Article
pubs.acs.org/Macromolecules
Yielding Criteria of Double Network Hydrogels
Takahiro Matsuda,† Tasuku Nakajima,*,‡ Yuki Fukuda,§ Wei Hong,∥,‡ Takamasa Sakai,⊥
Takayuki Kurokawa,‡ Ung-il Chung,⊥ and Jian Ping Gong*,‡
†
Graduate School of Life Science, ‡Faculty of Advanced Life Science, and §School of Science, Hokkaido University, N10W8, Kita-ku,
Sapporo, Hokkaido 060-0810, Japan
∥
Department of Aerospace Engineering, Iowa State University, 2271 Howe Hall, Ames, Iowa 50011-2271, United States
⊥
Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
S Supporting Information
*
ABSTRACT: Double network (DN) gels, consisting of a
brittle first and flexible second network, have been known to be
extremely tough and functional hydrogels. In a DN gel
subjected to force, the brittle first network breaks prior to the
fracture of the flexible network. This process, referred to as
internal fracture, dissipates energy and increases the energy
required to completely fracture DN gels. Such internal fracture
macroscopically appears as a yielding-like phenomenon. The
aim of this paper is to investigate the relationship between the
yield point and the first network molecular structure of DN
gels to more deeply understand the internal fracture mechanism of DN gels. To achieve this goal, we synthesized DN gels having
a tetra-PEG first network, which is known to be a nearly ideal and well-controlled network gel. We have found that yielding of the
DN gels occurs when the first network strands reach their extension limit (finite extensibility), regardless of their deformation
mode. This conclusion not only helps by further understanding the toughening mechanism of DN gels but also allows for the
design of DN gels with precisely controlled mechanical properties.
■
INTRODUCTION
In the 21st century, various mechanically robust hydrogels have
been created, which has enabled the practical use of hydrogels
in various fields.1−3 Double network (DN) gels, consisting of
two independent and asymmetric networks, are a representative
tough hydrogel system.4,5 Despite containing up to 90% water,
optimized DN gels show excellent strength, extensibility, and
toughness, similar to some kinds of industrial rubbers. Also,
these mechanical properties can be widely controlled by
altering the composition of the gels.4−8 DN gels are not only
tough but also have excellent functions like biocompatibility,9,10
bioactivity,11 low immune response,12 and low sliding friction.13
Based on these features, DN gels possess great potential,
especially as medical materials, such as substrates for cell
cultures,11 artificial articular cartilage,14,15 and substrates for in
vivo cartilage regeneration.16
The high toughness of DN gels mainly originates from their
contrasting double network structure. The first network of
tough DN gels must be brittle, dilute, and weak, while the
second network must be flexible, concentrated, and (relatively)
strong. This results in a system where extensibility and strength
of the first network are much poorer than those of the second
network. DN gels become tough regardless of their chemical
constituents if the two networks satisfy the above-mentioned
conditions.17 Such contrasting structures induce dramatically
large energy dissipation during stretching and fracture
tests.5,8,18,19 When a DN gel is deformed, the first network
© XXXX American Chemical Society
chains break internally at an early stage of deformation while
the second network maintains the integrity of the whole DN
gel. As a result, before the second network fracture, an
extremely large quantity of energy is dissipated due to the
internal fracture of the first network, which substantially
increases the toughness and other mechanical properties of
DN gels.
Such internal fracture of the first network visually appears as
a yielding-like phenomenon. Figure 1 shows the typical tensile
stress−strain curves of a tough DN gel. A clear yielding point,
Figure 1. Typical yielding-like and necking phenomena of a tough
double network hydrogel during a uniaxial tensile test.
