Coarse-Grained Simulations of Three-Armed Star Polymer Melts and

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Coarse-Grained Simulations of Three-Armed Star Polymer Melts and Comparison with Linear Chains Li Liu, Wouter Koenraad Den Otter, and Wim J. Briels J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b03104 • Publication Date (Web): 16 Oct 2018 Downloaded from http://pubs.acs.org on October 19, 2018

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The Journal of Physical Chemistry

Coarse-grained Simulations of Three-armed Star Polymer Melts and Comparison with Linear Chains ∗,†

Li Liu,

Wouter K. den Otter,

‡,¶

and Wim J. Briels

∗,‡,¶,§

†Department

of Information Science and Engineering, Dalian Polytechnic University, Dalian, China ‡Computational Chemical Physics, Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands ¶MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands §Forschungszentrum Ju¨lich, ICS 3, D-52425 Ju¨lich, Germany E-mail: [email protected]; [email protected]

Abstract Melts of three-armed star polymers have been simulated using a coarse-grained model parameterized by atomistic simulations of polyethylene. The bonds between the highly coarse grained, and hence soft, polymer beads are explicitly prevented from crossing by the TWENTANGLEMENT algorithm. Three melts of symmetric stars, diering in the lengths of the arms, are compared against ve melts of linear polymers with comparable dimensions, to study the impact of the branched architecture on the self-diusion and the bulk rheological properties. Dierently from the power-law relation between viscosity and molecular mass of linear chains, the star polymers in our 1

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simulations follow an exponential mass-viscosity relation and show qualitative agreement with the storage and loss moduli for stars with far longer arms from experiments. The stress relaxation dynamics of the stars are also compared with a theoretical analysis in terms of Rouse modes.

1

Introduction

Melts of star polymers behave dierently from melts of linear polymers made of identical monomers. They even obey distinctly dierent scaling laws. A well-known example is the relation between the zero-shear viscosity and the molecular mass: for linear polymers the viscosity scales as a power of the total mass, 1 while for symmetric star polymers the viscosity grows exponentially with the mass of an arm. 2,3 This marked dierence is a consequence of the polymer dynamics at the microscopic level. Long linear polymers perform reptation dynamics, i.e. they move predominantly along their own contour path, conned to the tube formed by their neighbours. 4 Star polymers with f long arms,where f is also known as the functionality, will have their arms conned to f distinct tubes for most of the time, thereby ruling out reptation dynamics. Instead, stars and other branched polymers are believed to move by retracting an arm from its tube through thermal motion, followed by extension of this arm into a new tube, which strongly reduces the dynamics of stars. 5,6 Insights into the dynamical properties of polymer melts, and the microscopic processes giving rise to the macroscopic properties, are gained in mutually supplementary studies by experiment, theory and computer simulation. 713 The long polymer or arm lengths required to see reptation or retraction dynamics, respectively, pose a challenge to computer simulations. Ideally, one would like to simulate polymers at the all-atom level by Molecular Dynamics (MD) to obtain a nearly exact agreement between simulations and experiments, but the long length and time scales involved make these computationally very demanding simulations. 1416 The group of Tanner reported united-atom simulations of C100 H202 polyethylene (PE) isomers, using non-equilibrium techniques to overcome the slow dynamics, in their study 2


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on the inuence of star architecture on the shear thinning characteristics. 17 To reduce the computational burden, one often lumps together several backbone atoms and their adjacent hydrogen atoms or small side groups into `beads'. These beads are typically modelled as Lennard-Jones (LJ) spheres, connected by nitely extensible nonlinear elastic (FENE) bonds. 18 In this bead-spring polymer model, the LJ and FENE potentials are tuned to guarantee that the chains cannot cross each other, thus retaining the key feature of polymer dynamics. The well-known Kremer-Grest model for linear polymers was also applied, by its developers, to study the structural features of star polymers, like the mass distribution, and their internal relaxation dynamics. 19,20 Continuing along the coarse-graining path of sacricing chemical details to gain access to longer time and length scales, one may group even more backbone atoms (and their side groups) together into mesoscopic blobs. The eective repulsive interaction between these blobs, as deduced from atomistic MD simulations, becomes softer with increasing blob mass,while the bonds between the blobs become longer. 21,22 Both trends are detrimental to the automatic prevention of bond crossings in the simulations, hence alternative schemes have to be introduced to re-instate entanglement eects. In slip-link models, the chains of particles are threaded through a sequence of rings which are connected to xed points in space by rigid constrains or harmonic potentials. 2325 The rings restrict the sideways motions of the chain, but not the reptational sliding of the chain through the rings; a series of rules govern the creation of new rings when loops or dangling ends grow too large, while vacated rings are removed. Padding and Briels developed the TWENTANGLEMENT algorithm that explicitly detects imminent bond crossings and inserts entanglement points to prevent the bonds from actually crossing. 26,27 The equations of motion of the blobs and entanglement points are coupled by mutual interactions; the chains are allowed to slip past an entanglement point, and several mechanisms allow for disentanglement. We have recently extended this algorithm to handle branched molecules. The rst simulations of a melt of symmetric three-armed stars already clearly highlighted the impact of the polymer architecture on the melt properties: the short-armed stars diuse like linear polymers of equal 3



mass, but their rheological properties resemble those of a linear polymer of equal end-to-end span. 28 In this work, we extend our previous study to symmetric three-armed stars with longer arms, thereby allowing the arms to get more thoroughly entangled. The dynamical and rheological properties of three melts are studied and compared against ve melts of matching linear polymers. A global overview of the TWENTANGLEMENT algorithm, the simulation set-up and the applied analysis methods are provided in Section II. The simulation results are presented and discussed in Section III, where the linear polymers are mainly discussed to facilitate the comparison. Our main conclusions are summarized in Section IV.

