Lecture 12: Beyond affine rubber elasticityMacromolecular Engineering: Networks and Gels Prof. Mark W. Tibbitt, 26. March 2020

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

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

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

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

Rubber Elasticity

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

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!!

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Rubber Elasticity

6Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt

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Lecture 12

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Modulus of the network:

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

density of network strands

number average molecular weight of the network strandAAAB8XicbVBNS8NAEJ34WetX1aOXxSJ4KokIeix68SJUsB/YhrLZTtqlm03c3RRK6L/w4kERr/4bb/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHDR2nimGdxSJWrYBqFFxi3XAjsJUopFEgsBkMb6Z+c4RK81g+mHGCfkT7koecUWOlx7suIx18SvmoWyq7FXcGsky8nJQhR61b+ur0YpZGKA0TVOu25ybGz6gynAmcFDupxoSyIe1j21JJI9R+Nrt4Qk6t0iNhrGxJQ2bq74mMRlqPo8B2RtQM9KI3Ff/z2qkJr/yMyyQ1KNl8UZgKYmIyfZ/0uEJmxNgSyhS3txI2oIoyY0Mq2hC8xZeXSeO84rkV7/6iXL3O4yjAMZzAGXhwCVW4hRrUgYGEZ3iFN0c7L8678zFvXXHymSP4A+fzByqKkJM=

AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK9gObUDbbSbt0s4m7m0Ip/RdePCji1X/jzX/jts1BWx8MPN6bYWZemAqujet+O4W19Y3NreJ2aWd3b/+gfHjU1EmmGDZYIhLVDqlGwSU2DDcC26lCGocCW+Hwdua3Rqg0T+SDGacYxLQvecQZNVZ69GVGfHzK+KhbrrhVdw6ySrycVCBHvVv+8nsJy2KUhgmqdcdzUxNMqDKcCZyW/ExjStmQ9rFjqaQx6mAyv3hKzqzSI1GibElD5urviQmNtR7Hoe2MqRnoZW8m/ud1MhNdBxMu08ygZItFUSaIScjsfdLjCpkRY0soU9zeStiAKsqMDalkQ/CWX14lzYuq51a9+8tK7SaPowgncArn4MEV1OAO6tAABhKe4RXeHO28OO/Ox6K14OQzx/AHzucPdMeQww==

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

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Phantom network

7Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt

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Lecture 12

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

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Phantom network

8Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt

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Lecture 12

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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

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Real Elastic Network Theory (RENT)

9Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt

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Lecture 12

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

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Real Elastic Network Theory (RENT)

10Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt

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Lecture 12

Zhong et al. Science 2016, 353, 1264–1268

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Real Elastic Network Theory (RENT)

11Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt

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Lecture 12

Zhong et al. Science 2016, 353, 1264–1268

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

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Entanglements

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

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Lecture 12

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

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Rubber Elasticity - summary

13Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt

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Lecture 12

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.

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