly strong nematic interactions in polybutadiene affect chain ordering whenever external fields promote segment orientation.

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METHODS AND RESULTS

It is easier to stretch and orient chains in a melt when nematic interactions are present. When chains are put under tension, backbone segments orient in the direction of applied force. In the presence of nematic coupling α, the orientation of a segment couples to the alignment field created by the orientation of surrounding segments. This amplifies the tendency of segments to orient in the direction of pulling and makes the chains easier to stretch. We combine SCFT and atomistic MD simulations of chains under tension to determine the nematic coupling α in polybutadiene using a method developed by Zhang et al.18 First, we perform atomistic MD simulations of an isotropic polybutadiene melt with an equal and opposite force applied at chain ends. The applied force induces a finite alignment in the simulated chains, quantified by the orientational order parameter q. Next, we compute analytically the order parameter at fixed force and different α values using selfconsistent field theory (SCFT) to describe a chain under an external tension within a melt of chains under the same tension. Comparing q from MD simulations to q(α) from SCFT, we determine α. With the calculated α and SCFT, now describing the alignment of free chains within an oriented melt, we obtain qprobe and qmelt, the order parameter of probe and host chains. We validate our results for qprobe with explicit simulations of free polybutadiene chains in a stretched melt, imitating the experiments. The ratio of qprobe to qmelt, denoted ϵ, is computed over a range of elevated temperatures (eq 1) and extrapolated to the experimental temperature (300 K). We simulate at high temperatures and extrapolate to take advantage of the faster equilibration well above 300 K. We find ϵ equals 0.8 at experimental temperature, in good agreement with the experimental value of 0.9, indicating a high correlation between the orientation of probe and host molecules. Atomistic MD Simulations of Polybutadiene. We start with building equilibrated isotropic polybutadiene melts as initial configurations for the simulations of chains under tension. Our polybutadiene melt contains 50 chains of cispolybutadiene with 20 monomers each. For ease of construction, these are first arranged as a regular array of parallel straight chains. The OPLS-AA potential was used for all bonded and nonbonded interactions.19 Then we perform NPT simulations at different temperatures (413, 513, 613, and 713 K) to obtain equilibrated melts (Figure 1b). The orientational order parameter q quantifies the average orientation of backbone tangents with respect to a specified direction and is given by

Figure 1. Snapshot of the simulation box at 413 K. Free polybutadiene: (a) chains folded into the box and (b) whole chains. Polybutadiene chains under tension: (c) chains folded into box and (d) whole chains. 1

q=

∫−1 dμ Π(μ)P2(μ)

(2)

Here μ = cos(θ), where θ is the deflection angle between a backbone tangent t (the direction of given backbone bond) and reference direction. Π(μ) is the probability distribution of backbone tangents, and P2(x) is the second-order Legendre polynomial. The order parameter for an equilibrated melt fluctuates about zero as shown in Figure 2a. This corresponds to an isotropic phase, in which chain segments are oriented randomly in all directions. Further we measure the autocorrelation time τ for order parameter fluctuations to estimate how long the system takes to equilibrate. Because τ measures the time required to obtain an uncorrelated value of q from the simulation, knowing τ allows us to estimate the error bar for the time-averaged orientation. The correlation times for the order parameter in our polybutadiene system ranged from 0.1 ns at 713 K to 2.5 ns at 413 K. For this reason, simulations were performed at elevated temperatures to accelerate the melt dynamics, minimize the autocorrelation time, and obtain good statistics with shorter simulation runs. We simulate until the total runtime is about 100 τ. Using trajectories from NPT simulations, we also determined the persistence length of polybutadiene. The persistence length is the characteristic chain length for exponential decay of tangent−tangent correlation function ⟨t0·ts⟩ equal to exp(−s/Np). A temperature-independent value of Np = 1.25 monomer units was obtained for polybutadiene from an exponential fit to ⟨t0·ts⟩ over the first six monomer units (Figure 3a). Our value for Np agrees well with the experimental value of 0.54 nm (1.15 monomer units).20 We expect the persistence length to be temperature independent as the backbone deflection angles are insensitive to temperature.21 We use the persistence length to determine the stiffness κ = kTNp of polybutadiene, treated as a semiflexible chain. κ is a key parameter in our SCFT. B