2

Model and methods

2.1 Coarse graining polymers In this study,we used the coarse-grain

twentanglement model originally developed by

Padding and Briels for the simulation of linear polymers, the algorithm was recently extended by us to study polymers with more complex architectures. 2628 The polymers are composed of coarse-grained particles (or blobs), each representing 20 consecutive backbone carbons of a polyethylene (PE) chain, see Fig. 1 for the representation of one star polymer in sketch.

Figure 1: Schematic picture of a star polymer in TWENTANGLEMENT model . Assuming complete separation of time scales between the eliminated and retained degrees 4


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of freedom, i.e. the positions Rn = {R1 , . . . , Rn } of the n blobs, and neglecting eects of inertia, the time evolution of the position of the ith blob is described by the rst order Langevin equation

√ 1 dRi = − ∇i Φdt + ξ

2kB T dt Θi . ξ

(1)

The friction coecient ξ in the rst term and the random displacements in the second term represent the inuence of the eliminated fast degrees of freedom on the retained slow degrees of freedom, with kB the Boltzmann constant, T the absolute temperature, and where Θi denotes a Markovian stochastic vector of zero mean and unit variance, devoid of correlations between its components and between particles, and without memory. The potential of mean force (PMF) is dened as Φ(Rn ) = −kB T ln P (Rn ), where the distribution function P of blob center of mass positions has previously been determined by grouping together 20 consecutive united atom CH2 particles in molecular dynamics simulations of a C120 H242 melt. 2629 For practical reasons, the PMF has been approximated as a sum of contributions,

Φ(Rn ) =

∑

ϕbond (li,i+1 ) +

i

∑

ϕang (θi,i+1,i+2 ) +

i

∑

ϕnb (Rij )

(2)

ij

with ϕbond the attractive bond potential between two consecutive blobs at a bond length

li,i+1 = |Ri+1 − Ri |, ϕang the bending potential between three consecutive blobs at an angle θi,i+1,i+2 , and ϕnb the repulsive non-bonded potential between any pair of blobs. A more elaborate discussion of the coarse grain model applied in this paper can be found in the appendix.

2.2 Uncrossability constraints As mentioned in the Introduction, when the number of backbone carbons lumped into a blob increases, the interactions between the blobs become softer and softer. The soft interactions between C20 blobs do not prevent the coarse grained (CG) polymers from crossing each other; this fundamental feature of polymers, with far reaching implications for the properties of a 5



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melt, is lost in the coarse-graining procedure. In order to restore proper polymer dynamics, these unrealistic bond crossing events are detected and prevented in the twentanglement algorithm. The crossing of the i to i + 1 bond with the j to j + 1 bond (where i and j can be blobs on the same polymer) is detected by a change in sign, during the time step dt, of the triple product

[ ] Vij = (Ri+1 − Ri ) × (Rj+1 − Rj ) · (Ri − Rj )

(3)

formed by the two bond vectors and their connecting vector. If the two bonds cross, an `entanglement point' X is introduced at the (approximate) position of the crossing event. The position of this entanglement point is updated every time step, using the procedure outlined below, until the two bonds disentangle; this annihilation event is detected by a procedure similar to that used to detect their creation. The collection of n blob positions and a variable number of entanglement points is advanced in time by the following sequence of steps. For a given conguration, the potential energy and the forces are calculated by means of Eq. (2), with the one exception that the bond length is not calculated as the direct distance between two consecutive blobs but as the contour length of the bond connecting these blobs:

∑

ϕbond (li,i+1 ) →

i

∑

ϕbond (Li,i+1 ),

(4)

i

where

Li,i+1 = |Ri − Xi,i+1 | + |X1 − Xi,i+1 − Ri+1 | | + . . . + |Xi,i+1 1 2 p

(5)

denoting the p entanglements along the bond connecting Ri to Ri+1 . with Xi,i+1 , . . . , Xi,i+1 1 p The positions of the entanglement points are then determined by numerically minimizing the total binding potential energy with respect to the entanglement points, subject to the given blob positions. The resulting bond forces on the blobs, in combination with the angular and

6


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non-bonded forces, are used to update the blob positions by means of Eq. (1). As the last step in the cycle, the system is scanned to detect creations of new entanglements and annihilations of old entanglements. Whereas the detection of bond crossing has been explained in Eq. (3) using straight blob-to-blob paths, the employed algorithm detects crossings of the meandering paths between the blobs by applying Eq. (3) to all straight segments connecting successive points (by using entanglement positions as well as blob positions) along a polymer chain. Since real polymers can slide past an entanglement point, the detection step also allows entanglement points to slip over a blob, e.g. Xi,i+1 becoming Xi+1,i+2 . Entanglements 1 p slipping over the last bead of a chain are annihilated. For details on the implementation of the

