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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tractionews.com
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:
AAAB+3icbVBNS8NAEN3Ur1q/Yj16WSyCp5KIoMdSD3qs0C9oQ9hsJ+3SzSbubool9K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5QcKZ0o7zbRU2Nre2d4q7pb39g8Mj+7jcVnEqKbRozGPZDYgCzgS0NNMcuokEEgUcOsH4du53JiAVi0VTTxPwIjIULGSUaCP5dvkO9+ExZRPcFyke+3Xc9O2KU3UWwOvEzUkF5Wj49ld/ENM0AqEpJ0r1XCfRXkakZpTDrNRPFSSEjskQeoYKEoHyssXtM3xulAEOY2lKaLxQf09kJFJqGgWmMyJ6pFa9ufif10t1eONlTCSpBkGXi8KUYx3jeRB4wCRQzaeGECqZuRXTEZGEahNXyYTgrr68TtqXVdepug9XlVo9j6OITtEZukAuukY1dI8aqIUoekLP6BW9WTPrxXq3PpatBSufOUF/YH3+AErJk0s=
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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
AAACG3icbVDLSsNAFJ34tr6iLt1cLEJdWJIi6EYQXehSwT6gKWEynejQySTOo1BC/sONv+LGhSKuBBf+jdPahbYeuNzDOfcyc0+Ucaa05305M7Nz8wuLS8ulldW19Q13c6uhUiMJrZOUp7IVYUU5E7Sumea0lUmKk4jTZtQ7H/rNPpWKpeJGDzLaSfCtYDEjWFspdGsXcAK98AxuoBIIAwcQJGYfgqBkOwT03rA+BLHEJK9Zv8jjInTLXtUbAaaJPyZlNMZV6H4E3ZSYhApNOFaq7XuZ7uRYakY4LUqBUTTDpIdvadtSgROqOvnotgL2rNKFOJW2hIaR+nsjx4lSgySykwnWd2rSG4r/eW2j4+NOzkRmNBXk56HYcNApDIOCLpOUaD6wBBPJ7F+B3GEbhLZxlmwI/uTJ06RRq/pe1b8+LJ+ejeNYQjtoF1WQj47QKbpEV6iOCHpAT+gFvTqPzrPz5rz/jM44451t9AfO5zfOQJ7j
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
http://tractionews.com
Real Elastic Network Theory (RENT)
11Macromolecular Engineering: Networks and Gels Instructor: Prof. Tibbitt
tractionews.com
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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