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All Gelatin Networks: 2. The Master Curve for Elasticity† Christine Joly-Duhamel,‡ Dominique Hellio,‡ Armand Ajdari,§ and Madeleine Djabourov*,‡ Laboratoire de Physique et Me´ canique des Milieux He´ te´ roge` nes, UMR ESPCI-CNRS 7636, and Laboratoire de Physico-Chimie The´ orique, 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 in the course of the sol-gel transition and during gel maturation, by performing dynamic measurements under very small deformations. Different concentrations and molecular weights were investigated. The solvent was mainly water, but mixed solvents containing water and glycerol were also considered. Blends of 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 1 of the series. The comparison between the storage modulus and the concentration of helices leads to a master curve valid for all the samples investigated. This master curve is analyzed in terms of the percolation regime, near the threshold and in terms of a homogeneous network far from the threshold. In the percolation regime, a critical exponent of 2 is found for the storage modulus versus the distance to the threshold, in agreement with previous results. The theoretical models for rigid networks are briefly presented. The fully developed network appears as an entangled assembly of rigid rods (triple helices) connected by flexible links. This study shows that gelatin networks under small deformations do not behave like rubber-like networks, where long random coils are connected by local cross-links or entanglements.
In the first paper of this series, gelatin samples from various sources were investigated and their environment in solution modified either by using additives (small molecules) soluble in water or by blending different types of gelatins. The thermal properties of the triple helices were extensively explored during cooling and heating of the solutions. These experiments demonstrate the large influence of all the parameters on the helix stability. Because helices create the network, they strongly modify the viscoelastic properties of the solutions. The work presented here deals with the rheology of the solutions measured in quiescent conditions, when oscillations of very small amplitudes were applied. Gelation is observed in all the cases reported here. The experimental conditions rigorously reproduce those of the optical rotation measurements presented in paper 1, as we have systematically tried to set up the correlation between the rheology and the helix formation. The paper contains the following sections: material and methods and results for single-component gels in aqueous solutions, then in mixed solvents, and finally for blends of gelatins in aqueous solutions. A discussion is then proposed which addresses the origin of the elasticity of the gelatin gels. I. Materials and Methods The gelatin samples of various sources were presented in detail in paper 1 with their molecular characteristics. A1 and A2 are the bovine gelatins of high and low molecular weights; B1 and B2 are pig skin gelatins of * To whom correspondence should be addressed. E-mail:
[email protected]. † This article is part of the special issue of Langmuir devoted to the emerging field of self-assembled fibrillar networks. ‡ Laboratoire de Physique et Me´canique des Milieux He´te´roge`nes, UMR ESPCI-CNRS 7636. § Laboratoire de Physico-Chimie The ´ orique, UMR ESPCICNRS 7083.
high and low molecular weights. The fish gelatins are from tuna, megrim, and cod skins. All samples were fully characterized in terms of their isoelectric point, molecular weight, polydispersity, and amino acid content. Rheology measurements were performed with an AR 1000 from TA Instruments operating in the oscillatory mode, with an imposed amplitude of deformation of 0.5% and a frequency of 1 Hz, during all experiments. Deformation was recorded at the same time as the shear modulus G′ and the loss modulus G′′. Temperature was controlled by a Peltier device. The device used was a cone/ plate geometry with a cone of 6 cm/2°. The protocols of temperature variation followed exactly those of the polarimeter and are recalled when necessary. II. Results Solutions containing more than approximately 1 or 2% of gelatin, cooled at low enough temperatures, lose their ability to flow and become soft solids or gels. The mechanical properties can be measured very precisely without disturbing the process if enough care is taken. As triple helices are stabilized by weak interactions (hydrogen bonds), when measurements are not performed in suitable conditions, they disrupt the links and thus modify the state of aggregation. The amplitude of the deformation must be kept as low as possible, especially at the beginning of the helix formation. The stress-controlled rheometer AR 1000 from TA Instruments exercises a rigorous control of the amplitude of deformation which is necessary for these experiments. The rheological experiments were performed in parallel with optical rotation measurements on identical samples and identical thermal histories. The experiments are presented in the following order: gelation of A type samples and of fish gelatins in aqueous solutions; gels with mixed solvents; gelation of blends in aqueous solutions. The correlation between rheology and helical content was systematically sought.