Twentanglement algorithm, we refer the reader to Padding and Briels. 26,27 When applied to three-armed star polymers, twentanglement must be extended in

two ways. A minor change is that the angular potential centered on the branching point reaches its minima at θ = ±120◦ , rather than at θ = 180◦ , as well as being stier than the bending potential centered on a blob in a linear chain. A major change is required to enable the slip of an entanglement point over a branching point, as this move now allows for more complex nal states than the slip of an entanglement point over a blob in a linear chain. In our previous paper 28 we presented and tested a pragmatic solution to this complication according to the following description. When an entanglement slips over a branch point, it has three possible states to end in: either the entanglement is annihilated, or it nds itself on one of the two remaining arms of the star over whose branch point it slips, or the slipping arm gets entangled with both of these arms. Usually the slipping arm belongs to another star than the one over whose branch point it slips. Other, complicated situations are possible, but need not be discussed separately, since the proposed solution applies to all cases. Our algorithm is designed such that in the rst two cases the exact nal state is obtained, including that it nds the correct arm in case two. In case three, when the sliding chain gets entangled with both remaining arms, we randomly choose one of the two to entangle with and ignore the other entanglement. 7



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The simulations were performed in cubic boxes with periodic boundary conditions, with the densities of all melts xed at ρ = 0.761 g/cm3 . The number of polymers in the box varied from 100 for the shortest polymer, B9 , to 80 for the longest chains, B19 and B(B6 )3 . Previous studies on linear polymers established an entanglement length of about 6 beads. A friction coecient ξ/M = 8 ps−1 was used for all polymers, with M being the mass of 20 CH2 groups. The thermostat implicit in Eq. (1) maintained a temperature of 450K. The timestep was 0.4 ps. Each melt was thoroughly equilibrated, followed by a production run covering about 12 µs.

2.3 Analysis methods The dynamics of polymers in the melt will be quantied by the diusion coecient, as extracted from the mean square displacement

D(t) =

⟩ 1 ⟨ cm 2 |Ri (t) − Rcm , i (0)| 6t

(6)

th with Rcm i (t) the center of mass position of the i polymer at time t, and the pointy brackets

denoting an average over all reference times and molecules. Stress relaxation moduli are calculated from equilibrium simulations by means of the auto-correlation function

G(t) =

V ∑ ⟨σαβ (t)σαβ (0)⟩, 6kB T αβ

(7)

where the sum runs over the six o-diagonal elements and the stress tensor is given by

σαβ = −

1 ∑ (Yi,α − Yj,α ) fij,β . V

(8)

⟨i,j⟩

Here ⟨i, j⟩ indicates that the sum runs through all pairs, with each pair occurring only once.

Yi,α denotes the α-component of the position Yi of the ith entity either a blob or an entanglement point in the box of volume V , fij,β is the β -component of the force exerted 8


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1

10

G(t) / [MPa]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

10

-1

10

-2

10

3

10

4

6

5

10

10

10

7

10

Time / [ps]

Figure 2: Stress relaxation modulus for B(B6 )3 on a double logarithmic scale. The markers present the raw data, red dots for positive values of G(t) and blue circles for negative values. ¯ The solid line is the block average G(t) calculated using Eq. (9). on this entity by entity j . The force on entanglements is calculated exactly as for beads. For example, if Yi is the position vector of an entanglement, it will appear in two dierent path-lengths Lk,k+1 , so for each of the two possible values of k , Yi = Xk,k+1 for some mk . mk k,k+1 The force fi,β on this entanglement is then −∂ϕbond /∂Lk,k+1 · ∂Lk,k+1 /∂Xm . For each k ,β

value of k , fi,β picks up two pair contributions. 27 A typical example of a calculated stress auto-correlation function is shown in Fig. 2. The plotted markers are randomly selected from the calculated G(t), showing all data points at low times but reaching a constant density for large times, with red points representing positive values and blue points for negative values. The distribution of these points gives a clear impression of the relatively low noise level obtainable by the

twentanglement method and of G(t) essentially having decayed

to zero for times exceeding a microsecond. The drawn line was obtained by block-averaging the raw data:

¯ = 1 G(t) 2ln∆

∫

lnt+ln∆

G(τ )dlnτ.

(9)

lnt−ln∆

This amounts to averaging with measure proportional to dlnτ = dτ /τ . Having established the stress relaxation modulus, the zero-shear viscosity is calculated 9



by the Green-Kubo integral

∫

∞

Page 10 of 28

G(t)dt,

(10)

sin(ωt)G(t)dt

(11)

cos(ωt)G(t)dt,

(12)

η0 = 0

while the Fourier transforms ′

∫

∞

G (ω) = ω ∫

0

′′

G (ω) = ω

∞

0

yield the storage and loss modulus, respectively. In the calculation of these integrals, the raw data were used up to the rst appearances of negative values, while for larger times G(t)

¯ . was approximated by a single decaying exponential tted to the block averaged curve G(t) We note that this smoothed tail contributes only ∼ 5% to the zero-shear viscosity, rendering the nal results insensitive to the details of the t.

3

Results and Discussion

It is well known that the properties of a melt are strongly aected by the architectures of the constituent polymers. 17,3032 Here, we study symmetric three-armed stars with arms approaching the entanglement length, up to B(B6 )3 . In this section the dynamical properties will be compared to those of matching linear polymers; for a B(Bn )3 star, these are the linear polymer of the same mass B3n+1 and the linear polymer with the same maximum end-to-end span, B2n+1 .

