Creep and Fracture of a Protein Gel under Stress

2.3 MB·

by  Mathieu Leocmach

Creep and fracture of a protein gel under stress
Mathieu Leocmach,1,∗ Christophe Perge,1 Thibaut Divoux,2 and S´ebastien Manneville1
1Universit´e de Lyon, Laboratoire de Physique, E´cole Normale Sup´erieure de Lyon,
CNRS UMR 5672, 46 All´ee d’Italie, 69364 Lyon cedex 07, France
2Centre de Recherche Paul Pascal, CNRS UPR 8641 - 115 avenue Schweitzer, 33600 Pessac, France
(Dated: August 21, 2014)
Biomaterials such as protein or polysaccharide gels are known to behave qualitatively as soft
solids and to rupture under an external load. Combining optical and ultrasonic imaging to shear
rheology we show that the failure scenario of a protein gel is reminiscent of brittle solids: after a
4
primary creep regime characterized by a power-law behavior which exponent is fully accounted for
1
bylinearviscoelasticity, fracturesnucleateandgrowlogarithmically perpendicularlytoshear,upto
0
2 thesuddenruptureofthegel. Asingleequationaccountingforthosetwosuccessiveprocessesnicely
captures the full rheological response. The failure time follows a decreasing power law with the
g
applied shear stress, similar to the Basquin law of fatigue for solids. These results are in excellent
u
agreement with recent fiber-bundlemodels that include damage accumulation on elastic fibers and
A
exemplify protein gels as model, brittle-likesoft solids.

9
PACSnumbers: 82.35.Pq,47.57.Qk,83.60.-a,83.80.Kn
1

]
t Biogelsformedthroughtheself-associationofpolysac-
f 1.6
o charidecoils,collagen,actinfilaments orattractiveglob-
s ular proteins play a major role in biochemistry and mi-
. 0.8
t crobiology [1], biological networks and cell mechanics [2]
a
as well as in food science [3]. These biomaterials all be-
m γ
have as elastic solids under small deformations but dis- 0.4
d- play remarkable nonlinear behavior generally featuring s) 110045
on srterveesrss-iboler rsutrpatiunr-esti[ff5,en6i]n.gI[r4r]evaenrdsibfrilaitcytusrteesmpsriforromtothire- 0.2 τ(f 111000123
c 102 σ(Pa) 103
existenceofanexternalcontrolparameter,e.g. tempera-
[ 0.1
tureorpHinthecaseofthermoreversibleoracid-induced 10−1 100 101 102 103 104 105

2 gels respectively. This makes such biogels fundamen- t (s)
v
tally different from other soft glassy materials such as
4
emulsions, colloidal gels and glasses that can be rejuve- FIG.1: (coloronline)Strainresponseγ(t)ina4%wt. casein
3
gel acidified with 1% wt. GDL for an imposed shear stress
2 nated by shear [7–10] or transient networks where frac-
σ = 200 (•), 300 (H), 400 ((cid:4)), 550 (N) and 1000 Pa ((cid:7))
8 tures spontaneously heal [11, 12]. So far, huge effort has
from right to left. The gap width is 1 mm. Gray dashes
1. been devoted to the design of protein gels with specific show γ = 1. Inset: failure time τ vs σ. The red line is
f
0 properties and textures at rest [13, 14]. However, their the best power-law fit τ = Aσ−β with β = 5.45±0.05 and
f
4 mechanical behavior deep into the nonlinear regime has A=(4.2±0.1)1017 s.Paβ.
1
only been partially addressed [15, 16] and several fun-
:
v damental issues remain unexplored such as the spatially
i resolvedrupturescenarioorthe physicalrelevanceofthe ation and growth of fractures leading to gel failure. Our
X
analogy with brittle failure in hard solids. resultsareinfullagreementwiththe predictionsofsome
r
a recentfiber-bundle models andhint touniversalfeatures
In this Letter we report on stress-induced fracture in of failure common to both soft and hard solids.
