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Lightweight and highly compressible, non-crystalline cellulose capsules Christopher Carrick, Stefan B Lindström, Per Tomas Larsson, and Lars Wagberg Langmuir, Just Accepted Manuscript • DOI: 10.1021/la501118b • Publication Date (Web): 28 May 2014 Downloaded from http://pubs.acs.org on June 7, 2014
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Lightweight and highly compressible, non-crystalline cellulose capsules By: Christopher Carrick,*a Stefan B. Lindström,b Per Tomas Larssoncd and Lars Wågberg*a,d
a
KTH Royal Institute of Technology, School of Chemical Science and Engineering,
Department of Fibre and Polymer Technology, SE-100 44 Stockholm, Sweden. b
Department of Management and Engineering, the Institute of Technology, Linköping
University, 581 83 Linköping, Sweden. c
Innventia AB, Drottning Kristinas väg 61, 114 86 Stockholm, Sweden.
d
KTH Royal Institute of Technology, School of Chemical Science and Engineering,
Wallenberg Wood Science Centre, WWSC, , SE-100 44 Stockholm, Sweden. † Electronic supplementary information (ESI) available:
Abstract We demonstrate how to prepare extraordinarily deformable, gas-filled, spherical capsules from non-modified cellulose. These capsules have a low nominal density, ranging from 7.6 to 14.2 kg/m3, and can be deformed elastically to 70% deformation at 50% relative humidity. No compressive strain-at-break could be detected for these dry cellulose capsules, since they did not rupture even when compressed into a disc with pockets of highly compressed air. A quantitative constitutive model for the large deformation compression of these capsules is derived, including their high-frequency mechanical response and their low-frequency force relaxation, where the latter is governed by the gas barrier properties of the dry capsule. Mechanical testing corroborated these models with good accuracy. Force relaxation measurements at a constant compression rendered an estimate for the gas permeability of air
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through the capsule wall, calculated to 0.4 ml µm/m2 days kPa at 50% relative humidity. These properties taken together open up a large application area for the capsules, and they could most likely be used for applications in compressible lightweight materials and also constitute excellent model materials for adsorption and adhesion studies.
Introduction Capsules with encapsulated gas are used in food and cosmetics to reduce the costs of i.e. pigments and increase the shelf-life of the products.1, 2 They are also used in functionalized materials to reduce weight, provide an acoustic bandgap, or enhance thermal insulation properties.3 It is also conceivable to create a material entirely from gas-filled capsules, which would then share many mechanical properties with closed foams, but with greater versatility owing to the highly controllable microstructures formed by the monodispersed capsules. Common gas-filled capsules are prepared by expanding polystyrene spheres (Styrofoam™). These spheres are merged together creating a lightweight foam material. Polystyrene foam can be compressed elastically up to approximately 5%. When further increasing the compressive strain, the material starts to buckle, yield and fracture, and the stress versus compressive strain levels out into a plateau regime at approximately 60% compression. After 60% compressive strain, the material reaches a densification regime where the cell walls are crushed together and the resistance against load increases rapidly.4 Another more compressible foam material has been prepared by cross-linking polyvinyl formaldehyde and glutaraldahyde. For this material, the stress versus compressive strain behavior is essentially repeatable during cyclic compression up to 70%.5 This material is, however, much more expensive than polystyrene, which will ultimately limit its commercial success. With this compressible foam application in mind, we aim to design a renewable, biodegradable and low-cost lightweight material based on cellulose capsules with low density and high
