Cooperative Effect of Stress and Ion Displacement on the Dynamics of

Biomacromolecules 2010, 11, 1571–1578

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Cooperative Effect of Stress and Ion Displacement on the Dynamics of Cross-Link Unzipping and Rupture of Alginate Gels T. Baumberger* and O. Ronsin INSP, UPMC Univ Paris 06, CNRS UMR 7588 140 rue de Lourmel, 75015, Paris, France Received February 22, 2010; Revised Manuscript Received May 3, 2010

We study the effect of nonbinding Na+ ions on the kinetics of rupture of alginate gels cross-linked by Ca2+. Wetting a crack tip with a saline solution at physiological concentrations is found to be able to induce a quasiinstantaneous, 10-fold velocity jump. This effect is analyzed with a phenomenological model for the rate-dependent fracture energy in physical gels, extended here to account for the role of ions on the rate of cross-link “unzipping”. Ionic interaction is found to act cooperatively with mechanical tension, leading to an enhanced rate of rupture. The kinetics turns out to be second order in counterion concentration. The definition of the reference state requires to take into account counterion condensation due to long-range interactions in the polyelectrolyte gel. Surprisingly, the contribution of the Na+ ions to the free energy of the activated state is essentially entropic, suggesting that the displacement of Ca2+ is primarily a steric process, electrostatic interactions being reduced to the constraint of charge conservation. This phenomenon may have important consequences on the rate of degradation of alginate based scaffolds for in vivo tissue regeneration.

Introduction An exciting and revolutionary approach to reconstructive medicine utilizes self-assembling biopolymers as temporary scaffolds for the delivery of cells right at the repair site.1-4 The design of the proper three-dimensional matrix is a critical step toward the achievement of an effective in vivo tissue or organ engineering. The task assigned to these biomaterials goes far beyond acting as a mere structural material. Certainly, an ideal scaffold must in the first place retain sufficient mechanical integrity to ensure cell immobilization under load-bearing conditions,3 but it is also crucial that the matrix mimics the elastic response of its natural environment to permit the cells to retain their native phenotype and produce their extracellular components.5 Moreover, it has been shown that optimum tissue quality is obtained when the polymer network degrades itself at a rate that matches that of tissue regeneration.4,6 Obviously, the numerous requirements regarding processing and injectability, mechanical compliance and strength, biocompatibility, and biodegradability of the scaffolds are strongly interrelated. They may even appear as antagonistic, considering for instance that strength and degradability of a biomaterial are most likely to compete. This pleads for investigating systematically the basic, physicochemical mechanisms at work during the whole life of a scaffold and their interplay. However, with respect to the plethora of always more complex scaffold candidates that are engineered at an ever increasing pace,1 such a fundamental approach remains marginal. Yet, some important progress has been made along these lines, using alginate gels as model systems.7-9,11 Alginates are polysaccharides derived from brown algae.12 They are linear binary copolymers of 1-4-linked β-D-mannuronic acid (M) and R-L-guluronic acid (G) organized along sequences, depending on the biopolymer origin. Homogeneous subsequences made of G-G residues promote interchain binding * To whom correspondence should be addressed. E-mail: tristan@ insp.jussieu.fr.

mediated by divalent cations, for example, Ca2+, resulting in the formation of gel networks cross-linked via extended “eggbox” structures.13,14 During the past decade, alginate-based biomaterials has been intensively studied for their potential applications in tissue, mostly cartilage, engineering due to their structural similarity to the natural extracellular matrix and because of their biocompatibility and low toxicity.4 In this respect, noncovalent, ionic binding is certainly advantageous because the toxicity and nondegradability of chemical crosslinkers is always an issue. As compared to chemical gels or elastomers, the study of the mechanical strength i.e. the resistance to rupture of physical gels is still in its infancy. Strong qualitative differences are observed, in particular in the case of alginate, which can be cross-linked either chemically or physically.9,10 An important characteristic of alginate gels is their rate-dependent fracture energy, a property shared by other physical gels, such as gelatin ones.15,16 More precisely, the dissipation of energy accompanying the propagation of a crack in the gel increases markedly with the crack tip velocity. This has been ascribed to the noncovalent nature of the cross-links, which act as mechanical fuses and prevent C-C bond rupture along the polymer backbone. Fracturing therefore proceeds via “unzipping” of the cross-link zones and pulling the subsequently free chains out of the gel matrix, at the expense of the viscous drag against the solvent. Both the plastic unzipping and the viscous pulling-out are intrinsically rate-strengthening mechanism. This means that, although not suitable for severe, static, load-bearing applications, alginate gels could well sustain transient, dynamic loading. Resistance to mechanical stresses is only one side of the issue raised by the need for controlling the structural integrity of the gel scaffold. Another one is the degradability of the polymer network in its physiological environment.4 In the case of ionically cross-linked alginates, there is a consensus about the crucial role played by nonbinding ions such as Na+ and Mg2+, which can “exchange” with the binding Ca2+ minority ions.6,7,17 This degradation process in vivo, unusual among biopolymer

