Biopolymer gels with “physical” cross-links: gelation kinetics, aging, heterogeneous dynamics, and macroscopic mechanical properties

Soft Matter

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aCMIC, Dipartimento di Chimica, Materia

Milano, 20133 Milano, Italy. E-mail: robertbDeFENS, Dipartimento per gli Alimenti, l

Milano, 20133 Milano, Italy. E-mail: laura.

Cite this: DOI: 10.1039/c3sm27153f

Received 18th September 2012Accepted 5th February 2013

DOI: 10.1039/c3sm27153f

www.rsc.org/softmatter

This journal is ª The Royal Society of

Biopolymer gels with “physical” cross-links: gelationkinetics, aging, heterogeneous dynamics, andmacroscopic mechanical properties

Eleonora Secchi,*a Tommaso Roversi,b Stefano Buzzaccaro,a Laura Piazzab

and Roberto Piazza*a

Alginate is a natural biopolymer that forms, in the presence of divalent cations, ionic-bound gels typifying a

large class of biological gels stabilized by non-covalent cross-links, and displaying a consistent restructuring

kinetics. We investigate the kinetics of formation and aging of alginate gels by slow permeation of a curing

CaCl2 agent by means of photon correlation imaging, a novel optical technique that allows obtaining the

microscopic dynamics of the sample, while retaining at the same time the spatial resolution of imaging

techniques. Specifically, the gelling kinetics displays a peculiar non-diffusive behavior, and the

subsequent restructuring of the gel structure shares several features in common with the aging of

colloidal gels, in particular for what concerns the occurrence of heterogeneous dynamics effects. A

comparative analysis of the gel macroscopic mechanical properties at different aging stages further

highlights distinctive effects arising from the non-permanent nature of the bonds.

1 Introduction

Understanding the connection between the microstructure andthe mechanical properties of polymer hydrogels, which consti-tute a basic structural component of living cells, of the extra-cellular matrix, and in general of most so biological tissues, isof primary importance in view of their increasing use in the eldof biomaterials.1 The most striking feature of these systems isthe gamut of elastic and rheological effects they exhibit understress, ranging from full uidization to strong reinforcement.2–4

The microscopic property commonly credited to be at the rootsof this rich behavior is due to the distinctive nature of the gelcross-links that, at variance with the case of simple polymergels, can usually break and re-form at a different location, eitherspontaneously or under the effect of minute changes of thesurrounding environment. The reversible binding of speciccross-linking proteins, allowing us to tune not only the gelmechanical properties but also the relevant time-scales inwhich these changes take place, has been for instance proposedas the basic mechanism mediating between uidization andreinforcement of the cytoskeleton subjected to a mechanicalstimulus.5–8

The opportunity of tuning the rheological properties of gelswith reversible cross-links, lying somehow in between chemicaland physical gels, is quite promising for technological

li e Ingegneria Chimica, Politecnico di

[email protected]

a Nutrizione e l'Ambiente, Universita di

[email protected]

Chemistry 2013

applications. For instance, in the eld of regenerative medicine,long-term application of cyclic strain has already been found toenhance the mechanical properties of engineered muscletissues.9 Exploiting in full the peculiar features of biologicalhydrogels requires however a clear understanding of the rela-tionship between microscopic cross-link dynamics and macro-scopic mechanical properties using simple model systems,which the complex hydrogels formed by protein biopolymers inliving systems rarely are. The main aim of this paper is to showthat substantial information can be obtained by studyingcommon polysaccharide hydrogels extensively used in the foodindustry to ameliorate texture, stability, or mouth feel, and inmore advanced applications such as the formulation of nutra-ceuticals providing health and medical benets.10 Among thiswide class of natural biopolymers, which include traditionalfoodstuff such as starch and pectins, an increasingly importantrole is taken by alginate, actually a family of unbranched poly-saccharides isolated from brown seaweeds, which is one of theextensively used materials in tissue engineering,11 for immobi-lizing living cells12 and, recently, to obtain, when mixed withpolyacrylamide, highly stretchable composite gels.13 Structur-ally, alginates are block copolymers composed of b-D-mannur-onic acid (M) and a-L-guluronic acid (G) arranged in a block-wise pattern where homopolymeric regions of M (M-blocks) andG (G-blocks) residues are interspersed by regions of alternatingstructures (GM-blocks).14 The addition of divalent cations suchas Ca2+, which can be hosted within molecular cavities (“egg-boxes”) formed by repeating G-blocks, can drive the formationof ionic gels15 by binding negatively charged oxygen atomsbelonging to two adjacent alginate chains. This simplied

Soft Matter

Table 1 Mannuronic (XM) and guluronic (XG) weight fractions in Algogel 6020,and weight fractions of the dimeric units XGG, XGM+MG and XMM. Data have been

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“egg-box”model has recently been questioned, since GM blockstoo are found to contribute to chain cross-linking, albeit to alesser extent.16 All evidence suggests anyway that alginate gelsare held together by non-covalent bonds primarily due to ionbridging, although hydrogen bonds and dispersion forces mayalso contribute to the gel strength.17 At variance with thechemical bonds in simple polymer gels, these “physical” cross-links are prone to break and re-form to relax local internalstresses that build up upon gelation. Such a non-permanent,reversible nature of the cross-links is very likely to be respon-sible for the thermo-reversibility of alginate gels and for theirpropensity to display creep behavior over long times.16,17

Moreover, the macroscopic properties of the gel are expected todepend not only on the density, strength, and lateral extent ofthe cross-links, but also on their restructuring kinetics.18,19

Rheological studies of alginate gels are most commonlyperformed in the linear region of small deformations, which isadequately described by usual viscoelastic models, and wherethe Young's modulus is typically used as a measure of gelstrength. Unfortunately, in most interesting applications thesamples are subjected to strains well beyond the linear regime,which can result in a very complex behavior brought in by theinterplay between structure and its evolution through junctionbinding/unbinding.20† Recent experimental ndings concern-ing cell biomechanics suggest that concepts borrowed from soglass rheology may fruitfully account for some basic features ofthe nonlinear rheological behavior of biopolymer gels. In thisconceptual framework, the system temporally evolves into acomplex energy landscape characterized by a large number oflocal minima, with a typical depth larger than the thermalenergy. Hence, moving in this complex landscape requires anactivation energy related to an “effective” or “noise” tempera-ture8 whose microscopic origin has been shown, in the case ofbundled cytoskeletal networks, to be related to frozen internalstresses promoting local rearrangement events.6

In this paper, through a detailed investigation performed byphoton correlation imaging, a novel light scattering techniqueparticularly suitable to investigate the microscopic dynamics ofspatially inhomogeneous systems, we highlight and discussseveral features of the gelation kinetics and the restructuringprocesses in alginate gels generated by the slow perfusion ofCaCl2.22 In particular, the advancement of the gelation front,accurately marked by a dramatic increase of dynamic timecorrelations, displays a very surprising, non-diffusive timebehavior, which can hardly be reconciled with the existinggelation model. Aer the formation of a dynamically quasi-arrested structure, the gel undergoes a time evolution charac-terized by a rst consistent restructuring and strengtheningaccompanied by a progressive slowing-down of the localdynamics, followed by a much longer period marked by drasticand abrupt global “uidization” events, which do not appar-ently lead to signicant changes of the gel structure, butpresumably contribute to a slow “creeping” behavior observed

† This is for instance the case of a recently developed method for controlledrelease from drug-loaded alginate hydrogels based on suitable control of thetime-response to applied stresses.21

Soft Matter

for the gel at very long time. Finally, the macroscopic mechan-ical properties of the gel show evident traces of the temporalevolution analyzed by PCI, giving experimental evidence thatalginate gels share many properties in common with attractivecolloidal glasses. Our results therefore suggest that the physicalmechanisms that are supposed to account for the mechanicaland rheological properties of so glassy systems may have awider generality, and may be shared by polymer gels withphysical cross-links.

2 Materials and methods2.1 Alginate samples

Sodium alginate (Algogel 6020) was obtained from Cargill Inc.,France, and used without further purication. The mannuronicand guluronic weight fractions within this specic alginate arereported in Table 1, which also shows the fractional amount ofthe homo- and hetero-bonds between the two monomer units,whose values strongly inuence the gelling process. Sizeexclusion chromatography23 yields a polymer number-averagedmolecular weight Mn x 110 kDa, corresponding to a degree ofpolymerization of about 200, and a weight-averaged molecularweight Mw x 330 kDa. Hence, this natural biopolymer displaysa very large degree of polydispersity Mw/Mn x 3. Alginatesolutions at a polymer weight fraction of 2% were prepared bydissolving solid Na+-alginate in the desired volume of deionizedwater containing 0.02% (w/w) sodium azide to prevent micro-biological growth, and gelation was induced by slowly perme-ating them with CaCl2 according to the protocols described inwhat follows.

