Difference in Concentration Dependence of Relaxation Critical Exponent

Biomacromolecules 2005, 6, 2150-2156

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Difference in Concentration Dependence of Relaxation Critical Exponent n for Alginate Solutions at Sol-Gel Transition Induced by Calcium Cations Lu Lu, Xinxing Liu, Lin Dai, and Zhen Tong* Research Institute of Materials Science, South China University of Technology, Guangzhou 510640, China Received February 19, 2005; Revised Manuscript Received April 8, 2005

The sol-gel transition in aqueous alginate solutions induced by chelation with calcium cations from in situ release has been investigated with viscoelastic methods. Two alginate samples having different molecular weights (MW) were used over the concentration CAlg of 2∼6 wt % with different mole ratio f of Ca2+ to the alginate repeat unit. The gel point fgel and relaxation critical exponent n were determined according to Winter’s criterion, the later agrees well with that obtained from the relaxation modulus. The results indicate that the power law is valid for the dynamic relaxation at the gel point and the critical gel possesses the self-similarity in structure. With increasing CAlg, fgel for the alginate with lower MW decreases dramatically and n is almost constant of about 0.71. In contrast, fgel for the higher MW alginate with is almost a constant and n decreases from 0.72 then levels off at 0.37 with increasing CAlg, indicating that the concentration dependence of n varies with MW of alginate in the starting solution. The fractal dimension df estimated from n suggests a denser structure in the critical gel of higher MW alginate. Either n or df has been found to follow one curve for the two samples if plotted against the number of cross-link junctions per polymer chain, which is proportional to the alginate MW. Introduction Alginate is a natural biomacromolecule isolated from seaweed and bacteria consisting of (1f4) linked β-Dmannuronate (M) and its C-5 epimer R-L-guluronate (G) residues. There are three types of dyad sequential blocks as MM, GG, and MG along the linear chain of alginate.1,2 These residue sequences endow alginate chains with different stiffness, consequently, the mean square end-to-end distance per uronate residue for the G component is 2.2 times larger than that for the M component.3 The amount and distribution of each dyad depend on the species, location, and age of seaweed from which the alginate is isolated.4 Alginate hydrogels induced by adding divalent cations (except Mg2+) have found many pharmaceutical and biomedical applications. A calcium cross-linked alginate hydrogel is the most typical one and has been used in many fields, such as drug delivery vehicles and cell transplantation matrixes.5-7 The G residue in alginate is in the alternate 1C4 conformation and paired G-sequences making a buckled 2-fold structure as the shape of “egg-box” form cavities to accommodate calcium cations.8,9 The binding occurs between calcium cations and G blocks possessing more than 20 GG residues on different polymer chains.10,11 Up to now, many studies have been devoted to reveal the gelation process and the gel property of alginate.12-15 The cross-linking density and mechanical property of the ionically cross-linked alginate gels can be readily manipulated by varying the M to G ratio and molecular weight of the alginate chain. Also some * To whom correspondence should be addressed.

modified alginate hydrogels have been studied to address the limitation of alginate, such as being not degradable, having limited mechanical properties and lacking functional groups required for cell interaction.16,17 However, only our previous studies concerned the critical behavior of the solgel transition in aqueous alginate solutions induced by adding divalent cations.18,19 The sol-gel transition is a common phenomenon in the nature, playing an important role in many fields such as biology and material engineering. It occurs in nearly all of the common polymer fabrication processes during solidification. The denaturalization of some proteins and the conformational transition in some biomacromolecules are always coupled with sol-gel transition.20 Many models have been proposed to describe cross-linking polymers at their gel point.21-27 Winter and Chambon found experimentally that the storage modulus G′ and loss modulus G′′ exhibited the power-law dependence on the angular frequency ω at the critical gel point28,29 G′(ω) ∼ G′′(ω) ∼ ωn

(1)

So that the relaxation modulus G(t) becomes G(t) ) St-n

(2)

where S is the gel strength and n is the relaxation critical exponent. The spectrum H(τ) of relaxation time τ is30 H(τ)d ln τ ∼ τ-n

(3)

The determination of the relaxation critical exponent n is

10.1021/bm050126u CCC: $30.25 © 2005 American Chemical Society Published on Web 05/05/2005

