# Revisiting the strain-induced softening behaviour in hydrogels

L. K. R. Duarte  Departamento de Física, Universidade Federal de Viçosa (UFV),
36.570-900, Viçosa, MG, Brazil.  Instituto Federal de Educação, Ciência e
Tecnologia de Minas Gerais, 35.588-000, Arcos, MG, Brazil.  L. G. Rizzi
Departamento de Física, Universidade Federal de Viçosa (UFV), 36.570-900,
Viçosa, MG, Brazil.

###### Abstract

Usually, the strain-induced softening behaviour observed in the differential
modulus $K(T,\gamma)$ of hydrogels has been attributed to the breakage of
internal structures of the network, such as the cross-links that bind together
the polymer chains. Here we consider a stress-strain relationship that we have
recently derived from a coarse-grained model to demonstrate that no rupture of
the network is needed for rubber-like gels to present such behaviour. In
particular, we show that, in some cases, the decreasing of $K(T,\gamma)$ as a
function of the strain $\gamma$ is closely related to the energy-related
contribution to the elastic modulus that has been experimentally observed,
e.g., for tetra-PEG hydrogels. Thus, our results suggest that, instead of the
breakage of structures, the softening behaviour can be also related to the
effective interaction between the chains in the network and their neighbouring
solvent molecules. Comparison to experimental data determined for several
hydrogels is included to illustrate that behaviour and to validate our
approach.

##  1 Introduction

Hydrogels, like many soft materials, can display a strain-induced softening
behaviour which is characterized by a decrease in the differential modulus at
intermediate deformations. Figure 1 illustrates such behaviour for a collagen-
based hydrogel, with the differential modulus defined as

$K(T,\gamma)=\left[\dfrac{\partial}{\partial\gamma}\sigma(T,\gamma)\right]_{T}~{}~{},$
(1)

where $\sigma(T,\gamma)$ is the stress for a given strain $\gamma$ at a given
temperature $T$. Investigating the origin of such effect in hydrogels is
crucial for gaining insights into a wide range of viscoelastic materials
Ferraro  _et al._ (2023); Richtering and Saunders (2014), which includes not
only biopolymer-based gels Lee  _et al._ (2010); Orakdogen, Erman, and Okay
(2010); Kurniawan, Wong, and Rajagopalan (2012); Kraxner  _et al._ (2021), but
also industry-related gels that are related to innovative therapies and
medical applications Annabi  _et al._ (2014); Thiele  _et al._ (2014); Kamata
_et al._ (2015); Caló and Khutoryanskiy (2015); Basu  _et al._ (2011); Wen
_et al._ (2012); Dai  _et al._ (2021).

In general, the softening phenomenon is attributed either to the breakage of
internal structures of the material, as in the case of reversibly crosslinking
in peptide gels Greenfield  _et al._ (2010) and polymer networks Xu and Craig
(2011); Burla  _et al._ (2019), or to the yielding, as in the case of soft
glasses Rogers, Park, and Lee (2019). Usually, the differential modulus
$K(T,\gamma)$ defined by Eq. (1) is evaluated from tensile experiments, but it
is worth noting that it can be also estimated from oscillatory experiments
Larson (1999), where in the later one has that
$K(T,\gamma)\approx\lim_{\omega\rightarrow 0}G^{\prime}(\omega,T,\gamma)$,
with $G^{\prime}$ being the storage modulus. Hence, one also has that
$G(T)=\lim_{\gamma\rightarrow 0}K(T,\gamma)$ corresponds to the usual elastic
(i.e., shear) modulus that characterizes the mechanical response of isotropic
semisolid materials at their linear viscoelastic (LVE) regime. Oscillatory
experiments can be particularly important since they can easily identify the
breakage of internal structures of the gel, which may become a liquid-like
material at high enough strains, so the loss modulus $G^{\prime\prime}$
surpasses the storage modulus $G^{\prime}$ (see, e.g., Refs. Greenfield  _et
al._ (2010); Rogers, Park, and Lee (2019); Yu  _et al._ (2021)).

Intriguingly, some hydrogels do display a strain-induced softening behaviour
as the one illustrated by Fig. 1 while keeping their semisolid character
unchanged, i.e., with $G^{\prime}>G^{\prime\prime}$. Examples include not only
the collagen-based hydrogel of Ref. Valero  _et al._ (2018), but also several
other gels, including those based on tetra-PEG polymers Kamata  _et al._
(2014, 2015) and some based on biofilaments Storm  _et al._ (2005); Motte and
Kaufman (2012); Oztoprak and Okay (2017). Indeed, numerical simulations of
sparsely connected networks Bouzid and Gado (2018) indicate that the softening
behaviour observed in the differential modulus can occur even without breaking
the bonds of the gel network. The caveat here is that the decrease in $K$
observed in Ref. Bouzid and Gado (2018) is obtained from simulations where
only enthalpic contributions were considered, which is the usual approach
assumed in many numerical studies of semiflexible polymer networks Heidemann
_et al._ (2015); Huisman, Storm, and Barkema (2010); Huisman and Lubensky
(2011). Given the significant presence of solvent within the network structure
of the hydrogels, the assumption of an athermal Rizzi (2022), i.e., purely
enthalpic, mechanical response is in clear contrast to the purely entropic
approaches, such as the ones considered in the classical theories Flory
(1953); James and Guth (1953); Flory (1977). Recent experimental Yoshikawa
_et al._ (2021); Sakumichi, Yoshikawa, and Sakai (2021); Aoyama and Urayama
(2023) and computational studies Shirai and Sakumichi (2023); Hagita  _et al._
(2023) have indicated, however, that purely entropic, e.g., rubber-based
Treloar (1975), theories are also insufficient for describing the mechanical
properties of hydrogels, and a significant energy-related contribution is
required to describe their elastic behaviour.

Figure 1: Differential modulus $K(T,\gamma)$ as a function of the strain
$\gamma$ for two different samples of a collagen-based hydrogel, i.e., with
and without crosslinking agent (transglutaminase II, TG2). Dash-dotted and
continuous lines correspond to the results obtained from Eq. (11) with $\Delta
E=0$ and $\Delta E>0$, respectively (see Sec. 3.3 for the values of all
parameters). Filled symbols correspond to the experimental data extracted from
Ref. Valero  _et al._ (2018).

