All Gelatin Networks: 2. The Master Curve for Elasticity †

All Gelatin Networks: 2. The Master Curve for Elasticity†

Christine Joly-Duhamel,‡ Dominique Hellio,‡ Armand Ajdari,§ andMadeleine Djabourov*,‡

Laboratoire de Physique et Mecanique des Milieux Heterogenes, UMR ESPCI-CNRS 7636,and Laboratoire de Physico-Chimie Theorique, UMR ESPCI-CNRS 7083,

10, Rue Vauquelin, 75231 Paris Cedex 5, France

Received February 15, 2002. In Final Form: May 16, 2002

The rheological properties of gelatin gels from various sources (fish and mammalian) were followed inthe course of the sol-gel transition and during gel maturation, by performing dynamic measurementsunder very small deformations. Different concentrations and molecular weights were investigated. Thesolvent was mainly water, but mixed solvents containing water and glycerol were also considered. Blendsof gelatins from fish and mammalian in aqueous solutions were prepared and rheologically characterized.A systematic comparison was established between these measurements and those presented in paper 1of the series. The comparison between the storage modulus and the concentration of helices leads to amaster curve valid for all the samples investigated. This master curve is analyzed in terms of the percolationregime, near the threshold and in terms of a homogeneous network far from the threshold. In the percolationregime, a critical exponent of 2 is found for the storage modulus versus the distance to the threshold, inagreement with previous results. The theoretical models for rigid networks are briefly presented. The fullydeveloped network appears as an entangled assembly of rigid rods (triple helices) connected by flexiblelinks. This study shows that gelatin networks under small deformations do not behave like rubber-likenetworks, where long random coils are connected by local cross-links or entanglements.

In the first paper of this series, gelatin samples fromvarious sources were investigated and their environmentin solution modified either by using additives (smallmolecules) soluble in water or by blending different typesof gelatins. The thermal properties of the triple heliceswere extensively explored during cooling and heating ofthe solutions. These experiments demonstrate the largeinfluence of all the parameters on the helix stability.Because helices create the network, they strongly modifythe viscoelastic properties of the solutions. The workpresented here deals with the rheology of the solutionsmeasured in quiescent conditions, when oscillations ofvery small amplitudes were applied. Gelation is observedin all the cases reported here. The experimental conditionsrigorously reproduce those of the optical rotation mea-surements presented in paper 1, as we have systematicallytried to set up the correlation between the rheology andthe helix formation.

The paper contains the following sections: material andmethods and results for single-component gels in aqueoussolutions, then in mixed solvents, and finally for blendsof gelatins in aqueous solutions. A discussion is thenproposed which addresses the origin of the elasticity ofthe gelatin gels.

I. Materials and Methods

The gelatin samples of various sources were presentedin detail in paper 1 with their molecular characteristics.A1 and A2 are the bovine gelatins of high and lowmolecular weights; B1 and B2 are pig skin gelatins of

high and low molecular weights. The fish gelatins arefrom tuna, megrim, and cod skins. All samples were fullycharacterized in terms of their isoelectric point, molecularweight, polydispersity, and amino acid content.

Rheology measurements were performed with an AR1000 from TA Instruments operating in the oscillatorymode, with an imposed amplitude of deformation of 0.5%and a frequency of 1 Hz, during all experiments. Defor-mation was recorded at the same time as the shearmodulus G′ and the loss modulus G′′. Temperature wascontrolled by a Peltier device. The device used was a cone/plate geometry with a cone of 6 cm/2°. The protocols oftemperature variation followed exactly those of thepolarimeter and are recalled when necessary.

II. Results

Solutions containing more than approximately 1 or 2%of gelatin, cooled at low enough temperatures, lose theirability to flow and become soft solids or gels. Themechanical properties can be measured very preciselywithout disturbing the process if enough care is taken. Astriple helices are stabilized by weak interactions (hydrogenbonds), when measurements are not performed in suitableconditions, they disrupt the links and thus modify thestate of aggregation. The amplitude of the deformationmust be kept as low as possible, especially at the beginningof the helix formation. The stress-controlled rheometerAR 1000 from TA Instruments exercises a rigorous controlof the amplitude of deformation which is necessary forthese experiments. The rheological experiments wereperformed in parallel with optical rotation measurementson identical samples and identical thermal histories.

The experiments are presented in the following order:gelation of A type samples and of fish gelatins in aqueoussolutions; gels with mixed solvents; gelation of blends inaqueous solutions.

The correlation between rheology and helical contentwas systematically sought.

