Shear-induced memory effects in boehmite gels
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Shear-induced memory effects in boehmite gels
Shear-induced memory effects in boehmite gels
Iana Sudreau,1, 2 Sébastien Manneville,2 Marion Servel,1 and Thibaut Divoux2
1)
IFP Energies nouvelles, Rond-point de l’échangeur de Solaize, BP 3, 69360 Solaize, France
2)
Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
(Dated: 29 March 2021)
Colloidal gels are formed by the aggregation of Brownian particles into clusters that are, in turn, part of
a space-spanning percolated network. In practice, the microstructure of colloidal gels, which dictates their
mechanical properties, strongly depends on the particle concentration and on the nature of their interactions.
Yet another critical control parameter is the shear history experienced by the sample, which controls the size
and density of the cluster population, via particle aggregation, cluster breakup and restructuring. Here we
investigate the impact of shear history on acid-induced gels of boehmite, an aluminum oxide. We show that
arXiv:2103.14476v1 [cond-mat.soft] 26 Mar 2021
following a primary gelation, these gels display a dual response depending on the shear rate γ̇p used to reju-
venate their microstructure. We identify a critical shear rate γ̇c , above which boehmite gels display a gel-like
viscoelastic spectrum upon flow cessation, similar to that obtained following the primary gelation. However,
upon flow cessation after shear rejuvenation below γ̇c , boehmite gels display a glassy-like viscoelastic spec-
trum together with enhanced elastic properties. Moreover, the nonlinear rheological properties of boehmite
gels also differ on both sides of γ̇c : weak gels obtained after rejuvenation at γ̇p > γ̇c show a yield strain
that is constant, independent of γ̇p , whereas strong gels obtained with γ̇p < γ̇c display a yield strain that
significantly increases with γ̇p . Our results can be interpreted in light of previous literature on shear-induced
cluster densification, which accounts for the reinforced elastic properties at γ̇p < γ̇c , while we rationalize the
critical shear rate γ̇c in terms of a dimensionless quantity, the Mason number, comparing the ratio of the
strength of the shear flow to the interparticle bond force.
I. INTRODUCTION of the present article. Indeed, depending on its inten-
sity, mechanical shear may either enhance or compete
with the attractive interactions driving the formation of
Soft Glassy Materials (SGM) are ubiquitous in our ev- clusters, which play a key role in the solid-like behav-
eryday life. They encompass a broad range of viscoelastic ior of the sample upon flow cessation19,20 . Moreover,
materials that are key to out-of-equilibrium living organ- upon partial yielding, gels often break down in a spatially
isms and omnipresent in major industries, e.g., foodstuff, anisotropic way21–23 , and such anisotropy gets frozen into
personal care, and building industry1–3 . SGMs are com- the gel microstructure during flow cessation24,25 . In that
posed of sub-units such as particles or polymers, whose framework, shear history strongly impacts the rheological
interaction and volume fraction control the macroscopic properties of colloidal gels, both in their fluidized state as
mechanical properties of these soft materials4–6 . In prac- weakly aggregated suspensions and upon flow cessation,
tice, SGMs display solid-like properties at rest and un- when they form a viscoelastic solid.
der small strains, whereas they yield and flow like liquids On the one hand, for stresses much larger than the
when submitted to a large enough strain7,8 . At rest, their yield stress, the gel microstructure is broken down into
solid-like behavior originates either from the presence of clusters of particles with attractive interactions. In such
attractive interactions between constituents, which form a suspension, the balance between hydrodynamic forces
a space-spanning network at low volume fractions, or and internal cohesive forces determines whether shear in-
from the jamming of the constituents for large enough duces aggregation, break-up or internal restructuring of
volume fractions. The former category of SGMs is re- the cluster26,27 , among which the latter two phenomena
ferred to as gels, whereas the latter category is referred are thermally activated28 . Starting from a fully dispersed
to as soft glasses8–10 . state, i.e., from a suspension of individual particles, in-
Beside volume fraction and particle interactions, an creasing shear intensity accelerates the growth and de-
additional control parameter of SGM properties at rest creases the size of clusters, whose steady-state proper-
is the route followed towards the solid-like state through ties depend on both the shear intensity and the particle
the shear history experienced during the liquid-to-solid concentration29,30 . More generally, for any configuration
transition. In soft glasses, oscillatory shear deformation in the fluidized state, i.e., partially or fully dispersed, the
of moderate amplitude can be used to mechanically en- gel response during transient flows depends on whether
code some memory into the material through local plas- the shear is increasing or decreasing. This effect leads
tic deformations11–13 . The imprint of the shear pro- to thixotropy and rheological hysteresis, which are ubiq-
tocol may subsequently be read out by a strain sweep uitously observed when measuring a constitutive equa-
from small to large amplitudes14 . Similar memory ef- tion, shear stress σ vs. shear rate γ̇ 31,32 . Such hysteresis
fects have been identified in colloidal gels15 , where shear loops have been characterized through their area, which
history significantly affects the microstructural and me- is maximum for a critical sweep rate of the shear, point-