Although models based on worm-like chains (WLC) are expected to encompass the
stress-strain relationship of biopolymer hydrogels Palmer and Boyce (2008);
MacKintosh, Käs, and Janmey (1995), in most cases they seem to be able to
describe only their hardening behaviour. In principle, the lack of the
softening effect can be rationalized in terms of the well-known force-
extension relation introduced by Marko & Siggia Marko and Siggia (1995), where
the force $f_{\text{WLC}}$ is a monotonically increasing function of the
deformation, with $\partial f_{\text{WLC}}/\partial z>0$ and
$\partial^{2}f_{\text{WLC}}/\partial z^{2}>0$ for any value of the deformation
$z$. Thus, the same strictly increasing behaviour is expected for its
differential modulus, and no softening behaviour occurs since $K_{\text{WLC}}$
is proportional to the first derivative of $f_{\text{WLC}}$. Alternatively,
one can access the response of hydrogels due to both enthalpic and entropic
contributions by considering a simple coarse-grained model that we have
recently introduced in Ref. Duarte and Rizzi (2023), which takes into account
the interaction of the chains in the network with their neighbouring solvent
molecules. Hence, the main idea of this study is to consider the possibility
that such model could explain the strain-induced softening behaviour that is
being observed in several hydrogels, including the one related to the results
displayed in Fig. 1.

The remainder of the paper is given as follows. Section 2 include the minimum
information about the model presented in Ref. Duarte and Rizzi (2023), and
also the relevant information about its connection to the experimental results
and the phenomenological model of Ref. Yoshikawa  _et al._ (2021). In Sec. 3
we derive the theoretical expression for the differential modulus
$K(T,\gamma)$ and other relationships between experimentally relevant
quantities. In that section we also present the main results and validate our
approach through comparisons to experimental data obtained from different
hydrogels, including several synthetic and biopolymer-based ones. Section 4
include some concluding remarks where we discuss the relevance and limitations
of our approach.

##  2 Previous results

###  2.1 Theoretical results from Duarte & Rizzi Duarte and Rizzi (2023)

As demonstrated in Ref. Duarte and Rizzi (2023), rubber-like hydrogels can be
described by a stress-strain relationship that is given by

$\sigma(T,\gamma)=\bar{n}\left\\{\Delta
E+k_{B}T\ln\left[\frac{\gamma_{b}(T)+\gamma}{\gamma_{s}(T)-\gamma}\right]\right\\}~{},$
(2)

where $\bar{n}=n_{e}(q_{\ell}^{\,}+1)/[2(q_{\ell}^{\,}-1)]$, with $n_{e}$
being related to the number density of elastic elements within the network,
i.e., chains which are connected by cross-links, and
$q_{\ell}=-\ell_{s}/\ell_{b}$. Equation (2) is obtained from a two-state
model, so that the parameters $\ell_{b}$ and $\ell_{s}>|\ell_{b}|$ correspond
to the end-to-end displacements of the blob (b) and stretched (s)
conformational states which are available for each of the $N$ segments that
compose a chain that connects two cross-links in the gel network (see Ref.
Duarte and Rizzi (2023) for further details). It is assumed that these two
possible states have energies $E_{b}$ and $E_{s}$, so that $\Delta
E=E_{s}-E_{b}$ corresponds to the energy difference between the $s$ and $b$
conformations. The temperature-dependent functions $\gamma_{s}(T)$ and
$\gamma_{b}(T)$ should also depend on the values of $q_{\ell}$ and $\Delta E$,
thus making Eq. (2) with just three parameters. Explicitly, these functions
can be written as Duarte and Rizzi (2023)

$\gamma_{b}(T)=\frac{q_{\ell}^{\,}+1}{q_{\ell}^{\,}-e^{\Delta E/k_{B}T}}~{},$
(3)

and

$\gamma_{s}(T)=\frac{q_{\ell}^{\,}+1}{q_{\ell}^{\,}e^{-\Delta
E/k_{B}T}-1}~{}.$ (4)

Each chain in the gel network may be stretched up to a maximum length
$\ell_{c}=N\ell_{s}$, which is approached when the applied stress becomes
large enough. In addition, the natural end-to-end distance of a chain in the
gel network is determined in the absence of stress and is given by Duarte and
Rizzi (2023)

$\ell_{0}^{\,}(T)=\dfrac{\ell_{c}}{q_{\ell}^{\,}}\left(\dfrac{q_{\ell}^{\,}-e^{\Delta
E/k_{B}T}}{1+e^{\Delta E/k_{B}T}}\right)~{}.$ (5)

The above expression indicates that $\ell_{0}^{\,}=\ell_{0}^{\,}(T)$ is a
fraction of its maximum contour length $\ell_{c}$, regardless if
$q_{\ell}^{\,}$ is positive or negative. With $|q_{\ell}|>1$, the distance
$\ell_{0}^{\,}>0$ allows one to define the strain
$\gamma=(\ell-\ell_{0}^{\,})/\ell_{0}^{\,}$, with $\ell>0$ denoting the mean
end-to-end distance between cross-links when stress is applied to the network
Duarte and Rizzi (2023). Moreover, one may infer that $\gamma_{s}$, as defined
by Eq. (4), is the maximum strain supported by a chain in the network. In
fact, if the integrity of the internal structures of the network are to be
preserved, $\gamma_{s}$ also corresponds to the maximum strain supported by
the hydrogel as a whole. Its role as the maximum strain is also confirmed by
Eq. (5), from where one can find that
$\gamma_{s}=(\ell_{c}-\ell_{0}^{\,})/\ell_{0}^{\,}$, which is precisely the
result one would get from expression (4).