protein gels by means of creep experiments coupled to Gelsarepreparedbydissolvingsodiumcaseinatepow-
optical and ultrasonic imaging. Gels formed by slow der (Firmenich) at 4% wt. in deionized water under
◦
acidification of a sodium caseinate solution display frac- gentle mixing at 35 C and 500 rpm. To induce gela-
tures under large strain at fixed low pH values [15, 17], tion, 1% wt. glucono-δ-lactone (GDL) in powder (Fir-
which makes them perfect candidates to quantify the menich) is dissolved in the solution and its hydrolysis
rupture of soft solids and tackle the above-mentionedis- progressively lowers the pH over the course of 8 hours
sues. We demonstratethatunderanexternalload,these [see Fig. 1(a) in the Supplemental Material, including
casein gels display brittle-like failure that results from Ref. [18]]. While still liquid, the solution is poured
two successive physical processes: (i) a primary creep into the gap of a polished Plexiglas concentric cylin-
regime where dissipation is dominated by viscous flow der (or Taylor-Couette, TC) shear cell immersed into a
through the gel matrix without any detectable macro- temperature-controlled water tank at 25.0±0.1◦C [19].
scopic strain localization and (ii) the irreversible nucle- Rheological data are recorded during gel formation by
2
sponse γ(t) is monitored. Images of the gel are recorded
104 a) 103 simultaneously to the rheology (Logitech Webcam Pro
P
(
103 ′′G 0.15 9000). The local velocity and strain fields can also be
γ/˙min102 ′G, 10210−2 10−1 100 101 iomusalgyedtoinrhtehoeloggryadbiyenat–cvuosrttoimcit-ymapdlaenuelt(rra,szo)nsiicmsuclatnanneer-
γ˙ −0.85 f (Hz) detailed in [20]. In this case, prior to acidification, the
101 sodium caseinate solution is seeded with acoustic trac-
ers here 3% wt. polyamide spheres (Orgasol 2002 ES3
100 (a) NAT 3, Arkema, diameter 30 µm, density 1.02) that do
10−4 10−3 10−2 10−1 100 not modify the final gel properties [Fig. 1(b) in the Sup-
t/τf plemental Material]. The radial position r is measured
4 from the inner rotating cylinder and the vertical axis z
pointsdownwardswiththe originabout15mmfromthe
105
γγ˙/˙min 23 τ(s)min110011301 103 105 tesotxrppUaeinonrfdimetγrhe(etna)tTdcrCeoisqnpcusletialralye.nsstFataoaiprlpuoprrbleeiuepsbdaterisetnhimageaefirrrerdessehvtpreeesrsanssimdbeσlpen,l,ceeetihan(ce-Fhsiiggtclu.roe.b1ea)p:l
τf(s)
γ(t) slowly growswith time up to γ ∼1 then accelerates
until the gel fails at a well-defined time τ . These three
1 f
(b) successivestepsarebetterhighlightedinFig.2byfocus-
0 0.2 0.4 0.6 0.8 1 ing on the globalshear rate γ˙(t). Figures 3 and 4 gather
t/τf theresultsfromlocalmeasurementsandarediscussedbe-
low together with eachof the successive regimes inferred
104 (c) from global data. Note that similar failure dynamics are
observed for other casein and GDL concentrations, for
103 different gap widths of the TC cell and in a different
n geometry namely a plate-plate cell (see Movie 1 in the
γ˙mi102 −1 Supplemental Material).
/
γ˙ AsseenintheinsetofFig.1,thefailuretimeτ sharply
101 decreases as a power law of σ with an exponentfβ ≃5.5.