chanical properties16–18 , which constitutes the subject ing towards a thixotropic timescale that is characteristicShear-induced memory effects in boehmite gels 2 of the sample33–35 , and relates to the sample memory of composed of complex building blocks most of which are flow. unbreakable aggregates of the original colloids. Following On the other hand, shear history also plays a key role this peculiar “primary” sol-gel transition, the properties on gels upon flow cessation. Indeed, high shear rates of boehmite gels are stable in time, and cycles of shear- fully break the structure of colloidal gels and lead, af- induced fluidization followed by gelation upon flow ces- ter shear cessation, to more homogeneous and stronger sation can be consistently repeated. Such “secondary” gels. In contrast, preshear at low shear rates creates sol-gel transitions are the topic of the present study in largely heterogeneous and much weaker gels with re- which we focus on the impact of the shear intensity ap- duced elasticity17 . Such a sensitivity to shear history plied during the shear-rejuvenation step. has often been presented as a limitation that needs to be overcome to measure SGM properties accurately. How- In practice, starting from a sample that has experi- ever, shear history can be used as a way to control the enced a primary sol-gel transition in quiescent conditions, liquid-to-solid transition and the resulting microstruc- we impose a shear-rejuvenation step at a shear rate γ̇p , ture of the gel formed during flow cessation. In that following which the flow is stopped and the sample left spirit, shear-assisted gelation of colloidal suspensions of at rest. The linear and nonlinear viscoelastic properties carbon black particles was controlled through the rate of the gel that eventually reforms are monitored before of flow cessation, which allows tuning the microstruc- imposing a new shear-rejuvenation step with a different ture of the gel continuously36 . Fast flow cessations lead shear intensity. We explore three decades in values of to strong gels with a highly connected microstructure, γ̇p to determine the impact of shear history on boehmite whereas slow flow cessations produce weaker gels com- gels over a broad range of Mason numbers. Following posed of poorly connected aggregates. These results were this protocol, we show that after a primary gelation, further confirmed by means of a rheo-impedance study boehmite gels can be rejuvenated reversibly by an ex- on more complex mixtures, e.g., carbon black suspen- ternal shear, whose intensity strongly impacts the sub- sions in a semisolid flow batteries solvent37 , and more sequent properties of the gels that reform following flow recently on carbon black suspensions in propylene car- cessation. In particular, we identify a critical shear rate bonate through shear rheology coupled with small angle γ̇c , above which boehmite gels display a gel-like viscoelas- neutron scattering38 . tic response upon flow cessation, similar to that obtained In the case of model spherical particles with short- following the primary gelation. However, for shear reju- range attractive interactions, the above observations venation such that γ̇p < γ̇c , boehmite gels display upon can be rationalized in terms of a dimensionless ratio, flow cessation a glassy-like viscoelastic spectrum together called the Mason number Mn, comparing the strength with enhanced elastic properties. The value of γ̇c is found of the shear flow to the interparticle bond force at to be independent of both the boehmite and acid concen- contact23,39–41 . At low Mason number, Mn < 10−2 , a tration over the range explored. gel behaves as a viscoelastic solid. For intermediate val- ues, 10−2 ≤ Mn ≤ 1, constant breakup and reformation In light of the framework built on the Mason num- of interparticle bonds results in a complex heterogeneous ber, we propose an interpretation of our observations in dynamics, whereas for Mn > 1 the gel is fluidized into a which gels sheared at γ̇p > γ̇c are fully rejuvenated and weakly aggregated suspension of clusters and particles42 . fluidized into individual unbreakable aggregates, which Yet, this framework, which holds for model spherical par- yield a gel upon flow cessation that shows similar me- ticles, might not account for particles with complex shape chanical and microstructural properties to that obtained and interactions, including chemical reactions and irre- after a primary gelation. On the contrary, shearing gels versible binding, thus calling for further experimental in- at γ̇p < γ̇c results in shear-induced aging and leads to vestigations of non-spherical colloidal gels of industrial the formation of marginally stable clusters of larger di- interest. mensions than in the primary gel. Moderate shear al- In the present paper, we study the impact of shear lows for the restructuring of these clusters into denser history on gels of boehmite, an aluminum oxyhydrox- and stronger structures that form an anisotropic network, ide (AlO(OH)), which is a precursor of gamma alumina and result in enhanced elastic properties. The latter sce- (γ–Al2 O3 ) widely used in industries to design catalyst nario, in which the gel is not fully rejuvenated but in- supports43,44 . These gels are part of a large family of stead encodes some memory of the flow history into its systems, whose primary sol-gel transition involves chem- microstructure, is a prototypical example of what was re- ical reactions, which result in the formation of strong cently coined “directed aging” in the literature48,49 . Fi- aggregates of anisotropic shape that are unbreakable, nally, we demonstrate that imposing stress ramps, in- even when submitted to extended period of high shear stead of abrupt flow cessation, allows us to continuously intensity. Note that aside boehmite gels, other examples tune the linear and non-linear rheological properties of include dispersions of silica colloids in presence of salt, boehmite gels between the two extreme cases discussed whose addition triggers the fast and irreversible aggrega- above. Our results show that shear history is a promis- tion of the colloids through the formation of permanent ing candidate for tailoring the microstructure of catalyst interparticle siloxane bonds45–47 . These systems are thus supports obtained from these soft precursors.
Shear-induced memory effects in boehmite gels 3
Primary Unbreakable Reversible cluster B. Experimental set-up and protocol
particle aggregate
Rheological measurements are performed in a smooth
cylindrical Couette geometry (height 58 mm, rotating in-
10 nm ner cylinder of radius 24 mm, fixed outer cylinder of ra-
100 nm dius 25 mm, gap e = 1 mm). The cell is topped with a
homemade lid to minimize evaporation, and the rotor is
connected to a stress-controlled rheometer (AR-G2, TA
Instruments). The Couette cell is immersed in a water
bath, allowing all the experiments to be performed at a
fixed temperature of (20.0 ± 0.1) ◦ C.
