Glass-like dynamics of the strain-induced coil/helix transition on a permanent polymer network

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by  O. Ronsin

Glass-like dynamics of the strain-induced coil/helix transition on a permanent
polymer network
O. Ronsin,1 C. Caroli,1 and T. Baumberger1
Institut des NanoSciences de Paris, CNRS, Sorbonne Universit´e – Pierre et Marie Curie, UMR 7588,
4 place Jussieu, 75005 Paris, France
We study the stress response to a step strain of covalently bonded gelatin gels in the temperature range
where triple helix reversible crosslink formation is prohibited. We observe slow stress relaxation towards
a T-dependent finite asymptotic level. We show that this is assignable to the strain-induced coil → helix
transition,previouslyevidencedbyS.Courty,J.L.GornallandE.M.Terentjev(PNAS,102,13453(2005)),of
6 a fraction of the polymer strands. Relaxation proceeds, in a first stage, according to a stretched exponential
1
dynamics, then crosses over to a terminal simple exponential decay. The respective characteristic times τ
0 K
and τ exhibit an Arrhenius-like T-dependence with an associated energy E incompatibly larger than the
2 f
activation barrier height for the isomerisation process which sets the clock for an elementary coil → helix

n transformation event. We tentatively assign this glass-like slowing down of the dynamics to the long-range
a
couplingsduetothemechanicalnoisegeneratedbythelocalelementaryeventsinthisrandomelasticmedium.
J

8
2
I. INTRODUCTION particle pinning in studies of point-to-set correlations in

glass-forming liquids6.
]
ft The slow relaxation dynamics of glass-forming deeply The experimental procedure consists in applying to
o supercooled liquids exhibits, following a thermal or me- this elastic solid a step shear strain which, as demon-
.s chanical quench, well-known characteristics1. It is in stratedin5, triggers,onthe stretchedpolypeptidic inter-
at general well described by the phenomenological Kol- crosslink strands, the coil→helix transition of successive
m rausch expression, namely a stretched exponential ∼ monomer segments. It is these elementary cis-trans iso-
- exp[−(t/τα)β], with 0 < β < 1. The α-relaxation time merisation processes which play the role of relaxation-
d τ exhibits,foravastmajorityofmaterials,theso-called promotinglocalevents. Any suchprocessinduces avari-
α
n ation of the tension of the strand on which it occurs,
fragile behavior, i.e. a faster than Arrhenius growth
o hence a set of stress signals in the embedding network,
as temperature T decreases towards the glass transition
c
which constitute the dynamical noise. We follow the re-
[ temperature Tg – a behavior which can be formulated
sponse ofthe systemby monitoring the relaxationof the
asthe paradoxicalincreaseuponcoolingofan”apparent
1 macroscopic shear stress σ.
activation energy”.
v
We observe a dynamics characterizedby the following
6 It is now recognized that the elementary relaxation
features :
0 events responsible for this dynamics are rearrange-
7 ments of clusters of a few (typically ≤ 10) structural • Relaxation of σ towards a finite asymptotic value.
7 units (atoms, monomers,...), which involve coordination The associated stress drop ∆σ grows non linearly with
0 changes2. Each such event thus necessarily gives rise to the applied strain γ0. At fixed γ0, it decreases with T,
1. a long range elastic field which propagates into the de- until the effect vanishes above a threshold temperature.
0 formableembeddingsolid,asevidencedinthesimulation Thisbehaviorisfullyconsistentwith,andthusconfirms,
6 resultsof3. Howandtowhichextentthecollectiveeffects the picture put forwardby Terentjev et al. of the role of
1 of the interplay between this ”dynamical noise” and the the strain-induced coil → α-helix transition in the me-
:
v elasticnon-affinityduetostructuraldisorderresultinthe chanicalresponseofnetworksofdenaturedpolypeptides.
i spectacularslowingdownobserveduponapproachingT • The relaxation dynamics itself exhibits two distinct
X g
remains for the moment an essentially open issue. stages: the earlier one is well fitted by a Kolrausch
r
a Here, in order to try and shed some further light on stretched exponential expression. From this we identify
this question, building upon recent results of Terentjev a relaxation time τK, many orders of magnitude larger
and coworkers4,5, we perform an experimental study of than the Rouse time of the inter-CL strands, which is
the stress evolution triggered by an applied strain in a here the relevant viscoelastic time, since the CL are ir-
highly deformable permanent elastic network, obtained reversible and the gelatin concentration is such that our
initial polymer solution is in the non-entangled regime.
by covalent cross-linking of biopolymer (gelatin) chains.