From the stress-strain relationship, Eq. (2), one can obtain the elastic
modulus of the hydrogel through its definition, that is,

$G(T)=\lim_{\gamma\rightarrow
0}\left[\frac{\partial}{\partial\gamma}\sigma(T,\gamma)\right]_{T}=n_{e}k_{B}T\phi(T)~{},$
(6)

where the temperature-dependent factor is given by

$\phi(T)=\left(\dfrac{1+e^{\Delta
E/k_{B}T}}{2}\right)\left(\frac{q_{\ell}^{\,}e^{-\Delta
E/k_{B}T}-1}{q_{\ell}^{\,}-1}\right)~{}.$ (7)

It is worth noting that, if there is no energy difference between the $s$ and
$b$ conformational states, i.e., $\Delta E=0$, then $\phi(T)=1$, at any
temperature. In this case, the model recovers the usual result obtained from
purely entropic cross-linked polymer networks Rubinstein and Colby (2003),
where the elastic modulus is equal to its positive contribution, i.e.,
$G(T)=G_{S}(T)=n_{e}k_{B}T$. Hence, by assuming that the elastic modulus is
given by $G(T)=G_{S}(T)+G_{E}(T)$ as in Refs. Yoshikawa  _et al._ (2021);
Sakumichi, Yoshikawa, and Sakai (2021), one have that the energy-related
contribution $G_{E}(T)$ is zero.

On the other hand, when $\Delta E>0$, the function $\phi(T)$ behaves
differently when describing distinct types of hydrogels. For
$q_{\ell}^{\,}>1$, Eq. (7) revealed the existence of a temperature
$T_{0}^{*}$, which defines a range of temperatures where the elastic modulus
$G(T)$ is positive, that is, $T>T_{0}^{*}$. In this case, the energy-related
contribution to the elastic modulus is no longer zero, and, in fact, it is
significantly negative, $G_{E}(T)<0$. As discussed in Ref. Duarte and Rizzi
(2023), for $q_{\ell}^{\,}>1$, the temperature $T_{0}^{*}$ is found to be
given by

$T_{0}^{*}=\dfrac{\Delta E}{k_{B}\ln q_{\ell}^{\,}}~{}.$ (8)

When $q_{\ell}^{\,}<-1$, however, one finds from Eq. (7) that no threshold
temperature exists, since $\phi(T)$ is positive for any temperature. In this
case, the energy-related contribution $G_{E}(T)$ may be either positive or
negative, depending on the temperature of the system. This later case, where
$q_{\ell}^{\,}<-1$, will be discussed in detail in Sec. 3.3, while the former
case, where $q_{\ell}^{\,}>1$, is directly related to the experiments and to
the phenomenological model presented in Ref. Yoshikawa  _et al._ (2021), which
are summarized in the following.

###  2.2 Connections to the results from Yoshikawa et al. Yoshikawa  _et al._
(2021)

In Ref. Yoshikawa  _et al._ (2021), Yoshikawa et. al. presented extensive
experimental evidence for a negative energy-related contribution to the
elastic modulus for tetra-PEG hydrogels, which are synthetically designed gels
with well established topology. In order to describe their data, they
introduced a phenomenological model where the stress-strain relationship is
expressed as Yoshikawa  _et al._ (2021)

$\sigma(T,\gamma)=a(T-T_{0})\,\gamma~{}~{},$ (9)

with $a$ and $T_{0}$ being system-dependent positive phenomenological
constants. The above expression indicates that the elastic modulus is simply
given by $G(T)=a(T-T_{0})$. Thus, by writing it as $G(T)=G_{S}(T)+G_{E}(T)$,
the authors of Ref. Yoshikawa  _et al._ (2021) attributed the term
$G_{S}(T)=aT$ to a positive “entropic” contribution, while the negative one,
$G_{E}(T)=G(T)-G_{S}(T)=-aT_{0}$, is assumed to be an “energetic” (or
“enthalpic”) contribution to the elastic modulus (here we refer the reader to
the Appendix of Ref. Duarte and Rizzi (2023) for a detailed discussion about
the correct meaning of the claimed “entropic” and “enthalpic” contributions).

Although the mechanical properties of tetra-PEG hydrogels were somewhat well
described in Refs. Yoshikawa  _et al._ (2021); Sakumichi, Yoshikawa, and Sakai
(2021), the theoretical basis for the phenomenological expression (9) was not
clear until we showed that it is a limiting case of the expression obtained
from the model we proposed in Ref. Duarte and Rizzi (2023). In particular, by
considering Eqs. (6) and (7) one can deduce the following expression for the
energy-related contribution to the elastic modulus

$G_{E}(T)=-\dfrac{\Delta E}{k_{B}T}\,\dfrac{q_{\ell}^{\,}+e^{2\Delta
E/k_{B}T}}{(q_{\ell}^{\,}-e^{\Delta E/k_{B}T})(1+e^{\Delta
E/k_{B}T})}\,G(T)~{}.$ (10)

Indeed, as one may check, by considering $e^{x}\approx 1+x$ and $q_{\ell}>1$,
the above expression is approximately equal to the simpler phenomenological
form, i.e., $G_{E}(T)=-aT_{0}$, with the temperature $T_{0}$ associated to the
temperature $T_{0}^{*}$ given by Eq. (8). Similarly, one can show that the
above expression obtained from our model leads to a $G_{S}(T)$ contribution
which can be approximated to a term that is proportional to the temperature
Duarte and Rizzi (2023), just as in the phenomenological model, i.e.,
$G_{S}(T)=aT$.