10.1021/la020190m CCC: $22.00 © 2002 American Chemical Society Published on Web 07/27/2002
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Figure 1. Storage modulus G′ at 1 Hz versus temperature for gels at concentrations c ) 4.5% g/cm3 measured in the same conditions as in Figure 4, of part 1. The lines are guides for the eye.
Gelation of A Type Samples (Bovine) and of Fish Gelatins in Aqueous Solutions. First of all, we display in Figure 1 the measurements of the storage moduli obtained for the same samples and thermal histories as in Figure 4 of paper 1, where the corresponding helix amounts were shown. As in Figure 4, large differences appear between the samples. When choosing one temperature, for instance 10 °C, the moduli vary between 0 for cod gelatin, 1500 Pa for A2, and 5000 Pa for B1. No shear modulus could be measured for the hydrolyzed sample of A type (very low molecular weight). The position of G′ versus temperature for the different samples follows the same trend as those for the helical content (Figure 4, paper 1). A more quantitative comparison can be performed by following the moduli versus time and temperature for all these systems, during gel formation or melting, 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 final temperature (10 °C) are shown in Figure 2. G′ and G′′ are plotted versus time, and the thermal history is also shown. An example of the instrumental control of the amplitude of deformation (strain) is shown, in Figure 2b, in the course of gelation showing that the measurements did not disturb gelation. The effect of the molecular weight on the kinetics is shown in Figure 3 for A1 and A2, at a concentration of c ) 4.5% g/cm3. A large difference of the shear moduli of the gels is observed with the two different molecular weights (between 1000 and 4000 Pa). Combining the rheological measurements and the helix amounts, one obtains in Figure 4, for A1, at a fixed concentration of 4.5% g/cm3 and at five different temperatures (during cooling and annealing and also during melting at the lowest temperature, 5 °C), a single curve for G′(χ). This plot provides evidence for a strong correlation between the storage moduli and the amount of helices, independent of the thermal histories, at a given concentration. The same correlation is also observed in Figure 4 at 10 °C for a lower molecular weight, A2, at the same concentration. The correlation thus holds for two molecular weights: when an equal amount of helices is present, the moduli of the gels are identical, independent of temperature. The minimum amount of helices required to form an elastic gel is close to χ ≈ 0.1 at a concentration of 4.5% g/cm3 for any temperature of gelation or molecular weight (A1 and A2) (Figure 4). The data are frequency independent for G′ > 10 Pa and slightly dependent on frequency for 1 < G′
Figure 2. (a) Kinetics of gelation at different concentrations: storage and loss moduli versus time measured during cooling and annealing of A1 solutions at three different concentrations: 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-controlled instrument.
Figure 3. The effect of molecular weight on the kinetics of gelation: G′ versus time for A1 and A2 at c ) 4.5% g/cm3.
< 10 Pa. We consider that they represent the static moduli of the gels. The scatter of the experimental data on this figure and others is due to minute differences of temperature between the two techniques: the Peltier device on the rheometer gives a very accurate temperature control, while the cell for optical rotation is controlled by an external bath and over long periods of time a slow drift
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Figure 4. Storage modulus G′ versus helix amount χ for A1 and A2 at a fixed concentration (c ) 4.5% g/cm3) and different thermal treatments. The helix amounts are extracted from measurements reported in paper 1.
Figure 5. Storage modulus G′ and loss modulus G′′ versus helical concentration chel for A1 samples at different gelatin concentrations.