In this system, relaxation proceeds via local transfor- This early stretched exponential response crosses over
mation events coupled together by the stress noise they to a simple exponential decay towards the above-
generate in the network on which they take place. As mentioned asymptotic stress drop ∆σ, thus defining a
cross-links (CL) are permanent, the network topology terminal relaxation time τf, which turns out to be com-
remains invariant: in contrast with the case of glass- parable with the cross-overtime τco.
formers, relaxation does not affect coordination. Note We find that, in the limited temperature range (see
thatthe associateddecimationofrelaxationpathsinthe Section II) accessible to our experiments, τ (T) and
K
energy landscape is reminiscent of the effect of selective τ (T) both exhibit an Arrhenius-like behavior, with the
f
2
same activation energy E. However, E turns out to be relatedcontributionto the stress response. Courty,Gor-
much larger than that, E , for the prolyl cis-trans iso- nallandTerentjev5 thenmeasuredtheopticalactivityof
act
merisation process which has been long ago recognized stretched thin plates of gelatin gels, which gives access
tosetthe clockforthe coil→α-helix transitioninvolved to the helix content, and showed that its non-monotonic
in the renaturation of gelatin into collagen7. dependenceonthestrainamplitudeisingoodqualitative
On the basis of the observation of (i) stretched ex- agreement with their theoretical predictions.
ponential relaxation on time scales way too large to be For our present purpose, it is important to note that
assignable to simple network viscoelasticity and (ii) an the system used in these experiments was a physical
apparent Arrhenius slowing down upon cooling much gel, obtained by cooling a gelatin-in-water solution be-
larger than expected from that of the elementary micro- low T ∼30◦C. Under such conditions, gelation occurs
gel
scopic relaxation events, we are led to conclude that the via the formation of segments of the original triple helix
elastic noise generatedby localrelaxationevents is suffi- collagenstructure interconnected by coiled strands. The
cienttobringtheresponsetoquenchofasolidexhibiting H-bonds which stabilize these cross-links (CL) are weak
frozenstructuraldisorderinto the class of glass-likeslow enoughforthegelstobethermoreversible. Thatis,asthe
dynamics. CLareabletorearrangeviapartialzipping/unzippingof
the triplehelix underthe combinedeffectofinternaland
applied stresses, gelatin gels obtained by cooling exhibit
II. EXPERIMENTAL slowrelaxationandglass-likeagingfeatures,suchaslog-
arithmic shear modulus strengthening13,14, sensitive to
applied stress15.
The system we study is a gelatin hydrogel. Gelatin
As developed in Section I, we are interested here in
is obtained from collagen, whose molecules are consti-
studying the dynamics of strain-induced relaxation in a
tuted of the wrapping into a right-handed helix of three
network where monomer ”transfer” between contiguous
left-handed single strand α-like (polyproline II) helices.
strands is prohibited. This makes physical gels unsuit-
Collagendenaturationresults,athightemperatures(typ-
ically & 30◦C), in solutions of single stranded chains in able, due to the fact that biased zipping/unzipping at
the ends of a triple helix segment results in strand slid-
the coil configuration.