2.2 Photon correlation imaging

Photon Correlation Imaging (PCI) is a recently introducedoptical method, suitable to investigate slow dynamics incolloidal glasses and gels by measuring the time-correlationfunction of the scattered light, as in standard Dynamic LightScattering (DLS), but with the major advantage of allowing forspatial resolution by investigating the local dynamics at distinctpoints within the scattering volume. We shall give only asummary description of PCI, mainly emphasizing the richamount of physical information that can be extracted using thispowerful technique, and referring to the seminal contributionsby L. Cipelletti and coworkers for a more detailed analysis.24,25

Basically, the experiment consists of forming on a multi-pixelcamera an image of the scattering volume, observed at a givenscattering angle w, through a suitably stopped-down optics (seeFig. 1a). In other words, the imaging optics is provided with apartially closed iris diaphragm placed in the focal plane of the

obtained according to the method described in ref. 23

XG XM XGG XGM+MG XMM

0.56 0.44 0.26 0.60 0.14

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Fig. 1 (a) Scheme of the PCI setup in a 90� scattering angle configuration. (b)Scheme of the optical cell used to perform gelation.

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lens that, besides accurately selecting the scattering wave-vectorq ¼ (4p/l)sin(w/2), causes the image to become “speckled”because the intensity at each given point on the image planeoriginates from the interference of the eld scattered by a nite-size region in the sample plane. The simultaneous measure-ment of the time-dynamics over many speckles yielded by themulti-pixel detector, besides providing a fast ensemble aver-aging of the intensity correlation function (crucial when inves-tigating samples with a very slow dynamics), allows us toidentify the occurrence and follow the temporal evolution ofthose dynamic effects associated with the presence of spatialheterogeneities in the investigated sample. Quantitatively, thelatter can be characterized as follows. First, the speckle patternis subdivided into “regions of interest” (ROIs) by groupingtogether a given number of adjacent pixels. Then, one intro-duces the so-called “degree of space–time correlation” (or“correlation index”) cI(s;t,r) between two images taken at times tand t + s as

cIðs;t;rÞ ¼�IpðtÞIpðtþ sÞ�

r�IpðtÞ

�r

�Ipðtþ sÞ�

r

� 1 (1)

where Ip is the scattered intensity measured by the pth pixel andh/ir denotes an average over all pixels within a ROI centeredaround r. Hence, cI(s;t,r) is simply related to the covariancebetween the intensity measured on the same speckle at twodifferent times, sampled over the given ROI and, in particular,cI(0;t,r) is the relative variance of the intensity in r at time t. Notethat cI is a function of the delay time s that, in samples dis-playing a restructuring and aging kinetics, depends para-metrically on the aging time t (and, for spatially inhomogeneoussamples like those considered in this work, also on the localposition r). In fact, as shown in what follows, it is exactly thistime-dependence that yields the basic features of the gelationprocess. Provided that the investigated kinetics is sufficientlyslow, however, the statistical accuracy can be enhanced by time-averaging cI(s;t,r) over a time window dt that is much shorterthan the characteristic evolution time of the investigatedkinetics, which allows us to reduce the statistical noise due tonite sampling on the limited number of pixels in a ROI. Thelocal dynamics can then be quantied by dening‡

g2(s) � 1 ¼ hcI(s;t,r)idt. (2)

Such a “coarse-grained” correlation index actually bears thesame information of the intensity correlation functionmeasured in a standard DLS experiment, yet with the crucialadvantage of a much better statistical accuracy due to pre-averaging over the speckles in a ROI, which allows averagingover a time window dtmuch shorter than the total measurementtime required in a DLS measurement.

In glassy or gelling systems, however, the correlation indexalso displays intrinsic uctuations, which are not just due to theinstrumental noise associated with nite-sampling, but to real

‡ To simplify this notation, we leave out the parametric dependence of g2(s) on t

and r, which is however to be understood.

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physical processes taking place in the system. A strikingmanifestation of these physical uctuations on which we shallparticularly focus is the occurrence of sudden “correlationbursts” due to structural rearrangements that rapidly propagateover large regions of the sample. Such a “dynamic heteroge-neity” can be investigated by considering the (coarse-grained)variance of cI(s;t,r) on a ROI

c(s) ¼ hcI(s;t,r)2idt � hcI(s;t,r)i2dt (3)

which quanties the temporal evolution of the restructuringevents taking place at a given location and aging time. As shownin the above equation, the dependence of c(s) on the delay sallows us to easily set apart slow changes of the local dynamicsassociated with aging from the aforementioned sudden globalrestructuring events. Whereas the former processes are indeedcharacterized by a peak in the local value of c occurring at adelay time s comparable to the relaxation time of g2(s) andgetting more pronounced the larger the size of the correlatedregion within the sample,26 the latter are highlighted by asudden increase of c(s) over the whole sample which is notcorrelated with the local dynamics.

What is also particularly interesting for the present study isthat PCI, besides being a sensitive probe of the dynamic slow-ing-down and kinetic arrest associated with gelation, alsoallows us to detect and map the presence of hydrodynamicmotion in the sample. Indeed, whereas standard DLS can detectonly relative particle motion (in other words, provided that thescattering volume is uniformly illuminated, the far-eld specklepattern generated by a system of scatterers in uniform motionremains stationary), in a PCI conguration the speckle motionon the image plane faithfully maps the local ow eld withinthe sample. As in standard Particle Imaging Velocimetry (PIV27)techniques this allows us to obtain via a FFT algorithm thespatial cross-correlation function of the speckle pattern at

Soft Matter

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different times t1 and t2. In this scheme, the local sampledisplacement in the time interval t2 � t1 is related to the posi-tion of the cross-correlation peak, whereas the height of thepeak itself decays on a time scale set by the Brownian dynamicsof the sample measured at a wave-vector q. This technique,which has been successfully used to probe the collapse andrestructuring of colloidal depletion gels,22 proves to beextremely useful for investigation of the detailed mechanism bywhich gelation proceeds.

To follow gelation, the alginate solution is conned to thelower, optically accessible part of a cuvette (VWR Interna-tional), by a porous membrane with 100 mm pore size, which ismounted on a custom-made plastic frame that tightly ts intothe wedge-shaped connection between the sample compart-ment and an upper reservoir (see Fig. 1b). This cell geometryallows us to laterally image a rectangular area S within thesample with a vertical extent L x 16 mm and a horizontalwidth W x 6.4 mm, which allows mapping the sample regionextending from the cell bottom down to about 3.5 mm over themembrane (in the following, we call z the vertical distance fromthe bottom of S). Gelation is started by gently pouring thecuring agent (a 0.1 M CaCl2 solution) in the reservoir atop themembrane, carefully avoiding the presence of air bubbles.The cell is then rapidly turned upside down and mounted onthe cell holder with the reservoir at the bottom to avoidconvection effects that may arise from the fact that the CaCl2solution is slightly denser than the alginate uid (see alsoSection 3.2). The illumination optics is composed of a 50 mWsemiconductor laser module operating at l ¼ 656 nm (SYLasiris SNF501L, StockerYale, New Hampshire, USA) thatincludes a holographic grating allowing us, with the aid of alens (f ¼ 300 mm), to shape the beam on the cell entrancewindow as a uniform vertical sheet of width w ¼ 20 mm (seeFig. 1a). An objective images the investigated sample region Son a rectangular area with a size of 300 px � 120 px of a CMOScamera sensor (Optikam Pro5, Ponteranica, Italy), whereas adiaphragm placed in the objective focus selects a scatteringangle w ¼ 90� (corresponding to a scattering wave-vector q x1.8 � 105 cm�1) with a speckle size of about 3 px � 3 px. Theexposure time is typically set to 1/15 s, with a time intervalbetween successive images of 30 s, and the sensor gain is tunedduring the gelation process so as to optimize the intensity ofthe speckle pattern within the gel phase. Since during thegelation kinetics the intensity and correlation index areexpected to depend only on the vertical position z, the specklepattern image was subdivided into 30 ROIs with a height of 10px (corresponding to about 530 mm in the sample) and laterallyextending over the whole width W of the investigated sampleregion (120 px). Each ROI contains therefore about 100speckles, a value which was found to yield a good verticalresolution with an acceptable statistical noise in the correla-tion function. If necessary, spatial resolution can nonethelessbe simply improved by changing the imaging lens and focusingon a smaller gel region. Finally, the space–time resolvedcorrelation index cI(s;t,r) is then computed for each ROI using acustom-made code running on ImageJ, a public domain imageprocessing soware.