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Sol-Gel Transition in Aqueous Alginate Solutions

thus a significant in comprehending dynamics of the incipient gel. The tangent of the loss angle δ can be derived from eq 1 as tan δ ) G′′/G′ ) tan(nπ/2)

(4)

Thus, ω independence of tan δ provides a convenient interpolation method to determine gel point and critical exponent n. A lot of experiments have been performed on various gelation systems in order to reveal whether the critical exponent n is universal or depending on the structure details of the system. It is usually demonstrated that the critical gelation of chemical gels cross-linked by covalent bonds is in fair agreement with the theoretical predictions.31-34 In contrast, the transient nature of the physical gelation crosslinked by intermolecular physical interactions makes it difficult both to describe the critical phenomenon and to determine the gel point with mechanical methods unambiguously. According to de Gennes,35 the physical gelation can either obey the universal law for “strong gelation” or be compared to a “glass transition” in the case of “weak gelation”. Some investigations on physical gelation illustrate that the critical exponent are predicted well with the percolation theory and independent of detailed conditions for cross-linking, such as pH, catalyst amount and precursor concentration.36-40 However, the reported critical exponent n for most of the physical gelation is not universal and varies from 0.11 to 0.8,41-43 depending on the stoichiometry, molecular weight, concentration, and cross-linking mechanism. Though the universal critical exponents are expected according to the phase transition nature of the gelation, experimental investigations cannot produce the universal values even for the chemical gelation.44-46 The percolation model, which is commonly used to describe the gelation, is based on a random growth of molecular clusters in a d-dimensional space. We suppose that the gelation may be classified into two catalogues referred to as growth and crosslink according to molecular size in the solution just before the gelation. In the former, the gelation occurs from small molecules or precursors via. completely independent crosslinking following the Markoff process, thus, the critical behavior can be described with the present percolation model. In the latter, however, the gelation starts from already existing macromolecular precursors and the cross-linking will be a high-order Markoff process correlated with the previously formed junctions, despite these junctions are of chemical or physical. In this way, the critical behavior of the sol-gel transition could be comprehensively understood consistently. The gelation of aqueous alginate solutions cross-linked with Ca2+ was chosen in the present study due to the plentiful applications and precisely measurable dynamic and relaxation moduli during the sol-gel transition. What we are expecting is to observe either of the above two gelation processes of growth and cross-linking with one gelation system. Experimental Section Alginate Samples. Two alginate samples (Kimitsu Chemical Industries Co., Japan) were purified as follows: the

