Dual origin of viscoelasticity in polymer-carbon black hydrogels a rheometry and electrical spectroscopy study

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Dual origin of viscoelasticity in polymer-carbon

black hydrogels: a rheometry and electrical

spectroscopy study

Gauthier Legrand,†S´ebastien Manneville,†,‡Gareth H. McKinley,¶and Thibaut

Divoux∗,†

†ENSL, CNRS, Laboratoire de physique, F-69342 Lyon, France

‡Institut Universitaire de France (IUF)

¶Hatsopoulos Microﬂuids Laboratory, Department of Mechanical Engineering, MIT, 77

Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

E-mail: Thibaut.Divoux@ens-lyon.fr

Abstract

Nano-composites formed by mixing nanoparti-

cles and polymers oﬀer a limitless creative space

for the design of functional advanced materi-

als with a broad range of applications in ma-

terials and biological sciences. Here we focus

on aqueous dispersions of hydrophobic colloidal

soot particles, namely carbon black (CB) dis-

persed with a sodium salt of carboxymethylcel-

lulose (CMC), a food additive known as cellu-

lose gum that bears hydrophobic groups, which

are liable to bind physically to CB particles.

Varying the relative content of CB nanoparti-

cles and cellulose gum allows us to explore a

rich phase diagram that includes a gel phase

observed for large enough CB content. We

investigate this hydrogel using rheometry and

electrochemical impedance spectroscopy. CB-

CMC hydrogels display two radically diﬀerent

types of mechanical behaviors that are sepa-

rated by a critical CMC-to-CB mass ratio rc.

For r < rc, i.e., for low CMC concentration,

the gel is electrically conductive and shows a

glassy-like viscoelastic spectrum, pointing to a

microstructure composed of a percolated net-

work of CB nanoparticles decorated by CMC.

In contrast, gels with CMC concentration larger

than rcare non-conductive, indicating that the

CB nanoparticles are dispersed in the cellu-

lose gum matrix as isolated clusters, and act

as physical crosslinkers of the CMC network,

hence providing mechanical rigidity to the com-

posite. Moreover, in the concentration range,

r > rcCB-CMC gels display a power-law vis-

coelastic spectrum that depends strongly on the

CMC concentration. These relaxation spectra

can be rescaled onto a master curve that ex-

hibits a power-law scaling in the high-frequency

limit, with an exponent that follows Zimm the-

ory, showing that CMC plays a key role in the

gel viscoelastic properties for r > rc. Our re-

sults oﬀer an extensive experimental characteri-

zation of CB-CMC dispersions that will be use-

ful for designing soft nano-composites based on

hydrophobic interactions.