300-1200 nm First, we determine the rheological properties of a pris-
tine or “primary” gel, which forms for the first time in
quiescent conditions, i.e., in the absence of any exter-
FIG. 1. Sketch of the structure of a reversible cluster (high- nal shear. In practice, a fresh sample is loaded in the
lighted by a light grey disk) composed of several unbreakable Couette cell right after its preparation and submitted to
aggregates of primary particles and formed during the pri-
the three following steps : (i) a period of rest of 3000 s
mary sol-gel transition.
during which the linear viscoelastic properties are moni-
tored under small-amplitude oscillatory shear (f = 1 Hz,
γ = 0.1%), followed by (ii) a frequency sweep of total du-
ration 156 s from f = 10 Hz to f = 0.1 Hz at γ = 0.1%.
Finally, we perform (iii) a strain amplitude sweep of to-
II. MATERIALS AND METHODS tal duration 275 s from γ = 0.01 % to γ = 100 % at
f = 1 Hz to determine the nonlinear response and the
yield strain of the sample.
Following the primary sol-gel transition, the properties
A. Preparation of boehmite gels of the samples are stable in time, provided that the sam-
ple does not suffer from evaporation, i.e., typically over
12 h with our experimental setup. The core of the present
Boehmite gels are prepared by adding a boehmite pow- study is performed on non-pristine samples, i.e., gels ob-
der (Plural SB3, Sasol) to an aqueous solution of ni- tained by “secondary” gelations, which experience multi-
tric acid50–52 . Unless specified otherwise, the boehmite ple sequences of shear and rest periods following their pri-
and acid concentrations are respectively 123 g.L−1 and mary gelation. In practice, a large batch of about 75 mL
14 g.L−1 . These concentrations were chosen to produce of pristine gel is prepared and kept at rest in a closed con-
gels, whose secondary gelation takes place within a few tainer of volume 100 mL for at least seven days. For a
seconds after flow cessation. In practice, the boehmite given series of measurements, 7.5 mL of gel is transferred
powder is dispersed into a solution of nitric acid, first into the Couette cell. Each sample is then rejuvenated
by mixing at 600 rpm during 20 min before mixing at by shearing at a constant shear rate γ̇p during 600 s. Af-
1800 rpm during 15 min. The result is a suspension ter shear rejuvenation, the secondary gelation process is
containing particles with a diameter of a few tens of characterized by the exact same three-step protocol (i),
nanometers53–55 , which is left at rest for at least one week (ii), and (iii) described above. Such shear rejuvenation
prior to any rheological test. Indeed, while the primary and secondary gelation are successively repeated for nine
sol-gel transition occurs after a couple of hours, it takes values of shear rates γ̇p ranging from 2 s−1 to 1000 s−1
several days for the pH of the suspensions to stabilize at and chosen in a random order.
pH = 3.5 due to the dissolution and the surface charging
of the alumina52,54 . The presence of the nitrate anion
NO− III. EXPERIMENTAL RESULTS
3 screens the electrostatic repulsive interactions be-
tween the positively charged surface hydroxyl groups55,56
and the boehmite particles eventually self-assemble via A. Impact of the shear-rejuvenation step on the gel linear
short-range van der Waals attractive interactions into un- viscoelastic properties
breakable aggregates of diameter54 Rag ' 100 nm. As
sketched in Fig. 1, these aggregates further assemble into Figure 2 shows the temporal evolution of the gel lin-
a percolated network of large clusters of typical diameter ear viscoelastic properties during the primary gelation,
Rcl ' 300–1200 nm as determined from light scattering and for a series of four subsequent secondary gelations,
measurements. The elastic properties of these gels are each preceded by a 600 s preshear of constant intensity
dominated by the fractal nature of the clusters, which γ̇p = 2, 10, 100 s−1 and 500 s−1 . During the primary gela-
results in a power-law increase of the gel storage modu- tion, the formation of a space-spanning colloidal network,
lus with the boehmite content51,57 . defined as a first estimate by the crossover point betweenShear-induced memory effects in boehmite gels 4
systematically explored the impact of γ̇p applied dur-
10 3 10 2
ing the shear-rejuvenation step preceding secondary gela-
tions. The results are gathered in Fig. 4, where we
10 2
report over three decades of γ̇p , the stress σp at the
10 1 end of the shear-rejuvenation step, the terminal value
of the gel storage modulus G00 determined 3000 s after
10 1 the end of the rejuvenation step, and the slope n char-
10 0
acterizing the logarithmic dependence of the loss factor
(a) (b) with f , i.e., G0 ∼ n log f . These three observables ex-
10 0
10 1 10 2 10 3 10 1 10 2 10 3 hibit two distinct behaviors separated by a critical shear
rate γ̇c = 30±10 s−1 . More specifically, the stress at the
end of the shear-rejuvenation step σp is roughly constant,
FIG. 2. Temporal evolution of (a) the storage modulus G0 and independent of γ̇p for γ̇p < γ̇c , whereas σp increases loga-
(b) the loss modulus G00 for a primary gelation (×), and for rithmically with γ̇p for γ̇p > γ̇c [Fig. 4(a)]. Moreover, sec-
subsequent secondary gelations all preceded by a 600 s shear ondary gelations following a rejuvenation step such that
rejuvenation under various shear rates γ̇p = 2 s−1 (N), 10 s−1 γ̇p < γ̇c lead to stronger gels, with G00 = 1250 ± 150 Pa,
(), 100 s−1 (•), 500 s−1 (). For secondary gelations, the ori-
whereas secondary gelations following a rejuvenation step
gin of time is taken at the end of the shear-rejuvenation step.