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a weak nematic order in the simulated melt. The force we apply is modest, corresponding to 2 kT per mean end-to-end distance. The end-to-end chain distance was measured in simulations of isotropic chains without tension. The backbone tangents tend to align in the direction of applied force (the zdirection), and an average alignment results (Figure 1d). The corresponding order parameter fluctuates about a nonzero value (Figure 2b). SCFT of a Chain under Tension. To determine the nematic coupling α, we construct a self-consistent field theory for a chain under a uniaxial tension, surrounded by chains under the same tension. For chains under tension f, the backbone segments tend to align in the direction of applied force, resulting in a quadrupolar alignment field Q = ⟨titj⟩ − δij/ 3. When the force acts along the z-direction, Q is a symmetric, traceless tensor with Qxx = −q/2, Qyy = −q/2, and Qzz = q, with q given by eq 2 The nematic coupling α couples a given backbone tangent to the mean quadrupolar field Q. Hence, the total potential acting on a backbone tangent t of size a is given by V (t) = −α t·Q·t − a f·t

(3)

The uniaxial symmetry of the nematic ordering allows us to express V in terms of μ = cos(θ) and Legendre polynomials Pn(x) as

Figure 2. Order parameter of polybutadiene melt in MD simulations: (a) free chain simulations and (b) chains under tension equivalent to 2 kT per end-to-end chain distance; T = 413 K (blue circles) and T = 713 K (red triangles).

V (μ) = −αqP2(μ) − afP1(μ)

(4)

The Hamiltonian for a chain of length S and stiffness κ under the aligning potential V can be written H0 =

∫0

S

ij κ dt s dsjjjj 2 d s k

2

yz + V (ts)zzzz {

(5)

Here the first term of the integrand corresponds to bending energy of a worm-like chain, s is the monomer index, and ts is the unit tangent vector of monomer s. The tangent distribution Π(μ) can be written in terms of the propagator Z(μ,S) as S

Π(μ) =

∫0 dn Z(μ , n)Z(μ , S − n) 1

∫−1 dμ Z(μ , S)

(6)

Here Z(μ,S) is defined as Z(μ , S) = AS

∫ dμ1dμ2 ...dμS e−βH δ(μ1 − μ) 0

(7)

Physically, the propagator Z gives the statistical weight for a chain of S + 1 monomers (hence S tangents) to have tangent orientation μ at one end. Π(μ) is proportional to the product of two propagators integrated over the chain length. The product Z(μ,n)Z(μ,S−n) gives the statistical weight for a chain where the nth monomer has orientation μ. The front half of the chain must arrive from S = 0 with orientation μ to the nth position, and the back half must arrive from S = N to the nth position again with orientation μ. The factor A normalizes the propagator with respect to isotropic chains. The propagator Z is obtained using a procedure described in Zhang et al., where it is expressed as an expansion of eigenfunctions of the angular diffusion equation, describing the loss of orientational memory along the semiflexible chain.18 Within SCFT, we determine the order parameter at fixed α and f self-consistently. When q input into single chain

Figure 3. Tangent−tangent correlation function of polybutadiene chains for (a) free chains at different temperatures and (b) for chains under different tensions at T = 513 K.

Using the isotropic melt as initial configurations, we apply an equal and opposite force to the chain ends. We thereby induce C

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The relation between the force f and the chain end-to-end length x in the direction of force is given by f = −κfx, where κf is the spring constant. As the chain length in the direction of force increases beyond a certain value, the force required to further stretch the chain increases nonlinearly, as chains cannot stretch indefinitely. This nonlinear increase in force is equivalent to an increase in the spring constant. It is easier to extend a less flexible chain. But the spring constant for extending a worm-like chain is κf = kT/Np. This is because the natural fluctuations in the mean square end-to-end distance are larger for a stiff chain than a flexible chain of the same arc length. Therefore, interpreting a slightly larger κf for a chain under tension in terms of Np, we find Np is slightly smaller. We use the persistence length corresponding to chains under tension in SCFT to determine the tangent distribution. At 513 K, the persistence length from tangent−tangent correlations decreases from 1.25 monomers for chains without tension to 1.11 monomers for chains under tension of 2 kT per end-toend chain distance (Figure 3b). SCFT results for Np = 1.1 agree well with the tangent distribution from simulations (Figure 5). Hence, we compare q from MD simulations and SCFT for Np = 1.1 to obtain α at different temperatures (Figure 6).

Hamiltonian H0 (eq 5) and q calculated using eq 2 agree, the self-consistency is satisfied. For polybutadiene, with force of order 2 kT per end-to-end chain distance applied at each temperature, the function q(α) reduces to a single curve at all temperatures (Figure 4). Comparing q(α) with q observed in MD simulations, we determine α.

Figure 4. Orientational order parameter versus nematic coupling constant α from SCFT.