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2.3 Mechanical tests

To investigate the macroscopic mechanical properties of algi-nate hydrogels, several gel samples were prepared by pouringthe liquid solution of alginate into dialysis tubes with a diam-eter of 21 mm (Sigma Aldrich, MWCO 12 000) and an overalllength of 200 mm, which were then sealed and immersed in awater bath containing a 0.1 M calcium chloride solution. Todetect the effects of microscopic restructuring observed by PCIon the gel macroscopic properties, the samples were extractedin sequence from the gelation bath aer a curing timeincreasing from 6 hours up to 6 days. Each cylinder was trans-versally cut into cylindrical sections, which were then separatelysubjected to uniaxial compression tests using a texture analyzerTA (XTplus, Stable Micro System, England) operating at T ¼23 �C. To eliminate any effect due to the non-ideal cutting ofthe cylinders, the samples are sandwiched between the upperand lower plates of the TA, and rst pre-loaded with a stress ofabout 2–3 kPa, so as to compress them to a length h0 which isabout 2% smaller than the unperturbed value. Measurementsthen consist of further compressing them at a minimumconstant speed of V ¼ 1 mm s�1 from an initial height h0,accurately measured aer the pre-stress has been applied, up torupture. To check for reproducibility, tests were repeated onseveral samples for each value of the time spent in the gelationbath. To obtain the stress relaxation curve behavior, similarcylindrical gel samples were submitted to a quasi-static uniaxialcompression at constant deformation, and the stress recordedas a function of time for a maximal duration of about 1 h. Theinitial deformation was applied at the very high strain rate of33 mm s�1, allowing us to consider it as quasi-instantaneous.

3 PCI analysis of the gel formation andaging3.1 General features of the gelation process

A preliminary qualitative description of the progress of thegelation process permits us to single-out its main features andset apart distinct time regimes, allowing us at the same time tofamiliarize with the kind of information that can be extractedfrom a comparative analysis of the intensity and degree ofcorrelation proles obtained by PCI. With a negligible delay,due to the time it takes to cross the separating membrane, thecuring agent that has been poured into the lower compartmentof the cell starts diffusing into the upper compartment, trig-gering the onset of alginate gelation and generating a gelationfront that propagates upwards, as shown by the three snapshotsin Fig. 2. Visually, this front is marked by a noticeable increaseof the scattered intensity in the cell region already invaded bythe CaCl2 solution.

This effect can be related to the microscopic dynamics of thegelling alginate solution by contrasting the time behaviour ofthe scattered intensity and the correlation index on a single ROIof the sample, located at z ¼ 5 mm from the cell bottom(namely, approximately 8 mm above the membrane). Fig. 3shows that at t0 x 24 h, roughly corresponding to the time ittakes for the curing solution to propagate from the membrane

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Fig. 2 Snapshots of the cell taken about 13 (I), 25 (II), and 45 (III) hours after thecuring agent has been poured into the reservoir. The z axis indicates the measure-ment coordinate, whereas the yellow box indicates the size and shape of a ROI.

Fig. 3 Time dependence of the intensity ratio I(t)/I(0) (open dots, right axis) andthe normalized degree of the space–time correlation cI(s,t,z) after a delay of s ¼30 s (bullets, left axis), measured for a ROI centered at z¼ 5mm. The full line is anexponential fit to the intensity curve for t > tg with a time constant tax 850 s. Theinset shows the normalized correlation index at delay times s ¼ 10 min and s ¼50 min.

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up to the ROI location, aer an initial dip the scatteringintensity starts to grow, reaching a plateau which is about 20times larger than its value at t ¼ 0. The rising part of theintensity is rather well tted by an exponential with a timeconstant ta x 850 s. Since an increase of the scattering intensitywitnesses a change in the sample structure factor,§ this meansthat structural rearrangements keep on taking place in the ROIfor a long period of time, quantied by the “aging” time ta.

The temporal evolution of the local microscopic dynamics is,however, very different. To highlight this contrasting behavior, itis useful to introduce a “normalized” degree of correlation by

§ Reecting a progressive increase of the size of the correlated scattering regions,provided that they do not get so large that the scattering pattern becomes stronglyforward-peaked.

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taking the ratio cI(s;t,z) ¼ cI(s;t,z)/cI(0;t,z) of cI(s;t,z) to its value ats ¼ 0.{ Let us rst concentrate on the behavior of cI for shortestdelay time s ¼ 30 s between two images of the ROI. Since thedecay time of the correlation function for a uid alginate solu-tion, measured by standard DLS at the same q-vector, is of about25 ms, it is not surprising that, for t < t0, the correlation indexbetween the two speckle patterns is basically zero. However, assoon as the scattered intensity starts to change, cI(s,t,z) shows arapid increase, reaching a value of about 0.9 in a fewminutes. Inother words, for t > t1 the speckle pattern is still fully correlatedaer a delay time that is more than two orders of magnitudelonger than the correlation time of the alginate solution. Hence,for t > t1, the whole alginate solution in the ROI is already a quasi-arrested gel, where however noticeable structural rearrange-ments, highlighted by the persisting increase of I(t), are stillgoing on. Noticing that t1� t0 is of the order of magnitude of thetime it takes for the Ca2+ ions to traverse the ROI, the conversionof the solution into a dynamically arrested structure can beassumed to take place almost instantaneously when the curingagent becomes available. Such a sharp dynamic arrest allows usto take tg ¼ (t0 + t1)/2 as a reasonable average time in the ROImarking the transition from a uid to a gelled system. By quasi-arrested, however, wemean that residual dynamic effects persistin the gel well beyond tg. The inset in Fig. 3 shows indeed that thedegree of correlation still decays over much longer delays.Indeed, aer a delay of s ¼ 50 min, cI(t) has decreased to zeroeven at a time t when I(t) has basically reached its plateau value.If at this stage structural changes are associatedwith such a long-time decay of the correlations, they are too weak to be mirroredby changes in the scattering intensity at the selected q-vector.Actually, we have not detected a fully arrested dynamics aer anyripening time. An investigation of the macroscopic stress relax-ation discussed in Section 4 shows that such a long persistenceof diffusive microscopic dynamics is arguably responsible for aslow macroscopic creeping of alginate gels.

3.2 Kinetics of the gelation front

The abrupt increase of cI suggests that dynamic arrest, or atleast a dramatic increase of the correlation time imputable tothe formation of a loose gel structure, is a quasi-instantaneousprocess, taking place as soon as Ca2+ comes into contact withalginate. The time-dependence of the advancement of thegelation front along the cell, tracked by plotting the “gellingtime” tg for each single ROI, is however quite surprising. Fig. 4shows that the gelation front advances indeed linearly with timewith a speed of vg ¼ z(tg)/tg x 0.3 mm h�1 over the wholeinvestigated range. Extrapolated to tg¼ 0, the linear t yields z¼�2.8 mm, which is consistent with the estimated position of themembrane. Note also that the time t1� t0 it takes for cI at a delaytime of 30 s to switch up to a value close to one is approximatelyconstant and in good agreement with the time it takes for thegelation front to cross a ROI at the constant speed vg.

{ This procedure may be reasonably expected to get rid of most of the statisticaluctuation associated with the nite pixel sampling and to better highlight thephysical contribution due to local restructuring processes.28

Soft Matter

Fig. 4 Kinetics of the gelation front, showing the position z corresponding to aspecific gelling time tg ¼ (t1 + t0)/2. The linear fit yields an advancing speed vg x0.3 mm h�1, and an intercept z(0) x �2.8 mm, approximately coincident to theposition of the separating membrane. The inset shows that t1 � t0 is almostindependent from the position in the cell, roughly amounting to the time it takesfor the gelation front to traverse a ROI at the constant speed vg (here full andbroken lines indicate the average value and standard deviation of t1 � t0).

k Further effects which are not provided for by the M–E model may be related tothe presence of an osmotic unbalance at the gel front, where the local ionicstrength shows a steep gradient. In colloidal suspensions, salinity gradients areknown to generate Donnan effects, which affect for instance particle diffusionor sedimentation.39 Whether similar effects occur during the formation of anionic polymer gels is surely a question that deserves further investigation.

** In PCI measurements, the latter is clearly highlighted by the observation ofslow convective rolls traced by residual dust particles in the liquid alginate phase.