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compressibility as load-bearing elements. The spherical geometry of the capsules also enables fundamental studies of colloid or surface science of cellulose, e.g., the swelling or shrinking behavior of differently charged cellulose colloids, or the characterization of macroscopic cellulose interfacial properties where the adhesion and compatibility between cellulose and other materials could be evaluated. As a first step to achieve this, we report a new way of preparing stable, gas-filled capsules with a controlled shell structure of cellulose, governed by e.g. cellulose concentration and gas pressure.6 These capsules with a regenerated cellulose shell are very stable in water since the cellulose hydrogel structure suppresses gas dissolution. The oxygen permeability is naturally a function of the relative humidity (RH) and when the water content is reduced from the wet to the dry state at 50% RH, the oxygen permeability decreases by a factor of 104.7
Cellulose is an abundant biopolymer since it is the main constituent of many living organisms and one of the components of plants and trees.8 Cellulose is frequently used as an energy source but it can also be converted into more sophisticated products such as clothing, paper and building materials. In all these applications, it is mainly the mechanical properties of the fibers that make the broad utilization of cellulose possible. Cellulose is not, however, considered to be an elastic polymer since it is crystalline and the elastic regime of cellulosebased products is typically less than 1% strain in the dry state.9 For this reason, there is a growing interest in preparing lightweight cellulose structures by freeze-drying dispersed nanofibrillated cellulose or nanocrystalline cellulose dispersions and thus creating a highly porous structure with a density in the range of 14 to 100 kg/m3.10 For comparison, expanded polystyrene foam and the cross-linked polyvinyl formaldehyde with glutaraldehyde have reported densities of about 30 to 120 kg/m3 and 73 kg/m3,11 respectively. The lowest reported
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density for cellulose materials is 7 kg/m3 which was achieved by freeze-drying microfibrillated cellulose, creating a brittle foam material with ultra-high porosity.12
Another way of using the excellent mechanical properties of cellulose would be to dissolve and regenerate cellulose into new shapes and structures, where the strength of the cellulose is efficiently combined with other functionalities. These new shapes must be stable during forming, drying and application. A critical property is then the ability of the capsules to retain gas over an extended period of time. In many materials made of gas-filled capsules, the gas tends to diffuse into the surrounding medium and the capsules subsequently coalesce due to the high gas permeability through the shell structure.13 Capsules with an extended lifetime have recently been prepared with different materials and novel methods where, for instance, the gas-filled capsules are coated with insoluble surfactants, phospholipids or biopolymers, which enhance the barrier properties as well as the buckling stability of the shell.2, 14 To further increase the stability, “reinforced” gas-filled capsules, enriched with colloidal particles, can be very stable due to the presence of a dense interfacial layer that precludes buckling and thus prevents gas dissolution.15 From these studies, it is evident that coating the capsules with polystyrene beads or poly(ethylene glycol) diacrylate alters the physiochemical properties of the capsule shell and suppresses gas dissolution.
The novelty of this paper is the material itself and its very special properties and we demonstrate how to prepare low-density, gas-filled cellulose capsules from native cellulose. Crucially, this can be achieved at a low cost by using only heat to dry the capsules, i.e. no freeze-drying process is needed to maintain the integrity of the capsule. Another novelty is a quantitative description of the mechanism governing its mechanical response of the formed capsules.
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Experiments Materials Cellulose-rich wood fibers (Domsjö Dissolving Plus) were provided by Aditya Birla Domsjö Fabriker, Domsjö, Sweden. They contained 93 wt% cellulose and had a degree of polymerization (DP) of about 780. N,N-dimethylacetamide (DMAc) and lithium chloride (LiCl) were purchased from Sigma-Aldrich and propane gas was purchased from AGA gas AB. The chemicals were used without further purification.
