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DOI: 10.1039/C7SM00364A
(Paper)
Soft Matter, 2017, Advance Article

Jurij Sablić^{a},
Matej Praprotnik*^{ab} and
Rafael Delgado-Buscalioni*^{cd}
^{a}Department of Molecular Modeling, National Institute of Chemistry, Hajdrihova 19, SI-1001 Ljubljana, Slovenia. E-mail: praprot@cmm.ki.si
^{b}Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
^{c}Departamento Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, Campus de Cantoblanco, E-28049 Madrid, Spain. E-mail: rafael.delgado@uam.es
^{d}Condensed Matter Physics Center, IFIMAC, Campus de Cantoblanco, E-28049 Madrid, Spain

Received
21st February 2017
, Accepted 26th May 2017

First published on 30th May 2017

This work analyses the rotation of star polymers under shear flow, in melts, and in good solvent dilute solution. The latter is modeled by single molecule Brownian hydrodynamics, while melts are modeled using non-equilibrium molecular dynamics in closed (periodic) boxes and in open boundaries. A Dissipative Particle Dynamics (DPD) thermostat introduces pairwise monomer friction in melts at will, in directions normal and tangent to the monomer–monomer vectors. Although tangential friction is seldom modeled, we show that it is essential to control hydrodynamic effects in melts. We analyze the different sources of molecular angular momentum in solution and melts and distinguish three dynamic regimes as the shear rate is increased. These dynamic regimes are related with the disruption of the different relaxation mechanisms of the star in equilibrium. Although strong differences are found between harmonic springs and finitely extensible bonds, above a critical shear rate the star molecule has a “breathing” mode with successive elongations and contractions in the flow direction with frequency Ω. The force balance in the flow direction unveils a relation between Ω and the orientation angle. Using literature results for the tumbling of rings and linear chains, either in melt or in solution, we show that the relation is general. A different “tank-treading” dynamics determines the rotation of monomers around the center of mass of the molecule. We show that the tank-treading frequency does not saturate but keeps increasing with . This is at odds with previous studies which erroneously calculated the molecular angular frequency, used as a proxy for tank-treading.

Star polymers have a wide range of technical applications such as medical and pharmaceutical applications, wetting, lubrication, coating, binders in toners for copying machines, oil industry, etc.^{11} The synthesis of monodisperse (in number of arms and degree of polymerisation) star molecules, such as polyisoprene and polybutadiene, by anionic polymerisation enabled the experimental study of static as well as collective and single-molecule dynamic properties of these substances using various microscopic techniques.^{2,5,11}

However, the number of theoretical studies and simulations of star molecules under shear flow is scarce. Gompper’s group has published several papers on the dynamics of star molecules in a sheared solution using the multiparticle collision dynamics (MCD). Ripoll et al. studied individual star molecules with f ∈ [5,50] and m = 30 under shear flow^{3} and then Singh et al. conducted a study of dilute to semidilute solution ranging from c = 0.19c* to c ∼ 2.5c*^{12,13} (here, c* is the overlap concentration). More recently, Xu and Chen^{7} have presented a numerical study of star polymers with a range of functionality f ∈ [3,60] and m = 20 in melts under shear, and Yamamoto et al. have revisited the problem of sheared star polymers in solution.^{14}

At large enough shear rates, any measure of the polymer extension in the flow plane reveals the onset of alternating extensions and contractions of the molecule, typically analyzed from the cross correlation of the molecular elongations in the flow and gradient directions C_{XY}(t).^{15,16} This behavior is quite general and it has also been studied in linear chains (either free^{17,18} or tethered^{16}) and in polymer rings under shear.^{15,19} But how these signals are interpreted as motion depends on the molecule’s architecture. Clearly, in shear flow, a rigid ellipsoid tumbles (turns around) with a precise frequency.^{20} A linear flexible chain also tumbles because one can assign a “head” and a “tail” and precisely determine a tumbling event (a turn around). However, this is not possible in star polymers and the determination of one tumbling event becomes somewhat arbitrary. A possible criterion is a zero crossing of the molecule’s orientation θ evaluated from its gyration tensor, as it was done for ring polymers^{15} where tumbling and tank-treading seem to coexist.^{15} Although a similar study has not been performed in the case of star molecules, for this architecture a discussion on the transition from tumbling to tank-treading dominated dynamics^{7,13} seems somewhat artificial. In any case, star molecules do not (or seldom) “tumble” but rather keep their flow orientation angle more or less stable, while the arm monomers rotate around the molecule’s CoM following a sort of “tank-treading” motion. The term tank-treading was invented to describe the dynamics of soft visco-elastic objects (e.g. red-blood cells) which rotate with frequency ω_{R} ∝ around their center of mass (CoM) but without tumbling, i.e. without essentially altering their orientation against the flow direction.^{21–23} Later, this term was extrapolated to star polymers.^{3} To determine the rotation of the monomers around the CoM previous works have used the molecular angular momentum L as a proxy for the angular frequency ω.^{24} This route is based on the rigid-body relation L = Jω_{L} and involves the inertia tensor J. The instantaneous rotation frequency would be strictly that of a rigid-body having the instantaneous configuration of the polymer, then one averages over configurations to obtain 〈ω〉.^{24} This analogy still makes sense for a linear chain^{24} which, after all, tumbles like a rigid body does (but certainly with different frequency and dynamics). In the case of star molecules, Ripoll et al.^{3} and subsequent works^{12,13} also measured ω_{L} and report that, in dilute and semidilute solutions, it reaches a plateau (ω_{L} → const.) at large shear rates. In melts, Xu et al.^{7} recently reported ω_{L} ∼ ^{α} where α ∈ [0.5 − 0.75] increases with the functionality f ∈ [3,60]. These works assign ω_{L} to the tank-treading frequency ω_{R} or, in other words, assume that ω_{R} = ω_{L}. However, if star molecules do not tumble (but rather keep their orientation stable), is it still judicious to interpret ω_{L} as the frequency of the monomer's rotation around the molecule’s CoM? Moreover, the tank-treading frequency in vesicles scales linearly with the shear rate^{21} instead of saturating to a constant shear-independent value. We believe these questions should be revised. Chen et al.^{15,19} also used MCD simulations in an attempt to discern tank-treading in ring polymers. They conclude that ring chains mostly tumble at large shear rates but have a significant probability to tank-tread (P_{tt} ≃ 0.2) with a frequency ω_{R} ∼ ^{0.6}. They obtain ω_{R} from the time correlation of the angle between the CoM-monomer direction and the direction of the chain tilt in flow. This measure captures the individual monomer rotation more precisely. Interestingly, Chen et al. found that tumbling and tank-treading scale similarly with .

