Viscosity of Semiflexible Chitosan Solutions ... - ACS Publications

Biomacromolecules 2002, 3, 342-349

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Viscosity of Semiflexible Chitosan Solutions: Influence of Concentration, Temperature, and Role of Intermolecular Interactions J. Desbrieres CERMAV (CNRS), affiliated with the Joseph Fourier University, Grenoble, BP 53, 38041 Grenoble Cedex 9, France Received October 8, 2001; Revised Manuscript Received December 11, 2001

The influence of polymer concentration and temperature on the rheological behavior of chitosan solution was studied. The threshold concentrations for the different viscometric regimes were determined and the different power laws exponents were calculated and compared with those predicted from models. Different observations and the high values of these exponents within the high concentration region lead to consideration of the presence of intermolecular interactions as soon as the polymer concentration is larger than the overlap concentration. The activation energy was determined as a function of the polymer concentration, and its evolution was compared with theoretical predictions. A gel-sol transition was demonstrated at high concentrations. Introduction Chitin is a structural polysaccharide in the exoskeleton of many arthropods and it is the second most abundant polysaccharide in nature. But its applications are limited due to its lack of solubility in major solvents. Its deacetylation, performed in basic medium, leads to chitosan, which is soluble in acidic medium due to the presence of amino groups which may be protonated. Chitosan is a linear polysaccharide consisting of β(1 f 4) linked 2-amino-2deoxy-D-glucopyranose and 2-acetamido-2-deoxy-D-glucopyranose units (Figure 1). It is characterized by its degree of acetylation (DA), which is the average molar ratio of N-acetyl-D-glucosamine units within the macromolecular chain. The potential uses of chitosan derive from its unique chemistry: it is a polycation in neutral and acidic solutions (due to the value of pKa equal to 6.01,2). Hence it is a biopolymer favorable for a broad variety of industrial and biomedical applications.3-7 The cationic property of chitosan is exploited in the recovery of proteins from food processing wastes8 and in the chelation of heavy metals from wastewater.9 Chitosan may also be modified and hydrophobic chitosans were prepared by grafting alkyl chains.10 These hydrophobic associating water-soluble polymers represent a new class of industrially important macromolecules. They possess unusual rheological characteristics which are thought to arise from the intermolecular association of neighboring hydrophobic substituents11 which are incorporated into the polymer molecule through chemical grafting or suitable copolymerization procedures. The hydrophobic associations give rise to a three-dimensional polymer network. Although there have been many reports in the literature dealing with dilute chitosan solution properties,12-16 studies on the rheological properties of more concentrated solutions

Figure 1. Chemical structure of chitin (a) and chitosan (b). DA is the degree of acetylation.

are rare.17-19 In this paper the influence of polymer concentration and temperature on the rheological properties was studied within a large range of these parameters. This will allow further determination of the influence of chemical modifications on the rheological behavior of solutions of chitosan derivatives with different degrees of acetylation and substituents, especially when hydrophobically modified chitosans will be considered. Experimental Section The chitosan sample was from Pronova (Norway). The degree of acetylation (DA) was determined by 1H NMR and considered to be the most sensitive technique using an AC300 Bruker spectrometer20 and found to be equal to 0.12. Its viscosity-average molar mass was determined to be equal

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Viscosity of Semiflexible Chitosan Solutions