Methods Preparation of cellulose solution The cellulose was dissolved in a mixture of 5 wt% LiCl in DMAc according to a previously described protocol.16 Water present in the solvent impairs the dissolution of cellulose17 and promotes the formation of polymer aggregates.18 Before adding cellulose to the solvent mixture, which is highly hygroscopic, the solvent was therefore heated to 105 °C for 30 min to remove traces of water. Different amounts of oven-dried pulp were then added to reach different cellulose concentrations. Since additional water may be introduced when the hygroscopic cellulose is added, the solution was finally re-heated to approximately 80 °C to further remove traces of water, and thus promote the dissolution of the cellulose. Preparation of cellulose capsules Cellulose capsules were formed by a solution regeneration method,19 which involves saturating the LiCl–DMAc cellulose solution with a soluble gas. In this work, propane gas was used. The cellulose solution with dissolved gas was then added drop-wise into a nonsolvent water, methanol or propanol solution, so that the each cellulose-containing drop
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regenerated as a cellulose gel capsule with a propane bubble in its center. These cellulose capsules were then dried in a 50 °C oven for at least 30 minutes. Also, damaged capsules were intentionally prepared by drying two capsules in contact with each other and then carefully pulling them apart to create a small hole across the wall membrane of one of those capsules. Mechanical testing The mechanical properties of the cellulose capsules were tested in compression with a Deben micro tensile tester (Judges Scientific plc., West Sussex, UK) using a 50 N load cell. One capsule was compressed at a time at a rate of 0.2 mm/min at 23 °C and 50% relative humidity (RH). The procedure was repeated for at least five capsules for each cellulose concentration. Long-time force relaxation tests were performed by compressing dry cellulose capsules individually to 1 N, 6 N and 10 N, or completely wet capsules to 1 N, fixing the level of compression and monitoring the load as a function of time. Structural characterization The wall thickness, dimensions and morphology of the cellulose capsules were determined from images collected with a field emission scanning electron microscope, FE-SEM (Hitachi S-4800) operating at high vacuum. The samples were fixed on metal stubs using carbon tape. The wall thickness was assessed on 10 cellulose capsules at 4 positions each, where the wall structure was exposed by crushing frozen capsules between two metal plates. The surface roughness was determined with a Dimension Icon atomic force microscope (Bruker CA, USA) operating in the tapping mode. SiO2 cantilevers (Bruker, CA, USA) with tips having a radius of curvature of about 8 nm and a spring constant of 40 N/m (according to the manufacturer) were oscillating at their fundamental resonance frequencies which ranged between 200 and 400 kHz. The atomic force microscopy measurements were made under ambient conditions.
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Solid state nuclear magnetic resonance Cross polarization (CP) magic angle spinning (MAS), carbon 13 nuclear magnetic resonance (13C-NMR) spectra were recorded with a BrukerAvance III AQS 400 SB instrument operating at 9.4 Tesla. This instrument was fitted with a double air-bearing two-channel probe head in order to determine the crystalline order of the cellulose in the dry capsules. Samples of the capsules were packed uniformly in a 4 mm zirconium oxide rotor. All measurements were performed at 296 (± 1) K. The MAS rate was 10 kHz. Acquisition was performed with a CP pulse sequence using a 2.95 microsecond proton 90 degree pulse, an 800 microsecond ramped (100 – 50%) falling contact pulse and a 2.5 second delay between repetitions. A small phase incremental alternation with 64 steps (SPINAL64) pulse sequence was used for 1H decoupling. The Hartman–Hahn matching procedure was performed on glycine and the chemical shift scale was calibrated with tetramethylsilane, by assigning the data point of maximum intensity in the α-glycine carbonyl signal a chemical shift of 176.03 ppm. Gas chromatography Head space gas chromatography, Hewlett Packard 6890 gas chromatography system (Germany), was used to analyze the residual propane gas in the dry capsules after preparation. Each experiment produced at least 300 dried cellulose capsules, corresponding to approximately 4 ml of gas-filled capsules. The capsule preparation included 2 hours regeneration in water, where the cellulose solvent was removed, and 30 minutes drying in a 50 °C oven. Immediately after the drying, the capsules were placed in a 20 ml chamber containing inert nitrogen gas, and the chamber was closed with a septum lid. For the analysis, the head space gas was injected 5 minutes, 1 hour, 2 hours, 5 hours and 24 hours after the start of the experiment. Each injection contained 10 µl of gas, and the composition of the displaced gas was analyzed. Experiments were also performed on severely damaged capsules in the closed chamber by pinching the capsules with a needle.