τ further allows us to rescale all the shear rate data
100 f
γ˙(t) onto the single master curve of Fig. 2(a) by plot-
100 10−1 10−2 10−3 10−4 ting γ˙/γ˙ vs t/τ , where γ˙ is the minimum shear
(τ −t)/τ min f min
f f rate reached at a time τ [see also Fig. 2 in the Sup-
min
plemental Materialfor the original γ˙(t) data]. On about
FIG. 2: (color online) Normalized shear rate responses
four decades for t . 0.1τ , the shear rate decreases as
γ˙(t)/γ˙min corresponding to the data of Fig. 1 and plotted so a power law γ˙(t) ∼ t−αfwith α = 0.85 ± 0.04. This
astoemphasizethethreesuccessiveregimes. γ˙min isthemin-
imumshearratereachedatτmin (seetextandSuppl. Fig.2). is strongly reminiscent of the primary creep observed in
The yellow line shows the master curve inferred from fitting solidsandreferredtoasAndradecreep[21–24]. Interest-
γ˙(t) by Eq. (1) with α = 0.85, leading to λ = 0.378±0.002 ingly, here, the exponent can be inferred from linear vis-
and µ = 0.187 ± 0.002. (a) Primary creep: γ˙(t)/γ˙min vs coelasticity. Indeed casein gels display a power-law rhe-
t/τf in logarithmic scales. Inset: Linear viscoelastic mod- ology G′(f) ∼ G′′(f) ∼ f0.15 [inset of Fig. 2(a)], which
′ ′′
uli G (top) and G (bottom) as a function of frequency f
correspondsto a compliance J(t)≡γ(t)/σ ∼t0.15 in the
for a strain amplitude of 0.1%. Red lines are power laws
Gin′li∼neGar′′sc∼alefs0..15G.ra(yb)daSsehceosndsharoywctrheeepm:inγ˙i(mt)u/mγ˙moinf vEsq.t/(τ1f) ltionetahredoebfsoerrmveadtiγo˙n(t)re∼gitm−e0.8[25.5]M. oTrheiosvenricuelltyracsoornriecspimonadgs-
reached at τmin =0.556τf. Inset: τmin vs τf. The red line is ing reveals velocity and strain fields averaged over the
τmin=0.56τf. (c)Tertiary creep: γ˙(t)/γ˙min vs(τf −t)/τf in verticaldirection z that linearly decreasewith the radial
logarithmic scales with a reversed horizontal axis. position r within the gap [Fig. 4(a-b)] with no slippage
at the Plexiglas walls [see arrows in Fig. 4(a)]. Together
with direct visualization[Fig. 3(a)], these local measure-
a stress-controlled rheometer through small amplitude mentsdemonstratethatduringprimarycreepthereisno
oscillatory shear at frequency f = 1 Hz (Fig. 1 in the macroscopic strain localization or fracture, although we
Supplemental Material). Gelation is complete when the cannot rule out rearrangements below the available spa-
elastic(G′)andviscous(G′′)modulireachaplateauwith tial (∼ 10 µm) and/or temporal (∼ 1 s) resolutions due
G′ ≫ G′′. A constant stress σ is then applied to the to limited signal-to-noise ratio at very low shear rates
sample from time t = 0 and the subsequent strain re- [Fig. 3(a, left)].
3
(a)t/τf=0.02 (b)t/τf=0.57 (c)t/τf=0.93 (d)t/τf=0.99
0 0 0 0
10 10 10 10
m)
m
z(20 20 20 20
30 30 30 30

0 1 2 0 1 2 0 1 2 0 1 2
r(mm)
−5 0 5 v(µm/s) −5 0 5 −20 0 20 −20 0 20
FIG.3: (coloronline)Ultrasonicimages(left)anddirectvisualizationoftheTCcell(right)recordedsimultaneouslyatvarious
times in the primary [(a) t/τf = 0.02], secondary [(b) t/τf = 0.57 ≃ τmin/τf] and tertiary [(c) t/τf = 0.93 and (d) 0.99]
creep regimes. Ultrasonic images are azimuthal velocity maps v(r,z,t) computed by averaging over 4 s and coded using the
linear color levels shown below the images. Their position relative to the picture of the cell reflects theactual arrangement of
the ultrasonic probe along the vertical direction z. The gap width is 2 mm and r is the radial distance to the inner rotating
cylinder. Experimentperformedona4%wt. caseingelseededwith3%wt. polyamidespheresandacidifiedwith1%wt. GDL
underσ=300 Pa. Seealso Movie 2 in theSupplementalMaterial.
For 0.1 . t/τ . 0.9, γ˙(t) departs from power-law the vicinity of the crack tip is indicative of strong com-
f
behavior and goes through a minimum value at time pression and recoil of the gel matrix. Finally, the frac-
τ =(0.56±0.04)τ independentlyoftheappliedstress ture length ℓ(t) is observed to grow logarithmically with
min f
[Fig.2(b)andinset]assimilarlyreportedformetals[26], (τ −t) upon approaching τ [Fig 4(d)]. In other words
f f
solid composite materials [23] and fiber-bundle models one has dℓ/dt ∼ γ˙(t) ∼ (τ −t)−1, which indicates that
f
(FBMs)[27,28]. Thislinearitybetweenτ andτ ,also the global shear rate is linked to fracture-induced dis-
min f
known as the Monkman-Grant relation [29], allows one placements.
