Journal of the American Chemical Society
Communication
2
1
.5, 10.6 ± 1.7, and 27.1 ± 1.6 J·m− for Gel-1, Gel-2, and
model for force-dependent bond scission under dynamic
4
Gel-3, respectively (Figure 3c).
loading, the rate constant of I at a loading rate of 10 nN/s is
5
−1
The difference in tearing energies is attributed to the only
significant difference in the gels: the single mechanophore
present in each linker. In strands made from 1 and 2, the
2
.6 nN for II and 3.8 nN for III are obtained using calculations
19,20
9
21
mechanical “weak bond”
is the cyclobutane mechano-
data to a power law dependence (Figure S4), Γ ∼ f
phore, whereas the site of scission in strands incorporating 3 is
less clear. We hypothesize that the likely scission pathway is α-
cleavage at the C−C bond next to the 1,4-disubstituted 1,2,3-
triazole ring, which has been reported to be a mechanochemi-
cally weak site, but other possibilities include bonds at highly
substituted centers such as the junctions of the 4-arm PEG and
the α carbon of isobutyrate ester.
The correlations highlight an important distinction. To date,
many molecular interpretations of polymer fracture are based
on the well-known Lake−Thomas theory, which connects
the fracture energy of the polymer network to the energy
stored per chain along the elastically active strand when it
breaks. Such interpretations of polymer fracture energies
default to the thermodynamics of the bond scission reaction
a
,
break
provides a value of a = 1.9 across the series of gels, in
comparison to a = 2 expected from the “coupled spring”
models described above. Given the assumptions involved in
this treatment, a more complete and quantitative examination
of the relationship between macroscopic tearing energy and
strand scission force is desirable. In particular, future efforts
2
1
22
will benefit from a detailed characterization of network
15,29
topology,
the effective strain rate at the crack tip, and
the precise rate−force dependence of strand scission for these
mechanophores.
Nonetheless, the results suggest a direct connection between
macroscopic and molecular behavior. Gel-1 and Gel-2 are
structurally indistinguishable, with the notable exception of the
difference in aryl versus methylene substituents at a single
moiety in the linker connecting tetra-PEG macromers. The
total aryl group content of Gel-1 is 0.55 wt % and immersed
within a majority mobile solvent phase, yet, as noted above, the
two gels are easily distinguished by their feel to the touch. The
difference in network properties that is actually being felt by
hand, in essence, is the connection between tearing energy and
force-coupled reactivities of the embedded mechanophores on
each strand.
(
e.g., total bond dissociation energy) as the relevant molecular
23,24
thermodynamic quantity.
The systems employed here
reveal limitations of this assumption, as the total BDEs
enthalpy of the cycloreversion reaction) for cyclobutane
scission in 1 is exothermic, so fracture would require no
energy. Instead, the relevant molecular parameters are
associated with the kinetics of reactivity, as captured in
SMFS measurements of force-coupled bond lifetimes for 1 and
. As noted above, the forces required for lifetimes of tens of
ms are roughly 1 and 2 nN for 1 and 2, respectively, whereas
extrapolating prior calculations by Smalø and Uggerud gives a
(
9
2
21
corresponding value of ∼3 nN for triazole α-cleavage in 3.
We note the likely origins of the underlying reactivity
differences. Although the cycloreversion mechanism of II has
not been studied explicitly, it is expected to have a diradicaloid
22
According to the Lake−Thomas theory, when the number
of broken chains is constant, the fracture energy Γ is
proportional to the energy stored per strand at rupture, U.
9
,30,31
transition state akin to similar cyclobutanes.
Relative to
2
5,26
On the basis of recent adjustments
to the Lake−Thomas
II, therefore, the lower activation force of I can be attributed in
theory, the energy U is approximately proportional to the
large part to the stabilization of the diradical transition
9
,31,32
square of the breaking tension of the bridging strands (U ∼
state
by the aryl substituents, in a large part through
2
f
), because the stored elastic energy of the bridging
break
quantum mechanical effects manifested as resonance stabiliza-
polymer chain is dominated by enthalpic deformation that is
assumed to have a springlike, linear force−displacement
dependence. A quadratic dependence is expected if the
majority of the stored energy is not held within the bridging
strands, but in the much softer (lower force constant) but still
springlike, entropic deformations of nonbridging strands
connected to the bridging strands in a treelike structure.
Assessing these molecular theories, however, requires knowl-
edge of the actual breaking forces of the strands during crack
propagation, which likely differ from those measured by SMFS,
because the loading conditions during fracture and SMFS are
not identical. Unfortunately, the precise single-chain loading
rates at the propagating fracture plane are not known, but we
offer a rough estimate based on fracture mechanics, SMFS
experiments, and mechanical tests, and simulated data.
33
tion. Ultimately, the differences in macroscopic behavior that
are easily felt by hand originate in these molecular-scale
electronic substituent effects.
Additional implications of these results include the
opportunity to expand the use of mechanophores as
quantitative probes of molecular fracture mechanisms.
Mechanochromic and mechanoluminescent mechanophores
provide molecular insights by enabling visualization of the
34,35
damaged zone,
and scissile mechanophores with known
force-coupled reactivity complement those tools by connecting
the structural observations to quantitative relationships
between macroscopic and molecular mechanical responses.
That opportunity motivates further characterizations of the
8
,36−41
force dependency of mechanochemical reactions
that are
suited to this purpose, noting that historical connections
between strand scission and bond dissociation energies are
prone to error and become less valid as strands break through
reaction mechanisms other than homolytic scission. Together
with a better understanding of the relevant loading rate and
associated reaction dynamics at the propagating crack tip, such
efforts might ultimately lead to quantitative, first-principles
prediction of macroscopic fracture behavior as a function of
network molecular structure.
On the basis of previously reported treatments of steady-
state fracture processes, the relevant strain rates at the process/
2
−1 27
plastic zone for Gel-1−3 are estimated to be ∼10 s . The
reported stretch stiffness of linear PEG is about 100 nN in the
enthalpic deformation regime at which strand scission
2
8
occurs, so the operative loading rate (product of strain rate
4
9
previously reported force dependence and applying the Bell
3
716
J. Am. Chem. Soc. 2021, 143, 3714−3718
Wang, Shu