Received: November 30, 2015
Revised: February 4, 2016
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To investigate the yielding criteria of DN gels, we will
systematically tune the conformation of the first TPEG
network. In particular, the original structure of the TPEG
network (such as connectivity, average molecular weight
between cross-linking points, etc.) is kept constant, while the
prestretching ratio of the TPEG network was widely varied by
the molecular stent method. As shown in Figure 2,
accompanied by macroscopic necking, can be found. The
mechanical character of DN gels dramatically changes at the
yield point, being dominated by the rigid network prior to and
the flexible network after yielding. Thus, the yield point can be
considered to represent the point of the first network internal
fracture. Based on these phenomena, the hypothesis that the
first network breaks into discontinuous fragments at the yield
point has been proposed.5,7,20,21
While it has been understood that yielding is an extremely
important phenomenon for the toughening and fracture
mechanisms of DN gels, one remaining problem is making a
connection between the fracture behavior and the molecular
structure of DN gels. To access this problem, comparing the
yield point of DN gels and the first network structure is
required, since the yield point is the representative point of the
internal fracture of the first network. To date, however, such
comparison has been especially difficult since the first network
of DN gels is typically synthesized by a free-radical
copolymerization of vinyl monomer and divinyl cross-linker.
This reaction is uncontrolled and results in an essentially
random structure of the first network, which prevents
correlation between the yield point and the first network
molecular structure in DN gels.
The aim of this study is to investigate the relationship
between the yield point and the first network molecular
structure of double network gels by using a well-defined first
network to further our understanding of the toughening
mechanism of DN gels. In recent years, tetra-PEG (TPEG) gels
have attracted great attention as model gels since they have a
well-defined and homogeneous network structure.22−25 TPEG
gels are prepared by the cross-end coupling of two mutually
reactive tetra-arm polymers. As polymerization degree of each
arm is well-controlled, the length distribution of network chains
is kept in a narrow range. Small-angle neutron scattering studies
showed that TPEG gels had practically no noticeable excess
scattering in the region below 200 nm.23 Infrared spectroscopy
revealed a reaction conversion of up to 0.9, which suggested
near absence of dangling chains.24 Mechanical testing of TPEG
gels suggested that formations of trapped entanglements or
elastically ineffective loops are negligible.25 These results
strongly suggest that TPEG gels constitute a near-ideal polymer
network. Thus, TPEG gels are the most suitable candidate for
creating well-defined first networks in DN gels.
We have previously synthesized tough DN gels using a tetraPEG first network.26 As TPEG gels have a relatively flexible and
condensed network in water, they cannot be directly used as
the brittle first network. Therefore, we introduced strong, linear
polyelectrolytes, called molecular stents, into the TPEG gels to
increase the overall osmotic pressure. As high osmotic pressure
increases their swelling ratio, TPEG network chains are
stretched to an extended conformation, and the properties of
the TPEG gels changed to become brittle and weak, which are
suitable properties for the first network of tough gels. These
swollen and brittle TPEG network-based DN gels, called StTPEG DN gels, also showed high toughness and yielding-like
behaviors like conventional DN gels with a random first
network, which indicates that homogeneity of the first network
structure does not affect the essence of the fracture mechanism
of DN gels. On the other hand, the total amount of internal
fracture of the St-TPEG gels before the yield point is much less
than that of conventional DN gels due to the first network
inhomogeneity of the latter.
Figure 2. Synthesis process of St-TPEG/PAAm DN gels having
various prestretching ratios of the first network, which is controlled by
the molecular stent (polyelectrolyte) concentration.
polyelectrolytes trapped in a gel (molecular stent) generate
extra osmotic pressure due to their counterions to swell the gel.
By tuning the polyelectrolyte concentration in the TPEG gel,
the swelling ratio of the gel can be varied, which permits us to
finely tune the prestretching ratio and the network density of
the Tetra-PEG network and investigate the effect of these two
structural factors on the yielding of St-TPEG/PAAm DN gels.
■
EXPERIMENTS
Materials. Tetra-amine-terminated PEG (TAPEG) and tetra-NHSglutarate-terminated PEG (TNPEG) (NOF Corporation) were used
as received. Average molecular weights of the two tetra-arm-PEGs
were 10 kg/mol. 2-Acrylamido-2-methylpropanesulfonic acid sodium
salt (NaAMPS, Toa Gosei, Co., Ltd.) was used as received. Acrylamide
(AAm, Junsei Chemicals, Co., Ltd.) was recrystallized from chloroform. N,N′-Methylenebisacrylamide (MBAA, Wako Chemical Industries, Co., Ltd.) was recrystallized from ethanol. 2-Oxogultaric acid (αketo, Wako Chemical Industries, Co., Ltd.) was used as received. Sodalime glass plates (thickness: 3 mm) were used for the gel synthesis.
