Supporting Information
Modeling Controlled Photodegradation in Optically Thick Hydrogels
Mark W. Tibbitt1, April M. Kloxin1,2,#, and Kristi S. Anseth1,2
1Department
of Chemical and Biological Engineering, University of Colorado Boulder,
Boulder, Colorado 80303
2Howard
Hughes Medical Institute, University of Colorado Boulder, Boulder, Colorado
80303
#Current
address: Departments of Chemical and Biomolecular Engineering and
Materials Science and Engineering, University of Delaware, Newark, Delaware 19716
Correspondence to: Prof. Kristi S. Anseth (E-mail: kristi.anseth@colorado.edu)
This file includes:
Supporting Text
Supporting Information References
Supporting Text
Description of the relationship between shear modulus and degradation kinetics
To calculate the kinetic rate of photodegradation from rheometric measurements,
it is necessary to relate the measured quantity (shear modulus, G) to the kinetic rate of
photodegradation, k. From Rubber Elasticity Theory, it is known that G is directly
related to the crosslinking density, xl, of a polymer network.1 This implies that the
normalized modulus is equivalent to the normalized crosslinking density.
𝐺′
𝐺0′
𝜌
= 𝜌 xl
Eq. S1
xl0
The crosslinking density at a given point in space and time, xl(z,t), is calculated by the
product of the initial crosslinking density, xl,0, and the fraction of intact crosslinks at the
same point in space and time, f1(z,t).
𝜌xl (𝑧, 𝑡) = 𝑓1 (𝑧, 𝑡) ⋅ 𝜌xl0
Eq. S2
As discussed in the main text of the manuscript, f1(z,t) can be calculated directly from
the probability that a crosslink is intact at the given point in space and time.
𝑓1 (𝑧, 𝑡) = (1 − 𝑃(𝑧, 𝑡))2
Eq. S3
Eq. S3 is quadratic as the PEGdiPDA crosslinker contains two NBE moieties that both
must remain intact for the crosslink itself to remain fully intact. P(z,t) is defined as the
probability of a given NBE moiety being cleaved at a given point in space and time,
which can be described as an exponential decay assuming that the intensity of light
remains constant during the course of the rheometry experiment and the gel is an
optically thin film in this experiment.2
[NBE](𝑧,𝑡)
𝑃(𝑧, 𝑡) = 1 − [NBE]
0 (𝑧,𝑡)
= 1 − exp(−𝑘𝑡)
Eq. S4
Combining, Eqs. S3 and S4:
𝑓1 (𝑧, 𝑡) = (exp(−𝑘𝑡))2 = exp(−2𝑘𝑡)
Eq. S5
Therefore, from Eqs. S1 and S2:
𝐺′
𝐺0′
𝜌
= 𝜌 xl = exp(−2𝑘𝑡)
Eq. S6
xl0
Finally, by taking the natural log of both sides:
𝐺′
ln 𝐺′ = −2𝑘𝑡
Eq. S7
0
Thus, the slope of a linear fit of a plot of the natural log of the normalized shear modulus
as a function of time can be used to calculate the kinetic rate of photodegradation, k.
The factor of two in Eq. S7 was introduced by the fact that the crosslinking molecule
contains two NBE moieties, resulting in Eq. S3 becoming quadratic.
Assumption of the number of crosslinks per kinetic chain, N
To determine the number of crosslinks per kinetic chain for a chain polymerized
hydrogel, such as the photodegradable gel presented in this manuscript, one can use
mass distributions of the cleaved species to determine the length of the kinetic chains.
That is, if the gel is eroded completely into soluble degradation products (i and ii from
Figure 1c) by cleaving every NBE moiety of the original network, chromatography can,
in principle, be employed to measure the mass of each degraded species from the gel.
Precise measurements can instruct the number of crosslinks in each of the polyacrylate
kinetic chains to measure a value of N, the number of crosslinks per kinetic chain.
However, gel permeation chromatography (GPC) data failed to provide reliable
calculations of the masses of the degradation products from these networks and this
method was not employed to calculate precisely the number of crosslinks per kinetic
chain, N. Instead, an estimation of N based on the time to reach reverse gelation in the
10 mol%:90 mol% PEGdiPDA:PEGA hydrogel was used to guide an assumption of N
for the gel systems used in this paper. The following is an explanation of the logic for
the assumption of N for this system. However, if N cannot be estimated easily it can be
used as a free parameter of the model to fit erosion predictions to experimental data.
A time to reach reverse gelation was estimated from rheometric measurements of
photodegradation of the 10 mol%:90 mol% PEGdiPDA:PEGA hydrogels. In these
experiments, the rheometer was no longer able to measure reliably a storage and loss
modulus for the gel system after ~ 7 minutes for I = 10 mW cm-2 at  = 365 nm.
Therefore, we estimated that the time to reach reverse gelation, trg, was ~ 420 seconds.
From Eq. S4,
𝑃(𝑧, 𝑡) = 1 − exp(−𝑘𝑡)
Eq. S8
From the main text of the manuscript the critical extent of NBE cleavage to reach
reverse gelation (Pc; Eq. 11) can be written as
𝑃𝑐 = 1 −
1
√𝑁−1
Eq. S9
where N is the average number of crosslinks per kinetic chain. By equating Eqs. S8 and
S9 at t = trg and solving for N, one can estimate the average number of crosslinks per
kinetic chain.
1
= exp(−𝑘𝑡𝑟𝑔 )
Eq. S10
√𝑁 − 1 = exp(𝑘𝑡𝑟𝑔 )
Eq. S11
√𝑁−1
𝑁 = 1 + exp(2𝑘𝑡𝑟𝑔 )
Eq. S12
The kinetic rate of photodegradation, k, was calculated to be 0.0035 s-1 at I = 10 mW
cm-2 and  = 365 nm, which correspongs to trg = 420 seconds. Solving for N with k =
0.0035 s-1 and trg = 420 s, leads to N  20. Therefore, the average number of crosslinks
per kinetic chain was assumed to be 20 for the gel systems in this work. Since the initial
assumption, N = 20, fit the data reasonably well for all gel systems in this work, further
refinement of N was not conducted in this work.
Supporting Information References
1. R.J. Young and P.A. Lovell Introduction to Polymers, Chapman & Hall, London,
U.K. 1991
2. A.M. Kloxin, M.W. Tibbitt, A.M. Kasko, J. Fairbairn, and K.S. Anseth. Tunable
Hydrogels for External Manipulation of Cellular Microenvironments Through
Controlled Photodegradation Adv Matls 2010 22: 61-66.