10.1021/bm1002015  2010 American Chemical Society Published on Web 05/25/2010

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gels,4 is again a scissionless mechanism which, unlike for example, enzymatic hydrolysis, leaves the chains intact but dismantles the network structure. This manifests itself by a slow decrease of the elastic stiffness and an enhanced swelling upon immersion of a gel sample in a saline bath.7,18 The slow and poorly controllable rate at which degradation proceeds in macroscopic samples17 can be ascribed to diffusion and exchange of ions, both slow, coupled processes. Therefore, assessing the sole kinetics of ion exchange in alginate gels is a difficult task, which has discouraged attempts since the careful study of Mongar and Wassermann,19 60 years ago, when these authors have concluded to a third order kinetics where a bound Ca2+ is exchanged against two nonbinding Na+. This simple, charge-conserving scenario has never been reassessed with respect to more recent advances in the understanding of polyelectrolyte gels,20 which put an emphasis on the role played by long-range electrostatic interactions between the densely charged polyanionic network and its counterions, leading to the “condensation” of a significant fraction of the latter.21 The present study enables us to address these questions through an apparently even more complex issue, namely the interplay between stress- and ion-induced cross-link unzipping. The basic physical idea underlying this study is that when a cross-link zone is on the verge of being unzipped, the free energy of the binding ions is strongly reduced.11 This in turn must favor ion displacement and lead to a tremendous enhancement of the exchange rate. Because the scaffold material is, by construction, subject to stresses induced by the growing tissues, in some cases large enough to initiate cracks,22 it is important in practice to consider mechanical and chemical deassembling as a whole. To disconnect crack nucleation from its subsequent propagation, we dealt with steadily propagating cracks that were previously initiated in alginate gels. As will be described in details, we observed a strong, quasi-instantaneous speeding-up of the crack in response to an environmental shock, consisting in the stepwise input of sodium ions at a physiological concentration in the crack tip. We will argue that sodium diffusion is not a seriously limiting process here and that the whole kinetic effect can be attributed to ion displacement. Hence, this unusual configuration is actually a powerful and straightforward tool for studying ion displacement kinetics. We will analyze our results within the framework of a minimal model for the rate-dependent fracture energy, which accounts for cross-link unzipping, incorporating thermal activation effects and viscous dissipation associated to chain pull-out.15 This phenomenological model, the main features of which are recalled in the next section, has been previously validated on alginate gels.11 Here, it is readily extended in order to account for the contribution of sodium ions to the activation free energy of the binding ions.

Theoretical Considerations Fracturing a solid material requires that the remotely applied stress is suitably amplified in order to reach a local value large enough to “break” bonds. In nominally intact samples, this is provided by defects or flaws. The tip of a man-made notch is also a suitable stress-amplifying locus from which a crack may propagate. Propagation itself requires that enough energy can flow to the tip region where it is ultimately dissipated.23,24 In the case relevant to the present study, namely, that of a quasistatic crack propagation where inertial effects are negligible, the amount of elastic energy relaxed in the sample by a unit advance of the crack must equal the amount of energy dissipated