The chain tangent distribution Π(μ) provides a more detailed description of tangent orientations in the melt than the order parameter q, which is just the average of P2(μ) over Π(μ). We validate our value of α by comparing the tangent distribution Π(μ) from the simulations with the SCFT predictions. We find that the tangent distributions from MD simulations are better represented by a slightly lower persistence length of Np = 1.1 than the value Np = 1.25, found by simulating without tension (Figure 5). To explore this, we performed simulations

Figure 6. Nematic coupling constant α at different temperatures obtained from simulations (blue circles) and critical nematic coupling constant αc calculated from SCFT for polybutadiene chains of stiffness βκ = 1.1 monomer units: 20 monomers long (red circles) and 1000 monomers long (black squares).

Determining ϵ. To compare with the experimental results of Ylitalo et al., we compute the orientation ratio ϵ (eq 1). We use the same α in the field theory for a free chain in a melt of stretched chains to compute the order parameter of free chains. In a melt of chains under tension with nematic coupling, the quadrupolar field Q affects the bonds of a free chain. The aligning field acting on a tangent in a free chain with orientation μ is

Figure 5. Tangent distribution function for polybutadiene chains under a tension equal to 2kT per end-to-end chain distance at T = 413 K predicted from simulations (red open circles) and from SCFT using Np = 1.1 monomers (black dashed line) and Np = 1.25 monomers (blue dashed line) simulations (red open circles).

for chains under tensions of 0.81 kT and 2 kT per end-to-end chain distance and determined the persistence length in each case (Figure 3a). The tangent−tangent correlation function decays slightly faster as the applied force increases, as if the persistence length were slightly lower for chains under tension (Figure 3b). Lower persistence lengths for chains under higher tension can be understood by considering how the spring constant of a chain might vary with tension. The spring associated with chain extension is anharmonic. When a force is applied at chain ends, the chain conformations extend in the direction of force.

V (μ) = −αqP2(μ)

(8)

The order parameter q in eq 8 corresponds to the melt of stretched chains. We compute the tangent distribution and mean order parameter for free chains qprobe (eq 6), while using V of eq 8 in the single chain Hamiltonian. To check our value of qprobe for free chains computed using SCFT, we also perform MD simulations of dilute free chains in a melt of chains under tension, imitating experiments. We simulate two such systems: (a) one free chain and 49 chains under tension and (b) 5 free chains and 45 stretched chains to D

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Locating TIN. Nematic coupling α and chain stiffness κ govern TIN, the isotropic-to-nematic transition temperature. Higher TIN values are observed for chains with stiffer backbones and stronger nematic coupling. Polybutadiene does not exhibit a nematic phase. Nevertheless, we compute TIN for polybutadiene from theory to find out how much larger α would need to be for nematic phase to appear above the glass transition Tg. We again use our single chain SCFT which provides a meanfield treatment of a polymer melt that may undergo an isotropic-to-nematic transition. We compute the system free energy F(q), which undergoes a first-order phase transition at a critical αc, where the nematic free energy F(qc) equals the isotropic free energy F(0). We find TIN by locating αc from the first-order transition and mapping it to α(T) determined from simulations of chains under external tension. The mean-field free energy, which corresponds to a variational upper bound, is given by

improve statistics while remaining dilute. Both free chains and chains in the melt are 20 monomers long. These simulations provide qprobe for a given qmelt and compared with the value of qprobe obtained from SCFT. The stretched chains have the same order parameter as for a system with all chains under tension. The free chains have a nonzero order parameter induced by the surrounding chains (Figure 7), consistent with SCFT predictions. The large error bars on q of free chains measured in simulations is due to limited statistics.

F(q) = ⟨H ⟩ − TS0

(9)

Here ⟨H⟩ is the multichain Hamiltonian and S0 is the model ensemble entropy. S0 can be calculated from the single chain free energy, F0 = ⟨H0⟩ − TS0. F0 is the single chain free energy, determined using the single chain model partition function Z0 = ∫ 1−1dμ Z(μ,S). To calculate S0(q), we consider a single free chain in an external quadrupolar field A. The aligning potential V on the chain is given by V(μ) = −AP2(μ). The order parameter q established in the system can be numerically determined to be a function of A using single chain Hamiltonian H0. For a given A, we can compute both q and F0. Hence, F0 can be expressed as a function of q. The model ensemble entropy S0 can be calculated from F0(q) as

Figure 7. Order parameter of free chains and pulled chains in polybutadiene. Open symbols: q obtained from simulations involving one free chain. Closed symbols: q obtained from simulations involving five free chains.

To compare with experimental results, we compute ϵ and extrapolate to lower temperatures. We fit a linear function to ϵ versus temperature and extrapolate to obtain ϵ = 0.8 at 300 K (Figure 8). This is in reasonable agreement with the experimental value of around 0.9 for small molecular weight probe chains (M < Me).2

−TS0(q) = F0(q) − F0(0) + Aq

(10)

The model ensemble entropy is calculated with isotropic system as the reference. The mean-field free energy F(q) with respect to isotropic system reduces to 1 F(q) = − αq2 − TS0(q) 2

Figure 8. Orientational coupling constant ϵ versus temperature T.