* To whom correspondence should be addressed. E-mail:[email protected].

† This article is part of the special issue of Langmuir devoted tothe emerging field of self-assembled fibrillar networks.

‡ Laboratoire de Physique et Mecanique des MilieuxHeterogenes, UMR ESPCI-CNRS 7636.

§ Laboratoire de Physico-Chimie Theorique, UMR ESPCI-CNRS 7083.

7158 Langmuir 2002, 18, 7158-7166

10.1021/la020190m CCC: $22.00 © 2002 American Chemical SocietyPublished on Web 07/27/2002

Gelation of A Type Samples (Bovine) and of FishGelatins in Aqueous Solutions. First of all, we displayin Figure 1 the measurements of the storage moduliobtained for the same samples and thermal histories asin Figure 4 of paper 1, where the corresponding helixamounts were shown. As in Figure 4, large differencesappear between the samples. When choosing one tem-perature, for instance 10 °C, the moduli vary between 0for cod gelatin, 1500 Pa for A2, and 5000 Pa for B1. Noshear modulus could be measured for the hydrolyzedsample of A type (very low molecular weight). The positionof G′ versus temperature for the different samples followsthe same trend as those for the helical content (Figure 4,paper 1). A more quantitative comparison can be per-formed by following the moduli versus time and temper-ature forall thesesystems,duringgel formationormelting,as we did for the helix amounts.

The kinetics of gelation for three concentrations 2, 4.5,and 8% g/cm3 of A1 at the same cooling rate and finaltemperature (10 °C) are shown in Figure 2. G′ and G′′ areplotted versus time, and the thermal history is also shown.An example of the instrumental control of the amplitudeof deformation (strain) is shown, in Figure 2b, in the courseof gelation showing that the measurements did not disturbgelation. The effect of the molecular weight on the kineticsis shown in Figure 3 for A1 and A2, at a concentration ofc ) 4.5% g/cm3. A large difference of the shear moduli ofthe gels is observed with the two different molecularweights (between 1000 and 4000 Pa). Combining therheological measurements and the helix amounts, oneobtains in Figure 4, for A1, at a fixed concentration of4.5% g/cm3 and at five different temperatures (duringcooling and annealing and also during melting at thelowest temperature, 5 °C), a single curve for G′(ø). Thisplot provides evidence for a strong correlation betweenthe storage moduli and the amount of helices, independentof the thermal histories, at a given concentration. Thesame correlation is also observed in Figure 4 at 10 °C fora lower molecular weight, A2, at the same concentration.The correlation thus holds for two molecular weights: whenan equal amount of helices is present, the moduli of thegels are identical, independent of temperature. Theminimum amount of helices required to form an elasticgel is close to ø ≈ 0.1 at a concentration of 4.5% g/cm3 forany temperature of gelation or molecular weight (A1 andA2) (Figure 4). The data are frequency independent for G′> 10 Pa and slightly dependent on frequency for 1 < G′

< 10 Pa. We consider that they represent the static moduliof the gels. The scatter of the experimental data on thisfigure and others is due to minute differences of temper-ature between the two techniques: the Peltier device onthe rheometer gives a very accurate temperature control,while the cell for optical rotation is controlled by anexternal bath and over long periods of time a slow drift

Figure 1. Storage modulus G′ at 1 Hz versus temperature forgels at concentrations c ) 4.5% g/cm3 measured in the sameconditions as in Figure 4, of part 1. The lines are guides for theeye.

Figure 2. (a) Kinetics of gelation at different concentrations:storage and loss moduli versus time measured during coolingand annealing of A1 solutions at three different concentra-tions: 2, 4.5, and 8% g/cm3. The thermal history is also shown.(b) Strain control during gelation. The strain imposed was 0.5%during gelation, which was achieved with the stress-controlledinstrument.

Figure 3. The effect of molecular weight on the kinetics ofgelation: G′ versus time for A1 and A2 at c ) 4.5% g/cm3.

All Gelation Networks Langmuir, Vol. 18, No. 19, 2002 7159

of the temperature inside the cell may appear sometimes(no more than 0.1 °C).

