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DOI: 10.1039/C7SM00616K
(Paper)
Soft Matter, 2017, Advance Article

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

Received
27th March 2017
, Accepted 30th August 2017

First published on 5th September 2017

The Eckart co-rotating frame is used to analyze the dynamics of star polymers under shear flow, either in melt or solution and with different types of bonds. This formalism is compared with the standard approach used in many previous studies on polymer dynamics, where an apparent angular velocity ω is obtained from the relation between the tensor of inertia and angular momentum. A common mistake is to interpret ω as the molecular rotation frequency, which is only valid for rigid-body rotation. The Eckart frame, originally formulated to analyze the infrared spectra of small molecules, dissects different kinds of displacements: vibrations without angular momentum, pure rotation, and vibrational angular momentum (leading to a Coriolis cross-term). The Eckart frame co-rotates with the molecule with an angular frequency Ω obtained from the Eckart condition for minimal coupling between rotation and vibration. The standard and Eckart approaches are compared with a straight description of the star's dynamics taken from the time autocorrelation of the monomer positions moving around the molecule's center of mass. This is an underdamped oscillatory signal, which can be described by a rotation frequency ω_{R} and a decorrelation rate Γ. We consistently find that Ω coincides with ω_{R}, which determines the characteristic tank-treading rotation of the star. By contrast, the apparent angular velocity ω < Ω does not discern between pure rotation and molecular vibrations. We believe that the Eckart frame will be useful to unveil the dynamics of semiflexible molecules where rotation and deformations are entangled, including tumbling, tank-treading motions and breathing modes.

We have recently completed a series of works on star polymer dynamics.^{19,26,27} This series started by a study of the effect of open boundaries compared with closed systems in the rheology of melts under shear (simulations using Open Boundary Molecular Dynamics (OBMD)^{26–28} permit fixing the pressure load and shear stress, instead of the density and shear velocity). As a continuation of such work, we studied the dynamics of stars in solution and melt^{19} and observed that the tank-treading frequency of monomers around the molecule's CoM, ω_{R}, was completely different from the “apparent” angular velocity obtained from the standard (lab-frame) analysis, ω. We also noted that the origin of such strong differences was not explained in previous works. Motivated by these observations, we decided to tackle the problem of soft molecule rotational dynamics using an old and robust formalism, which apparently has been largely forgotten by the soft matter community: the Eckart co-rotating frame.

The Eckart frame formalism, derived in 1935,^{29} uses a non-inertial frame, which rotates with the molecule. It allows disentangling translation, rotations, and vibrations. Aside from vibrations without angular momentum contribution (which can be detected in the inertial frame), the non-inertial frame allows revealing vibrations with angular momentum. These are the displacements with respect to a purely rotating (rigid-body) reference configuration. The Eckart condition determines the rotation frequency of the reference configuration by minimizing the coupling between vibrational angular momentum and pure rotation.^{30} The calculus of the so called “Eckart angular velocity” Ω can be carried out by the Eckart frame formalism and has been mostly used to study the infrared and Raman spectra of small molecules^{31,32} as well as in a variety of other applications, such as structural isomerization dynamics of atomic clusters^{33} or molecular dynamics (MD) integration.^{34–37} The “apparent” angular velocity ω extracted from the total angular momentum in the inertial frame mixes up pure rotation and vibrational angular momentum. A misinterpretation of this apparent angular velocity had, as a consequence, some large discrepancies in the polymer literature on shear flow.^{11,16}

While the Eckart formalism is traditionally used in equilibrium states, here we use it to describe a situation which is far away from equilibrium. Although the Eckart condition is first-order accurate, we show that it is robust enough to capture the correct physics. In particular, we show that the Eckart frame is independent of the reference configuration chosen (see Appendix) and that, for any shear rate, the resulting frequency Ω equals within error bars the monomer rotation frequency about the molecule’s CoM, ω_{R}. Star polymers are particularly interesting for this sort of study because of their rich dynamics in shear flow (with tank-treading and breathing modes^{19}) and also because they represent a bridge between the physics of polymers and colloids.^{38–40} More generally, we expect this work will foster the use of the Eckart frame as another useful tool in the analyses of flowing macromolecules' dynamics.

