Supporting Information
Interpenetrating Polymer Networks with Gradient Mechanical Properties
for Hydrogel Based tendon-Muscle Systems
Sina Naficy1, Shota Kawakami2, Sasha Sadegholvaad1, Minato Wakisaka2 and Geoffrey M.
Spinks1
1
ARC Centre of Excellence in Electromaterials Science and Intelligent Polymer Research Institute,
AIIM Faculty, Innovation Campus, University of Wollongong, Squires Way, North Wollongong,
NSW, 2522, Australia.
2
Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology, 2-4
Hibikino, Wakamatsu-ku, Kitakyushu, 808-0196, Japan.
Correspondence to: Geoffrey Spinks (E-mail: gspinks@uow.edu.au )
Swelling of Hydrogels
The modulus of a polymer network can be expressed as:
𝐺 = 𝜐 × 𝐹𝑒𝑙
S1
where 𝜐 is the number of elastically effective polymer chains per unit volume and 𝐹𝑒𝑙 is the elastic
free energy per chain. In a simplified model, the number of elastically effective polymer chains per
unit volume 𝜐 can be determined as a function of polymer network volume fraction (𝜙) and the
volume of a polymer strand connecting two adjacent crosslink points: 𝜐 = 𝜙/𝑁𝑏 3 , where 𝑁 is the
number of building units of a polymer strand, and 𝑏 is the size of the building units. Based on ideas of
Panyukov, the elastic free energy of a single chain in a network swollen in a solvent can be related to
the chain’s mean-square end-to-end distance (𝑅 2) and the mean-square end-to-end distance of an
2
identical free chian in that solvent (𝑅𝑟𝑒𝑓
):1
𝑅2
𝐹𝑒𝑙 ≈ 𝑘𝑇 𝑅2
S2
𝑟𝑒𝑓
If the end-to-end distance of polymer strands in the as-prepared network is 𝑅𝑜 , with the affine
deformation assumption the end-to-end distance of that network after swelling in the solvent is: 𝑅 2 =
(𝜆𝑅𝑜 )2 . Here, 𝜆 is the linear deformation of strands of the unconfined network upon swelling in the
solvent. 𝜆 can be determined based on the network volume fraction in the solvent 𝜙 and the asprepared network volume fraction (𝜙𝑜 ): 𝜆 = (𝜙𝑜 /𝜙)1/3 . By replacing the above parameters in
equations S1 and S2 the resulting modulus is:
𝜙
𝐺(𝜙) = 𝑁𝑏3 × 𝑘𝑇
(𝜆𝑅𝑜 )2
2
𝑅𝑟𝑒𝑓
S3
Swelling in θ Solvents
2
In a θ solvent the mean-square end-to-end distance of a free chain 𝑅𝑟𝑒𝑓
is independent of the
concentration, and is equal to the mean-square end-to-end distance of a random coil:
Supporting Information
2
𝑅𝑟𝑒𝑓
≈ 𝑅𝑜2 ≈ 𝑁𝑏 2
S4
2
By substituting 𝑅𝑟𝑒𝑓
and 𝑅𝑜2 in equation S3 using equation S4 the network modulus is obtained as a
function of network volume fraction upon swelling 𝜙, as-prepared network volume fraction 𝜙𝑜 and
network structure (i.e. 𝑁):2
𝑘𝑇
2/3
𝐺(𝜙) ≈ 𝑁𝑏3 𝜙𝑜 𝜙1/3
S5
The network volume fraction in equation S5 is related to the network swelling ratio (𝑞) by: 𝑞 = 1/𝜙.
However, it is important to emphasise that the swelling ratio here is not necessarily the equilibrium
swelling ratio (𝑄). The equilibrium swelling ratio can be determined by assuming that equilibrium is
reached when the network modulus 𝐺𝑒𝑞 is equal to the osmotic pressure Π𝑒𝑞 : 𝐺𝑒𝑞 ≈ Π𝑒𝑞 .3 The meanfield prediction for the osmotic pressure of a polymer solution with the polymer concentration of 𝜙 in
a θ solvent is: Π =
𝑘𝑇 3
𝜙 .
𝑏3
Using this expression for the osmotic pressure along with equation S5 the
equilibrium swelling ratio 𝑄 is obtained as:
𝑁 3/8
𝑄≈
1/4
𝜙𝑜
S6
At equilibrium, 𝜙𝑒𝑞 = 1/𝑄. Thus, equation S5 can be rewritten using equation S6 to obtain the
network modulus as a function of equilibrium swelling ratio:
𝐺(𝑄) ≈
𝑘𝑇 −3
𝑄
𝑏3
S7
Swelling in Athermal Solvents
In an athermal solvent the chains are more extended compared to a θ solvent. In the athermal solvent
the end-to-end distance of a polymer strand (i.e. 𝑅𝑜 or 𝑅𝑟𝑒𝑓 ) is a function of polymer volume fraction
(i.e. 𝜙𝑜 or 𝜙):4
𝑅 = (𝑏𝑁 1⁄2 )𝜙 −(𝑚−1⁄2)/(3𝑚−1)
S8
Using equations S3 and S8, the modulus of a network swollen in an athermal solvent is:
𝐺(𝜙) ≈
𝑘𝑇
1⁄[3(3𝑚−1)] (9𝑚−4)⁄[3(3𝑚−1)]
𝜙
𝜙
𝑁𝑏3 𝑜
S9
The osmotic pressure of a polymer solution in an athermal solvent can be considered to be: Π =
𝑘𝑇 3𝑚⁄(3𝑚−1)
𝜙
.
𝑏3
Again, the equilibrium swelling ratio is obtained by 𝐺𝑒𝑞 ≈ Π𝑒𝑞 , using equation S9:
𝑄=
𝑁 3(3𝑚−1)⁄4
1/4
𝜙𝑜
S10
The exponent 𝑚 is assumed to be around 𝑚 ≅ 0.588,5 resulting in 𝐺(𝜙)~𝑁 −1 𝜙𝑜0.44 𝜙 0.56 and
𝑄~𝑁 0.57 𝜙𝑜−0.25. At equilibrium, the network modulus can be expressed as a function of equilibrium
swelling ratio by substituting equation S10 in equation S9 for 𝜙𝑒𝑞 = 1/𝑄:
𝐺(𝑄) ≈
𝑘𝑇 −2.3
𝑄
𝑏3
S11
Supporting Information
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1976.
5. Rubinstein, M.; Colby, R. H. In Polymer Physics; Oxford University Press: Oxford, 2003.