Table 1. Characterization of Alginate Samples sample MLGH MHGH

Mw × 10-4 polydispersity M/G 62 330

7.3 9.2

FG

FM

FGG

FMM

FGM

0.60 0.63 0.38 0.45 0.20 0.18 0.91 0.52 0.48 0.44 0.40 0.08

aqueous solution of alginate samples of about 3 wt % was first dialyzed in distilled water using cellulose tubular membranes (the cutoff molecular weight is about 14000) until the conductivity of water outside became constant before and after refreshing. Then, the solution was filtered and freeze-dried to produce purified dry samples. Molecular weight Mw and polydispersity of the sample were determined by gel permeation chromatography (GPC) with a Waters apparatus, using 0.1 M Na2SO4 aqueous solution as the elution and narrowly distributed PEO as the standard. The mole ratio of mannuronate (M) to guluronate (G) residues (M/G) and the mole fractions of GG, MM, and GM (MG) dyad sequences FGG, FMM, and FGM were determined by 1H NMR (nuclear magnetic resonance) according to Grasdalen’s procedure47,48 in D2O of 14 mg/mL at 70 °C. The characterization results are summarized in Table 1. Preparation of Ca-Alginate Gels by in Situ Release of Ca2+. We define the stoichiometric mole ratio f ) [Ca2+]/ [COO- in alginate] as a structure parameter which controls the gelation process with the assumption that the ratio f is proportional to the fraction of inter- and intramolecular crosslinks formed. To prepare homogeneous Ca-alginate gels at room temperature, the method of in situ release of calcium cations from ethylenediamine tetraacetic acid (EDTA, standard reagent, Tianjin Institute for Chemicals) chelate was adopted through lowering the pH with slow hydrolysis of D-glucono-δ-lactone (GDL, Sigma) as described in the literature.49,50 Ca-alginate gels were prepared by blending 3 mL of concentrated alginate stock solution with 0.8 mL of Ca-EDTA solution with different concentrations at pH ) 7, followed by adding the required amount of GDL freshly dissolved in 0.2 mL of H2O. The total system achieved the desired alginate concentration CAlg and f according to the calibration curve for the concentration relation of added GDL and released Ca2+. All samples were homogeneous and transparent after magnetic stirring for 15 min at room temperature and incubated at 12 °C for 48 h prior to the rheology measurement. Rheology Measurement. The viscoelastic measurement was conducted in three steps. First, the shear strain γ dependence of the complex modulus G* was measured at 10 rad/s to determine the linear viscoelasticity region. Second, the dynamic viscoelastic spectra were measured over the angular frequency ω of 10-1∼102 rad/s. Third, the stress relaxation modulus G(t) was measured on the samples very close to the sol-gel transition threshold by applying a constant shear strain within the linear viscoelasticity region. The measurements were performed on a RFS-II rheometer (Rheometrics Ltd) at 25 ( 0.1 °C with different fixtures, including a 25 mm diameter cone plate, a 50 mm diameter cone plate, and a 25 mm diameter parallel plate. The angle of the cones was 0.04 rad.

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Figure 1. Shear strain γ dependence of absolute value of complex modulus G* for Ca-MLGH of the CAlg ) 4 wt % system with indicated f values at 25 °C and ω ) 10 rad/s.

Figure 2. Angular frequency ω dependence of storage modulus G′ (solid symbol) and loss modulus G′′ (open symbol) for Ca-MLGH of the CAlg ) 4 wt % sample with indicated f. The data have been vertically shifted by a factor of 10a with given a to avoid overlapping.

Results The shear strain γ dependence of complex modulus G* for the Ca-MLGH aqueous system of concentration CAlg ) 4 wt % is shown in Figure 1 with various f values below and above the gelation threshold at 25 °C as an example for confirming the linearity. The absolute value of G* is independent of strain γ over the range from 1 to 100% and increases with f. The other systems with different alginate concentrations and the MHGH sample with higher molecular weight also show the similar strain and f dependence for G* during the sol-gel transition. All of the viscoelastic measurements were carried out within the linear region. Figure 2 shows the angular frequency ω dependence of storage and loss moduli G′ and G′′ for the Ca-MLGH aqueous system of CAlg ) 4 wt % at various f values. The data are vertically shifted by a factor of 10a to avoid overlapping. Some G′ values at low frequency with low f were too small to be measured accurately. At low f values, G′ and G′′ are proportional to ω1.5-1.6 and ω0.9-1, respectively,

Lu et al.