to 193 000 g/mol from the Mark-Houwink parameters previously found21 (the intrinsic viscosity in 0.3 M acetic acid (AcOH)/0.2 M sodium acetate (AcONa) at 298 K was equal to 780 mL/g). It was purified by solubilization in 0.5 M acetic acid solution. This solution was filtered through sintered glass filters and then cellulose nitrate membranes with average pore sizes of 3, 1.2, 0.8, and 0.45 µm successively. The filtrate was neutralized with NaOH up to a pH of 8.5, and the chitosan precipitates. It was then washed with water until reaching the conductivity of distilled water. Finally it was washed with ethanol before drying. The solvent used for rheological experiments was 0.3 M AcOH/0.05 M AcONa. This electrolyte concentration was chosen to screen electrostatic repulsions without favoring hydrophobic interactions promoted by electrolytes. The determination of the intrinsic viscosity as a function of the temperature was performed with an Ubbelohde capillary viscometer (φ ) 0.58 mm). At a temperature of 278 K it is equal to 1013 mL/g. A low shear viscometer (LS40 from Contraves) was used for low polymer concentration solutions (up to 1 g/L). A cone-plate rheometer (AR1000 from TA Instruments) was used for high polymer concentration solutions and steady flow or dynamic experiments. The cones we have used have 2, 4, or 6 cm diameter and angles of 1° (for the 6 cm cone) and 3°59′ (for the other cones) with a cap to avoid vaporization. The oscillatory shear measurements were performed within the linear viscoelastic regime where the storage (G′) and loss (G′′) moduli are independent of the strain amplitude. The temperature range was 278-353 K. Results and Discussion Within the concentration range, different regimes may be considered related with the evolution of the mean-square endto-end distance of the chain with the polymer concentration. In the dilute regime (the polymer concentration is lower than C*, the overlap concentration) the macromolecular chains are isolated toward other ones. The concentrated regime, in which the chain dimensions are independent upon the polymer concentration, was defined for concentrations higher than C**. In the dilute regime the chain dimensions may be characterized from the intrinsic viscosity ([η]). The latter was generally determined from the Huggins relation ηred )

ηsolution - ηsolvent ) [η] + kHC[η]2 ηsolventC

τR ∼ ηsp ∼

(1)

where ηsolution is the zero shear rate viscosity of the polymer solution at C concentration and ηsolvent is the viscosity of the solvent. kH is the Huggins constant and depends on the form of the molecules and the quality of the solvent. The variation of the viscosity of a polymer solution as a function of the shear rate presents, in general, two domains. At low shear rates the viscosity is independent upon the shear rate; the solution is Newtonian. For higher values than a critical shear rate (γ˘ c), the viscosity decreases and tends toward a power law variation: η ) K(γ˘ )n

The beginning of this pseudoplastic behavior, characterized by γ˘ c, depends on the molecular weight of the polymer and on the concentration of the solution. γ˘ c is related to the longest relaxation time of the solution, τR. There are different theories allowing the calculation of this value. The most employed models are Rouse22 or Zimm23 models (according to whether the hydrodynamic interactions are negligible or not) and the reptation de Gennes model,24 but other models are described in monographs by Doi and Edwards25 or Lapasin and Prial.26 1. Influence of Concentration. The results at a temperature of 278 K were presented but similar conclusions may be presented at other temperatures in the range 278-328 K for polymer concentrations up to 50 g/L. The influence of the concentration on the rheological curves is presented in Figure 2. For low concentrations the solutions exhibit a Newtonian behavior, but increasing the polymer concentration leads to the appearance of a non-Newtonian behavior. The zero-shear rate viscosity η0 and the critical shear rate (related with the relaxation time) at which the Newtonian behavior disappears were determined from the plots in Figure 2. Different procedures may be used for the determination of this shear rate. The high shear rate viscosity data were fit to a power law in shear rate, and extrapolation of this power law to η0 gives our definition of the relaxation time τR as the reciprocal of this critical shear rate, γ˘ c.27,28 This relaxation time may be assigned to the time of disengagement for entangled chains. As the final exponent of the power law is not well defined, the critical shear rate may also be determined as being the shear rate for which the viscosity is a given fraction of the viscosity on the Newtonian plateau.29 It was observed that the evolution of the critical shear rate with the polymer concentration was similar whatever the chosen procedure. The scaling prediction for the specific viscosity and the relaxation time for polyelectrolytes in salt excess are shown below for both semidilute entangled and unentangled solutions.30 These predictions are identical with the scaling predictions for uncharged polymers in good solvent31 because the electrostatic interactions are thought to be analogous to excluded volume:32

(2)

{ {

C1/4 semidilute unentanged C3/2 semidilute entanged C5/4 semidilute unentanged C15/4 semidilute entanged

Other scaling laws exist, and these theoretical predictions are available for very long monodisperse chains. Polydispersity smears out the sharpness of the transitions between the solution regimes. Different plots are considered, according to authors, to determine the different solution regimes from rheological data. As the specific viscosity (ηsp) (η0 ηsolvent)/ηsolvent) calculated from viscosities at zero-shear rate (or on the Newtonian plateau) may be expressed by the relation33,34 ηsp ) C[η] + kH(C[η])2 + b(C[η])n

(3)

the overlap concentration C* will be determined as the value

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Desbrieres

Figure 2. Influence of the shear rate and the concentration on the rheological curves of the chitosan solutions (solvent, 0.3 M AcOH/0.05 M AcONa; temperature, 278 K).