Baumberger and Ronsin

at the tip. The latter is termed fracture energy G expressed by unit crack area. It is generally rate-dependent, that is, it is a function of the crack tip velocity V. In a typical experiment G is imposed by the remote stress level. For suitable setup geometries, it can be fixed at a prescribed value, uniformly over the sample. The crack velocity V then adjusts itself to fulfill the energy balance criterion. As long as the “process zone” where energy is dissipated remains much smaller than the sample size, the fracture energy can be considered as an intrinsic characteristics of the dissipative process itself, whatever the loading geometry. The G(V) characteristics is therefore a sensitive probe of the underlying physical mechanisms. Usually, however, the resistance to rupture of a hydrogel is quantified by the maximal stress or strain a nominally intact sample with a given geometry can sustain before catastrophic breaking or crushing occurs.25-28 This is certainly useful for the sake of comparison between different materials and for assessing the “ultimate” characteristics of a given load-bearing sample. It is, however, not suitable for physical modeling because the remote, macroscopic stress does no account for stress amplification and is therefore not characteristic of the local “bond-breaking” mechanism. Even if the total work-to-failure is computed,18,28 because the actual crack path is usually not well-known, the fracture energy cannot be defined unambiguously. Moreover, macroscopic failure results from crack nucleation and its subsequent growth, and it is difficult to disentangle both ratedependent processes.29,30 Accordingly, a significant G(V) expression can only be determined using an experimental geometry in which the crack path and the crack velocity are unambiguously defined and where steady-state conditions make crack nucleation irrelevant. Systematic experimental studies along these lines have been carried out in our group.11,15,16 On this basis, we have proposed a minimal theoretical model for scissionless fracture of a wide class of physical hydrogels, including alginate and gelatin ones. The ability of the model to provide a predictable framework for describing complex dynamical situations has been recently demonstrated.31 Hereafter we recall its main constitutive hypotheses (we explicitly refer to the “egg-box” model for calcium bounded alginate gels, although our model does not depend on the details of the physical self-assembling mechanism leading to crosslinking). (i) Advancing the fracture plane by a unit area requires pulling Σ0 chains out at a velocity ϑ with respect to the gel matrix. ϑ is proportional to the crack tip velocity V: ϑ ) RV, with R a geometric parameter indicating the degree of “blunting” of the crack tip at its very end. (ii) The stress level required for deassembling the cross-links is such that in the tip vicinity the network must be stretched taut perpendicularly to the fracture plane. (iii) The pull-out velocity is controlled by the average rate ν(f0) of unzipping of the most remote cross-link near a chain end that plays the role of ultimate mechanical fuse. By “unzipping” we mean the progressive cleavage of the “eggbox” zone through sequential opening of the chelating units. Because the chains are pulled-out taut, they move rigidly at the velocity ϑ ) aν, with a = 0.9 nm, the length of a G-G block. Here, ν-1 is the average time required for opening a chelating unit. (iv) Due to the viscous drag against the solvent of viscosity η, the mechanical tension along a single chain decreases from its (average) value ftip at the tip down to its minimal or “ground” level f0 near its ends. Assuming a linear tension drop: f0 ) ftip - βΛηϑ, with Λ the contour length of the chain and β a geometric factor accounting for the details of the chain/solvent

Dynamics of Unzipping and Rupture of Alginate Gels


flow. (v) A chain cannot be pulled out unless f0 is very close to the threshold value U/a = 5 × 10-11 N, with U = 12kBT, the binding energy of a chelated Ca2+ ion.11 These chains, which will ultimately dissipate energy, define the process zone. We assume implicitly that no significant dissipation occurs in the bulk of the gel as stresses build up. Indeed, it has been found15 that dissipation is strongly localized in the tip region, over a depth d j 100 nm. This order of magnitude results from an experiment performed on gelatin gels where viscous dissipation can be modified locally through diffusion of glycerol between the gel and a drop of solvent that wets the tip. The diffusion skin depth being inversely proportional to the crack velocity, the depth of the process zone can be estimated from the velocity above which the effect of the wetting drop fades out. Due to the topological similarity of gelatin and alginate gels, it will be assumed that the sizes of their process zones are of the same order of magnitude. (vi) Egg-box unzipping is a thermally activated process over the barrier U corresponding to the activated state of a Ca2+ ion released from its chelating box. This barrier is biased by the mechanical tension sustained by the cross-link, which favors unzipping:32 ν ) ν0 exp[-(U f0a)/kBT]. Tension is supposed to be large enough for rebinding events to be irrelevant. This does not attempt to account for any cooperativity of adjacent boxes in the unzipping process. (vii) The local tension of the chains bridging the ultimate gap in the tip opening is related to the fracture energy through an expression derived by Dugdale24 assuming a cohesive zone at the crack tip sustaining a constant (plastic) stress, here σtip ) ftipΣ0, until full decohesion, corresponding to a maximal tip opening δmax: G ) σtipδmax. Here, δmax = Λ the average chain contour length, because the chains must be stretched taut for the stress level required by cross-link unzipping to be reached. (viii) Finally, within the framework of the present study, we assume that the interaction between the binding Ca2+ and the nonbinding counterions Na+ can be taken into account via a contribution to the free energy of the activated state, namely, U ) U0 -∆U(x), with x as the counterion concentration in the process zone.