and G′′ is always higher than G′ in the whole frequency range without any plateau appearing in G′ vs ω curves. This is the typical characteristic of a viscoelastic fluid according to the Rouse-Zimm theory.30 At high f values, G′ becomes higher than G′′ with a plateau appearing in the G′ vs ω curve in the low-frequency range. This indicates the formation of viscoelastic gels. At a moderate value of f, there is a transition region from solution to gel, where the ω dependent curve of G′ becomes parallel to G′′ on the logarithmic coordinate over a wide frequency range, such as the curves of f ) 0.06 in Figure 2. The slope of the curves G′(ω) and G′′(ω) gives the power-law (eq 1) exponent n. The curves for these two samples at different alginate concentrations very close to the gel point are summarized in Figure 3. Similar linear dependence on log ω of log G′ paralleling to log G′′ is also the case for all of the systems, which is considered as the viscoelastic characteristic for the critical gel proposed by Winter and Chambon.28,29 This power-law behavior suggests that the critical gel should possess a self-similar fractal structure over a wide spatial scale.51,52 In this work, the gel point is determined by fgel, at which the stress relaxes following eq 2, so that the viscosity approaches infinite and the network spreading whole sampling space begins to appear in the system. The values of fgel and n for the sol-gel transition in aqueous alginate solutions induced by adding Ca2+ have been determined using the Winter’s criterion of eq 4. Figure 4 depicts the f dependence of tan δ obtained from the data in Figure 2 at several frequencies. The fgel and n values for all gelation systems were evaluated similarly and listed in Table 2. The deviation in the n value was produced from the determination of the intersecting point in the tan δ vs f plot. The alginate concentration dependence of fgel and n for the two alginate samples are compared in Figures 5 and 6, respectively, for direct impression. For the alginate sample Ca-MLGH with lower molecular weight, fgel in Figure 5 decreases with increasing alginate concentration dramatically. In contrast, fgel for the Ca-MHGH with higher molecular weight is almost a constant for all concentrations. At the same concentration, the fgel value is always higher for Ca-MLGH than that for Ca-MHGH. This fact means that more Ca2+ cations are required to cross-link short alginate chains into infinite networks and this requirement becomes more evident at lower concentrations, because the network of infinite molecular clusters must be formed at the gel point. Figure 6 depicts that the critical exponent n varies with alginate concentration in significantly different manners for these two samples. For Ca-MLGH, the critical exponent n is almost independent of alginate concentration and close to the de Gennes’ prediction of n ) 0.71 based on the analogy to a random electrical network.53-55 On the other hand, the n for Ca-MHGH decreases first from 0.72 to about a half for the Ca-MLGH and then levels off with an increase in alginate concentration. This behavior will be discussed later on. The relaxation modulus G(t) was measured under constant strain within the linear viscoelasticity region on the alginate systems at the vicinity of the gel point with the same f values as those indicated in Figure 3 for the corresponding system, where log G′ is almost parallel to log G′′ against log ω.



Figure 3. Power-law viscoelastic behavior of Ca-MLGH and Ca-MHGH samples with indicted alginate concentrations and f values close to the gelation thresholds. The data have been vertically shifted by a factor of 10a with given a to avoid overlapping (G′: solid symbol; G′′: open symbol).

Figure 4. tan δ at indicated ω plotted against f for Ca-MLGH of the CAlg ) 4 wt % sample to determine the gel point fgel and critical exponent n. The fgel and n are evaluated from the intersecting point as 0.06 and 0.71, respectively.

Figure 5. Alginate concentration dependence of fgel for Ca-MLGH and Ca-MHGH samples.

Table 2. Critical Exponent n Obtained from Dynamic Moduli and Relaxation Modulus sample

CAlg/wt %

fgel

na

nb

Ca-MLGH

2 3 4 5 6 2 3 4 5 6

0.098 0.081 0.060 0.052 0.044 0.040 0.039 0.037 0.030 0.030

0.72 ( 0.01 0.71 ( 0.02 0.71 ( 0.02 0.70 ( 0.02 0.68 ( 0.01 0.72 ( 0.01 0.50 ( 0.01 0.37 ( 0.01 0.37 ( 0.01 0.37 ( 0.01

0.63 ( 0.09 0.69 ( 0.04 0.75 ( 0.04 0.72 ( 0.02 0.71 ( 0.05 0.78 ( 0.06 0.45 ( 0.05 0.31 ( 0.06 0.34 ( 0.03 0.37 ( 0.01

Ca-MHGH

a

b

The critical exponent from dynamic moduli using Winter’s criterion. The quasi-critical exponent from relaxation.

Though the observed relaxation time t is limited due to the sampling-rate and transducer sensitivity of the rheometer, the linear relaxation demonstrated in Figure 7 confirms the validity of eq 2 and provides the relaxation quasicritical exponent n from the slope (Table 2). For every sample, the

Figure 6. Alginate concentration dependence of the critical exponent n for Ca-MLGH and Ca-MHGH samples.

relaxation modulus G(t) was measured at three constant strains within the linear viscoelastic region and the listed quasicritical exponent n values are the mean value with the standard deviation. The f value (related to Ca2+ concentration) used for the relaxation experiment is not exactly the fgel at the gel point determined with Winter’s criterion, owing to

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Figure 7. Time dependence of the relaxation modulus G(t) for CaMLGH (A) and Ca-MHGH (B) samples at the f values where log G′ is almost parallel to log G′′ as shown in Figure 3.

the difficulty in precisely controlling the Ca-EDTA release by the hydrolysis of GDL. Therefore, the n so estimated from the relaxation, called quasicritical exponent, is probably slightly different from that determined with Winter’s criterion. The consistency of n values determined from these two methods is reasonable as seen from Table 2, indicating that the power-law dynamic behavior originates from the critical gel structure in the aqueous alginate systems.