Importantly, the experiments done in Refs. Yoshikawa  _et al._ (2021);
Sakumichi, Yoshikawa, and Sakai (2021) revealed that the negative contribution
to the elastic modulus can be as high as $|G_{E}(T)|\approx 2G(T)$, so that
$0<G(T)<G_{S}(T)$, highlighting the significance of this energy-related
contribution to the elastic modulus $G(T)$. We note that Eq. (10) is valid for
both $q_{\ell}^{\,}>1$ and $q_{\ell}^{\,}<-1$, however, since both $G(T)$ and
$T_{0}^{*}$ are assumed to be positive constants for tetra-PEG hydrogels, one
can argue that $G_{E}(T)$ will be negative and $T_{0}^{*}$ (Eq. (8)) will be
positive only if $\Delta E>0$ and $q_{\ell}>1$. Indeed, these were the ranges
we have considered for those parameters in Ref. Duarte and Rizzi (2023) in
order to describe the whole set of experimental data obtained for tetra-PEG
hydrogels presented in Refs. Yoshikawa  _et al._ (2021); Sakumichi, Yoshikawa,
and Sakai (2021). It is worth recalling that $\Delta E$ and $q_{\ell}$ are,
respectively, the energy difference and the ratio between distances for the
two possible conformational states, $s$ and $b$, of a segment of a chain that
connects cross-links in the gel network. By considering positive values for
these two parameters, one is assuming not only that the blob state has a
smaller energy, $E_{b}<E_{s}$, but also that $\ell_{b}$ is negative, which
mean that in such state the segment of the chain is folded around itself (see
the schematic drawing in Ref. Duarte and Rizzi (2023) for a clearer picture).
While the positive value of $\Delta E$ is supported by many numerical
simulations which assume effective interactions between the chains and the
solvent Zierenberg, Marenz, and Janke (2016), the negative value of $\ell_{b}$
can be interpreted as a somewhat non-affine behaviour Boyce and Arruda (2000)
of the segments within the chains in the gel network, since $b$ states tend to
decrease the distance between cross-links at the mesoscopic scale. Indeed,
such non-affine behaviour can be observed from experiments and numerical
simulations of single PEG chains Kolberg  _et al._ (2019) and it can be
clearly related to the effective interaction between the polymer chain and the
solvent molecules, which corroborated our approach. Even so, one should note
that our model does not require that $\ell_{b}<0$ and $q_{\ell}>1$. In fact,
we will show later in this study that some hydrogels can be only described by
Eq. (2) if $\ell_{b}>0$ and $q_{\ell}<-1$.

Finally, it is worth emphasizing that, despite the agreement between the
phenomenological model and our approach as a limiting case, if the model of
Ref. Yoshikawa  _et al._ (2021) is considered, the stress-strain relationship
given by Eq. (9) leads to a strain-independent differential modulus, that is,
$K(T,\gamma)=a(T-T_{0}^{*})$. Hence, such phenomenological model is not able
to describe the strain-induced softening behaviour illustrated in Fig. 1. In
the following we show that is not the case of the stress-strain relationship
obtained from our model, previously discussed in Sec. 2.1.

##  3 Results

Clearly, as we have discussed above, the WLC-based models Palmer and Boyce
(2008) and the phenomenological model of Ref. Yoshikawa  _et al._ (2021) are
incapable of describing the strain-induced softening behaviour observed for
hydrogels like the one illustrated by the data displayed in Fig. 1. In order
to investigate that issue, we now consider the stress-strain relationship
given by Eq. (2) and evaluate the differential modulus through its definition,
i.e., Eq. (1), which yields

$K(T,\gamma)=\dfrac{G(T)}{\left[1+\gamma/\gamma_{b}(T)\right]\left[1-\gamma/\gamma_{s}(T)\right]}.$
(11)

Here, $G(T)$ is the elastic modulus given by Eq. (6), and $\gamma_{b}(T)$ and
$\gamma_{s}(T)$ are defined as in Eqs. (3) and (4), respectively. Accordingly,
as $\gamma$ approaches zero, the above expression indicates that the
differential modulus will tend to the elastic modulus, i.e.,
$G(T)=\lim_{\gamma\rightarrow 0}K(T,\gamma)$. On the other hand, when the
strain approaches $\gamma_{s}(T)$, the differential modulus will increase,
indicating that the hydrogel is becoming harder.

Below we first discuss what happens for intermediate values of the strain and
relate the mechanical response of the hydrogel with the non-linear strain-
induced softening behaviour similar to what is shown in Fig. 1. Besides
comparing our theoretical results to the data extracted from experiments to
validate our approach, we also include a discussion about the hardening
behaviour in view of known results presented in the literature.

###  3.1 Strain-induced softening behaviour

Importantly, from Eq. (11) one observes that the value of the differential
modulus will reach a minimum for finite values of $\gamma$, where the
denominator of $K(T,\gamma)$ reaches a maximum. This corresponds to a value of
strain that is given by

$\gamma_{\textrm{min}}(T)=\gamma_{s}(T)\left(\dfrac{1-e^{-\Delta
E/k_{B}T}}{2}\right)~{}.$ (12)

At this strain, the minimum value of the differential modulus is given by

$K_{\min}(T)=G(T)\,\text{sech}^{2}\left(\dfrac{\Delta E}{2k_{B}T}\right).$
(13)

In addition, by replacing the value obtained for $\gamma_{\min}(T)$ in the
stress-strain relationship given by Eq. (2), one finds the corresponding
stress

$\sigma_{\min}=\bar{n}\Delta E.$ (14)

Interestingly, the above result indicates that the stress which minimizes the
differential modulus is independent of the temperature. In fact,
$\sigma_{\min}$ will only depend on the number density $\bar{n}$ (which, in
turn, depends on $n_{e}$ and $q_{\ell}^{\,}$), and on the energy difference
$\Delta E$. Obviously, all the mentioned parameters are characteristic to the
hydrogel itself (see Ref. Duarte and Rizzi (2023) for further details).

####  3.1.1 Case $\Delta E=0$

If $\Delta E=0$, then $\textrm{sech}(0)=1$, thus Eq. (13) indicates that the
minimum value of the differential modulus is just $K_{\textrm{min}}(T)=G(T)$.
In that case, Eq. (11) becomes a strictly increasing function of $\gamma$
which is given by $K(T,\gamma)=G(T)\\{1-[\gamma/\gamma_{s}(T)]^{2}\\}^{-1}$
(since $\gamma_{b}(T)=\gamma_{s}(T)$, as established by Eqs. (3) and (4)).
This implies that $K(T,\gamma)\geq G(T)$ for any strain in the range
$0<\gamma<\gamma_{s}(T)$, and at any temperature $T>0$, since from Eq. (8) one
has that $T_{0}^{*}=0$ when $\Delta E=0$. Besides, Eqs. (12) and (14) indicate
that $\gamma_{\min}(T)=0$ and $\sigma_{\min}=0$, consequently, our model
predicts that there is no strain-induced softening behaviour in the cases
where $\Delta E=0$. As it may be checked, this also corresponds to gel
networks with a purely entropic response, because, when the energy difference
$\Delta E$ is zero, it follows from Eq. 10 that $G_{E}(T)=0$, thus
$G(T)=G_{S}(T)$.