ν ) ν0e∆U(x)/kBTe-[(U-f0a)/kBT]

(1)

These minimal assumptions define a closed set of equations which yield an expression for the rate-dependent fracture energy G(V). The structure of the gel is lumped into a small number of parameters (Σ0, a, Λ, U0, ν0) and the whole complexity of the actual stress field ahead of the tip is incorporated into R. The set of equation can be solved for G. It exhibits two contributions, one linear in V corresponding to viscous dissipation and dominant at high velocities, one logarithmic reminiscent of premature unzipping events due to thermal activation. This is qualitatively what is observed experimentally. A more quantitative discussion on the basis of reasonable values for the model parameters can be found in ref 11. For the present study, it is convenient to express G(V, x) with respect to a reference state (V0, x0). With ∆U(x0) ) 0, the fracture energy reads

(( )

G(V, x) ) G(V0, x0) + C ln

V - V0 V ∆U + V0 kBT V*

)

(2)

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Figure 1. Multiscale schematic representation of an alginate gel in the crack-tip region. (a) Sample-scale showing the crack-opening. (b) Tip scale showing the alginate chains at different stages of the pulling-out process. (c) Cross-link scale showing the “egg-box” structure. (d) Binding-unit size, showing a Ca2+ ion chelated by a GG-GG unit. Mechanical tensions ftip and f0 are defined in the text. Reprinted with permission from Baumberger, T.; Ronsin, O. J. Chem. Phys. 2009, 130, 061102. Copyright 2009, American Institute of Physics.

where C and V* are independent of the reference state and can be readily measured. The velocity V* marks the crossover between a thermally activated and a viscous dominated crack dynamics.

Experimental Section Materials and Methods. Hydrogel Formation. Sodium alginate isolated from Lessenia nigrescens was provided by Kalys S.A. (Grenoble, France) and used as purchased. The nominal structural j ) 216 kDa; fraction of characteristics are: average molar mass M mannuronic acid: FM ) 0.55. Gel samples were prepared as described previously.31,33 Alginate powder was thoroughly dissolved in deionized water and blended with calcium carbonate powder made of 50% standard micro particles (Prolabo, France) and 50% nanoparticles, average size 90 nm (American Elements, Los Angeles, CA). Freshly prepared solution of slowly hydrolyzing D-glucono-δ-lactone (GDL from Sigma) was quickly incorporated into the solution ([GDL]/[CaCO3] ) 2) just prior to pouring it into a mold. The final, nominal composition of 100 mL of pregel was 1.5 g of alginate, 1.5 × 10-3 mol of Ca2+ (15 mM), and 7.6 × 10-3 mol of Na+ (76 mM). The latter is the concentration Cp of counterions arising from the sample itself, that is, without added sodium salt. The mold consists of a rectangular metal frame sandwiched between two plates covered with Mylar films. The longest sides of the frame are rigid bars designed to provide a suitable grip for stretching the gel plate. For that purpose, their inner faces are covered with the curly parts of adhesive Velcro tapes which are impregnated by the pregel solution and remain subsequently embedded into the gel itself. Pregel solutions were cured for 15 h at room temperature (=20 °C) prior to measurements. Mechanical Testing. After gelling was completed, the mold was clamped to the mechanical testing setup (see Figure 2); the removable pieces were then taken off, leaving the L × h0 × e (300 × 30 × 10 mm3) gel plate firmly fixed to its grips, ready to be stretched. The Mylar films were left in position to prevent solvent evaporation and peeled off just before running the experiments. A crack was initiated by notching one edge of the sample with a sharp blade. The plate was then stretched by translating one grip bar with a precision motorized stage. The new width of the plate was then h ) h0 + ∆h, with ∆h ranging between 3 and 7 mm. For these values, a crack developed from the notch and propagated steadily along the midplane of the plate. The tip positions were tracked and recorded with a video camera. A subsequent treatment yielded the crack velocity V.

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Figure 2. Experimental setup showing a nonwetted crack. The length of the sample, L ) 300 mm, gives the scale. The crack is made to propagate downward vertically so as to be easily wetted by a drop of saline solution.

Figure 3. Response of a crack to a drop of saline solution. The crack velocity V ) dx/dt is calculated by numerical derivation of the filtered x(t).

Thanks to the aspect ratio of the samples (L . h), G remains constant along the crack path. This fails only on approaching the plate free edges. Measurements were carefully restricted to the homogeneous zone, where G is a function of ∆h solely. This was determined in a preliminary experiment by stretching an unnotched sample while measuring the tensile force F through a load cell attached to one grip. The area below the F(∆h) curve yielded the stored elastic energy W from which the fracture energy was estimated according to G(∆h) ) W/(Le). “Salted” crack experiments were performed by setting a drop (=50 µL) of a NaCl solution (concentration c ) 50 mM-1 M) into the crack opening. The drop remained stuck to the crack tip due to both gravity (the crack was running downward) and capillarity. In such “salt limited” experiments, no further input of salt was provided. We also performed alternative “steady flow” experiments in which the saline solution was poured steadily into the crack opening to have the crack tip constantly soaked with fresh solution. In all experiments, crack velocities were measured before and after salting while keeping ∆h constant, that is, we characterized the response to a local environmental perturbation under the constraint of constant fracture energy G.