to “predict” the failure time from the intermediate-time Finally,weemphasizethatthe Andrade-likecreepand
response. During this secondary creep regime, regularly- the crack growth are two physical processes that effec-
spacedcracksnucleatefromthetopandbottomedgesof tively superimpose to yield the global rheological re-
the cell and start growing perpendicular to the applied sponse. Indeed, as seen from the yellow line in Fig. 2,
stress [Fig. 3(b), see also Movie 2 in the Supplemen- the master curve γ˙(t)/γ˙ vs t/τ is perfectly fitted by:
min f
tal Material]. These macroscopic fractures are invaded
−α
with water expelled from the surrounding gel matrix. γ˙(t) t µ
=λ + , (1)
Although fractures have not yet entered the ultrasonic γ˙ (cid:18)τ (cid:19) 1−t/τ
min f f
region of interest, velocity maps become heterogeneous
with only two adjustable parameters λ and µ once α =
along the z direction [Fig. 3(b, left)] and display inter-
0.85 is fixed. The remarkable collapse of the whole data
mittent fluctuations [Fig 4(c,d)]. The level of these fluc-
set to such a simple equation allows us to interpret the
tuations, which can also be seen on the global response
secondarycreepregimeasamerecrossoverfromcreepto
for the same applied stress [H in Fig. 2(b)], is poorly
crack growth.
reproducible and stress-dependent. Such intermittency
Letusnowsummarizeanddiscussthemostprominent
mayarisefromcrackgrowthoutsidetheregionofinterest
results of this Letter. First we have shown that casein
which induces long-range elastic deformation propagat-
gels display a remarkable failure scenario similar to that
ing acrossthe sample mainly alongthe verticaldirection
ofbrittle solids andcharacterizedby the same three suc-
[see the lower levelof fluctuations along the radialdirec-
cessive creep regimes. Here, the power-law exponent of
tion in Fig 4(d)].
the primary creep [γ˙(t) ∼ t−α with α = 0.85 ± 0.04]
In the tertiary creep regime, for t & 0.9τ , the shear is fully accounted for by linear viscoelasticity. Such a
f
rate increases by more than four orders of magnitude link between creep and viscoelasticity is shared by other
and diverges as (τ −t)−1 [Fig. 2(c)]. This finite-time biopolymer gels with power-law rheology [30, 31] as well
f
singularity corresponds to the final growth of the frac- as hard-sphere-like colloidal glasses [10]. To better un-
tures along the vorticity direction z as they accelerate derstand this link, additional creep and recovery tests
and eventually meet in the middle of the cell at time were performed within the primary regime (see Figs. 3
τ [Fig. 3(c-d)]. Ultrasonic velocity maps directly corre- and4inthe SupplementalMaterial,includingRef. [32]).
f
late withthe crackgrowthandrevealthe complexstruc- Successive loading/unloading sequences on a fresh sam-
ture of the displacement field at the tip and around the pleshowthat(i)thestrainisnotfullyrecoveredbyafew
fracture. In particular, Fig. 3(d, left) suggests that the percents, (ii) the viscoelastic properties of the material
fracture is initiated at the inner cylinder at r = 0 and areunalteredand(iii)strainresponsesareindistinguish-
the presence of large positive and negative velocities in ablefromonetesttoanother. Thisstronglysuggeststhat
4
15 (a) (c)
v(µm/s)
−2 −1 0 1 2
/s) 10 0 0.5 (e)
µv(m 5 (mm) 1200 0.4
z
0 30 0.3
0.3 (b) 1 (d) H
/
oc 0.2 m/s) 00..68 ℓ 0.2
γl µ
0.1 v( 0.4 0.1
δ
0.2
0 0 0
0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 10−1 10−2 10−3
r (mm) t/τf (τf−t)/τf
FIG.4: (color online) (a) Local velocity hv(r,z,t)iz and(b)local strain fieldhγloc(r,z,t)iz averaged overthevertical direction
z at various times during primary creep: t/τ = 1.910−3 (•), 1.710−2 (H) and 0.15 ((cid:4)). Solid lines are linear profiles. The
f
arrows in (a) indicate the velocity of the inner cylinder inferred from the current shear rate. (c) Spatiotemporal diagram of
thelocal velocity hv(r,z,t)ir averaged over theradial direction r and plotted in linear color levels as a function of z and t/τf.
