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Notes Elastic Modulus of Microfibrillar Cellulose Gels Reghan J. Hill Department of Chemical Engineering and McGill Institute for Advanced Materials, McGill University, Montreal, Quebec H3A 2B2, Canada Received May 5, 2008 Revised Manuscript Received July 17, 2008
1. Introduction Scientific and commercial interests in cellulosesthe most abundant biopolymer1sare being rejuvenated by demands for renewable, recyclable, and biodegradable alternatives to petroleum. Cellulose is principally derived from terrestrial plants (e.g., soft-woods, ramie, flax, and cotton) but is also synthesized by bacteria (e.g., Acetobacter xylinum2), photosynthetic algae (e.g., Valonia macrophysa and Oocystis apiculata2), fungi (e.g., Microdochium niVale3), and marine animals (e.g., Halocynthia roretzi4,5). Cellulose has high intrinsic strength, low toxicity, and biocompatibility. Moreover, the high chemical purity and structural homogeneity of bacterial cellulose6,7 has already led to successful commercialization in a variety of niche markets, including health foods, high-end audio, specialty papers, and wound care.8,9 This has undoubtedly motivated efforts to synthesize microfibrillar cellulose from the abundance of macroscopic fibers presently available for the manufacture of paper and other cellulose derived textiles. Note also that lignocellulosic byproduct from agro-industries has also been pursued as the starting material for microfibrillar cellulose.10 Pa¨a¨kko¨ and co-workers11 recently demonstrated that careful control of enzymatic hydrolysis1 and mechanical stress applied to wood-pulp fibers produces high-aspect-ratio microfibrils. These have comparable diameters to microcrystalline cellulose from acid hydrolysis,12 ca. 5-30 nm, but with lengths on the order of several micrometers. The microfibrils entangle to form elastic hydrogel networks with potential applications as reinforcing for multicomponent mixtures and templates for functional nanostructures.11 Moreover, aerogels with useful optical, sieving, structural, and thermal-insulating properties can be synthesized from microfibrillar cellulose hydrogels with the aid of solvent exchange drying.13 Pa¨a¨kko¨ and co-workers synthesized microfibrillar gels starting from bleached sulfite softwood pulp. This was mechanically refined, incubated with endoglucanase enzyme (50 °C for 2 h), refined again, and then homogenized several times. Gels with varying fiber concentration were prepared from a 2% w/w sample by filtration and dilution and dispersed with a highintensity mixer. Transmission electron microscopy (TEM), atomic force microscopy (AFM), and NMR spectroscopy and spectral fitting established a fibril diameter ca. 5-6 nm, with fibril aggregate diameters ca. 10-20 nm. The degree of crystallinity from NMR was reported to be ca. 8-12%. Dynamic rheology was performed with strain amplitudes in the regime of linear viscoelasticity. The samples, which were sealed with silicone oil to avoid drying at elevated temperatures, were
allowed to rest for 5-10 min before performing frequency sweeps in the range 0.01-100 Hz. The experimental data used in this note provide the concentration dependence of the storage modulus at a frequency of 1 Hz (at room temperature). Given the unusually high stiffness and strength of cellulose,14 it is important to understand how the macroscale stiffness of its networks relates to the intrinsic physical properties of the fibers. Indeed, the connection between macroscale rheology and microstructure is central to the fields of polymer physics and materials science.15,16 Interestingly, the elastic modulus of Pa¨a¨kko¨ and co-workers’ microfibrillar gels is unusually sensitive to the fiber concentration. The power-law exponent, suggested by the authors to be about three, is considerably larger than theoretical expectations. Models for fibrous gels yield a storage modulus that scales with the second and slightly higher powers of concentration,17–19 and theories for semiflexible biopolymers exist that yield exponents of 2.2 and 2.5, depending on the fiber concentration.20 Note that de Gennes scaling theory21 and Doi-Edwards theory22 for swollen gels and entangled polymer melts yield an exponent of 2.25. This note presents a new scaling theory that exhibits the experimentally observed transition from a power-law exponent 11/3 ≈ 3.67 at low fiber concentrations to 7 at high concentrations. Also noteworthy is that the unknown dimensionless constants in the theory are ascertained from the experiments to be order-one quantities. This supports the underlying physical basis of the theory, where the elastic energy of cellulose fibers is attributed to bending. Entanglement imparts a predominantly elastic rheology that is insensitive to frequency over a wide range of frequencies,11 suggesting a microstructure in which Brownian forces (entropic free energy) and microstructural reconformation are weak.