When the concentration of the solutions is varied, onemay calculate the concentration of helices, chel, at anymoment, which is the product of the amount of helices, ø,and of the concentration of gelatin in solution, in units ofg/cm3: chel ) øc. Figure 5 displays the shear moduli versusconcentration of helices chel for the three gels of A1 atdifferent gelatin concentrations. A single curve is obtainedfor G′(chel): one point on this curve, such as at chel ) 0.01g/cm3, can represent either c ) 2% g/cm3 and ø ) 0.5, c) 4.5% g/cm3 and ø ) 0.22, or c ) 8% g/cm3 and ø ) 0.125,independent of time or temperature. Provided that thehelix concentration is the same, so is the storage modulus.However, the loss moduli G′′(chel) are not identical. Theloss moduli may reflect the presence of dangling ends,loops attached to the network, or free chains, whichcontribute to dissipation of energy by friction and not tothe elastic modulus. They increase with gelatin concen-tration at a fixed helix content.

Using the same procedure and the appropriate rangeof temperatures, we measured the storage moduli versustime for the cod and tuna samples, which are two extremesfor the imino acid compositions. Two cod solutions wereprepared: c ) 4.5% g/cm3 cooled at 1.2 °C and c ) 8% g/cm3

cooled at 0.8 °C and kept for 2 h. Two tuna gelatin solutionswere prepared, same concentrations c ) 4.5% g/cm3 andc ) 8% g/cm3, both were cooled to 10 °C and kept for 5 h.Measurements were taken continuously with the opticaland rheological techniques. The plot of the static storagemoduli versus the helix concentration for the three typesof gelatins (for the two fish gelatins and for the mammaliangelatin A1) is reported in Figure 6. One notices again thatall the data are remarkably superimposed, given thediversity of molecular composition, molecular weight,gelatin concentration, time, or temperature. The thresholdfor gelation is about chel ) 0.0035 g/cm3 in all the casesreported.

Single Gelatins, Mixed Solvents. When gelatin isdissolved in mixed solvents (water + glycerol) the gelationtemperatures are shifted. As for helix formation, thetemperature shift is related to the amount of glycerol inthe binary solvent. The effect of solvent composition ongelation is shown in Figure 7. Again, we tested thecorrelation between the storage moduli and the concen-tration of helices. The agreement with the previous resultsis illustrated in Figure 8. To quantify the shift of the“apparent” gelation temperature with the solvent com-position, we applied the following procedure: solutionswere prepared with different gelatin concentrations andwere submitted to identical cooling ramps. Optical rotationangles were recorded, and the concentration of heliceswas calculated. The temperature at which the concentra-tion of helices reached the threshold chel ) 0.0035 g/cm3

was defined as the “apparent gelation temperature”, thedesignation “apparent” meaning that it undoubtedlydepends on the cooling rate. In Figure 9 we compare the“apparent” gelation temperatures in water to mixtures ofwater and glycerol for various gelatin samples, A1, tuna,and cod. The apparent gelation temperature increaseswith gelatin concentration: for the more concentratedsolutions, the threshold is reached at higher temperatures,whereas for dilute solutions one needs lower temperaturesto form the necessary amount of helices. By adding glycerolto water, the apparent gelation temperatures increasedwith concentration, in parallel to pure water. The differ-ence in temperatures mainly depends on the amount ofglycerol. For 30 wt % glycerol, it reaches 1.5-2 °C for A1,2.5-3 °C for tuna, and 2.5 °C for cod. For 50 wt % glycerolthe shift was 3.5 °C for A1 compared to water.

Blended Gels. Figure 10 shows G′ versus helixconcentration chel in bends of A1 and tuna gelatinscontaining a total concentration of 8% g/cm3 with 3% A1and 5% tuna. The beginning of the plot corresponds tohelices arising from A1 mainly, because they are formedfirst; the second part of the curve is a blend containinghelices from both species. Here again the master curve ofFigure 6 holds. Similar data were obtained (not shown)with mixtures of cod and A1 gelatins. The contribution ofcod is however less visible in this case as the secondnetwork (cod gelatin) is formed when the A1 network isalready fully developed. There is no noticeable departurefrom the master curve.

The rheological experiments reported here cover thedifferent classes of gels described in detail in paper 1. Thediscussion will focus now on the existence of the mastercurve and its meaning. The experimental data provideevidence for the existence of a master curve which relatesthe storage modulus and the helix concentration. Byselecting the data obtained with the most rigoroustemperature control for both techniques, we will refer inour analysis to the “master curve” represented Figure 11.

Figure 4. Storage modulus G′ versus helix amount ø for A1and A2 at a fixed concentration (c ) 4.5% g/cm3) and differentthermal treatments. The helix amounts are extracted frommeasurements reported in paper 1.

Figure 5. Storage modulus G′ and loss modulus G′′ versushelical concentration chel for A1 samples at different gelatinconcentrations.