Results The effect on the crack velocity of a drop of 200 mM NaCl solution at an opening ∆h ) 5 mm is shown in Figure 3. The main feature is a quasi-instantaneous speeding-up of the crack. More precisely, V is doubled on a time scale that cannot be resolved with a video frame grabbing period of 100 ms. This is followed by a slow deceleration of the crack which usually ends up with the vanishing of the drop as all the solution has been spread over the wettable crack surfaces.


Figure 4. Fracture energy G vs crack velocity V for a dry crack (squares) and a crack steadily wetted by a flow of NaCl solution, 200 mM (circles). The lowest value of G corresponds to a crack opening ∆h ) 3 mm. Successive data correspond to a stepwise increase of the opening by 0.5 mm. The upper curve is a fit according to G ) G0 + C[ln V + V/V*] (see text). The characteristic velocity V* marks the crossover between thermally activated and viscous dominated crack propagation. The lower curve is a mere vertical translation of the upper one by an amount ∆G.

Apart from the decrease of the drop volume, it is obvious that the amount of salt in the drop decreases steadily as salt diffuses into the gel. To circumvent this effect, we have performed steady flow experiments with a 200 mM saline solution at different crack openings. The velocity remained constant after the initial positive step. The results are displayed in Figure 4. For the sake of comparison we have plotted the steady state velocities measured for both the unwetted and the salted cracks at different levels of fracture energy. The unwetted crack characteristic can be fitted with the expression of eq 2, yielding a crossover velocity V* ) 1.3 mm · s-1. The speeding-up effect of a saline environment evidenced on the salt-limited experiment is confirmed here at all openings. Moreover, the data strongly suggest that, within experimental errors, the salted crack G(V) characteristic can be obtained from the unwetted crack one by a mere translation along the energy axis by constant amount ∆G. Though preferable in principle, the steady flow experiments are not free of any potential artifact due, for instance, to the dripping of saline solution along the lateral surfaces, which can affect remote regions of the gel by diffusion well before the crack reaches them. We have therefore chosen to exploit salt limited experiments systematically. We define the reference velocity V0 of the crack as measured in its unwetted state. The velocity V is determined just following wetting. The data are gathered in Figure 5. It is found that the velocity jump amplitude increases nonlinearly with the salt concentration and is already important for a physiological environment (c = 150 mM). In an attempt to compare data for different crack openings (hence, different unwetted crack velocity levels) we have plotted in Figure 6 the ratios V(c)/V0 as a function of c. The values of V0 being measured just before wetting (therefore accounting for eventual small inhomogeneities along the crack path). There remains an explicit dependence on V0, namely, for a given c, the lower V0, the higher the relative jump. Interestingly, the data for the smallest opening is markedly detached from the other ones. Note that the corresponding dry crack dynamics is essentially thermally activated (V0 = 200 µm · s-1 , V*). A 10-fold velocity jump is easily obtained by salting the tip of such a sluggish crack.



Figure 5. Crack velocities, measured just after wetting the tip by a drop of saline solution of concentration c. ∆h ) 3 (b), 4 ((), 5 (9), 6 (1), 7 mm (2), the corresponding G values can be read in Figure 4. Arrows indicate the unwetted crack velocities V0.

Figure 6. Same data as Figure 5 divided by the value of the dry crack V0, as measured just before wetting. The curves are calculated according to to eq 5 (see text).

Analysis Our purpose in this section is to make a heuristic use of eq 2 to transform and analyze the results displayed in Figures 4 and 5 in a way that is suitable for a physical discussion. We first notice that ∆G as defined in Figure 4 reads C∆U/kBT. Thus, within the theoretical framework, the existence of ∆G is a mere macroscopic manifestation of the shift ∆U of the activation barrier for cross-link unzipping. We have to keep in mind, however, that ∆U depends on x, which is the concentration of counterions into the process zone, while c is the concentration of the outer saline solution. A crucial task, which we postpone to the Discussion section, will be to establish a relation between x and c, namely, to justify that, for most of the data reported here, x = c. At the present stage, let us consider a data set V(c) corresponding to a fixed crack opening, hence, to a given value of the fracture energy. This includes the velocity V0 measured on the unwetted crack, that is, in the reference state characterized by x0 corresponding to the stoichiometric concentration [Na+ ] = 76 mM. Equating G(V, c) and G(V0, x0) yields

( )

V - V0 ∆U V + ) ln kBT V0 V*

(3)

Because V* is known, the V(c) data can readily be transformed into the corresponding ∆U(c), as displayed in Figure 7. First of all, it appears that the global scattering of the resulting data is not larger than the scattering of the individual sets,

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Figure 7. Calculated shift ∆U (in kBT units) of the activation free energy for unzipping (see text). The dashed line has a slope 2. Same data as Figure 5.

