Article pubs.acs.org/Biomac
Micromechanics and Poroelasticity of Hydrated Cellulose Networks P. Lopez-Sanchez,*,† Mauricio Rincon,‡ D. Wang,† S. Brulhart,‡ J. R. Stokes,‡ and M. J. Gidley† †
ARC Centre of Excellence in Plant Cell Walls, Centre for Nutrition and Food Sciences, Queensland Alliance for Agriculture and Food Innovation, and ‡ARC Centre of Excellence in Plant Cell Walls, School of Chemical Engineering, The University of Queensland, Brisbane, 4072, Australia S Supporting Information *
ABSTRACT: The micromechanics of cellulose hydrogels have been investigated using a new rheological experimental approach, combined with simulation using a poroelastic constitutive model. A series of mechanical compression steps at different strain rates were performed as a function of cellulose hydrogel thickness, combined with small amplitude oscillatory shear after each step to monitor the viscoelasticity of the sample. During compression, bacterial cellulose hydrogels behaved as anisotropic materials with near zero Poisson’s ratio. The micromechanics of the hydrogels altered with each compression as water was squeezed out of the structure, and microstructural changes were strain rate-dependent, with increased densification of the cellulose network and increased cellulose fiber aggregation observed for slower compressive strain rates. A transversely isotropic poroelastic model was used to explain the observed micromechanical behavior, showing that the mechanical properties of cellulose networks in aqueous environments are mainly controlled by the rate of water movement within the structure.
■
INTRODUCTION The growing interest in cellulose networks arises not only from a biological point of view, but also from a material science perspective, since cell walls are the raw material for industries such as food, paper, and, more recently, biofuels. The primary plant cell wall is a complex composite material, which in dicotyledonous and some monocotyledonous plants can be described as two interacting but independent networks of highly organized cellulose fibrils and hemicelluloses embedded in a pectin network.1 The plant cell wall also contains some structural proteins and up to 90% water. A bottom-up approach is taken here and elsewhere that involves construction of a fiber network to mimic the cellulose scaffold of the cell wall using cellulose producing bacteria such as Gluconacetobacter xylinus.2−5 Cellulose produced by this organism has a high degree of purity, and it is possible to produce a solid water-rich network, or hydrogel, with long cellulose fibers entangled with each other in water. In plant cells, the cellulose microfibrils are the main determinants of anisotropy in cell wall mechanical properties; this is reflected by the fact that the wall is much less extensible in a direction parallel to the cellulose fibers than perpendicular to them.6−10 A degree of anisotropy in bacterial cellulose is also inferred from the way it is produced,11 with cellulose fibrils exhibiting some degree of organization within layers in a similar fashion to the plant cell wall.12 Plant cell walls experience mechanical stresses from the increase in pressure that arises due to osmotic flow of water in/ out of the cell. In plant tissues, water movement occurs not only across the plasma membrane of cells (i.e., the symplast), © XXXX American Chemical Society
but also along the cell wall and intercellular spaces (i.e., the apoplast).13 In the apoplast, pressure gradients are the only driving force for water movement due to the permeable nature of the cell wall. Growth and motion of plants are thus regarded as “mechano-hydraulic” processes, since they are driven by both water movement6,13 and cell wall mechanics.14 In plant cells15 and cell wall analogues,4,5,16,17 cellulose has been identified as the main load bearing component. Rheological studies show that it is not only the stiffness of the cellulose itself that contributes to the mechanical properties of cell wall analogues, but also the physical entanglements and the orientation of the fibers.5 Chanliaud et al. simulated the mechanical effects of turgor pressure using bulge-testing and found that, under tension, bacterial cellulose exhibits a linear elastic behavior at low strains and “strain-softening” at large strains and subsequent failure.5,17 The failure zone in uniaxial tension was shown to be associated with large-scale fiber alignment,18 consistent with this being the main load-bearing component. The mechanical response of cellulose hydrogels has been modeled as a linear elastic material and found to have a Young’s modulus in the range of 200 to 500 MPa in biaxial tensile tests16 and 1 to 10 MPa in uniaxial tensile tests.18 It is suggested that this difference in moduli could be due to the rearrangement of cellulose fibers during the uniaxial tensile testing, a mechanism that is not possible with the biaxial tensile testing.18 One limitation of previous mechanical deformation Received: March 17, 2014 Revised: April 30, 2014
A
dx.doi.org/10.1021/bm500405h | Biomacromolecules XXXX, XXX, XXX−XXX
Biomacromolecules
Article
bacterial strain, kept in Hestrin and Schramm (HS) agar medium, was inoculated onto HS agar plates and incubated for 72 h at 30 °C for revival and subculture. To make the primary inoculum, the medium was adjusted to pH 5 with 0.1 M HCl and glucose solution (50% w/v) was added aseptically. The primary inoculum was prepared by adding a few colonies from the HS agar plate to 20 mL of HS medium. The inoculum was then incubated for 72 h at 30 °C under static conditions. After this time, the cellulose had grown on the surface of the HS medium. For scale-up, 9 mL of HS broth medium was inoculated with 1 mL of primary inoculum in a 70 mL container with a diameter of approximately 41 mm and was incubated for 72 h at 30 °C. To dislodge the bacteria from the samples, they were agitated on an orbital platform shaker (KS 260 IKA-Werke, Staufen, Germany) for 5 min at 350 rpm. The samples were harvested from the medium with forceps and washed 6 times in ice cold deionized water under agitation on the orbital platform shaker at 150 rpm. Discs with an average diameter of 41.2 ± 0.5 and 2.5 ± 0.2 mm thicknesses were obtained. The cellulose samples (glucose content 97.8 ± 1% dry weight, average of three batches) were stored in 0.02% NaN3 to avoid any contamination and microbiological growth. Micromechanical Testing and Rheology. Measurements were carried out on an advanced rotational rheometer (HAAKE Mars III Rheometer, Thermo Fisher Scientific, Karlsruhe, Germany) with a heat adjustable Peltier element at a temperature of 25 °C. A 60 mm diameter plate−plate geometry made of titanium was used. No constraint to fluid movement during confinement of the samples was used, other than the top and bottom plates, and thus, water could flow out of the samples radially during compressive deformations. The samples had a diameter smaller than that of the plates; therefore, both the shear and stress factors were adjusted appropriately.29 The gap (the distance between top and bottom plate) was always zeroed at a normal force of 4 N before each test to correct for zero gap errors, as described elsewhere.30 The samples were handled and loaded into the rheometer with tweezers. The surface tension was used to pull out the excess water by holding the sample against the storage container wall. The cellulose samples were placed at the center of the parallel plates with the help of a stencil. For each experiment, the gap was adjusted according to the sample thickness, which slightly varies due to the biological nature of the samples. Prior to the experiments, small-amplitude oscillatory shear tests were performed in order to determine the linear viscoelastic region of samples compressed to different degrees. A first test was carried out at a low frequency of 0.1 rad/s and a second test at a high frequency of 10 rad/s over a stress range from 1 to 100 Pa. To characterize the effect of confinement and mechanical pressures on the micromechanics and rheology of the