Solvent Effect on the Diffusion of Unentangled Linear Polymer Melts

Article pubs.acs.org/Langmuir

Solvent Effect on the Diffusion of Unentangled Linear Polymer Melts Binghui Deng, Liping Huang, and Yunfeng Shi* Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180, United States ABSTRACT: We conducted molecular dynamics (MD) simulations to study how solvent chains affect the diffusion of linear polymers in the unentangled regime. For monodisperse solvent chains, the self-diffusivity of a tagged chain scales with its chain length. The solvent chain length affects both the prefactor and the exponent, the latter of which ranges from −0.79 to −0.85. The scaling exponent here deviates from −1 as predicted by the Rouse model, which may suggest that the friction coefficient increases with the solvent chain length. In addition, we carried out diffusion simulations on two polydisperse melts, one with the Flory−Schulz distribution and the other with the Gaussian distribution. The measured diffusivity as a function of the tagged chain length agrees with a simple proposed model accounting for the heterogeneous medium. friction coefficient ξ. However, the friction coefficient takes exactly the same value for all of the segments regardless of their positions in the chain. An explicit illustration of the chain ends effect is the depression of the glass-transition temperature for polymer melts composed of short chains because chain ends experience a larger amount of free volume than central monomers.28 Only a few experimental studies have been carried out to investigate how solvent chains affect the self-diffusion of tagged chains. Smith et al.29,30 studied the diffusion of dyed and photobleached poly(propylene oxide) chains in the poly(propylene oxide) melts with molecular weight both below and above (up to 4 times) the critical molecular weight Me for entanglement. The diffusivity was found to scale with the solvent molecular weight (Ms) as D ∼ Ms−(0.7−0.93) in the unentangled regime and D ∼ Ms−(0.25−0.93) in the entangled regime, depending on the molecular weight of tagged chains. Green et al.31 studied the solvent effect on diffusion of long deuterated polystyrene chains in undeuterated polystyrene melts ranging from very short unentangled chains to long highly entangled chains by forward-recoil spectrometry. In their study, the diffusivity of tagged chains is independent of sufficiently long solvent length in the highly entangled regime (Ns ≫ Ne), strongly depends on the solvent chain length (D ∼ Ns−3) in the intermediate range (Ns > Ne), and is inversely proportional to the much shorter solvent chain length in the unentangled regime. Such great discrepancies might be caused by different measurement approaches and polymer systems.

1. INTRODUCTION The dynamics of polymer melts have been extensively studied over the last several decades because of their unusual viscoelastic behaviors characterized by the wide spectrum of relaxation time.1−3 Unlike the common Newtonian liquids,4 the movement of polymer chains is subject to extremely complicated topological constraints5 and excluded volume interactions.6,7 The scaling of relaxation time strongly depends on the chain length N in a manner that can be qualitatively characterized by Rouse8 and reptation9 models applicable to chain length below and above the entanglement length Ne, respectively. Ideally, the diffusivity D scales with chain length N as D ∼ N−1 in the unentangled regime and D ∼ N−2 in the highly entangled regime. However, deviations from these two scaling laws have been widely reported in both experimental work10−14 and simulation studies15−19 over the last few decades. Of particular interest here is the consistent observations of subdiffusive behavior of tagged polymer chains with exponent α varying from −0.75 to −0.85 in the unentangled regime.13,16,19−25 Such a deviation is generally attributed to effects from chain stiffness, local packing, and local friction variation.26 The effect of the solvent chains on polymer diffusion is of great significance in understanding the reported deviation from the Rouse model in many studies. One intuitive justification for the solvent effect is the considerable difference in the overall population of chain ends for solvents with different chain lengths. It is obvious that chain ends have different intermolecular environment from other segments along the chain. For instance, chain ends have on average more first nearest intermolecular neighbors as compared to those in the center of the chain.27 On the contrary, the Rouse model assumes that all of the intermolecular interactions are reduced to a stochastic and frictional force, which is characterized by the © XXXX American Chemical Society