of the temperature inside the cell may appear sometimes (no more than 0.1 °C). When the concentration of the solutions is varied, one may calculate the concentration of helices, chel, at any moment, which is the product of the amount of helices, χ, and of the concentration of gelatin in solution, in units of g/cm3: chel ) χc. Figure 5 displays the shear moduli versus concentration of helices chel for the three gels of A1 at different gelatin concentrations. A single curve is obtained for G′(chel): one point on this curve, such as at chel ) 0.01 g/cm3, can represent either c ) 2% g/cm3 and χ ) 0.5, c ) 4.5% g/cm3 and χ ) 0.22, or c ) 8% g/cm3 and χ ) 0.125, independent of time or temperature. Provided that the helix concentration is the same, so is the storage modulus. However, the loss moduli G′′(chel) are not identical. The loss moduli may reflect the presence of dangling ends, loops attached to the network, or free chains, which contribute to dissipation of energy by friction and not to the elastic modulus. They increase with gelatin concentration at a fixed helix content. Using the same procedure and the appropriate range of temperatures, we measured the storage moduli versus time for the cod and tuna samples, which are two extremes for the imino acid compositions. Two cod solutions were prepared: c ) 4.5% g/cm3 cooled at 1.2 °C and c ) 8% g/cm3
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cooled at 0.8 °C and kept for 2 h. Two tuna gelatin solutions were prepared, same concentrations c ) 4.5% g/cm3 and c ) 8% g/cm3, both were cooled to 10 °C and kept for 5 h. Measurements were taken continuously with the optical and rheological techniques. The plot of the static storage moduli versus the helix concentration for the three types of gelatins (for the two fish gelatins and for the mammalian gelatin A1) is reported in Figure 6. One notices again that all the data are remarkably superimposed, given the diversity of molecular composition, molecular weight, gelatin concentration, time, or temperature. The threshold for gelation is about chel ) 0.0035 g/cm3 in all the cases reported. Single Gelatins, Mixed Solvents. When gelatin is dissolved in mixed solvents (water + glycerol) the gelation temperatures are shifted. As for helix formation, the temperature shift is related to the amount of glycerol in the binary solvent. The effect of solvent composition on gelation is shown in Figure 7. Again, we tested the correlation between the storage moduli and the concentration of helices. The agreement with the previous results is illustrated in Figure 8. To quantify the shift of the “apparent” gelation temperature with the solvent composition, we applied the following procedure: solutions were prepared with different gelatin concentrations and were submitted to identical cooling ramps. Optical rotation angles were recorded, and the concentration of helices was calculated. The temperature at which the concentration of helices reached the threshold chel ) 0.0035 g/cm3 was defined as the “apparent gelation temperature”, the designation “apparent” meaning that it undoubtedly depends on the cooling rate. In Figure 9 we compare the “apparent” gelation temperatures in water to mixtures of water and glycerol for various gelatin samples, A1, tuna, and cod. The apparent gelation temperature increases with gelatin concentration: for the more concentrated solutions, the threshold is reached at higher temperatures, whereas for dilute solutions one needs lower temperatures to form the necessary amount of helices. By adding glycerol to water, the apparent gelation temperatures increased with concentration, in parallel to pure water. The difference in temperatures mainly depends on the amount of glycerol. 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 % glycerol the shift was 3.5 °C for A1 compared to water. Blended Gels. Figure 10 shows G′ versus helix concentration chel in bends of A1 and tuna gelatins containing a total concentration of 8% g/cm3 with 3% A1 and 5% tuna. The beginning of the plot corresponds to helices arising from A1 mainly, because they are formed first; the second part of the curve is a blend containing helices from both species. Here again the master curve of Figure 6 holds. Similar data were obtained (not shown) with mixtures of cod and A1 gelatins. The contribution of cod is however less visible in this case as the second network (cod gelatin) is formed when the A1 network is already fully developed. There is no noticeable departure from the master curve. The rheological experiments reported here cover the different classes of gels described in detail in paper 1. The discussion will focus now on the existence of the master curve and its meaning. The experimental data provide evidence for the existence of a master curve which relates the storage modulus and the helix concentration. By selecting the data obtained with the most rigorous temperature control for both techniques, we will refer in our analysis to the “master curve” represented Figure 11.
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Figure 6. Storage modulus versus the helical concentration for A1, cod, and tuna gelatins.
Figure 7. Effect of solvent composition on gelation of A1 gelatins. Upon addition of glycerol, gelation temperatures are shifted progressively toward higher values.
III. Discussion Before presenting our own analysis of the elasticity of the gelatin gels, it is worthwhile to mention some of the different models which have been proposed in the literature: Gelatin is described as a “physically cross-linked elastomer” in an article in the Journal of Chemical Education by Henderson et al.,1 1985, and in the textbook by L. H. Sperling.2 On the basis of the assumption that gelatin gels obey the rubber elasticity theory, the Young modulus of the gels allows the authors to determine the cross-linking density of the gel. Hydrogen bonds are assumed to create the junctions of the network. The authors arrive at the unrealistic evaluation of 0.6 hydrogen bonds per molecule independently of the concentration. (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; John Wiley and Sons, Inc.: New York, 1985.
Figure 8. Storage modulus versus helical concentration for A1 samples containing different solvents: water and glycerol 30 and 50 wt %.