cellulose networks, axial compression was applied on the samples followed by a smallamplitude oscillatory shear test. During axial compression the cellulosic hydrogels were compressed by 100 μm at a constant speed while measuring the normal force. Three different compression speeds were chosen (0.33, 3.3, 33 μm/s) to cover a range of deformation time scales that may be found in plants.31 The zero normal force was set for a gap of 2500 μm, where the upper plate was in close contact with the top surface of the samples; therefore, it was considered as the thickness of the uncompressed sample. After each compression step, a small amplitude oscillatory deformation (SAOS) test was performed at a frequency of 0.016 rad/s (1 Hz) and at a low constant stress of 1 Pa, chosen from the linear viscoelastic region, for 60 s. The normal force Fn development was recorded during the relaxation stage simultaneously with the SAOS test. SAOS allows the linear viscoelastic properties of the samples to be measured including the storage (elastic) modulus, G′. A sequence of compression− relaxation/SAOS tests were carried out until a narrow gap of approximately 500 μm was reached that corresponds to the limit of the normal force transducer. At least three samples per compression speed were measured and a high degree of reproducibility was found. Scanning Electron Microscopy (SEM). The microstructure of uncompressed and compressed bacterial cellulose hydrogels was observed by scanning electron microscopy (SEM). Compressed
studies performed on water-rich bacterial cellulose networks is that the flow of water during deformation has not been considered. In addition, these cellulosic hydrogels have not been characterized under mechanical stresses that are similar to the normal stresses that arise at the cell wall due to turgor pressure. Water movement in cellulose networks and plant cell walls can be studied by considering the physics of flow through a porous elastic material. The poroelastic or biphasic theory19 was originally developed to describe the mechanical behavior of hydrated soft tissues, notably articular cartilage.20−22 In this theory, the tissue is modeled as a mixture of two distinct, homogeneous, immiscible, and intrinsically incompressible phases: an elastic solid and an inviscid fluid. Fluid flow and solid deformation are coupled through a momentum exchange term, linearly proportional to the relative velocity of the two phases, which is responsible for the description of the timedependent (viscoelastic) behavior. Further development has incorporated the intrinsic viscoelasticity and plasticity of the solid phase into the original theory,23−26 leading to the prediction of the short relaxation times and very high peak-toequilibrium forces observed in strain-relaxation tests for some tissues, but invariably requiring additional parameters. Articular cartilage and plant cell walls are strikingly similar from a structural standpoint: a fiber-reinforced composite swollen by water (for articular cartilage the fibrillar network is made of collagen, cross-linked by aggregates of proteoglycans). In both cases, water movement through the matrix interstices limits the speed of volumetric changes, avoiding violent deformations due to fluctuations in mechanical forces. The similarity between the two structures and the success of the poroelastic model in the realm of articular cartilage27 suggests that this model can provide a reasonable representation of hydrated cellulosic networks and potentially the plant cell wall itself. The present work investigates the micromechanical responses of cellulose hydrogels using a new rheologicalbased experimental approach combined with simulation using a poroelastic constitutive model. The experimental technique consists of a series of compression and relaxation steps, along with simultaneous measurement of the linear viscoelastic storage modulus during the relaxation stage. In this way we obtain insights into both the water flow from the structure and the mechanical properties of the cellulose network, as it is confined to a relatively thin structure by deforming it under a constant external mechanical pressure, at different compressive strain rates. We model the sequence by assuming that the bacterial cellulose hydrogel behaves as a linear poroelastic matrix and obtain material properties and permeability. Furthermore, we investigate how the strain rate-dependence behavior is related to the flow of water through the cellulosic network. This is performed to provide insights into the mechanics and poroelasticity of cellulose hydrogels. This study also aims to provide a benchmark for developing a biomimetic approach to elucidating the role of noncellulosic polymers on the micromechanics of plant cell walls; this will involve studying the effect of incorporating pectin and hemicelluloses in the bacterial cellulose hydrogels
■
MATERIALS AND METHODS
Production of Bacterial Cellulose Hydrogels. Bacterial cellulose hydrogels were produced following the method described by Mikkelsen et al.28 using Gluconacetobacter xylinus (ATCC 53524 American Type Culture Collection, Manassas, VA, U.S.A.). The B
dx.doi.org/10.1021/bm500405h | Biomacromolecules XXXX, XXX, XXX−XXX
Biomacromolecules
Article
samples of different thicknesses (1500, 1000, and 500 μm) were prepared for SEM immediately after the compression test without further storage. Samples were prepared using a critical point dryer (Autosamdri-815, Tousimis, Rockville, Maryland 20852, U.S.A.) to replace the water following a series of dehydration steps. The samples were first cut into smaller pieces (2 × 2 mm) and directly immersed in liquid nitrogen for about 10 s in order to snap freeze the sample. The samples were then transferred to a solution of 3% glutaraldehyde in methanol for 24 h at −20 °C followed by another 24 h at −20 °C in 100% methanol. The samples were warmed up to 25 °C in anhydrous ethanol before drying in the critical point dryer. Samples were coated with approximately 10 nm of platinum (Bal-tec coater, Leica microsystems, Wetzlar, Germany) and examined using a JSM 6300F scanning electron microscope (JEOL, Tokio, Japan) at 5 kV and 10 mm working distance. Images were taken from at least three different locations of each sample and six images were taken from each position, with magnification increasing from 1000×, 5000×, 10000×, 25000×, 50000×, to 100000×. All images were taken from the top side of the sample, which is the one in contact with air during production. Solid State NMR. To examine changes in cellulose crystallinity following compression, 13C CP/MAS NMR experiments were performed at a 13C frequency of 75.46 MHz on a Bruker MSL-300 spectrometer. Uncompressed samples were blotted dry, and compressed samples were used without modification and packed in a 4 mm diameter, cylindrical, PSZ rotor with a KelF end-cap. The rotor was spun at 5 kHz at the magic angle (54.7°). The 90° pulse width was 5 μs and a contact time of 1 ms was used for all samples with a recycle delay of 3 s. The spectral width was 38 kHz, acquisition time 50 ms, time domain points 2 k, transform size 4 k, and line broadening 50 Hz. At least 2400 scans were accumulated for each spectrum. Spectra were referenced to external adamantane. Dry Weight Measurement. The solids content was determined in triplicate by drying the samples in a vacuum oven at 80 °C and a pressure of 4 KPa for 12 h. The sample weight was measured before and after drying for 12 h. Samples were placed back in the oven for three more hours in order to confirm the previous measurement. Means and standard deviation were calculated. Modeling of the Micromechanical Behavior Using Poroelastic Theory. An extended biphasic theory27 for a linear transversely isotropic case was used to model the mechanical behavior of the cellulose networks. Here, the plane perpendicular to the 3-axis is isotropic (Figure 1), and the material can be fully characterized by five
σn(t ) = E3ε0̇ t0 + E1
ε0̇ R2 C11k
⎧ ∞ exp(− α 2C kt /R2) − exp[− α 2C k(t − t )/R2] ⎫ n 11 n 11 0 ⎬ × Δ3⎨∑ αn2[αn2Δ22 − Δ1/(1 + ν21)] ⎩ i=1 ⎭ ⎪