3.1 Diusion To make the comparison with the linear polymers more explicit, the three curves are redrawn in separate subplots of Fig. 3 along with the D(t) of the matching linear polymers. It is readily observed that, of the three polymers per sub-gure, the linear polymer of matching end-to-end span has the largest diusion coecient. For small times, the diusion coecients 10


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of the stars closely resemble those of the linear polymers of equal mass. The smallest star, even matches the D(t) of the equal mass linear polymer over the entire time range. With increasing arm length, however, the eect of the topological dierence becomes apparent as the stars converge to smaller terminal diusion coecients than the matching equal mass linear polymers. Notice however, that the curves for the equal mass linear polymers have not fully converged yet. From experience with smaller linear polymers we know that their diusivity in general rst develops a minimum at some late time, while the long time values need much longer runs to settle at this minimum value or only slightly below that. We must consider the diusion coecients of the longer linear polymers therefore as slightly overestimations of the real values. To indicate details of diusion property, the mean square displacement of blob and center of mass for B(B5 )3 and B(B7 )3 are presented in Supporting Information.

3.2 Stress relaxation moduli To facilitate the comparison, the stress relaxation functions of the three stars are plotted in the three subgures to Fig. 4 together with the curves of their matching linear polymers. For the smallest star, B(B4 )3 , the G(t) closely follows that of the linear polymer of equal maximum span, B9 . With increasing arm length, the tail of G(t) decays slower for the star than for the linear polymer of equal span, and creeps toward the G(t) of the polymer of equal mass. This slackening of the relaxation process at late times is attributed to the architecture of the star. It is interesting to notice that these ndings seem to be conrmed by the experimental results of Lohse et. al. 33 (see their Fig. 9), although their data do not extend to small enough molecular masses to be denitive. The storage and loss moduli of the star polymers are presented in Fig. 5. At the low frequency side, the G′ ∝ ω 2 and G′′ ∝ ω tails are seen to shift to lower frequencies with increasing molecular mass, reecting an increase of the zero-shear viscosity. 34 At the high frequencies, the storage and loss moduli of the three stars nearly coalesce to straight lines 11



B(B4)3 B9 B13

10

-5

B(B5)3 B11 B16

2

D / [nm /ps]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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10

-5

B(B6)3 B13 B19

10

-5

10

3

10

4

10

5

10

6

Time / [ps] Figure 3: Time-dependent diusion coecients for three melts of three-armed star polymers (black solid lines), compared against the diusion of linear polymers of equal mass (green dashed lines) and of equal end-to-end span (red solid lines). in the double logarithmic plot, with power laws tted to the individual curves yielding exponents in the 0.35 to 0.55 range. The storage and loss moduli of the star polymers are being compared against the moduli of the linear polymer of equal mass in Fig. 6. For every melt, the frequencies have been scaled by the frequency at the cross-over point, ωc ∼

10−4 ps−1 , and the moduli have been scaled by the modulus at the cross-over point, G′ (ωc ) = G′′ (ωc ) ∼ 1 MPa. The plot shows that the rescaled moduli of the smallest star closely adhere in the two decades following the cross-over point to those of the linear polymer of equal mass, closer than to those of the linear polymer of equal end-to-end span, suggesting that the polymer topology barely aects the dynamics of these short-armed stars in this frequency range. With increasing arm length, the moduli of the stars are seen to deviate

12


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B(B4)3 B9

10

/ [MPa]

10

G(t)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


B13

0

-1

B(B5)3 B11

10

10

B16

0

-1

B(B6)3 B13

10

10

B19

0

-1

10

3

10

4

10

5

10

6

Time / [ps] Figure 4: Stress relaxation moduli for three melts of three-armed star polymers (solid black lines), compared agains the moduli of linear polymers of equal mass (red solid lines) and of equal end-to-end span (green dashed lines). from the moduli of the linear polymers in the two decades following the cross-over. The main feature to be noted is the `wedge' between the storage and loss moduli, i.e. the gradual growth with increasing frequency of the dierence between the solid and dashed curves. This wedge is observed for the melts of linear polymers, and opens up with approximately the same angle for all three melts in Fig. 6. An identical wedge is seen for the star with the shortest arms, B(B4 )3 . For the stars with one additional bead per arm, however, the angle has reduced considerably. The wedge has essentially closed for the largest star, B(B6 )3 , where the storage and loss moduli are seen to nearly coalesce for the rst decade and a half following the cross-over. (Note that for this melt the curves were rescaled relative to the cross-over point at the low frequency end of the overlap region). Although the polymers are evidently

13



1

G′ / [MPa]

10

B(B4)3 B(B5)3 B(B6)3

0

10

-1

10

-2

10

-7

10

-6

10

-4

-5

10

10

-3

10

-1

ω / [ps ] 1

10

G″ / [MPa]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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B(B4)3 B(B5)3 B(B6)3

0

10

-1

10

-2

10

-7

10

-6

10

-4

-5

10

10

-3

10

-1

ω / [ps ]

