Article pubs.acs.org/Macromolecules
Finding the Missing Physics: Mapping Polydispersity into LatticeBased Simulations Nicholas A. Rorrer and John R. Dorgan* Department of Chemical and Biological Engineering, Colorado School of Mines 1500 Illinois Street, Golden, Colorado 80401, United States ABSTRACT: Polydispersity plays an important role in polymer physics influencing both processing approaches and final properties. Despite the obvious physical importance of polydispersity, studies usually simulate monodisperse chains or occasionally, a few different chain lengths. This work presents a comprehensive methodology for mapping various molecular weight distributions onto a finite number of lattice chains. The use of a lattice in the present derivation enables a variety of lattice-based simulations to incorporate polydispersity. Examples are provided by extending the cooperative motion (COMOTION) algorithm for polydispersity to create the “polydisperse cooperative motion algorithm” (p-COMOTION). The dynamic version of p-COMOTION captures the dynamics of entangled polydisperse melts. For the same weight-averaged molecular weight, polydispersity gives a lower Rouse time and introduces a broadening of the reptation transition. In addition, when adapted into the cooperative motion with flow (COMOFLO) algorithm the resulting p-COMOFLO algorithm captures important polydispersity effects including the suppression of wall slip. Equilibrium results for confined geometries provide evidence that polydispersity modifies wall effects through reduced perturbation of chain dimensions and legth-based migration effects. These and other important effects associated with including the missing physics of polydispersity can be addressed using the newly developed formalism.
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thermodynamics and phase behavior,13,14 in the microphases formed in copolymer systems,15 in polymer rheology,16 in the phenomena of die drool,17,18 and in processing instabilities.19,20 In fact, there are relatively few polymer features that do not depend on the molecular weight distribution. Because of its clear physical significance, incorporating polydispersity into molecular simulations is an important undertaking that has surprisingly been badly neglected. Relatively little simulation work on polydispersity effects has been reported previously. An exception is the work by Boyd and co-workers21−23 where simulation moves are composed of cutting and reconnecting chains; this aphysical approach evolves the system to a particular polydispersity rather than providing the ability to specify the distribution of chain lengths. Some simulations use two chain lengths to simulate polydispersity.24−26 One prevalent technique for introducing polydispersity is the end-bridging Monte Carlo technique developed by Harmandaris et al.27 In this work, the authors prescribe a new way of rearranging polymer chains in Monte Carlo simulations in which polymer chains are sometimes “cut” and rejoined. This results in chains of varying length and produces modest polydispersity. The end-bridging move can be incorporated into other Monte Carlo simulations1,2,9 to simulate coarse grained polymer melts. Dodd and Jayaraman28 use Monte Carlo to simulate polydisperse chains grafted onto a
INTRODUCTION Computer simulations offer an important alternative to laboratory scale experiments. A distinguishing feature is that they provide unambiguous molecular scale details. Most detailed molecular simulations fall into one of two categories; they are either molecular dynamics (MD) or Monte Carlo (MC) simulations.1,2 MD simulations typically offer a high level of atomistic detail but are computationally limited to relatively small time scales, especially for large system sizes. Coarsegrained approaches sacrifice atomistic detail in order to gain access to longer time-scale phenomena and lattice-based simulations are the least detailed but fastest in the hierarchy of simulation techniques. For example, it is possible to design a set of Monte Carlo moves on a lattice that mimic polymer motion1−10 instead of solving Newton’s equations of motion at each time step as is done in MD. Some, but not all, of these molecular motion mimicking Monte Carlo methods are constructed on a lattice in order to save computation time. For example, a comparison of simulating polymer melt rheology with the present approach11 versus a coarse-grained MD approach12 suggests the dynamic MC method is ten thousand times faster. Accordingly, coarse grained lattice models fill an important space within the hierarchy of simulation methods. Almost all polymer simulations appearing in the literature assume that all chains are the same length. This monodisperse assumption is a severe oversimplification as it is widely recognized that polydispersity can have dramatic effects. For example, polydispersity is known to be important in polymer © XXXX American Chemical Society