Aside from this tank-treading or monomer rotation about the molecule’s CoM, another collective motion is observed in star molecules: the overall shape of the star fluctuates in time indicating alternating extensions and contractions with concomitant fluctuations in the instantaneous tilt angle. Following the vesicle dynamics analogy, these would correspond to “breathing” and “swinging” motions,^{22,23} rather than tumbling. In the case of stars, we have seen that breathing and swinging (tilt fluctuations) have the same origin. We will use the term “breathing” to indicate global molecular expansion/contraction cycles, which are different from tank-treading dynamics. The relation between the breathing frequency (or tumbling in ring and linear chains) Ω and the molecular architecture is also a question that has been posed in the literature.^{15} Do ring, linear, and star polymers fluctuate in extension in essentially different ways due to their different form? We present a general argument based on the average force balance in the flow direction, showing that the compression/expansion frequency is just determined by the ratio of average molecular extensions in the flow and gradient directions. Thus, the physical origin of tumbling in linear chains and rings and that of “breathing” in stars are the same, and the difference is almost a question of nomenclature. The gradient-to-flow ratio of average extensions is closely related to the tilt angle θ, which decreases with and depends on the environment (solution quality), bond type (the harmonic versus the FENE) and hydrodynamics, being also quite different in melts.^{7} Thus, for the expansion/contraction frequency, differences in the environment and bond type are more determinant than the architecture.

Another question that this work treats concerns the role of friction in melts. Comparison between single molecules in solution and melts led us to the conclusion that tangential friction enhances the hydrodynamic character of the melt. We checked the “microscopic” origin of such coincidence and found that tangential friction increases the screening length of momentum spreading in the melt. This observation strengthens the message that friction should be an essential part of any coarse-graining model of polymer melts,^{6,25} and not just a way to extract heat from a non-equilibrium simulation.

We start by presenting the methods used in Section 2 and calibrate the models' relaxation times in equilibrium in Section 2.1. Then, in Section 3, we present the results on the molecular orientation angle θ in flow. Sections 4 and 5 present an analysis of the molecule’s breathing frequency Ω and the “rotation” frequency ω_{L}. Comparison of both frequencies leads to a discussion of the dynamic regimes in Section 6. Although we defer the study of the tank-treading frequency for future work, in Section 6.2, we show that, contrary to the common assumption, ω_{L} does not represent the “tank-treading” frequency. This observation leads to our concluding remarks in Section 7.

Hydrodynamic interactions (HI) between monomers in dilute solution (single chain limit) are introduced implicitly, using a standard Brownian hydrodynamic scheme (overdamped Langevin dynamics) equipped with the Rotne–Prager–Yamakawa mobility matrix.^{26,27} The hydrodynamic radius of each monomer is set to a = 0.5σ and the Fixman’s method is used to approximate the square root of the mobility appearing in the noise term of the overdamped Langevin equation.^{27} The monomer diameter σ and WCA energy ε are the same as in melt simulations. The time integration scheme is explicit, using the Euler scheme with the time step 0.01τ. The viscosity is set to η = 0.25, the temperature T = 4, the monomer bare diffusion coefficient D_{0} = kT/ξ_{m} = 0.703 with ξ_{m} = 6πηa = 5.69. In terms of the reference monomer diffusion time τ = a^{2}/D_{0} = 2.07, the time step is dt = 0.01τ. Some of the simulations of single stars in solution are carried out without hydrodynamic interactions (mobility matrix set to a scalar), so as to clearly observe HI effects by comparison. In solution, HI are long ranged (1/r) and induce an Oseen-like perturbative back-flow which is superimposed onto the mean shear. In particular, under shear, HIs expand the chain in the neutral direction and tend to reduce its extension along the flow direction. We shall see that the latter effect has consequences in its rotation dynamics.

In the melt, hydrodynamics arises from the fully resolved, momentum conserving, molecular dynamics (MD). In melts, hydrodynamic interactions are exponentially screened;^{28} however, we shall see that the back-flow exists and also introduces inter-chain interactions; its shape crucially depends on the monomer–monomer friction forces. In fact, an essential part of a coarse-grained model, like the present one, requires implementation of friction forces between monomers. Here, such friction is introduced in a pair-wise fashion by a DPD thermostat so as to preserve momentum conservation. We consider two pair-wise friction models: one where friction acts only along the vector joining two monomers (normal friction) and another model adding friction also along the perpendicular direction (tangential friction). These generalized DPD thermostats were initially developed by Español^{29} and then used in ref. 6, 30 and 31. Their relevance in coarse-graining modeling was justified from rigorous bottom-up theory by Hijon et al.^{25} The DPD thermostat used in this work is explained in ref. 6 and here we just recall the relevant details. In units of m_{0}/τ_{0} (where m_{0} = 1 is the monomer mass and τ_{0} = σ_{r}(m_{0}/ε_{r})^{1/2} is the reference time), the normal friction coefficient γ_{∥} and the tangential one γ_{⊥} are γ_{∥} = 1.0 and γ_{∥} = 0 for normal friction and γ_{∥} = 1.0 and γ_{⊥} = 1.0 for the tangential friction case. In both models, the friction coefficient is constant within a cut-off distance R_{DPD} (we used 2^{7/6}σ and 1.5 × 2^{1/6}σ) and vanishes beyond (i.e. Heaviside friction kernel).

Melt simulations are performed in a closed periodic environment using the SLLOD dynamics and Lees–Edwards boundary conditions and also in an open (non-periodic) environment using Open Boundary Molecular Dynamics (OBMD).^{32} In the OBMD setup, the simulation box is open in at least one direction. The system can thus exchange mass with the surroundings. Moreover, OBMD also enables the imposition of the external boundary conditions (e.g. constant normal load and shear flow) on the system, without the modification of Newton’s equations of motion. We refer to ref. 6 for a comparison between these two boundary conditions in terms of rheological properties and pressure–density relations. In melt simulations, the box is 390 × 117 × 117σ_{r}^{3} and the f = 12, m = 6 star polymers occupy a volume fraction of Φ = 0.2, corresponding to a molecular concentration above overlapping c = 1.42c*, where c* = (4πR_{g}^{3})^{−1} and the gyration radius R_{g} = 7.65 (recall that the monomer WCA diameter is σ = 2.415). The equations of motion are integrated by the velocity-Verlet algorithm with time step 0.01τ_{0} for small and moderate shear rates, and 0.005τ_{0} for high shear rates.