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Heat release during compression An infra-red camera (FLIR SC6000 MWIR) was used to monitor the infra-red radiation emitted from the cellulose capsule surface during large, cyclic compression between 0% and approximately 90% of the capsule diameter. The capsules were compressed between two IRtransparent glass plates. Three cellulose capsules were monitored at the same time during at least five deformation cycles. The experiment was performed under ambient conditions.
Results Preparation of gas-filled cellulose capsules Propane gas was dissolved in the LiCl-DMAc cellulose solution. This solution was then regenerated with the aid of a non-solvent/precipitating agent. Since propane gas has different solubilities in water, methanol and propanol, it was possible to prepare different types of capsules by altering the composition of the non-solvent (Fig. 1). When water was used to regenerate the cellulose, swollen, propane-gas-filled capsules formed. When the polarity of the non-solvent was reduced using 1:1 water/methanol mixture, methanol and propanol, less propane gas was contained within the cellulose capsules formed.
Figure 1.Cellulose capsules prepared from a 1 wt% cellulose solution with dissolved propane gas four hours after regeneration in from left to right a) water, b) 1:1 mixture of water and methanol, c) methanol and d) propanol.
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The capsules that were investigated in this study had an external diameter of approximately 3 mm with a thin cellulose shell as shown in Fig. 2a. The wall thickness was controlled by the cellulose concentration in the solution. Two species of cellulose capsules, CC1 and CC2 (see table 1), were used in the further experiments, where water was used as non-solvent: The first batch (CC1) was prepared from a 1.0 wt% cellulose solution, and these capsules had a wall thickness h = 4.0 µm and a wall thickness-to-diameter aspect ratio of 1:750. The second batch (CC2) was prepared from a 1.5 wt% cellulose solution, resulting in h = 7.5 µm and an aspect ratio of 1:400. The capsule density was estimated to be as low as approximately 8 kg/m3 for the CC1 capsules, which is similar to ultra-high porosity and light-weight cellulose foam structures produced from cellulose nanofibrils using a freeze-drying process.12 Furtheremore, the wall density, 𝜌! was then calculated as: 𝜌! =
𝑚 4𝜋 ! 𝑎 − (𝑎 − ℎ)! 3
(1)
where m is the mass, a is the radius, and h is the wall thickness of the cellulose capsule. This wall density was approximately 1200 kg/m3 which is similar to densely packed cellulose films with high gas barrier properties.7 The wall porosity of the dry cellulose capsules was taken to be 1 − 𝜌! /𝜌! , where ρc = 1582 kg/m3 is the density of the one-chain, triclinic crystal structure of cellulose.20 The cellulose wall is however swollen in the wet state and the gas diffusion rate through the cellulose shell structure is then higher. When the contents of newly prepared dried cellulose capsules were explored by gas chromatography, no traces of propane gas could be found, which suggests that the propane gas is replaced by air during the drying process.
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Figure 2. SEM micrograph of air-dried cellulose capsules where a) represents a CC1 capsules, b) the wall cross-section of a CC1 capsule and c) the wall cross-section of a CC2 capsule. The capsules have been cut with a razor blade to show the macrostructure of the thin capsule wall.
Table 1. Effects of the initial cellulose concentration on capsule and capsule wall properties, of capsules regenerated in water. Capsule type Cellulose concentration 3
Wall density (kg/m ) 3
Capsule density (kg/m ) Wall thickness (µm) Surface roughness, Rq (nm) Porosity (void%)
CC1
CC2
1%
1.5%
1201.6
1205.6
7.6 ± 0.6
14.2 ± 0.9
4.04 ± 0.7
7.54 ± 2.4
5.9 ± 0.6
6.4 ± 1.0
24.0
23.8
Structures of the cellulose capsule The dried cellulose capsule wall structure was analysed with SEM and solid state NMR (Figs. 2 and 3). The SEM micrographs of the capsule wall indicate an orthotropic wall structure, with thin layers of cellulose and a small amount of voids. The solid state NMR spectra recorded for a water-soaked cellulose II reference material and for dry cellulose capsules are shown in Fig. 3. The spectrum for the dry cellulose capsules lacks the discernable NMR signal maxima at characteristic positions typical for crystalline cellulose, cellulose I (89 ppm to 88 ppm and 65 ppm) and cellulose II (107 ppm and 89 ppm to 86 ppm), and it is therefore
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concluded that the β-(1-4) D-glucan polymers in the cellulose capsules do not exist in any of these crystal forms and that the polymers are most probably in a non-crystalline form. 1200 1000 800 600 400
Intensity
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200 0 120
110
100
90
80
70
60
50
ppm Figure 3. Solid state NMR spectra of dried cellulose capsules prepared from 1 wt% cellulose solution (solid line) and of crystalline cellulose II (dashed line).