Figure 8. Best fit according to eq 5 of data c(V) corresponding to ∆h ) 5 mm.

confirming that within the reproducibility level of our experiments, the tip environment effect can legitimately be ascribed to the existence of a shift ∆U(c). Moreover, the semilogarithmic representation of Figure 7 suggests that

∆U = kBT ln(c/c0)β

(4)

with β = 2 and c0 is a constant commensurate with (though smaller than) the stoichiometric concentration [Na+ ] in the gel bulk. This can be tested in more detail on considering a single set, selected for the small scattering of its data. The above expression with β ) 2 plugged into eq 3 predicts that V would be given implicitly by

[ (

c ) c0

V - V0 V exp V0 V*

)]

1/2

(5)

The result of a best fit of c(V) for ∆h ) 5 mm (G ) 3 J · m-2) according to the above expression is displayed in Figure 8. When V0, V* and c0 are left as free parameters, the agreement is remarkable. Fitting values for V0 (3.1 mm · s-1) and V* (1.17 mm · s-1) do not differ significantly from their experimental ones. An important physical output to be discussed later is c0 ) 37 mM. This extrapolated value appears as the concentration of a wetting solution that would mimic an unwetted tip, that is, corresponds to ∆U ) 0. Finally, we compare in Figure 6 the relative jump data with the predictions of eq 5, using the value of c0 determined above.

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As already indicated in the previous section, and shown in Figure 6, the curves become very sensitive to the value of the parameter V0 when it is smaller than V*. This can be understood in view of eq 5 rewritten for the relative jump V˜ ) V/V0

c ) c0√V˜ exp

[

]

V0 (V˜ - 1) 2V*

(6)

Indeed, finite values of c/c0 are incompatible with the large jumps induced by the exponential divergence of the right-hand term unless mitigated by a coefficient V0/V* , 1. It is worth noting that the fitting expression eq 6 has no free parameter because c0, V*, and the average value of V0 for each crack opening are known. As can be seen in Figure 6, the overall agreement is very satisfactory and gives confidence in the previous analysis.

Discussion The functional form taken in eq 4 by the ion-induced shift ∆U of the activation free energy responsible for cross-link unzipping is strongly suggestive of an entropic origin. If, anticipating on the following discussion, we admit that in our experiments the salt concentration c in the drop equals the concentration x of mobile Na+ counterions in the process zone of the tip, the opening rate (eq 1) of a load-bearing cross-link in this zone would read

ν ) ν0

() [ x x0

2

exp -

U0 - fa kBT

]

(7)

where x0 stems from the counterion concentration in the reference state of the gel. On setting eq 7, we have forced exponent β to its nearest integer value β ) 2. This choice, fully compatible with the previous analysis, is dictated by the picture of a charge-conserving “ion exchange” process between a Ca2+ and two Na+. This picture is unexpectedly simple in view of the apparent complexity of the ionic configuration at the crack tip, if only because of the strongly charged polyelectrolyte nature of alginate gels. In the following we address some qualitative aspects of this issue. In the first place we have to deal with the process of ion diffusion at the moving liquid/gel interface. Manning Condensation and Donnan Equilibrium. The static equilibrium between a saline solution (here the drop) and a polyelectrolyte (here the gel), both sharing the same counterion (here Na+), is an instance of a Donnan equilibrium: it is primarily ruled by the fact that, unlike the co-ion Cl-, the negatively charged polyelectrolyte network cannot be displaced. Hence, electroneutrality of the drop forces both salt ions to diffuse in concert. As a result, the concentration of counterions (respectively, co-ions) at equilibrium is larger (respectively, smaller) in the gel than in the drop. A further complication arises from the fact that long-range electrostatic interactions in the presence of large enough quasi-linear charge distributions (the alginate chains) lead to localization of the counterions along the chains, a phenomenon often referred to as Manning condensation. One has therefore to distinguish between the free ions, which can diffuse, and the condensed ones, which can only move along the chains. Counterion condensation is controlled by the dimensionless parameter ξe defined as21