⎪
⎪
⎪
t > t0
(2) ̂ Here, k represents the apparent permeability (k = k/μ), where k̂ is the intrinsic permeability (fully controlled by the pore structure), μ is the fluid viscosity, R is the sample radius, and αn corresponds to the roots of the transcendental equation 2 ⎛ ⎞ 1 − ν31 E1/E3 ⎟xJ0 (x) = 0 J1(x) − ⎜ 2 ⎝ 1 − ν21 − 2ν31E1/E3 ⎠
(3)
where J1 and J0 are Bessel functions of the first kind. Furthermore 2 Δ1 = 1 − ν21 − ν31 E1/E3
(4)
2 Δ2 = (1 − ν31 E1/E3)/(1 + ν21)
(5)
Δ3 = Δ2 /Δ1
(6)
The so-called aggregation modulus characterizing confined compression, C11, is given by
■
2 C11 = E1(1 − ν31 E1/E3)/[(1 + ν21)Δ1]
(7)
RESULTS AND DISCUSSION Pressures Achieved During Mechanical Deformation at Different Strain Rates. High mechanical pressures were generated in the hydrated cellulose networks in an attempt to mimic the confinement conditions encountered in plant cell walls under turgor pressures. High normal pressures were achieved by reducing the gap between two parallel plates in the rheometer. As the gap is reduced, the samples are compressed between the plates, thus the gap size corresponds to the sample thickness. The increase in the normal force Fn as the gap is reduced followed by the total or partial recovery of normal force recorded during the oscillation step is shown in Figure 2. Samples were compressed from a gap of 2500 μm, which
Figure 1. Representation of the cellulose samples geometry, where 1− 2 is the plane of transverse isotropy.
material functions: the Young’s modulus and Poisson’s ratio in the transverse plane (E1 and ν12 = ν21) and out of plane (E3 and ν31 = ν13 E1/E3) and the out of plane shear modulus, G31. The normal stress σn(t) resulting from a ramp displacement in the z direction at a constant strain rate ε̇ during t0 seconds, followed by a relaxation stage at constant strain is σn(t ) = E3ε0̇ t + E1 0 < t < t0
ε0̇ R2 ⎧ 1 Δ3⎨ − C11k ⎩ 8 ⎪ ⎪
∞
∑ i=1
Figure 2. Typical normal force Fn dependence on gap. Several cycles are shown from a gap of 2500 μm, corresponding to the thickness of the uncompressed sample, to a narrow gap of 500 μm. Each cycle shows an increase in Fn during the compression step (in this case 33 μm/s) followed by a drop in Fn recorded during the oscillation step. Inset represents one cycle.
⎫ ⎬ 2 2 2 αn [αn Δ2 − Δ1/(1 + ν21)] ⎭ exp(− αn2C11kt /R2)
⎪ ⎪
(1)
and C
dx.doi.org/10.1021/bm500405h | Biomacromolecules XXXX, XXX, XXX−XXX
Biomacromolecules
Article
sample is allowed to expand freely), which is the case in our experiments. This seems to be the case of bacterial cellulose hydrogels, which showed hardly any deformation in the direction orthogonal to the compression and collapsed in the direction of the applied load. A recent study of the deformation mechanics of cellulose networks using Raman spectroscopy37 estimated a slightly negative value of the Poisson’s ratio for bacterial cellulose ν = −0.1. This value was found to be the best fit to the Raman shift rate observed during tensile deformation tests, in which the samples were rotated from 0 to 90°. For the fitting, bacterial cellulose was considered as a 2D uniformly dispersed material. This assumption and the fact that the samples were studied under tension-rotation could explain the small disagreement between the estimated small negative Poisson’s ratio and the positive near zero value measured in our compression studies. Alternatively this discrepancy in Poisson’s ratio could be due to different orientations of the cellulose structure, which has been shown to lead to positive and negative values for a variety of networks.33,38 For example, cork, which has an anisotropic structure, presents ν ≤ 0 in the direction of applied tensile stress and ν > 0 orthogonal to this direction.39 A semire-entrant honeycomb structure has been proposed to model zero Poisson’s ratio materials.40 Microstructural Changes: Strain Rate Dependence. In order to associate the mechanical behavior to the underlying microstructure and molecular properties, changes in the cellulose networks as a result of applied pressure were investigated by SEM. The uncompressed sample (Figure 3)
corresponds to the thickness of the uncompressed sample, to 500 μm. Cellulose samples could not be compressed to gaps below 500 μm under the conditions used due to the large normal forces achieved, which were out of the range of the instrument. Differences in the profile of the normal force development during confinement were observed as a function of the strain rate (0.33, 3.3, and 33 μm/s). The faster the strain rate, the larger the normal forces that were achieved. The maximum normal forces achieved were about 18, 28, and 54 N from low to high strain rate, respectively. The normal force can be converted to a normal stress or pressure σn. The normal stress is defined as the normal force divided by the surface area of the samples (area = 1.32 × 10−3 m2). Normal stress values of about 0.01, 0.02, and 0.04 MPa were reached for 0.33, 3.3, and 33 μm/s strain rate, respectively. We proposed that the normal stresses experienced by the cellulosic hydrogels in our study approximate those stresses experienced in cell walls from appressed cells due to turgor pressure, with values of 0.5−1 MPa.32 Poisson’s Ratio of Hydrated Cellulose Networks. Interestingly, and in contrast with most biopolymer networks, the bacterial cellulose hydrogels collapsed in the direction of the applied pressure, showing hardly any expansion in the radial direction for any of the strain rates studied. Furthermore, the samples never recovered their initial thickness once they were compressed. This irreversible compression may arise due to the weakness or lack of vertical support, leading to a structure that is strong in the radial direction and weak in the compressive direction. There were no significant differences (Student t test p = 0.111 (>0.05), Minitab statistical software) in the diameter of the samples before compression (41.2 ± 0.5 mm) and after compression (41.6 ± 0.4 mm). We calculated the Poisson’s ratio ν, which is defined as the negative of the ratio of the radial strain to the axial strain, from the change in the sample dimensions during compression. The radial strain is then the change in the average diameter divided by the initial diameter and, for small deformations, the axial strain is the change in the average thickness divided by the initial sample thickness ν13 = −dεradial/dεaxial. A near zero Poisson’s ratio of 0.01 ± 0.01 was obtained for bacterial cellulose hydrogels using this definition. During the compression, no obvious slip of the samples was observed; nevertheless, to account for error due to the boundary conditions, a set of experiments was performed, coating the plates with sand paper leading to similar Poisson’s ratio (data not shown). In general, for isotropic materials, the Poisson’s ratio has values between 1 ≤ ν ≤ 0.5, including most known solids, such as polymers and ceramics. Zero or near zero Poisson’s ratios are found for more compressible materials such as glasses and minerals;33 polyurethane foams with an open-cell structure that showed a near zero Poisson’s ratio at 50% compression34 and cork35 v = 0 ± 0.05. Under uniaxial compression, cork shows no lateral expansion, similar to the behavior of bacterial cellulose hydrogels in our experiments. This behavior was explained based on the bending and buckling of cells in the cork tissue, with no radial expansion occurring because the cells fold up.35 Materials with stiff arms in the direction normal to the applied load such as honeycomb structures will resist transverse contraction and exhibit ν ≈ 0. This behavior is also characteristic of some fibrillar tissues such as cartilage,27,36 which display the same residual force when compressed under lateral confinement (the sample is not allowed to expand) and confinement-free conditions (the
Figure 3. Scanning electron micrograph of the uncompressed cellulosic network. Similar microstructures were observed for samples compressed to 1500 and 1000 μm for all strain rates used (images not shown).