The rest of the paper is structured as follows: first, we describe the standard (laboratory frame) analysis and the Eckart frame. Then, we describe our working models (star polymer in melt and solution under shear flow). Results and discussion are then presented, followed by Conclusions.

ṙ_{α} = ṙ_{cm} + ω × (r_{α} − r_{cm}) + ṽ_{α}.
| (1) |

ω = J^{−1}·L.
| (2) |

(3) |

A common mistake is to interpret ω as the molecular angular velocity. However, ω does not describe the pure rotational component of the molecule and in fact, it is called the apparent angular velocity in the literature dealing with the Eckart formalism.^{32} Only in the case of rigid-body motion (ṽ = 0) does ω coincide with the rotational angular velocity. The reason will become clear in the next section.

The first step of the Eckart frame formalism is to choose some rigid molecular configuration, which is taken as the reference one.^{31} The Eckart frequency and kinetic energy are, however, independent of the rigid reference configuration chosen. This fact is illustrated in the Appendix, where we compare three different reference configurations. Once the reference configuration is chosen, we introduce the initial internal coordinate system, defined by the three right-handed base vectors f_{1}, f_{2}, and f_{3} with the origin in the CoM of the molecule. The initial internal coordinate frame (f_{1}, f_{2}, f_{3}) can be chosen arbitrarily, i.e. its initial orientation is arbitrary. The components of the position vector of the α-th monomer of the reference configuration expressed in the initial internal coordinate system are denoted as c^{α}_{i}, i = 1, 2, 3. Once defined, the c^{α}_{i}s remain constant during the computation of the angular velocity and fulfill the equation:^{29,31}

(4) |

(5) |

(6) |

(7) |

The reference components c^{α}_{i}s are in the instantaneous Eckart frame given as:

(8) |

The rotation of the polymer is defined by the rotation of the base vectors of the Eckart frame:

ḟ_{i} = Ω × f_{i}.
| (9) |

ċ_{α} = Ω × c_{α}.
| (10) |

The reference positions of every monomer in the laboratory frame are computed as:

d_{α} = r_{cm} + c_{α},
| (11) |

ρ_{α} = r_{α} − d_{α}.
| (12) |

(13) |

Fig. 1 (top) A sketch of internal and laboratory frames. The unit base vectors f_{1}, f_{2}, and f_{3} span the internal coordinate system, i.e. the Eckart frame, which translates and rotates together with the molecule. The laboratory frame's base vectors are e_{1}, e_{2}, and e_{3}. The arrows indicate the rotation of the molecule. (bottom) The sketch gradually introduces the different types of displacements resolved by the Eckart formalism. The black line corresponds to pure rigid rotation (monomer velocity Ω × δr) which does not introduce molecular deformation. The blue line (velocity ṽ) introduces vibrations without angular momentum contribution (e.g. compression and expansion) and the red line introduces vibrations with angular momentum (fluctuations with velocity u) which deform the molecule's shape (e.g. due to Brownian diffusion). Note that u·Ω × δr < 0 (Coriolis term). The different velocities are explained in the text (see e.g. eqn (16) and (17)). |

The angular velocity of the Eckart's coordinate system is given by:

(14) |

(15) |

The velocity of a given monomer α can be written as:^{30,34}

ṙ_{α} = ṙ_{cm} + Ω × (r_{α} − r_{cm}) + Δv_{α}.
| (16) |

Δv_{α} = ṽ_{α} + u_{α},
| (17) |

u_{α} = (ω − Ω) × δr_{α},
| (18) |

According to, eqn (1) the kinetic energy of any rotating molecule can be written as:

(19) |

T = T_{trans} + T^{lab}_{rot} + T^{lab}_{vib},
| (20) |

On the other hand, using the Eckart frame, the velocity of each monomer is expressed by eqn (16) and the kinetic energy of a molecule is decomposed as^{32}

(21) |

T = T_{trans} + T^{Eck}_{rot} + T^{Eck}_{vib-non-ang} + T^{Eck}_{vib-ang} + T^{Eck}_{Cori}.
| (22) |

T^{lab}_{vib} = T^{Eck}_{vib-non-ang},
| (23) |

T^{lab}_{rot} = T^{Eck}_{rot} + T^{Eck}_{vib-ang} + T^{Eck}_{Cori}.
| (24) |

T_{Ω} ≡ ½Ω·J·Ω = T^{Eck}_{rot},
| (25) |

(26) |

(27) |

In the next section, we resort to the Eckart frame formalism in the analysis of the rotational and vibrational behavior of star polymers in solution and melt. Differences with respect to the laboratory frame will be highlighted.

We have so far introduced two frequencies (i.e. ω and Ω) describing rotation in polymers. In what follows, we will introduce two additional frequencies and for the sake of clarity and reference, we list them all in Table 1.

To present the results in non-dimensionalized form, we use the Weissenberg numbers Wi and Wi_{rot}. The latter is based on the rotational diffusion time of the star. This is defined as Wi_{rot} = τ_{rot} where τ_{rot} is the time for rotational diffusion, τ_{rot} = R^{2}/D_{r}, of the molecule in equilibrium (see ref. 19 and 27 for details). The Wi, on the other hand, is based on the largest relaxation time (τ_{rel}) of the molecule, i.e. Wi = τ_{rel}. It has to be said that the molecular rotational diffusion is the slowest relaxation process for stars with harmonic bonds, while for star molecules with FENE bonds, the slowest relaxation is the process of arm disentanglement. The corresponding relaxation times (rotational and arm-disentanglement) for simulations in solution and in melt are given in Table 2.

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

Closed melt: γ_{∥} = 1.0, γ_{⊥} = 1.0 |
710 ± 40 | 390 ± 10 |

Open melt: γ_{∥} = 1.0, γ_{⊥} = 1.0 |
700 ± 40 | 390 ± 10 |

Solution harmonic bonds | 270 ± 20 | 180 ± 20 |

Solution FENE bonds | 370 ± 30 | 950 ± 90 |

Before entering into details, some general comments on the star molecule model are due. The main purpose of this work is to illustrate how the Eckart formalism can be used to provide information about the dynamics of soft molecules under rotation. The same type of analysis could be applied to any other star-molecule model or to other soft molecules (vesicles, semi-flexible linear polymers, or rings, etc.). Here, we have just deployed some relatively simple star molecule models (with short arms m = 6) in two quite different scenarios (melt and solution). It has to be said that some properties of this star molecule model depend on the arm length m, particularly at large Weissenberg numbers. As shown in Fig. 1 of the paper in ref. 11 by Ripoll et al., the elongational parameter G_{11}(Wi)/G_{11}(0) − 1 decreases with m at large shear rates. However, it is relevant to say that the resistance parameter m_{G}, which is directly related to the breathing rotational frequency (see ref. 19 and 27), exhibits a master relation, independent of m (see Fig. 3 of Ripoll et al.). Concerning the simulations in melt, we have recently shown that the type of friction between monomers somewhat alters the melt rheology. The calculation of the “proper” friction kernel representing a particular atomistic model requires expensive Green–Kubo type evaluations of force–force cross-correlations.^{41} Here, we use a given friction kernel, a la DPD, and notice that the results should slightly vary with the kernel type. The reader is referred to ref. 19 and 27 where we present some studies on how the rheology and dynamic properties of this model of star molecules' melt vary with the friction kernel.