Following the pioneering study by Flory and Weaver8, ing. We circumvent this difficulty by making use of gels
resulting from the covalent bonding, via an enzymatic
considerableefforthasbeendevotedtoanalyzingthe de-
tailsofthekineticpathleadingtocollagenrenaturation9. route, of gelatin chains in their high temperature coil
state. We then study, in the temperature range where
It is now agreed that the kinetics is limited by cis-trans
physical triple-helix CL formation does not occur, the
isomerisationoftheprolylpeptidebondspresentinlarge
full time-dependent stress response to a step strain.
proportion along the chains. The associated activation
energyE0 ≃72kJ/molisquitelarge,sothattheclock-
act
setting time for the dynamics of the coil to helix transi-
tion τ lies in the unusually slow range of a few ten A. Gel preparation and characterization
c/h
seconds10.
This transition is in general triggered by cooling. Be- Inordertopreparecovalentlybondedgelatinnetworks
sides, several theoretical works11,12 have predicted that withagivenshearstoragemodulus G,we proceedasde-
imposingafiniteend-toendextensionRtoamoleculeof scribed in detail in16. In short, we dissolve gelatin (300
a polymer exhibiting a coil/helix transition results in a Bloom, type A from porcine skin, Sigma) in deionized
displacement of the transition at the expense of the coil water et 65◦C. After total dissolution of the polymer,
state. Indeed, as R is increased, due to the decrease of the solution is quickly mixed at 40◦C with a Tgase en-
the coil entropy, the free energy balance is increasingly zyme solution (microbial transglutaminase, Activa-WM,
biased in favor of the helix state. So, for example, at a AjinomotoFoodsEuropeSAS)soastoreachafinalcom-
temperature T slightly larger than the free chain transi- position of 5 wt% gelatin and 2.6 nmol of Tgase (cor-
tion one T , one expects a finite helix fraction to appear responding to an enzymatic activity of 2U). We have
0
beyondsomeextensionthresholdwhichdecreasesasT is checked that, for this concentration, the gelatin solution
lowered. This fraction is predicted to grow with R until is in the semi-dilute, non entangled regime17. The solu-
the moleculebecomes fully helical,beyondwhichfurther tion is then poured into the temperature-controlled cell
extension results in its decrease. Such effects have in ofa stress-controlledrheometer(MCR 501,AntonPaar)
particular a bearing on single molecule force-extension equipped with a cone-plate, sand-blasted cell. The sam-
curves11. ple is protected against solvent evaporation by a paraf-
More recently, Kutter and Terentjev4 have extended a fin oil rim. The cell temperature is controlled to within
simplified version of the theory to the case of polymer 0.1◦C with the help of a thermoelectric device.
◦
networks. In this latter case, since the end-to-end dis- Gelation then proceeds at T = 40 C. At this tem-
set
tancesofthepolymerstrandsareimposedbyneighboring perature, chosen well above T ≃ 30◦C, no triple-helix
gel
crosslinks,submitting a gel to a mechanical deformation reversible cross-link can form whereas Tgase catalyzes
must induce both a variation of the helix content and a actively the formation of inter-chain covalent bonds be-
3
tween two specific residues. The advancement of the 278
cross-linkingprocessismonitoredbymeasuringthestor- 0
age shear modulus G at f = 1 Hz with a 1% strain am- 276 dt
plitude. When the target G level is reached, we quickly /aw-0.5
heat the sample up to 70◦C and maintain it at this tem- a) 274 σdr
peraturefor10min,afterwhichitiscooleddownbackto P dt, -1
( 272 G/
the working temperature T >T . After such a heating w d
stage, the enzyme is known togbele inactivated, i.e. fur- σra 270 ε -1.5
35 36 37 38 39 40 41
ther covalent cross-linking is fully inhibited. Indeed, as T (°C)
illustrated on Figure 1, the gel shear modulus no longer 268
increases.
266
0 1 104 2 104 3 104
1000 80 t (s)
800 70
760 FIG. 2. Stress response σraw(t) of a gel with initial shear
modulus G = 1400 Pa following a step strain of amplitude
a) 600 a)758 60 T ǫ = 20%, at T = 35◦C. Dashed line: asymptotic linear de-
G (P 400 G (P756 50 (°C) cslroepaese(.dσInraswet/:dtT)te→m∞pe(rraetdursequdaerpeesn)daenndceosfotfhethdeecaasyymǫdpGto/tdict
754 (black dots) due to the protease-induced modulus decrease
6 103 104 1.4 104
t (s) (vertical unit: 10−4 Pa.s−1). The error bars correspond to
200 40
thescattering of the results from three different samples.