consisted of a mat of apparently random oriented cellulose fibers with an approximate diameter of 75 ± 17 nm (ImageJ41), similar to the diameters previously reported.37 No changes in microstructure were observed down to a gap of 1000 μm, with no apparent increase in densification of the cellulose network, independent of the compression speed used. However, a marked change in the cellulose network was observed in samples compressed to 500 μm, depending on the strain rate (Figure 4). Fast compression speed (33 μm/s) produced similar structures to the uncompressed sample, with no apparent changes in cellulose fibers and an overall similar open microstructure with pores of different sizes (Figure 4a,b). In contrast, when the samples were compressed at a slow speed (0.33 μm/s), the cellulose fibers appeared to be thicker, as if D
dx.doi.org/10.1021/bm500405h | Biomacromolecules XXXX, XXX, XXX−XXX
Biomacromolecules
Article
Figure 4. Scanning electron micrographs of bacterial cellulose (A) compressed at a speed of 33 μm/s, (B) large magnification image of the same sample illustrating the open structure of individual cellulose fibers, (C) compressed at a speed of 0.3 μm/s, and (D) large magnification image of the same sample showing the aggregation of the cellulose fibers into thicker bundles.
they were merging with each other, showing a clear increase in density of the cellulose network (Figure 4c,d). This fiber aggregation was confirmed by image analysis, which revealed that the mean diameter of the fibrils of the slowly compressed samples was 115 ± 34 nm, whereas that of the rapidly compressed networks was 75 ± 17 nm, similar to the uncompressed sample. Cross-polarization/magic angle spinning (CP/MAS) 13C NMR spectroscopy42,43 showed no change in the ratio of crystalline to amorphous cellulose in the cellulosic networks as a function of the strain rate, with values of 13 ± 2% amorphous and 87 ± 2% crystalline both before and after compression. Furthermore, the ratio Iα and Iβ was unchanged after compression with values of 63 ± 2% and 37 ± 2% for each allomorph, respectively. This suggests that there are no changes at a molecular level and that only physical interactions such as fiber entanglements or hydrogen bonds arise as a result of the slow compression process. We propose that water plays a key role during the deformation of these hydrated cellulose networks. While a fast compression speed will generate a large amount of internal pressure from the water which will keep fibers apart from each other, during a slow compression the water will have time to radially flow out of the network bringing cellulose fiber layers close to each other and increasing contact points between them, allowing the formation of new fiber entanglements and resulting in fiber aggregates as observed in the SEM micrographs. In slow compression, the forces holding the fibers together are able to compete against the fluid pressure, which is keeping them apart, producing slow, targeted reorientation of the fibers that leads to thicker bundles. During fast compression, the initial water pressure gradient is higher, drawing the fibers apart followed by rapid water flow (∝ −∇p) that induces fast, random fiber reorientation during the relaxation stage. It is interesting to note that even long after relaxation has been reached for samples undergoing fast compression, the fibrils do not tend to merge, indicating that merging occurs during the compression stage and the initial
part of the relaxation. Our results indicate that down to thickness of 1000 μm the pressures applied on the samples result in only water flowing out of the system with no further changes in the cellulose network. These structural observations seem to be in agreement with a near zero Poisson’s ratio (which suggests that the cellulose networks are very weak in the axial direction), reflected by the layers of cellulose fibers collapsing on top of each other, with hardly any deformation in the radial direction. Micromechanical Behavior: Confinement and Strain Rate Dependence. To identify the effect of confinement on the micromechanical behavior, we compared samples compressed at large gaps of 1500 to 1400 μm and narrow gaps of 600 to 500 μm at three compression rates (Figure 5). During compression, samples were confined to narrower gaps (or sample thickness), and water was released leading to a change in cellulose concentration. The cellulose concentration was calculated for each gap from the dry mass of the samples 0.023 ± 0.001 g and the theoretical water loss at each gap as follows. The water volume (ρ = 1 g/cm3) is determined by subtracting the dry mass from the sample weight. Since the radius does not change during compression, volume changes at each gap are due to water volume loss. The cellulose concentrations of the samples were 1% (w/w) for 1500 μm gap and 6% (w/w) for the narrow 500 μm gap, the latter approaching those concentrations expected in a plant cell wall.1 The experimental data shows that the normal stress increases during compression but relaxes to a finite value with time upon cessation of compression. The graphs in Figure 5 are plotted, with any residual normal stress being set to zero at the start of the test. At the large gap and slow/medium compression speed (Figure 5a,b), the apparent behavior of bacterial cellulose hydrogels under pressure is that of a viscoelastic material; there is a linear-elastic region at low strains that is followed by an apparent plastic deformation and time-dependent relaxation. The elastic region is extended up to a strain of about 0.04 as the deformation rate increases. At narrow gaps, the elastic behavior E
dx.doi.org/10.1021/bm500405h | Biomacromolecules XXXX, XXX, XXX−XXX
Biomacromolecules
Article
Figure 5. Stress relaxation curves for large gaps 1500−1400 μm (a−c) and narrow gaps 600−500 μm (d−f) at three strain rates. Time is expressed at t/t0, where t is the experimental time and t0 is the time at which compression was stopped (t0 = 303 s for (a) and (d); t0 = 30.3 s for (b) and (e); and t0 = 3.03 s for (c) and (f)). The open symbols represent the experimental data, and the solid line represents the prediction from the poroelastic model. Note different scales on both axes.