Table 1 summarizes the models considered in this work.

Melt (M) | Bonds | Boundaries | Friction |
---|---|---|---|

MN-op | Harmonic | Open | γ_{∥} = 1.0, γ_{⊥} = 0.0; R_{DPD} = 1.5σ_{0} |

MN-cl | Harmonic | Closed | γ_{∥} = 1.0, γ_{⊥} = 0.0; R_{DPD} = 1.5σ_{0} |

MT-op | Harmonic | Open | γ_{∥} = γ_{⊥} = 1.0; R_{DPD} = 2σ_{0} |

MT-cl | Harmonic | Closed | γ_{∥} = γ_{⊥} = 1.0, R_{DPD} = 2σ_{0} |

Solution (S) | Bonds | Interactions | f–m |
---|---|---|---|

SH+HI | Harmonic (H) | Hydrodynamics (+HI) | 12–3; 12–6; 12–11 |

SH−HI | Harmonic | No-hydrodynamics (−HI) | 12–6 |

SF+HI | FENE (F) | Hydrodynamics | 12–6 |

Star polymers have several relaxation mechanisms,^{1} which are still under debate.^{5} We consider three main relaxation mechanisms, which determine the rotation of the whole molecule τ_{rot}, the single-arm length decorrelation τ_{arm}, and the arm–arm decorrelation τ_{dis}. The latter time has also been called disentanglement time; however in this work, the arms are short and entanglements are negligible. In fact, τ_{dis} is more properly related to the collective or coordinated motion of the arms, which is known to be enhanced in stars with short arms.^{5} The time correlation of these mechanisms is expressed in eqn (1)–(3). And their characteristic times can be estimated from the integral of the corresponding normalized autocorrelation function (ACF), via :

(1) |

(2) |

(3) |

System | τ_{rot} |
τ_{arm} |
τ_{dis} |
---|---|---|---|

MT, 12–6 | 700 ± 40 | 60 ± 6 | 390 ± 10 |

MN, 12–6 | 59 ± 5 | 3 ± 1 | 33 ± 2 |

SH+HI, 12–3 | 95 ± 10 | 10 ± 1 | 67 ± 6 |

SH+HI, 12–6 | 270 ± 20 | 20 ± 2 | 180 ± 20 |

SH+HI, 12–11 | 700 ± 70 | 55 ± 5 | 490 ± 50 |

SH−HI, 12–6 | 500 ± 30 | 25 ± 2 | 260 ± 20 |

SF+HI, 12–6 | 370 ± 30 | 11 ± 1 | 950 ± 90 |

In the case of SH+HI stars, the rotational diffusion for different arm lengths scales like τ_{rot} ≃ σ^{3}m^{1.65}. This exponent is consistent but a bit smaller than 3ν = 1.74, which would result from the relation τ_{r} = R_{g}^{2}/D_{rot} ∼ R_{g}^{3} ∼ m^{3ν} and D_{rot} ∼ k_{B}T/(ηR_{g}) the rotation diffusion coefficient.

We now define the Weissenberg number for our molecules. Following the standard protocol, we use the longest relaxation time of the molecule (and quote it as τ_{rel}) to define the Weissenberg number as Wi = τ_{rel}. Here, ^{−1} represents the time needed to deform a fluid element in shear flow, which is compared with the time needed for the polymer to relax back to its equilibrium shape. We warn, however, that the polymer relaxation time τ_{rel} does not have the same physical meaning in the case of harmonic and FENE stars. The stiffer arms of the FENE star present a much larger arm–arm decorrelation time τ_{dis}, compared with those observed in harmonic bonds (see Table 2). This large τ_{dis} is consistent with the enhanced cooperative motion of short arms recently discussed in ref. 5. Thus, we use τ_{rel} = τ_{dis} for the FENE model while τ_{rel} = τ_{rot} is used for harmonic bonds. This is the reason why in some of the graphs below the FENE model data are shifted towards larger values of Wi (about 3 times larger), compared with those cases modeled with harmonic springs. We also plot some of the graphs against Wi_{ROT} = τ_{rot} to reveal this fact.

(4) |

Because the case of star molecules has been relatively less studied, the present results provide new insights into the important relation θ = θ(Wi). These results are shown in Fig. 1 for both solution and melt, for which we have also analyzed the raw data in ref. 7. Let us first consider the polymer dilute solution. The first clear observation is that hydrodynamic's mutual drag forces tend to “compact” the monomers' distribution by increasing the flow-gradient coupling G_{12}, reducing G_{11} − G_{22}, and also increasing G_{33} (see below). This leads to larger tilt angles and larger orientational resistances m_{g} = Witan(2θ). The trend for θ with Wi is also seen to depend on the presence of HI and the type of bonds. In the case of SH+HI (hydrodynamics included) we observe θ ≃ Wi^{−1/3} for a broad range of Wi >1. Under no-HI, we find θ ∼ Wi^{−1/2}. However, if the bonds are modeled by harmonic springs, above Wi > 100 cases with HI also converge to the 1/2 exponent. This indicates that at large Wi HIs are reduced along with the increase of the average intermonomer distance (large molecular elongations). By contrast, in the FENE stars the trend is approximately θ ∼ Wi^{−1/4} and it is comparatively less sensitive to hydrodynamics. The larger exponent is a consequence of the stronger excluded volume interactions (the FENE bonds make less compressible molecules).