Received: January 15, 2014 Revised: March 12, 2014
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addition, ci is a proportionality constant which sets the length of chains in group i relative to the target “average length”, No. (measured in number of lattice segments). Substitution of eqs 5 and 6 into the numerator of eq 1 and eq 3 into its denominator gives
nanoparticle. In their work, there is a relatively small number of chains and a small range of chain lengths. In two papers by Escobedo,29,30 the author develops an in depth method for studying confined polydisperse systems using the Massieu function; this formulation is elegant in nature but somewhat complex in implementation. In addition, like the end-bridging Monte Carlo method, this method starts with an initial system which evolves to a particular polydispersity. In real polymer melts, there is a given molecular weight distribution that is governed by the polymerization technique used and influenced by processing when either degradation or trans-reaction between chains occurs. The present approach provides a universal method for mapping a molecular weight distribution onto a set of chain lengths for use in lattice-based molecular simulations. This mapping allows the specification of a predetermined weight-averaged molecular weight and polydispersity. Since the polymer chains are not reconstructed by cutting and reconnecting, the approach offers many advantages. The predominant advantage is that arbitrary molecular weight distributions can be studied as a means of understanding how such distributions can be designed to realize processing strategies and desired properties.
Nw = N0 =
Nw = N0 =
Nsites = Nonk T
Nn =
Nn =
∑ niNi
(9)
Nsites nk T
(10)
∑ (ciN0)(bink T) N nk ∑ cibi = o T = No ∑ cibi nk T nk T (11)
Combining eqs 10 and 11 gives the result Nsites = Nonk T
(1)
(2)
(3)
ni =
where Nsites is the total number of occupied sites on the lattice. Furthermore, Σni, ni, and Ni are expressed as,
∑ ni = nk T
(4)
ni = bink T
(5)
Ni = ciNo
(6)
∑ cibi
(12)
Equations 9 and 12 represent constraints on the constants ci and bi that must be satisfied for any constructed distribution. It is possible to relate bi to an arbitrary probability distribution. Referring to Figure 1, let pi represent the area under the normalized probability distribution function centered around one of the selected chain lengths in the model. This value represents the probability of picking a lattice site at random and having it belong to a chain of group i. Since this probability is also related to the fraction of lattice sites that belong to a chain in group i, it can also be represented in terms of the number of chains. That is, the total number of chains in group i can be expressed as
In both cases the sum ΣniNi is subject to the constraint that Nsites =
∑ ci 2bi
Note that eq 10 is just an alternative definition of the number-averaged molecular weight; the number of segments occupied divided by the number of chains. Alternatively substitution of eqs 5 and 6 into eq 2 gives
where No serves as the target “average length”. In eq 1, Ni is the length of chains measured in number of segments that are of type i, and ni is the number of chains of type i (that is, the number of chains having length Ni). To be clear, i denotes only the number of a given subgroup of chains in the distribution (i.e., i = 1 can refer to all chains of length N1 = 32 segments). Similarly, Nn is defined in analogy to the number-average molecular weight as,
∑ niNi ∑ ni
(8)
Similarly, substitution of eqs 3 and 4 into the numerator and denominator of eq 2 provides,
2
Nn =
No 2nk T ∑ ci 2bi Nsites
Rearrangement of eq 8 gives the following result
METHODOLOGY The weight-averaged, Mw, and number-averaged, Mn, molecular weights are often combined in order to characterize a molecular weight distribution. The quotient of these two numbers is equal to the polydispersity index, PDI. In a lattice-based coarse grained model there is no explicit weight, chains are represented by connected sequences of lattice segments. Accordingly, for the purpose of the mapping, the length Nw (measured in number of lattice-segments) is defined in an analogous manner to Mw as ∑ niNi ∑ niNi
(7)
The total number of chains, nkT, and target average lengths are independent of the sum over i and can be factored out to give eq 8.