7160 Langmuir, Vol. 18, No. 19, 2002 Joly-Duhamel et al.

III. Discussion

Before presenting our own analysis of the elasticity ofthe gelatin gels, it is worthwhile to mention some of thedifferent models which have been proposed in the litera-ture: Gelatin is described as a “physically cross-linkedelastomer” in an article in the Journal of ChemicalEducation by Henderson et al.,1 1985, and in the textbookby L. H. Sperling.2 On the basis of the assumption thatgelatin gels obey the rubber elasticity theory, the Youngmodulus of the gels allows the authors to determine thecross-linking density of the gel. Hydrogen bonds areassumed to create the junctions of the network. Theauthors arrive at the unrealistic evaluation of 0.6 hydrogenbonds per molecule independently of the concentration.

More recently, a paper by Normand et al.3 dealt withthe kinetics of gelation of extracts of various molecularweights of bovine bone gelatin. The authors base theirinterpretation on the model of Pearson and Graessley,4which makes the assumption of a phantom network. Thisassumption is better fulfilled at low concentrations andat cure temperatures close to the critical gelation tem-perature. This interpretation consequently refers to gelswith low shear moduli. The model allows calculation ofthe “fraction of structural units participating directly inthe cross-links” (the “R” parameter in the paper) assumingthat each chain has one cross-link on average at the gelpoint. The model is used for the lowest molecular weightextract and the gel times were found in good agreementwith the experiments. The molecular weight of the primarychains and the number of “structural units per chain” arefixed parameters which serve to compute the “R” param-

(1) Henderson, G. V. S., Jr.; Campbell, D. O.; Kuzmicz, V.; Sperling,L. H. J. Chem. Educ. 1985, 62, 269-270.

(2) Sperling, L. H. Introduction to Physical Polymer Science; JohnWiley and Sons, Inc.: New York, 1985.

(3) Normand, V.; Muller, S.; Ravey, J. C.; Parker, A. Macromolecules2000, 33, 1063-1071.

(4) Pearson, D. S.; Graesseley, W. W. Macromolecules 1978, 11, 528-533.

Figure 6. Storage modulus versus the helical concentration for A1, cod, and tuna gelatins.

Figure 7. Effect of solvent composition on gelation of A1gelatins. Upon addition of glycerol, gelation temperatures areshifted progressively toward higher values.

Figure 8. Storage modulus versus helical concentration forA1 samples containing different solvents: water and glycerol30 and 50 wt %.


eter. The early stages of the kinetics of gelation were alsointerpreted for solutions of larger concentrations. In this

case, the authors limit their analysis to the moment when“the growth of the cross-links begins”.

Gilsenan and Ross-Murphy5 interpret the meltingtemperatures of gelatin gels versus concentration usingthe Eldridge-Ferry6 model which determines the enthalpyof melting of the cross-links. The effect of molecular weightis put in evidence. The enthalpy determined from theseplots varies strongly with the temperature of gelation and/or the source of gelatin. There is no evidence for a simplerelation between these values. They certainly reflect thepresence of different amounts of helices which in turnvary with the whole history of the sample. The approachproposed by te Nijenhuis7,8 is also based on the Eldridge-Ferry model. The work of Eldridge and Ferry6 originallystates that there is a relation between the concentrationof the gel and its melting temperature, for a givenmolecular weight and a given temperature of gelation. Alinear relation between the ln c and 1/Tm is expected whichallows determination of the enthalpy of the junctions,“provided that the length of triple helices is independentof temperature”, adds te Nijenhuis. We showed in ourexperiments that the melting temperatures of the helicesdo not depend on the concentration of the solutions (givena molecular weight) but on the temperature of gelation,certainly time of maturation, origin of gelatin, etc. Toobtain a straight line in these plots, one needs to have aconstant “enthalpy of cross-linking”. There is actually anenthalpy associated with the helix formation and melting,but there is no direct relation between this enthalpy andconcentration. Besides, from a practical point of view the“melting temperatures” derived from “rheology” are dif-ferent from those derived from direct measurement of the“amount of helices”, which is the usual definition of themelting of a structure. The model6 also predicts thedependence of the elasticity with the square of concentra-tion and assumes that the proportion of cross-linking sitesis “always small”.