Glass molds were prepared from two glass plates spaced by silicone
rubber (thickness: 0.5−2 mm).
Synthesis of the Tetra-PEG Network. 74.6 mg/mL of TAPEG
was dissolved in phosphate buffer (pH 7.4), and 74.6 mg/mL of
TNPEG was dissolved in phosphate−citric acid buffer (pH 5.8) to
obtain c* solutions.27 Equal volumes of the TAPEG and the TNPEG
c* solutions were mixed together, poured into the glass mold, and kept
for 12 h at room temperature to obtain platelet TPEG gels. The
resulting average molecular weight of a network strand in the TPEG
gels is 5 kg/mol, which corresponds to an average polymerization
degree of 114. The same TPEG gels were used for synthesis of all the
St-TPEG DN gels.
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Synthesis of the Molecular Stent. The TPEG gels were then
immersed in 0.2−2.5 M of 2-acrylamido-2-methylpropanesulfonic acid
sodium salt (NaAMPS) and 0.1 mol % of α-keto aqueous solutions for
at least 1 day. The gels were then sandwiched by the two glass plates,
wrapped by a plastic film, and moved into an argon blanket. 365 nm
UV (4 mW/cm2) polymerization was then carried out for 3 h to
synthesize linear PNaAMPS within the first TPEG network. TPEG
gels containing PNaAMPS are called St-TPEG gels.
Synthesis of the Second Network. The St-TPEG gels were
immersed in the second network precursor aqueous solutions
containing 2−6 M of acrylamide (AAm), 0.01 mol % of MBAA, and
0.01 mol % of α-keto for about 12 h and then sandwiched with the two
glass plates. After that 365 nm UV (4 mW/cm2) polymerization was
carried out for 9 h to synthesize the second PAAm network within the
St-TPEG gels in an argon blanket. A part of the obtained St-TPEG/
PAAm DN gels was immersed in pure water before measurements
(called swollen samples) while the rest was directly used for testing
(called as-prepared samples).
One-Dimensional Swelling Ratio Measurement. The asprepared TPEG gel was chosen as the reference state. The onedimensional swelling ratio of the DN gels, α, was defined as α = t/t0,
where t is the thickness of the DN gel and t0 is at reference state.
Thickness of the gel was measured with calipers.
Uniaxial Tensile Test. Uniaxial tensile tests were performed on
dumbbell-shaped samples standardized as JIS K 6261-7 (12 mm in
length, 2 mm in width, 0.5−2.7 mm in thickness) with an Instron 5965
tensile tester accompanied by a noncontact extensometer AVE
(Instron Co.). Tensile velocity was fixed at 100 mm/min. Nominal
tensile stress, σtens, was defined as the force divided by the original
cross-sectional area. The deformation ratio, λtens, was defined as the
length of the deformed sample divided by that of the sample in the
relaxed state. The tensile yield point was defined as where the slope of
the stress−strain curve is zero. Tensile yield stress, σy‑tens, and yield
deformation ratio, λy‑tens, were defined as σtens and λtens at the yield
point, respectively.
Uniaxial Compression Test. Uniaxial compression tests were
performed on cylinder-shaped gels (17 mm in diameter, 2.5−6.2 mm
in thickness) with a Tensilon RTC-1310A (Orientec Co) tensilecompressive tester. The samples were compressed by two parallel
metal plates. Silicone oil was added between the sample and the metal
plates to prevent adsorption and friction between them. The
compression rate was fixed at 0.1 min−1. The definitions of σcomp
and λcomp are the same as those of the tensile test, so λcomp = 1
represents the undeformed state and the compression proceeds with
decreasing λcomp. The compressive yield point was defined as a flection
point of the stress−strain curves. Compressive yield stress, σy‑comp, and
yield deformation ratio, λy‑comp, were defined as σ and λ at the yield
point, respectively.
Figure 3. One-dimensional swelling ratio, α, of the St-TPEG/PAAm
DN gels with various concentrations of molecular stent in the asprepared and swollen states. Unless visible, error bars are smaller than
the symbol size (number of experiments, n = 3).