0 30
0 4 103 8 103 1.2 104
time (s) checked to lie within the linear elastic response regime),
the network weakening obviously contributes to the de-
creaseoftheshearstressσ atarateǫdG/dt. Asshown
raw
FIG. 1. Build-up of the shear modulus G(t) (dark curve) on Fig.2 (inset) this contribution fully accounts for the
in response to the thermal history (light red curve). Inset:
Blow-up of the late stage of shear modulus evolution. asymptotic decrease of σraw, since (dσraw/dt)t→∞ and
ǫdG/dt are equal within experimental error. In order to
correctfor this linear additive drift, we define the stress:
However, closer inspection (see inset of Figure1) re- σ(t)=σraw(t)−t(dσraw/dt)t→∞
veals that G(t) systematically exhibits a very slow and
linear decrease. This feature is clearly assignable to Itisthiscorrected,intrinsic,stresswhichisdealtwithin
the presence in the enzyme preparation of traces of the following sections.
protease16, the effect of which is to weaken the gel by
catalyzing the scission of gelatin network strands. This
is consistent with the fact that (see inset of Fig.2), the III. RESULTS
higher the temperature, the steeper the modulus de-
crease. Figure3showsatypicalstressresponseσ(t), obtained
at T = 35◦C for a gel of shear modulus G = 1400 Pa
under the applied strain ǫ = 20%. After gradual relax-
B. Stress relaxation experiments ationextendingoverthousandsofsecondsthestressdrop
σ0−σ(t)saturatesatthevalue∆σ =σ0−σ∞. Notethat,
Stress relaxation experiments are performed accord- although ∆σ/σ lies in the 10−2 range, the stress drop
0
ing to the following protocol. After quenching at a rate value remains much larger than the noise level.
of 15◦C/min from 70◦C to the working temperature T, Figure4displaystheevolutionof∆σwithtemperature
we shear the sample at the rate 0.1 s−1 until we reach in the range where physical cross-linking upon cooling
the strain level ǫ = 20%. From this instant which we does not occur. Note that for G = 1400 Pa this range
choose as the time origin, we record the shear stress sig- extends down to the value T = 25◦C lower than T .
gel
nal σ (t). Indeed we have checked that, as suggested by previous
raw
As can be seen on Figure 2, which displays a typical results18,theshearmodulusofthesampleremainedcon-
recording,thestressrelaxes,andexhibitsalinearasymp- stant over the duration (∼ 105 s) needed for full stress
toticdecrease. Ontheotherhand,wehaveseenthatpro- relaxationatthistemperature. ∆σ(T)isseentodecrease
tease trace contamination results in a linear decrease of steeply downto a negligible level, reachedfor T &45◦C.
the gelshearmodulus G withtime. Since the gelsample Inorderto assessthe degreeofthermalreversibilityof
isundertheconstantsheardeformationǫ(whichwehave thephysicalprocessunderlyingrelaxationinoursystem,
4
278 finalrelaxationtime τf. However,this late stagefit does
σ not account for the steeper decrease observed at earlier
0 times,whichwe findto be verywellfitted bya stretched
exponential of the form
)
a 274 ∆σ
P δσ(t)=∆σexp −(t/τ )β (1)
( K

σ (cid:2) (cid:3)
These two regimes exhibit a rather narrow crossover
σ
270 ≈ ∞ about a time τ significantly larger than τ .
co K
10
0 5 103 1 104 3 104
t (s)
FIG. 3. Intrinsic stress response σ(t) = σraw(t) − ) 1
a
t(dσraw/dt)t→∞. Same data as Fig.2. P
(

σ
δ 0.1
60
0.01
40
) 0 2 103 4 103 6 103
a
P
t (s)
(
σ
∆ 20
FIG. 5. Circles: Semi-logarithmic plot of δσ(t)= σ(t)−σ∞
for the data set of Fig.3. Full line: Stretched exponential
0 fit (see eq.1) of the initial decay, performed over 2103 s and
25 30 35 40 45 50 extrapolatedtolatetimes. Dashedline: asymptoticexponen-
T (°C) tialdecayextrapolatedtoearlytimes. Thefitparametersare
listed in thethird line (T =35◦C) of Table I.