continues to higher strains going up to 0.04 for the slow deformation rate until a pure elastic behavior is observed at the higher deformation rates of 3.3 and 33 μm/s. It seems reasonable to explain these differences in the micromechanical response based not only on differences in the cellulose microstructure, as shown by SEM, but also to the rate at which water is squeezed out of the structure during the compression; this is a source of the apparent plasticity. When the compression is faster than the rate at which water can move through the cellulose network, the samples become stiffer. At narrow gaps and high strain rate, bacterial cellulose hydrogels showed only a linear elastic region, likely the result of the
restricted movement of water within the time scale of the experiment. At large gaps (Figure 5a−c), the samples recovered almost completely for all of the strain rates applied. During the first few microseconds, the normal stress recovers very quickly and is followed by a slower recovery; the stress was almost completely recovered after only 1 s. However, at narrow gaps (Figure 5d− f), the normal force did not relax completely during the relaxation stage, indicating a significant increase in stiffness in the compression direction. The normal force can be decomposed into three phases: the initial force Fi that results as soon as the gap starts to decrease, the peak force Fp that F
dx.doi.org/10.1021/bm500405h | Biomacromolecules XXXX, XXX, XXX−XXX
Biomacromolecules
Article
water adsorbed within the structure thoroughly connected to the water film wetting the external surface. The actual curvature of the water−air interface is likely to be much smaller than that for an idealized micrometer-sized capillary, making the equilibrium pressure difference closer to zero. As an alternative way to extract E3 from the stress-relaxation curve we considered that for a small compression speed, it is possible to extend the ramp time, t0, while still having a small deformation with respect to the relaxed sample. For α21t0 ≫ tg = C11k, eq 1 yields, after sufficiently long time, ε0̇ tg f (t ) = E3ε0̇ t + E1 Δ3 tg /α12 < t0
(10)
This accounts not only for capillary effects, but also for the fact that water diffusion causes aggregation and reorientation of the fibers, at a rate that would reasonably be of the same order of magnitude as that of water transport ∼tg/α1. This effect could be more accurately captured by taking E3 as a viscoelastic modulus following, for example, a Maxwell formulation. However, this procedure adds further fitting parameters to the scheme and, since the Maxwell constitutive G
dx.doi.org/10.1021/bm500405h | Biomacromolecules XXXX, XXX, XXX−XXX
Biomacromolecules
Article
Table 1. Mechanical Parameters of the Bacterial Cellulose Hydrogels Obtained through Fitting of the Experimental Data with the Linear Transversely Poroelastic Model axial modulus E3 (kPa)
equilibrium axial modulus (kPa) E∞ 3 = E3 + F*/ε̇0t0
radial modulus E1 (kPa)
sample thickness (μm)
0.33 μm/s
3.3 μm/s
33 μm/s
0.33 μm/s
3.3 μm/s
33 μm/s
0.33 μm/s
3.3 μm/s
33 μm/s
1500 1000 500
2.7 9.5 39.3
21 46 102
26 132 129
78 189 268
87 156 172
50.4 125 212
1.2 3.8 9
3.2 6.1 10.2
2 3 3.2
speeds, the observed increase is considerably more pronounced ∞ for the slow compression speed, with E∞ 3 (500 μm)/E3 (1500 μm) = 7.5. For medium and fast compressions, E∞ (500 μm)/ 3 E∞ (1500 μm) is 3.2 and 1.6, respectively. Therefore, the 3 structural changes undergone by the slowly compressed sample (Figure 4) significantly increase its axial elasticity. The radial modulus is nearly 2 orders of magnitude higher than E∞ 3 at all compression rates and gaps, in agreement with the hypothesis that the main mechanical role of the fibril network is to provide biaxial stiffness.16,47 It is also far below the modulus of individual fibers, estimated to be around 80 GPa from atomic force microscopy measurements.48 Permeability and Poroelastic Times. The permeability k of bacterial cellulose hydrogels was calculated from compression data using the poroelastic model as described above. It is assumed that the water movement through the cellulose structure follows Darcy’s law for the flow of fluid through a porous medium. Table 2 contains the permeability at three
equation is ultimately empirical, it does not add much more substance to the understanding of the problem. Memory “loss” (i.e., viscoelasticity) is known to be significant in networks of long rods that reorientate upon deformation and interstitial flow.4,45,46 However, rigorous tackling of such effects has generally required the introduction of additional moduli and relaxation times that are usually hard or impossible to determine uniquely. In the present case, F* can simply be obtained as F* = b1 − Δf in Figure 6. For the linear solution, friction with the plate is ignored. Furthermore, relaxation of the normal force is assumed to be unaffected by SAOS in the plane of isotropy. While full sliding seems a contradictory boundary condition for SAOS (where the top surface of the sample is meant to deform at the same angular velocity of the moving plate in contact with it), finite element analysis shows that eq 1 and 10 are very accurate for a fully adhesive plate as long as the compression speed is small (the influence of friction is described in detail in the Supporting Information). The predicted mechanical behavior for these hydrated cellulose networks is represented by the solid lines in the plots in Figure 5, and the predicted parameters are shown in Table 1. The good agreement between experiments and model allows insights to be obtained into the structural mechanics of these and similar highly hydrated fibrous system. For fast (33 μm s−1) and medium compression speeds (3.3 μm s−1) at 500 μm gap, direct observation of the experimental curves reveals that the poroelastic time tp (which has the same order of magnitude as the relaxation time) is comparable to or higher than the ramp time t0, which means that the simplifications stated in eqs 