Deformation of a gas-filled capsule in compression Suppose that a sphere, with wall thickness h and external radius a, is compressed between two rigid plates separated by a variable distance x. The compressive strain is defined as: 𝜀 = 1 − so that
𝑥 (2) 2𝑎
> 0 for the case of compression. As described earlier, the cellulose capsules had a
wall thickness between 4.0 and 7.5 µm, depending on the cellulose concentration prior to capsule formation. The diameter of the capsules was about 3 mm.
To assess the mechanical response in compression, normal force measurements were made on capsules of types CC1 and CC2 under compressive load. These measurements were made at 50% RH for both dry and wet capsules. The force versus compressive strain data are plotted in Fig. 4. The mechanical responses of the dry CC1 and CC2 capsules are essentially the same
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despite the rather large difference in wall thickness and capsule mass. This behavior is in contrasted to that of the wet capsules, where a larger force was needed to compress the wet CC2 capsules with a thicker wall than the wet CC1 capsules.
The wet cellulose capsules are initially weaker than the dry capsules but follow the same trend after approximately
= 0.9 deformation. The dry CC1 and CC2 capsules were compressed to
= 0.98 without any sign of rupture. The wet CC1 capsules ruptured at approximately = 0.92 whereas the wet CC2 capsules remained intact throughout the entire compression process into a cellulose disc.
Figure 4. Force–deformation relationship under compression for wet and dry CC1 and CC2 capsules. The inset is a magnification of the wet capsule force–deformation relation, showing the wall-thickness-dependence in the wet state.
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Compressibility of dry cellulose capsules The compressive force was measured for dry CC1 capsules under cyclic compressive deformation with increasing peak strain
= [0.1; 0.3; 0.5; 0.7; 0.9], as shown in Fig. 5. Cyclic
compression measurements were also made on CC2 capsules and these gave the same qualitative result (not shown). According to Fig. 5 and a video recording of the compression cycle (see SI), the dry capsule showed a remarkable elastic behavior during cyclic compression to very large deformations. The data from consecutive compression cycles almost overlapped each other when the peak deformation was increased from
= 0.1 to
= 0.7, indicating that the capsule shape recovered when the capsule was unloaded, but hysteresis was observed in each strain cycle, indicating recoverable losses due to, e.g. friction or viscoelasticity. Owing to the small wall thickness-to-diameter ratio (1:750), the capsules buckle upon compression. This buckling was associated with plastic deformation at large levels of compressive strain, residual wrinkles were observed on the capsules after the release of a compressive load giving a strain of
= 0.9 [Fig 5. and video in SI].
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Figure 5. Compression cycling of a CC1 capsule. The capsule was compressed to a peak strain of
= 0.1 (green), 0.3 (purple), 0.5 (blue), 0.7 (red) and 0.9 (black) in consecutive
cycles. Micrographs show 1) a never-deformed capsule, 2) a capsule deformed to 3) a capsule after the release of a compressive load to
= 0.9 and
= 0.9.