C(t) = A^{2}cos(ω_{R}t + ψ)exp(−Γt),
| (28) |

Fig. 3 (top) Autocorrelation function of position of the final monomers of each polymer's arm in the gradient direction, fitted by eqn (28) with parameters: A = 0.93, ω_{R} = 0.35, Γ = 0.0084, and Ψ = 0.0025. (middle panels) The tank-treading frequency ω_{R} and decorrelation rate Γ obtained from the fits (stars in solution and in melt). (bottom) The quality factor q = ω_{R}/Γ for the dynamics of monomer rotations, comparing our 12-6 stars with linear FENE chains (N = 60) (with excluded volume interactions) and dumbbells, from ref. 53. |

Solution harmonic bonds | Solution FENE bonds | ||||||||
---|---|---|---|---|---|---|---|---|---|

Wi | T | T_{Ω} |
T_{ṽ} |
T_{u} |
Wi | T | T_{Ω} |
T_{ṽ} |
T_{u} |

13.25 | 1102 | 412 | 1028 | −338 | 9.5 | 875 | 303 | 794 | −222 |

53 | 1117 | 517 | 1042 | −442 | 95 | 876 | 365 | 793 | −282 |

106 | 1135 | 464 | 1058 | −387 | 570 | 912 | 498 | 808 | −394 |

424 | 1363 | 1189 | 1275 | −1101 | 1520 | 1086 | 763 | 919 | −596 |

Stars with harmonic bonds in solution seem to reach the scaling ω/ ∼ Wi^{−1} (i.e. ω → cte) as the shear rate is increased (although, in fact, at very large , ω decreases). This apparent scaling was attributed in ref. 11 (and subsequent citations) to a universal limiting trend for tank-treading rotation of star polymers. However, although the apparent angular velocity ω reaches a maximum value, the tank-treading frequency, ω_{R}, keeps increasing with , like ω_{R} ∼ Wi^{α} with α = 0.5 ± 0.02. This is shown in Fig. 4 where one can see that ω and ω_{R} differ significantly.

Finally, in melts (bottom panel of Fig. 4), we observe that the molecular rotational frequencies are similar in the open and closed environments. This is in agreement with our previous studies (ref. 19 and 27) and indicates that the rheological differences measured in open and closed environments are of thermodynamic origin (density decreases when an open polymer enclosure is sheared).

(29) |

The energy of vibrations with angular momentum corresponds to deformations of the arms away from pure rigid body rotation (see eqn (18)). In solution, these motions arise from Brownian diffusion so we expect that the kinetic energy |T_{u}| is proportional to ΓD_{arm} where D_{arm} is the diffusion coefficient of the center of mass of one star's arm (which is independent of the shear rate). The scaling this hypothesis predicts is validated in Fig. 5 where |T_{u}| (normalized with its value at zero shear rate) is compared with Γτ_{rot} for increasing Weissenberg number. Results for different types of star polymers (harmonic and FENE bonds) confirm that both magnitudes are proportional and indicate that our intuition contains physical insight. In melts, however, both quantities differ significantly (see Fig. 5 bottom panel) indicating that, in this case, molecular deformations are also determined by other (non-Brownian) mechanisms, like inter-molecular collisions.

Fig. 5 The absolute value of kinetic energy related to the vibrational non-angular momentum |T_{u}| compared with the rate of decorrelation (Γ) of the monomer pure rotation around the molecule’s center (see Fig. 3). Both quantities are normalized with their values at zero shear rate. Top and middle panels show results for solution and the bottom panel, for the melt. |

Fig. 6 (a) The angular momentum free vibrational kinetic T_{ṽ} (symbols) and the fit T_{ṽ}(Wi) = T_{ṽ}(0) + aWi^{β} with ΔT_{ṽ} = aWi^{β} the coherent part and T_{ṽ}(0) the thermal contribution (results for stars in solution). (b) The coherent contribution ΔT_{ṽ} is compared with the breathing mode energy estimated as , where G_{1} is the principal eigenvalue of the gyration tensor and Ω_{B} is the breathing frequency, reported in ref. 19 (results for solution). (c) The same as (b) but for the melt case, and . |