8 and 9 cannot be used. Therefore, all parameters (E3, E1, v21, and k) were fitted for each particular scenario. The equilibrium modulus in Table 1, E∞ 3 = E3 + F*/ε̇0t0 = σ∞ n /ε̇0t0 corresponds to the stress-to-strain ratio when the normal stress has reached its residual value σ∞ n . F* is always negative, which means that it corresponds to an aggregation force. While general consistency is found between most parameters throughout the different compression rates, this is not the case with the axial modulus E3, which appears to be an order of magnitude lower for slow compression compared to the other compression rates. However, the actual axial modulus is viscoelastic in nature and will be influenced by the deformation history of the sample, including the rate of strain. In this sense, E3 reported in Table 1 is a suitable elastic modulus to fit the compression curve. Fast compression resembles step deformation, for which the instantaneous response is mostly elastic,29 producing the highest values for E3. The decay in E3 is fully assigned to F*, which is naturally larger (in absolute value) the larger the compression rate is. On the other hand, E∞ 3 is purely elastic and independent of history. At 1500 μm gap, all samples start off with a similar value for this modulus. While its value increases with decreasing gap at all
Table 2. Permeability and Poroelastic Times of the Bacterial Cellulose Hydrogels at Three Different Thicknesses, Obtained through Fitting of the Experimental Data with the Linear Transversely Poroelastic Model permeability, k × 1014 m2
poroelastic time, τp/s
sample thickness (μm)
0.33 μm/s
0.33 μm/s
3.3 μm/s
33 μm/s
3.3 μm/s
33 μm/s
1500 1000 500
4.0 1.0 0.5
4.0 1.0 0.5
5.9 1.5 0.5
59 139 250
17.6 62 106
4.96 17.7 29.8
different sample thicknesses, while Figure 7 depicts the evolution of permeability as a function of apparent strain for
Figure 7. Variation of permeability (empty circles) and poroelastic time (filled triangles) with apparent strain at slow compression rate. The fitting line for the permeability corresponds to 1015k = 3.0 + 35.2 × e−4.8ε. H
dx.doi.org/10.1021/bm500405h | Biomacromolecules XXXX, XXX, XXX−XXX
Biomacromolecules
Article
the previously reported linear elastic region of bacterial cellulose.54
slow compression. The exponential decay of permeability with strain has been found in several poroelastic biological structures49,50 and is due to a decrease in pore size and increase in tortuosity. The micrographs in Figures 3 and 4 show that the pore size is in the order of 1−3 μm, approximately 1000× larger than the estimated values for the cell wall pore size about 4−5 nm.51 For an ideal network of cylindrical pores with average radius ⟨r⟩, the permeability is given by
⟨r ⟩2 (11) 8τ 52 τ is the tortuosity coefficient. For τ = 3 (typical of regular lattices) and ⟨r⟩ = 1 μm, k = 4.1 × 10−14 Pa·s, which is quantitatively representative of the permeability values obtained using the poroelastic model (where no details of the single pore dimensions or network tortuosity are incorporated). In the context of plant cell walls, the measured permeabilities fall in the range 10−20 ≤ k ≤ 10−19 m2,53 which suggests that the critical pore size for water transport is not imposed by the cellulose network, but by the interstices between the other macromolecules present in the wall. With k = 10−19 m2, eq 11 predicts ⟨r⟩ ∼ 5 nm when τ = 3. However, eq 11 makes sense only in the hydrodynamic regime. This suggests that, in the plant cell wall, water diffusion occurs either in the molecular regime, where water−hemicellulose or water−pectin interactions dominate over water−water interactions, or in the transition regime. Table 2 shows no significant differences in water permeability between the different compression rates. This is understandable since, though structural changes take place, these would not greatly affect the pore size. It is believed that plants mostly rely on water and their cellulosic walls to move. Given this, the diversity of movements and time scales observed in plants is quite remarkable.31 We calculated the poroelastic time τp ∼ L2/D ∼ μR2/KC11 for the movement of water within bacterial cellulose hydrogels as a function of gap size and deformation rate. L is a characteristic length and D is the gel diffusion time. The faster the compression speed, the lower the values of poroelastic time obtained (Table 2). As depicted in Figure 7 for the slowest compression rate, the poroelastic time increases with increasing strain as a consequence of the decreasing permeability, although the effect is slightly dampened by the increase in the aggregation modulus C11 (given in eq 7), water plays a key role in the deformation mechanics of these systems and potentially in cell wall micromechanics. The calculated poroelastic times (50−250 s) are in agreement with those found during swelling−shrinking behavior of the cell wall,31 indicating that the time scale of the deformation in the experiment is reasonable for our future studies investigating bacterial cellulose composites with noncellulosic polymers as a means to obtain insights into the micromechanics of the plant cell wall. Furthermore, the poroelastic time clearly correlates with the relaxation times in Figure 5 and determine how fast volumetric changes can occur in the structure. Small Amplitude Oscillatory Shear Behavior. The linear viscoelastic region of uncompressed cellulose hydrogels was identified to be between 1 and 10 Pa at both frequencies tested using SAOS (data not shown). Above about 10 Pa, both G′ and G″ decrease, indicating that the structure is perturbed and that there is a nonlinear relationship between stress and strain. In the linear viscoelastic regime, G′ was typically greater than the loss modulus G″ (Figure 8), showing that bacterial cellulose behaves as a strong gel. These results were in agreement with k=
Figure 8. Evolution of the elastic modulus G′ and loss modulus G″ as a function of gap size, or sample thickness, at different strain rates.