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This striking experimental evidence can hardly be reconciledwith the existing models of alginate gelation. The most widelyused approach, due to Mikkelsen and Elgsaeter (M–E model29),is indeed based on assuming that the local concentration ofboth calcium ions and free alginate behaves diffusively,decreasing at the same time because of the formation of thepolymer gel. The latter is conversely assumed to be immobile,with a local concentration that steadily grows because of twodifferent mechanisms, the inclusion of an additional monomerto an already formed network and the novel associationbetween two free chains. For a generic dependence of thediffusion coefficients on the concentration of the three species,these assumptions lead to a set of nonlinear equations thatmust be solved numerically.29 However, within the (strongly)simplifying assumption of constant diffusion coefficients ofalginate and calcium ions, the M–E model becomes just aparticular case of a reaction–diffusion process, which has beenextensively investigated in the literature because of its largeinterest both in chemical and biological pattern formation30

and in the context of diffusion limited aggregation31 withsophisticated methods.32–35 When applied to gelation, theseanalytical solutions yield a gelation front position zf(t) that, forinitially separated reagents at concentrations a0 and c0, behavesdiffusively, zf 2ðtÞ ������!

t/NDf t, with a long-time front diffusion

coefficient Df that can be evaluated by equating the uxes of thereactants at the position of the gelation front.35

Whereas a suitable dependence of the diffusion coefficientson the concentration of the reacting species may lead to a timebehavior which is not strictly diffusive, as observed in otherexperiments,36,37 it is much harder spotting a clear physicalmechanism yielding a gelation front that, instead of a diffusing,propagates at a constant speed along the cell. A possible

Soft Matter

explanation may be hinted by the initial moderate decrease inintensity which is detected just in the region where cI starts togrow (namely, close to the leading edge of the gelation front, seeFig. 3), which is observed to a greater or lesser extent for all theROIs. Such a dip in the scattered intensity may reasonably beattributed to the presence of a thin layer before the advancinggel, with a thickness of the order of a few hundred microns,where the concentration of alginate is sensibly lower than in thebulk. The physical origin of such a “depletion layer”, which hasapparently been observed in other measurements of alginategelation,37,38 is not easy to spot. Tentatively, we suggest that itcould stem from a mechanism that is not included in the M–Emodel: syneresis, namely a progressive shrinking over longtimes of the structure by solvent ejection, which is a commonand extensively studied phenomenon in polymer gels. In theinvestigated sample, syneresis of the alginate matrix is alreadyhighlighted for times slightly exceeding the gelation time of thewhole sample by the onset of a slow uniform translation of thespeckle eld, witnessing a backward motion of the whole gelstructure towards the negative z direction, which eventuallyleads, at much longer times, to gel detachment from the cellbottom and lateral walls. However, gel shrinking is very likely totake place even before,while the gelation front advances and thegel gets progressively reinforced by the increase in the cross-linkdensity supplied by the incoming Ca2+ ions. Since the diffusionconstant of alginate is very low, this might presumably leave apolymer-depleted region just on top of the gel.k

Whether or not this physical mechanics is plausible, thepresence of the depletion layer implies that there is a regionwithin the sample where the density prole of the solution getsinverted, regardless of whether the alginate or the curing solu-tion is initially placed on top. Such an inverted density prole ishydrodynamically instable, unavoidably leading to micro-convective effects. The latter, however slow, would noticeablymodify the spatial distribution of the curing agent, leading ingeneral to a quasi-uniform concentration prole within thedepletion layer. The presence of convective ow is commonlyexpected and oen experimentally observed to acceleratetransport processes with respect to their purely diffusive limit.40

Micro-convection within the depletion layer may thus lead to avery different time behavior of the gelation front.38 To supportthis view, we tried to enhance convective effects by repeating thesame experiment, but without turning the cell upside down, sothat the heavier curing solution lies on top of the alginatesolution. Fig. 5, where the times t2 required for the scatteringintensity to double its initial value (which is just slightly longerthan tg) are compared, shows that in the presence of macro-scopic convection,** the time it takes for the scattered intensity

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Fig. 5 Time t2 required for the scattered intensity to double its initial value, as afunction of the position in the cell, for the sample discussed in the text (bullets),and for an identical sample placed in the cell mounting with the reservoir con-taining the curing agent at the top (squares). The inset shows the dependence ofthe aging time ta, obtained from an exponential fit to the growth of the scatteredintensity curve, on the square of the measuring coordinate z.

Fig. 6 Correlation functions g2(s) � 1 obtained in a fixed ROI located at z ¼ 0.5mm, measured at different aging times t� tg: the semilogarithmic plot in the insetshows that the correlation functions decay roughly exponentially, with a decayconstant sr that substantially grows with t.

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to double its initial value linearly depends on the position z, yetthe front advances with a speed that is more than three timeslarger than in the hydrostatically stable case. This evidence,although far from being conclusive, shows that convectiveeffects, which are not considered in the simple diffusion/reac-tion model, may drastically change the time behaviour of agelation process. This observation may be relevant in mostpractical applications of alginate, where the adopted geometryimplies horizontal concentration gradients, unavoidably leadingto convection.

Nonetheless, effects related to the spreading of theadvancing front of the curing agent can be detected by lookingat the aging time ta, dened like in Section 3.1 as the timeconstant of the exponential growth of the scattered intensity at agiven position z. The inset in Fig. 5 shows indeed that ta growsquadratically with z from aminimum value ta0 x 11.5 h at z¼ 0.Such an increase can be reasonably attributed to a delay in thesupply of the additional Ca2+ ions required for the loosepercolating structure formed at tg to “ripen”. Fitting the curve as

zðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~Dðta � ta0Þ

qactually yields a t parameter ~D x 1.7 �

10�5 cm2 s�1, which has the typical order of magnitude of thediffusion coefficient of small ions. This rather strange combi-nation of a linearly advancing gelation front with a gel agingthat shows evident traces of the ion permeation of an alreadygelled structure points out the need for further studies of thekinetics of gelation processes in gels with ionic cross-links.

3.3 Gel aging

The progressive change in the gel dynamics during therestructuring period following tg can be quantitatively analyzedby means of the correlation functions dened in eqn (2), using acoarse-graining time interval of dt ¼ 100 min that, as shownbelow, is still much shorter than the aging time scale of the gel.

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Fig. 6 shows that all correlation functions g2(s) � 1 obtained ona ROI located at z ¼ 0.5 mm eventually decay to zero regardlessof the aging time, conrming that the gel dynamics is never fullyarrested. The semi-logarithmic plot in the inset shows that thedecay of g2(s)� 1 is reasonably well tted by a single exponentialdecay, with a relaxation time sr that grows with the aging time,increasing by more than two order of magnitudes beforepossibly starting to level off. A qualitatively similar behavior isfound analyzing different ROIs.

In order to compare the aging behavior at long times fordifferent sample locations, it is useful to evaluate the dynamicscounting the time, for each single ROI, from the moment whenthe gel has locally reached a comparable restructuring time. Thelatter can be consistently stated as the sum of the time tg whenthe dynamic arrest takes place with the subsequent character-istic aging time ta. Fig. 7 shows that the results for sr at differentROIs, when plotted as a function of the shied time Dt(z) ¼ t �[tg(z) + ta(z)] at the ROI position z, rescale on a single mastercurve. This means that, to a good approximation, the dynamicevolution of the gel, when referred to Dt, does not depend on theposition along the sample. This is further conrmed by the insetin the gure, showing that the full correlation functionsobtained for different ROIs at the same value Dt superimposequite well. These observations suggest that, once the gelrestructuring reaches a nal stage, the microscopic dynamics isspatially homogeneous along the sample and depends only onthe “local” age of the gel.

At rst sight, a simple exponential form of the correlationfunctions in a complex system like a gel looks rather surprising.Actually, when colloidal or polymer gels are probed over lengthscales, set by the inverse scattering wave-vector q�1, which aresmaller than the mesh size x, g2(s) usually displays a verycomplex behavior basically reecting internal relaxationprocesses, which share many aspects in common with thecorrelation functions observed in glassy systems. However, weprobe the gel structure over a length scale of q�1 x 55 nm,

Soft Matter

Fig. 7 Microscopic relaxation time sr as a function of Dt(h) measured withrespect to a time-origin tg(z) + ta(z), given for each ROI an approximate indicationof the instant when the gel has reached an almost stationary structure. Thepositions along the cell corresponding to the different symbols are specificated inthe legend. The inset shows that the full correlation functions obtained fordifferent ROIs at the same value Dt almost superimpose.

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which is quite larger than the typical mesh size x ¼ 5–15 nm ofalginate gels.41 In fact, internal modes, which have typicalrelaxation times in the range of hundreds of ms42 (hence 5 ordersof magnitude shorter than the rst delay s1 ¼ 30 s measured byPCI), are responsible only for the tiny reduction with respect tothe ones in the apparent intercepts of the correlation functionsshown in Fig. 6. Notice that the latter, which is already as smallas 5% for a relatively “young” gel (t � tg ¼ 15 h), becomes totallynegligible as the gel ages.

The very slow dynamics detected by PCI has therefore aradically different physical origin. In the past few years, severalexperimental and theoretical investigations have indeed sug-gested that weak colloidal gels and so glassy systems mayexhibit distinctive dynamical features, driven by the accumu-lation of internal stresses43–45 generated on a microscopic scaleby bond formation and rupture. These stresses relax via local-ized microcollapses creating a long-range elastic deformationeld and constituting the elementary steps leading to aprogressive gel compaction. On the much longer time scales ofsuch a microscopic stress relaxation, the DLS correlation func-tions are predicted to behave very differently, both in terms ofshape and q-dependence. Whereas the superposition of internalmode relaxations yield a short-time dynamics which is oenwell described by a stretched exponential,†† g2(s) may display acompressed exponential form, namely, the instantaneous decayrate increases with time. A general model based on the relaxa-tion of stress dipoles44 shows that, provided that the micro-collapses take place very rapidly, the behavior of g2(s) is set by acharacteristic time scale ~tq x h(qx)/G0, where G0 is the gel

†† For a discussion of this functional behavior, see Section 4.2.