Figure 5: Storage (top) and loss (bottom) moduli for melts of three-armed star polymers. too short to attain the large numbers of entanglements required for reptation dynamics, the simulations nevertheless agree qualitatively with experiments on long polymers in yielding a wider wedge for the linear polymers than for the star polymers. 3539 In the experimental studies the storage and loss moduli of star polymers continue to rise after the cross-over point, while for linear polymers G′ crosses the local maximum of G′′ . The steeper slopes in Fig. 6 - and the absence of a local maximum in G′′ for the linear polymers - reect the limited length of the simulated polymers. The viscosities calculated from the stress relaxation curves in Fig. 4 are plotted in Fig. 7 against the molecular mass M . Also included is the small B(B3 )3 star of our previous study. 28 Fitting the viscosities of these ve melts with a power law yields the red line in Fig. 7, with a power of 1.6 ± 0.2. Whereas the melts of linear polymers follow a power law with a markedly 14


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10

G′/G′c and G"/G"c

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3

10 B(B4)3 B13

B(B5)3 B16

B(B6)3 B19

3

1

0.3 -1 10

1

10

0

10

1

10

2

10

0

ω / ωc

10

1

10

-1

10

0

10

1

0.3 2 10

Figure 6: Storage (solid lines) and loss moduli (dashed lines) for melts of three-armed star polymers (black) and linear polymers of equal mass (red). For comparison purposes, the frequencies and moduli have been scaled by the frequency and moduli at their cross-over points, ωc , G′c and G′′c respectively, of the respective melts. higher power, the star polymers reveal a systematic deviation from a power law in Fig. 7. The viscosities are better tted by an exponential, η ∝ eνM with ν = (4.2±0.1)×10−4 mol/g, as shown by the green line in Fig. 7. Assuming an entanglement mass Me of six beads 26 ′

yields η ∝ eν (Ma /Me ) with Ma the mass per arm; the coecient ν ′ ≈ 2.12 overestimates the experimental value of 0.6. 33 Experiments on stars with many and/or long arms point at a power-exponential scaling law, 35 but ts of the simulation viscosities with a function of this type converge to exponents close to zero. In order to better appreciate the validity of our claim of exponential behavior, we have included error bars in Fig. 7. To this end we have divided each run into blocks that were long enough to calculate viscosities, and next calculated mean values and errors in the mean values. For the larger stats the errors are appreciably. Nevertheless the plot shows that there is proof for exponential increase of the viscosity with increasing mass of the stars.

3.2.1

Rouse modes

To get a better appreciation of the role of entanglements in the stress relaxation modulus of the star polymers, we have calculated the G(t) as expected for star polymers obeying Rouse-like dynamics with relaxation times extracted from the simulations. In Rouse theory, 15


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η0 / [Pa s]


Page 16 of 28

this work power law exponential

0.1

0.03 3

4

3×10

1×10

Molecular Mass / [g/mol]

Figure 7: Zero shear viscosities with error bars for ve melts of three-armed star polymers, B(B3 )3 through B(B7 )3 , plotted against the molecular mass. An exponential function (green line) ts the data better than a power law (red line). the blob coordinates Ri are mapped onto Rouse coordinates Xk by a linear transformation that decouples the motions of the Rouse coordinates.The relaxations of these modes are well described by stretched exponentials,

Ck (t) =

⟨Xk (t) · Xk (0)⟩ ∗ β = e−(t/τk ) k , 2 ⟨Xk ⟩

(13)

with τk∗ the relaxation time (deviant from the conventional Rouse time obtained for βk = 1) and βk the stretching exponent. For a collection of Nc uncorrelated polymers, each on average adhering to the above relaxation process, it is shown in Supporting Information that N −1 Nc kB T ∑ −2(t/τk∗ )βk G(t) ≈ e , V k=1

(14)

with N the number of beads per chain. Hence, the N − 1 relaxation curves Ck (t) were calculated from the blob coordinates stored during the simulations of the stars, and both τk∗ and

βk were extracted by tting the initial decay till Ck (t) ≈ 10−1.5 by a stretched exponential. The resulting relaxation times show the same degeneracy pattern as the theoretical Rouse times derived in Supporting Information; the stretching exponents mostly lie between 0.7

16


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B(B4)3 Rouse

10

10

G(t) / [MPa]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

-1

B(B5)3 Rouse

10

10

0

-1

B(B6)3 Rouse

10

10

0

-1

10

3

10

4

10

5

10

6

Time / [ps] Figure 8: Stress relation moduli of the simulated star polymers (black) compared against the theoretical predicted curves for Rouse-like star polymers with the same transient relaxation dynamics. and 1. Inserting the tted values into Eq. (14) from the blob coordinates stored during the simulations of the stars, and both τk∗ and βk were extracted by tting the initial decay till

Ck (t) ≈ 10−1.5 by a stretched exponential. The resulting relaxation times show the same degeneracy pattern as the theoretical Rouse times derived in Supporting Information; the stretching exponents mostly lie between 0.7 and 1. Inserting the tted values into Eq. (14) produces the red curves in Fig. 8, plotted alongside the actual stress relaxation functions in black. The pairs of curves show a satisfactory agreement for short times, certainly in view of the assumptions made in deriving Eq. (14), while at large times the slowing down of the relaxation dynamics due to the entanglements becomes visible again. The storage and loss moduli following the Rouse model with stretched exponentials are

17



B(B4)3

10

10

G′ and G″ / [MPa]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Rouse

1

0

B(B5)3

10

10

Rouse

1

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B(B6)3

10

10

Rouse

1

0

10

-5

10

ω

-4

10

-3

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-2

-1

/ [ps ]

Figure 9: Storage modulus G′ (solid lines) and loss modulus G′′ (dashed lines) of three-armed star polymer (black) compared with stretch Rouse modulus (red). shown in Fig. 9. In agreement with the direct analysis of the star polymer simulations, the wedge between the storage and loss moduli decreases with increasing arm length. The angle of the wedge, however, is considerably wider than in the direct analysis.