For isotropic materials, Poisson’s ratio can be defined as ν = E/2G − 1, where E and G are the elastic and shear moduli, respectively. This expression should be valid in the plane of isotropy (12-plane) for these cellulosic networks, with E, G, and v corresponding to the in-plane shear modulus (G12), Young’s modulus (E1), and Poisson’s ratio (v12), respectively. The values of G12 obtained using this expression are reported in Table 3 for Table 3. Out-of-Plane Shear Modulus G′ and Predicted InPlane Shear Modulus G12 out-of-plane measured shear modulus G′ (kPa)
in-plane predicted shear modulus G12 (kPa) G12 = E1/2(ν12 + 1)
sample thickness (μm)
0.33 μm/s
3.3 μm/s
33 μm/s
0.33 μm/s
3.3 μm/s
33 μm/s
1500 1000 500
16.9 47.8 73.0
14.5 50.7 80.6
16.5 55.3 53.8
19.5 47.2 67
14 39 75.5
12.6 26.3 28
three different sample thicknesses, along with the values of G13 obtained from the oscillatory shear measurements. The predicted G12 is very close to G13 for both slow and medium speed compression at all sample thicknesses, indicating the sample is mostly isotropic with respect to shearing. Since shearing does not involve water loss but only small displacements of water within the structure, the response is not greatly affected by the fluid-dynamics; that is, it is mostly a function of the solid structure. The measured loss modulus G″ was always an order of magnitude below the storage modulus G′, a remarkable behavior for a material that is more than 99.5% w/w water (and approximately 99% v/v water if the skeletal density of cellulose is taken as 1.64 g cc−1), reflecting the very weight efficient structuring in cellulose networks. It is possible that the shear modulus in the direction perpendicular to a given surface is more greatly influenced by the number of knots (connections between microfibrils) per unit area than the orientation of the microfibrils per se. This hypothesis requires further investigation. For fast compression, significant differences are I
dx.doi.org/10.1021/bm500405h | Biomacromolecules XXXX, XXX, XXX−XXX
Biomacromolecules
Article
observed between G′ and G12; however, some superposition was observed between the relaxation and the oscillation dynamics, affecting the experimental storage modulus G′. Moreover, the model produces relatively poor prediction of the relaxation in fast compression, which means that the predicted E2 is not quantitatively accurate. Still, G′ and G12 present similar orders of magnitude. In the next section, we discuss the behavior of G′ with cellulose concentration using viscoelastic theory. Viscoelasticity Dependence on Cellulose Concentration. In the present work, the dependence of the elastic modulus on cellulose concentration was studied considering that decreasing the gap corresponds to an increase in cellulose concentration since only water is lost from the system under compression. Figure 8 shows the evolution of the elastic and loss moduli (values selected at the end of the SAOS test) as a function of the gap size. As the samples were compressed to narrower gaps, or sample thickness, a steep increase in both shear moduli was measured. Similar behavior was observed for all three compression speeds studied, indicating that, despite the differences in the final microstructure observed as a result of the time scale of the deformation (Figure 4), those differences did not have an effect on the small deformation behavior. The modulus/concentration dependence has been analyzed using the method of Clark and Ross-Murphy.55 The observed behavior is similar to that of other biopolymer gels which have been successfully fitted to the cascade model.55−58 This is a percolation theory that predicts a critical concentration for the network formation. Figure 9 shows the scaled elastic modulus
G′ increased dramatically in response to a small increase in cellulose concentration whereupon it starts leveling off toward a plateau. The small deformation behavior of these bacterial cellulose networks has been proposed to be dominated by the number of fibril entanglements.5 Our results suggest that, as the samples are compressed, the effective number of entanglements per unit volume increases, either because cellulose fibers are getting closer to each other or they are being reoriented during compression, and thus G′ increases. The number of possible new entanglements decreases as sites are less available and it becomes increasingly difficult for the stiff cellulose fibers to reorient to accommodate additional cross-links; this is reflected in a smaller slope until no more entanglements are being formed and G′ is approaching a plateau.
■
CONCLUSIONS A poroelastic model has been successfully applied to explain the micromechanical behavior of cellulose hydrogels of bacterial origin. The mechanical response in these bacterial cellulose hydrogels is not just determined by the mechanical properties of the cellulose network, but also the relative time scale of the deformation process and that for which water flows out of the structure. The anisotropy present in bacterial cellulose must be considered. In this case, the extension of the biphasic theory by Cohen et al.27 for the case of transversely isotropic tissues provides good agreement with experiments when the imposed strain rate is sufficiently low. The extracted poroelastic times (50−250 s) for the compressed bacterial cellulose networks agree with predictions by Skotheim and Mahadevan31 for expansion/contraction processes in different plants. Microstructural changes occurred in the bacterial cellulose hydrogels that were a function of the strain rate applied during compression, a slow compression leading to cellulose fiber aggregation through increased physical interactions. A near zero Poisson’s ratio was measured, indicating that the hydrogel is structurally weak in the direction of the compression and strong in the radial expansion direction which is the direction in which the cellulose fibers are orientated. Small amplitude oscillatory shear tests showed that these cellulosic hydrogels have similar modulus-concentration dependence to biopolymer hydrogels. We conclude that this combined experimental and computational approach provides insights into the micromechanics of cellulose hydrogels and is suitable for studying other poroelastic and poro-viscoelastic materials, including human cartilage models. In addition, we consider that our results support the use of this approach and the bacterial cellulose hydrogel model for future studies that will probe a biomimetic cellulose hydrogel that is compositionally similar to the plant cell wall. This will allow, for example, insights to be gained on the role of noncellulosic polymers and structural anisotropy on the micromechanics of plant cell walls.
Figure 9. Scaled elastic modulus G′ as a function of cellulose concentration. Closed symbols represent experimental data. The predicted behavior for the Cascade model is also included, represented by the solid line.
■
ASSOCIATED CONTENT
* Supporting Information
plotted as a function of the cellulose concentration. Compressed samples showed a concentration dependence of the storage modulus of the form G′/G′scale = (c/c0 − 1)n, where c0 is the critical concentration of polymer for gelation, G′scale is a scaling modulus, and n is the exponent. Although a number of parameter combinations can lead to different exponents, the best fit was found for n = 1.8. Exponents in the range n ∼ 2 are found for other biopolymer networks such as agar, carrageenan, and pectin gels.58
S
Detailed description of the poroelastic modeling approach and effect of friction on the mechanical behavior. This material is available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Notes
The authors declare no competing financial interest. J
dx.doi.org/10.1021/bm500405h | Biomacromolecules XXXX, XXX, XXX−XXX
Biomacromolecules
■
Article
(35) Gibson, L. J.; Easterling, K. E.; Ashby, M. F. Proc. R. Soc. A 1981, 377, 99−117. (36) Soulhat, J.; Buschmann, M. D.; Shirazi-Adl, A. J. Biomech. Eng. 1999, 121, 340−347. (37) Tanpichai, S.; Quero, F.; Nogi, M.; Yano, H.; Young, R. J.; Lindstrom, T.; Sampson, W. W.; Eichhorn, S. J. Biomacromolecules 2012, 13, 1340−1349. (38) Hall, L. J.; Coluci, V. R.; Galvao, D. S.; Kozlov, M. E.; Zhang, M.; Dantas, S. O.; Baughman, R. H. Science 2008, 320, 504−7. (39) Silva, S. P.; Sabino, M. A.; Fernandes, E. M.; Correlo, V. M.; Boesel, L. F.; Reis, R. L. Int. Mater. Rev. 2005, 50, 345−365. (40) Grima, J. N.; Oliveri, L.; Attard, D.; Ellul, B.; Gatt, R.; Cicala, G.; Recca, G. Adv. Eng. Mater. 2010, 12, 855−862. (41) Schneider, C. A.; Rasband, W. S.; Eliceiri, K. W. Nat. Methods 2012, 9, 671−675. (42) Yamamoto, H.; Horii, F. Macromolecules 1993, 26, 1313−1317. (43) Atalla, R. J.; VanderHart, D. L. Science 1984, 223, 283−286. (44) Bartell, F. E.; Ray, B. R. J. Am. Chem. Soc. 1952, 74, 778−783. (45) Seifzadeh, A.; Wang, J.; Oguamanam, D. C. D.; Papini, M. J. Biomech. Eng. 2011, 133, 081004−081008. (46) Purslow, P. P.; Wess, T. J.; Hukins, D. W. L. J. Exp. Biol. 1998, 201, 135−142. (47) Schopfer, P. Am. J. Bot. 2006, 93, 1415−1425. (48) Guhados, G.; Wan, W.; Hutter, J. L. Langmuir 2005, 21, 6642− 6646. (49) Argoubi, M.; ShiraziAdl, A. J. Biomech. 1996, 29, 1331−1339. (50) Li, L. P.; Soulhat, J.; Buschmann, M. D.; Shirazi-Adl, A. Clin. Biomech. 1999, 14, 673−682. (51) Carpita, N. C. Science 1982, 218, 813−814. (52) Feng, C. F.; Kostrov, V. V.; Stewart, W. E. Ind. Eng. Chem. Fundam. 1974, 13, 5−9. (53) Molz, F. J.; Ikenberr, E. Soil Sci. Soc. Am. J. 1974, 38, 699−704. (54) Torres, F. G.; Troncoso, O. P.; Lopez, D.; Grande, C.; Gomez, C. M. Soft Matter 2009, 5, 4185. (55) Clark, A. H.; Ross-Murphy, S. B. Br. Polym. J. 1985, 17, 5. (56) Clark, A. H.; Gidley, M. J.; Richardson, R. K.; Rossmurphy, S. B. Macromolecules 1989, 22, 346−351. (57) Clark, A. H.; Farrer, D. B. J. Rheol. 1995, 39, 1429−1444. (58) Stokes, J. R. In Food Materials Science and Engineering; Bhandari, B., Y.H, R., Eds.; Wiley-Blackwell: Chicester,2012; pp 151−176.