We expect that the coherent part of the vibrational energy ΔT_{ṽ}(Wi) comes out from a collective “oscillation” of the molecular shape. Such type of collective vibration was discussed in a previous work on star polymers,^{19} and was referred to as a “breathing mode”. The dynamics of the breathing mode is revealed in the time correlation of the components of the gyration tensor (G_{ij}), given by,^{3,16,19}

(30) |

(31) |

(32) |

In this work, we just consider star polymers with f = 12 arms and m = 6 monomers per arm. According to a recent analysis,^{58} star molecules become chain-like for f < 6, so our stars are within the “colloidal-like” regime. But, what would be the dynamics of more massive stars? While this question is open to future works, we have good reasons to believe that they will be quite similar to that found for f = 12, m = 6. In fact, several computational works on star polymers under dilute^{11} and semidilute^{16} conditions, covered a relatively large range of values of f ≤ 50 and m < 50 and (by defining the proper Weissenberg number) they found that all data for ω collapse in a master curve, indicating that the length of the arms or the functionality was not essentially changing the polymer dynamics. The dynamics would surely change in the case of a semidilute solution (or melt) if the stars had very long arms (m > 100), because entanglements should play a major role in distorting their rotation dynamics. However, we emphasize that the Eckart framework would be still applicable in such regimes and provide valuable dynamic information.

It also has to be noted that the present analysis can be complementary to the more detailed normal mode analysis of vibrations, within the framework of the theory of molecular vibrations.^{29} In the latter, each internally rotating part of the molecule would require the introduction of additional internal coordinate systems inside the translating and rotating Eckart frame.^{34–37,59–63} Presently, we leave this discussion for future work, since the main aim of this paper is the separation of rotations from vibrations, or the consequences such decomposition brings up in the interpretation of molecular rotations. The objective of this work is to show that the Eckart frame, successfully and routinely used to describe infrared and Raman spectra of small molecules, is also a robust and useful tool to investigate the complex dynamics of soft, semiflexible macromolecules.

In all three described definitions of c^{α}_{i}s, the unit base vectors of the internal coordinate system f_{1}, f_{2}, and f_{3} and the origin of the Eckart frame, defined by r_{cm}, are different in every snapshot of the sampled trajectory. Only the reference components c^{α}_{i}s remain constant throughout the whole trajectory in (i) and (ii), while in (iii) also c^{α}_{i}s change in time, as described above. Molecules rotate in the flow-gradient plane. Therefore, the only component of the molecules' angular velocity with non zero-average is in the neutral direction and we denote, ω = (ω_{1}, ω_{2}, ω_{3}) and Ω = (Ω_{1}, Ω_{2}, Ω_{3}), where indices 1, 2, and 3 denote the flow, gradient, and neutral direction, respectively.

To determine the optimal way to define c^{α}_{i}s, we plot, in Fig. 8, angular velocities obtained by the Eckart formalism using definitions (i), (ii), and (iii) for solution of star polymers with 12 arms of 6 monomers (connected by Hookean springs). Plots for the melt are qualitatively similar and are not shown here. We observe that approach (iii), in which c^{α}_{i}s change every τ_{w}, gives the angular velocity surface that at the shortest τ_{w} corresponds to the standard approach (i.e. using eqn (2) and (3)). With increasing τ_{w}, it approaches the values obtained by approaches (i) and (ii). At a certain value of τ_{w}, we observe a sharp crossover in angular velocity of polymers at very high shear rates, which results in qualitatively different dependencies Ω/(Wi) emerging only due to the different reference frames. A similar crossover is also observed in melts, but is more prominent in solutions. Furthermore, we observe that this crossover occurs at higher τ_{w} for the star polymers with longer arms.

Importantly, we find that definitions (i) and (ii) yield basically the same results, which are also similar to the results obtained by definition (iii) after the crossover. Therefore, in the manuscript, we present only results obtained by definition (ii).

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