4

Conclusions

The diusive and rheological properties of linear and three-armed polyethylene polymers of various lengths have been studied by computer simulations using the highly coarse-grained TWENTANGLEMENT technique. Comparing the simulated symmetric star polymers with linear polymers, matching in mass or end-to-end span, highlights the impact of the architecture on the dynamical properties of the melt: Stars are seen to diuse slower than both 18


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matching linear polymers, while the stress relaxation curve lies between those of the matching linear polymers. The viscosity of the star polymers show exponential scaling instead of power law in linear polymers. A qualitative agreement with experiments is observed for the wedge between the storage and loss moduli, with linear polymers yielding a larger wedge than star polymers. The quantitative validation of the star simulations has to wait till experimental data on three-armed stars with short arms become available.

Acknowledgement

The authors acknowledge nancial support from the National Natural Science Foundation of China (Grant: 21704010), the Natural Science Foundation of Liaoning Province (Grant: 2015010217-301), and the European Union's Seventh Frame-work Programme (FP7) through the Marie Curie initial training network DYNamics of Architecturally Complex Polymers (Grant: 214627).

A

Remarks about coarse graining

From a physical point of view coarse graining consists in eliminating details from the molecular description, while retaining those elements that cause the phenomena that one wants to address. It is obvious that a procedure like this will always lead to the loss of phenomena that are mainly dened in terms of the eliminated degrees of freedom. Still one wants to create a model that describes the remaining physics as well as possible. In the present application to star polymers coarse graining amounts to using a set of coordinates {R1 , q1 , R2 , q2 , ..Rn , qn }, where Rn stands for the position of the branch point of the n'th star and qn for its internal degrees of freedom. The mathematical procedure consists of averaging over all possible initial values of the internal degrees of freedom, with the correct Boltzmann weight of course. When doing so, we nd the equation of motion for

19



Page 20 of 28

the remaining degrees of freedom n ∫ t n ∑ ∑ d2 Ri dRj M 2 =− ζ (Rj − Ri ; t′ ) ′ (t − t′ )dt′ − ∇i Φ + Fijr (t) dt dt j=1 0 j=1

(15)

< Fijr (t)Fijr (0) >= kB T ζ(Rj − Ri ; t).

(16)

with

Here M is the mass of the bead, and Φ(Rn ) = −kB T lnP (Rn ) as in the main text. Suppose we know Φ(Rn ) and ζ (Rj − Ri ; t), and simulate these equations we will have all properties which can be dened in terms of Rn and these functions correct. At this point one has to make approximations. First, one needs a tractable expression for the potential of mean force

Φ. In the paper we have taken a rather simple expression, which ignores many body interactions between the simulated beads. The parameters that we have used are the ones from a previous study by Padding and Briels. 26 They were obtained by simulating a full atomistic system of C120 H242 chains in the melt state and calculating the probability distribution of the centers of mass of 'blobs' of 20 consecutive CH2 groups, and next taking minus kB T times its logarithm. The blobs in this description constitute the beads in our model. The next approximation is that we assume that the time range over which ζ (Rj − Ri ; t) diers substantially from zero is short compared to the time needed before the velocity

dRj /dt has changed substantially. This allows us to put the velocity of the j'th particle in the equation above in front of the integral. This approximation is what was referred to in the main text as assuming complete separation of time scales. Next, we perform the time ∫∞ integral and call the result 0 ζ (Rj − Ri ; t′ ) dt′ = ξij . Apart from some details we have now arrived at the DPD level of description. Since we apply our method to a melt, we do not expect hydrodynamic interactions to be important. On the atomistic scale, the local motion of one chain element is not noticed by another chain element at large distance. Exchanged momentum is immediately dissipated along the backbones of the involved chains. Therefore, in the sum in Eq. 15 no j contribute for which |Rj − Ri | is large. The main cause of the 20


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friction term are the uctuations of the internal degrees of freedom and their instantaneous interactions with those of neighboring blobs, as opposed to the average interactions which are taken care of in the potential of mean force. We therefore expect that friction forces can eectively be described by single particle frictions, which removes the sum in the right hand side above altogether, i.e. out of all ξij we only retain ξii . In this paper we use ξii /M = 8 ps−1 . This result was obtained by Padding and Briels in the paper mentioned above by tting the mean square displacements of the coarse model to those of the atomistic model, and later by integrating the random force correlations from atomistic simulations. 27 Our nal approximation is to assume that the blobs hardly displace during time lapses large enough for the velocities of the blobs to equilibrate, i.e. to completely forget about their velocities at the beginning of the time step. Performing the standard procedure 40 for this approximation we arrive at equation 1. Broadly speaking, leaving out some subtleties, we set the left hand side of Eq. 15 equal to zero. The friction in Eq. 1 is dened as ξ = ζii M . The procedure described above, in principle conserves all thermodynamic properties of the system, since integration of the Boltzmann factor obtained with the potential of mean force gives the correct free energy. This only holds true, of course, provided an exact representation of the potential of mean force has been used as input for the coarse simulation. Usually, as in the present paper, the potential of mean force is only represented in an approximate way, so no claim as to the thermodynamic properties is made. Long time dynamic properties are conserved, provided the correct friction factors are being used. These were obtained by Padding and Briels. 27 The mean square displacement of the blobs is conserved and thereby all mean squared displacements dened by the blobs. Information atomistic mean square displacements at short times is lost. At long times these hardly dier from those of the corresponding blobs. Information about the shear relaxation modulus at short times is lost. In a fully atomistic model the fast dynamics of the degrees of freedom that have been eliminated in the coarse model will cause large values of the shear relaxation modulus, that decay over very short times, much shorter than the ones studied in the present case. 21