2. Scaling Theory The characteristic radius and length of the fibers are denoted a and l, respectively, with fiber volume fraction φ ∼ nla2, where n is the fiber number density. Note that semidiluteness is generally achieved when nl3 ∼ φ(l/a)2 > 1 with φ , 1, so (a/ l)2 < φ , 1. However, semidiluteness is not sufficient for entanglement. Rather, as demonstrated belowsand supported by Pa¨a¨kko¨ and co-workers’ experiments11sa sufficient condition for entanglement of microfibrillar cellulose is that φ J a/l. The microfibrils in Pa¨a¨kko¨ and co-workers’ experiments are not monodisperse. Therefore, the single radius and length adopted here are suitably averaged values, details of which are well beyond the scope of present scaling methodology. Nevertheless, the bulk elastic modulus is ultimately independent of the fiber radius and length, so the fiber radius and length distributions are unlikely to affect the bulk modulus when the fibers are sufficiently long and entangled. In a random network of entangled fibers, the average number of contacts that each fiber makes with other fibers in the network scales as
N ∼ φ(1 + Rφ)l/a
(1)
Here, the intrinsic semiflexibility of the fibers has been drawn
10.1021/bm800490x CCC: $40.75 2008 American Chemical Society Published on Web 09/13/2008
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Notes
upon to justify neglecting the possibility of fibers intersecting themselves and establishing more than one contact with any other fiber. The first term in eq 1 reflects a perfectly random probability, whereas the second term accounts for the influence of an interaction potentialsdue to electrostatic and van der Waals forces and hydrogen bonding, for examplesthat modulates fiber intersection probability. The magnitude of R indicates the reciprocal solid volume fraction above which two-body interactions become important. This is expected to be positive (negative) if the interaction potential is attractive (repulsive), and it should increase with electrolyte concentration, since electrolyte screens fixed charge on the fibers. The average length of fiber between contact points is
l ∼ l/N
(2)
Moreover, the elastic modulus of a network scales as a bond energy density23
G ′ ∼ k/ξ
(3)
where ξ-3 ∼ nN is the bond number density and k is the average elastic spring constant, i.e., k is the second partial derivative of the bond energy with respect to bond stretching. From eq 1 and φ ∼ nla2, it follows that
ξ/a ∼ φ-2/3(1 + Rφ)-1/3
(4)
In this model, the elastic energy is attributed to the bending of fibers between contact points. It is well-known that a point force applied at the midpoint of an elastic “simply supported” beam is f ∼ ζEI//l 3, where ζ is the displacement at the midpoint, l is the length of the beam, E is its Young’s modulus, and I is the moment of inertia.24 For a circular cross section, I ) (1/4)πa4, so
k ∼ EI/l ∼ Ea /l 3
4
3
(5)
Finally, combining the five scaling relationships above gives
G ′ ∼ E(a3/l 3)φ2/3(1 + Rφ)1/3 ) βEφ11/3(1 + Rφ)10/3 (6) where β is an order-one constant. Noteworthy is that G′ is independent of the fiber radius and length, provided, of course, that the fibers are sufficiently long and entangled. More specifically, a necessary condition for entanglement is that N J 1, so Eqn. (6) is valid when
φ J γa/l
(7)
where γ is a constant related to the average number of contacts (per fiber) required to form a space-filling elastic network. Therefore, with l ∼ 2 µm and 2a ∼ 5 nm, for example (see Figure 2b of the work of Pa¨a¨kko¨ and co-workers11), entanglement is expected when φ J 10-3. Equations 6 and 7 are the principal results of the present scaling theory. Experimentally, G′ has a power-law scaling with respect to the fiber mass fraction φm when φm j 0.005. This range is in excellent agreement with the theoretical expectation φm J 0.002. Here, φm ≈ sφ, where s ≈ 1.5 is the specific gravity of cellulose.25 