Lu et al.

Figure 8. Alginate concentration dependence of df for Ca-MLGH and Ca-MHGH samples calculated from n using eq 5 (A) and eq 6 (B).

between the critical exponent n and the fractal dimension df has been proposed for monodisperse polymers.58,59 Muthukumar further investigated the effect of polydispersity on cross-linking using a hyperscaling relation from the percolation theory.60 When hydrodynamic interactions are completely screened out and the excluded volume effect is dominant in the cluster n)

Discussion For the gelation in the present alginate solutions induced by Ca2+, the dynamics described by the power law of eq 1 and the relaxation process of eq 2 have been found at the gel point, which suggests that the self-similarity does exist in the critical gel.51,52 It is worth noting that the critical exponent n for Ca-MHGH depends on alginate concentration, whereas n for Ca-MLGH is almost a constant of about 0.71 over the observed concentration range, which agrees well with the percolation prediction.55,56 These results imply that the concentration dependence of the critical exponent n varies with molecular weight of alginate in the starting solution. We reveal further the structure of the critical gel in the framework of fractal. Based on the fractal behavior of the critical gel, several relationships between the critical exponent n and the fractal dimension df have been proposed. Muthukumar derived a theory predicting the frequency dependence of the complex viscosity in terms of the arbitrary fractal dimension of the molecular cluster.57 Then, the relationship

d df + 2

(5)

If excluded volume effect as well as hydrodynamic interaction can be completely screened out, n is expressed by n)

d dhf + 2

)

d(d + 2 - 2df) 2(d + 2 - df)

(6)

where dhf is the fractal dimension of the polymer where the excluded volume effect is fully screened and d is the space dimension. If only partial screening is realized, the fractal dimension takes a value somewhere between df and dhf. According to this theory, a looser structure will lead to a lower value of df and a higher value of n. We have calculated the df value from n for our critical gels using eqs 5 and 6 and plotted in Figure 8, parts A and B, as a function of alginate concentration CAlg, respectively. The unreasonable result of df beyond 3 was obtained from eq 5 (Figure 8A) for the Ca-MHGH. This seems due to the reduction in the excluded volume effect with increasing alginate concentra-



value the gelation will deviate from the percolation mechanism of random growth and deduce a denser structure for the critical gel. Concluding Remarks

Figure 9. Critical exponent n and fractal dimension df of Ca-alginate samples vs CAlgMw.

tion, because the chain excluded volume in dilute polymer solutions decreases gradually to approach the unperturbed state with increasing concentration. Consequently, we rely the further discussion on the df values in Figure 8B evaluated from eq 6. The higher df value for the critical gel of Ca-MHGH sample compared with that for Ca-MLGH sample at the same CAlg indicates that the cluster space in the Ca-MHGH gel is more densely filled than that in the Ca-MLGH gel at the critical state. These different structures can be attributed to the difference in the alginate chain length before cross-linking, for the FGG value (Table 1) is almost the same for these two samples, which is responsible to the cross-linking with Ca2+. For the Ca-MHGH sample with increasing concentration, fgel is nearly a constant but the critical exponent n decreases significantly to a value much lower than the theoretical prediction. This indicates that with increasing alginate monomer concentration the chain length between two neighboring junctions formed by Ca2+ does not change so much as known from only slight change in fgel, which is regarded as the mole ratio of cross-linker to alginate monomer, but the critical network becomes denser as indicated with an increasing df,. We consider that this denser structure comes from increase in the chain density due to increasing CAlg when fgel is a constant. The density of formable cross-links is proportional to CAlg2 and the density of original polymer chains is proportional to CAlg/Mw. Consequently, CAlgMw is proportional to the number of crosslink junctions per polymer chain separated with a fixed interval determined by f. Figure 9 displays the critical exponent n and fractal dimension df varying with CAlgMw so calculated. It is interesting that the data obtained from two alginate samples can follow one curve and can be separated into two groups according to the CAlgMw value. When CAlgMw is lower than 7 × 104, n is higher and agrees well with the percolation prediction, and df is lower, suggesting a loose structure for the critical gel. When CAlgMw is higher than 12 × 104, n becomes lower, deviating from the theoretical prediction, and df becomes higher, indicating the formation of a denser structure in the critical gel. The present results reveal that the original chain length determines the possible junction number along one polymer chain with a fixed interval, and when this junction number is beyond a certain