The different models which have been used so far aimat deriving the cross-linking parameter (“R” parameter,enthalpy of junctions, ...) from the elasticity of the gelsunder certain assumptions and limitations. The presentwork clearly identifies the relevant parameter for elastic-ity, which is the helix concentration. Despite the largedifferences that arise between gelation and meltingproperties of gelatins from various sources and withvarious thermal treatments, a unique master curve wasfound. The simplicity of this result shows that one mayignore the details which have been taken previously intoaccount (source, molecular weight, solvent, time, history,...) in order to interpret the origin of elasticity. The mastercurve stretches from the threshold of gelation to the lastmeasurable points for the helical concentration of solu-tions. There is no specific restriction for time or concen-tration. The following picture arises: for each sample theamino acid residues are distributed between flexible coilsand rigid triple helices. The random coils have a persis-tence length9 of 2 nm, while the persistence length knownfor native collagen10 is about lp ) 170 nm. The mastercurve strongly suggests that G′ is determined by the solehelix concentration chel. It is thus necessary to envisionthe structure controlling the elasticity as a network of

(5) Gilsenan, P. M.; Ross-Murphy, S. B. Food Hydrocolloids 2000,14, 191-195.

(6) Eldridge, J. E.; Ferry, J. D. J. Phys. Chem. 1954, 58, 992-995.(7) te Nijenhuis, K. Colloid Polym. Sci. 1981, 259, 1017.(8) te Nijenhuis, K. Adv. Polym. Sci. 1996, 130.(9) Pezron, I.; Djabourov, M.; Leblond, J. Polymer 1991, 32, 3201-

3210.(10) Nestler, F. H.; Hvidt, S.; Ferry, J. D. Biopolymers 1983, 22,

1747-1758.

Figure 9. The “apparent gelation temperature” versus con-centration for mammalian and fish gelatins at various solventcompositions. The lines are guides for the eye.

Figure 10. G′ versus helix concentration for a blend of A1 andtuna with a total gelatin concentration of 8% g/cm3. Comparisonwith the single component A1 at the same total concentration.

Figure 11. The master curve for the storage modulus G′ versushelix concentration chel for all gelatin samples investigated.


semiflexible triple helices, interconnected by flexiblestrands. In our view, the helices build the network itselfand are not considered as the cross-links of the long flexiblecoils, as the traditional pictures tend to represent thegelatin network. G′′ is an increasing function of the coilconcentration (Figure 5), suggesting that there aredangling chains and loops which contribute to the dis-sipation but not to the network elastic strength. Withincrease of helix concentration, the master curve describesthree successive regimes: (i) for chel < ccrit there is nomacroscopic network; (ii) for ccrit < chel < 2ccrit a percolationregime appears with an elastic structure which spreadsover the sample and with a steep increase of the modulus;(iii) for chel > 2ccrit a stronger network is formed. We shallnow comment upon the two latter regimes: the vicinityof the gel point and the dense gel.

The Vicinity of the Threshold: A Weak Network.Experimentally, the threshold for appearance of therelaxed storage modulus (gel point) was found around thecritical value ccrit ) 0.0035 g of helices/cm3. At thethreshold, the volume fraction occupied by the helices(assuming a density of the protein of 1.44) is only Φc ) 2.4× 10-3. The steep increase of the storage modulus withhelix concentration and the presence of a well-definedthreshold make the sol-gel transition the analogue of apercolation transition. Previous experiments11 performedon a limited range of temperatures and with one con-centration of gelatin, already suggested this analogy,which is now much better documented in this paper.

The percolation threshold in networks made of highlyanisotropic particles, such as fibers or rods, depends ontheir aspect ratio (the ratio l/a between the length l andthe diameter a of the rods). Scaling relations were recentlyderived for homogeneous and heterogeneous networks.Experimental work12,13 has been performed on the con-ductivity of random networks of carbon fibers, withvariable volume fractions and aspect ratios. Balberg etal.14 simulated homogeneous percolation in a system ofrandomly oriented rods with uncorrelated contacts andestablished the relation Φcl/a ) 0.7 (with l/a . 1), whichindicates that the contact number (average number ofneighbors in contact with a certain particle) at thethreshold is of the order of 1 (actually 1.4).

Gelation of rodlike macromolecules investigated bySinclair et al.15 shows a threshold for the gel to soltransition at a low polymer concentration, Φc ) 0.05 wt%, in agreement with the excluded volume of the molecule.Gels of colloidal rods were also investigated experimentallyby Philipse and Wierenga.16,17 These authors also analyzethe case of heterogeneous networks, which lead to verylow critical volume fractions at the threshold, whenparticles form fractal clusters. They also observed ex-perimentally this regime.