Page 22 of 28

Their inuence on the slow decay of G(t) is captured by the friction factors and the potential of mean force. Since viscosities are given by the integral of G(t), their values will be slightly underestimated, only slightly since the integrals are dominated by the extremely long times involved in the slow decays (notice the logarithmic time scale in G(t)) plots. Taken literally, the relation of our model to the underlying atomistic model is obtained by replacing the beads by 20 CH2 groups. In rheology it is well known that in the entangled regime systems with similar values of mass divided by entanglement mass behave similar in their scalings. Supporting Information.

1.Stress relaxation modulus

2.Mean square dis-

placement of blob and of center of mass

References

(1) Pearson, D. S.; Fetters, L. J.; Graessley, W. W. Viscosity and Self-Diusion Coecient of Hydrogenated Polybutadiene.

Macromolecules 1994, 27, 711719.

(2) Bartels, C. R.; Jr., B. C.; Fetters, L. J.; Graessley, W. W. Self-diusion in Branched Polymer Melts.


(3) Pearson, D. S.; Helfand, E. Viscoelastic Properties of Star-shaped Polymers.

Macro-

molecules 1984, 17, 888895. (4) Doi, M., Edwards, S. F., Eds.

The Theory of Polymers Dynamics ;

Oxford Science

Publications: Oxford, U. K., 1986. (5) Shull, K. R.; Kramer, E. J.; Fetters, L. J. Eect of Number of Arms on Diusion of Star Polymers.

Nature 1990, 345, 790791.

(6) Milchev, A.; Müller, M.; Klushin, L. Arm Retraction Dynamics and Bistability of a Three-arm Star Polymer in a Nanopore.


22


Page 23 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(7) Peters, L. B.; Salerno, M. K.; Agrawal, A.; Perahia, D.; GrestS, G. S. Coarse-Grained Modeling of Polyethylene Melts: Eect on Dynamics. J.

Chem. Theory Comput. 2017,

13, 28902896. (8) Ghelichi, M.; Eikerling, M. H. Conformational Properties of Comb-Like Polyelectrolytes: a Coarse-Grained MD Study.

J. Phys. Chem. B 2016, 20, 28592867.

(9) Maurel, G.; Schnell, B.; Goujon, F.; Couty, M.; Malfreyt, P. Multiscale Modeling Approach Toward the Prediction of Viscoelastic Properties of Polymers.

J. Chem. Theory

Comput. 2012, 8, 45704579. (10) Panizon, E.; Bochicchio, D.; Monticelli, L.; Rossi, G. MARTINI Coarse-Grained Models of Polyethylene and Polypropylene.

J. Phys. Chem. B 2015, 119, 82098216.

(11) Zheng, F.; Goujon, F.; Mendonça, A. C.; Malfreyt, P.; Tildesley, D. J. Structure and Rheology of Star Polymers in Conned Geometries: a Mesoscopic Simulation Study.

Soft Matter 2015, 11, 85908598. (12) Ramirez-Hernandez, A.; Müller, M.; de Pablo, J. J. Theoretically Informed Entangled Polymer Simulations: Linear and Non-linear Rheology of Melts.

Soft Matter

2013,

9,

20302036. (13) Snijkers, F.; Pasquino, R.; Olmsted, P.; Vlassopoulos, D. Perspectives on The Viscoelasticity and Flow Behavior of Entangled Linear and Branched Polymers.

J. Phys.

Condens. Matter 2015, 27, 473002. (14) Harmandaris, V. A.; Mavrantzas, V. G.; Theodorou, D. N. Atomistic Molecular Dynamics Simulation of Polydisperse Linear Polyethylene Melts.

Macromolecules

1998,

31, 79347943. (15) Tjornhammar, R.; Edholm, O. Reparameterized United Atom Model for Molecular

23



Page 24 of 28

Dynamics Simulations of Gel and Fluid Phosphatidylcholine Bilayers. J.

Chem. Theory

Comput. 2014, 10, 57065715. (16) Swope, W. C.; Carr, A. C.; Parker, A. J.; Sly, J.; Miller, R. D.; Rice, J. E. Molecular Dynamics Simulations of Star Polymeric Molecules with Diblock Arms, a Comparative Study.

J. Chem. Theory Comput. 2012, 8, 37333749.

(17) Jabbarzadeh, A.; Atkinson, J. D.; Tanner, R. I. Eect of Molecular Shape on Rheological Properties in Molecular Dynamics Simulation of Star, H, Comb, and Linear Polymer Melts.


(18) Kremer, K.; Grest, G. S. Dynamics of Entangled Linear Polymer Melts: a Moleculardynamics Simulation.

