Lecture 12: Beyond affine rubber elasticity

Macromolecular Engineering: Networks and Gels

Lecture 12: Beyond affine rubber elasticity Prof. Dr. Mark W. Tibbitt, 1. April 2021

Macromolecular engineering of networks and gels

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Polymeric precursor Complementary polymer

10 nm scaleSolution of polymeric species

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12

Macromolecular engineering of networks and gels

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Polymer network or gel

10 nm scaleViscoelastic insoluble network or gel

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12

Macroscale properties are controlled by molecular details

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Network architecture

Molecular details

+k

Viscoelasticity

DiffusivitySurface

chemistry

Macroscale Properties

Macromolecular details inform material properties and provide a tunable handle in their design.

Swelling

Degradation

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12

Rubber Elasticity

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Consider the thermodynamics of the network under deformation

Affine network model - each polymer chain deforms in the same manner that the whole network deforms

Other assumptions - Gaussian chains - constant volume - flexible chains (T > Tg) - no crystallization at large strain - energetic component of the free energy = 0

Entropy dominates again!!

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12

Rubber Elasticity

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Modulus of the network:

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density of network strands

number average molecular weight of the network strand<latexit sha1_base64="4Uki4P0T/qTyQlRoShTKJqDn4dI=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSJ4KokIeix68SJUsB/YhrLZTtqlm03c3RRK6L/w4kERr/4bb/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHDR2nimGdxSJWrYBqFFxi3XAjsJUopFEgsBkMb6Z+c4RK81g+mHGCfkT7koecUWOlx7suIx18SvmoWyq7FXcGsky8nJQhR61b+ur0YpZGKA0TVOu25ybGz6gynAmcFDupxoSyIe1j21JJI9R+Nrt4Qk6t0iNhrGxJQ2bq74mMRlqPo8B2RtQM9KI3Ff/z2qkJr/yMyyQ1KNl8UZgKYmIyfZ/0uEJmxNgSyhS3txI2oIoyY0Mq2hC8xZeXSeO84rkV7/6iXL3O4yjAMZzAGXhwCVW4hRrUgYGEZ3iFN0c7L8678zFvXXHymSP4A+fzByqKkJM=</latexit>

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Modulus scales with temperature and is inverse with molecular weight between crosslinks.

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12

Phantom network

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Consider that the junction points in the network can fluctuate but the mean locations are affine. In addition, these fluctuations are Gaussian and independent of the strain.

Chains are connected to an elastic, non-fluctuating background

Polymer Physics Rubinstein and Colby

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12

Phantom network

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Decrease in expected modulus as compared to affine network model

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density of crosslinks

Polymer Physics Rubinstein and Colby

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12

Real Elastic Network Theory (RENT)

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In reality, many loops can be present in the network. Should

account for these also!

Zhong et al. Science 2016, 353, 1264–1268Networks aren’t always ideal

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12

Real Elastic Network Theory (RENT)

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tractionews.comZhong et al. Science 2016, 353, 1264–1268

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12

Real Elastic Network Theory (RENT)

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tractionews.comZhong et al. Science 2016, 353, 1264–1268

A further decrease in expected modulus from the phantom network model because of network defects!

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12

Entanglements

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Consider a ‘second’ network that is cross-linked by entanglements

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number average molecular weight between entanglements

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Total modulus for the system that is the sum of the contributions from cross-

links and entanglements

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12

Rubber Elasticity - summary

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The mechanical properties of networks and gels are imparted mostly by changes in entropy of network strands upon deformation.

The modulus G of a network or gel is proportional to the number density of elastically active network strands with each strand contributing kT.

When calculating modulus it is important to take into account network topology such as loops and entanglements.

Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt Lecture 12