Figure 3. Dependence of the specific viscosity as a function of (a) the overlap parameter, C[η], and (b) the function C[η] + kH(C[η])2 for chitosan solutions (solvent, 0.3 M AcOH/0.05 M AcONa; temperature, 278 K; ηsolvent ) 1.58 mPa‚s; kH ) 0.45).

for which the variation of ηsp with {C[η] + kH(C[η])2} deviates from the slope 1 of Huggins law (Figure 3, curve b). By reporting on curve a of Figure 3 the value of C*[η] is around 1.07, and considering the intrinsic viscosity of the polymer in these conditions (equal to 1013 mL/g at 278 K) the overlap concentration can be determined and is found equal to 1.05 g/L. The curves were drawn in Figure 3 using the experimental value of kH equal to 0.45, which is typical of a good solvent. The viscosity at C* should be roughly twice the solvent viscosity,35 which is the case as ηsp is around 1. But as practically C* is, in most cases, a regime from around 0.5/[η] to 2/[η], this assertion (ηsp at C* is around 1) is not so specific. Different definitions allow an estimate of the theoretical value of C*[η]. Graessley,36 using the Flory Fox relation and assuming that C* is the concentration at which the average distance between two chains is equal to 2R0 (R0 being the gyration radius at zero concentration), estimates this value to 0.77. If it is assumed an hexagonal arrangement of undeformable spherical coils with a radius equal to the gyration radius, it is found to be 1.1 or

1.5 in the case of a complete occupation of the volume by deformable spheres. Values around 1.3 were found in the literature for rigid molecules such as xanthan,37 wellan,38 or succinoglycan.39 For other polymers Morris et al.40 have found a crossover point between the two linear domains of the curve a in Figure 3 which was independent of the studied polymers and a value of C[η] equal to 4 but it is not to be considered as an estimation of the overlap concentration C*. The C** concentration corresponding to the semidilute entangled regime may be determined from the value at which the final linear domain begins. The overlap parameter is found to be equal to ∼13.5, and hence C** is equal to 13.4 g/L. In this semidilute entangled solution, ηsp ∼ C5.2. This exponent is larger than those found for neutral polymers41-43 or some polyelectrolytes44 and much larger than the reptation prediction (ηsp∼ C15/4) for semidilute entangled polyelectrolytes in salt excess30 and semidilute entangled solutions of neutral polymers in good solvents.31,45 This high value of the exponent may be considered as the expression of a high number of contacts between macromolecular chains related with the presence of interchain interactions as already observed for galactomannans.46,47 This presence of interactions within chitosan solutions was already mentioned by Amiji48 or Anthonsen et al.49 The nature of the interactions was discussed by Philippova et al.50 The chitosan is able to form hydrophobic intermolecular domains at concentrations larger than the overlap concentration C* (around 1 g/L). This was also reported from static49 and dynamic51,52 scattering experiments. Using these techniques which are more sensitive than rheology, the presence of aggregates was observed even in dilute solutions. These domains are stable and only scarcely affected by heating or by the addition of salt, urea, or ethanol. The nature of the stabilization of aggregates causing an heterogeneous structure of the polymer solution is still under discussion. It is suggested that neither hydrogen bonds nor hydrophobic interactions are responsible for the


Figure 4. Dependence of the specific viscosity as a function of the overlap parameter, C[η], and comparison with the representation from Kwei (ref 54) (solvent, 0.3 M AcOH/0.05 M AcONa; temperature, 278 K).

Figure 5. Influence of the polymer concentration on the relaxation time (solvent, 0.3 M AcOH/0.05 M AcONa; temperature, 278 K).