Figure 4. (a) Tensile stress−deformation ratio curves of the asprepared St-TPEG/PAAm DN gels with various α. The curves with an
asterisk mean that the measurements were stopped before sample
failure due to the measurement limitation of the video extensometer.
(b) Tensile stress−deformation ratio plot highlighting the small
deformation region. (c, d) Yield deformation ratio, λy‑tens, and yield
stress, σy‑tens, dependence on α of the St-TPEG/PAAm DN gels. Error
bars are less than the size of the symbols (number of experiments n =
3), except for a sample of α = 2.1 (n = 1 due to the experimental
difficulty). Dashed lines represent the observed power-law relationships.
■
RESULTS
Swelling Properties. Figure 3 shows the one-dimensional
swelling ratio, α, of the St-TPEG/PAAm DN gels with various
concentrations of molecular stent, PNaAMPS. α increased with
the NaAMPS feed concentration as the large number of
counterions of PNaAMPS induces high osmotic pressure. α of
the DN gel was successfully controlled over a very wide range
from 1.0 to 3.9 (in the as-prepared state) and 1.6 to 5.4 (in the
swollen state). At small α, the first network TPEG chains are in
a coiled conformation. With increasing α, the chains are
gradually prestretched to their extended state.
Tensile Yield Point of the St-TPEG DN Gels. Figure 4a
shows the tensile stress−deformation ratio curves of the asprepared St-TPEG/PAAm DN gels with various α. Mechanical
properties of the gels dynamically changed with varying α. In
general, the DN gels with larger α showed lower stress than
those with smaller α. This occurs because decreasing α results
in dilution of the first network, which mainly bears stress in DN
gels.
Figure 4b shows the tensile stress−deformation ratio curves
of the as-prepared St-TPEG/PAAm DN gels at small
deformation. Most of the DN gels showed a yield point except
those with very small α. The yield deformation ratio, λy‑tens, and
yield stress, σy‑tens, monotonically decrease with an increase in
α. In order to obtain a quantitative relationship, λy‑tens and σy‑tens
of both the as-prepared and swollen St-TPEG/PAAm DN gels
were plotted against α with logarithmic axes as shown in Figure
4c,d. Two definite scaling relationships, which are
λ y‐tens ∝ α −1
(1)
and
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σy‐tens ∝ α −2
similarity implies that the yield point of DN gels corresponds to
the ultimate stretching point of the first network chains in the
tensile direction. However, the value of L (= 9.0) is slightly
larger than the theoretical value of 7.6. Three possible reasons
can be raised. One is the change of bond angle or length in the
highly stretched PEG chains. It is well-known that when
covalent bonds are stretched, their bond angle and length
significantly change.29,30 This idea is also applicable to PEG
chain fracture. Oesterhelt et al. have proposed that when PEG
chains reach their maximum extension, their bond angle and
length should change, based on AFM measurements of a PEG
single chain.31 Also, Heymann et al. have implied similar
assumptions based on their theoretical study on PEG.32
Another possible reason is fluctuation of the relative position
of the network chain-ends, which is not assumed in the affine
network model. The phantom network model is the simplest
network model which takes into account positional chain-end
fluctuations in a network.33,34 It has been found that the
phantom network model is more applicable than the affine
network model for the tetra-PEG gels synthesized at relatively
low concentration.27 In the phantom network model, a single
network strand having N monomers with chain-end fluctuation
can be considered as a longer chain having Nf/(f − 2)
monomers without fixed chain-ends, where f is the functionality
of the network.32 In this work, f is 4 and therefore Nf/(f − 2) =
2N. Simply assuming a Gaussian chain, extensibility of such a
single chain is bN1/bN0.5 = N0.5, where b is the Kuhn length and
should be constant. Thus, a theoretically phantom network
chain can be stretched (2N)0.5/N0.5 = 1.44 times larger than the
affine network chain. While this idea (applying the phantom
network model) can explain the larger experimental L, a
problem of this idea is whether chain-end fluctuation still exists
at such highly stretched states or not. The other possible reason
is imperfect network connectivity. Defects in the network lead
to a decrease of substantive cross-linking density, which may be
a reason for the difference between experimental and
theoretical L. Indeed, infrared spectroscopy studies have
shown that the conversion of the cross-end coupling reaction,
p, of the TPEG gels used in this work was 0.91, which is less
than 1.35 However, our previous experiments and simulations
imply that p does not affect to the maximum extensibility
dramatically when p is significantly high (>0.7).28,35 Thus, at
this time, the authors believe the first effect (bond angle/length
change) is most probable.