FIG. 4. Temperature dependence of the stress drop ∆σ =
σ0 −σ∞ (see Fig. 3) for gels with G = 1400 Pa. Except
for the T =25◦C datum, the error bars are smaller than the
The above described behavior is systematically ob-
dots.
served, for G = 1400 Pa, for temperatures ranging from
45 to 25◦C. The corresponding values of the fit param-
eters are listed in Table I, together with similar data
we have performed the following control experiment17. obtained with a G = 700 Pa gel on a much narrower
After the system has reached its asymptotic state under temperature range.
strain at T =35◦C, we rapidly reheat it under the same
strain up to T = 55◦C, a value chosen so that stress
relaxationvanish. Wefindthatσ recovers,withinexper- TABLE I. Fit parameters for the early and late relaxation
regimes.
imental error, the value GT=55◦ ×ǫ corresponding to a
purelyelasticresponseatthistemperature. Besidesprov-
ing the absence of wall slip in our experiments at least G T(◦C) β τK(s) τco/τK τf(s)
up to the 50% strain level used in this experiment, this 1400 45 0.53 70 7.1 320
result demonstrates the full reversibility of the physical
1400 40 0.50 170 3.5 680
process responsible for the observed stress evolution.
1400 35 0.46 250 5.3 1850
Wenowturntotheanalysisofthefulltimedependence
1400 30 0.48 1850 3.7 7300
of
1400 25 0.41 3200 3.7 16500
δσ(t)=σ(t)−σ∞ 700 40 0.34 920 3.1 5800
700 38 0.29 1600 2.5 10500
which characterizes relaxation towards the final, equi-
700 36 0.40 4100 2.3 16500
librium state. As immediately appears from the semi-
logarithmic plot of a typical set of data (see Fig.5), the
terminal decay of δσ is a mere exponential, defining a
5
IV. DISCUSSION Since our experiments are performed at temperatures
chosentoliecloseabovethetransitiontemperatureT ≃
0
A. The relaxation process Tgel of the free chain, we will from now on specialize to
the case where θ is a small positive number.
We first need to identify the physical process respon-
sible for the observed stress relaxation. n /N
h
First of all, it is clear that it cannot be assigned to
1
standard viscoelasticity. Indeed, from the values, on the
order of 1 kPa, of their shear modulus, we can estimate
the average mesh size of our gels ξ ≃ (k T/G)1/3 to
B
lie in the 10 nm range. The corresponding Rouse times
τ , on the order of 10−5s, are thus fully negligible on
R
the relevant time scale, which ranges from hundreds to
thousands of seconds.
We must also exclude the relevance of the so-called x γ x x
“slow mode”, observed in some scattering experiments φ – +
andassociatedwithfrictionalslidingofentanglements19.
Indeed, we have checked17 that the 5% gelatin pre-gel φ
solution which we use is in the semi-dilute, non entan- +
gled regime. (Note, moreover,that the slow mode, when
present,correspondstoacharacteristictimeof,typically,
at most 103τ ∼10−2s).
R
These remarks lead us to conclude that the stress de-
cay triggered by a step strain in our system originates B
from a structural transformation which we identify as
φ
thestrain-inducedtransition,brieflysketchedoutinSec-
–
tionIIabove,ofatleastpartofthepolymerstrandsfrom C
A
the coilto the helix configuration. In order to show that
suchatransitiondoesindeedlead,followingfastloading,
x γ x x
to a stress drop, let us now very briefly summarize the – +
simplified model formulated by Kutter and Terentjev4.