aggregation of chitosan. The aggregates are concentration dependent, and a small fraction of the total weight of the sample contributes to the high molecular weight aggregates. This clustering of macromolecules above C* was already discussed by Burchard.53 More recently Kwei54 has experimentally fitted the representation of specific viscosity as a function of the overlap parameter using the relation ηsp ) C[η] {1 + k1C[η] + k2(C[η])2 + k3(C[η])3} (4) with k1 ) 0.4, k2 ) k1/2!, and k3 ) k1/3! for different polymers. These are the first terms of the series expansion of the function C[η] exp(kC[η]), which was proposed earlier by Martin55 and relation 3 is a generalization of such an expression. When this relation is considered, a good agreement is obtained for our chitosan sample (Figure 4) but the experimental curve deviates from the theoretical one for C[η] values larger than 16. This confirms the presence of interactions causing the large exponent mentioned previously. For the most concentrated solutions, relaxation times were determined from the onset of shear thinning and are plotted in Figure 5. For polymer concentrations higher than 15 g/L, it was found that τR ∼ C3.7, as obtained by Burchard for cellulose in metal complexing aqueous solutions Cd-tren,53 which is a stronger concentration dependence than the semidilute entangled prediction (τR ∼ C3/2) or the 2.08 exponent predicted by the renormalization group theory for


Figure 6. Dependence of the slope of the viscosity power law with shear rate as a function of the polymer concentration (solvent, 0.3 M AcOH/0.05 M AcONa; temperature, 278 K).

nonassociating linear chains.25 It was observed from different authors a disagreement between theory and experiments either for polyelectrolytes27,28 or uncharged polymers in good solvent.41-43 But in our case the discrepancy between experimental values and theoretical ones is very large and is attributed to the interactions between the macromolecular chains as discussed previously and admitted by Burchard in the cellulose solutions.53 When the shear rate is larger than the critical shear rate, there was observed to be a dependence of the viscosity with the shear rate as (γ˘ )n. The slope is a function of the polymer concentration (Figure 6). This slope increases with the polymer concentration and has to tend toward a limit which is around 0.8 as predicted by Graessley.36 According to Ferry56 this limit value would be obtained for C[η] equal to 20. Below this the slope, as the relaxation time, is found to be strongly influenced by the nature of the polymer-solvent interactions.57 Generally the strong variations of the slope observed for C[η] values smaller than 10 are generally attributed to a large decrease of hydrodynamic interactions when the concentration increases. This behavior was observed in solutions of flexible polymers (as polystyrene in toluene)57 or rigid polysaccharides (as xanthan in aqueous solutions).37 For chitosan solutions the variation of the slope (even if it is not the final one) with the polymer concentration is smaller than that for other polysaccharides and hence confirms the presence of intermolecular interactions even for concentrations smaller than 20 g/L. Associations between macromolecular chains decrease the mobility of the chains toward others and hence the variation of viscosity with shear rate. When the polymer concentration is larger than 45 g/L (as it will be discussed later), a gel is observed and the slope of the variation of the complex viscosity with the pulsation decreases due to interactions. At a polymer concentration of 52.85 g/L the storage and loss moduli are shown as a function of the frequency in Figure 7. At the gel point G′ and G′′ curves become parallel to each other and the power law behavior (G′ = G′′ ∼ ωm) in frequency was observed. This behavior was observed for polymer concentrations larger than 45 g/L. Hence this latter may be considered as an estimation of the gelation concentration, in the same range as mentioned previously18 for

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Desbrieres

angle X-ray scattering experiments on solutions of chitosan (with a degree of acetylation of 21%) in 2 wt % acetic acid aqueous solution. The reason for this difference in fractal dimension may be related with the fact that the heterogeneous structure was very sensitive to the degree of acetylation of the chitosan sample. 2. Influence of Temperature. Newtonian viscosity η0 may be written as containing at the same time a structure sensitive factor S and a mobility factor Fm61 η0 ) S(M,φv) Fm(φv,T)

Figure 7. Mechanical moduli as a function of the frequency (polymer concentration, 52.85 g/L; solvent, 0.3 M AcOH/0.05 M AcONa; temperature, 278 K).

Figure 8. Influence of chitosan concentration on the variation of tan δ ()G′′/G′) at different frequencies (solvent, 0.3 M AcOH/0.05 M AcONa; temperature, 278 K).

chitosans with similar physicochemical characteristics. To determine more accurately this value, the evolution of tan δ (equal to G′′/G′) as a function of the polymer concentration (at several frequencies) is given in Figure 8. These curves intercept at the gelation concentration, and for higher concentrations as the G′(ω) and G′′(ω) curves are parallel, the tan δ values are independent of the frequency. The found gelation concentration is equal to 43 g/L. The value of m (0.41) observed for the system at a concentration of 52.85 g/L is considerably smaller than that (0.7) predicted for percolating networks.45,58 To rationalize values deviating from the percolation value, Muthukumar59 developed a theoretical model in which it is assumed that the variations in the strand length between cross-linking points of the incipient network gives rise to changes in the excluded volume interactions. The surmise is that increasing strand length will enhance the excluded volume effect. If the excluded volume interaction is fully screened, the relaxation exponent for a polydisperse system may be written as56 m ) d(d + 2 - 2df)/2(d + 2 - df)