Subsequently, we will discuss the origins of the relationship
σy‑tens ∝ α−2. Figure 5b shows auxiliary illustrations for the
following discussion. Based on the proposed assumption that
yielding of DN gels is due to finite extensibility of the TPEG
network in the tensile direction, yield stress, σy‑tens, is ideally
expressed as
(2)
were found. Interestingly, the results of the as-prepared DN
gels and the swollen DN gels overlapped, although their second
network structure should be different due to swelling of the
latter. This fact implies that the first network structure is the
dominant factor of the yielding criterion of the St-TPEG/
PAAm DN gels.
Next the origins of these yield-related relationships and the
coefficients were studied. First, we will discuss the origins of the
relationship of λy‑tens ∝ α−1. This equation can be rewritten as
αλy‑tens = L, where L is a proportional constant. L can be
determined from the slope of the plot of λy‑tens vs α−1 as 9.0.
Here, let us revisit the physical meaning of α and λtens. Figure 5a
Figure 5. Schematic illustrations for determining factors of (a) the
yield deformation ratio and (b) the yield stress of DN gels.
shows auxiliary illustrations for the following discussion. α is
the one-dimensional swelling ratio of the St-TPEG/PAAm gels
compared to the as-prepared TPEG gels. The TPEG network
in the St-TPEG/PAAm DN gels is already prestretched α times
by swelling. On the other hand, λtens is the uniaxial deformation
ratio of the St-TPEG/PAAm DN gels possessing a prestretched
TPEG network. Thus, the total deformation ratio of the TPEG
network in the stretched St-TPEG DN gels compared to the asprepared TPEG gels is α × λtens. Considering these relationships, the equation αλy‑tens = L means that yielding of the StTPEG/PAAm DN gels occurs when the total deformation ratio
of the TPEG network in the tensile direction reaches a certain
constant value, L. This means that the yield point of the StTPEG/PAAm DN gels is determined only by the deformation
ratio of the TPEG network in the tensile direction, regardless of
the deformation in the other two directions.
Then, what happens to the TPEG network at the point
αλy‑tens = L? We simply consider the finite extensibility of a
single polymer chain and the affine deformation model. It is
assumed that average structure of an ideal TPEG network with
perfect connectivity is a diamond lattice structure. By using the
preparation condition and lattice parameter, the original endto-end distance of one TPEG network strand in the reference
state is calculated as 5.4 nm.28 At a theoretical fully stretched
state, each TPEG network strand reaches its contour length,
which also can be calculated as 41 nm without consideration of
bond length or bond angle change (for details, see Supporting
Information). By dividing the latter by the former, the
theoretical limit extension ratio of a single chain is calculated
to be 7.6. This value is almost consistent with L = 9.0. This
σy‐tens = Fd
(3)
where F (N) is the force required to break one TPEG network
strand by stretching and d (m−2) is the area density of the
TPEG chains in the relaxed DN gels in the tensile direction. As
F is theoretically constant, σy‑comp should be proportional to the
area density, d. d decreases with increasing one-dimensional
swelling ratio, α, of the gel with a relationship of d ∝ α−2. By
substitution of this relationship into eq 3, we obtain σy−tens ∝
α−2.