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Nw = N0 =
∑ (ciN0)2 (bink T) Nsites
pN i sites Ni
=
pN i sites ciN0
(13)
where eq 6 has been used to introduce the constants ci. Using expression 5, bi represents the probability of selecting one chain out of the total number of chains and can be written as, pN i sites
pN i sites
p /ci n cN ciN0 bi = i = i 0 = = i pN ∑ pi /ci nk T nk T ∑ ic Nsites i 0
where nkT is the total number of chains in the simulation. Accordingly, bi is just the fraction of the total number of chains in group i; that is the fraction of chains having length Ni. In
(14)
The probability pi is defined in terms of the probability density function, f(x |μ, σ), where μ is the mean of the B
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simple example is readily extended to systems having multiple chain lengths. This method is tested in conjunction with the cooperative motion algorithm and its extensions.5,6,11,31 This family of algorithms offers many advantages including being able to simulate high density systems, maintain chain connectivity, capture realistic dynamics, and simulate flow phenomena. This algorithm is able to obtain the proper dynamics and scaling of the zero shear viscosity with molecular weight in under 2 months on a single core processor, which is a significant speed up compared to coarse grained molecular dynamic simulations that took a year on a 64 processor cray architecture.12 Accounting for Moore’s Law, this represents a 10 000 times speed up! The simulation results that are presented here were conducted on a fully occupied (φ = 1.0) face centered cubic (fcc) lattice. An fcc lattice is implemented because of its high coordination number, which has been shown to be in agreement with MD simulations and experiment.11,32 Hard athermal walls (χ = 0) are usually specified in x-direction while periodic boundaries are present in the y and z direction. To minimize confinement effects the x-direction walls are separated by at least six radii of gyration. Initially the polymer chains are laid in an all trans configuration on the lattice. This configuration is subsequently melted by performing over one million closed loops of cooperative motion (representing 106 Monte Carlo time units). Attempted moves are accepted as long as excluded volume and a constant bond length are preserved. Equilibration is monitored by tracking the ensemble average radii of gyration and equilibrated ensembles are then sampled over an additional million closed loops to obtain chain properties (e.g., binned number-average radii of gyration and center of mass density) and dynamic correlation functions. Equilibrated ensembles can also be studied under shear (as outlined in Dorgan et al.11) Further details can be found in previous works1,11,33 Results for polymer melts that have one, three, and five discrete chain lengths corresponding to PDI values of 1.00, 3.20, and 1.42 respectively are presented.
Figure 1. Example of mapping a set of chain lengths to a probability distribution. Three discrete chain lengths (N1,=128, N2 = 256, and N3 = 384) are mapped to the distribution. The shaded region represents the value of pi for chains of length 256.