Consider a polymer strand, with fixed end-to-end dis-
FIG. 6. Upper panel: Equilibrium helix fraction on a strand
tance R, containingN monomersoflength a arrangedin of N monomers vs. reduced end-to-end distance x = R/Na.
twoconsecutiveblocks,namelyahelicalsegment,aligned Lower panel: Corresponding equilibrium strand tension φ
with the end points, comprised of n monomers, each of (full black curve). The arrows correspond to the evolution
whichoccupiestheeffectivelengthγa(withγ <1)along of φ for thestrain-induced path ABC (see text).
the helix axis. The second block is a Gaussian coil of
end-to-end distance (R−γna) formed by the remaining
(N −n) monomers. The strand free energy then reads: The variationof neq with x at constant T (fixed value
of θ) is shown on Fig.6.a. Several regimes appear: at
small extensions x < x− = γ − γ2−θ, no helix is
3k T
F =n∆f + 2(N −Bn)a2 (R−γna)2+∆fint (2) plinreesaernlyt o,nutnhteilstthraenfdu.llB-heeyloixndsttahtiespnthres=hoNld,inserqegarcohweds
eq
for x = γ. Further extension results in its symmetric
The free energy gained per monomer in the helix (h)
linear decrease, up to the upper threshold x = γ +
+
configuration ∆f = C(T −T ) vanishes linearly at the
0 γ2−θ above which the helix is completely unwound.
transitiontemperatureT ofthefreepolymerchain. The
0 Aps T increases, the thresholds x± move toward γ, so
second term in the r.h.s. is the elastic cost associated
that, for θ ≥ γ2, any strand is fully coiled whatever its
withimposingtheend-to-enddistanceofthecoiledblock.
extension.
The third one accounts for the presence one helix/coil
The tensionφ =(∂F/∂R) , shownonFig.6.b. For
and one helix/cross-link interfaces. For a given mate- eq neq
rial (given value of γ), the number of monomers n en- smallx<x− itisgivenbythestandardcoilelasticform:
eq φc (x) = 3k Tx/a. In the (h/c) coexistence regime ex-
gagedinthehelixsegmentatequilibrium,obtainedfrom eq B
∂F/∂n = 0, depends on the two dimensionless parame- tending fromx− to γ,itremains constantatthe plateau
ters: value φ− = φceq(x−), then jumps to the upper plateau
φc (x ) in the unwinding coexistence regime, etc..
eq +
Let us now sketch the evolutionof a strandsubmitted
R 2∆f 2C(T −T ) to a step increase of its end-to-end extension, from the
0
x= , θ = = (3)
Na 3kBT 3kBT initial value xA to xB =xA(1+ǫ). In order to fix ideas,
6
we specialize to the case xA <x−, where the unstrained B. Relaxation dynamics
strand is a mere coil. Since the loading rate is very fast
onthetimescaleτc/h oftheisomerisationprocessneeded Canwenowunderstandthestressrelaxationdynamics
for an elementary configurational change of order a few itself on the basis of the above framework - namely, as
tenseconds),theevolutionofthestrandfrompointsAto resulting fromthe evolutiontowardstheir equilibriumof
B(seeFig.6.b)correspondingtothestrainstepoccursat asetofmechanicallyindependentstrainedinter-crosslink
theconstanthelixcontentneq(xA)=0,wherethetension strands? Forthispurpose,weneedtobuildadynamical,
increase is ruled by the instantaneous stiffness: extended version of the static Kutter-Terentjev model4.
On the basis of previous works on the helix-coil transi-
tion, we may reasonably assume20 that the strand dy-
namicsisaFokker-Planckone,namelythatthedistribu-
∂φ 3k T
B
κ=Na = (4) tionf(n,t|x)ofthenumberof(h)monomersonastrand
(cid:20)∂x(cid:21) Na2
n=neq(xA) of dimensionless extension x evolves according to:
Two cases are then possible. ∂f ∂ ∂f 1 ∂F(n,x)
=D + f (5)
• If xB >x−, neq(xA) 6=neq(xB), the strand configu- ∂t ∂n(cid:18)∂n kBT ∂n (cid:19)
rationthereforerelaxesuntilthehelixcontentreachesits
where F is defined by expression 2 and the diffusion co-
equilibrium value at the strained extension x , and the
B
efficient
strand tension decreases from φ to φ , the amplitude
B C
of this relaxation increasing with (xB −x−).