In our system, d can be calculated since the TPEG network
structure has been well-determined. Area density, d, of the
TPEG network is calculated as
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d=
⎛ v0 ⎞2/3
⎜
⎟
⎝ α3 ⎠
(4)
where v0 is the number density of elastically effective network
strands in the TPEG network in the reference state (1/m3). v0
can be calculated by adopting the theory of a treelike structure
for tetrafunctional networks, such that
3
⎛
⎛1
⎞1/2 ⎞⎛ 3
⎛1
⎞1/2 ⎞
1
3
3
v0 = n0⎜⎜ + ⎜ − ⎟ ⎟⎟⎜⎜ − ⎜ − ⎟ ⎟⎟
4 ⎠ ⎠⎝ 2
4⎠ ⎠
⎝p
⎝p
⎝2
(5)
where n0 is the number concentration of the tetra-arm polymer
in the TPEG pregel solution and p is the reaction conversion
(fraction of connected bond).36,37 For this study, n0 was
determined from the preparation condition as 8.4 × 1024 m−3
and p is 0.91. Thus, v0 can be calculated as 6.9 × 1024 m−3. On
the other hand, F is determined by the weakest bond in a
TPEG network strand, which consists of ether bonds and an
amide bond.22 Our previous study shows that activation
enthalpies of the ether bond (∼86 kJ/mol) and the amide
bond (∼89 kJ/mol) are very similar, suggesting forces required
for cleavage of these bonds are also similar.38 For an ether bond
in PEG, Aktah et al. have calculated with the DFT method that
the force required for cleavage of a PEG chain in water is
around 3 nN.39 Thus, this time we roughly adopt this value as
F. By using these values, the experimental yield stress, σy‑tens,
and the ideal yield stress, Fd, can be compared. As a result, in all
cases, the ideal value, Fd, was much larger than the
experimental value σy‑tens. For example, in the case of α = 2.3,
calculated Fd is 21 MPa, whereas measured σy‑tens was only 0.83
MPa, which is 25 times smaller than the ideal value. This result
means that not all the TPEG chains are fully stretched and bear
stress at the yield point of the DN gels. The large difference
between σy‑tens and Fd can be explained by the assumption that
stress is concentrated on a part of the TPEG chains in the StTPEG/PAAm DN gels. The MALDI-TOF mass spectrum of
10k TAPEG (see Figure S1 of Supporting Information) implies
that even though the dispersity (Mw/Mn) of tetra-PEG is very
small (1.05), a distribution in molecular weight exists. In
addition, abrupt strain hardening of a PEG chain only occurs
when the chain approaches its finite extensibility.31 Thus, it is
easy to imagine that stress is concentrated on the relatively
short TPEG chains, and such chains mainly bear the stress at
the yield point of the St-TPEG/PAAm DN gels.
Compressive Yield Point of the St-TPEG DN Gels.
Figure 6a shows the compressive stress−deformation ratio
curves of the St-TPEG DN gels having various α. Clear flection
points can be found in the curves. As the gels show softening at
this point, such flection points may correspond to yielding.
Thus, in this paper, such points are called the compressive yield
point. Figure 6b,c shows a scaling plot of the compressive yield
deformation ratio, λy‑comp, and yield stress, −σy‑comp, against α.
Two definite scaling relationships, which are
λ y‐comp ∝ α 2
Figure 6. (a) Compressive stress−deformation ratio curves of the asprepared St-TPEG/PAAm DN gels with various α. The measurements
were stopped at σcomp ∼ −43 MPa due to limitations of the load cell.
(b) Compressive stress−deformation ratio plot highlighting the area
near the yield points. (c, d) Yield deformation ratio, λy‑comp, and yield
stress, σy‑comp, dependence on α of the St-TPEG/PAAm DN gels.
Error bars are smaller than the size of the symbols (n = 3). Dashed
lines are observed power-law relationships. The solid line in (d) is the
theoretical power-law relationship.
yield point may also be associated with TPEG chain rupture in
the first network when finite extensibility is reached. A sample
under axial compression also extends in the radial direction. For
an incompressible gel, the radial stretch λr and the axial stretch
λz are related as λzλr2 = 1. Substituting λz for the yield stretch,
λy‑comp, in eq 6, we arrive at the scaling relation in terms of the
radial stretch at the yield point, λy‑r
λ y‐r ∝ α −1
This relationship is consistent with the relationship derived in
the tensile case (eq 1). Furthermore, a calculated proportionality coefficient of eq 8, which corresponds to L in the tensile
case, is 8.9, which is very close to the obtained L of 9.0. Such
consistency suggests that the same yielding criterion may be
applied to the DN gels regardless of deformation mode: the
finite extensibility of the first network chains.