probability distribution and σ is the standard deviation. The mathematical formulation is given by b
pi =
∫a
a=
Ni − 1 + Ni 2
f (x | μ , σ ) d x
(15) (16)
Ni + Ni + 1 (17) 2 where a and b are the limits of integration. If the chain is the shortest chain in the model (N1) then the a-bound is 0 and if the chain is the longest chain in the model then the b-bound is ∞. By preselecting a target weight-averaged chain length (the N0 value), a discrete number of chain lengths each with a finite length (a set of ci values), a desired probability distribution f(x |μ, σ), and a lattice size (Nsites) the representative length distribution can be constructed. An illustrative example is now provided. For simplicity, consider a three chain system that is centered on N0 = 256. For this example chains of length N1 = 32, N2 = 256, and N3 = 512 are preselected so that eq 6 gives c1 = 1/8, c2 = 1, and c3 = 2. Adopting a Gaussian distribution with a standard deviation of 256 (μ = 256 and σ = 256), eq 16 and 17 give three sets of integration limits (a,b) of (0,144), (144,384), and (384, ∞). Integration according to eq 15 provides the resulting values of p1 = 0.32, p2 = 0.36, and p3 = 0.32. Application of eq 14 yields bi values of b1 = 0.847, b2 = 0.107, and b3 = 0.046. Using a lattice that is 128 × 64 × 64 in its xyz dimensions (giving 262144 occupied sites on the face centered cubic lattice constructed by filling every other site) and eq 9 (with the constraint that Σcibi = Σci2bi = 0.304) gives the total number of chains as nkT = 3362 with nk1 = 2848, nk2 = 360, and nk3 = 154. This yields values of Nw = 256 and Nn = 78. On the lattice site basis the distribution was centered about 256 and the standard deviation of the data set is σ = 215 lattice units. This value is slightly lower than the one prescribed by the probability distribution, but it arises because of truncation and rounding errors in this example. This b=
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RESULTS AND DISCUSSION During implementation, several variables are calculated including the binned radii of gyration and the center of mass distribution for each chain length group. The squared radius of gyration, s2, for a chain is defined as the average distance from the chain center of mass or, s2 =
1 N
N
∑ (ri − rcm)2 i=1
(18)
Where ri is the position of the ith monomer, rcm is the position of the center of mass, and N is the total number of segments in the chain. The chain length can be averaged across the whole simulation box or in a binned fashion as a function of the xdirection. The center of mass density, ρcom,i(x), is defined as ρcom, i (x) =
nki(x) ni /L
(19)
where nki(x) is the number of chains in group i that have a center of mass in a given x-plane and L is the number of lattice sites between the hard walls. When athermal monodisperse polymer melts are confined between two neutral walls there are wall effects present within two radii of gyration of the wall.34 The binned root-meanC
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square radii of gyration near the walls are lower than its bulk value due to reflection folding the chain back upon itself. The near-wall polymers also align their end-to-end vectors parallel to the walls. Introducing polydispersity reduces these wall effects. The modified wall-effects are shown in Figure 2, which plots the number-averaged radii of gyration as a function of
Figure 3. Center of mass density simulated between two hard walls that are ten radii of gyration apart. A three chain model centered around Nw = 256 is simulated for the PDI = 3.2. The polymer chains are evenly distributed in the bulk but the shorter chains are present at higher normalized concentrations closer to the wall. Figure 2. Binned number-average radii of gyration for chains between two hard walls that are ten radii of gyration apart; L* is the normalized plate spacing (L* = x/L). A three chain model centered around Nw = 256 is simulated for the Nw = 256, PDI = 1.0 case. The polydisperse melts possess a lower ensemble average radius of gyration than the monodisperse melts. The wall effect is present for both melts but is less pronounced in the polydisperse melt.
With this definition, every time a polymer segment is displaced the time counter is incremented by tu so that when the time counter is equal to 1 every segment has moved or had the chance of moving. This is physically equivalent to what happens in an MD simulation at every time step. The unit of time so defined is referred to as a Monte Carlo time step (mct). By comparing diffusion coefficients between the simulation and laboratory data, the Monte Carlo time can be assigned a value in real time;35 this procedure yields the value of a 1.0 × 10−11 s/ mct when applied to polyethylene. Once Monte Carlo time is defined, dynamics of polymers can be interrogated through correlation functions. In these functions ri(t) is the position of a given segment at a certain time and rcm(t) gives the position of the center of mass of the overall chain. The mean-squared displacement of central monomers is given by