Therefore, for hydrogels where there is no energy difference, one can only
observe the hardening behaviour for any value of the strain, i.e.,
$0<\gamma\lesssim\gamma_{s}(T)$. This is precisely what is shown in Fig. 1 for
the sample of a collagen-based hydrogel with TG2 crosslinking agent, where the
differential modulus displays a strictly increasing behaviour up to relatively
large strains, i.e., $\gamma<0.45$. Clearly, for even higher values of the
strain, i.e., $\gamma\gtrsim 0.45$, the experimental data displayed in Fig. 1
indicate that there is a softening behaviour for both samples, but we argue
here that such behaviour is, in fact, due to the breakage of the internal
structures of the gel network, so it is beyond the scope of our model.

Figure 2: Top panels display the stress-strain curves: (a)
$\sigma\times\gamma$ in linear scale and (b) $\sigma/G\times\gamma/\gamma_{s}$
in log-log scale; while bottom panels display the differential modulus: (c)
$K\times\gamma$ in linear scale and (d) $K/G\times\gamma/\gamma_{s}$ in log-
log scale. Filled symbols correspond to the experimental data extracted from
Ref. Kamata  _et al._ (2014), where PEG hydrogels with different hydrophobic
ratios $r$ were studied. Continuous lines correspond to the stress-strain and
differential modulus expressions given by Eqs. (2) and (11), respectively,
evaluated with the parameters listed in Table 1.

####  3.1.2 Case $\Delta E>0$

Because $\operatorname{sech}(x)\leq 1$ for $x\neq 0$, Eq. (13) establishes
that, if $\Delta E\neq 0$, the minimum value of the differential modulus will
correspond to a fraction of the elastic modulus, with
$K_{\textrm{min}}(T)<G(T)$. Besides, Eq. (11) indicates that $K(T,\gamma)$
will decrease for $0<\gamma<\gamma_{\textrm{min}}(T)$, reaching its minimum
value when $\gamma=\gamma_{\textrm{min}}(T)$. From this value up to
$\gamma_{s}(T)$, one observes that the differential modulus increases
indefinitely, indicating then a hardening behaviour. Interestingly, it can be
shown from Eq. (11) that $G(T)=K(T,2\gamma_{\textrm{min}}(T))$, which is an
identity that may be used to better determine the value of
$\gamma_{\textrm{min}}(T)$ in the cases where $K_{\textrm{min}}(T)$ cannot be
clearly identified from the experimental data. For
$\gamma<2\gamma_{\text{min}}(T)$ the gel is softer than before being deformed
(i.e., $\gamma=0$), and, when $\gamma>2\gamma_{\text{min}}(T)$, it is harder.
So we consider that the strain-induced softening behaviour happens for
$0<\gamma<\gamma_{\min}(T)$, while the hardening behaviour occurs in the range
$\gamma_{\min}(T)<\gamma\lesssim\gamma_{s}(T)$. This kind of response is
illustrated by the data presented in Fig. 1 for the collagen-based hydrogel
without TG2 crosslinking agent. One can easily identify $\gamma_{\min}(T)=0.1$
since, at $\gamma=0.2=2\gamma_{\min}(T)$, the differential modulus is equal to
the elastic modulus, i.e., $G(T)=K(T,2\gamma_{\min}(T))\approx 0.25\,$kPa
(dashed horizontal line). Thus, we assume there that the strain-induced
softening behaviour occurs for $\gamma<0.1$, whereas the hardening behaviour
is observed for $\gamma>0.1$ (obviously excluding the range of even higher
strains, i.e., $\gamma\gtrsim 0.45$, as indicated in the last subsection).

Table 1: Here, $n_{e}$, $q_{\ell}^{\,}$ and $\Delta E$ are parameters obtained from a fit procedure of Eq. (2) to the experimental data extracted from Ref. Kamata  _et al._ (2014), where the stress-strain of samples of PEG hydrogels with different hydrophobic ratios $r$ are presented. The values of $\gamma_{s}$, $G$, $\gamma_{\textrm{min}}$, and $K_{\textrm{min}}$ were obtained from these three parameters through Eqs. (4), (10), (12), and (13), respectively. Expressions (8) and (10) were used to estimate the temperature $T_{0}^{*}$ and the ratio $G_{E}/G$, respectively. $r$ |  $n_{e}\,$[nm-3] | $q_{\ell}^{\,}$ |  $\Delta E\,$[pN.nm] | $\gamma_{s}$ |  $G\,$[kPa] | $\gamma_{\textrm{min}}$ |  $K_{\textrm{min}}\,$[kPa] | $G_{E}/G$ |  $T_{0}^{*}\,$[K]
---|---|---|---|---|---|---|---|---|---
$0.2$ | $0.00666$ | $13.50$ | $5.44$ | $5.198$ | $14.512$ | $1.87$ | $9.934$ | $-0.73$ | $151.31$
$0.3$ | $0.00761$ | $6.58$ | $4.52$ | $5.887$ | $14.554$ | $1.92$ | $11.146$ | $-1.09$ | $173.81$
$0.4$ | $0.00868$ | $4.18$ | $3.66$ | $6.678$ | $15.519$ | $1.92$ | $12.711$ | $-1.36$ | $185.53$
$0.5$ | $0.01121$ | $3.20$ | $3.13$ | $7.801$ | $18.070$ | $2.03$ | $15.848$ | $-1.60$ | $195.18$

###  3.2 Application to tetra-PEG hydrogels

Now, in order to validate our approach, we first consider the experimental
data obtained from synthetically designed hydrogels with well controlled
topology. As discussed in Refs. Kamata  _et al._ (2014, 2015), the mechanical
properties of tetra-PEG hydrogels are sensitive to changes of the polymeric
chains about their affinity to water. It is shown in Ref. Kamata  _et al._
(2014) that, by increasing the molar fraction $r$ of the hydrophobic precursor
molecules in the hydrogel, it is possible to control the swelling behaviour of
such materials.