•If,onthecontrary,xB <x−,neq(xB)=neq(xA)=0, D =τ−1 (6)
pointsBandCcollapseandnoretardedtensionevolution c/h
occurs.
is the inverse of the cis-trans isomerisation time.
This sketch is easily extended to various other possi- One then immediately checks that the terminal relax-
ble cases: for instance, if n (x ) and x both lie in ation of the average helix content < n > from its initial
eq B B
the [x−,γ] interval, after its step increase φ exhibits full value neq(xi) to the strained equilibrium one neq(xf =
relaxation to its unstrained value. x (1+ǫ) is exponential with, in the case of interest here
i
Let us now come back to our stress relaxation experi- (xi <x− and x− <xf <γ >), the characteristic time:
ments. The gelrandomnetworkcontains gelatinstrands
with all possible orientations, and R values distributed k T ∂2F −1 N(γ−x )
around the average mesh size R¯ = ξ ∼ 10 nm. In or- τ = B =τ f (7)
der to evaluate its response, we have extended17 to the strand D (cid:20)∂n2(cid:21)neq(xf) c/h(γ2−θ)3/2
imposed simple shear deformation geometry the highly
simplified network theory of ref.4, which assumes inde- whosevariationwith temperatureis primarilycontrolled
by the Arrhenius dependence of τ . As already men-
pendent strands with a gaussianx-distribution, centered c/h
at x¯ = R¯/Na, and of width ∆x. Note that the mea- tioned, the associated activation energy from existing
biochemical results E ≈0.75 eV .
sured fractional stress drop ∆σ/σ is small, typically of act
0
Thisistobecomparedwiththemeasuredstressrelax-
order a few percent. This indicates that, in the T-range
ationtime τ ofthe gel. TheArrheniusplotofτ forthe
of our interest, x¯+∆x . x−, i.e. that the helix con- f f
system with G=1400 Pa, shown on Fig.7, is indeed lin-
tent in the unstrained gel, carried by a tiny minority
earoverthe (rathernarrow)exploredT-range. However,
of strands in the tail of the x-distribution, is negligibly
the associated energy
small. Strain drives beyond x− a larger fraction of the
strandsorientedclosetothestretchingprincipalaxis,the
evolutionofwhichtowardstheirequilibriumgivesriseto
d(lnτ )
f
the observed stress relaxation. As T grows, so does the E =kB =1.7±0.05eV (8)
d(1/T)
threshold x−, and ∆σ/σ0 gradually vanishes, the blur-
ring of the transition reflecting the distributed character
turns out to be considerably larger than E . This
ofintercross-linkspacings. Thisbehavior(seeFig.4of17) act
discrepancyentailsanimportantconclusion. Indeed,one
is in qualitative agreement with the experimental stress
could a priori be tempted to interpret the observedtwo-
drop results.
stage relaxationdynamics as resulting fromindependent
This analysis, together with the full mechanical re- activatedjumpsacrossenergybarrierswithawideheight
versibilityobservedwhenreheatingunderconstantstrain distribution due to disorder. If such is the case, τ must
f
afully relaxedsystem,providesstrongconfirmationthat beviewedastheactivationtimeassociatedwiththemax-
the stress relaxation we observe does originate from the imum barrier E, of the form τ = τ exp(E/k T). Now,
f 0 B
partial transition to the helix configuration of stretched for G = 1400 Pa, we measure (see table I) τ = 7300 s
f
gelatin strands triggered by the applied strain. at T =30◦C. At this temperature, exp(E/k T)∼ 1030,
B
7
which would lead to an utterly unphysical value for the analyzed in terms of a growing apparent activation en-
“microscopic” time prefactor, of order 10−26 s! ergy (often fitted by the Vogel-Fulcher-Tamman expres-
Fromthisargument,weconcludethatcollectiveeffects sion E = AT/(T −T )) which may become much larger
0
play a prominent part in the dynamics of our system. than that for an elementary local event.