normalized plate spacing. Polydisperse melts possess a lower average radii of gyration due to the high number of shorter chains present. As with the monodisperse case, for polydisperse systems there is still a lower radii of gyration near the wall as compared to the bulk. In the monodisperse case the wall effect persists for approximately two radii of gyration. In the polydisperse case the effect is less exaggerated and only accounts for 5% at each wall for the same lattice dimensions. In the case of the monodisperse melts this perturbation is upward of two bulk radii of gyration while the polydisperse perturbation is only about one box-averaged radius of gyration. The reduction of the wall effect arises due to a migration of chains having varying lengths. Longer polymer chains possess a larger radius of gyration, thus making it more difficult for them to reside close to the wall due to the greater perturbation in their coil size. In addition, previous work illustrates that chain ends are entropically favored at the wall.31,34 The combination of these two factors leads the shorter polymer chains to reside closer to the walls. The center of mass density is reported in Figure 3 which clearly shows the migration phenomena where the normalized density of the shortest chains is highest near the walls. The result being that the perturbation of the wall on the mean coil size is less significant for the polydisperse case in comparison to the monodisperse case. Dynamics of both monodisperse and polydisperse polymer melts are also studied. In order to study the dynamics a distinct time step must be defined. This time step, tu, is defined as3 1 tu = Nsites (20)
g1(t ) = ⟨[ri(t ) − ri(t = 0)]2 ⟩,
for i =
N 2
(21)
where N is the total length of a given chain. The mean-squared displacement of central monomers relative to the center of mass is given by g2(t ) = ⟨[(ri(t ) − rcm(t )) − (ri(t = 0) − rcm(t = 0))]2 ⟩, for i =
N 2
(22)
The mean-squared displacement of the center of mass is g3(t ) = ⟨[rcm(t ) − rcm(t = 0)]2 ⟩
(23)
The mean-squared displacement of the chain ends is given by g4 (t ) = ⟨[ri(t ) − ri(t = 0)]2 ⟩,
for i = 1 and N
(24)
Finally, the mean-squared displacement relative to the center of mass is given by D
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Figure 5 presents a side by side comparison of the g1 correlation function between the monodisperse and poly-
g5(t ) = ⟨[(ri(t ) − rcm(t )) − (ri(t = 0) − rcm(t = 0))]2 ⟩, for i = 1 and N
(25)
This set of five correlation functions provides great detail on the dynamics of polymers in the melt.25 Unentangled polymer melts will exhibit Rouse dynamics for which the g1 and g4 correlation functions scale as t at short times, t1/2 at intermediate times, and t at long time scales. However, most polymer melts of technical interest are entangled and exhibit reptation like dynamics. At very short times, the g1 and g4 correlation functions scale as t1/2. At intermediate times (between the entanglement and Rouse time), there is the hallmark scaling of g1 ∼ t1/4. At longer times (between the Rouse and disentanglement time), g1 scales as t1/2. Finally, at very long time scales (above the disentanglement time), g1 scales as t. In addition, it is important to note that the g3 correlation function scales as 6Dt at long time scales. There are many theories and models36,37 that describe the various characteristics of entangled polymer melts, but their scaling is always described by these fundamental exponents. Previous work has accurately captured the dynamics of the polymer melts for monodisperse polymer melts.11 At chain lengths above the entanglement length (N > 150) reptation dynamics are reported by the observation of the characteristic t1/4 scaling in the g1 and g4 correlation functions. In the case of monodisperse polymer melts, the transition between all regimes of scaling is always continuous and not abrupt. This effect arises because there is a spectrum of relaxation times (e.g., chains near the wall relax at a different rate than chains in the bulk). Polydispersity broadens the transitions due to the increased relaxation spectrum associated with chains of varying lengths. The dynamics of a polymer melt with modest polydispersity are presented in Figure 4. This Figure shows that polymer melts still exhibit reptation like dynamics when a slight degree of polydispersity is introduced. The t1/4 scaling is still present and still demonstrates the continuous transition.