Figure 2 includes the experimental data of Ref. Kamata  _et al._ (2014), where
the stress-strain curves were determined for hydrogels synthetized with
several values of $r$ at $T=37\,^{\text{o}}$C. As one may observe from Fig.
2(a) as well as from Fig. 2(b), where $\sigma(T,\gamma)/G(T)$ is plotted as a
function of $\gamma/\gamma_{s}(T)$, our theoretical expression for the stress,
Eq. (2), represents well the experimental data for the different values of
$r$. Besides, Fig. 2(c) indicates that the differential modulus $K(T,\gamma)$
numerically determined from the experimental data is also reasonably well
described by Eq. (11). In addition, from Fig. 2(d), where it is plotted
$K(T,\gamma)/G(T)$ against the reduced strain $\gamma/\gamma_{s}(T)$, one can
clearly observe that the curves for $K(T,\gamma)$ present minima for finite
strains. For clarity, we show in Figs. 2(a), (c), and (d) only the
experimental data obtained for two values of $r$, however, in Table 1 we
include the values determined for the parameters so that the expressions (2)
and (11) can be consistently used to describe all the data extracted from
Refs. Kamata  _et al._ (2014, 2015).

As indicated by the values of $G(T)$ and $K_{\min}(T)$ listed in Table 1, the
softening-induced behaviour becomes more pronounced as the ratio $r$ decreases
and the energy difference $\Delta E$ increases. This is more evident, for
instance, when the sample of PEG hydrogel with $r=0.2$ (green squares) is
compared to the one with $r=0.5$ (purple circles). In fact, that result
corroborates the behaviour somewhat expected from expression (13), since
$K_{\min}(T)/G(T)$ is a decreasing function of $\Delta E$.

The values presented in Table 1 also indicate that, as the ratio $r$
increases, the value of the maximum strain supported by the hydrogel
$\gamma_{s}(T)$ also increases. This behaviour can be explained through Eq.
(4), even though it is not apparent from theory how the parameters $n_{e}$,
$q_{\ell}^{\,}$, and $\Delta E$ should change as functions of $r$. One can
empirically observe that $q_{\ell}$ increases with increasing values of
$\Delta E$, with both quantities becoming larger as the ratio $r$ decreases.
Consequently, one can infer from Eq. (5) that, by decreasing the fraction of
hydrophobic chains in the network (i.e., decreasing $r$), the natural end-to-
end distance of the cross-links $\ell_{0}^{\,}(T)$ tends to increase. Hence,
the increasing of $r$ reflects on reducing the values of $\gamma_{s}(T)$, that
is, $(d\gamma_{s}(T)/d\Delta E)\propto-(1/\ell_{0}^{2})(d\ell_{0}^{\,}/d\Delta
E)$, since one has that $\gamma_{s}=(\ell_{c}-\ell_{0}^{\,})/\ell_{0}^{\,}$.
This result corroborates the fact the smaller values for $\ell_{0}(T)$ are the
source of large swelling processes observed for the PEG hydrogels with higher
$r$ ratios described in Refs. Kamata  _et al._ (2014, 2015). In fact, this
behaviour is mentioned in Ref. Kamata  _et al._ (2014) as a definition, but
here one sees that our model allows a quantitative discussion.

#### 3.2.1 Analysis of the negative contribution

Since the material discussed here is a PEG hydrogel like the one studied in
Refs. Yoshikawa  _et al._ (2021); Sakumichi, Yoshikawa, and Sakai (2021), the
existence of a negative energy-related contribution $G_{E}(T)$ to the elastic
modulus is somewhat expected Duarte and Rizzi (2023). Indeed, as the values
presented in Table 1 indicate, the value of the ratio $G_{E}(T)/G(T)$
increases in absolute value as the ratio $r$ of hydrophobic precursor
molecules in the gel network increases. Besides, one finds from Eq. (8) that
the values determined for the characteristic temperature $T_{0}^{*}$ are all
positive, with higher values of $r$ leading to higher temperatures. As
discussed before, Eq. (10) confirms $T_{0}^{*}>0$ as is an indicator of
$G_{E}(T)<0$. Interestingly, even though $\Delta E$ decreases with increasing
values of $r$, the parameter $q_{\ell}^{\,}$ decreases faster, making
$T_{0}^{*}$ an effectively increasing function of $r$.

#### 3.2.2 Hardening effect

As already discussed from the data shown in Fig. 1, the description of the
hardening effect is a limitation of our model, particularly when what is being
observed in experiments is the rupture of the internal structures of the gel
network after being subjected to a certain deformation. Yet, one can interpret
$\gamma_{s}(T)$ as the maximum strain (on average) which the material may
support before rupture, and that is the regime we are interested to describe
here.

Besides, it is worth noting that, while presenting the differential modulus,
many studies in the literature Bertula  _et al._ (2019); Licup  _et al._
(2015); Wang  _et al._ (2020); Martikainen  _et al._ (2020); de Almeida  _et
al._ (2019) include plots of $K$ as a function of the stress $\sigma$ instead
of the strain $\gamma$. In order to investigate how those physical quantities
are related in our model, we consider Eq. (2), which can be rearranged as

$\dfrac{1}{[1-\gamma/\gamma_{s}(T)][1+\gamma/\gamma_{b}(T)]}=\dfrac{\cosh^{2}\left[\left(\sigma/\bar{n}-\Delta
E\right)/2k_{B}T\right]}{\cosh^{2}\left(\Delta E/2k_{B}T\right)}~{},$ (15)

so that the differential modulus given by Eq. (11) can be rewritten in terms
of the stress $\sigma$ as

$K(T,\sigma)=K_{\textrm{min}}(T)\cosh^{2}\left(\dfrac{\sigma-\sigma_{\min}}{2\bar{n}k_{B}T}\right)~{}.$
(16)

Accordingly, the above expression indicates that the differential modulus
$K(T,\sigma)$ will reach its minimum value $K_{\min}(T)$ when $\sigma$ is
equal to the stress $\sigma_{\min}$ given by Eq.(14).

