# Fluctuating Entanglements in Single-Chain Mean-Field Models

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## Abstract

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## 1. Introduction

**Consistency with multi-chain models.**The single-chain model should predict the same magnitude for the elastic modulus as multi-chain models and therefore estimate the same entanglement density when comparing with rheological data. Cross-linked network calculations have already established the importance of node fluctuations for this agreement [30,31,32,33,34]. In the present paper, we verify only qualitatively whether the effect of ESFs on the plateau modulus in single-chain mean-field models is consistent with that observed in multi-chain models. Specifically, the plateau modulus should decrease when the size of ESFs increases.**Consistency with nonequilibrium thermodynamics.**Equilibrium thermodynamics and statistical mechanics can be applied in a straightforward way to cross-linked systems [52]. Similarly one can apply nonequilibrium thermodynamics to entangled melts, using either GENERIC or a simple virtual-work argument to derive the stress tensor. For cross-linked systems we simply require that the entropy of the universe not decrease when reversibly deforming a network.**Consistency with the stress-optic rule (for moderate deformations).**It is well established experimentally [38] that polymer melts show a linear relationship between the stress tensor and the refractive index tensor when the chains are Gaussian [53,54]. The dependence of the refractive index tensor on chain conformations has a straightforward derivation [55], so we assume that it is correct. The refractive index tensor should be compared to the stress tensor from criterion 2.**Consistency between Green–Kubo predictions and the relaxation modulus predicted for infinitesimal deformations.**Linear response theory requires that the relaxation of stress from small external perturbations should be the same as relaxation from small fluctuations that arise at equilibrium [56]. Both expressions require as input an expression for the stress tensor, and this expression should come from criterion 2.

## 2. Existing Entangled Melt Models and the Four Criteria

#### 2.1. Doi–Edwards Model and Öttinger’s Generalization

**Table 1.**Several models/simulations and their agreement with the four criteria, which are numbered the same as in Section 1: ESFs = entanglement spatial fluctuations, NETD = nonequilibrium thermodynamics, SOR = stress-optic rule, OC = overcounting, GKR = Green–Kubo relation, NA = not applicable. The last three entries are cross-linked network models. The others are entangled melt models/simulations.

Model ↓ | Criterion → | 1: ESFs | 2: NETD | 3: SOR/OC | 4: GKR |
---|---|---|---|---|---|

Doi–Edwards model [1,2,3,4,5] | ✕ | ✓ | ✓ | ✓ | |

Öttinger model [63] | ✓ | ✓ | ? | ✓ | |

Associating polymer model I [41] | ✕ | ✓ | ✕ | ✓ | |

Associating polymer model II [41] | ✓ | ✕ | ✓ | ✓ | |

Slip-link model [10,64] | ✕ | ✓ | ✓ | ✓ | |

Slip-spring simulation (2005) [65] | ✓ | ✕ | ✓ | ✕ | |

Slip-spring simulation (2007) [39] | ✓ | ✕ | ✓ | ✓ | |

PCN simulation [66] | ✓ | ✕ | ✓ | ✓ | |

Heinrich–Straube–Helmis model [57] | ✓ | ✓ | ✓ | NA | |

Rubinstein–Panyukov model [58,59] | ✓ | ✓ | ✓ | NA | |

Everaers model [60] | ✓ | ✓ | ✓ | NA |

#### 2.2. Associating Polymers

#### 2.3. Slip-Link Model

#### 2.4. Slip-Spring Simulation

#### 2.5. PCN Simulation

## 3. Proposed Virtual-Spring Dynamics of Ronca and Allegra

**Figure 1.**Sketch of single-strand network model on detailed level of description. The strand (thick black line) is attached to cross-links (red dots) connected to affinely moving anchors (blue crosses) by virtual springs (thin black lines). Position vectors and connector vectors are indicated by black and colored arrows, respectively.

**Figure 2.**Sketch of proposed slip-link model on detailed level of description. The chain (thick black line) passes through slip-links (red circles) connected to affinely moving anchors (blue crosses) by virtual springs (thin black lines). Position vectors and connector vectors are indicated by black and colored arrows, respectively.

## 4. Examples: Single-Strand Mean-Field Unentangled Network Models

#### 4.1. Detailed Single-Strand Unentangled Network Model

#### 4.1.1. Stress Tensor and Stress-Optic Rule

#### 4.1.2. Green–Kubo Relation and Infinitesimal Deformations

#### 4.2. Coarse-Grained Single-Strand Unentangled Network Model

#### 4.2.1. Stress Tensor and Stress-Optic Rule

#### 4.2.2. Green–Kubo Relation and Infinitesimal Deformations

#### 4.2.3. Consistency with the Multi-Strand Front Factor

## 5. Proposed Slip-Link Models

#### 5.1. Detailed Mobile Slip-Link Model

#### 5.2. Coarse-Grained Mobile Slip-Link Model

## 6. Conclusions

## Acknowledgements

## Appendix

## A. Differential Formulation of the Molecular Stress Function Model

## B. Details of the Single-Strand Mean-Field Unentangled Network Models

#### B.1. Derivation of the Mean Strand Conformation

#### B.2. Coarse-Graining

#### B.3. Change of Free Energy and Mean Path Under Infinitesimal Deformation

## C. Coarse-Graining of the Detailed Mobile Slip-Link Model

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## Share and Cite

**MDPI and ACS Style**

Schieber, J.D.; Indei, T.; Steenbakkers, R.J.A. Fluctuating Entanglements in Single-Chain Mean-Field Models. *Polymers* **2013**, *5*, 643-678.
https://doi.org/10.3390/polym5020643

**AMA Style**

Schieber JD, Indei T, Steenbakkers RJA. Fluctuating Entanglements in Single-Chain Mean-Field Models. *Polymers*. 2013; 5(2):643-678.
https://doi.org/10.3390/polym5020643

**Chicago/Turabian Style**

Schieber, Jay D., Tsutomu Indei, and Rudi J. A. Steenbakkers. 2013. "Fluctuating Entanglements in Single-Chain Mean-Field Models" *Polymers* 5, no. 2: 643-678.
https://doi.org/10.3390/polym5020643