Figure 5. g1 correlation function for an Nw = 512 polymer melts between two hard walls spaced 10 radii of gyration apart. The polydisperse melts still exhibit the reptation like signature and distinctions in the characteristic Rouse and disentanglement times are observed. The 95% standard deviation is less than the symbol size.
disperse melts. The difference between the Rouse time (the transition from t1/4 to t1/2) and the entanglement time is smaller in the polydisperse case than it is in the monodisperse situation. Polydisperse melts possess lower Rouse and disentanglement times. This phenomenon arises because of the spectrum of chain lengths (specifically the shorter chains), each possessing their own characteristic times. Doi et al.36,37 postulate and show that, for chains of similar lengths, the characteristic times obey simple mixing rules, which is demonstrated by the results presented here. The polymer melt data presented in Figure 4 and Figure 5 contains five different chain lengths (Ni = 128, 256, 512, 1024, 2048) and was mapped to a Gaussian probability distribution centered on Nw = 512. The resulting bi values are b1 = 0.2726, b2 = 0.2716, b3 = 0.4201, b4 = 0.0310, and b5 = 0.0047, respectively. In the monodisperse case, these individual polymer chains possess Rouse times of approximately 3.5 × 104 mct, 5.0 × 104 mct, 7.0 × 104 mct, 1.0 × 105 mct, and 1.4 × 105 mct, respectively. By applying a simple number-averaged mixing rule to this polymer melt the resulting Rouse time is approximately 5.6 × 104 mct, this value is in agreement with Figure 5. Accordingly, the results of the present investigation support Doi’s postulate that mixing rules can be applied to understand polydispersity effects on the Rouse time. Polydisperse polymer melts have also been subjected to shear flow. Shear flow is modeled in the method described previously in Dorgan et al.11 in which the movement of polymer segments is biased depending on their position on the lattice. This method is based of work done by Katz et al.38 for biasing the Ising model subject to an external field. The same biasing has also been implemented by Glieman et al.31 and Jzeska and Pakula39 to model parabolic and shear flow. When the polymer melts are subject to shear flow their rheological properties are calculated by using Kramer’s bead and spring model40 in which
Figure 4. Reptation-like dynamics for a Nw = 512 PDI = 1.42 polymer melt between two hard walls spaced 10 radii of gyrtation apart. Reptation dynamics are hallmarked in the t1/4 scaling that is present in g1 and g4 correlation functions. The inset is of the g1 correlation function and also demonstrates the t1/4 scaling. E
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the stress tensor, σ, is calculated as the dyadic product of the bond vectors, ⟨ rr ⟩. This formalism can be expressed41 as σ = 2vkTβ 2⟨ rr ⟩
response curve exhibits nonmonotonic behavior where two shear rates correspond to the same shear stress. When modest polydispersity is present the nonmonotonic region is reduced in severity. At higher degrees of polydispersity the nonmonotonic region completely vanishes. These results are consistent with decades of observations on the flow behavior of polymer melts where it is well-known that broadening polydispersity improves processability and decreases flow instabilities.20
(26)
where ν is the number of chains per unit volume, k is Boltzmann’s constant, T is the temperature, and β2 is 3/ (2Nbk2), where bk is the length of the Kuhn segment and N is the number of segments in the chain. Equation 26 can be rewritten as 3⟨ rr ⟩ 3⟨ rr ⟩ σ = = 2vkT 2Nb k 2 2⟨R2⟩
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CONCLUSIONS This work provides a formalism that can be used to map molecular weight distributions to a set of chain lengths on a lattice. The set of equations derived allows the distribution of chain lengths, target molecular weight, and the number of representative chains to be set in a predetermined way. The chain collection so constructed can either be used in lattice based simulations or as an initial configuration for molecular dynamics simulations. The method is applicable to both the study of equilibrium and dynamic systems. When this method is implemented with the COMOTION algorithm polydispersity reduces wall effects. The near-wall region over which chain dimensions are perturbed is reduced as measured by the system average radius of gyration; this change is due to migration of the shorter chain lengths to the wall. So in another sense wall effects are more profound in polydisperse systems as they lead to chain migration based on differing molecular weights. Polydispersity also affects dynamics. Reptation-like dynamics are obtained for a polydisperse system and are hallmarked by the presence of a t1/4 scaling in the g1 and g4 correlation functions. Polydisperse melts possess lower characteristic times and a broader transition between the different scaling regimes. For narrow Gaussian distributions, the Rouse time follows a simple component mixing rule. In addition when polydisperse melts are subject to flow, the shear stress response is dramatically affected. As the PDI is increased, the region of the nonmonotonic behavior decreases and is eventually eliminated. These results are in post facto agreement with theory and experiments. Polydispersity is important for understanding molecular details under equilibrium conditions, in the dynamics of polymers, and for polymers under flow. The present study shows the importance of including the often missing physics of polydispersity into molecular scale simulations of polymers.
