Lecture 12: Beyond affine rubber elasticity
Transcript of 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