Soft Matter

compression modulus, and h is the dynamic viscosity. Speci-cally, ln[g(s)] decays as:

ln½gðsÞ� �(ðqsÞ3=2 s � ~sq

q3=2s s[~sq(4)

The experimental results for a very weak colloidal gel, with acompression modulus of G0 x 0.1 mPa, actually support boththe compressed exponential form of g(s) and the q�1 depen-dence of its decay rate predicted by the upper expression in eqn(4).43,45 As we shall see in Section 4.2, however, the elasticmoduli of the alginate gels we investigate are hugely larger, inthe range of 102 kPa. Hence it is not surprising that the decay ofg2(s) is very well tted by a simple exponential, as predicted byeqn (4) for s[~tq. Note that, even in this case, the decay rateshould still display a non-diffusive q3/2 behavior. Unfortunately,our experimental setup cannot be simply adapted to investigatethe q-dependence of g2(s), which might yield an interesting testof the stress-dipole. Nevertheless, when ln[g(s)] � q3/2s, thestress-dipole relaxation model predicts the experimental decaytime of g2(t) to grow linearly with the gel aging time. Fig. 7 showsthat indeed, to a good approximation, sr increases linearly withDt in all ROIs, giving an indirect support to the model.‡‡

3.4 Dynamic heterogeneity

The aging of colloidal gels, made of particles linked by short-range attractive interactions, is usually also characterized by astrongly heterogeneous dynamics.28,46 This means that the localdegree of correlation at a given time shows strong spatial uc-tuations. Moreover, on top of the micro-scale collapses that aresupposed to yield the overall decay of the local correlationfunctions, according to the model described in the formersection, the gel aging is characterized also by structural rear-rangements witnessed by abrupt decreases of the degree ofcorrelation (“spikes”) that extend over wide regions of thesample. The result presented in what follows show that a verysimilar behavior is shared by alginate polymer gels with phys-ical cross-links. Studying gels characterized by an aging stagevarying along the z direction allows us to investigate by PCI howthe degree spatial and temporal heterogeneity qualitatively andquantitatively depends on aging.

Striking evidence of the sudden rearrangements character-izing the heterogeneous dynamics of aging alginate gels can beobtained by focusing on a single structural rearrangement eventvisualized by means of a so-called “dynamical activity map”,where the spatial distribution of the correlation index is rep-resented by a false-color map.47 In order to clearly point out thatdynamic heterogeneity occurs not only along the aging direc-tion, but also at constant z (namely, within gel regions that aresupposed to be structurally homogeneous, at least on theaverage), it is useful to re-dene the subdivision of the sampleinto regions of interest as follows. A rst ROI in the specklepattern is constructed as a 15 px � 15 px square. Then, all theother ROIs are generated by shiing the latter by 5 px alongeither the z-axis or the horizontal coordinate x. Besides yielding

‡‡ Note that, when ln[g(s)] � (qs)3/2, one would rather expect sr � (Dt)2/3.

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a much clearer visualization, this method, although intro-ducing some correlation between adjacent partially overlappingROIs, can be shown to ensure better spatial resolution. To eachof these regions is then attributed the average value of thenormalized correlation index cI(s;t,r) within the ROI, evaluatedat the delay time s ¼ 30 s. Fig. 8 shows a typical sequence ofmaps referring to a single rearrangement event. In the rst map,referring to a starting time t ¼ t* just before the rearrangementoccurs, a fully correlated region (in red), where the sample isalready dynamically arrested, can be clearly told apart from afully uncorrelated zone (in blue) close to the top of the obser-vation region, where gelation has not yet taken place. Moreover,a decrease of the degree of correlation by increasing z witnessesthe different aging stages within the gel phase. In the secondactivity map, taken at starting times t* + s, a rearrangementevent generating close to the gel interface is seen to turn thewhole gel into a dynamically active structure that almostcompletely looses correlation within a delay time s, showingthat the events rapidly propagate over macroscopically largelength scales. Subsequently, correlation is recovered startingfrom the bottom, more “aged” part of the gel, until, at tx t* + 3sthe gel reverts to a state that does not substantially differ fromthe original one. The possibility that these correlation drops arenot artifacts due to a rigid motion of the speckle eld,28

generated for instance by a sudden detachment of the gel fromthe cell walls as a consequence of a macroscopic shrinkage, canbe easily ruled out by an image correlation velocimetry analy-sis.22 As already mentioned, gel shrinking effects can converselybe clearly detected aer gelation has taken place over the wholesample.

These results, together with those discussed in Section 3.3,suggest that there are two kinds of dynamic processes takingplace in an aging alginate gel: a progressive slowing down of thelocal microscopic dynamics, witnessed by the increase of thestructural relaxation time sr, and an intermittent sequence ofrestructuring “bursts”, involving the whole sample but notleaving any signicant trace on the gel structure, at leastapparently. These two different processes are better appreciatedby considering the time behavior of the uctuations in cI,

Fig. 8 Sequence of activity maps, obtained as described in the text, referring to asingle restructuring event taking place at a time t* x 40 h after sample prepa-ration. The scale to the right displays the color scale for the normalized degree ofcorrelation cI(30 s;t,r).

This journal is ª The Royal Society of Chemistry 2013

expressed by its (coarse-grained) variance c(s). As discussed inSection 2, this amounts to evaluating how much the degree ofcorrelation aer a delay time s uctuates over the pixels in agiven ROI, taking than the average of the standard deviation ofcI over a short time window dt. In the two panels of Fig. 9, whichrefer to two ROIs at different vertical positions z, the behavior ofc as a function of s is represented by a single vertical line ofthickness dt with a color code that varies from blue to red withan increasing value of c(s), whereas the way c(s) changes withthe time t aer sample preparation is monitored on the hori-zontal axis. On both ROIs, the structural rearrangementsleading to the slowing down of the local microscopic dynamicsare associated with a strong increase of c(s) at short delays s,lasting for a signicant fraction of the aging time ta. Since bothtg and ta depend on z, both the temporal location and theduration of this process are considerably different for the twoROIs. A more detailed analysis shows that c(s) is peaked aroundthe value of the structural relaxation time sr at time t, whereasthe amplitude of the peak progressively decreases with s.Conversely, aer the main structural rearrangement on spatialscales of the order of q�1 has completed (namely, for t > ta whenthe local dynamics has reached the quasi-homogenous stageshown in Fig. 7), sudden increases of c(s) lasting for a time t #dt, but persisting for much longer delays s, take place. Sincethese events correspond to de-correlation bursts rapidly prop-agating over the whole sample, they take place at the same timeon the two ROIs, namely, their timing is not dictated by the localaging stage, but only on the absolute time t.

Such a temporal coincidence of the de-correlation bursts ondifferent sample regions is strikingly highlighted by Fig. 10,where a time window of about 20 h, beginning at t > tg + ta, isshown for ve different ROIs along the sample. Notice that thetime interval between two consecutive drops increases with theaging time. The evolution of the gel towards a more homoge-neous state, witnessed for t � tg < ta both by the behavior of thelocal dynamics and by the fading of the uctuations in cI, is thenalso accompanied by a later progressive decrease of the frequencyof global rearrangements persisting for a very long time.Although the timing of the global de-correlation events is the

Fig. 9 Dependence on the delay time s (vertical axis) of the standard deviationc(s) of the correlation index, shown in the color code, as a function of time t. Thetwo graphs refer to different ROIs displaying the indicated gelation and agingtimes tg, ta. The arrow points at a typical abrupt de-correlation burst.

Soft Matter

Fig. 10 Coincident “spiking” behavior of cI(s;t,z) for five different ROIs over atime window of about 20 h. The inset shows the superposition of the spikingevents occurring between the two grey lines in all the ROIs.

Fig. 11 Upper-left inset: uniaxial stress–strain relationship for alginate gelsmade by permeation of a 0.1 M calcium chloride solution for the periods of time tshown in the legend, obtained at a compression speed of 1 mm s�1. Vertical barsindicate the maximum compression that each sample can bear before gelrupture. Each curve is obtained as the average of at least four samples obtainedunder the same conditions. Main body: expanded linear view. The boxed region,extending up to a maximum compression of about 20%. Lower-right inset: linearbehaviour of the apparent yield stress s0 with the gel's Young's modulus E.

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same over the whole gel, the amplitude and duration of a singlespike on a specic ROI depends on the aging stage of the latter.The inset in Fig. 10 shows indeed that the bursts get stronger andlonger with z. Thismeans that the sample regionswhere the gel isrelatively “younger” are more affected by the rearrangement andtake more time to revert to their original state.