(27)
2
where o is the unperturbed mean-squared end-to-end vector in the bulk. Equation 27 provides a dimensionless stress. The deviatoric stress tensor is defined as, τ=σ −p (28) where p is the pressure tensor and is defined as the total stress tensor, σ, under no flow conditions. Previous work demonstrated that monodisperse entangled polymer melts exhibit failure at high shear rates.35 This failure corresponds to weak and strong slip, and is heralded by discontinuities in the velocity profile. For monodisperse chains, when the chain length reached Nw = 128, the first evidence of a nonmonotonic stress response is observed. However, as polydispersity is introduced the severity of the nonmonotomic stress response is slowly reduced and eventually eliminated. This is shown in Figure 6 which presents a comparison of the
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AUTHOR INFORMATION
Corresponding Author
*(J.R.D.) E-mail:
[email protected].
Figure 6. Shear stress response curves for three Nw = 128 polymer melts simulated between two hard walls 10 radii of gyration apart. Apparent shear rates are calculated as a linear fit to the velocity profile.
Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.
stress vs strain rate relationship for three different polydispersities. At low shear rates all polydispersities scale linearly with apparent shear rate and have the same slope corresponding to the same zero shear viscosities. It is known from the experimental literature that polymer melts possessing the same weight-average molecular weight will demonstrate the same zero shear viscosity.41 As the shear rate is increased the shear stress continues to increase but the proportionality between shear stress and shear rate is no longer constant, corresponding to shear thinning behavior. When the monodisperse melt is subject to high shear rates the stress
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Dr. Timothy Kaiser for support of the Blue M supercomputing platform at the Colorado School of Mines on which some of the present results were generated. This work is funded by the Fluid Dynamics Program of the National Science Foundation under Grant CBET-1067707. F
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(37) Doi, M.; Graessley, W. W.; Helfand, E.; Pearson, D. S. Macromolecules 1987, 20 (8), 1900−1906. (38) Katz, S.; Lebowitz, J. L.; Spohn, H. J. Stat. Phys. 1984, 34 (3−4), 497−537. (39) Jeszka, J. K.; Pakula, T. Polymer 2006, 47 (20), 7289−7301. (40) Kramer, H. A. The behavior of macromolecules in inhomogeneous flow. Physica 1944, 11, 1. (41) Larson, R. G. Constitutive Equations for Polymer Melts and Solutions; Butterworth-Heinemann: New York, 1988; p 368.