Now let us turn to the compressive yield stress. Because of
volume incompressibility, the yielding phenomenon (or the
rupture of the first network) should not be affected by the
hydrostatic stress state of the sample. By adding to a sample a
hydrostatic tension, sr = sθ = sz = s, which equals the axial
compression in magnitude, the uniaxial compression state is
converted to an equivalent state of equal biaxial tension:
sz = 0, sr = sθ = s
(9)
It should be noted that the hydrostatic stress, s, is the true stress
measured with respect to the deformed geometry. The true
stresses (force per unit deformed area) are related to the
nominal stress (force per unit original area) by the simple
geometric relations
(6)
and
−σy‐comp ∝ α −4
(8)
(7)
can be found. The power indexes are different from those of the
tensile tests.
Let us look further into the origin of the compressive yield
point. In analogy to the tensile yield point, the compressive
sr = λrσr ,
sθ = λθ σθ ,
sz = λzσz
(10)
In terms of the nominal stresses, the equal biaxial tension takes
the form
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σr = σθ = −
s
= − λz s
λr
in our case, the two curves are almost overlapping. It may be
attributed to a significant strain hardening effect of the first
network. We think such large strain hardening effectively
“hides” the effect of the other two axes. For further discussion
on this topic, an in-depth study of biaxial stretching of DN gels
is required.
Normalization of the Stress−Strain Curves. Based on
the above-mentioned discussion about the extraordinary strain
hardening effect of the first network, it can be expected that
stress−deformation behavior of DN gels is almost only
dominated by the deformation ratio of the first network chains
in the stretching axis to its reference state and their density. To
confirm this idea, we normalized all the stress−deformation
ratio curves of the St-TPEG/PAAm DN gels shown in Figures
4a and 7a based on the first network structure. In particular, σ
was normalized by initial area density of the first network
chains (normalized σ: σα2), and λ is normalized by the
deformation ratio of the first network polymer strand length to
its reference state (normalized λ: λα). Figure 8 shows the
(11)
where s is related to the nominal compressive stress, σz, by s =
λσz.
Consequently, at the yield point, the equivalent biaxial stress,
σy‑biax, is related to the compressive yield stress, σy‑comp, as
σy‐biax = −λ y‐comp3/2σy‐comp
(12)
Under consideration of the above-mentioned yielding criterion
of DN gels, if we assume that yielding under biaxial tension
follows the same scaling relation as that of uniaxial tension,
σy‑biax ∝ α−2, the corresponding compressive yield stress is
−σy‐comp ∝ α −5
(13)
This scaling relation is very close to that of eq 7. The difference
may be attributed to technical difficulties in experiments such as
friction between plates and gels, which may induce nonideal
compression (sometimes called barreling problem), although
silicone oil was used to prevent it. On the basis of the above
results, we can conclude that the proposed yielding criterion of
DN gels, which is that a DN gel yields when the first network
strands reach their extension limit, is general and can be applied
to both tensile and compressive states.
Comparison of Uniaxial and Equal Biaxial Stretching.
Ideally, the deformation modes of a uniaxial compression tests
are equivalent to that of the equal biaxial tensile tests. Based on
the above-mentioned idea shown in eq 11, compressive stress,
σcomp, can be converted to equal biaxial (radial) stress, σr. Also,
the compressive deformation ratio, λcomp, can be converted to
the equal biaxial (radial) deformation ratio, λr. We calculated
the equal biaxial tensile stress−deformation ratio curves of the
St-TPEG/PAAm DN gels from the compression test results as
shown in Figure 7a. Interestingly, the calculated curves have all
Figure 8. Normalized stress−deformation ratio curves, both uniaxial
and equal biaxial stretching, of the St-TPEG/PAAm gels with various
α.
normalized stress−deformation ratio curves (both uniaxial and
equal biaxial) of the St-TPEG/PAAm DN gels. As we expected,
all the curves collapse almost perfectly. Amazingly, they not
only show similar yield points but also similar strain hardening
behavior started from the normalized deformation ratio of
around 7. This overlapping is also proof of the dominancy of
the first network structure on the mechanical behavior of DN
gels.