Introduction

Carbon black (CB) particles are colloidal soot

particles produced from the incomplete com-

bustion of fossil fuels. These textured nanopar-

ticles of typical size 0.5 µm are made of per-

manently fused “primary” particles of diameter

20-40 nm.1Cheap and industrially produced

at large scale, they are broadly employed for

their mechanical strength, high surface area,

and their electrical conductive properties, even

arXiv:2210.03606v1 [cond-mat.soft] 7 Oct 2022

at low volume fractions. Applications include

pigments for ink,2reinforcing ﬁllers in tires

and other rubber products,3,4 electrically con-

ductive admixture in cement,5conductive ma-

terials for supercapacitors,6biosensors,7,8 and

electrodes for semi-solid ﬂow batteries.9–13 Be-

ing much cheaper than carbon nanotubes or

graphene, CB nanoparticles appear promising

for applications in energy storage,14 including

ﬂow electrodes for which the ultimate goal is

to maximize the conductivity, while minimiz-

ing the shear viscosity of the material. In that

framework, ﬂow batteries based on aqueous dis-

persion of carbon black nanoparticles have re-

cently received an upsurge of interest.10,15–18

Due to their hydrophobic properties, carbon

black particles are easily dispersed in aprotic

solvents such as hydrocarbons,19 where parti-

cles interact only via van der Waals forces20

that correspond to a short-range attractive po-

tential, whose depth is typically about 30 kBT

in light mineral oil.21 As a result, carbon black

dispersions organize into space-spanning net-

works even at low volume fractions, and behave

as soft gels22–24 characterized by a yield stress

at rest and a highly time-dependent mechanical

response under external shear, which involves

delayed yielding, heterogeneous ﬂows,25–28 and

shear-induced memory eﬀects.29–31

In contrast, untreated CB particles are dif-

ﬁcult to disperse in water where they tend

to ﬂocculate rapidly, before creaming or sed-

imenting.16,32 Stabilizing aqueous dispersions

of CB nanoparticles requires keeping the par-

ticles apart, either by electrostatic repulsion

or by steric hindrance. In practice, this is

achieved in three diﬀerent ways: (i) surface

oxidation yielding acidic functional groups,33

(ii) functionalization of CB particles with poly-

mers, i.e., polymer grafting chemically onto

their surface34–40 or CB encapsulation through

emulsion polymerization,41,42 and (iii) physi-

cal adsorption of a polymer dispersant. The

latter method allows reaching CB mass frac-

tions in water as large as 20%, and the disper-

sants investigated include polyelectrolytes,43

ionic surfactants44,45 such as sulfonate surfac-

tants,46–49 sulfonic acids,50,51 cetyltrimethylam-

monium bromide (CTAB)48,52–54 and chloride

(CTAC),55 non-ionic surfactants48,56 such as

silicone surfactants57 or block copolymers sur-

factants,56,58,59 as well as biopolymers such as

Arabic gum,16,60 or polysaccharides.61–64 From

a structural point of view, dispersants ad-

sorb as monolayers onto the surface of CB

nanoparticles due to hydrophobic interactions,

whose strength depends on the molecular struc-

ture and weight of the dispersant.49,63 Irre-

spective of the nature of the dispersant, such

stabilized CB dispersions behave as shear-

thinning ﬂuids.62,64,65 However, depending on

the formulation, carbon black polymer mixtures

may present a solid-like behavior at rest with

weakly time-dependent properties,51,59,63 sug-

gesting the presence of a percolated network of

the CB nanoparticles.66

The large variety of dispersants used so far in

CB dispersions is in stark contrast with the lim-

ited knowledge regarding the link between the

microstructure and rheological properties of the

resulting materials. Among the open questions,

it remains to disentangle the respective contri-

butions of the CB and the dispersant to the

macroscopic mechanical properties of the mix-

ture, and whether the CB nanoparticles form a

percolated network on their own, or if they are

bridged by the polymeric chains.

Here we perform such an in-depth study with

a semi-ﬂexible anionic polysaccharide disper-

sant, namely carboxymethylcellulose (CMC).

CMC is a water soluble cellulose ether, which is

commonly used as a water binder and thickener

in pharmacy, cosmetics, food products,67 and as

a dispersing agent in semi-solid ﬂow batteries,62

while showing a great potential for biomedical

applications.68 The solubility and overall prop-

erties of CMC are set primarily by its molec-

ular weight, and to a lesser extent by its de-

gree of substitution (DS).69,70 The latter is de-

ﬁned as the number of hydrogen atoms in hy-

droxyl groups of glucose units replaced by car-

boxymethyl, and varies typically between 0.4

and 3.67 Over a broad range of concentrations,

CMC aqueous solutions are shear-thinning vis-

coelastic ﬂuids.71,72 However, weakly substi-

tuted CMC, i.e., with DS values lower than

about 1, display hydrophobic interactions that

favor inter-chain association in aqueous solu-

tion, yielding larger viscosities, and eventually

leading to a sol-gel transition for large enough

polymer concentrations.73–78

In the present work, we take advantage of

such hydrophobic regions on CMC molecules to

use this polymer as a dispersant of CB nanopar-

ticles. Varying the contents of CB and CMC,

we unravel a rich phase diagram, which we

characterize by rheometry and electrochemical

impedance spectroscopy. The outline of the

manuscript is as follows: after presenting the

materials and methods, we introduce the phase

diagram and focus speciﬁcally on the hydrogel

phase. We then discuss the impact of the CMC

concentration on the gel elastic properties be-

fore turning to the role of the CB content. Our

results allow us to identify two diﬀerent types of

hydrogels, whose microstructures are sketched

and extensively discussed before concluding.

Materials and methods

Samples are prepared by ﬁrst dissolving sodium

carboxymethyl cellulose (Sigma Aldrich, Mw=

250 kg.mol−1and DS = 0.9) in deionized water.

Stock solutions up to 5% wt. are prepared and

stirred at room temperature for 48 hours until

homogeneous, before adding the CB nanopar-

ticles (VXC72R, Cabot). Samples are placed

in a sonicator bath for two rounds of 90 min

separated by a period of 24 h under mechanical

stirring. The samples are ﬁnally left at rest for

another 24 h before being tested. The CMC so-

lution is considered to be the solvent, while CB

particles is the dispersed phase; hence we de-

ﬁne the CMC weight concentration as cCMC =

mCMC/(mCMC +mwater) and the CB weight frac-

tion as xCB =mCB/(mCB +mCMC +mwater),

where mCB,mCMC, and mwater are respectively

the mass of CB, CMC, and water in the sample.