Force relaxation measurement When dry CC1 capsules were held under compression at a constant strain between two plates with an initial compressive force 1 N, 6 N or 10 N, the load decreased rapidly within the first few minutes. After this initial relaxation, the force decreased very slowly (Fig. 6a). A similar trend was seen for all dry capsules held under compression. In absolute figures, the rate of load decrease was higher for the capsules compressed to a higher initial load. When the normalized force is plotted against log(time), a straight line appears for the initial response lasting about 20 hours (Fig. 6b). This type of relaxation process is typical of the primary creep of glassy polymers below the glass transition temperature.21 Since the NMR results show that the cellulose is non-crystalline, it is expected to be in a low crystallinity, non-equilibrium state and to exhibit glassy dynamics. Fig.6b shows that the relative force relaxation decreased linearly with the logarithm of time for approximately 20 hours. After 20 hours, the force decreased more rapidly than expected from an extrapolation of the initial relaxation, suggesting that another deformation process becomes more dominating after longer relaxation times.
For the wet CC1 capsules compressed under a force of 1 N and then kept at a constant strain, a much faster rate of force relaxation was observed than for the dry capsules. The force dropped to a small fraction of its initial value within the experimental time-scale, as shown in Fig. 6c. When these data were plotted in a semilogarithmic diagram with a logarithmic
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vertical axis, the decrease was linear (Fig. 6d). The load thus decreased exponentially which is indicative of a viscoelastic response of a Maxwell-type material, which suggests that the cellulose capsule wall has gone through a solvent-induced glass transition between its dry and wet state.
Figure 6. Force relaxation of CC1 capsules. A) Lin-log plot of dry capsules at 50% RH and 23 C compressed under an initial load of 1 N (square with crosses), 6 N (solid triangles) and 10 N (unfilled circles), B) lin-log plot of the same dry relaxation. C) Lin-lin plot for wet capsules compressed to an initial load of 1 N, and D) log-lin plot of the wet capsule relaxation.
Discussion Structure It was found that the structure of the cellulose capsule depends strongly on the concentration of dissolved gas in the cellulose solution, as well as the solubility of the gas in the
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regenerating non-solvent. The concentration of dissolved gas in the cellulose solution was investigated by dissolving three different gases, nitrogen (N2), carbon dioxide (CO2) and propane, in ambient conditions for one hour. It was found that 0.039, 1.37 and 1.89 g gas/kg was dissolved of the respective gases in the cellulose solution. The low amount of dissolved N2 resulted in a small encapsulated gas volume and a large cellulose wall thickness of 350 µm. When instead using the more soluble CO2 gas, the encapsulated gas volume increased and the wall thickness decreased to 130 µm. When further slightly increasing the gas solubility, by using propane gas, the encapsulated gas volume increased and the wall thickness decreased to 4 µm. This relatively large difference in capsule wall thickness when comparing the CO2 and propane gas can be explained by the solubility of CO2 in the water non-solvent which is 1.5 g/l,22 while the solubility of propane in water is only 0.040 g/l.23 This means that CO2 can easily escape into the non-solvent during regeneration, while propane becomes trapped inside the cellulose capsule. Consequently, the relative gas solubility in the cellulose solution and the non-solvent is very important when designing the capsule’s dimensions.
As shown in the capsule wall cross-section in Fig. 2bc, the material is oriented in the plane of the membrane, presumably due to the elongational flow field created during the formation of the capsule. The wall density is relatively high according to Table 1, but since the density of pure cellulose exceeds 1500 kg/m3 the porosity of the capsule wall is at least 20%. In Fig. 3 it is also clear that the cellulose in the capsule wall has a very low crystallinity, i.e. the capsule wall has an orthotropic, non-crystalline structure.
Deformation
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The results of the mechanical testing of the capsules show that both the gas pressure inside the capsule and the mechanical properties of the non-crystalline, orthotropic shell influence the mechanical properties of the capsule. To separate their importance, we have investigated two different situations, with an essentially impermeable capsule and with a punctured capsule where the gas pressure has a negligible effect.