4 Mechanical properties of the gels

The alginate gels whose macroscopic properties we investigatedwere prepared according to a procedure which had to neces-sarily be quite different from the way gelation was obtained inthe PCI experiments. Here gelation proceeds from themembrane inwards, hence the longitudinal compressionmeasurements performed at a given aging time reect only theaverage mechanical properties of gels whose structure variesalong the radial direction. This amounts to testing simulta-neously a set of parallel “springs” whose elastic constantsphysically correspond to gels at diverse aging stages. Thecompression and stress relaxation tests discussed in thissection are not therefore meant to provide a full mechanicalcharacterization of alginate gels, but rather to support qualita-tively our microscopic PCI investigation. Nonetheless, ourpreliminary data already bear clear evidence of the progressiveoverall aging of the gel structure, and yield further hints on thenature and kinetics of cross-links in alginate gels.

4.1 Uniaxial compression tests

We rst discuss the results of uniaxial compression testsobtained as described in Section 2 for alginate solutions kept inthe gelation bath for a period of time ranging between 6 h,roughly corresponding for the selected experimental geometryto the minimum time required to obtain a macroscopic weakgel, and 144 h. The samples were compressed at constant speed

Soft Matter

from an initial length h0 to a nal length h, and the uniaxialcompression stress sm in excess of the pre-stress value denedin Section 2 was measured as a function of the strain l ¼ 1 �h/h0. The experimental excess stress sm was then corrected forthe increase of gel cross-section at a constant volume to obtainthe “true” stress s ¼ sm(1 � l). The double-log plot in the insetof Fig. 11 shows that the overall stress–strain relationships fordifferent aging times t progressively merge into a single curve,characterizing the region where a consistent strain hardening isobserved. With the exception of the samples kept in the gelationbath for 6 h, the maximum compression that the gels cansustain before rupture is almost independent of t. An expandedlinear view of the different curves of s(l) for a compressionalstrain not exceeding lx 0.2, shown in the main body of Fig. 11,rst displays a linear increase of s(l) (full lines), correspondingto a Young's modulus E ¼ s/l that steadily increases from E x150 kPa to E x 350 kPa with increasing gelation time. Thislinear elastic region (region I) is limited however to compres-sion values not exceeding about 5%, and is followed by a roll-offrapidly leading to a behavior which is still linear, but corre-sponds to an apparent elastic constant E0 x 120 kPa which isalmost independent of sample aging (region II). The lower-rightinset shows that the intercept s0 of the linear ts (dotted lines)with the l ¼ 0 axis, corresponding to an apparent yield stress,grows roughly linearly with the gel Young's modulus Emeasured at small compression values. While the behavior atsmall l shows that the gel stiffens with t, for a sufficiently largedeformation the time spent in the gelation bath seems to affectthe alginate gels only for what concerns s0, whereas theirapparent compression modulus does not change.

The latter observation, together with the merging of all theresponse curves in the strain hardening region, suggests thatthe gel elastic response at small l may stem from the

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concurrence of more microscopic mechanisms, whereas largerdeformations are controlled by a single mechanism, notaffected by t. A similar behavior has been recently observed ingels made of synthetic alginates where the fractional content ofGM alternating sequences can be carefully controlled by Mørchet al.,48 who pointed out that the gels display a degree of plas-ticity that increases with the length of the alternating GMblocks, very probably due to the restructuring of the junctionsinvolving GM sequences. On the basis of this evidence, theobserved stress–strain behavior can be accounted for byassuming that the crosslinks between GG sequences form at theearly stages of the gelation process and are weakly affected bothin number and strength by gel ripening, whereas additionallinks due to GM sequences increase in number and strengthwith aging, increasing the overall stiffness of the gel. Becausethe latter cross-links are weaker, they are more easily subject tobreaking under stress than pure guluronate cross-links, so that,by increasing l, the gel evolves from a state where its elasticresponse is due by both kinds of links to an intermediatepseudo-plastic state where the apparent Young's modulus isdetermined by the stiff GG bonds alone. Moreover, the separa-tion between two different elastic regimes should be muchmore pronounced for more aged gels, in agreement with theexperimental evidence (in fact, it is almost negligible at t ¼ 6 h,when the inner part of the gel has barely formed).

Such a different behavior of the two kinds of cross-links forwhat concerns their rates of formation and breaking can argu-ably be justied by the specic features of calcium binding toalginate, which takes place in three separate steps:49 Ca2+ ionsrst interact with a single guluronate unit to form ion–monomercomplexes, which then pair to generate egg-box dimers even-tually associating into bond sequences. Whereas the kinetic rateof the rst step clearly does not depend on the kind of link thatis formed, in the case of GG blocks, the following ones arehowever strongly cooperative due to the specic stereochemistrydescribed by the egg-box model. In fact long sequences of pairedGG blocks can form at a much faster rate.48 Hence, the GGbinding sites presumably get saturated as soon as the gelationfront runs into them, while the subsequent gel ripening entails aslow spatial rearrangement of the GG junctions, but not theirincrease in number and strength. Conversely the cross-linksformed by alternating GM sequences, which are known to bemuch less stereospecic,18 grow at a much slower rate,progressively strengthening a basic framework mostly made ofGG bonds. In this picture, rather large GG junctions are thenseen as stiff, permanent blocks embedded into an elasticmedium whose global elastic properties evolve however with thegel aging, because they depend on the number and strength ofGM bonds too. The latter, which break and restructure muchmore easily than the large GG cross-link sequences,18 actuallyprovide an efficient way to relax local stresses in the gel. Thepresence of this weaker, easily restructuring links, is then likelyto play a crucial role in determining the peculiar mechanicalproperties of a physical polymer gel like alginate.13

Let us briey comment on the eventual strain-hardeningbehaviour of alginate gels at very large values of l, which isactually shared by many other biopolymer gels. In the case of

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thin and exible F-actin and intermediate lament networks,strain hardening may be accounted for by a nonlinear force-extension behavior of the individual laments leading to auniform affine deformation of the structure.1,50 Conversely, theresponse of a network consisting of stiff thick bers, such ascollagen and bundled actin, seems to be governed by collectivebending deformations.51–54 Because it contains very large cross-link regions, oen exceeding the size of the chain segmentsbetween two junctions, an alginate gel may be regarded as akind of composite, simultaneously displaying the presence ofstiff and so regions.55 The non-linear elastic behavior of thiscomposite polymer network has started to be investigated onlyrecently.13 When the volume fraction of the stiff component is solow that it does not form a percolating network, strain hard-ening can be accounted for by modeling the system as a ber-reinforced elastic composite, where the cross-link regions areregarded as stiff rods randomly and isotropically distributed in asoer matrix.56–58 Such a composite model, however, does nottake into account the most peculiar property of polymer gelscharacterized by physical cross-links with a nite lifetime suchas alginate, namely, that the network topology is unceasinglyuctuating because the cross-links continuously break and re-form at different locations. The experimental consequences ofthis ceaseless network dynamics is discussed in the next section.

4.2 Stress relaxation

At a compression speed of 1 mm s�1, therefore, the gel respondsto a uniaxial compression with a nite stress, displaying anoverall behavior that, in regions I and II, resembles that of a“pseudo-plastic” solid. However, both the full decay of thecorrelation functions measured by PCI and the persistence oflarge scale restructuring bursts, witnessing that themicroscopicdynamics never stops, suggest that the gel may actually creepmacroscopically over a very long time, so that its mechanicalresponse should be regarded as truly plastic. In other words,one may wonder whether the gel would actually yield whendeformed at an innitesimally slow rate. This question is betteranswered by testing the value aer a delay s of the stress s(s)aer a given compression has been rapidly applied at s ¼ 0, forsamples kept in the gelation bath for increasingly longerperiods of time t. Here we present only the results obtained inthe limits of a very short or very long t and to a single value l ¼0.1 of the initially applied strain, which roughly correspond tothe initial part of region II in Fig. 11, where, according to ourtentative interpretation, most of the GM cross-links havealready opened up. This does not mean, however, that theweaker GM bonds should not contribute to stress relaxation, fornovel cross-links of this kind may re-form and break while thesample relaxes.

A rst clear evidence that can be extracted from the stressrelaxation measurements shown in Fig. 12 is that the stress s(s)(made dimensionless by normalizing to its initial value) alwaysdecays asymptotically to zero, even for the largest curing timet ¼ 144 h. Thus, in the long time limit, all gels yield and complyto the applied strain, which actually implies a fully plasticbehavior at vanishingly small compression rates. Hence, the

Soft Matter

Fig. 12 Main body: normalized stress s, as a function of the delay time s after acompression of 10%was rapidly applied, for samples kept in the gelation bath for6 h and 144 h. Inset: negative logarithm �ln[ss(s) � ss(N)] of the short-timerelaxations, obtained as discussed in the text, fitted with stretched exponentials inthe range 50 s# s# 500 s. The best-fit parameters are b¼ 0.47, sss¼ 115 s for t¼6 h, and b ¼ 0.49, sss ¼ 150 s for t ¼ 144 h.