Viscoelastic properties are measured under small-amplitude such that γ̇p > γ̇c lead to gels of elasticity comparable
oscillatory shear (f = 1 Hz, γ = 0.1 %). to that of a primary gel, i.e., G00 = 500 ± 150 Pa vs
G00 = 580 Pa, respectively [Fig. 4(b)]. Remarkably, the
storage modulus is roughly independent of the shear in-
G0 and G00 , occurs after about t ' 100 s. The gelation dy- tensity on both sides of γ̇c . Finally, a similar binary
namics are noticeably faster for the subsequent gelations, outcome is found on the slope n of the logarithmic scal-
where the crossover takes place within the first 20 s, in- ing of the loss factor tan δ with frequency [Fig. 4(c)]. For
dependently of the shear rate γ̇p applied during the reju- γ̇p < γ̇c , n is negative, whereas for γ̇p > γ̇c , n is posi-
venation step. Moreover, for all the secondary gelations, tive similarly to what is observed for a primary gelation
the storage modulus G0 increases over time and reaches [Fig. 3(c)]. This last result shows that on both sides
a plateau at t ' 1000 s, which value G00 depends on of γ̇c , the gels display opposite relaxation behaviors58 ,
the applied shear rate γ̇p during the rejuvenation step. which points to significant differences in the microstruc-
A first important result is that the terminal values of ture of gels reformed after shear rejuvenation above or
G00 and G000 reached at the end of a secondary gelation below γ̇c .
following strong shear rejuvenation (i.e., γ̇p = 100 s−1
and 500 s−1 ) are comparable with those obtained after
the primary gelation. On the contrary, gels obtained af- B. Impact of the shear-rejuvenation step on the gel
ter shear rejuvenation at a low intensity show stronger nonlinear viscoelastic properties
viscoelastic properties, by a factor of about 2.5 for the
storage modulus and 4 for the loss modulus. We now turn to the impact of the shear intensity γ̇p
The terminal state of the four secondary gelations are on the nonlinear response of boehmite gels and on their
further characterized by measuring the linear viscoelas- yielding scenario. Figure 5(a) shows the evolution of G0
tic spectrum over two decades of frequency [Fig. 3(a) and and G00 during a strain sweep for gels obtained after two
3(b)]. For both the primary and secondary gelations, the rejuvenation steps performed at low and high shear in-
storage modulus shows a logarithmic increase with the tensity, respectively γ̇p = 2 s−1 and 500 s−1 . At low
frequency, whose slope is larger in the case of gels formed strain amplitude, i.e., γ . 1 %, both gels display a strain-
after a rejuvenation step of low shear intensity, whereas independent response, which is mainly elastic, i.e., such
the loss modulus hardly depends on f . Concomitantly, that G00 G000 . Upon increasing the strain amplitude γ,
the loss factor tan δ allows us to split the data into two the storage modulus decreases monotonically and departs
groups: the loss factor is an increasing function of the from the linear regime at a strain γ0 defined arbitrarily
frequency for the primary gel and secondary gels formed as G0 (γ0 ) = 0.95G00 . Concomitantly, the loss modulus
after a rejuvenation step of high shear intensity, whereas increases and shows a maximum in the vicinity of the
the loss factor is a decreasing function of f for secondary yield strain γy , defined as the locus of the crossover be-
gels formed after a rejuvenation step of low shear inten- tween G0 and G00 , before decreasing for γ > γy . Through-
sity [Fig. 3(c)]. This second remarkable result suggests out the linear regime and above γ0 , the amplitude σ of
that gels formed after a low shear intensity display a dif- the oscillatory stress response increases up to the yield
ferent microstructure than the primary gel and secondary point, beyond which it becomes roughly strain indepen-
gels formed after a rejuvenation step of high shear inten- dent [Fig. 5(b)]. Finally, the loss factor shows a similar
sity. monotonic trend for both gels, which is indicative of the
To get a better sense of the impact of the shear in- transition from a solid-like response to a liquid-like one
tensity applied during the rejuvenation step, we have beyond γy [Fig. 5(c)]. To summarize, the monotonic de-Shear-induced memory effects in boehmite gels 5
1500 60 0.05
0.04
40
1000
0.03
20
0.02
500
(a) (b) (c)
0 0.01
-1 0 1
10 10 10 10 -1 10 0
10 1
10 -1 10 0
10 1
FIG. 3. Frequency dependence of (a) the storage modulus G0 , (b) the loss modulus G00 , and (c) the loss factor tan δ following a
primary gelation of 3000 s (×) or a secondary gelation including a rest period of 3000 s preceded by a 600 s shear-rejuvenation
step under a shear rate γ̇p = 2 s−1 (N), 10 s−1 (), 100 s−1 (•), 500 s−1 (). Viscoelastic spectra measured with a strain
amplitude γ = 0.1 %.
crease of G0 (γ) and the non-monotonic shape of G00 (γ), G0 beyond the yield point is all the more steep than γ̇p
which are prototypical of the nonlinear response of nu- is large, i.e., νG0 increases with γ̇p , whereas the corre-
merous yield stress fluids including colloidal gels59–61 and sponding exponent for G00 is poorly sensitive to the shear
glasses62 , dense emulsions, and foams63,64 , are qualita- intensity γ̇p . Interestingly, for γ̇p < γ̇c , the ratio νG0 /νG00
tively insensitive to the shear intensity γ̇p of the shear- is roughly constant, with νG0 /νG00 ' 2 in agreement with
rejuvenation step. predictions from Mode Coupling Theory applied to soft
We note, however, that both the strain γ0 that marks glassy systems65 . For γ̇p > γ̇c the ratio continuously in-
the end of the linear regime and the yield strain γy are sig- creases up to νG0 /νG00 ' 3 for the largest explored shear
nificantly smaller when the boehmite gel is rejuvenated rates, which shows that shear rejuvenation of large in-
at low shear intensity. This result is general and holds tensity yields gels with a strikingly different decrease of
for the whole range of shear intensities γ̇p under study, the structural relaxation time with increasing strain am-
as illustrated in Figs. 6(a) and 6(c). More precisely, sec- plitude during the yielding transition65,66 .
ondary gelations following shear rejuvenation at γ̇p < γ̇c
yield gels where γ0 and γy increase for increasing γ̇p ,
C. Impact of flow cessation following shear rejuvenation
whereas γ̇p > γ̇c leads to gels with constant strain val- on the gel viscoelastic properties
ues, i.e, γ0 ' 3.5% and γy ' 38%, independent of γ̇p .