Fig. 1 The molecular orientation angle (in degrees) given by eqn (4) for stars in solution (top panel) and in melt (bottom panel). For melts, we compare our results with those of ref. 7. The longest relaxation time was not provided in ref. 7, but just the arm relaxation. For comparison, we multiply the shear rate in ref. 7 by 5500, which is about 10 times the arm elastic relaxation reported therein. |

In melts, the trend for θ is highly dependent on the number of arms, as revealed by analysis of the data presented in the recent work of Xu and Chen.^{7} Our results for f = 12 and m = 6 are consistent with these data (approximately θ ∼ ^{−1/4}). Singh et al.^{13} have also presented results for the molecular orientation θ of star polymers (with harmonic bonds) in dilute and semidilute solutions c/c* ∈ [0.2,2.4] obtained by multiparticle collision dynamics (MCD) simulations. Their results fit into a master curve upon scaling the shear rate with a molecular relaxation time given by τ_{rel} = ϕ(f)τ_{arm}, where ϕ(f) ∼ f^{−2/3} is a functionality dependent function (τ_{rel} is proportional to rotation relaxation). They found a scaling θ ∼ Wi^{−0.43} at large shear rates, which is not quite different from that in Fig. 1 (top, in solution). We have verified that the data of Xu and Chen^{7} for melts (Fig. 1) do not obey this scaling law: their results for θ (or tan(2θ)) cannot be set in a master curve by scaling with the relaxation time. This disagreement might be due to solvent effects, although one might expect that a dense enough solution should converge to a melt. This disagreement indicates that further revision of the results for melts and dense solutions of star molecules in shear is needed. Another important conclusion is that the tilt angle in stars is very much dependent on hydrodynamics, excluded volume, bond stiffness and environment (solution or melts). Understanding how molecular orientation changes with the shear rate requires more detailed analyses which should incorporate force balances in the gradient and normal directions, taking into account all these contributions. Such analysis will be the subject of our future work.

ṙ_{i} = ṙ_{cm} + ω × (r_{i} − r_{cm}) + ṽ,
| (5) |

ω_{L} = J^{−1}L.
| (6) |

(7) |

The average effective “rigid body” rotation frequency 〈ω_{L}〉 can be obtained either from 〈ω_{L}〉 = 〈J^{−1}L〉 or from 〈J^{−1}〉 〈L〉. The average is a temporal average. As discussed by Aust et al.^{24} for the case of linear chains, both evaluations are quite close, reflecting the scarce correlation between J and L. This has also been observed for stars.^{12} We use the first one, but their differences were found to be negligible. In what follows, we will skip the brackets to alleviate notation and indicate that ω_{L} → 〈ω_{L}〉. We are interested in the component of ω_{L} along the neutral (vorticity) direction ω^{(3)}_{L}, as ω^{(1)}_{L} = ω^{(2)}_{L} ≃ 0. Therefore, we will write ω_{L} to indicate ω^{(3)}_{L}.

The angular velocity ω_{L} for a single chain has been analyzed in many works,^{12,13,24,39–43} in the search for a connection between the chain structure and its dynamics. An approximate relation between ω_{L} and the components of the molecule’s gyration tensor (here r_{i} = x_{i,α}ê_{α}),

(8) |

(9) |

(10) |

From its very derivation, ω_{G} in eqn (10) should be equal to ω_{L} in eqn (6) if polymer–polymer friction and hydrodynamic couplings between monomers were absent (free draining or Rouse regime). However, ω_{G} ≃ ω_{L} also holds in the case of linear chains in solution with hydrodynamics.^{24} And, as shown in Fig. 2, we also observe an excellent agreement between ω_{G} and ω_{L} in the case of sheared star polymers in solution. This indicates that hydrodynamic interactions do not seem to directly contribute to the molecular angular momentum. Note, however, that HIs indirectly modify ω_{L} because they modify the gyration tensor G_{αα} (see eqn (10)). In particular, the effective (rigid body) rotation is faster when HIs are activated [see Fig. 2(a)] because the molecule is made shorter in the flow direction (smaller G_{11}). This fact [shown in Fig. 3(a)] is due to the increased monomer friction induced by mutual hydrodynamic interactions. The other effect of HIs on the molecular shape is to increase its dimension in the vorticity direction G_{3} (see Fig. 3). This is due to the carrier fluid incompressibility, a condition which is present in the Oseen mobility tensor. However, the expansion in the neutral direction has no direct consequence in the rotation dynamics.

Fig. 2 The frequency ω_{L} defined in eqn (6) compared with that of eqn (10). Panel (a) corresponds to star polymers in solution, and the results corresponding to the melt are shown in (b) and (c). In (a–c), the symbols correspond to ω_{L} and the lines to ω_{G} ≡ G_{22}/(G_{11} + G_{22}). Panel (b) corresponds to our results for [f,m] = [12,6] stars in the melt (polymer volume fraction ϕ = 0.2 and T = 4ε) (the trends for the same star in solution have been included for comparison). Panel (c) corresponds to the results of Xu and Chen^{7} for melts of stars with different number of arms f and arm length m = 20. Panel (d) compares the relative difference Δ_{ω} = (ω_{L} − ω_{G})/ω_{L} obtained at large shear rates (Wi ∼ 30) for different cases in solution and melt. |

Fig. 3 The eigenvalues of the gyration tensor defined in eqn (8) for a single star molecule in solution with harmonic bonds (a) and with FENE bonds (b) and in melt (c). In (a and b), we compare hydrodynamic interactions (+HI) with the free draining limit (−HI). In melts (c), we compare the cases where monomers bear only normal friction (MN) and added tangential friction (MT). |

A systematic, although small, increase in the deviation of ω_{G} and ω_{L} is observed in passing from a purely Brownian polymer (without HI) to the same molecule (the harmonic bonds) with HI. This can be seen in Fig. 2(d), where we plot the relative deviation Δ_{ω} ≡ (ω_{L} − ω_{G})/ω_{L} for Wi ∼ 30. In the SH−HI case (harmonic bonds and no HI), one expects ω_{L} = ω_{G}, so the value of Δ_{ω} might reflect the noise contribution; meanwhile, cases with HI show small deviations, yet above this noise level. The somewhat smaller deviation found for the FENE bonds (SF+HI), compared with the harmonic ones (SH+HI), might be consistent with the stiffer (more rigid-like) molecular structure although, admittedly, this is a hand-waving argument.

The case of melts is illustrated in Fig. 2(b). We observe that the effective rotation of molecules with added tangential friction (via the DPD thermostat, see above and ref. 6) is faster than that of molecules with just normal friction. In ref. 6, it was also shown that tangential friction has some measurable effect on the polymer rheology: tangential friction contributes to decrease the melt viscosity at similar Wi (a counter-intuitive effect related to the reduction of the monomer kinetic stress when tangential friction is added). The present observation for ω_{L} speed-up goes in the same line. Interestingly, the trend for tangential friction [MT in Fig. 2(b)] resembles the trend obtained in solution with hydrodynamics, while the MN (normal friction) case seems to follow the no-HI trend in solution. Motivated by this coincidence, we revise the trend for the eigenvalues of the gyration tensor in solution (with and without HI) and melt (with normal and tangential friction). Such comparisons can be analyzed in Fig. 3(a)–(c). Interestingly, the dominant eigenvalue of G (approximately the extension in the flow direction, G_{1}) becomes smaller if tangential friction is added, but this is similar to what the HI induce in a dilute solution. In melts, tangential friction also increases the extension of the molecules in the vorticity direction G_{3}, and the molecules breath in the “gradient” direction G_{2}. And this is exactly what the mutual hydrodynamic mobility brings up in solution, as seen in Fig. 3(a) and (b) (note that the HI-induced increase of G_{2} is particularly relevant for the FENE bonds).