The equivalent aspect ratio which can be derived at thethreshold for gelatin is of 290. This value indicates anaverage length of the sequences close to the native collagenrod 290 nm (the full length of a molecule in a helicalconformation), the diameter of a collagen rod being 1 nm.

However, this is just an indication of an order ofmagnitude. The type of percolation which is involved inthe formation of the gelatin network is not exactly the onedescribed by independent rods with a given length or adistribution of lengths. The rods (triple helices) areinterconnected by strands shared between two or three ofthem. The concentration of helices gives informationneither on the typical length of the helical sections nor ontheir polydispersity. A given concentration of helices cancorrespond in principle to a few long helices or to manyvery short segments.

The gel network, on electron microscopy pictures, showslong, linear filaments of triple helices18 for gelatinconcentrations of 2% g/cm3 in the percolation regime (seebelow). Thus, obviously, the sol-gel transition also differsfrom a classical vulcanization reaction which is a molecularcross-linking between entangled flexible coils.

The network is built step by step, by increase of thehelix amount with time, at a fixed temperature (annealing)or during cooling. The triple helices are the “bricks” of thenetwork. When a new “brick” is added to the network,there are two possibilities, either it is stuck in prolongationwith an already existing one, and in this case the sequencegrows linearly with the same bundle of three coils wrappedtogether, or the coils are involved into separate sequencesand in this case branched structures appear. The poorthermal stability of the gels indicates that the growth ofthe triple helices involves loops, mismatches, and defectswhich have dramatic effects on the thermal stability ofthe sequences but which are not directly related to therigidity of the network. When helices grow, two types ofbonds are created: a “rigid type inteeconnection” betweenbricks is established when linear growth of the triple helixproceeds creating rigid rods and a “flexible interconnec-tion” when the new sequences are deviated and separatedby a few monomers in coil conformation, creating cross-links. We do not mean, therefore, that each chain consistsof one single triple helix sequence. Branching is necessaryto create the network. Despite the great diversity of thechemical composition, we found no noticeable variation ofthe threshold, indicating the presence of some regulatingmechanisms in the buildup of the network (growth andbranching/connection of helical sequences). Because thenetwork appears mainly fibrilar, linear growth predomi-nates.

In the vicinity of the threshold we find a power law

with a critical exponent t≈2. Such an exponent is expectedin lattice percolation models with scalar forces betweenreacted bonds, following an analogue of de Gennes19

between elasticity of percolating networks of Hookeansprings and conductivity of percolating resistor networks.Kantor and Webman20 have also considered the case ofthe vector nature of elasticity which obeys a differentuniversality class, with a higher critical exponent for thestorage modulus, close to 4. The extent of the criticaldomain in percolation is necessarily

By taking the maximum range up to

(11) Djabourov, M.; Leblond, J.; Papon, P. J. Phys. (Paris) 1988, 49,333-343.

(12) Carmona, F.; Prudhon, P.; Barreau, F. Solid State Commun.1984, 51, 255-257.

(13) Carmona, F.; Barreau, F.; Delhaes, P.; Canet, R. J. Phys. Lett.1980, 41, L531-L534.

(14) Balberg, I.; Binenbaum, N. Phys. Rev. A 1987, 35, 5174.(15) Sinclair, M.; Lim, K. C.; Heeger, A. J. Phys. Rev. Lett. 1983, 51,

1768-1769.(16) Philipse, A. P.; Wierenga, A. M. Langmuir 1998, 14, 49-54.(17) Wirenga, A. M.; Philpise, A. P.; Lekkerkerker, H. N. W. Langmuir

1998, 14, 55-65.

(18) Djabourov, M.; Bonnet, N.; Kaplan, H.; Favard, N.; Favard, P.;Lechaire, J. P.; Maillard, M. J. Phys. II 1993, 3, 611-624.

(19) de Gennes, P. G. Scaling Concepts in Polymer Physics; CornellUniversity Press: Ithaca, NY, 1979.

(20) Kantor Y.; Webman I. Phys. Rev. Lett. 1984, 52, 1891-1894.

G′ ∼ (chel - ccrit)t (1)

(chel - ccrit)/ccrit , 1


we define the largest domain for the critical behavior.Figure 12, one can see that experimentally the power lawholds in this range. The exponent was more carefullydetermined in a previous publication (ref 11) for one typeof gelatin, in a narrow range of temperatures. Theexperiments presented here confirm this determination.No specific work was undertaken in here to furthervalidate the critical behavior by studying, for instance,the frequency dependence of the moduli. The modelassumes that the data are rigorously independent of thefrequency in the range investigated, which probably needssome justification around 1 Pa.