These values are comparable to those measured follow-
ing a primary gelation, namely γ0 ' 5% and γy ' 29%. In all the experiments reported above, the 600 s shear-
rejuvenation step was performed at a fixed intensity γ̇p ,
We now focus on the stress σ0 at the onset of the non- and followed by an abrupt flow cessation by imposing
linear regime and the yield stress σy , which respectively γ̇ = 0 s−1 . In the present section, we explore the
correspond to the amplitudes of the stress responses mea- impact of flow cessation when performed over a finite
sured for strain amplitudes γ0 and γy [see Fig. 5(b)]. duration ∆t by means of a decreasing ramp of shear
The dependence of σ0 and σy with the shear rate γ̇p ap- stress. In practice, after 600 s of shear rejuvenation at
plied during the shear-rejuvenation step is reported in γ̇p = 500 s−1 , the flow is switched to stress-controlled by
Figs. 6(b) and 6(d). Except perhaps for a weak maxi- imposing σ = σp ' 50 Pa, the latest stress value recorded
mum at γ̇p ' γ̇c , the two characteristic stresses σ0 and at the end of the 600 s rate-controlled period. The stress
σy do not show any clear systematic trend with γ̇p . Note is then swept down from σp to 0 Pa by discrete steps of
that the evolutions of σ0 and σy are very similar to amplitude 1 Pa over a total duration ∆t ranging from
those of G00 γ0 and G00 γy , respectively [see grey squares in ∆t = 0 s to ∆t ' 13000 s
Figs. 6(b) and 6(d)], which suggests that σ0 and σy are As shown in the previous two sections, shear rejuvena-
essentially governed by the product of the elastic modu- tion at γ̇p > γ̇c followed by an abrupt flow cessation yields
lus and the corresponding characteristic strain. a gel of storage modulus G0 ' 500 Pa. When the flow
Finally, beyond the yield point, the decrease of the cessation is performed over a finite duration of increas-
storage and loss modulus are both well fitted by power- ing value, the storage modulus increases up to a value
law decays G0 ∼ γ −νG0 and G00 ∼ γ −νG00 [see Fig. 8 in G0 ' 1000 Pa that is comparable to that obtained af-
Appendix A]. The dependence of the exponents νG0 and ter shear rejuvenation at γ̇p < γ̇c followed by an abrupt
νG00 with γ̇p is reported in Fig. 6(e), while that of the flow cessation [see black arrow in Fig 4(b)]. Concomi-
ratio νG0 /νG00 is reported in Fig. 6(f). The decrease in tantly, the loss factor, which is a decreasing function ofShear-induced memory effects in boehmite gels 6
70
(a)
10 3
60
10 2
50
10 1
0 0 y y (a)
40
1500
(b)
10 2 y
0
1000 1
10
103
500 102
10 0
101
100
100 101 102 103 (b)
0
10 1
10 (c)
5
10 0
0
-5 10 -1
-10 (c)
10 -2
10 0 10 1 10 2 10 3 10 -1 10 0 10 1 10 2
FIG. 4. Dependence on the shear rate γ̇p used for shear reju- FIG. 5. Strain dependence of (a) the storage modulus G0 (full
venation of (a) the stress σp measured at the end of the shear- symbols) and the loss modulus G00 (empty symbols), (b) the
rejuvenation step, (b) the storage modulus G0 measured after stress amplitude σ, and (c) the loss factor tan δ of a boehmite
3000 s of rest period following the shear-rejuvenation step, gel obtained in a secondary gelation following a rejuvenation
and (c) the slope n of the linear regression of the loss fac- step of intensity: γ̇p = 2 s−1 (N) and 500 s−1 (•). Strain sweep
tor tan δ versus log f . The shear-rejuvenation step is either experiment performed at f = 1 Hz. In (a)–(c), the vertical
stopped abruptly (γ̇ = 0 s−1 , red symbols) or stopped over a dashed and dashed-dotted lines respectively mark the strain
finite duration ∆t using a linear decreasing ramp of stress fol- γ0 beyond which the gel response becomes nonlinear and the
lowing a 600 s shear rejuvenation at γ̇p = 500 s−1 (other col- yield strain γy defined as the crossover of G0 and G00 .
ored symbols). The duration ∆t of the ramp is ∆t = 0 s (),
∆t = 54 s (), ∆t = 270 s (), ∆t = 540 s (), ∆t = 3240 s
() and ∆t = 12960 s (). The inset in (b) shows the stor-
age modulus G0 vs γ̇p measured at different points in time t arrow in Fig 4(c)]. A similar trend is visible on the non-
following the abrupt cessation of shear rejuvenation [t = 6 s linear properties of the boehmite gel when increasing the
(), 13 s (), 26 s (), 63 s (), 300 s (), and 3000 s duration of the flow cessation: the strain γ0 at the on-
()]. Error bars in (a) and (b) represent one standard de- set of the nonlinear regime and the yield strain γy both
viation computed on three to eight independent tests. Error decrease for increasing ∆t [see black arrow in Figs. 6(b)
bars in (c) represent the root-mean-square error of the linear and 6(d)]. Moreover, the exponent νG0 of the power-law
regression. The grey rectangle highlights the critical shear decrease of G0 (γ) beyond the yield point, and the ratio
rate γ̇c = 30±10 s−1 . νG0 /νG00 both decrease to values comparable to those ob-
tained by shear rejuvenation at γ̇p < γ̇c followed by an
abrupt flow cessation [see black arrow in Figs. 6(e) and
frequency in the reference case of shear rejuvenation at 6(f)].