ABBREVIATIONS MD, molecular dynamics; MC, Monte Carlo; DMC, dynamic Monte Carlo; BD, Brownian dynamics; PDI, polydispersity index; mct, Monte Carlo time
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REFERENCES
(1) Pakula, T. Chapter 6 In Simulation Methods for Polymers, 1st ed.; Kotelyanskii, M., Theodorou, D. N., Eds., Marcel Dekker: New York, 2004; pp 147−176. (2) Binder, K.; Paul, W. J. Polym. Sci., Part B: Polym. Phys. 1997, 35 (1), 1−31. (3) Mansfield, K. F.; Theodorou, D. N. Macromolecules 1989, 22 (7), 3143−3152. (4) Pant, P. V. K.; Theodorou, D. N. Macromolecules 1995, 28 (21), 7224−7234. (5) Pakula, T. Macromolecules 1987, 20 (3), 679−682. (6) Geyler, S.; Pakula, T.; Reiter, J. J. Chem. Phys. 1990, 92 (4), 2676−2680. (7) Gauger, A.; Weyersberg, A.; Pakula, T. Makromol. Chem.Theory Simul. 1993, 2 (4), 531−560. (8) Pakula, T. Comput. Theor. Polym. Sci. 1998, 8 (1−2), 21−30. (9) Binder, K. Monte Carlo and Molecular Dynamics Simulations in Polymer Science, 1st ed.; Oxford University Press: New York, 1995; p 587. (10) Carmesin, I.; Kremer, K. Macromolecules 1988, 21 (9), 2819− 2823. (11) Dorgan, J. R.; Rorrer, N. A.; Maupin, C. M. Macromolecules 2012, 45 (21), 8833−8840. (12) Kroger, M.; Hess, S. Phys. Rev. Lett. 2000, 85 (5), 1128−1131. (13) McMaster, L. P. Macromolecules 1973, 6 (5), 760−773. (14) Thompson, R.; Matsen, M. Phys. Rev. Lett. 2000, 85 (3), 670. (15) Bendejacq, D.; Ponsinet, V.; Joanicot, M.; Loo, Y. L.; Register, R. A. Macromolecules 2002, 35 (17), 6645−6649. (16) Minoshima, W.; White, J. L.; Spruiell, J. E. Polym. Eng. Sci. 1980, 20 (17), 1166−1176. (17) Lee, B. L.; White, J. Trans. Soc. Rheol. 1975, 19 (3), 481−492. (18) Musil, J.; Zatloukal, M. Chem. Eng. Sci. 2012, 81 (0), 146−156. (19) Denn, M. M. Annu. Rev. Fluid Mech. 2001, 33 (1), 265−287. (20) Hatzikiriakos, S. G. Prog. Polym. Sci. 2012, 37 (4), 624−643. (21) Boyd, R. H. Macromolecules 1989, 22 (5), 2477−2481. (22) Boyd, R. H.; Pant, P. V. K. Macromolecules 1991, 24 (14), 4078−4083. (23) Boyd, R. H.; Gee, R. H.; Han, J.; Jin, Y. J. Chem. Phys. 1994, 101 (1), 788−797. (24) Baschnagel, J.; Paul, W.; Tries, V.; Binder, K. Macromolecules 1998, 31 (12), 3856−3867. (25) Hernandez-Ortiz, J. P.; Ma, H. B.; Pablo, J. J.; Graham, M. D. Phys. Fluids 2006, 18 (12), 123101. (26) Khare, R.; Graham, M. D.; de Pablo, J. J. Phys. Rev. Lett. 2006, 96 (22), 224505. (27) Harmandaris, V. A.; Mavrantzas, V. G.; Theodorou, D. N. Macromolecules 1998, 31 (22), 7934−7943. (28) Dodd, P. M.; Jayaraman, A. J. Polym. Sci., Part B: Polym. Phys. 2012, 50 (10), 694−705. (29) Escobedo, F. A. J. Chem. Phys. 2001, 115 (12), 5642−5652. (30) Escobedo, F. A. J. Chem. Phys. 2001, 115 (12), 5653−5661. (31) Gleiman, S. S.; Dorgan, J. R. J. Chem. Phys. 2000, 112, 6073− 6083. (32) Downey, J. P.; Crabb, C. C.; Kovac, J. Macromolecules 1986, 19 (8), 2202−2206. (33) Gleiman, S. S.; Dorgan, J. R. J. Chem. Phys. 2000, 112 (13), 6073−6083. (34) PaiPanandiker, R. S.; Dorgan, J. R.; Pakula, T. Macromolecules 1997, 30 (20), 6348−6352. (35) Dorgan, J. R.; Rorrer, N. A. Phys. Rev. Lett. 2013, 110 (17), 176001. (36) Doi, M.; Edwards, S. F. Theory of Polymer Dynamics; Oxford Science Publications: Oxford, U.K., 1988 G
dx.doi.org/10.1021/ma5001207 | Macromolecules XXXX, XXX, XXX−XXX