Rheological measurements are performed

with a stress-controlled rheometer (MCR 302,

Anton Paar) equipped with a cone-and-plate

geometry (angle 2◦, radius 20 mm). For very

soft samples with elastic moduli lower than

10 Pa, we use a larger cone-and-plate geom-

etry (angle 2◦, radius 25 mm), whereas for

the stiﬀest samples with elastic moduli larger

Figure 1: Frequency dependence of the re-

sistance Z0(•) and reactance Z00 (◦) of a

CMC-CB dispersion. Inset: Nyquist plot Z00

vs. Z0for the same data. Measurement per-

formed in AC mode by ramping down the fre-

quency from 106Hz to 10−2Hz. Each point is

averaged over two cycles. The red continuous

curves in both the main graph and the inset

show the best ﬁt of the data for Z0and Z00 si-

multaneously by Eq. (1), which corresponds to

the electrical circuit sketched in the main graph

with Rion = 90 Ω, RCB = 1.0 kΩ, n= 0.8,

and Q= 2.1×10−5Ω−1.sn. Data obtained on

a sample composed of cCMC = 0.15% wt. and

xCB = 8% wt.

than 10 kPa, we use a parallel-plate geome-

try (gap 1 mm, radius 20 mm). For all the

geometries used, the stator is smooth and the

rotor is sandblasted. Finally, to ensure a re-

producible initial state following the loading

step into the shear cell of the rheometer, each

sample is shear-rejuvenated at ˙γ= 500 s−1

(or ˙γ= 50 s−1for samples with elastic mod-

uli larger than 10 kPa), before being left at

rest for 1200 s, during which we monitor the

linear viscoelastic properties through small am-

plitude oscillatory shear (γ0= 0.03-0.3%, and

f= 1 Hz, see supplemental Fig. S1).

Electrical measurements are performed in AC

mode in a cylindrical cell made of Teﬂon (thick-

ness of about 1 mm, and surface of about

0.5 cm2). The inner sides are made of metal

act as electrodes that are connected to a multi-

frequency impedance analyzer (SP-300 Poten-

1 cm

(a) (b) (c)

(d) (e)

Figure 2: (a) Phase diagram of aqueous CMC-CB dispersions as a function of the CB solid

weight fraction xCB and the CMC weight fraction cCMC. Pictures of (b) demixed phase [blue region

and ×symbols in (a)], (c) the viscoelastic liquid phase [grey region and ◦symbols in (a)], (d) the

viscoelastic solid phase [red region and •symbols in (a)], (e) brittle paste phase [yellow region and

symbols in (a)].

tiostat, Biologic). A decreasing ramp of fre-

quency allows determining the frequency de-

pendence of the sample impedance Z∗(ω) =

Z0−iZ00 , which real and imaginary part, Z0and

Z00 respectively, are shown in Fig. 1 for a sam-

ple containing cCMC = 0.15% wt. and xCB = 8%

wt. The resistance Z0shows a decreasing step

shape, while the reactance Z00 displays a bell-

shaped curve. Overall, the complex impedance

measured experimentally can be well ﬁtted by

the following equation:

Z∗(ω) = Rion +RCB

1 + RCBQ(iω)n(1)

that corresponds to the simple circuit sketched

in Fig. 1. Equation (1) accounts for the addi-

tive contributions of the ions in solution, and an

electronically-conductive percolated network of

CB particles. More precisely, the circuit com-

prises a resistance Rion accounting for the ionic

conductivity of the sample in series with two el-

ements modeling the percolated network of CB

particles, namely a resistance RCB in parallel

with a constant phase element characterized by

a dimensionless exponent n, and a parameter Q.

The latter element, which was ﬁrst introduced

in the context of particulate suspensions79 re-

lates to fractal interfaces and corresponds to an

imperfect capacitor.80,81 Although such a mod-

eling fails to describe the dependence of Z00(ω)

at very low and very high frequencies, poten-

tially due to some parasitic inductance, it does

accounts very well for Z0(ω) over eight orders

of magnitude. The low- and high-frequency

limits of Z0(ω) allow us to determine the re-

sistances Rion and RCB. The latter parame-

ter can be converted into the electrical conduc-

tivity σCB =k/RCB, with kthe cell constant

determined by independent measurements on

KCl solutions of diﬀerent concentrations (here

k= 0.38±0.01 cm−1, see supplemental Fig. S2).