γ̇p > γ̇c followed by an abrupt flow cessation, becomes To conclude, varying the duration of the flow cessation
a increasing function of frequency, i.e., n < 0 [see black following shear rejuvenation at γ̇p > γ̇c allows us to tuneShear-induced memory effects in boehmite gels 7
5 50 3
(a) (c) (e)
4 40 2.5
3 30 2
2 20 1.5
1 10 1
0 0 0.5
35 3.5
(b) 100 (d) (f)
30
80 3
25
20 60
2.5
15
40
10 2
20
5
0 0 1.5
10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3
FIG. 6. (a) Strain amplitude γ0 at the onset of the nonlinear regime and (b) corresponding stress amplitude σ0 vs the shear
rate γ̇p applied during the shear-rejuvenation step. (c) Yield strain γy and (d) yield stress σy vs γ̇p . (e) Exponent νG0 (filled
symbols) and νG00 (open symbols) characterizing the power-law decrease of G0 and G00 beyond the yield point vs γ̇p and (f)
ratio νG0 /νG00 vs γ̇p . Grey squares in (b) and (d) respectively show G00 γ0 and G00 γy /3.3. Same symbols, color code, and error
bar estimations as in Fig. 4.
continuously the viscoelastic properties of boehmite gels. that can be repeated numerous times on the same sample
Moreover, slow enough flow cessation following shear re- provided that it does not suffer from solvent evaporation.
juvenation at γ̇p > γ̇c provides the boehmite gel with the Here we have shown that the terminal properties of the
same properties as shear rejuvenation at γ̇p < γ̇c followed gel depend on the exact value of the shear intensity γ̇p ap-
by an abrupt flow cessation (see Fig. 9 in Appendix B). plied during the shear-rejuvenation step. We have iden-
This result shows that the key parameter controlling the tified a critical shear rate γ̇c that delineates two different
terminal properties of the gel at rest is not the largest macroscopic viscoelastic properties upon flow cessation.
shear rate imposed to the gel, but rather the period of This suggests that shear rejuvenation leads to two differ-
time during which the sample was sheared at γ̇ < γ̇c . ent sample microstructures on both sides of γ̇c .
On the one hand, a boehmite gel obtained by an abrupt
flow cessation following shear rejuvenation at γ̇p > γ̇c is
IV. DISCUSSION characterized by a storage modulus G0 ' 500 Pa com-
parable to that of the primary gel, and by a viscoelastic
spectrum where both the storage and loss moduli are
Let us now summarize and discuss the results of
increasing functions of frequency. Moreover, the loss fac-
the present study. The primary sol-gel transition of a
tor tan δ also increases with frequency in the same way as
boehmite suspension leads to the aggregation of colloidal
the primary gel. Such a viscoelastic spectrum is typical
particles, whose diameter is a few tens of nanometers,
of a weak colloidal gel 67,68 , in which the fastest relaxation
into aggregates of radius about 100 nm. These aggregates
modes (associated with high frequencies) dominate58 .
are unbreakable under shear and act as the new building
These observations strongly suggest that the sample is
blocks for any subsequent gelation. The primary gel is
fully fluidized for γ̇p > γ̇c , and that the microstructure
therefore constituted of a space-spanning percolated net-
that reforms upon flow cessation is similar to that of the
work of clusters of such aggregates. Under external shear
primary gel, i.e., a percolated network of small clusters
the gel yields and turns into a suspension of these clus-
of a few hundred nanometers.
ters, which reorganize under flow and reassemble upon
flow cessation, through a secondary gelation. Such a sec- On the other hand, a boehmite gel obtained by an
ondary gelation yields consistent and reproducible results abrupt flow cessation following shear rejuvenation atShear-induced memory effects in boehmite gels 8
which the flow direction changes periodically, we may in-
terpret the decreasing trend seen in γy upon decreasing
γ̇p as the signature of an increasing level of anisotropy.
In that framework, we can propose a microscopic ori-
gin for the critical shear rate γ̇c . Assuming that a gel
rejuvenated at γ̇ < γ̇c displays a microstructure involv-
ing clusters made of several unbreakable aggregates of
the primary gel, the transition at γ̇c = 30 s−1 corre-
2
sponds to a critical Mason number Mnc = 6πηs Rcl γ̇c /F ,
where ηs = 1 mPa.s is the solvent viscosity, here water,
Rcl ' 300–1200 nm the typical cluster radius, and F
the interaction force between two aggregates in a clus-
ter. We may further estimate F as the van der Waals
force between two aggregates of radius Rag , namely
(a) (b) F = −AH Rag /(24h2 ), with AH the Hamaker constant
and h the typical distance between the two aggregates76 .
FIG. 7. Sketch of the gel network microstructure obtained This leads to Mnc = 144πηs h2 Rcl 2
γ̇c /(AH Rag ). Taking
after a shear-rejuvenation step at (a) γ̇p < γ̇c yielding large AH = 5.2kT and h ' 10 nm for boehmite particles77,78 ,
and dense clusters, and at (b) γ̇p > γ̇c , yielding smaller clus- we get Mnc = 0.06–1 in good agreement with a transition
ters forming a more open network. controlled by the Mason number42 .