(5)

where d (d ) 3) is the spatial dimension and df is the fractal dimension which relates the mass of a molecular cluster to its radius of gyration by Rdf ∼ M. The value of the fractal dimension, calculated from eq 5 is equal to 2.1. This value is significantly higher than that (1.3) observed60 from small-

where M is the molar mass, φv the polymer volume fraction, and T the temperature. S is related to the characteristics of the molecule and especially its structure when Fm depends on the relative movement of this molecule to the solvent. The last term includes ξ, the friction coefficient per monomer unit. In all the rheological theories (Rouse,22 Zimm,23 de Gennes24) this friction coefficient is concerned. It is governed by the solvent in the dilute regime and by polymeric molecules in semidilute or concentrate regimes. All theories concerning the variation of the Newtonian viscosity with the polymer concentration were developed for constant T and ξ parameters. To compare rheological experimental results with theories, the experimental viscosities have to be corrected with the mobility effect, i.e., to work at constant ξ and study only topological effects. This correction may be obtained from the variation of ξ with the polymer concentration. Indeed the influence of the temperature on the viscosity may be expressed using the relation56 ln η0 ) A + Ea/RT + B/(T - T∞)

(6)

where Ea, B, and T∞ represent respectively the activation energy, a parameter depending of the expansion coefficient of the free volume, and the Vogel temperature.62 B is not relevant to the dependence of the viscosity on the temperature for the concentration range studied. The Ea/RT factor depends on the concentration and contains, if it exists, the variation of the friction coefficient ξ with the concentration. To perform the corrections due to the variations of Fm with the polymer concentration, it is necessary to represent (ln η0 Ea/RT) as a function of ln C knowing that Ea varies with the polymer concentration. This correction is valid only if the structure factor, that is to say the conformation, does not depend on the temperature. For the first time the evolution of the rheological curves was studied as a function of the polymer concentration, the shear rate, and the temperature as presented in the first part of this paper. If there is no modification of the conformation, it shall be possible to define a master curve when the specific viscosity is plotted as a function of C[η] (Figure 9), which is the case. From this figure it may be concluded that all the conclusions we have given in the first part are still valid in the studied temperature domain. The conformation of chitosan molecules in solution is usually assumed to be a wormlike chain,63 and the persistence length was found to be equal to around 110 Å indicating a semirigid macromolecular chain.64 To determine the activation energy at a given concentration, the zero shear rate viscosity was studied as a function



Figure 9. Master curve for chitosan solutions in 0.3 M AcOH/0.05 M AcONa at different temperatures.

Figure 10. Arrhenius dependence of the zero shear rate viscosity of chitosan solutions (solvent, 0.3 M AcOH/0.05 M AcONa).

Figure 11. Variation of the activation energy with the polymer concentration (solvent, 0.3 M AcOH/0.05 M AcONa).

of the temperature according to the Arrhenius law. In Figure 10 the variation of η0 as a function of 1/T at constant polymer concentration is plotted. The Newtonian viscosity follows an Arrhenius law, and the slope allows the determination of the activation energy. The variation of the activation energy

with the polymer concentration is plotted in Figure 11. These apparent activation energies are of the same order of magnitude than those described by Wilke65 for polyethylene and by Westra66 for carboxymethylcellulose but lower than polystyrene ones (75.2 kJ/mol).66 Wang17 has determined for a chitosan concentration of 20 g/L (in 0.2 M AcOH/0.1 M AcONa) that the activation energy varies from 25 kJ/mol when DA is 9% to 15 kJ/mol when DA is 25%. These values are slightly smaller than ours, but we have no explanation for this. To interpret this experimental curve, it is necessary to determine the activation energy of the solvent: as the relative viscosity ηrel is defined as the ratio between the viscosity of the solution (ηsolution) and the viscosity of the solvent (ηsolvent) it may be written R∂ ln ηrel ∂T and