More recently, a paper by Normand et al.3 dealt with the kinetics of gelation of extracts of various molecular weights of bovine bone gelatin. The authors base their interpretation on the model of Pearson and Graessley,4 which makes the assumption of a phantom network. This assumption is better fulfilled at low concentrations and at cure temperatures close to the critical gelation temperature. This interpretation consequently refers to gels with low shear moduli. The model allows calculation of the “fraction of structural units participating directly in the cross-links” (the “R” parameter in the paper) assuming that each chain has one cross-link on average at the gel point. The model is used for the lowest molecular weight extract and the gel times were found in good agreement with the experiments. The molecular weight of the primary chains and the number of “structural units per chain” are fixed parameters which serve to compute the “R” param(3) Normand, V.; Muller, S.; Ravey, J. C.; Parker, A. Macromolecules 2000, 33, 1063-1071. (4) Pearson, D. S.; Graesseley, W. W. Macromolecules 1978, 11, 528533.
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Figure 9. The “apparent gelation temperature” versus concentration for mammalian and fish gelatins at various solvent compositions. The lines are guides for the eye.
Figure 10. G′ versus helix concentration for a blend of A1 and tuna with a total gelatin concentration of 8% g/cm3. Comparison with the single component A1 at the same total concentration.
Figure 11. The master curve for the storage modulus G′ versus helix concentration chel for all gelatin samples investigated.
eter. The early stages of the kinetics of gelation were also interpreted for solutions of larger concentrations. In this
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case, the authors limit their analysis to the moment when “the growth of the cross-links begins”. Gilsenan and Ross-Murphy5 interpret the melting temperatures of gelatin gels versus concentration using the Eldridge-Ferry6 model which determines the enthalpy of melting of the cross-links. The effect of molecular weight is put in evidence. The enthalpy determined from these plots varies strongly with the temperature of gelation and/ or the source of gelatin. There is no evidence for a simple relation between these values. They certainly reflect the presence of different amounts of helices which in turn vary with the whole history of the sample. The approach proposed by te Nijenhuis7,8 is also based on the EldridgeFerry model. The work of Eldridge and Ferry6 originally states that there is a relation between the concentration of the gel and its melting temperature, for a given molecular weight and a given temperature of gelation. A linear relation between the ln c and 1/Tm is expected which allows determination of the enthalpy of the junctions, “provided that the length of triple helices is independent of temperature”, adds te Nijenhuis. We showed in our experiments that the melting temperatures of the helices do not depend on the concentration of the solutions (given a molecular weight) but on the temperature of gelation, certainly time of maturation, origin of gelatin, etc. To obtain a straight line in these plots, one needs to have a constant “enthalpy of cross-linking”. There is actually an enthalpy associated with the helix formation and melting, but there is no direct relation between this enthalpy and concentration. Besides, from a practical point of view the “melting temperatures” derived from “rheology” are different from those derived from direct measurement of the “amount of helices”, which is the usual definition of the melting of a structure. The model6 also predicts the dependence of the elasticity with the square of concentration and assumes that the proportion of cross-linking sites is “always small”. The different models which have been used so far aim at deriving the cross-linking parameter (“R” parameter, enthalpy of junctions, ...) from the elasticity of the gels under certain assumptions and limitations. The present work clearly identifies the relevant parameter for elasticity, which is the helix concentration. Despite the large differences that arise between gelation and melting properties of gelatins from various sources and with various thermal treatments, a unique master curve was found. The simplicity of this result shows that one may ignore the details which have been taken previously into account (source, molecular weight, solvent, time, history, ...) in order to interpret the origin of elasticity. The master curve stretches from the threshold of gelation to the last measurable points for the helical concentration of solutions. There is no specific restriction for time or concentration. The following picture arises: for each sample the amino acid residues are distributed between flexible coils and rigid triple helices. The random coils have a persistence length9 of 2 nm, while the persistence length known for native collagen10 is about lp ) 170 nm. The master curve strongly suggests that G′ is determined by the sole helix concentration chel. It is thus necessary to envision the 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, 32013210. (10) Nestler, F. H.; Hvidt, S.; Ferry, J. D. Biopolymers 1983, 22, 1747-1758.