In our case,as discussedabove,independent activated
105 local events cannot account for the relaxation dynam-
ics. This leads us to conclude that, in our system it
is the combination of local relaxation events and of the
104 elastic noise which they generate in the random network
which is, most likely, responsible for the emergence of
) slow glassy dynamics.
s 103
( Let us finally emphasize a conspicuous difference be-

τ
tween our dilute random network and supercooled liq-
102 τ uids,namely: inourgel,theKolrauschregimeunambigu-
f
ously crosses over to a simple exponential decay with a
τ
K relaxationtimeτ proportionaltoτ ,while,indeeplysu-
f K
101 percooled glass-formers, no termination of the stretched
3.1 3.2 3.3 3.4 exponential behavior has been observed1.
1000/T (K-1) A hint about the origin of this difference could possi-
bly be provided by the following remark. We have pre-
viously studied the aging behavior of gelatin networks18
FIG. 7. Arrhenius plot of the Kolrausch time τK and of the containing a hybrid population of thermoreversible and
terminal relaxation time τf for gels with G=1400 Pa (data covalent crosslinks (CL). As is well known, the fully re-
from Table I). Lines: independentbest exponential fits.
versible gels exhibit a logarithmic growth of their shear
modulus, which has not been observedto saturate13, in-
dicatingthat the upper limit ofthe relaxationspectrum,
Clearly,thedynamicalversionoftheKutter-Terentjev
if any, lies far beyond times of order months. We have
model misses an essential feature of the relaxation pro-
found that the presence of an increasing fraction of ir-
cess. Indeed, the independent strand assumption over-
reversible CL results in the exponential decrease of the
looks an important physical point: consider an elemen-
”aging slope” dG/d(lnt). As compared with the triple
tary (c) → (h) event occurring on the strand connecting
helix physical CL, which are able to zip/unzip (hence to
nodes (i) and (j). Before the event, the tension forces on
slide),covalentbondsprohibittheexchangeofmonomers
eachnodeareequilibrated;thetransitioninducesajump
between neighboring strands, thus reduce the possibility
δφ ofthestrandtension,i.e. apairofextraforces±δφ
ij ij of monomer long range motion. We conjecture that this
on(i)and(j). Thisforcedipoleresultsinanelasticdefor-
dynamical restriction constitutes the essential difference
mationoftheembeddingnetwork,andthecorresponding
long-ranged strain field, decaying as r−3, in turn shifts betweenglass-formersandourcovalentrandomnetworks.
The tentative interpretation which we propose for our
the end-to-enddistance R ofallother strands. Agiven
kl resultsclearlyasksforquantitative tests,whichcanonly
strandthusreceivesasetofsuchsignals,whichconstitute
be provided by numerical studies. We note in this re-
a self-generated ”mechanical noise”, the effects of which
gardthat,incontrastwithreversiblegels,thankstotheir
combine with those of the thermal one to determine the
frozen CL topology, the fully covalent networks studied
relaxation dynamics.
here appear amenable to realistic simulations.
This phenomenologyis stronglyreminiscent ofthe de-
scription of relaxation in deeply supercooled liquids in
terms of local structural rearrangements and of the long
ACKNOWLEDGMENTS
rangeelasticEshelby fields whichthey induce inthe em-
bedding medium3. Further support in favor of a close
connectionbetweendeeplysupercooledglass-formersand We are grateful to M. Djabourov for an enlightening
of our helix-forming gels is lent by the nature of the discussion. We thank Ana¨el Lemaˆıtre for a helpful criti-
early stage stress relaxation, which we find to be unam- cal reading of the manuscript.
biguously of the Kolrausch stretched exponential type
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