(d) Standard deviation δzv(t) of hv(r,z,t)ir taken over the vertical direction z (thick black line) together with corresponding
standarddeviationδrv(t)computedovertheradial directionr onthez-averagehv(r,z,t)iz (thinredline). (e)Fracturelength
ℓ(t) vs (τ −t)/τ as inferred from direct visualization (•, average over 6 different fractures, error bars show the standard
f f
deviation) and from ultrasonic imaging (◦) and normalized by the height H of the TC cell. Gray dots show the visualization
dataforthelongest fracturewhichleadstothefailure ofthesampleat τ . Redlinesarethebestfitsℓ(t)=a+blog(1−t/τ )
f f
to thevisualization data. Same experiment as in Fig. 3 and SupplementalMovie 2.
the small irreversibility observed in the primary regime elongational load predict three successive creep regimes
mainly stems from viscous dissipation (which accounts exactly alike Fig. 2 with similar proportionality between
′′
for the nonzero viscous modulus G ) due to the solvent τ and τ and finite-time singularity [23, 28]. This
min f
flow within the gel fibrous matrix and involves only a highlights the relevance of FBMs in the context of creep
minute amount of local damage of the gel network. Pro- in protein gels, whose microstructure indeed appears to
teingelsthusappeartoexperienceakineticallyreversible be formed of strands [38, 39], but also urges to study
primary creep, i.e. dominated by viscous dissipation FBMs in shear geometries to check whether they would
rather than by plasticity. be able to predict fracture growth as observed here.
To conclude, the present time- and space-resolved
The logarithmic fracture growth in the tertiary creep
study exemplifies protein gels as model, brittle-like soft
regime constitutes our second important result. Such an
solids. Promptedbythe remarkablesimplicityofEq.(1)
evolution is also commonly reported in disordered solid
that encompasses Andrade-like creep, the Monkman-
materials displaying brittle rupture and interpreted in
Grant relation and finite-time singularity within a sin-
the framework of Griffith-like models based on global or
gle equation, future modeling will undoubtedly focus on
local energy barriers [33]. However these approaches all
the microscopic ingredients needed to predict quantita-
predict exponential scalings for τ (σ) while our data are
f
best fitted by the decreasing power law τ ∼ σ−β with tively the Monkman-Grant prefactor and the Basquin
f
exponent β. The next experimental step consists in a
β = 5.45±0.05 [inset of Fig. 1]. This last key result
statistical study of the fluctuations associated to crack
suggests that thermally activated crack growth does not
nucleation and growth as well as a systematic investiga-
playanyprominentrole. Ratherthepower-lawscalingis
tionofothersystemsinorderto checkforuniversalityin
strikinglyreminiscentoftheBasquinlawoffatiguefound
theirreversiblecreepruptureofsoftsolidsingeneraland
for avariety ofheterogeneousorcellular materialsunder
ofbiogelssuchasactin,alginateoragargelsinparticular.
cyclic deformation [34–36]. Basquin law has also been
Such a study is expected to have important implications
recently predicted for creep experiments by FBMs that
in understanding the behavior of biomaterials under ex-
combine elastic fibers with a local yield strain and take
treme stress conditions.
intoaccountdamageaccumulation[34,37]. Interestingly
assumingdamageaccumulationtobe proportionaltoσγ TheauthorsthankM.Alava,T.Gibaud,S.Lindstr¨om,
directlyleadstoBasquinlawwithβ =γ andlargevalues G. H. McKinley and N. Taberlet for fruitful discussions,
of γ, typically larger than 5 as found here for β, lead to M. Kal´ab for kindly providing us with unobtainable ref-
macroscopic cracksdue to the simultaneous rupture of a erences, and A. Parker at Firmenich for precious advice
large number of fibers [37]. More generally FBMs under and for providing the casein and GDL. This work was
5
fundedbytheInstitutUniversitairedeFranceandbythe and gap 1 mm with an inner rotating cylinder of ra-
EuropeanResearchCouncilunderthe EuropeanUnion’s dius 24 mm. Optical and ultrasonic imaging coupled to
SeventhFrameworkProgramme(FP7/2007-2013)/ERC rheometry (Figs. 3 and 4) were performed on an ARG2
rheometer(TAInstruments)inaTCcellofheight60mm
grant agreement No. 258803.
and gap 2 mm with an inner rotating cylinder of radius
23 mm. We checked that the commanded shear stress σ
is reached within 20 ms. A detailed, quantitative study
of theinfluenceofthegap widthandofthetemperature
on thefailure dynamics is left for future work.
∗
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