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semiflexible triple helices, interconnected by flexible strands. In our view, the helices build the network itself and are not considered as the cross-links of the long flexible coils, as the traditional pictures tend to represent the gelatin network. G′′ is an increasing function of the coil concentration (Figure 5), suggesting that there are dangling chains and loops which contribute to the dissipation but not to the network elastic strength. With increase of helix concentration, the master curve describes three successive regimes: (i) for chel < ccrit there is no macroscopic network; (ii) for ccrit < chel < 2ccrit a percolation regime appears with an elastic structure which spreads over the sample and with a steep increase of the modulus; (iii) for chel > 2ccrit a stronger network is formed. We shall now comment upon the two latter regimes: the vicinity of the gel point and the dense gel. The Vicinity of the Threshold: A Weak Network. Experimentally, the threshold for appearance of the relaxed storage modulus (gel point) was found around the critical value ccrit ) 0.0035 g of helices/cm3. At the threshold, 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 with helix concentration and the presence of a well-defined threshold make the sol-gel transition the analogue of a percolation transition. Previous experiments11 performed on a limited range of temperatures and with one concentration of gelatin, already suggested this analogy, which is now much better documented in this paper. The percolation threshold in networks made of highly anisotropic particles, such as fibers or rods, depends on their aspect ratio (the ratio l/a between the length l and the diameter a of the rods). Scaling relations were recently derived for homogeneous and heterogeneous networks. Experimental work12,13 has been performed on the conductivity of random networks of carbon fibers, with variable volume fractions and aspect ratios. Balberg et al.14 simulated homogeneous percolation in a system of randomly oriented rods with uncorrelated contacts and established the relation Φcl/a ) 0.7 (with l/a . 1), which indicates that the contact number (average number of neighbors in contact with a certain particle) at the threshold is of the order of 1 (actually 1.4). Gelation of rodlike macromolecules investigated by Sinclair et al.15 shows a threshold for the gel to sol transition 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 experimentally by Philipse and Wierenga.16,17 These authors also analyze the case of heterogeneous networks, which lead to very low critical volume fractions at the threshold, when particles form fractal clusters. They also observed experimentally this regime. The equivalent aspect ratio which can be derived at the threshold for gelatin is of 290. This value indicates an average length of the sequences close to the native collagen rod 290 nm (the full length of a molecule in a helical conformation), the diameter of a collagen rod being 1 nm. (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.
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However, this is just an indication of an order of magnitude. The type of percolation which is involved in the formation of the gelatin network is not exactly the one described by independent rods with a given length or a distribution of lengths. The rods (triple helices) are interconnected by strands shared between two or three of them. The concentration of helices gives information neither on the typical length of the helical sections nor on their polydispersity. A given concentration of helices can correspond in principle to a few long helices or to many very short segments. The gel network, on electron microscopy pictures, shows long, linear filaments of triple helices18 for gelatin concentrations of 2% g/cm3 in the percolation regime (see below). Thus, obviously, the sol-gel transition also differs from a classical vulcanization reaction which is a molecular cross-linking between entangled flexible coils. The network is built step by step, by increase of the helix amount with time, at a fixed temperature (annealing) or during cooling. The triple helices are the “bricks” of the network. When a new “brick” is added to the network, there are two possibilities, either it is stuck in prolongation with an already existing one, and in this case the sequence grows linearly with the same bundle of three coils wrapped together, or the coils are involved into separate sequences and in this case branched structures appear. The poor thermal stability of the gels indicates that the growth of the triple helices involves loops, mismatches, and defects which have dramatic effects on the thermal stability of the sequences but which are not directly related to the rigidity of the network. When helices grow, two types of bonds are created: a “rigid type inteeconnection” between bricks is established when linear growth of the triple helix proceeds creating rigid rods and a “flexible interconnection” when the new sequences are deviated and separated by a few monomers in coil conformation, creating crosslinks. We do not mean, therefore, that each chain consists of one single triple helix sequence. Branching is necessary to create the network. Despite the great diversity of the chemical composition, we found no noticeable variation of the threshold, indicating the presence of some regulating mechanisms in the buildup of the network (growth and branching/connection of helical sequences). Because the network appears mainly fibrilar, linear growth predominates. In the vicinity of the threshold we find a power law
G′ ∼ (chel - ccrit)t
(1)
with a critical exponent t ≈ 2. Such an exponent is expected in lattice percolation models with scalar forces between reacted bonds, following an analogue of de Gennes19 between elasticity of percolating networks of Hookean springs and conductivity of percolating resistor networks. Kantor and Webman20 have also considered the case of the vector nature of elasticity which obeys a different universality class, with a higher critical exponent for the storage modulus, close to 4. The extent of the critical domain in percolation is necessarily
(chel - ccrit)/ccrit , 1 By taking the maximum range up to (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; Cornell University Press: Ithaca, NY, 1979. (20) Kantor Y.; Webman I. Phys. Rev. Lett. 1984, 52, 1891-1894.