Special Issue: Tribute to Keith Gubbins, Pioneer in the Theory of Liquids Received: August 16, 2017 Revised: September 19, 2017 Published: September 20, 2017 A

DOI: 10.1021/acs.langmuir.7b02901 Langmuir XXXX, XXX, XXX−XXX

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Langmuir Thus, a comprehensive understanding of the solvent chains effect, even for polymer melts with tagged and solvent chains both in the unentangled regime, is still lacking. Existing studies have exclusively focused on the monodisperse (very low polydispersity index in experimental studies) solvent polymer melts. However, polydispersity is ubiquitous and affects the kinetic and thermodynamic properties of polymer melts and their processing.32−34 Doi et al.35 demonstrated that the characteristic time scale for relaxation in polydisperse melts would follow simple mixing rules using the proper average molecular weight for narrow molecular weight distributions. However, such mixing rules fail to work for broader distributions. Picu et al.36 used a coarse-grained model to study the diffusion in entangled bidisperse polymer melts. They found that the presence of long chains slowed down the dynamics of the short ones but observed the opposite effect on the dynamics of long chains and further attributed these effects to constraint release. Theodorou et al.37 carried out MD simulations to investigate the dynamic properties of a few linear polydisperse polyethylene melts with different mean chain lengths but the same dispersity (D̵ ) of 1.09. The results showed that the friction coefficient ξ involved in the Rouse model depended on the chain length up to a certain threshold (system-dependent) above which could be considered a constant. These studies aimed at different aspects of polymer melt dynamics using well-designed simple chain length distributions and provided valuable insights. However, it is yet unclear how to understand the diffusion of tagged chains in a heterogeneous diffusion medium such as an arbitrary polydisperse polymeric solvent. Here we carried out extensive molecular dynamics (MD) simulations to study the effect of monodisperse and polydisperse solvent chains on the diffusion of tagged chains in the unentangled regime using a coarse-grained model developed in our previous work.38 The results indicate that the diffusivity of tagged chains can be written as a power law form as a function of its chain length for monodisperse solvent chain systems. Both the prefactor and the scaling exponents are dependent on the solvent chain length. In addition, chainlength-resolved diffusivity in two distinct polydisperse polymer melts was calculated, which agrees with a simple harmonic mean of diffusivities measured in monodisperse solvents.

smoothed LJ form, ϕLJsmooth(r ) by the inclusion of a repulsive bump term b(r) outside of the potential well. ϕmLJ(r) is formulated as b ⎧ ϕ smooth(r ), r < rαβ ⎪ LJ ⎪ b ϕmLJ(r ) = ⎨ ϕLJsmooth(r ) + b(r ), r ≥ rαβ ⎪ c ⎪ 0, r ≥ rαβ ⎩

(1)

is the species-dependent cutoff (α, β denotes species of A c or B). The bump term b(r) is applied only in the range of rαβ c rαβ

b and rαβ . To ensure that the potential energy and its first c , we use a smoothed LJ function derivative are continuous at rαβ

ϕLJsmooth(r ) as follows, c c c ϕLJsmooth(r ) = ϕLJ(r ) − ϕLJ(rαβ ) − (r − rαβ )ϕLJ′ (rαβ ) 6 ⎞ ⎛ σ 12 σαβ αβ ϕLJ(r ) = 4εαβ ⎜⎜ 12 − 6 ⎟⎟ r ⎠ ⎝r

(2)

εαβ and σαβ give the energy and length scales, respectively. The bump term b(r) is formulated as b(r ) = b εαβ

b εαβ εAB

⎛

c rαβ −r ⎞ ⎟ sin ⎜π c b ⎟ ⎝ rαβ − rαβ ⎠ 2⎜

(3)

is the magnitude of the repulsive bump. Because the bump

b c and rαβ , term itself and its first derivative are zero at both rαβ the overall potential is still smooth after the inclusion of b(r). Note that the bump term is exclusively applied to heterogeneous bonds (i.e., A−B bonds). Although this polymeric coarse-grained model is reactive (i.e., capable of describing polymerization and depolymerization reactions), the potential energy bump is sufficiently high that reactions (chain joining/breaking) are negligible in the following simulations described here. The chain ends are also constructed as the same type to avoid chain joining. The energy and length units are set up as εAB and σAB, respectively. All of the particles have the same mass of m0 and the time unit t0 = σAB m0/εAB . Additionally, the temperature unit is εAB/kB(kB is the Boltzmann constant). The potential parameters are listed in Table 1. We use the LAMMPS40