§§ A better estimate of the overall decay constant of the SE relaxation is given by

its time integral hsi ¼ðN0

ds exp½�ðs=sss Þb� ¼ b�1Gðb�1Þsss ; G being the gamma

function, yielding hsi x 260 s at t ¼ 6 h and hsi x 310 s at t ¼ 144 h. The factthat hsi is more than one order of magnitude smaller than ssl conrms a clearseparation of the time scales for the short and long time relaxations.

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stress–strain relationship depends on the compression rate, sothat alginate gels behave in fact as thixotropic materials, albeittheir creeping behavior is extremely slow. The semi-log plotshows that, at long delay time s, the normalized stress displays asimple exponential decay

sðsÞ ������!t/N

slðsÞ � exp�� s=ssl

�; (5)

where the long-time stress decay constant ssl increases fromabout 2100 s for the weak gel kept in the gelation bath for just 6h, to 5600 s for the most aged gel. Quite interestingly, thesevalues are of the same order of magnitude of the microscopicrelaxation time sr of the correlation functions in aged gels. Themicroscopic stress-relaxation model we used in Section 3.3 toaccount for the time behavior of g2(s) suggests a physical pictureof the long-time creeping behavior of the gel expressed by eqn(5), possibly providing an explanation of the close similaritybetween ssl and sr. The decrease in s(s), which is in fact adecrease in the pressure felt by the compressing plate due to gelyielding, is of course related to a progressive reduction of theoverall gel strain l(t) through the stress–strain experimentalrelationship shown in Fig. 11, which deviates only moderatelyfrom linearity. If we assume that the macroscopic strain relax-ation results from the addition of a large number of uncorre-lated microscopic collapses occurring at random points in thegel, l(t) would be a Poisson process, exponentially decaying withthe same time constant of the local correlation.

At short time, however, the strain relaxation curve is signif-icantly non-exponential, displaying in particular an extremelysteep decay for s / 0, witnessing that, at short time, very fastrelaxation mechanisms contribute to s(s) too. Stress relaxationcurves are commonly tted with an arbitrary superposition ofexponentials that, however, are very hard to be attributed to

Soft Matter

specic physical mechanisms. It is conversely tempting to try togive an overall description of s(s) that, albeit approximate, maybe related to a general physical picture. As a rst guess, the longtime contribution to the stress relaxation given by eqn (5) maybe tentatively factorized out by writing

s(s) ¼ sl(s)ss(s). (6)

This amounts to assuming that the mechanism responsiblefor the long-time creeping of the gel is totally uncorrelated withthe microscopic mechanisms contributing to the fast initialdecay of s(s). This is a rather crude approximation, whichhowever yields a short-time contribution that of course does notdecay to zero, but to a nite value ss(N)x 0.4 for both curves, abehavior actually expected for a fully arrested gel. Interestingly,by subtracting from ss(s) its long-time plateau value both curvesbehaves very similarly and can be accurately tted, except at verysmall s, by a “stretched-exponential” (SE) function

ss(s) � ss(N) ¼ exp[�(s/sss)b] (7)

where sssis a characteristic time constant, and the exponent

b # 1 accounts quantitatively for the amount of “stretching”,lower values of b corresponding to a function that at short timedecay much faster than a simple exponential with the samedecay constant sss

, displaying however a much longer long-timetail. This behavior is better appreciated by plotting the (nega-tive) logarithm of ss(s) � ss(N). A pure stretched exponentialyields indeed a power-law behavior �ln[ss(s) � ss(N)] ¼ (s/sss

)b,which on a double log plot appears as a straight line with a slopeb. As shown in the inset in Fig. 12), this is actually the case fors T 30–50 s. Moreover, both curves show an almost identicalstretch exponent bx 0.5, and close values of sss

x 100–150 s.§§SE relaxations are observed frequently in a multitude of

different physical situations ranging from the microscopicdynamics of supercooled liquids and spin glasses,59,60 to theresponse of critical mixtures and polymer solutions to externalelds.61,62 One of the reasons for the ubiquity of this functionalbehavior is because a SE behavior naturally emerges as the long-time asymptotic behavior of the superposition of parallelrelaxation mechanisms that depend on a continuous parameter‘ and individually yield an exponential decay with a timeconstant s(‘), provided that the probability distribution of thesedifferent mechanisms has the very general form P(‘) � exp[�(‘/‘0)

p].63 In particular, when s(‘) scales as ‘q, the stretchexponent is given by b ¼ p/(p + q). Since in our general picturethe gel, initially subjected to an external stress, relaxes by thesequential breaking of cross-links of the alginate chains thatcan have widely different lengths ‘ and bond strengths, anoverall SE relaxation of ss(s) is not too surprising. Borrowing

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some basic concepts commonly used in describing stressrelaxation in biopolymer bundles,6,64 we assume that breaking abond of size made of ‘ individual Ca2+ links requires an acti-vation energy Ea(‘) linearly proportional to ‘ and the locallongitudinal compressional force F the bond feels,64 which isdirectly related to the macroscopically applied stress. In a rstapproximation, the time sb it takes to cross the barrier andbreak the bond is given by the product of the Boltzmann factorof Ea(‘) times the equilibrium lifetime of a single link seqb in theabsence of an external load,6 sb(‘) ¼ seqb exp(F‘/kBT). Hence,assuming that bonds of a specic size ‘ lead to a single expo-nential decay of the stress proportional to exp[�s/sb(‘)] (so thatq ¼ 1), a stretched exponential decay with b x 1/2 would resultfrom an exponentially decaying (p ¼ 1) probability distributionfor the bond size ‘ or, approximately, even if P(‘) is a Poissondistribution with a low average value h‘i. Although ratherheuristic, this simple superposition model provides some cluesfor the origin of the short-time SE behavior of the stress relax-ation in terms of local rearrangements.

5 Conclusions and perspectives

Our results show that polymer networks like those formed byalginate by the addition of a bivalent electrolyte, inducing thecooperative formation of non-covalent, reversible bonds madeby a variable number of ionic links, share many features incommon with physical colloidal gels. Specically, the geldisplays a complex aging behavior, characterized by a stronglyheterogeneous microscopic dynamics involving both localrearrangements persisting over very long time, and collectiverestructuring effects evidenced by sudden de-correlation bursts.At variance with simple colloidal gels, however, the specicionic nature of the cross-links in alginate yields a network whichis never fully arrested, always showing a fully decaying behaviorof the local time-correlation functions. Moreover, the slow gelformation induced by the perfusion of the curing agent displaysa peculiar non-diffusive advancement of the gelation front,which might possibly be due to micro-convective effects, fol-lowed by a slow reinforcement of the network, ascribable to theformation of novel bonds and the restructuring of the existingones. A basic investigation of the macroscopic mechanicalresponse of the gel, both as a function of the applied strain andas a function of time following an initial rapid deformation,supports the basic picture derived from the investigation of themicroscopic dynamics. In particular, the slow creeping behaviorof the gel observed aer a rst rapid stress relaxation, which canbe attributed to the concurrence of many superpositions of localrelaxation modes yielding an overall SE decay, is consistent withthe persistence of a local microscopic diffusive dynamics overlong times and with the occurrence of intermittent globalrearrangements.

The simultaneous presence of stronger, stereospecicbonds, provided according to the egg-box model by the GGsequences, with the weaker links, which breaks and re-formsmuch more easily, provided by the alternate GM sequences,seems to be crucial to induce the long-time creeping behavior ofalginate gels. A detailed analysis of the distinct role played by

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these two kinds of links and their specic contribution to thedynamics and macroscopic rheology of alginate gels would bebetter performed by investigating synthetic alginates, where theamount and distribution of the GG and GM sequences can becarefully controlled, as described in the seminal study made byMørch et al.48 Nonetheless, a basic test of their different rolescan also be performed by gelling alginate in the presence of aconsistent amount of a monovalent, neutral electrolyte likeNaCl. Whereas the presence of such an additive would notinuence too much the strongly stereospecic bounds providedby the GG sequences, unless at very high ionic strength, the Na+

cations would indeed favorably compete with calcium inbinding to the GM sites, strongly hindering their propensity toform cross-links.18 Several experimental investigations of themacroscopic properties of alginate gels in the presence ofneutral electrolytes seem to support this view.36,65 Preliminarygelation experiments actually show that, in the presence of anamount of NaCl as low as 0.1 M, the microscopic dynamics isdrastically modied. In particular, the gel gets fully arrested, therestructuring kinetics also changes. Consistently, the macro-scopic creeping of the gel is totally suppressed. A full analysis ofelectrostatic effects on the microscopic dynamics of alginategels will be the subject of forthcoming publications.