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(chel - ccrit)/ccrit ∼ 1
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(2)
we define the largest domain for the critical behavior. Figure 12, one can see that experimentally the power law holds in this range. The exponent was more carefully determined in a previous publication (ref 11) for one type of gelatin, in a narrow range of temperatures. The experiments presented here confirm this determination. No specific work was undertaken in here to further validate the critical behavior by studying, for instance, the frequency dependence of the moduli. The model assumes that the data are rigorously independent of the frequency in the range investigated, which probably needs some justification around 1 Pa. A Stronger Network at Larger Concentrations. At higher concentrations chel > 2ccrit (2ccrit ≈ 0.007 g of helices/cm3), one leaves the critical domain and enters progressively the homogeneous network. Using the assumption of randomly distributed filaments oriented and located randomly in the volume, one can derive the average distance between filaments18
d)
[ ] 1 2Lv
-1/2
(3)
where Lv 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 been proposed in the past, as mentioned before. To provide a first discrimination, we start by a simple estimate. For helix concentrations in the range 0.01 < chel < 0.05 g/cm3, the distance between the semiflexible strands (mesh size of the network of helices) (eq 3) yields 7 nm > d > 3 nm, which is much smaller than the persistence length of collagen, lp ≈ 170 nm (ref 10). The existence of the master curve suggests that the helices are primarily responsible for the elasticity, and we have just estimated that they form an intricate structure at scales such that they have to be considered as rigid objects (d , lp), a situation comparable to that of actin gels, for example. Therefore we rule out here again the picture of a rubber-like network for which the strands between cross-links or entanglements are random coils. We thus consider a network of semiflexible strands of persistence length lp, of length l and 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 modulus is controlled by the bending elasticity of the rods (bending constant κ ) kBTlp). Let us estimate the elastic energy per unit volume E of a slightly deformed gel. When a rod of length l is bent at one end producing a deflection δl, its curvature is constant and of the order of δl/l2. Integrating the bending energy over the total contour length l of the rod, one finds the elastic energy stored ) 1/2lκ(δl/l2)2. In a cube of size l3, there are l2/d2 rods, so that the energy stored per unit volume E is
E ∼ 1/2l -3(l2/d2)lκ(δl/l2)2 ∼ κl-2d-2(δl/l)2 where δl/l is the strain. The energy per unit volume is related to the storage modulus by the simple relation
E ) 1/2G′(δl/l)2 Then the storage modulus scales like
Figure 12. Percolation regime. The critical exponent for the storage modulus is close to t ) 1.9 and the critical threshold is 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)
where B1 is a constant of the order of 1, which depends on the geometry of the network. In the second one, the links are flexible and thermal agitation induces fluctuations of the semiflexile strands around their minimal energy configuration. Deformation of the network implies the stretching of strands. Stretching them is then opposed by entropic forces. To stretch by an amount δl, a semiflexible strand of length l requires a free energy (MacKintosh et al.21) ∼ kBT(lp/l)2(δl/l)2, which leads to the expression of the storage modulus
G2′ ) B2kBTlp2l-3d-2
(5)
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 may exist which would modify the picture, bringing the modulus closer to G1′. A third picture is that of absolutely rigid rods connected by very loose links. The elasticity of such a network is then purely entropic and due to the constrained thermal agitation of the rods. Such a case has been considered by Jones and Marques22 (freely hinged network)
G3′ ) B3kBTl-1d-2
(6)
lp is logically absent here as the strands are taken infinitely stiff (relative to the floppy links). The adimensional constants B1, B2, and B3 depend on the specific geometry and connectivity of the network. To test the concentration dependence, further assumptions must be made as to the geometry of the network (topology, connectivity) and in particular as to the scaling of the strand length l with helix concentration chel. In the absence of such information we can make the simplest postulate, i.e., that the length of the rods l and their typical distance d scale alike. Since d ∼ chel-1/2, this leads to (21) MacKintosh, F. C.; Ka¨s, J.; Janmey, P. A. Phys. Rev. Lett. 1995, 75, 4425-4428. (22) Jones, J. L.; Marques, C. M. J. Phys. (Paris) 1990, 51, 11131127.