2. SIMULATION METHODOLOGY We utilized a coarse-grained polymer model developed and detailed in our previous work.38 The model is briefly described below for the sake of completeness. Note that we have carried out extensive studies to demonstrate the model from various aspects, including the static, kinetic, dynamic, and mechanical properties in ref 38. This coarse-grained model consists of bifunctional A and B particles (identical particle attributes except for the particle type) with pairwise interactions. The heterobonds (A−B) are attractive, whereas homobonds (A−A or B−B) are generally repulsive due to relatively large σAA and σBB values. The bifunctionality of each particle is achieved by adjusting the length and magnitude of the homobond repulsion. In this way, each particle can only potentially bond up to two opposite-type particles. Thus, a typical polymer chain is abstracted as −A−B−A−B−A−B−A−. More importantly, the homobonds have relatively weak bond strengths to simulate the weak interchain van der Waals interactions. The pairwise interaction is described by a modified Lennard-Jones (mLJ) potential ϕmLJ (r),38,39 which differs from the conventional

Table 1. Parameters of the mLJ Force Field for the CoarseGrained Polymer Model A−A B−B A−B

εαβ/εAB

σαβ/σAB

εbαβ/εAB

rsαβ/σAB

rcαβ/σAB

0.1 0.1 1.0

2.1 2.1 1.0

0 0 2.85

0 0 1.2

2.9 2.9 1.8

package with a customized force field to carry out all of the simulations. Newton’s equations of motion are numerically integrated using the velocity Verlet41 algorithm with a time step of 0.005t0. The temperature and pressure are controlled via a Nose-Hoover42,43 thermostat and barostat, respectively. OVITO44 visualization software is used to generate simulation snapshots. B

DOI: 10.1021/acs.langmuir.7b02901 Langmuir XXXX, XXX, XXX−XXX

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Langmuir We first constructed 16 different diffusion systems of a mixture of tagged chains in monodisperse solvent chains. The tagged chain lengths are 7, 15, 21, and 33. The solvent chain lengths are 11, 21, 33, and 55. Both tagged chains and solvent chains are within the entanglement chain length Ne (estimated to be around 50 in ref 38). The population of tagged chains and solvent chains of each of the 16 diffusion systems can be found in Table 2. Note that we used multiple tagged chains instead of

follows a Gaussian chain length distribution with a mean of 28 and a standard deviation of 3.4. The mean and standard deviation are chosen so that the molar fraction of the longest chain, 55, is identical for the two polydisperse systems. Snapshots of the polydisperse systems are shown in Figure 1c,d. The molar distributions of both systems are shown in Figure 1e, with the detailed population of each constituent listed in Table 3. Note that the tagged chains are essentially the constituents in these systems. Therefore, only 11 chain lengths ranging from 3 to 55 are included in the polymer melts to ensure that there are a sufficient number of chains of identical length in the system for diffusivity measurements. To be comparable to the existing literature and follow the convention in ref 38, all samples were equilibrated at a temperature of 0.15εAB/kB with a pressure of 0.03εAB/σAB3 for a duration of 25 000t0. The pressure was chosen such that the volume fraction of these samples is roughly 40%, very close to the well-known Kremer model.46 To measure the diffusivity (D), mean-square displacements (MSDs) were calculated for equilibrated samples using the NVT ensemble for a sufficiently long time to achieve the diffusion regime. The approach has been widely used to study the transport properties of many complex systems at the simulation time scale.47−49 Three independent samples for each system were simulated to obtain statistically reliable data.