ξe )

lB bj

(8)

with bj representing the average distance between charges along a chain and lB ) e2/εkBT representing the Bjerrum length (lB ) 7.14 Å for water at 20 °C). For quasi-1D charge distributions, ξe ) 1 is a critical value above which condensation of monovalent counterions should occur. Considering lengths of 4.35 and 5.17 Å for guluronic and mannuronic acid repeating units, one computes the average length bj = 4.8 Å, hence, ξe = 1.5. According to Manning’s simple picture, counterions will condense so as to reduce the average charge density down to its critical value (ξe ) 1). Binding ions must be treated separately.20 Due to the stoichiometry of the ionic binding and to the splicing geometry of the egg-box structure (see Figure 1d), cross-links behave as negatively charged rods with the same ξe as unbounded alginate chains. This remains true even if some calcium ions are involved in stacking boxes together, as suggested in the original scenario.13 In the “calcium limited” regime of interest here, we assume that all the calcium ions are chelated. We define F ) [Ca2+ ]bound/Cp, with Cp representing the stoichiometric concentration in negative charge carried by the polyions. We have F ) (15/76 mM) = 1/5. The equilibrium fraction of condensed counterions φ ) [Na+ ]condensed/Cp can be determined as follows: First note that, upon binding, due to splicing, the total network contour length Λnet. along which charges distribute, transforms as Λnet. f (1 - 2F)Λnet.. Both binding and condensation contribute to reducing the total negative charge as Qnet. f (1 2F - φ)Qnet.. According to Manning, the charge density must transform as Qnet./Λnet. f ξ-1 e Qnet./Λnet.. Finally, one obtains

φ ) (1 - 2F)(1 - ξ-1 e )

(9)

hence, x0 ) [Na+ ]free = 61 mM. We take one step further and treat the equilibrium between the drop and the gel within the minimal “limiting law” framework, as discussed at length by Manning in its seminal paper.21 The drop is considered as a salt reservoir at a fixed concentration c. Inside the gel, counterions and co-ions diffuse in equal quantity, so that the concentration of free counterions x ) [Na+ ]free is equal to its initial value x0 remaining after condensation, augmented by the concentration in co-ion [Cl- ]. Because both ions must diffuse together, it is the total chemical potential of the salt species that must be continuous across the interface. The resulting law of mass action reads 2 (x)x(x - x0) ) c2 γ(

with γ( representing the average activity coefficient of the salt ions in the gel that stems for the free energy of interaction between the salt ions and the polyion. Within the Debye-Hu¨ckel approximation one obtains the limiting law expression

[

c2 ) x(x - x0) exp -

1 2x/x0 - 1

]

(10)

Note that the electrostatic origin of the activity coefficient is hidden into x0 via its ξe dependence. The reported experiments being performed with c . x0, it is legitimate to expand eq 10 to leading orders: x = c + (3/4)x0.


As expected, in this limit there is no salt-exclusion effect of the polyelectrolyte and x ∼ c. The other limit of interest is that of infinite salt dilution of the drop. Counterions remain trapped into the gel by the neutrality requirement and x ∼ x0. The crossover between both regimes occurs smoothly around c ) x0/4 = 15 mM. This can be taken as a lower estimate of parameter c0. In view of the numerous simplifying assumptions underlying the previous analysis, we find it satisfactory that the measured value c0 ) 37 mM lies somewhere in between x0/4 and x0. Diffusive Skin Depth and Related Transients. So far we have discussed the equilibrium state between the drop and the gel. Yet, because the crack tip is moving, the system is clearly out-of-equilibrium. Let us note that DNa+ represents the effective diffusion coefficient of counterions in the gel. We are looking for the steady state concentration profile of free Na+ ions, which sets up in the gel, in front of a crack tip propagating at a velocity V. At infinity, the concentration is x0. In the immediate vicinity of the interface, the concentration must be the equilibrium one, namely, x(c), as estimated in the preceding paragraph. Ahead of the tip apex, a steady concentration field sets up as resulting from a balance between the diffusive flux and the advective one. Along the propagation axis, where both fluxes are headon, concentration decays from x(c) to x0 over the diffusive length DNa+/V. A conservative estimate for the salt diffusion coefficient is DNa+ = 10-9 m2 · s-1. For the experimental velocity range 10-4 < V < 10-2 m · s-1, this yields DNa+/V > 100 nm. It is therefore likely that the diffusion skin encompasses the whole process zone and that, for all purposes, cross-link unzipping proceeds in a saline environment in equilibrium with the drop. It may be worth evaluating the range of time during which the process zone is effectively soaked by a saline solution at the nominal concentration c. One limitation is the time required for setting the steady state profile. It is of the order DNa+/V2 > 10-1 s. The other limitation is due to the decrease of the drop concentration through osmotic salt pumping across the drop/ gel interface. This transient corresponds to the slow decay of the tip velocity following the jump, as displayed in Figure 3. The decay time depends not only on the velocity through the steepness of the concentration gradient across the diffusion skin, which controls the flux, but also on geometrical features such as the volume of the drop and the area of the exchange surface with the gel. We have found that for all purposes the characteristic decay time is at least of a few seconds. Thus, there is a large bracket of time during which diffusion of the counterions from the reservoir drop is not a limiting process. Then, the physical mechanisms at work in the process zone are remotely controlled by the composition of the wetting drop and the tip dynamics can legitimately be considered as a reliable probe of the interplay between stress- and ion-induced unzipping. Transient Role of Na+ in Unzipping. Strikingly, eq 7 does not feature an enthalpic contribution of the counterions to the free energy of the activated state. Namely, whereas the ionic strength is varied over more than one decade, the activation coefficient of the Na+ ions can be set to unity in the rate of opening of a chelating box. The concentration in counterions therefore appears as a mere front factor in the activated rate expression. Obviously, the binding Ca2+ and the nonbinding Na+ do not play symmetric roles in the ion displacement process. During an activated unzipping event, the calcium ion has to be removed and kept away from the chelating box. Two sodium ions would possibly help in forming a transient and weak