The analysis leading to eq 6 is similar to that of Satcher and Dewey.18 However, they established that l ∼ aφ-1/2 with ξ ∼ l, which is appropriate for three-dimensional ordered networks. Accordingly, their analysis, and those of others, e.g., Kroy and Frey,19 leads to a bulk modulus G′ ∼ Eφ2. The quadratic scaling with concentration has been successful for interpreting the bulk elasticity of open foams18 and Actin networks with low filament and cross-link densities.26 It is close to the scaling G′ ∼ Eφ11/5 established for fiber networks when the elasticity is entropic.20,26 For semiflexible protein assemblies (e.g., gelatin gels), van der
Linden and Parker27 find G′ ∼ φ1.4 at high concentrations, with G′ ∼ (φ - φp)1.94 near the percolation threshold concentration φp ∼ a/l. Note also that single wall carbon nanotube suspensions28 yield G′ ∼ (φ - φp)2.3, which is in good agreement with theories highlighted in the introduction. In striking contrast, the much more rapidly varying bulk modulus established by eq 6 reflects a lower density of fiber intersections at a given fiber concentration, e.g., eqs 1 and 2 give l ∼ aφ-1 as φ f 0. Therefore, while the present theory predicts a weaker bulk modulus at a given fiber density, the modulus is remarkably sensitive to the fiber concentration. Noteworthy is that the exponent from eq 6 at low concentrations (11/3 ≈ 3.67) is close to the value 3.75 ( 0.11 ascertained by Sahimi and Arbabi29 from simulations of elastic percolation networks with bonds that resist stretching and rotation; moreover, networks with freely rotating, inextensible bonds yield an exponent of 2.1 ( 0.2 that is representative of the foregoing theories for Actin, gelatin, and carbon nanotubes. Sahimi and Arbabi’s simulations (valid for low concentrations near the percolation threshold) therefore corroborate the present theory, which does indeed attribute all the elastic energy of Pa¨a¨kko¨ and co-worker’s microfibrillar cellulose gels to microscale bending. In addition, the present theory accounts for an entangled, random microstructure in which the fiber persistence length is longer than the average distance between contact points. Accordingly, the bulk stiffness increases rapidly with decreasing distance between contact points, with the finite volume of the fibers modulating the probability of forming contact points.
3. Interpretation of Experiments To expedite a quantitative comparison with Pa¨a¨kko¨ and coworkers’ experiments, it should be noted that their data when φm J 0.005 is well represented by the semiempirical relationship (emerging from eq 6)
G ′ ≈ 2.25φm11/3(1 + 21.0φm)10/3 GPa
(8)
This is plotted with the experimental data in Figure 1. The transition from a power-law exponent 11/3 when 0.005 < φm , 1/21.0 ≈ 0.0465 to 7 as φ ) φm/s f 1 provides a remarkably good fit to the data, which is not well-characterized by a single power-law exponent [Fitting a single power law to the data with φm > 0.005 yields a power-law exponent of 4.57]. Furthermore, the fitting parameters (constants), which are examined in detail below, are physically acceptable, and the power-law exponents (spanning the range 11/3 ≈ 3.67-7) are much more appropriate than the single exponent 3 suggested by Pa¨a¨kko¨ and co-workers. The intrinsic elastic modulus of cellulose fibers depends on the degree of crystallinity.14 In particular, the highest elastic modulus measured for highly crystalline cellulose fibers is widely acknowledged to be Ec ≈ 137 GPa,14,30 and the value for amorphous cellulose is generally taken to be the value for glassy polymer Ea ≈ 5 GPa.14 Therefore, from the foregoing fit to the experiments (eq 8), we may write
G ′ ) βEφ11/3(1 + 31.5φ)10/3
when φ J γa/l
(9)
where γ ≈ 2.5 and βE ≈ 1.8E ) 2.0Ea ≈ 10 GPa, with E ≈ 5.5 GPa Young’s modulus for cellulose with 10% crystallinity;14 recall that Pa¨a¨kko¨ and co-workers reported ca. 8-12% crystallinity.