In other words, shear rejuvenation at γ̇p > γ̇c leads
to the complete separation of all clusters into unbreak-
γ̇p < γ̇c is characterized by a storage modulus larger than able aggregates that subsequently percolate at rest into
that of the primary gel by a factor of up to 3. This result an isotropic microstructure similar to that of the primary
strikingly contrasts with previous observations on deple- gel. For γ̇p < γ̇c , however, moderate shear allows for the
tion gels, in which preshear at “low” rates yielded weaker growth of larger and denser clusters that encompass mul-
gels with reduced elasticity due to a largely heteroge- tiple aggregates, as pictured in Fig. 7. The microstruc-
neous microstructure inherited from the that acquired ture of a gel built from these clusters is shaped by the
during shear17 . Moreover, the viscoelastic spectrum in- external shear and is likely to show spatial anisotropy,
volves a storage modulus that is an increasing function of hence bearing some memory of the rejuvenation step. In
frequency, whereas the loss modulus is independent or a this context, stronger gels are obtained when the sam-
mildly decreasing function of frequency, similar to aging ple is sheared for a sufficient period of time at γ̇p < γ̇c ,
systems including gels69,70 and soft glasses71 . The loss which suggests that (i) shear rejuvenation at γ̇p < γ̇c
factor tan δ decreases with increasing frequency, which followed by an abrupt flow cessation, and (ii) shear reju-
is reminiscent of glass-like behavior, where the slowest venation at γ̇p > γ̇c followed by a slow decreasing ramp of
relaxation modes dominate58 . This suggests that the stress should lead to similar microstructures and mechan-
boehmite gel sheared at γ̇p < γ̇c displays a microstruc- ical properties, in agreement with the results discussed in
ture composed of large clusters, much larger than those section III C.
constituting the microstructure of the primary gel. These The above mentioned scenario is mostly based on the
clusters become denser under shear and form in turn a shear-induced aggregation of clusters, which form non-
percolated network responsible for the enhanced elastic- Brownian aggregates that percolate into an anisotropic
ity of the gel obtained upon flow cessation. Such a con- structure. As such, this scenario should neither depend
clusion is supported by previous experimental and nu- on the boehmite concentration, nor on the acid concen-
merical observations that moderate shear during the sol- tration. To check this hypothesis, we have repeated the
gel transition of a colloidal suspension favors the growth measurements reported in sections III A and III B for four
of denser and thus stronger aggregates72–74 . Concomi- different gel compositions. The results summarized in
tantly, the shear-assisted growth of such a microstructure Figs. 10 and 11 in Appendix C confirm the robustness of
should result in some structural anisotropy, which could the phenomenology reported in the main text and that
also contribute in reinforcing the gel elasticity. In our the critical shear rate γ̇c does not change over a broad
experiments, an indirect proof of such anisotropy could range of boehmite and acid concentrations.
lie in the decrease of the yield strain for decreasing shear Our results on boehmite gels show some apparent sim-
rejuvenation intensity [Fig. 6(d)]. Indeed, the anisotropy ilarities with previous studies on shear-induced floccu-
encoded in the gel microstructure by a continuous shear lation in colloidal suspensions destabilized by salt. For
is expected to strengthen the material along that specific instance, a critical shear rate γ̇c ' 100 s−1 was previ-
direction of shear32,75 . Moreover, a more anisotropic mi- ously identified, while monitoring the aggregate size in
crostructure should also result in a lower resistance of the flocculating suspensions of latex particles under shear in
gel to a change in the shear direction61 . Since the yield a Taylor-Couette cell79 . For γ̇ > γ̇c , the aggregate size
strain is determined by a strain amplitude sweep during increases monotonically in time to reach a steady-stateShear-induced memory effects in boehmite gels 9
value, whereas for γ̇ < γ̇c , the aggregates size displays
an overshoot, characterized at long times by a substan-
tial decrease at constant mass, hence corresponding to a 10 3
restructuring and a densification of the aggregates. As
reviewed in Ref.27 , similar results were reported in differ-
ent geometries, e.g., a mixing tank equipped with various 10 2
types of impellers80,81 , and for non-Brownian particles,
e.g., precipitated calcium carbonate82,83 ). These reports
from the literature indicate that γ̇c ∼ 50 − 100 s−1 for
a broad variety of systems and experimental configura- 10 1
tions. This suggests that the restructuration of the ag- 10 -1 10 0 10 1 10 2
gregates under shear in the limit of low shear intensity,
i.e., low Peclet number, is generically responsible for the
strengthening of the elastic properties observed in gels
rejuvenated at γ̇p < γ̇c . FIG. 8. Storage modulus G0 and loss modulus G00 vs strain
amplitude γ during a strain sweep performed at frequency
Nonetheless, the comparison between the literature f = 1 Hz on a boehmite gel following shear rejuvenation
on shear-induced flocculation and our results cannot be at γ̇p = 2 s−1 for 600 s, and a rest period of 3000 s. The
pushed further for none of these previous studies have continuous and dashed lines respectively correspond to the
characterized the rheological properties of the gels that best power-law fits of G0 and G00 for γ ≥ γy .
form upon flow cessation after shear rejuvenation below
or above γ̇c . Moreover, previous cluster size measure-
ments were performed on fully dispersed suspensions,
whereas our experiments are performed on boehmite gels Appendix B: Evolution of the loss factor depending on the
flow cessation
in their solid-like state. Therefore, comparisons with
previous studies should be considered with caution. In
order to fully unveil the restructuration scenario below Figure 9 illustrates the impact of the rate of flow ces-
γ̇c , future experiments will focus on (i) measuring the sation on the frequency dependence of the loss factor.
flow profiles during shear rejuvenation, in order to deter- Abrupt flow cessation from γ̇p > γ̇c yields a gel such that
mine whether the flow remains spatially homogeneous or tan δ increases with f . Decreasing the rate of flow cessa-
whether it displays heterogeneities such as shear bands, tion yields gels whose loss factor is less dependent on f .
wall slip or fractures, and (ii) characterizing the mi- Finally, gels obtained after a slow flow cessation show a
crostructural differences induced by the shear history loss factor that is a decreasing function of f and resem-
thanks to small-angle light and x-ray scattering measure- bles that of gels prepared by an abrupt flow cessation
ments. from γ̇p < γ̇c .