J. Chem. Phys. 1990, 92, 50575086.

(19) Grest, G. S.; Kremer, K.; Milner, S. T.; Witten, T. A. Relaxation of Self-entangled Many-arm Star Polymers.


(20) Grest, G. S.; Kremer, K.; Witten, T. A. Structure of Many Arm Star Polymers: a Molecular Dynamics Simulation.


(21) Pérez-Aparicio, R.; Colmenero, J.; Alvarez, F.; Padding, J. T.; Briels, W. J. Chain Dynamics of Poly(ethylene-alt-propylene) Melts by Means of Coarse-grained Simulations Based on Atomistic Molecular Dynamics.

J. Chem. Phys. 2010, 132, 024904.

(22) Lee, M. T.; Mao, R. F.; Vishnyakov, A.; Neimark, A. V. Parametrization of Chain Molecules in Dissipative Particle Dynamics.

J. Phys. Chem. B 2016, 120, 49804991.

(23) Likhtman, A. E. Single-Chain Slip-Link Model of Entangled Polymers: Simultaneous Description of Neutron Spin-Echo, Rheology, and Diusion.

Macromolecules 2005, 38,

61286139. (24) Schieber, J. D.; Andreev, M. Entangled Polymer Dynamics in Equilibrium and Flow Modeled Through Slip Links.

Annu. Rev. Chem. Biomol. Eng. 2014, 5, 367381. 24


Page 25 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(25) Steenbakkers, R. J.; Schieber, J. D. Derivation of Free Energy Expressions for Tube Models from Coarse-grained Slip-link Models.

J. Chem. Phys. 2012, 137, 034901.

(26) Padding, J. T.; Briels, W. J. Uncrossability Constraints in Mesoscopic Polymer Melt Simulations: Non-Rouse Behavior of C120 H242 .

J. Chem. Phys. 2001, 115, 28462859.

(27) Padding, J. T.; Briels, W. J. Time and Length Scales of Polymer Melts Studied by Coarse-grained Molecular Dynamics Simulations.

J. Chem. Phys. 2002, 117, 925943.

(28) Liu, L.; Padding, J. T.; den Otter, W. K.; Briels, W. J. Coarse-grained Simulations of Moderately Entangled Star Polyethylene Melts.

J. Chem. Phys. 2013, 138, 244912.

(29) Reith, D.; Pütz, M.; Müller-Plathe, F. Deriving Eective Mesoscale Potentials from Atomistic Simulations.

J. Comput. Chem. 2003, 24, 16241636.

(30) Pasquino, R.; Vasilakopoulos, T. C.; Jeong, Y. C.; Lee, H.; Rogers, S.; Sakellariou, G.; Hadjichristidis, N.; Richter, D.; Rubinstein, M.; Vlassopoulos, D. Viscosity of Ring Polymer Melts.

ACS Macro Lett. 2013, 2, 874878.

(31) Xu, X.; Chen, J.; An, L. Simulation Studies on Architecture Dependence of Unentangled Polymer Melts.

J. Chem. Phys. 2015, 142, 074903.

(32) Pyckhout-Hintzen, W.; Allgaier, J.; Richter, D. Recent Developments in Polymer Dynamics Investigations of Architecturally Complex Systems.

Eur. Polym. J. 2011, 47,

474485. (33) Lohse, D. J.; Milner, S. T.; Fetters, L. J.; Xenidou, M.; Hadjichristidis, N.; Mendelson, R. A.; García-Franco, C. A.; Lyon, M. K. Well-Dened, Model Long Chain Branched Polyethylene. 2. Melt Rheological Behavior. Macromolecules

2002,

35, 3066

3075. (34) Onogi, S.; Masuda, T.; Kitagawa, K. Rheological Properties of Anionic Polystyrenes. I.

25



Page 26 of 28

Dynamic Viscoelasticity of Narrow-Distributution Polystyrenes. Macromolecules

1969,

3, 109116. (35) Fetters, L. J.; Kiss, A. D.; Pearson, D. S.; Quack, G. F.; Vitus, F. J. Rheological Behavior of Star-Shaped Polymers.


(36) Graessley, W. W.; Roovers, J. Melt Rheology of Four-Arm and Six-Arm Star Polystyrenes.


(37) Kapnistos, M.; Vlassopoulos, D. Rheological Master Curves Of Crystallizing Polymer Mixtures.

Appl. Rheol. 2006, 16, 132135.

(38) Milner, S. T.; McLeish, T. C. B. Reptation and Contour-Length Fluctuations in Melts of Linear Polymers.

Phys. Rev. Lett. 1998, 81, 725728.

(39) Schausberger, A.; Schindlauer, G.; Janeschitz-Kriegl, H. Linear Elastico-viscous Properties of Molten Standard Polystyrenes. I. Presentation of Complex Moduli; Role of Short Range Structural Parameters.

Rheol./ Acta 1985, 24, 220227.

(40) Briels, W. J.; Sutmann, G.; Grotendorst, J.; Gompper, G.; Marx, D. Responsive Particle Dynamics for Modeling Solvents on the Mesoscopic Scale.

Computational Trends

in Solvation & Transport in Liquids: IAS series 28, Springer 2015,

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η0 / [Pa s]

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Coarse-Grained Simulations of Three-Armed Star Polymer Melts and

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