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G1′ ∼ chel2 G2′ ∼ chel2.5 and
G3′ ∼ chel1.5 The concentration dependence in our experiments does not provide a clear-cut exponent, the apparent slope in the log-log plots decreasing between 2.3 and 1.6, as shown in Figure 13, as chel varies from 0.009 to 0.04 g/cm3. The range of concentrations of helices is limited to less than 1 decade in this regime, and thus the behavior of the elasticity might be affected by a crossover from the neighbor percolation regime. For instance, a harmonic interpolation between G′ ∼ (chel - ccrit)1.9 and G′ ∼ chel1.5 provides a reasonable fit accounting for the progressive decrease of the effective exponent (Figure 13). We have chosen, in Figures 13 and 14, the experimental data of the master curve, Figure 11. Trying to discriminate between the models on the basis of the absolute values is not totally satisfactory either; a simple attempt is proposed in the Appendix, taking all prefactors to be 1 leads to G2′ > G1′ > G3′. With chel ) 0.04 g/cm3 one finds
Figure 13. Beyond the percolation regime, the homogeneous network progressively builds up. The power law for the storage modulus versus helical concentration decreases along the experimental curve roughly from 2.3 to 1.6.
G1′ ) 3.2 × 106 Pa G2′ ) 1.4 × 108 Pa and
G3′ ) 7.4 × 104 Pa The experimental value is G′ ) 8 × 103 Pa. However, if the physical attachment between helices occurs every five contacts, then roughly l/d ∼ 5, then with the same value for chel one finds
G1′ ) 1.3 × 105 Pa
Figure 14. Harmonic interpolation between a percolation regime and an homogeneous network of entangled rods connected by flexible links.
G2′ ) 1.2 × 105 Pa and
G3′ ) 15 × 103 Pa The moduli are extremely sensitive to the ratio l/d. One obtains a much better agreement with the experimental value when the helices are considered longer that the mesh size d. To end this discussion we propose in Figure 15 a schematic drawing of the structure corresponding to the third model. It is an entangled network of rigid rods, which can be deformed through the flexible links, without bending of the rods. The analysis presented here stands for the linear regime or small deformations. At this stage, the reservoir of coils does not seem to play an important role. However this network, although made of rigid strands, can support large deformations without breaking or bending of the rods provided the cross-links are flexible. The large deformation regime is the next interesting step for this analysis to be continued from both experimental and theoretical points of view, which should also help in refining this picture.
Figure 15. Schematic representation of the fully developed gelatin network. The average length of the rods l and the typical distance between the rods d are shown. In this case, l > d.
Conclusion In this study we examined closely the linear elasticity of the gelatin networks in a large range of concentrations
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and for various thermodynamic conditions. We have evidenced a remarkable correlation between the storage modulus and concentration or volume fraction or total length per unit volume of triple helices, regardless of the presence of large coils and loops and of the nature of the sample, the thermal history, the solvent, etc. This correlation results in a master curve displayed in Figure 11. It suggests that the helix concentration controls the elasticity, possibly as rigid rods connected by flexible links. The model proposed needs to be refined in order to predict quantitatively the amplitude of the storage moduli with the helical content.
Joly-Duhamel et al.
) 1. With kBT ) 4 × 10-21 J and lp ) 170 nm, this leads to
G1′ ) (2.1 × 109)chel2 G2′ ) (4.6 × 1011)chel2.5 G3′ ) (9.3 × 106)chel1.5
Appendix Elastic Moduli in the Three Models. From eq 3 we have established that d-2 ) (1.76 × 1018)chel with chel in g/cm3 and d in meters. We thus retain d-2 ) Achel, with A ) 1.76 × 1018. The three models correspond to the moduli
G1′ ) B1kBTlpA2(l/d)-2chel2 G2′ ) B2kBTlp2A5/2(l/d) - 3chel5/2
Acknowledgment. This study was performed in the context of the European Contract FAIR CT 97-3055. We wish to thank all partners for numerous and fruitful exchanges in the progress of the work. M.D. wishes to thank Lucilla De Arcangelis, Franc¸ ois Carmona, Hans Herrmann, and Albert Philipse for very stimulating discussions during the preparation of the manuscript. We thank a reviewer for critical reading of the manuscript and for the valuable suggestions.
G3′ ) B3kBTA3/2(l/d)-1chel3/2 We can use the simplest assumption that Bi ) 1 and l/d
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