Is there any mechanical justification for such coincidence between HIs in solution and tangential friction in melts? To inspect this issue, we calculate the mutual mobility between molecules in the melt by evaluating the ensemble average of the relative velocity between a tagged molecule 1 and another one “i” at a relative position R_{1i}. This is, 〈V_{1i}〉_{R}_{i1}, with V_{1i} = V_{i} − V_{1} the relative center of mass velocities. The result for a case in equilibrium (no shear flow) is shown in Fig. 4. We see that the mutual (molecular) mobility is more “hydrodynamic like” if tangential friction is present leading to larger screening distances (about a factor two). The larger hydrodynamic coupling between molecules is the reason for the faster rotation in melts with added tangential-monomer-friction.

In melts, inter-chain collisions leading to friction forces introduce a significant amount of angular momentum to each star which reflects in a deviation of ω_{L} from ω_{G}. The intermolecular drag contributes to fasten up the effective rotation frequency ω_{L} with respect to the mean flow approximation ω_{G}. One expects that such difference Δ_{ω} > 0 should be reduced in compact stars (large f) because of the reduction of the surface-to-volume ratio, while being larger in melts of linear chains. In passing, we note that such inequality Δ_{ω} > 0 is also observed in solution, see Fig. 2(d). As Fig. 2(b) illustrates, the deviation between ω_{G} (lines) and ω_{L} (filled symbols) increases with the shear rate, which is to be expected, due to increasing inter-molecular friction. An interesting point is that Δ_{ω} does not substantially depend on tangential friction. We could extract another interesting conclusion by analyzing the data presented in a recent work by Xu and Chen^{7} on star polymer melts. The results, presented in Fig. 2(c) and (d), permit us to further explore the implications of eqn (10) for architectures with increasing “compactness”, in terms of number of arms f and arm length m. Stars are known to behave like colloids in the limit of large functionality f and transit to “polymeric” behavior for small f. As expected, the values of Δ_{ω} obtained for Wi ∼ 30 in melts indicate an increasing deviation (Δ_{ω} > 0) for small f, as the stars become more polymer-like and less colloid like. Even with f as few as 10, the relative deviation Δ_{ω} remains at just about ten percent, while for f = 3, it suddenly jumps to more than 50 percent. Interestingly, the value of f, where this jump takes place (f ≃ 6), is consistent with that reported in recent work on the transition from polymer to colloidal behavior of star molecules.^{9} Our results with m = 6 and f = 12 are consistent with those of Xu and Chen,^{7} with a much larger arm size m = 20. This indicates that the important variable here is the number of arms, rather than the arm length (however, stars with very long arms probably behave differently in this respect).

Fig. 5 Effective rotation frequency ω_{L} (defined in eqn (6)) scaled with the shear rate. We compare the results of multiparticle collision dynamics (MCD) in dilute solution with the present results. The results of stars with f = 50 and m = 30 obtained via MCD are extracted from Ripoll et al.^{3} (single star molecule) and from Singh et al.^{12} (dilute solution, c = 0.19c*). The solid lines correspond to ω_{G} in eqn (10) and symbols to ω_{L} in eqn (6). Note that we scale the shear rate with the molecular rotational relaxation time Wi_{ROT} = τ_{rot}. |

(11) |

Above a certain shear rate, C_{12}(t) becomes an underdamped signal, i.e. presents negative anticorrelation peaks. The physical meaning is clear: a large fluctuation of the polymer breath in the gradient direction induces an increase in the overall flow drag ξX_{2} (we denote X_{α} as the average chain breath in the α direction), which has the consequence of a subsequent large fluctuation (elongation) in the flow direction. As it elongates in the flow direction, the molecule becomes less exposed to the flow drag and, at some stage, it coils back owing to the entropic penalty of being elongated above its equilibrium shape (entropic compression). This “cycle” is not purely periodic but has a characteristic frequency, often called tumbling frequency in the case of free linear chains.^{18,37} In fact, the same type of dynamics has also been observed in linear chains tethered to a surface and exposed to shear flow.^{46} In these cases (linear chains), the onset of tumbling takes place above a certain shear rate, which is about the inverse of the rotation relaxation time of the chain. Star molecules, however, have different characteristic times related to different relaxation mechanisms. These times are indicated in Table 2 for our molecular models: τ_{rot} molecular rotation, τ_{dis} arm disentanglement and τ_{arm} individual arm relaxation. It is thus pertinent to ask what is the relaxation mechanism which is altered upon shearing, eventually leading to the onset of expansion/contraction cycles (here we call them “breathing”).

Such a question requires a precise determination of the “transition” based on the form of time-correlation of the gyration tensor components. Fortunately, the time correlation of the gyration tensor components in eqn (11) offers a clear distinction of such transition, which is illustrated in Fig. 6 (corresponding to stars with harmonic bonds). At small shear rates (Wi < 10), the molecule is slightly strained by the flow drag and according to the decay of C_{12}(t) is able to exponentially relax its shape. Above Wi > 10, we observe an oscillatory component in C_{12}(t) with a pronounced maximum at t_{−} < 0 and minimum negative correlation at t_{+} > 0.^{15,17} The lapse τ_{t} = 2(t_{+} − t_{−}) should be interpreted as the characteristic time for one “breathing” event, in other words the average time between two stretched molecular configurations (stretched-contracted-stretched). The corresponding angular frequency is noted as Ω = 2π/τ_{t}. We can now answer the question we posed above; according to our results the transition to breathing dynamics takes place for τ_{rot} ∼ 10. A glance of Table 2 reveals that this corresponds to τ_{arm} > 1 and this indicates that the molecular expansion and contraction cycle starts to happen when the shear rate is faster than the arm relaxation time τ_{arm} (needed by elastic entropic forces to recover its most probable configuration).