0.05
ACKNOWLEDGMENTS
0.04
The authors kindly acknowledge fruitful discussions 0.03
with Catherine Barentin, Eric Freyssingeas, Thomas
Gibaud, Eric Lecolier, and Wilbert Smit.
0.02
0.01
10 -1 10 0 10 1
Appendix A: Power-law decay of the viscoelastic modulus
beyond the yield point
FIG. 9. Frequency dependence of the loss factor tan δ. The
The method used to fit the decay of the storage and loss 600 s shear rejuvenation at γ̇p = 500 s−1 is stopped over a
finite duration ∆t using a linear decreasing ramp of stress.
moduli beyond the yield point defined by γy is illustrated
The duration ∆t of the ramp is ∆t = 0 s (), ∆t = 54 s
in Fig. 8. Both G0 and G00 are fitted for γ ≥ γy with (), ∆t = 270 s (), ∆t = 540 s (), ∆t = 3240 s ()
power laws whose exponents nG0 and nG00 respectively, and ∆t = 12960 s (). Viscoelastic spectra measured with a
are reported in Fig. 6(e) in the main text. Note that the strain amplitude γ = 0.1 %.
exact range of strain fitted to determine these exponents
may vary by one or two points between data sets.Shear-induced memory effects in boehmite gels 10
TABLE I.
Gel Boehmite concentration (g.L−1 ) Acid concentration (g.L−1 ) γ̇c (s−1 ) G0c (Pa)
[B = 93, A = 14] 93 14 30 ± 10 330 ± 25
[B = 123, A = 14] 123 14 30 ± 10 760 ± 60
[B = 147, A = 14] 147 14 30 ± 10 1380 ± 70
[B = 123, A = 13] 123 12 10± 10 990 ± 60
[B = 123, A = 18] 123 17 30 ± 10 860 ± 60
2 6 2 6
1000 1000
600 5 600 5
1.5 1.5
200 200
4 4
80 120 160 12 14 16 18
1 3 1 3
2 2
0.5 0.5
1 1
(a) (c) (a) (c)
0 0 0 0
15 60 15 60
10 50 10 50
40 40
5 5
30 30
0 0
20 20
-5 10 -5 10
(b) (d) (b) (d)
-10 0 -10 0
10 -1 10 0 10 1 10 2 10 -1 10 0 10 1 10 2 10 -1 10 0 10 1 10 2 10 -1 10 0 10 1 10 2
FIG. 10. Impact of boehmite concentration. (a) Normalized FIG. 11. Impact of the acid concentration. (a) Normalized
storage modulus G00 /G0c measured 3000 s after the abrupt storage modulus G00 /G0c measured 3000 s after the abrupt
cessation of a 600 s shear rejuvenation performed at γ̇p , (b) cessation of a 600 s shear rejuvenation performed at γ̇p , (b)
slope of the linear regression of the loss factor versus log f , slope of the linear regression of the loss factor versus log f , (c)
(c) strain γ0 at the onset of the nonlinear regime, and (d) strain γ0 at the onset of the nonlinear regime, and (d) yield
yield strain γy as a function of γ̇p /γ̇c , where γ̇c is the critical strain γy as a function of γ̇p /γ̇c , where γ̇c is the critical shear
shear rate (see Table I). Symbols stand for gels with differ- rate (see Table I). Symbols stand for gels with different acid
ent boehmite concentrations and the same nitric acid concen- concentrations and the same boehmite concentration [B =
tration [B = 93, A = 14] (•), [B = 123, A = 14] () and 123, A = 12] (H), [B = 123, A = 14] () and [B = 123, A = 17]
[B = 147, A = 14] (). In (a), G0c is the value of G00 extrap- (N). In (a), G0c is the value of G00 extrapolated at γ̇p = γ̇c
olated at γ̇p = γ̇c and the inset shows G00 vs the boehmite and the inset shows G00 vs the acid concentration. The grey
concentration. The grey rectangle highlights the transition rectangle highlights the transition region around γ̇c .
region around γ̇c .
The storage modulus G00 of boehmite gels increases
Appendix C: Impact of the gel chemical composition steeply with the concentration of boehmite [inset in
Fig. 10(a)], whereas it is roughly insensitive to a change
On top of the sample with 123 g.L−1 of boehmite and of the acid concentration for a given boehmite content
14 g.L−1 of nitric acid reported in the main text, we have [inset in Fig. 11(a)]. Yet, for all gel compositions, the
investigated the four supplementary compositions gath- storage modulus G00 can be rescaled onto a master curve
ered in Table I by following the same shear rejuvenation that shows two distinct values on both sides of a criti-
and characterization protocols. The impact of boehmite cal shear rate γ̇c . Shear rejuvenation at γ̇p < γ̇c always
and acid concentrations on the various linear and non- yields a higher storage modulus than shear rejuvenation
linear rheological observables is reported in Figs. 10 and at γ̇p > γ̇c [Figs. 10(a) and 11(a)]. Moreover, γ̇c ap-
11, respectively. For the sake of clarity, the storage mod- pears to be independent of the boehmite and acid con-
ulus G00 is normalized by G0c , defined as the value of G00 tents over the range of compositions under study [see
extrapolated at γ̇p = γ̇c , while the shear rate γ̇p used for Table I]. These results confirm the robustness of the con-
shear rejuvenation is normalized by γ̇c . clusions presented in the main text in Fig. 4(b).Shear-induced memory effects in boehmite gels 11
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