A Stronger Network at Larger Concentrations.At higher concentrations chel > 2ccrit (2ccrit ≈ 0.007 g ofhelices/cm3), one leaves the critical domain and entersprogressively the homogeneous network.

Using the assumption of randomly distributed filamentsoriented and located randomly in the volume, one canderive the average distance between filaments18

whereLv is the total length of triple helices per unit volume.The question arises of the origin of the rigidity/elasticity

of this network, for which several models have beenproposed in the past, as mentioned before. To provide afirst discrimination, we start by a simple estimate. Forhelix concentrations in the range 0.01 < chel < 0.05 g/cm3,the distance between the semiflexible strands (mesh sizeof the network of helices) (eq 3) yields 7 nm > d > 3 nm,which is much smaller than the persistence length ofcollagen, lp ≈ 170 nm (ref 10). The existence of the mastercurve suggests that the helices are primarily responsiblefor the elasticity, and we have just estimated that theyform an intricate structure at scales such that they haveto be considered as rigid objects (d , lp), a situationcomparable to that of actin gels, for example. Thereforewe rule out here again the picture of a rubber-like networkfor which the strands between cross-links or entangle-ments are random coils. We thus consider a network ofsemiflexible strands of persistence length lp, of length land linear density 1/d2.

Three simple models can be considered:In the first one, the links between the strands are rigid

and thermal agitation is negligible, so that the modulusis controlled by the bending elasticity of the rods (bendingconstant κ ) kBTlp). Let us estimate the elastic energy perunit volume E of a slightly deformed gel. When a rod oflength l is bent at one end producing a deflection δl, itscurvature is constant and of the order of δl/l2. Integratingthe bending energy over the total contour length l of therod, one finds the elastic energy stored ) 1/2lκ(δl/l2)2. Ina cube of size l3, there are l2/d2 rods, so that the energystored per unit volume E is

where δl/l is the strain.The energy per unit volume is related to the storage

modulus by the simple relation

Then the storage modulus scales like

where B1 is a constant of the order of 1, which dependson the geometry of the network.

In the second one, the links are flexible and thermalagitation induces fluctuations of the semiflexile strandsaround their minimal energy configuration. Deformationof the network implies the stretching of strands. Stretchingthem is then opposed by entropic forces. To stretch by anamount δl, a semiflexible strand of length l requires afree energy (MacKintosh et al.21) ∼ kBT(lp/l)2(δl/l)2, whichleads to the expression of the storage modulus

where B2 is another numerical constant of the order of 1.Note that this calculation applies to tensionless filaments,whereas in a real network pre-established strains mayexist which would modify the picture, bringing themodulus closer to G1′.

A third picture is that of absolutely rigid rods connectedby very loose links. The elasticity of such a network isthen purely entropic and due to the constrained thermalagitation of the rods. Such a case has been considered byJones and Marques22 (freely hinged network)

lp is logically absent here as the strands are taken infinitelystiff (relative to the floppy links).

The adimensional constants B1, B2, and B3 depend onthe specific geometry and connectivity of the network.

To test the concentration dependence, further assump-tions must be made as to the geometry of the network(topology, connectivity) and in particular as to the scalingof the strand length l with helix concentration chel. In theabsence of such information we can make the simplestpostulate, i.e., that the length of the rods l and their typicaldistance d scale alike. Since d ∼ chel

-1/2, this leads to

(21) MacKintosh, F. C.; Kas, J.; Janmey, P. A. Phys. Rev. Lett. 1995,75, 4425-4428.

(22) Jones, J. L.; Marques, C. M. J. Phys. (Paris) 1990, 51, 1113-1127.

(chel - ccrit)/ccrit ∼ 1 (2)

d ) [ 12Lv]-1/2

(3)

E ∼ 1/2l-3(l2/d2)lκ(δl/l2)2 ∼ κl-2d-2(δl/l)2

E ) 1/2G′(δl/l)2

Figure 12. Percolation regime. The critical exponent for thestorage modulus is close to t ) 1.9 and the critical thresholdis ccrit ) 0.0034 g/cm3. The solid line is the best fit to G′ ∼ (chel- ccrit)1.9 with (chel - ccrit)/ccrit e 1.