Table 2. Population of Tagged Chains and Surrounding Solvent Chains for Four Monodisperse Systems with Chain Lengths of 11, 21, 33, and 55, Respectivelya N

Ns = 11

Ns = 21

Ns = 33

Ns = 55

7 15 21 33

300/2520 175/2520 125/2520 75/2520

300/1326 175/1326 125/1326 75/1326

300/756 150/756 100/756 75/756

300/504 150/504 125/504 75/504

a

The total numbers of particles constituting solvent chains are 27 720, 27 846, 24 948, and 27 720, respectively.

a single one for each system to get better statics in calculating the MSD. Figure 1a,b shows snapshots of two representative systems with N = 7, Ns = 55 and N = 33, Ns = 11, respectively. Next, we constructed two additional polydisperse samples to study chain diffusion in polydisperse melts. The longest chain lengths of both systems is set to be 55. The first polydisperse sample follows the Flory−Schulz45 distribution. The molar fraction for the Flory−Schulz distribution is formulated as

Ni = (1 − p)pi − 1

3. RESULTS AND DISCUSSION 3.1. Diffusion in a Monodisperse Solvent. The MSD was calculated following the definition as follows

(4)

g (t ) =

where i is the chain length and p denotes the degree of monomer conversion in polymerization in the range of 0 to 1. We set p = 0.94 in this case. The second polydisperse sample

1 ⟨|ri(t + t0) − ri(t0)|2 ⟩ N

(5)

where ri(t) is the position of tagged particle i at time t and the average is taken over all of the tagged particles. The MSD

Figure 1. (a) Snapshot of a monodisperse system with N = 7 and Ns = 55. (b) Snapshot of a monodisperse system with N = 33 and Ns = 11. Snapshots of two polydisperse systems with a Flory−Schulz chain length distribution (c) and a Gaussian distribution (d). (e) Molar fraction of each chain length for these two polydisperse systems with total numbers of particles of 28 262 and 27 379, respectively. Polymer chains are color coded with the chain length from blue to red. C

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Langmuir Table 3. Population (Number of Chains) of Each Constituent for Two Polydisperse Samples N

3

7

11

15

21

27

33

39

43

49

55

Flory−Schultz Gaussian

529 17

420 35

324 63

256 100

169 160

121 192

81 176

56 120

45 81

32 35

21 12

curves, based on the chain center of mass (COM), chain center (the three innermost particles), chain ends, and all particles, have great distinction at intermediate times but eventually overlap when they reach the diffusion regime as demonstrated in ref 38. For all of the simulations in the study, we ensure that all of the MSD curves reach the diffusion regime. The selfdiffusivity D can therefore be evaluated according to the Einstein equation, 1 D = lim g (t ) (6) t →∞ 6t Figure 2 shows the diffusivity of tagged chains immersed in a series of monodisperse polymer melts with solvent chain Figure 3. Diffusivity as a function of solvent chain length Ns for four different tagged chain length polymer melts with N = 7, 15, 21, and 33, respectively. The slopes for the linear fittings are −0.36 ± 0.017, −0.35 ± 0.033, −0.42 ± 0.018, and −0.40 ± 0.023, respectively.

Figure 2. Diffusivity as a function of tagged chain length for four different monodisperse polymer melts with Ns = 11, 21, 33, and 55. The slopes for the linear fittings are −0.79 ± 0.029, −0.80 ± 0.077, −0.82 ± 0.026, and −0.85 ± 0.048, respectively.