ACKNOWLEDGMENTS The authors would like to thank Deirdre Mikkelsen for helpful advice during the production of bacterial cellulose and Bernadine Flanagan for 13C NMR analysis of the samples. Rajesh Ranjan is gratefully acknowledged for performing the image analysis.
■
REFERENCES
(1) Somerville, C.; Bauer, S.; Brininstool, G.; Facette, M.; Hamann, T.; Milne, J.; Osborne, E.; Paredez, A.; Persson, S.; Raab, T.; Vorwerk, S.; Youngs, H. Science 2004, 306, 2206−2211. (2) Whitney, S. E. C.; Brigham, J. E.; Darke, A. H.; Reid, J. S. G.; Gidley, M. J. Plant J. 1995, 8, 491−504. (3) Whitney, S. E. C.; Brigham, J. E.; Darke, A. H.; Reid, J. S. G.; Gidley, M. J. Carbohydr. Res. 1998, 307, 299−309. (4) Astley, O. M.; Chanliaud, E.; Donald, A. M.; Gidley, M. J. Int. J. Biol. Macromol. 2003, 32, 28−35. (5) Whitney, S. E. C.; Gothard, M. G. E.; Mitchell, J. T.; Gidley, M. J. Plant Physiol. 1999, 121, 657−663. (6) Hamant, O.; Traas, J.; Boudaoud, A. Curr. Opin. Genet. Dev. 2010, 20, 454−459. (7) Lloyd, C.; Chan, J. Nat. Rev. Mol. Cell Biol. 2004, 5, 13−22. (8) Baskin, T. I. Annu. Rev. Cell. Dev. Biol. 2005, 21, 203−222. (9) Cosgrove, D. J. Nat. Rev. Mol. Cell Biol. 2005, 6, 850−61. (10) Marga, F.; Grandbois, M.; Cosgrove, D. J.; Baskin, T. I. Plant J. 2005, 43, 181−190. (11) Iguchi, M.; Y, S.; Budhiono, A. J. Mater. Sci. 2000, 35, 9. (12) McCann, M. C.; Wells, B.; Roberts, K. J. Cell Sci. 1990, 96, 323− 334. (13) Dumais, J.; Forterre, Y. Annu. Rev. Fluid Mech. 2012, 44, 453− 478. (14) Routier-Kierzkowska, A. L.; Smith, R. S. Curr. Opin. Plant Biol. 2013, 16, 25−32. (15) Probine, M. C.; Preston, R. D. J. Exp. Bot. 1962, 13, 17. (16) Chanliaud, E.; Burrows, K. M.; Jeronimidis, G.; Gidley, M. J. Planta 2002, 215, 989−996. (17) Chanliaud, E.; Gidley, M. J. Plant J. 1999, 20, 25−35. (18) McKenna, B. A.; Mikkelsen, D.; Wehr, J. B.; Gidley, M. J.; Menzies, N. W. Cellulose 2009, 16, 1047−1055. (19) Mow, V. C.; Kuei, S. C.; Lai, W. M.; Armstrong, C. G. J. Biomech. Eng. 1980, 102, 73−84. (20) Armstrong, C. G.; Lai, W. M.; Mow, V. C. J. Biomech. Eng. 1984, 106, 165−173. (21) Huang, C. Y.; Mow, V. C.; Ateshian, G. A. J. Biomech. Eng. 2001, 123, 410−417. (22) Mow, V.; Guo, X. E. Annu. Rev. Biomed. Eng. 2002, 4, 175−209. (23) Hoang, S. K.; Abousleiman, Y. N. J. Eng. Mech. 2009, 135, 367− 374. (24) DiSilvestro, M. R.; Zhu, Q. L.; Wong, M.; Jurvelin, J. S.; Suh, J. K. F. J. Biomech. Eng. 2001, 123, 191−197. (25) DiSilvestro, M. R.; Zhu, Q. L.; Suh, J. K. F. J. Biomech. Eng. 2001, 123, 198−200. (26) Garcia, J. J.; Altiero, N. J.; Haut, R. C. J. Biomech. Eng. 2000, 122, 1−8. (27) Cohen, B.; Lai, W. M.; Mow, V. C. J. Biomech. Eng. 1998, 120, 491−496. (28) Mikkelsen, D.; Gidley, M. J. In Plant Cell Wall: Methods and Protocols; Popper, Z. A., Ed.; Humana Place, Springer: New York, 2011; Vol. 715, pp 197−208. (29) Macosko, C. W. Rheology: Principles, Measurements and Applications; Wiley-VCH: New York, 1994. (30) Davies, G. A.; Stokes, J. R. J. Rheol. 2005, 49, 919−922. (31) Skotheim, J. M.; Mahadevan, L. Science 2005, 308, 1308−10. (32) Loescher, W. H.; Nevins, D. J. Plant Physiol. 1973, 52, 248−251. (33) Greaves, G. N.; Greer, A. L.; Lakes, R. S.; Rouxel, T. Nat. Mater. 2011, 10, 823−837. (34) Widdle, R. D.; Bajaj, A. K.; Davies, P. Int. J. Eng. Sci. 2008, 46, 31−49. K
dx.doi.org/10.1021/bm500405h | Biomacromolecules XXXX, XXX, XXX−XXX