Factors controlling the compression of a gas-filled capsule When a capsule is compressed, d /dt ≥ 0, between two plane, rigid surfaces, its shape is distorted yielding a strain energy We( ) in the capsule wall. At the same time, the enclosed volume V( ) of gas decreases. Assuming the compression of the gas is a polytropic process, we have22 𝑝! 𝑉(𝜀) ! = , (3) 𝑝(𝜀) 𝑉! where γ is a dimensionless constant, p( ) is the internal pressure, V( ) is the internal volume, 𝑝! = 𝑝(𝜀 = 0) is the initial atmospheric pressure and 𝑉! = 𝑉(𝜀 = 0) is the interior volume of the undeformed capsule. The work required to compress the gas22 is then ! !
𝑊! (𝜀) = −
!!
𝑝! 𝑉 ! 𝑉 !! d𝑉 = −
𝑝! (𝑉 𝜀 1−𝛾
!!!
!
𝑉! − 𝑉! ). (4)
An isothermal process is described by taking the limit 𝛾 → 1, whereas an isentropic process is represented by taking γ to be the ratio of specific heats, which is 1.40 for dry air at 20 C23, neglecting dissipative deformation processes and assuming that the shape of the capsule evolves so that the total potential energy 𝑊! + 𝑊! is minimized, subject to the geometrical constraints imposed by the plane surfaces.
Compression with negligible gas pressure
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The compression of an elastic spherical shell between two plane surfaces has been investigated theoretically24, 25 and experimentally24 for the case of negligible internal pressure, i.e. 𝑊! ≪ 𝑊! . With increasing compression, several distinct regimes of deformation were observed for the spherical shell: (i) At a very early stage of compression, the shell retains a finite curvature at the points of contact.25 (ii) Subsequently, the shell forms a contact disc against the flat surface.24, 25 (iii) The shell then delaminates from the surface and buckles into an inverted spherical cap, while retaining axial symmetry.24, 25 (iv) Axial symmetry is lost, and the deformations localize at an integer number of vertices that increases with .25 The critical compressive strains of the transitions between these regimes are ℎ ℎ ℎ 𝜀!!!! ≈ , 𝜀!!!!!! ≈ 2.5 , 𝜀!!!!!" ≈ 13 , (5) 𝑎 𝑎 𝑎 respectively.25 The force–deformation relation is 𝐹 ∝ 𝜀 ! in regime (i), 𝐹 ∝ 𝜀 !/! in regime (ii)24, 25, and 𝐹 ∝ 𝜀 !/! in regime (iii), derived by differentiation of the elastic energy of the fold.24 A corresponding scaling relation for regime (iv) is difficult to obtain due to the increasing complexity of the geometry. However, since the geometry of asymmetric buckling is less constrained than that of symmetrical buckling, it is reasonable to assume that the force is limited above by the extrapolated behavior 𝐹 ∝ 𝜀 !/! of regime (iii). Because the thickness-toradius ratio of the wall is small for the capsules, ℎ/𝑎 ≈ 3 ⋅ 10!! for CC1, the regime of symmetric buckling (iii) is reached at about buckling is predicted to occur when more slowly than
1/2
= 0.008, and the transition to asymmetric
> 0.04, at which point the force is predicted to increase
.
The damaged capsules, which cannot be affected by gas pressure due to the puncture, display a 𝐹 ∝ 𝜀 !/! scaling (Fig. 7), as expected for buckling in the axisymmetric, delaminated regime
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Langmuir
(iii). Moreover, for a large compression, 𝜀 > 0.2, the scaling is limited by 𝐹 ∝ 𝜀 !/! as predicted in the previous discussion for regime (iv). The damaged capsules did not expand back to their original size after the compressive strain was released, as manifested by a sharp drop in the load, creating a buckled disc-like material. It is also clear from this experiment that the buckling force was small for the damaged cellulose capsules.
Figure 7. Force–compression relation for dry and damaged CC1 capsules. (a) Lin–lin plot. (b) Log–log plot where the dash-dotted grey line has a slope of 0.5.