Results and discussion

Phase diagram

We ﬁrst discuss qualitatively the outcome of

dispersing carbon black (CB) nanoparticles in

a solution of carboxymethylcellulose (CMC). In

practice, we observe four diﬀerent phases that

are summarized in the phase diagram reported

in Fig. 2. For low CB and CMC concentra-

tion, the aqueous dispersion is unstable and the

CB nanoparticles sediment within about 20 min

[Fig. 2(b)]. On the one hand, increasing the

CMC concentration beyond about 10−2% al-

lows stabilizing the CB nanoparticles, yielding

a viscoelastic liquid [Fig. 2(c)]. On the other

Figure 3: Hydrogel region of the phase di-

agram of aqueous CMC-CB dispersions

as a function of the CB solid weight fraction

xCB and the CMC weight fraction cCMC. Color

levels code for the loss factor tan δ=G00/G0of

the gel phase determined by small amplitude

oscillatory shear at f= 1 Hz. The green curve

corresponds to r=rc, and separates two re-

gions in the gel phase with samples of diﬀerent

microstructures.

hand, increasing the content in CB nanopar-

ticles confers gel-like properties upon the sam-

ples, i.e., the sample shows a solid-like behavior

at rest and for small deformations, while it ﬂows

for large enough stresses [Fig. 2(d)]. Finally, for

CB content larger than about 10% wt., the sam-

ple behaves as a strongly elastic paste, with a

fragile behavior to the touch [Fig. 2(e)].

In the present work, we focus on the hydrogel

phase, which is observed over the entire range of

CMC concentrations explored, and for CB con-

tent ranging between a few % wt. and about

12% wt. In order to quantify the gel rheologi-

cal properties, we measure its linear viscoelas-

tic properties through small amplitude oscil-

latory shear, as detailed above. The ratio of

the viscous to the elastic modulus measured at

f= 1 Hz, i.e., G00 /G0= tan δ, also known as

the loss factor, is reported in Fig. 3 (see supple-

mental Figs. S3 and S4 for a similar representa-

tion of G0and G00, respectively). Such a phase

diagram built upon the loss factor highlights

two diﬀerent regions, which correspond to sam-

ples that mainly diﬀer by their CMC concen-

tration. Samples with a lower CMC concentra-

tion display relatively less viscous dissipation

(tan δ.0.1) than samples with the highest

CMC concentration (tan δ&0.1). This obser-

vation suggests that CMC-CB hydrogels come

in two diﬀerent ﬂavors, depending on the poly-

mer content. We quantify these qualitative re-

sults by studying the scaling of the viscoelastic

and electrical properties of the samples, with

respect to the CMC concentration and the CB

content.

Impact of CMC concentration on

the gel elastic properties

In this section, we ﬁrst discuss the impact of the

CMC concentration at ﬁxed CB content, which

corresponds to a vertical cut in the phase dia-

gram reported in Fig. 2(a). The dependence of

G0with the CMC concentration is pictured in

Fig. 4(a), as a function of the mass ratio r=

mCMC/mCB =cCMC (1 −xCB)/xCB, which rep-

resents the eﬀective number of CMC molecules

per CB nanoparticles. At low CMC concen-

trations, we observe that the elastic modulus

G0is constant G0=G0

p, independent of rover

about two decades of CMC concentration, i.e.,

10−46r610−2. For r > 10−2, the elastic

modulus drops abruptly by about 3 orders of

magnitude for increasing CMC concentration

within a narrow range of rvalues, reaching a

minimum value at r=rc'0.037. Finally,

for r > rc, increasing the CMC concentration

translates into an increase of G0, which scales

roughly as a power-law function of r. This evo-

lution of G0over the whole range of ris robust,

as evidenced by the data reported in Fig. 4(a)

for three diﬀerent CB content, namely xCB = 6,

8 and 10%, as emphasized in Fig. S5 in the sup-

plemental material.

These observations unambiguously conﬁrm

the trends determined thanks to the loss fac-

tor and show that the linear elastic properties

of CMC-CB hydrogels have two distinct origins

depending on the relative content in CB and

CMC. For r < rc, the gel elastic properties are

set by the amount of CB nanoparticles [see in-

set in Fig. 4(a)] and independent of the CMC

concentration, whereas for r > rc, the elastic

modulus is an increasing function of the CMC

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Dualoriginofviscoelasticityinpolymer-carbonblackhydrogels:arheometryandelectricalspectroscopystudyGauthierLegrand,ySebastienManneville,y,zGarethH.McKinley,{andThibautDivoux,yyENSL,CNRS,Laboratoiredephysique,F-69342Lyon,FrancezInstitutUniversitairedeFrance(IUF){HatsopoulosMicrouidsLaboratory,Depart...

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Dual origin of viscoelasticity in polymer-carbon black hydrogels a rheometry and electrical spectroscopy study

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