lengths (Ns) of 11, 21, 33, and 55 as a function of tagged chain length N. Not surprisingly, D decreases with increasing N for each system. The exponent of D dependence on N varies from −0.79 to −0.85 for different Ns systems. Thus, the diffusivity can be roughly written as a power law form as D ≈ N−(0.79−0.85). Note that the diffusivity of the same tagged chains decreases as the length of solvent chains increases. This is probably due to the fact that systems with shorter solvent chains have more chain ends (and thus larger degrees of freedom and slightly lower density as shown in ref 38) compared to systems with longer solvent chains. Similarly, Figure 3 shows the diffusivity of tagged chains immersed in different monodisperse polymer melts. The diffusivity for each length of tagged chain consistently decreases as the length of surrounding solvent chains increases from 11 to 55. The exponent varies from 0.36 to 0.42, and the scaling law can be roughly formulated as D ∼ Ns−(0.36 − 0.42). These results show that the diffusivity of the polymer chain not only depends strongly on the tagged chain length but also depends on the length of the solvent chains. We formulate the diffusivity in a simple power law form as DN = D0(Ns)Nα(Ns) because the diffusivity depends more sensitively on the tagged chain length. Both the prefactor D0 and the exponent α depend on the solvent chain length, as

Figure 4. Fitting of measured D0 and α as a function of the solvent chain length Ns.

shown in Figure 4. The prefactor can be formulated as a simple power law, and the exponent follows a linear relation such that D0 = 0.4Ns−0.31 and α = −0.773 − 0.00138Ns. The above formulation for diffusion in monodisperse solvent is essential to predicting diffusion in a polydisperse solvent. 3.2. Diffusion in a Polydisperse Solvent. Here we investigated the polymer diffusion in a polydisperse solvent. Figure 5 shows that the diffusivity decreases significantly as the tagged chain length increases. For tagged chains with the same length, the self-diffusivity in the solvent with the Flory−Schultz distribution is higher than that in the solvent with the Gaussian distribution. This is reasonable because there are more long chains in the latter distribution. To understand the diffusion properties measured above and ultimately predict the diffusivity for a solvent with an arbitrary chain length distribution, we developed a simple model to account for diffusion in a heterogeneous medium. Consider first a random walk on a three-dimensional lattice with diffusivity D

DOI: 10.1021/acs.langmuir.7b02901 Langmuir XXXX, XXX, XXX−XXX

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Langmuir

literature. On the basis of our MD simulation results, the diffusivity of monodisperse solvent can be written as a power law form of the tagged chain length, with both the prefactor and exponent varying with the solvent chain length. Thus, the deviation from the Rouse model observed here could arise from the friction coefficient that is solvent-dependent. However, other factors, such as chain stiffness and the local packing effect, could also play a role, which will be investigated later. Finally, on the basis of the diffusivity formula for a monodisperse solvent, a simple model was developed to calculate the diffusivity of a polydisperse solvent, which agrees well with the measured diffusivity in MD simulations.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Yunfeng Shi: 0000-0003-1700-6049

Figure 5. Comparison of calculated D for each constituent with values predicted by the proposed model for both polydisperse systems.

Notes

The authors declare no competing financial interest.

■

1 Γλ 2 , 6

where Γ is the jumping frequency and λ is the D= jumping distance. The heterogeneity of the diffusion medium is represented by a different jumping frequency with the same jumping distance. Thus, the diffusivity among different diffusion 1 media can be written as Di = 6 Γiλ 2 . For each lattice site, the residence time ti is inversely proportional to the jumping 1 frequency such that ti ≈ Γ . The average residence time is an

ACKNOWLEDGMENTS We acknowledge the support from the National Science Foundation (NSF) under grant DMR-1207439. We are grateful for stimulating discussions with Ed Palermo, Pawel Keblinski, Rahmi Ozisik, and Linda Schadler from Rensselaer. The MD simulations were carried out in the Center for Computational Innovations (CCI) at Rensselaer.

■

i

arithmetic mean of the residence time of all sites. Thus, the average jumping frequency of the entire diffusion medium is essentially the harmonic mean,

Γ̅ =

Nsite Nsite 1 Γi

∑i

(7)

Consequently, D̅ =

Nsite Nsite 1 Di

∑i

REFERENCES

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(8)

within a patch of uniform medium Di ∼ Γiλ , where Γi is the jumping frequency in a particular patch. For a given chain length distribution ρ(i), eq 8 becomes 1 D̅ = N 1 ∑i ρ(i) D (9) i 2

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DOI: 10.1021/acs.langmuir.7b02901 Langmuir XXXX, XXX, XXX−XXX