G1′ ) B1κl-2d-2 ) B1(kBTlp)l-2d-2 (4)

G2′ ) B2kBTlp2l-3d-2 (5)

G3′ ) B3kBTl-1d-2 (6)


and

The concentration dependence in our experiments doesnot provide a clear-cut exponent, the apparent slope inthe log-log plots decreasing between 2.3 and 1.6, as shownin Figure 13, as chel varies from 0.009 to 0.04 g/cm3. Therange of concentrations of helices is limited to less than1 decade in this regime, and thus the behavior of theelasticity might be affected by a crossover from theneighbor percolation regime. For instance, a harmonicinterpolation between G′ ∼ (chel - ccrit)1.9 and G′ ∼ chel

1.5

provides a reasonable fit accounting for the progressivedecrease of the effective exponent (Figure 13). We havechosen, in Figures 13 and 14, the experimental data ofthe master curve, Figure 11.

Trying to discriminate between the models on the basisof the absolute values is not totally satisfactory either; asimple attempt is proposed in the Appendix, taking allprefactors to be 1 leads to G2′ > G1′ > G3′. With chel ) 0.04g/cm3 one finds

and

The experimental value is G′ ) 8 × 103 Pa. However,if the physical attachment between helices occurs everyfive contacts, then roughly l/d ∼ 5, then with the samevalue for chel one finds

and

The moduli are extremely sensitive to the ratio l/d. Oneobtains a much better agreement with the experimentalvalue when the helices are considered longer that the meshsize d.

To end this discussion we propose in Figure 15 aschematic drawing of the structure corresponding to thethird model. It is an entangled network of rigid rods, whichcan be deformed through the flexible links, withoutbending of the rods. The analysis presented here standsfor the linear regime or small deformations. At this stage,the reservoir of coils does not seem to play an importantrole. However this network, although made of rigidstrands, can support large deformations without breakingor bending of the rods provided the cross-links are flexible.The large deformation regime is the next interesting stepfor this analysis to be continued from both experimentaland theoretical points of view, which should also help inrefining this picture.

ConclusionIn this study we examined closely the linear elasticity

of the gelatin networks in a large range of concentrations

G1′ ∼ chel2

G2′ ∼ chel2.5

G3′ ∼ chel1.5

G1′ ) 3.2 × 106 Pa

G2′ ) 1.4 × 108 Pa

G3′ ) 7.4 × 104 Pa

G1′ ) 1.3 × 105 Pa

G2′ ) 1.2 × 105 Pa

G3′ ) 15 × 103 Pa

Figure 13. Beyond the percolation regime, the homogeneousnetwork progressively builds up. The power law for the storagemodulus versus helical concentration decreases along theexperimental curve roughly from 2.3 to 1.6.

Figure 14. Harmonic interpolation between a percolationregime and an homogeneous network of entangled rods con-nected by flexible links.

Figure 15. Schematic representation of the fully developedgelatin network. The average length of the rods l and the typicaldistance between the rods d are shown. In this case, l > d.


and for various thermodynamic conditions. We haveevidenced a remarkable correlation between the storagemodulus and concentration or volume fraction or totallength per unit volume of triple helices, regardless of thepresence of large coils and loops and of the nature of thesample, the thermal history, the solvent, etc. This cor-relation results in a master curve displayed in Figure 11.It suggests that the helix concentration controls theelasticity, possibly as rigid rods connected by flexible links.The model proposed needs to be refined in order to predictquantitatively the amplitude of the storage moduli withthe helical content.

AppendixElastic Moduli in the Three Models. From eq 3 we

have established that d-2 ) (1.76 × 1018)chel with chel ing/cm3 and d in meters.

We thus retain d-2 ) Achel, with A ) 1.76 × 1018. Thethree models correspond to the moduli

We can use the simplest assumption that Bi ) 1 and l/d

) 1. With kBT ) 4 × 10-21 J and lp ) 170 nm, this leadsto

Acknowledgment. This study was performed in thecontext of the European Contract FAIR CT 97-3055. Wewish to thank all partners for numerous and fruitfulexchanges in the progress of the work. M.D. wishes tothank Lucilla De Arcangelis, Francois Carmona, HansHerrmann, and Albert Philipse for very stimulatingdiscussions during the preparation of the manuscript. Wethank a reviewer for critical reading of the manuscriptand for the valuable suggestions.

LA020190M

G1′ ) B1kBTlpA2(l/d)-2chel

2

G2′ ) B2kBTlp2A5/2(l/d) - 3chel

5/2

G3′ ) B3kBTA3/2(l/d)-1chel3/2

G1′ ) (2.1 × 109)chel2

G2′ ) (4.6 × 1011)chel2.5

G3′ ) (9.3 × 106)chel1.5


Documents

All Gelatin Networks: 2. The Master Curve for Elasticity †