Effect of the Second Network on the Yield Point. We
also studied the effect of the second network on the yield point
of the St-TPEG/PAAm DN gels. The feed concentration of
acrylamide for the second network was controlled from 2 to 6
M while α was kept constant (2.4−2.5). Figure 9 shows the
Figure 7. (a) Calculated equal biaxial stress−deformation ratio curves
of the St-TPEG/PAAm DN gels with various α. (b) Normalized
stress−deformation ratio curves of both uniaxial and equal biaxial
tensile tests.
the features of the stress−deformation ratio curves of uniaxial
tensile tests, which are a clear yield point, strain hardening (of
the first network) before yielding, and a stress plateau region
after the yield point. Further, the uniaxial and equal biaxial
tensile curves of the St-TPEG/PAAm DN gels with the same α
roughly overlap each other as shown in Figure 7b. This
phenomenon is quite unusual for rubbery materials. The
stress−deformation behavior of ideal rubbery materials obeying
entropic elasticity depends not only on the deformation ratio of
the stretching axis but also that of the other two axes. At the
same strain, the equal biaxial stress of rubbery materials is
theoretically always larger than the uniaxial stress.40 However,
Figure 9. Effect of the second network concentration on the stress−
deformation ratio curves of the St-TPEG/PAAm gels.
F
DOI: 10.1021/acs.macromol.5b02592
Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
*Tel & Fax +81-11-706-4815; e-mail gong@mail.sci.hokudai.ac.
jp (J.P.G.).
stress−strain curves of the as-prepared St-TPEG/PAAm gels
with various AAm concentrations. Although these DN gels had
different concentrations of the second network, their yield
points did not change significantly. This fact also clearly shows
that the yielding behavior of DN gels, which corresponds to the
fracture of the first network, is dominated strongly by the first
network and hardly depends on the second network.
Nevertheless, yield stress appears to slightly increase with
increasing concentration of the second network. Although the
fully stretched first network mainly bears the stress at the yield
point, the soft second network also bears some stress. As a
result, the apparent yield stress is the sum of the stress borne by
the first and the second networks at the yield point. The former
does not change, but the latter increases with increasing second
network concentration. This might be the reason for the slight
increase in the yield stress with increasing second network
concentration. Also, the yield deformation ratio slightly
decreases with increasing second network concentration.
Entanglement between the two networks might slightly reduce
the extensibility of the first network.
Note that this is not meant to suggest that the second
network is not important for toughening of DN gels. Existence
of the second network is necessary for the occurrence of
yielding of DN gels. If the relative amount of the second
network to the first network is not enough, the DN gels
become so brittle that they break before reaching the yield
point.8
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
This research was funded by a Grant-in-Aid for Scientific
Research (S) (No. 124225006) from the Japan Society for the
Promotion of Science (JSPS) and by the ImPACT Program of
Council for Science, Technology and Innovation (Cabinet
Office, Government of Japan). The authors thank Toa Gosei
Co., Ltd., for the supply of NaAMPS (ATBS-Na) and Dr. Yuki
Akagi (the University of Tokyo) for provision of the MALDITOF results. J. P. Gong thanks Prof. Michael Rubinstein
(University of North Carolina) for his precious suggestions. T.
Nakajima thanks Dr. Daniel R. King (Hokkaido University) for
his careful proofreading.
■
■
CONCLUSIONS
We first clarified the yielding conditions of the double network
gels by using St-TPEG/PAAm DN gels with a well-defined first
network structure. We have found the following yielding
criteria: (1) the yield point of DN gels is determined by the
finite extensibility of the first network chains; (2) the yield
stress (in the stretching direction) is determined by the area
density of the first network; (3) the effect of the second
network on the yield point is negligible. These important
findings could not be realized without using TPEG gels with
well-controlled network strand length and density. This study
gives us clear insight into the yielding and toughening
mechanism of DN gels and related tough soft materials having
multiple network structure.41,42 Additionally, the yield point is
sometimes considered to be the maximum allowable
deformation of materials for practical use since materials
begin deforming irreversibly at this point. Thus, clarification of
the yield point of DN gels, done in this work, helps in the
design of DN gels suitable for various applications such as
medical and industrial ones.9−11,14,43
■
ASSOCIATED CONTENT
S Supporting Information
*
The Supporting Information is available free of charge on the
ACS Publications website at DOI: 10.1021/acs.macromol.5b02592.
Calculation of contour length of the TPEG network
strands and MALDI-TOF measurements of the TPEG
prepolymer (PDF)
■
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Corresponding Authors
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(T.N.).
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