4. Discussion The dimensionless prefactors (constants) in eq 9 (β ≈ 1.8 and γ ≈ 2.5) both attest to the robustness of the scaling analysis.
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G ′ ≈ 1.8E[φ11/3/(1 - )8/3][1 + 31.5φ/(1 - )]10/3 when φ j 2.5a/l (10) Finally, fitting eq 10 to the experimental data in Figure 1 with φm < 0.005 gives ≈ 0.578 (i.e., 57.8% voids with a fiber volume fraction of 2.4φ in the continuous phase). Equation 10 is plotted in Figure 1 (dash-dotted line) with ) 0.578 when φm j 0.005. As expected by the theory, and as corroborated by the experiments, the storage modulus does indeed scale as Eφ11/3 at these asymptotically small values of φ. A more robust test of this theory clearly requires physical and three-dimensional computational experiments in which the fiber aspect ratio and intrinsic modulus can be varied while maintaining a similar degree of (semi)flexibility and varying the fiber concentration over several decades. Such a capability is presently unknown to this author and, therefore, seems to pose an interesting challenge for future work in the area.
Figure 1. Storage modulus G′ (at 1 Hz) as a function of fiber mass fraction φm: experimental data (circles) reported by Pa¨a¨kko¨ and coworkers;11 proposed theory (solid and dashed lines) G′ ≈ 2.25φm 11/3 (1 + 21.0φm)10/3 GPa with constants fitted to the experimental data in the apparent entanglement regime where φm J 0.005 (solid line). Power-law exponents at asymptotically small (φ f 0) and large (φ f 1) fiber volume fractions are indicated. The dash-dotted line is eq 10 with a constant void fraction ≈ 0.578 when φm j 0.005.
The constant γ ≈ 2.5, which identifies the regime of entanglement, indicates that the minimum number of contacts required to bend a fibersin the manner that underlies the theory [e.g., eq 1]sis about three; this is clearly the minimum number of contacts required to bend a simply supported elastic member in the absence of an external torque. Next, the order-one magnitude of β ≈ 1.8, which is due, in part, to taking the intrinsic elastic modulus of the fibers to be E ≈ 5.5 GPa, supports the assumptions adopted in the scaling analysis and seems to corroborate the low degree of crystallinity ascertained by Pa¨a¨kko¨ and co-workers from NMR spectroscopy. Finally, the magnitude of the parameter R ≈ 31.5 indicates that twobody interactions are important at fiber volume fractions φ J R-1 ≈ 0.0317. Recall that this value is likely to change with the solvent, electrolyte concentration, and surface charge. Let us briefly consider the discrepancy between the measurements and theory (see Figure 1) at very low fiber concentrations (φ j 2.5a/l). In this regime, there are too few fiber contacts to form a space filling network if the fibers are randomly and homogeneously distributed. One might then expect the elastic modulus in this regime, if one existed, to be lower than predicted by the theory. Clearly, the measured storage modulus is significantly higher than expected by extrapolating the theory to zero volume fraction (dashed lines). This suggests that the fibers reorganize to form a microstructure that has voids dispersed in a continuous matrix whose elastic modulus is given by the previous theory for homogeneous networks with φ J 2.5a/l. Denoting the void volume fraction by , the fiber volume fraction in the continuous phase must be φ′ ) φ/(1 - ), so the storage modulus of the continuous phase [eq 9 with φ replaced by φ′] is Gc′ ≈ 1.8E[φ/(1 - )]11/3[1 + 31.5φ/(1 - )]10/3. Next, assuming the storage modulus of the composite is approximately G′ ≈ (1 - )Gc′ gives
Acknowledgment. Support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canada Research Chairs program (Tier II) is gratefully acknowledged, as are travel funds from SENTINEL, the NSERC Bioactive Paper Network of Canada. Dr. S. Holappa (Helsinki University of Technology, Department of Forest Products Technology) is also acknowledged for kindly bringing this problem to the author’s attention.
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