-1

)

R∂ ln ηsolution ∂T

-1

-

R∂ ln ηsolvent ∂T-1

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Desbrieres

Ea,relative ) Ea,solution - Ea,solvent The activation energy of the solvent was measured and found equal to 14.5 kJ/mol, in a same order of magnitude as data from Roure67 in aqueous medium. When the specific viscosity was expressed by the relation ηsp ) C[η] + kH(C[η])2 + b(C[η])n the relative viscosity was found as ηrel ) 1 + ηsp ) 1 + C[η] + kH(C[η])2 + b(C[η])n (7) When we are in the dilute regime the relation becomes Figure 12. Continuous temperature ramp for a chitosan concentration of 52.85 g/L (solvent, 0.3 M AcOH/0.05 M AcONa, ω ) 0.1 Hz).

ηrel ) 1 + C[η] + kH(C[η])2 and hence ∂ ln ηrel ∂T-1

)

∂ ln(1 + C[η] + kH(C[η])2) ∂T-1

(8)

For concentrated solutions the relation can be written as ηrel ) b(C[η])n and ∂ ln ηrel ∂T

-1

)n

∂ ln[η] ∂T-1

(9)

Hence when 1 + C[η] + kH(C[η])2 is negligible toward b(C[η])n, the relative activation energy has to tend toward to a maximal value equal to Ea,rel max ) Rn

∂ ln[η] ∂T-1

(10)

Ea,rel max ) nEa,cf0

(11)

i.e.

For C[η] values larger than 15, it was determined from Figure 9 that an average value of n is equal to 5.2 and b has an average value of 7.96 × 10-4. As the average value of the Huggins constant within the temperature domain was 0.45, the b(C[η])n term is predominant toward (1 + C[η] + kH(C[η])2) when C[η] values are larger than 15. The Ea,cf0 was defined from the variation of the intrinsic viscosity of the chitosan with the temperature and found to be equal to 5.1 kJ/mol. Hence the maximal relative activation energy will be calculated as equal to 26.8 kJ/mol. The ∂ ln[η]/∂T -1 value was found equal to 740 and comparable with cellulose diacetate values (645).68 This value increases when the molecular weight decreases69 indicating an increase of the stiffness with the decrease in degree of polymerization. Experimentally the maximal value of the relative activation energy was found equal to 23 kJ/mol, which is in a rather good agreement with the calculated one (26.8). The increase of the activation energy is in agreement with the behavior based on the temperature dependence of the chain dimensions when the intrachain hydrogen bonds control the conformation and hence the viscosity.70 As a conclusion the variation of

Figure 13. Comparison of the rheological behavior of chitosan solutions at 278 and 353 K. Evidence of sol-gel transition (polymer concentration, 52.85 g/L; solvent, 0.3 M AcOH/0.05 M AcONa).

the friction coefficient with the temperature is negligible toward that of the dimensions in this domain and for polymer concentrations up to 40 g/L. Considering the gel at a concentration of 52.85 g/L, taking into account the complex viscosities at 0.02 Hz an activation energy equal to 25.4 kJ/mol was measured in the 278-323 K temperature range when experiments were carried out for discrete temperature values. This value is smaller than the maximal value found previously equal to 38 kJ/mol. As already mentioned for other polysaccharides,67 it may be the consequence of an increase of the stiffness (reflected by the persistence length) of the macromolecular chain related with the increase of intermolecular interactions when the polymer concentration increases, these interactions limiting the mobility of the chains or the increase of intrachain hydrogen bonds. But when a continuous temperature increase was performed on the gel from 278 to 353 K at a frequency of 0.1 Hz, the derivated curve ln η* ) f(1/T) is shown in Figure 12. Two extremal domains are observed and dynamic rheological experiments show that the change in the slope at high temperature corresponds to the gel-sol transition during the temperature increase (Figure 13). This transition temperature was also observed from differential scanning calorimetry experiments. When the slopes within the extremal temperature domains are measured, activation energies of 23 and 35.6 kJ/mol are respectively obtained within the gel and the solution domain. The latter is very close to the maximal value obtained for high concentration chitosan solution. The gel



and the solution do not have similar behaviors against temperature due to the presence of intermolecular interactions.

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Acknowledgment. The author wishes to thank M. Rinaudo and A. R. Khokhlov for fruitful discussions.

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