References

1 C. Storm, J. Pastore, F. MacKintosh, T. Lubensky andP. Janmey, Nature, 2005, 435, 191–194.

2 Y. Fung, Biomechanics: mechanical properties of living tissues,Springer, 1993, vol. 12.

3 O. Chaudhuri, S. Parekh and D. Fletcher, Nature, 2007, 445,295–298.

4 X. Trepat, L. Deng, S. An, D. Navajas, D. Tschumperlin,W. Gerthoffer, J. Butler and J. Fredberg, Nature, 2007, 447,592–595.

5 O. Lieleg, M. Claessens and A. Bausch, So Matter, 2010, 6,218–225.

6 O. Lieleg, J. Kayser, G. Brambilla, L. Cipelletti and A. Bausch,Nat. Mater., 2011, 10, 236–242.

7 A. Bausch and K. Kroy, Nat. Phys., 2006, 2, 231–238.8 C. Semmrich, T. Storz, J. Glaser, R. Merkel, A. Bausch andK. Kroy, Proc. Natl. Acad. Sci. U. S. A., 2007, 104, 20199.

9 B. Kim, J. Nikolovski, J. Bonadio and D. Mooney, Nat.Biotechnol., 1999, 17, 979–983.

10 W. Pilnik and F. Rombouts, Carbohydr. Res., 1985, 142,93–105.

11 A. D. Augst, H. J. Kong and D. Mooney, Macromol. Biosci.,2006, 6, 623.

12 O. Smidsrød, J. Chem. Soc., Faraday Trans., 1974, 57, 263.13 J. Sun, X. Zhao, W. Illeperuma, O. Chaudhuri, K. Oh,

D. Mooney, J. Vlassak and Z. Suo, Nature, 2012, 489, 133–136.14 E. Morris, D. Rees, D. Thom and J. Boyd, Carbohydr. Res.,

1978, 66, 145–154.15 K. Potter, B. Balcom, T. Carpenter and L. Hall, Carbohydr.

Res., 1994, 257, 117–126.16 I. Donati, S. Holtan, Y. Mørch, M. Borgogna, M. Dentini and

G. Skjak-Bræk, Biomacromolecules, 2005, 6, 1031–1040.

Soft Matter

Soft Matter Paper

Dow

nloa

ded

by E

TH

-Zur

ich

on 1

9 M

arch

201

3Pu

blis

hed

on 2

7 Fe

brua

ry 2

013

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C3S

M27

153F

View Article Online

17 Y. Zhao, F. Hu, J. Evans and M. Harris, Chem. Eng. Sci., 2011,66, 848–858.

18 I. Donati, Y. Mørch, B. Strand, G. Skjak-Bræk and S. Paoletti,J. Phys. Chem. B, 2009, 113, 12916–12922.

19 M. Mancini, M. Moresi and R. Rancini, J. Food Eng., 1999, 39,369–378.

20 L. Wolff, P. Fernandez and K. Kroy, New J. Phys., 2010, 12,053024.

21 K. Lee, M. Peters, K. Anderson and D. Mooney, Nature, 2000,408, 998–1000.

22 G. Brambilla, S. Buzzaccaro, R. Piazza, L. Berthier andL. Cipelletti, Phys. Rev. Lett., 2011, 106, 118302.

23 H. Grasdalen, Carbohydr. Res., 1983, 118, 255–260.24 L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta and S. Mazoyer,

J. Phys.: Condens. Matter, 2003, 15, S257.25 S. Maccarrone, G. Brambilla, O. Pravaz, A. Duri, M. Ciccotti,

J. Fromental, E. Pashkovski, A. Lips, D. Sessoms, V. Trappe,et al., So Matter, 2010, 6, 5514–5522.

26 L. Berthier, G. Biroli, J. Bouchaud, L. Cipelletti, D. El Masri,D. L'Hote, F. Ladieu and M. Pierno, Science, 2005, 310, 1797–1800.

27 P. Tokumaru and P. Dimotakis, Exp. Fluids, 1995, 19, 1–15.28 A. Duri, H. Bissig, V. Trappe and L. Cipelletti, Phys. Rev. E:

Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2005, 72,051401.

29 A. Mikkelsen and A. Elgsaeter, Biopolymers, 1995, 36, 17–41.30 R. Liesegang, Naturwiss. Wochenschr., 1896, 11, 353.31 T. Witten Jr and L. Sander, Phys. Rev. Lett., 1981, 47, 1400–

1403.32 G. Barkema, M. Howard and J. Cardy, Phys. Rev. E: Stat. Phys.,

Plasmas, Fluids, Relat. Interdiscip. Top., 1996, 53, 2017–2020.33 L. Gal and Z. Racz, Phys. Rev. A, 1988, 38, 3151.34 S. Cornell, Z. Koza and M. Droz, Phys. Rev. E: Stat. Phys.,

Plasmas, Fluids, Relat. Interdiscip. Top., 1995, 52, 3500.35 Z. Koza, J. Stat. Phys., 1996, 85, 179–191.36 B. Thu, O. Gaserød, D. Paus, A. Mikkelsen, G. Skjak-Bræk,

R. Toffanin, F. Vittur and R. Rizzo, Biopolymers, 2000, 53,60–71.

37 T. Braschler, A. Valero, L. Colella, K. Pataky, J. Brugger andP. Renaud, Anal. Chem., 2011, 83, 2234–2242.

38 J. Thumbs and H. Kohler, Chem. Phys., 1996, 208, 9–24.39 R. Piazza, S. Buzzaccaro and E. Secchi, J. Phys.: Condens.

Matter, 2012, 24, 284109.40 A. Kiselev and L. Ryzhik, Ann. Henri Poincare C, 2001, 18,

309–358.41 J. Klein, J. Stock and K. Vorlop, Eur. J. Appl. Microbiol.

Biotechnol., 1983, 18, 86–91.

Soft Matter

42 K. A. Strand, A. Bere, P. S. Dalberg, T. Sikkeland andO. Smidsrød, Macromolecules, 1982, 15, 570–579.

43 L. Cipelletti, S. Manley, R. C. Ball and D. A. Weitz, Phys. Rev.Lett., 2000, 84, 2275–2278.

44 J. Bouchaud and E. Pitard, Eur. Phys. J. E, 2001, 6, 231–236.45 A. Duri and L. Cipelletti, Europhys. Lett., 2006, 76, 972–978.46 H. Bissig, S. Romer, L. Cipelletti, V. Trappe and

P. Schurtenberger, PhysChemComm, 2003, 6, 21–23.47 A. Duri, D. Sessoms, V. Trappe and L. Cipelletti, Phys. Rev.

Lett., 2009, 102, 085702.48 Y. Mørch, S. Holtan, I. Donati, B. Strand and G. Skjak-Bræk,

Biomacromolecules, 2008, 9, 2360–2368.49 Y. Fang, S. Al-Assaf, G. Phillips, K. Nishinari, T. Funami,

P. Williams and L. Li, J. Phys. Chem. B, 2007, 111, 2456–2462.

50 M. Gardel, J. Shin, F. MacKintosh, L. Mahadevan,P. Matsudaira and D. Weitz, Science, 2004, 304, 1301–1305.

51 O. Lieleg, M. Claessens, C. Heussinger, E. Frey andA. Bausch, Phys. Rev. Lett., 2007, 99, 88102.

52 D. Chen, Q. Wen, P. Janmey, J. Crocker and A. Yodh, Annu.Rev. Condens. Matter Phys., 2010, 1, 301–322.

53 M. Sheinman, C. Broedersz and F. MacKintosh, Phys. Rev. E:Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2012, 85,021801.

54 P. Onck, T. Koeman, T. van Dillen and E. van der Giessen,Phys. Rev. Lett., 2005, 95, 178102.

55 J. Zhang, C. Daubert and E. Allen Foegeding, J. Food Eng.,2007, 80, 157–165.

56 E. Huisman, C. Heussinger, C. Storm and G. Barkema, Phys.Rev. Lett., 2010, 105, 118101.

57 C. Broedersz, C. Storm and F. MacKintosh, Phys. Rev. Lett.,2008, 101, 118103.

58 M. Das and F. MacKintosh, Phys. Rev. Lett., 2010, 105,138102.

59 W. Gotze and L. Sjogren, Rep. Prog. Phys., 1992, 55, 241.60 B. Castaing and J. Souletie, J. Phys., 1991, 1, 403.61 R. Piazza, T. Bellini, V. Degiorgio, R. Goldstein, S. Leibler

and R. Lipowsky, Phys. Rev. B: Condens. Matter, 1988, 38,7223.

62 V. Degiorgio, R. Piazza, F. Bellini, T. Mantegazza andR. E. Goldstein, Phys. Rev. Lett., 1990, 64, 1043.

63 M. A. Continentino and A. P. Malozemoff, Phys. Rev. B:Condens. Matter, 1986, 33, 3591.

64 R. Vink and C. Heussinger, J. Chem. Phys., 2012, 136, 5102.65 M. LeRoux, F. Guilak and L. Setton, J. Biomed. Mater. Res.,

1999, 47, 46–53.

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