The present observation on the sol-gel transition of aqueous alginate systems induced by the cross-link of Ca2+ chelated with the G sequence illustrates that the critical gel structure follows the power-law relaxation with the selfsimilarity. When the molecular weight of alginate is high, the critical exponent n, however, decreases to a value much lower than the percolation prediction with increasing alginate concentration. This finding reveals that the critical structure at the gel point formed by cross-linking existing long macromolecules is different from that formed from short chains. The concept of universal critical exponents fails for cross-linking existing macromolecules, because it will deviate from the Markoff process, as restricted by the G sequence already involved in the neighboring junction for the present case. However, the threshold length of original chains before gelation, beyond which the critical gelation follows the crosslink mechanism and deviates from the Markoff process, is unclear yet. We can image that this threshold original chain length should depend on the detailed structures of original chains and formation of the junctions. Acknowledgment. The authors are grateful to the NSF of China for the sponsorship to this work (Nos. 90303019 and 90206010). References and Notes (1) Moe, S. T.; Dragel, K. I.; Smidsrød, O. Alginates In: Stephen, A. M. editor. Food Polysaccharides and Their Applications; Marcel Dekker: New York, 1995. (2) Johnson, F. A.; Craig, D. Q. M.; Mercer, A. D. J. Pharm. Pharmacol. 1997, 49, 639. (3) Kawai, M.; Matsumoto, T.; Masuda, T.; Nakajima, A. J. Jpn. Soc. Biorheol. 1992, 6, 87. (4) Ji, M. The Chemistry of seaweed; Science Publishing House: Beijing, 1997. (5) Smidsrød, O.; Skjak-Braek, G. Trends Biomaterials 1997, 18, 71. (6) Matthew, I. R.; Browne, R. M.; Frame, J. W.; Millar, B. G. Biomaterials 1995, 16, 275. (7) Eiselt, P.; Yeh, J.; Latvala, R. K.; Shea, L. D.; Mooney, D. J. Biomaterials 2000, 21, 1921. (8) Grant, G. T.; Morris, E. R.; Rees, D. A. Smith, P. J. C.; Thom, D. FEBS Lett. 1973, 32, 195. (9) Morris, E. R.; Rees, D. A.; Thom, D.; Boyd, J. Carbohydr. Res. 1978, 66, 145. (10) Draget, K. I.; Bræk, G. S.; Smidsrød, O. Int. J. Biol. Macromol. 1997, 21, 47. (11) Oates, C. G.; Lucas, P. W.; Lee, W. P. Carbohydr. Polym. 1993, 20, 189. (12) Matsumoto, T.; Kawai, M.; Masuda, Y. J. J. Chem. Soc., Faraday Trans 1992, 88, 2673. (13) Wang, Z. Y.; Zhang, Q. Z.; Kono, M.; Saito, S. Chem. Phys. Lett. 1991, 186, 463. (14) Zheng, H. H.; Zhang, Q.; Jiang, K.; Zhang, H.; Wang, J. J. Chem. Phys. 1996, 105, 7746. (15) Kong, H. J.; Wong, E.; Mooney, D. J. Macromolecules 2003, 36, 4582. (16) Lee, K. Y.; Rowley, J. A.; Eiselt, P.; Moy, E. M.; Bouhadir, K. H.; Mooney, D. J. Macromolecules 2000, 33, 4291. (17) Bouhadir, K. H.; Hausman, D. S.; Mooney, D. J. Polymer 1999, 40, 3575. (18) Liu, X. X.; Qian, L. Y.; Shu, T.; Tong, Z. Polymer 2003, 44, 407. (19) Lu, L.; Liu, X. X.; Qian, L. Y.; Tong, Z. Polym. J. 2003, 35, 804.

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Difference in Concentration Dependence of Relaxation Critical Exponent

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