Compression dominated by gas pressure Due to the small capsule wall thickness and possibly the anisotropy of the wall, the capsule wall is flexible and easily buckles during deformation, while being relatively stiff in response to tensile loads in its plane. This suggests that the capsule wall could be modeled as an inextensible membrane with negligible flexural rigidity. The system is then dominated by the work of gas compression: 𝑊! ≫ 𝑊! , and the energy of the system is then minimized by maximizing the enclosed volume V( ). This is achieved if the capsule can essentially retain its spherical shape between the moving surfaces, while the spherical caps that would have extended across these boundaries buckle and conform to the plane surface, as shown in Fig. 8. This qualitative picture is supported by our video recording of a compressed dry capsule (Supporting materials).
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Figure 8: Geometry of a cellulose capsule under compression when the system is dominated by gas pressure. The interior volume is maximized by folds in the inextensible membrane.
The volume of this “decapitated” sphere is:
𝑉 =
!!! ! !
−
!! !
! !
𝑎−!
sphere
!
2𝑎 + ! − 4𝜋𝑎! ℎ (6)
caps
wall
Insertion of the definition of compressive strain 𝜀 = 1 − 𝑥/2𝑎 leads to 𝑉 𝜀 =
4𝜋𝑎! 𝜀! 3 − 𝜀 3ℎ 1− − . (7) 3 2 𝑎
Since the capsule wall is thin, ℎ ≪ 𝑎, the volume of the undeformed capsule is expressed as 4𝜋𝑎! 3ℎ 4𝜋𝑎! 𝑉! = 1− ≈ (8) 3 𝑎 3 Insertion of these expressions for V0 and V( ) into Eq. (4) gives the work of polytropic gas compression: 4𝜋𝑎! 𝑝! 𝑊! 𝜀 = − 3 1−𝛾
𝜀! 3 − 𝜀 3ℎ 1− − 2 𝑎
The corresponding compressive force is
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!!!
− 1 . (9)
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Langmuir
𝐹! 𝜀 =
dW! d𝑊! d𝜀 d𝑊! 1 = = − d𝑥 d𝜀 d𝑥 d𝜀 2𝑎
2𝜋𝑎! 𝑝! d 𝜀! 3 − 𝜀 3ℎ = 1− − 3 1 − 𝛾 d𝜀 2 𝑎
!!!
− 1
𝜀! 3 − 𝜀 3ℎ = 𝜋𝑎 𝑝! 𝜀(2 − 𝜀) 1 − − 2 𝑎
!!
!
. (10)
This force diverges when 2 − 𝜀 ! 3 − 𝜀 →
6ℎ , (11) 𝑎
which corresponds to the capsule being compressed into a disc.
We can also estimate the in-plane stresses that develop in the capsule wall under the assumption of an inextensible membrane. The in-plane stress of the curved part of the membrane, not in contact with the converging boundaries, is that of a spherical shell with the transmural pressure p( ) – p0: 𝜎(𝜀) =
𝑎 𝑝 𝜀 − 𝑝! (12) 2ℎ
Using Eq. (3) with a thin-wall assumption, h ≪ a, gives 𝑎𝑝! 𝜎(𝜀) ≈ 2ℎ
𝜀! 3 − 𝜀 3ℎ 1− − 2 𝑎
!!
− 1 . (13)
Since the principal stresses of the bulging capsule wall are 𝜎! = 𝜎! = 𝜎(𝜀) and 𝜎! = 𝑝! − 𝑝(𝜀) ≪ 𝜎(𝜀), the von Mises stress is 𝜎! =
1 2
(𝜎! − 𝜎! )! + (𝜎! − 𝜎! )! + (𝜎! − 𝜎! )! ≈ 𝜎(𝜀) (14)
where the approximation holds for h
a. Hence, the von Mises yield criterion is σ( ) = σy,
where σy is the yield stress of the solid. This implicitly defines a critical compressive strain of yielding
y
, obtained by solving 𝜎(𝜀! ) = 𝜎! , which leads to
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Langmuir
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ɛ!!
3 − ɛ!
2ℎ𝜎! 6ℎ =2− −2 +1 𝑎 𝑎𝑝!
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!!/!
. (15)
This equation has one unique analytically available solution in the interval 0