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complex with the two G-G moieties. The probability of formation of such a complex depends primarily on the number of available Na+ in the box vicinity. It is clearly an activated state, the enthalpy of which is dominated by the binding energy of the calcium ion. Therefore, the displacement of a Ca2+ ion by two Na+ ones would be primarily a steric process, electrostatic interactions being reduced to the constraint of charge conservation, which imposes the order of the kinetics. Beyond activation, the equilibrium state may correspond to either (i) both sodium ions condensed onto the chain and calcium passing into the drop or (ii) free sodium ions and the calcium one forming a new bond somewhere in the gel. The latter scenario is more likely to occur in a “calcium limited” situation where the stoichiometric ratio of Ca2+ and G-G is low enough for a significant fraction of the chelating sites to remain free. Such is the case of the present study (we have checked that increasing the amount of calcium available led to an increase of the gelatin stiffness, hence of the density of cross-links). Note that we have been led to propose in a previous work11 that the displacement of ionic bonds could be the self-toughening process responsible for the surprisingly high absolute level of fracture energy observed in alginate gels.

Conclusion We have shown that the process zone of a crack tip, progressing steadily in an alginate hydrogel, behaves as a unique “reactor” for ion-displacement experiments. In contradistinction to standard ion exchange experiments where the gel is used as a “permutite”,7,18,19 here we directly probe the rate of box opening that is both ion- and tension-aided. The bias provided by the mechanical tension prevents rezipping events to occur and concurs to enhance degradation. Actually, the effect of ions on fracture dynamics appears quasi-instantaneous when compared to ion-exchange, as probed through its effect on equilibrium properties (modulus, swelling ratio, ...), the latter occurring on an hour-to-day basis. This study has natural continuations. For instance, we shall quantify the role of nonbinding divalent cations, such as Mg2+, and check whether the displacement kinetics effectively turns out to be first order in [Mg2+]. We have shown that the polyelectrolyte nature of alginate gels have to be taken into account, through counterion condensation, to quantify the effect of salt concentration on crack speeding up. This could be readily tested by changing the alginate and calcium content of the gels. We are not able at that stage to conclude whether the nonequilibrium, unzipping process is kinetics of diffusion controlled. This is included in the bare rate ν0. Making use of counterions of different sizes (e.g., K+) may provide some insight into the role of diffusion. Finally, when dealing with scaffold design, it is important to keep in mind that stress and ion-displacement act cooperatively to degrade the gel under load-bearing conditions. This is especially efficient when the fracture process in the reference state is very slow. Then, a modest environmental change, compatible with physiological conditions, may turn a subcritical flow into an unstable crack, leading to catastrophic rupture of the gel. Acknowledgment. We are indebted to I. Donati for insightful comments.

References and Notes (1) Langer, R. AdV. Mater. 2009, 21, 3235–3236. (2) Lee, K. Y.; Mooney, D. J. Chem. ReV. 2001, 101, 1869–1879. (3) Drury, J. L.; Mooney, D. J. Biomaterials 2003, 24, 4337–4351.

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Cooperative Effect of Stress and Ion Displacement on the Dynamics of

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