In order to illustrate the behaviour of Eq. (16), we include in Fig. 3 the
differential modulus $K$ as a function of the stress $\sigma$ evaluated from
the experimental data obtained for PEG hydrogels of Ref. Kamata  _et al._
(2014) (which is also presented in Fig. 2). Clearly, expression (16) describes
reasonably well the experimental data, as can be verified from the continuous
lines determined for the hydrogels prepared with $r=0.2$, $0.4$, and $0.5$.

Now, by assuming relatively large values for the stress, i.e.,
$\sigma\gg\sigma_{\min}+2\bar{n}k_{B}T$, one has that the asymptotic behaviour
of the differential modulus given by Eq. (16) can be written as

$K(T,\sigma)\approx\frac{1}{4}K_{\textrm{min}}(T)\left[2+e^{(\sigma-\sigma_{\textrm{min}})/\bar{n}k_{B}T}\right]~{}.$
(17)

As illustrated in Fig. 3 by the curve determined for the hydrogel sample with
$r=0.2$ (i.e., dash-dotted line), the above asymptotic expression is suitable
for describing $K(T,\sigma)$ for large values of stress. Interestingly, a
similar asymptotic exponential behaviour, i.e., $K\sim e^{c\sigma}$, with $c$
being a positive real constant, is also observed to occur for a model of rigid
filaments connected by flexible cross-links, which is described by the free-
energy density of the WLC model Heidemann  _et al._ (2015).

Figure 3: Hardening effect observed from the differential modulus
$K(\sigma,T)$ as a function of the stress $\sigma$. Filled symbols correspond
to the differential modulus computed from the experimental data extracted from
Ref. Kamata  _et al._ (2014) for PEG hydrogels with different $r$; continuous
lines correspond to Eq. (16) evaluated using the parameters of Table (1). The
exponential behaviour observed for large stresses is also described by the
asymptotic expression (17), which is included only for the sample with $r=0.2$
(dash-dotted line).

Figure 4: Differential modulus $K(T,\gamma)$ given by Eq. (11) (continuous
lines) for (a) keratin gel at $25^{\circ}$C with $n_{e}=1.05\times
10^{-5}\,$nm-3, $q_{\ell}=-1.55$ and $\Delta E=10.9\,$pN.nm; (b) silk fibroin
gel at $25^{\circ}$C with $n_{e}=4.3\times 10^{-6}\,$nm-3, $q_{\ell}=-2.1$ and
$\Delta E=3.9\,$pN.nm; (c) neurofilament network at $37^{\circ}$C with
$n_{e}=1.67\times 10^{-6}\,$nm-3, $q_{\ell}=-18$ and $\Delta E=7\,$pN.nm; and
(d) collagen-based hydrogel at $37^{\circ}$C with $n_{e}=6.0\times
10^{-6}\,$nm-3, $q_{\ell}=-1.95$ and $\Delta E=2.33\,$pN.nm. Filled symbols
denote the experimental data extracted from Refs. Valero  _et al._ (2018);
Storm  _et al._ (2005); Elbalasy  _et al._ (2022); Oztoprak and Okay (2017).

###  3.3 Application to biopolymer-based hydrogels

Next, we consider the experimental data obtained for some biopolymer-based
hydrogels studied in the literature in order to illustrate what other systems
can be described by our model. First, it is worth exemplifying how one could
relate the three parameters of the model (i.e., $n_{e}$, $q_{\ell}$, and
$\Delta E$), to match the experimental data by using the above theoretical
results. In particular, one can use the differential modulus $K$ to
graphically estimate its minimum value, $K_{\textrm{min}}(T)$, and also the
elastic modulus, which is given either by $G(T)=\lim_{\gamma\rightarrow
0}K(T,\gamma)$ or by $G(T)=\lim_{\sigma\rightarrow 0}K(T,\sigma)$. Thus, from
Eq. (13), one has that the energy difference is given by $\Delta
E=2k_{B}T\operatorname{sech}^{-1}(\xi)$, with
$\xi=\sqrt{K_{\textrm{min}}(T)/G(T)}$. In addition, by estimating the value
$\gamma_{\min}(T)$, one can use Eq. (12) in a consistent manner to determine
the maximum value of the strain, that is,
$\gamma_{s}(T)=2\gamma_{\textrm{min}}(T)/(1-e^{-\Delta E/k_{B}T})$. Then, by
using Eq. (4), one can estimate the ratio
$q_{\ell}=[\gamma_{s}(T)+1]/[\gamma_{s}(T)e^{-\Delta E/k_{B}T}-1]$. Finally,
Eq. (7) can be used to determine the factor $\phi(T)$, so that the number
density $n_{e}$ is obtained from Eq. (6), that is, $n_{e}=G(T)/k_{B}T\phi(T)$.

Figure 4 shows the experimental data obtained for (a) keratin gel Elbalasy
_et al._ (2022), (b) silk fibroin gel Oztoprak and Okay (2017), (c)
neurofilament-based hydrogel Storm  _et al._ (2005), and (d) collagen-based
hydrogel Valero  _et al._ (2018). By considering the above procedure, we found
that the experimental data can be correctly described only if one assumes
positive values for the energy difference, i.e., $\Delta E>0$, for all the
biopolymer-based hydrogels considered. Even so, differently from what is
observed for the PEG hydrogels discussed in Sec. 3.2, our analyses yielded
only negative values for $q_{\ell}$ for the hydrogels displayed in Fig. 4.
Consequently, since this ratio is defined as $q_{\ell}=-\ell_{s}/\ell_{b}$,
both $\ell_{s}$ and $\ell_{b}$ must be positive parameters, so that the
contributions of both end-to-end displacements are supposed to be in the same
direction Duarte and Rizzi (2023).

