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A Thermodynamic Model of Physical Gels

Hanqing Jiang's picture

Physical gels are characterized by dynamic cross-linksthat are constantly created and broken, changing its state between solid andliquid under influence of environmental factors.  This restructuring ability of physical gelsmakes them an important class of materials with many applications, such as indrug delivery.  In this article, wepresent a thermodynamic model for physical gels that considers both the elasticproperties of the network and the transient nature of the cross-links.  The cross-links’ reformation is capturedthrough a connectivity tensor M at themicroscopic level.  The macroscopicquantities, such as the volume fraction of the monomer f,number of monomers per cross-link s,and the number of cross-links per volume q,are defined by statistic averaging.  Amean-field energy functional for the gel is constructed based on thesevariables.  The equilibrium equations andthe stress are obtained at the current state. We study the static thermodynamic properties of physical gels predictedby the model.  We discuss the problems ofun-constrained swelling and stress driven phase transitions of physical gelsand describe the conditions under which these phenomena arise as functions ofthe bond activation energy Ea,polymer/solvent interaction parameter c, andexternal stress p.

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Lianhua Ma's picture

 Dear Hanqing,

Thanks for posting your interesting paper where a thermodynamic phenomenological model of gels has been well described at
the microscopic level.

In comparison with Wei’s paper (,
the elastic energy of polymer networks  does not include the
logarithmic term Log(detF) in your work.. I have noted that opinion is divided
regarding the presence of the logarithmic term (Flory, 1976, Rubinstein and Colby(2003)). Because a gel can undergo
large deformations (when the solvent migrates into a polymer network, the
network expands. det F>>1), the presence or absence of the logarithmic
term Log(detF) in the elastic energy function has a great effect on the gel’s volumetric
deformation.  So my question is: which
one is more reasonable to describe the diffusion-induced deformation of gels?



As seen in the literature (Han, W.H., Horkay, F., McKenna, G.B., 1999. Mechanical
and swelling behaviors of rubber: A comparison of some molecular models with
experiment. Mathematics and Mechanics of Solids 4(2), 139-167.
), the elastic energy function without logarithmic term Log(detF) is called Phantom model where the conformation of each chain depends only on the
position of its ends and is independent of the conformations of the surrounding

In this model, the junctions in the network are assumed to be free to fluctuate around their mean positions,
then “the deformation of the mean positions of the end-to-end vectors is not
affine in the strain”

The energy function with Log(detF) is called Affine model, “it is assumed that the
displacement of the mean positions of the junctions should transform linearly
in the macroscopic strain(affine deformation) and, hence, that the transformation
of the distribution of the end-to-end vectors of the chains should likewise be


From the descriptions of the two models, I think the Affine model is more reasonable to characterize the elastic energy of polymer networks in the swelling and shrinking of gels, because the junctions
can change their positions and transform with the increasing of the macroscopic
strain when the network swells by absorbing solvent molecules.


I am not very clear about the two models. 

Could you help me out with the
question about the presence or absence of 







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