Fig. 6 Cross-correlation of polymer extension in the flow (1) and gradient (2) directions, C_{12}(t) defined in eqn (11) for several values of Wi and the SH+HI 12–6 case (see Table 1). The arrows correspond to the first extremes of C_{12} from which we obtain the time lapse τ_{t}/2 (see text). The breathing frequency is defined as Ω = 2π/τ_{t}. |

Fig. 6 indicates that the non-equilibrium dynamics have several characteristic times. A better determination of these dynamics might require fitting the time correlation of the gyration tensor components with at least two mechanisms (recovery and damping), using for instance C_{12}(t) ∼ exp[−Γt]cos[Ωt + ψ] (as some of us used in linear chains^{17}). Such simple fit (which, by the way, fails in the case of stars) would reveal the characteristic frequency Ω and another non-equilibrium relaxation rate Γ related to dissipation under flow.^{17} Such dissipative mode is already present before the “tumbling” or “breathing” transition (see Fig. 6 for Wi < 10). Here, we will just analyze the oscillatory component of the time-correlation of the gyration tensor, Ω, and defer its decorrelation envelope Γ (related to the quality factor of the corresponding power spectra) for future work.

First, a relation between the average extension in the flow X_{1} and gradient directions X_{2} can be determined from the balance of the drag force ξX_{2} and the molecular tension KX_{1},

KX_{1} ∼ ξX_{2}
| (12) |

ξẊ_{1} ∼ ξΩX_{1} ∼ KX_{1}, |

(13) |

(14) |

Before falsifying eqn (14) against simulation results, it is interesting to consider the force-balance in the molecular frame coordinates, given by the eigenvectors of the gyration tensor. In this frame, the molecular elongations can be taken as the square root of the eigenvalues of G, and we note them as X_{α}′ = G_{α}^{1/2}. The eigen-directions in the flow-gradient plane are rotated by an angle θ with respect to the laboratory frame (x_{1}′,x_{2}′) = _{θ}(x_{1},x_{2})^{T} with _{θ} being the rotation unitary orthogonal matrix and the primes denoting coordinates in the molecular frame. Second rank tensors transform as G′ = R_{θ}GR^{T}_{θ} and the average molecular tilt θ in eqn (4) is precisely the angle providing G_{12}′ = 0.

We find that approximately for θ < 0.4 rad the tilt is roughly proportional to the width-to-length ratio X_{2}/X_{1}. In particular, (G_{22}/G_{11})^{1/2} ≃ 1.8θ. A similar relation is found for the eigenvalues of the gyration tensor, (G_{2}/G_{1})^{1/2} ≃ 2.0θ. In fact, for any shear rate both ratios (G_{2}/G_{1} and G_{22}/G_{11}) are close and proportional to each other (G_{2}/G_{1} ≃ 0.88G_{22}/G_{11} for the FENE while the constant is 0.85 for the harmonic chains). This indicates that the force-balance in the longest molecular direction x_{1}′ has also the form of eqn (12). A simple geometrical argument supporting this claim is given in the Appendix.

Ω ≃ c_{θ}θ(Wi)Wi.
| (15) |

Fig. 8 Validation of eqn (15) for the tumbling frequency Ω = 2πf_{tb} in (a) ring polymers in solution and (b) linear (FENE) chains in melt. The results in (a) were analyzed from the multiparticle collision dynamics (MCD) simulations of Chen et al.^{15} on ring polymers with n = 40 and n = 80 beads in shear flow. And the results in (b) are from Xu et al.^{47} corresponding to a melt of FENE linear chains at volume fraction ϕ = 0.45 (non-equilibrium MD using a DPD thermostat). In (b) we plot the tumbling frequency and the resistance parameter m_{g} (see text). In both cases, the agreement is excellent. |

This result is important because it means that the differences in tumbling frequencies observed between different polymer architectures, polymer type and environment (excluded volume, hydrodynamics, melt vs. solution, etc.) can be explained from the force balance in the flow direction, being ultimately determined by the group (G_{2}/G_{1})^{1/2}. This group scales like the orientation resistance m_{g}. We consistently observe that

Ω = c_{m}m_{g} |

Linear (FENE) | Solution | Melt | |||
---|---|---|---|---|---|

Rings | Star-harmonic | Star-FENE | Linear ϕ = 0.45 | Star ϕ = 0.2 | |

0.4N^{−1/3} |
1.20 | 1.41 | 1.57 | 2.6 | 1.73 |

The present findings shed light on recent literature discussions^{3,13,15,19} about the different polymer architectures concerning tumbling dynamics. The conclusion is that such structure–dynamic relation is controlled by the aspect ratio G_{2}/G_{1}. A deeper understanding of this ratio (or equivalently m_{g}) requires unveiling the force balance in the gradient and neutral directions in different polymers and environments.

(16) |

6.1.1 Intermediate shear rates, τ_{arm} > 1. Approximately for τ_{arm} > 1 a transition to a “breathing” dynamics (expansion/contraction of the gyration tensor) takes place. Note that τ_{arm} is the arm elastic relaxation time, given in Table 2 (for a linear chain case f = 2 this transition would correspond to the chain “tumbling”). Using the relation ω_{L} ∼ /2 valid for immediately smaller shear rates in eqn (16) one gets

so that Ω and ω_{L} are roughly proportional. As shown in Fig. 9, we consistently find ω_{L}/Ω ≃ c_{f}^{−1}.

Ω ≃ 2c_{f}ω_{L} ≃ c_{f} |

6.1.2 Large shear rates, τ_{lag} > 1. At large shear rates, /ω_{L} ≫ 1 (see eqn (10) for G_{11} ≫ G_{22}) and eqn (16) becomes

This means that the expansion/contraction cycle becomes faster than ω_{L}.

Ω ≃ c_{f}(ω_{L})^{1/2}.
| (17) |

At larger shear rates, the molecule becomes highly elongated G_{11} ≫ G_{22} so that ω_{L} ∼ G_{22}/G_{11}. In the particular case of harmonic bonds, we observe that G_{22}/G_{11} ∼ ^{−1}. As a consequence, ω_{L} ∼ G_{22}/G_{11} reaches a plateau (see Fig. 2(a)) and, consistent with eqn (16), Ω ∼ ^{1/2}. This is observed for both HI and non-HI simulations, indicating that above a certain molecular elongation the effect of HIs becomes negligible.