(11)

The tangents with mean orientational order q interact with each other through a nematic coupling α. The factor 1/2 prevents double counting of the interactions between tangents. On determination of F(q) at different α values, the critical αc corresponding to a first-order transition from isotropic-tonematic phase can be determined when a second minimum of F(q) equal to zero is observed (Figure 9). We determine the critical nematic coupling constant αc for polybutadiene using the procedure described above. Using α(T) from simulations, we determine the temperature at which the critical αc can be attained by extrapolation (Figure 6). The critical nematic coupling constant αc is reached at extremely low temperatures. Thus, this transition is well embedded in the glassy state (Tg = 170 K), precluding us from ever observing a nematic state.22 Within SCFT, we can readily consider longer chains; we find that increasing the chain length is not enough to shift the IN transition out of the glassy state.

Ylitalo et al. observed that the orientational correlations dropped when the length of probe molecule exceeded the entanglement length.2 In experiments, for probe molecules with molecular weight M < Me, ϵ was constant around 0.9. But ϵ dropped to 0.4 for M > M*(M* ∼ 5Me). On theoretical grounds, this is hard to explain. Nematic coupling is an equilibrium phenomenon, and entanglement is relevant to dynamics, not equilibrium. In principle, ϵ could vary with M for reasons unrelated to entanglement; simply the chain ends are different from the middle. In any case, our study was undertaken for polybutadiene chains with M < Me. We observe high correlations between the orientation of probe and host molecules, consistent with experiments. For M > M* increasingly long simulation runs would be required to equilibrate and good statistics. The entanglement slows polymer dynamics, with correlation time proportional to M3, which restricts our study to short chains. E

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observable nematic phase is not predicted for polybutadiene, despite the sizable coupling. Our results support the existence of significant orientational correlations in polybutadiene, despite it being a flexible polymer.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Enrique D. Gomez: 0000-0001-8942-4480 Scott T. Milner: 0000-0002-9774-3307 Figure 9. Variational free energy versus orientational order parameter q for polybutadiene 20-mers with nematic coupling constant: α = 8.69 kT (blue), α = 8.697 kT (green), and α = 8.704 kT (red).

Notes

CONCLUSION In this paper, we investigate nematic interactions in polybutadiene by combining atomistic simulations and selfconsistent field theory for chains under tension. Ylitalo et al. probed nematic interactions in polybutadiene by measuring ϵ, the ratio of order parameters of “guest” molecules and “host” molecules in a melt under tension. Their results indicate a strong correlation between probe and host molecule orientations for molecular weight M < Me, with ϵ about 0.9. But the correlations drop, with ϵ decreasing to 0.4 for probe molecules with M > M*. On theoretical grounds, entanglement is to be irrelevant to equilibrium properties like orientation coupling, so that ϵ should be the same for M < Me or M < M*. These considerations were the motivation for our study. We combined atomistic molecular dynamics simulations and self-consistent field theory (SCFT) of chains under tension to estimate the coupling constant α. The tangent distribution and the orientational order parameter q(α) were computed selfconsistently using SCFT for a chain under an external tension and surrounded by a sea of chains under the same tension, with inputs of persistence length and monomer size of polybutadiene. Correspondingly we induced a weak nematic order in polybutadiene melt in MD simulations and measured the resulting order parameter q. By comparing q from MD simulations with q(α) from SCFT, we determined the nematic coupling constant α. We found that the apparent persistence length of the chains decreases slightly by increasing applied tension. Applied tension increases the spring constant for an anharmonic spring such as a finitely extensible chain. This manifests as reduced persistence length of chains in the simulations. With a slightly reduced persistence length, SCFT predicts the tangent distribution Π(μ) for chains under tension, which agrees well with simulation results. We compute the orientational ratio ϵ and extrapolate to lower temperatures to compare with experimental results. Using our value of α, we compute with SCFT the order parameter qprobe for probe molecules in a stretched melt. We also measured qprobe in simulations of free probe molecules in a melt under tension, in good agreement with predictions. At 300 K, our extrapolated value of ϵ = 0.8 agrees well with the experimental value of ϵ = 0.9. Finally, the isotropic-to-nematic transition temperature, TIN, was determined from the mean-field free energy. The critical nematic coupling constant αc was calculated at the first-order transition and mapped onto α(T) to determine TIN, which lies well below the glass transition for polybutadiene. Hence, an

ACKNOWLEDGMENTS Financial support from the National Science Foundation under Award DMREF-1629006 is acknowledged.

The authors declare no competing financial interest.

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REFERENCES

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