In addition, because $q_{\ell}<-1$, one cannot determine a threshold
temperature as $T_{0}^{*}$ given by Eq. (8), so these hydrogels will not
present any positive temperature $T$ where the elastic moduli $G(T)$ reaches
zero Duarte and Rizzi (2023), as in the case where $q_{\ell}>1$. Even so,
since $q_{\ell}<-1$, the energy-related contribution $G_{E}(T)$ given by Eq.
(10) can be either positive or negative, depending on the temperature. By
setting $q_{\ell}=-|q_{\ell}|$ in Eq. (10) one finds that, if
$T>T^{\dagger}_{0}$, then $G_{E}(T)<0$, while for $T<T^{\dagger}_{0}$, it
yields $G_{E}(T)>0$, with such temperature given by

$T^{\dagger}_{0}=\dfrac{2\Delta E}{k_{B}\ln(-q_{\ell}^{\,})}$ (18)

Interestingly, this last result suggests a way of testing our model
experimentally, since the sign of $G_{E}(T)$ would change at the predicted
temperature $T^{\dagger}_{0}$ given by Eq. (18). The problem is that, in some
cases, $T^{\dagger}_{0}$ might be estimated as a very high temperature, so
that the samples would be degraded when subjected to the hypothetical
experiment. That is the case of the keratin ($T_{0}^{\dagger}\approx
3331^{\circ}$C), silk fibroin ($T_{0}^{\dagger}\approx 489^{\circ}$C) and,
possibly, neurofilaments ($T_{0}^{\dagger}\approx 78^{\circ}$C) gels, which
are presented in Fig. 4. Yet, we believe that it is possible to find hydrogels
from which the experiments can be performed without either damaging the gel
network or boiling the solvent. One example is the collagen-based hydrogel
that provided the results displayed in Fig. 4(d) (which is the same data
presented in Fig. 1), for which the temperature is found to be
$T_{0}^{\dagger}\approx 17^{\circ}$C.

Here, it is worth noting that, because the experimental data presented in Fig.
4 were obtained either at room or physiological temperatures, the collagen-
based hydrogel is the only one for which the energy-related contribution to
the elastic modulus is negative, i.e., $G_{E}(T)/G(T)=-0.008$. For the keratin
hydrogel one has the most significant positive energy-related contribution,
being over twice its total elastic modulus, i.e., $G_{E}(T)/G(T)=2.21$, while
for silk fibroin and neurofilaments gels we found $G_{E}(T)/G(T)=0.233$ and
$G_{E}(T)/G(T)=0.096$, respectively.

In addition, we found for the biopolymer-based hydrogels that $\gamma_{s}=0.5$
(keratin), $\gamma_{s}=0.61$ (silk fibroin), $\gamma_{s}=3.77$
(neurofilaments), and $\gamma_{s}=0.45$ (collagen). Although Eq. 11 is not
able to describe the experimental data at large strains (possible due to the
breakage or to the heterogeneous character Rizzi (2020) of the internal
structures of the networks), these values present the opposite trend observed
for tetra-PEG hydrogels (see Table 1), i.e., the higher the value of
$|q_{\ell}|$, the larger the value of $\gamma_{s}$.

##  4 Concluding remarks

In this work we have investigated the possibility that the coarse-grained
model introduced in Ref. Duarte and Rizzi (2023) could be used to describe the
strain-induced softening behaviour observed in many hydrogels. Despite its
simplicity, the two-state model led to the stress-strain relationship given by
Eq. (2), from where we were able to determine exact expressions for the
differential modulus $K$ as a function of both the strain, Eq. (11), and the
stress, Eq. (16).

We showed that those expressions can be used to fully describe the mechanical
response and the thermal behaviour of several hydrogels with just three
physically meaningful parameters, i.e., the number density of chains which
connect cross-links, $n_{e}$, the ratio between end-to-end distances,
$q_{\ell}^{\,}=-\ell_{s}/\ell_{b}$, and the energy difference between the two
states, $\Delta E=E_{s}-E_{b}$.

The expressions obtained for the differential modulus indicate that, if
$\Delta E=0$, the model describes purely entropic hydrogels which do not
display the strain-induced softening behaviour, so only the hardening effect
is observed. When $\Delta E>0$, the model predicts that both the strain-
induced softening and the hardening behaviours will occur. In that case, our
results indicate that the softening behaviour can be observed for hydrogels
with both positive and negative values of $q_{\ell}^{\,}$, as exemplified by
the data presented in Figs. 2 and 3 for PEG hydrogels, where
$q_{\ell}^{\,}>1$, and in Fig. 4 for biopolymer-based gels, where
$q_{\ell}^{\,}<-1$.

By finding that the energy difference $\Delta E$ is positive, we were able to
infer the importance of effective interactions between the chains in the gel
network and their neighbouring solvent molecules. This energy difference not
only leads to the presence of the energy-related contribution $G_{E}(T)$, as
discussed in Ref. Duarte and Rizzi (2023), but it also induces, as we show
here, a solvent-related strain-induced softening behaviour. Although solvent-
induced effects have been previously considered in Ref. Yoshikawa  _et al._
(2021), our results revealed that their phenomenological model, like those
based on the WLC model Palmer and Boyce (2008), is not capable of displaying a
softening behaviour.

It is worth emphasizing that, in contrast to purely entropic Flory (1953);
James and Guth (1953); Flory (1977) and to purely enthalpic Bouzid and Gado
(2018); Heidemann  _et al._ (2015); Huisman, Storm, and Barkema (2010);
Huisman and Lubensky (2011) approaches, our statistical mechanical model takes
into account both contributions. Even so, our model is limited to situations
where there is no rupture or yielding of the internal structure of the gel
network, since it obviously involves, as already mentioned, a different
solvent-induced mechanism for the softening and hardening behaviours. As
discussed along the manuscript, the hardening behaviour that is observed at
large strains from our model corresponds to an exponential behaviour, i.e.,
Eq. (17), just like the asymptotic behaviour found in Ref. Heidemann  _et al._
(2015) from a completely different approach.

Finally, we note that the theoretical basis of our model may serve as a first
approximation for modelling approaches that aim to describe more complex
hydrogels related to biomedical applications Guidetti  _et al._ (2022); Greco
_et al._ (2023); Nicolle  _et al._ (2017).

The authors thank the funding agencies FAPEMIG (APQ-02783-18), CNPq
(312999/2021-6), and CAPES.

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