However, we stress that this regime with ω_{L} → cte (III in Fig. 9, left panel) is not observed in the FENE chains (neither in melts), so contrary to that claimed in ref. 3, the saturation of the monomer’s angular rotation (ω_{L} ∼ cte) is not a property of the star molecule’s architecture, but rather depends on the bond stiffness, excluded volume and environment (melt or solution).

We tried to understand what molecular time characterizes this large shear-rate regime, by thinking over the fastest molecular response time. The sequence of motion of the molecule in shear flow is first to elongate in the velocity gradient direction and after some time lapse τ_{lag} to stretch in the flow direction. This response time, τ_{lag}, is defined as the phase-lag between the gradient–gradient and flow–flow time-correlations of the gyration components [see ref. 17 and Fig. 10(a)] and it has been related with dissipation.^{17} It is quite sensitive to molecular material properties: a rigid molecule presents τ_{lag} = 0 while softer molecules have a finite response time. But, how this lag time changes with the shear rate? Fig. 10 illustrates the problem and presents the definition of τ_{lag} obtained from the phase difference between C_{22}(t) and C_{11}(t) which lags behind (more precisely they were obtained from τ_{lag} = t_{1} − t_{2} with C_{ii}(t_{i}) = c, and we tried c = {0.1,0.2,0.3} to verify that τ_{lag} is not sensitive to the cut-off c). The response time τ_{lag} is seen to increase from zero at small shear (obviously τ_{lag} = 0 for the molecule in equilibrium, as there is no cause preceding consequence), up to a maximum value τ^{max}_{lag}. Interestingly, we find that the ratio τ_{lag}/τ_{rot} converges at large Weissenberg number to a certain value which only depends on the molecule’s “stiffness”. For SH stars (harmonic bonds), we find τ_{lag}/τ_{rot} → 0.01 [see Fig. 10(b)], while for the FENE bonds (SF) this value is much smaller. The monomer relaxation time is τ_{m} = ξ_{m}/k_{0} and equals 0.28 for harmonic bonds with spring constant k_{0} (see Section 2). Scaling the lag time with this monomer time, one gets τ_{lag}/τ_{m} = 17 for SH−HI 12–6 stars and τ_{lag}/τ_{m} = 8 for SH+HI 12–6 stars. Assuming that one of the 6 linear chains forming two arms, which cross the star through its center (having N_{l} = 2m = 12 beads), has a relaxation spectrum of τ_{p} = (N_{l}/p)^{ν}τ_{m}, with ν = 2 for free-draining chains (no hydrodynamics) and ν = 3/2 when hydrodynamic interactions are present, one concludes that in both cases (with or without HI), τ_{lag} corresponds to a mode with p = 4. This means that the limit value of τ_{lag} at large shear rates is roughly equal to the relaxation time of 12/4 = 3 consecutive monomers of one arm. In the case of the SH+HI 12–3 star, we get τ_{lag}/τ_{m} ≃ 3.6 and p ≃ 2, also resulting in a mode with 6/2 = 3 monomers. Although these are gross estimations, the good agreement indicates that at some large enough shear rate, the flow strain might become even faster than the response time of a few consecutive monomers. In such a case, above a certain Wi, τ_{lag} saturates to the fastest possible molecular response time (a Rouse mode for consecutive monomers). The saturation of τ_{lag} is illustrated in Fig. 10(b) and the limiting shear rate can be deduced from the condition τ_{lag} > 1 illustrated in Fig. 10(c). For τ_{lag} > 1, monomers are advected collectively and the length distribution of the individual harmonic bonds shows strong deviations from the equilibrium value r_{eq}. The condition τ_{lag} = 1 occurs at Wi ≃ 100 (SH+HI 12–6 case) and at Wi ≃ 45 (for SH−HI 12–6). Above these values [indicated with vertical lines in Fig. 9(a)], ω_{L} reaches a plateau and Ω ∼ ^{1/2}. Around Wi ∼ 400, (G_{22}/G_{11}) reaches a maximum value [see SH cases in Fig. 2(a)]. In fact, at even larger shear rates (Wi > 10^{3}) we observed that ω_{L} ∼ ^{−1} (not shown). The reason is the following (see eqn (10)): for these extreme shear rates, the highly deformed molecular bonds weakly connect the molecular displacements in flow with contractions and expansions in the gradient direction. The molecule then approaches the limit of a Gaussian polymer under shear flow, for which G_{22} is constant and G_{11} ∝ ^{2}. Thus, the conclusion ω_{L} → cte for large Wi, made in previous works (ref. 3 and 12–14), is not exact and it is due to a limited window of observation. Here we present results for Wi < 400 (which is also the range of Wi analyzed in ref. 3) and defer an exploration of larger Wi for future work.

Fig. 10 (a) Time correlation of the star extensions in the flow–flow (C_{11}) and gradient–gradient (C_{22}) directions indicating the meaning of the lag time τ_{lag}: the time between a fluctuation in molecular extension along the gradient direction and the subsequent elongation in the flow direction. (b) The lag time scaled with the longest relaxation (rotational) time. (c) The lag time τ_{lag} scaled with the shear rate τ_{lag} against the Weissenberg number for several types of stars in solution. The Weissenberg numbers at which τ_{lag} = 1 are indicated with vertical arrows in Fig. 9. |

In any case, when bonds of finite extensibility (FENE) are considered (SF+HI case) ω_{L} keeps increasing with Wi: consistent with the picture above [see Fig. 10(b)], the flow is not able to strain faster than its fastest (“bond”) response (i.e. τ_{lag} < 1 probably for any shear rate; here, we reached Wi ∼ 10^{3}). A similar comment could be made in the case of melts; however, in this case, we could only reach Wi ∼ 100 due to strong heat dissipation.

This issue is relevant because in many previous papers^{3,7,12–14} ω_{L} appearing in eqn (6) has been taken to represent the monomer rotation frequency around the soft molecule’s CoM and thus related to “tank-treading” dynamics. This interpretation is incorrect and comes out from the fact that ω_{L} in eqn (6) (and its approximation ω_{G} in eqn (10)) corresponds to the tumbling frequency of a rigid spheroid having the average shape of the soft molecule. But the tumbling motion of a rigid spheroid and the tank-treading rotation of the soft molecule are completely different! The “real” rotation frequency of the monomers around the molecule’s CoM can be measured, for instance, from the time correlation C_{rot}(t) in eqn (2) which considers the vector between the center monomer and the end-monomer of each arm. This vector rotates with the monomers with a frequency ω_{R} which can be measured from a fit to a decaying sinusoidal function. We used a similar function (using the projection in the flow direction) to show that ω_{R} differs a lot from ω_{L}, particularly as the shear rate is increased. This fact is illustrated in Fig. 11 for the SH+HI case, but it is observed in all cases (either solution or melt). Fig. 11 compares Ω, ω_{R} and ω_{L} for the SH+HI case, showing that ω_{G} < ω_{R} < Ω. Unlike that stated in ref. 3 and more recent works,^{12–14} the “tank-treading” frequency does not saturate at large , but keeps monotonically increasing. Here, we focus on the origin of the breathing frequency Ω, so it has to be stressed that, so far, the tank-treading frequency of star molecules ω_{R} remains to be theoretically or experimentally studied. Such study should probably reveal strong similarities between tank-treading in star-molecules, in ring polymers^{15,19} and also in vesicles,^{21,23} even at large shear rates (see the comparison made in ref. 12).

Fig. 11 The frequency of monomer rotations around the star molecule’s center (tank-treading) ω_{R}, obtained from the time autocorrelation of the end-monomer-center distance in the flow direction X_{i} = (x_{1,i} − x_{cm}). We compare ω_{R} with ω_{L} and ω_{G} in eqn (6) and (10) and also with the breathing frequency Ω and c_{m}m_{g} where m_{g} is the orientational resistance, see Table 3. |

In view of this fact, ω_{L} should be just taken as a proxy for the angular momentum imparted by the mean flow to the molecule and not as a way to understand how a soft molecule rotates. Indeed, the soft-body analogous of ω_{L} appearing in eqn (5) cannot be deduced from the simple relation in eqn (6) but would require a special theoretical framework for soft rotating bodies, which will be the focus of our future work.

We analyzed the different dynamic regimes as the shear rate increased, and related the different dynamics with the cross-over of the shear rate above each one of the equilibrium rates of the star molecule (see Table 2). These relaxation times are star rotation τ_{rot}, arm elastic relaxation τ_{arm} and τ_{lag} related to the fastest response of consecutive bonds. For τ_{rot} > 1, the molecule begins to rotate with a frequency smaller than about ω_{L} ∼ /2. For τ_{arm} > 1, the molecules start to expand/contract in an underdamped fashion, with an (angular) frequency Ω. At larger shear rates, τ_{lag} > 1, the flow strains faster than the bond–bond interaction time. Hydrodynamic interactions delay the onset of the large regime. In this high-shear regime, the aspect ratio of the molecule scales like G_{22}/G_{11} → ^{−1}, the molecular tilt as θ ∼ ^{−1/2} and the molecular angular momentum seems to saturate ω_{L} → cte, while Ω ∼ (ω_{L})^{1/2}. The apparent “saturation” of ω_{L} is, however, only observed for stars with Hookean springs, but certainly not when using the FENE bonds. Moreover, the plateau of ω_{L} is not real because, at even larger shear rates, a star with harmonic springs behaves like a Gaussian polymer, for which ω_{L} ∼ ^{−1}. The situation in melts is much more complex. We analyzed the recent results of Xu and Chen^{7} along with those in the present work to show that ω_{L} ∼ ^{α} with exponents increasing with the functionality f (α ∈ [0.52,0.79] for f ∈ [3,60]). A definitive explanation of rotation and molecular breathing, stemming from the molecular architecture and mechanical properties, should take into account the force balance in the gradient and normal directions, including excluded volume and hydrodynamics.

The second part of the work concerned the expansion/contraction dynamics of the star molecules in shear. A relation for Ω was derived based on the force balance in the flow direction, leading to Ω ≃ c_{f}(G_{22}/G_{11})^{1/2} (with G_{ij} the components of the gyration tensor and 1-flow and 2-gradient directions). We also showed that this relation can be expressed in terms of the molecular tilt angle θ and, alternatively, in terms of the orientational resistance parameter Ω = c_{m}m_{g}, with m_{g} = Witan(2θ). Using our results in solution and melt along with those in the recent literature, we suggest that this relation is general, as it holds excellently well in stars, rings and linear chains. The ratio c_{m} = Ω/m_{g} is of order one and slightly depends on polymer properties and environment (melt or solution, bond type, molecular architecture). In view of the present results, it would be interesting to study collective vibrations in dendrimers under shear flow. Our guess is that their dynamics would be similar to the “breathing” mode of star molecules, but the higher coordination required for “arms” and monomer displacements in dendrimers might somehow modify these collective dynamics. Another aspect to have in mind is that this study focuses on star molecules with relatively short arms, where entanglement is irrelevant. Previous studies^{3} indicate that entanglements do not qualitatively alter the star structure or dynamics at least for arms with m ≤ 30 Kuhn lengths. However, it would be interesting to study if the entanglements created in star molecules with very long arms m > 10^{2} (and large enough functionality) might lead to qualitatively different dynamics under shear flow, at very dilute concentration (below overlap).

As a final relevant contribution, we showed that, contrary to what is stated in relatively recent works,^{3,7,12–14} the angular frequency ω_{L} evaluated from the mean-flow approximation of a “rigid-body-like” equation for the molecular angular momentum L is not related with the rotation frequency of the monomers around their center of mass, ω_{R}. This frequency, ω_{R}, is the one determining the tank-treading motion of star-polymers and also ring chains.^{15} The error in previous works was to directly relate the rotation equation for a rigid-body with the much more complex rotation of soft-bodies, which would require a more elaborate framework.

The flow drag pushes in the x_{1} (lab) direction (F_{f} = F_{f}x_{1}) but its components in the molecular frame are F_{f}′ = F_{f}[cos(θ)x_{1}′ − sin(θ)x_{2}′] (see Fig. 12). When observed from the (tilted) molecular frame, the drag force stretches the molecule in the x_{1}′ direction and compresses it in the x_{2}′ direction. Such compression reduces the polymer width G_{2} according to another force balance, to be studied in a subsequent work. Here, we consider the balance in the most elongated molecular x_{1}′-direction, where the average elastic force KX_{1}′ should be balanced by the projected drag force X_{2}cos(θ). Now, if X_{2} ∼ X_{2}′/cos(θ), one concludes that

KX_{1}′ ∼ ξX_{2}′, |

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