Mechanics of Emulsion Electrospun Porous Carbon Fibers as Building

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Functional Nanostructured Materials (including low-D carbon)

Mechanics of Emulsion Electrospun Porous Carbon Fibers as Building Blocks of Multifunctional Materials Yijun Chen, Jizhe Cai, James G. Boyd, William Joshua Kennedy, and Mohammad Naraghi ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.8b10499 • Publication Date (Web): 12 Oct 2018 Downloaded from http://pubs.acs.org on October 14, 2018

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ACS Applied Materials & Interfaces

Mechanics of Emulsion Electrospun Porous Carbon Fibers as Building Blocks of Multifunctional Materials Yijun Chena, Jizhe Caia, James G. Boyda, William Joshua Kennedyb and Mohammad Naraghi*a aDepartment

of Aerospace Engineering, Texas A&M University, College Station, Texas 77843, USA

bMaterials

and Manufacturing Directorate, Air Force Research Laboratory - WrightPatterson Air Force Base, Ohio, 45324, USA E-mail: [email protected]

Key words: Carbon nanofibers, porous, stress concentration, structural energy storage, emulsion electrospinning, multifunctional

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1. Abstract Many multifunctional composite structures incorporate porosity at various length scales in order to increase the available surface area of a functional component. One material system of particular interest is activated or porous carbon fibers and nanofibers that can serve as structural reinforcement as well as providing active surface for added functionality. A key question in the design and manufacturing of these fibers is to what degree the induced pore affects the mechanical properties by inducing discontinuities in the material. To address this problem, mechanics of porous carbon nanofiber (CNF) was studied for the first time as a function of their porous structure. Hollow CNF with porous shell was prepared by coaxial electrospinning a Polyacrylonitrile (PAN)/Poly(methyl methacrylate) (PMMA) blend shell with a PMMA core. PMMA was removed by thermal decomposition during pyrolysis to form pores. Solid shell CNF was prepared as a control with no PMMA in the shell. Results show that the modulus and strength of the porous shell CNF with a porosity of 19.2  1.3% was 65.0  6.2 GPa and 1.28  0.14 GPa respectively, 13.9  2.1% and 35.5  4.9% lower than the solid shell CNF. Finite element analysis models were developed to decouple the effect of stress concentration and reduced load bearing area in porous CNFs on their mechanical properties. The model predictions were in general agreement with the experimental results and were used to identify the most critical parameters which can affect load bearing in porous nanofibers. 2


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Considering the comparison of the experimental and modeling results, the intrinsic material strength does not seem to be affected by inducing pores, thus, fiber and pore geometries might be developed where the load paths are designed for even less of a strength loss.

2. Introduction Carbon nanofibers (CNF), fabricated through carbonization of polymeric nanofibers, have drawn tremendous attentions in recent decades due to their attractive properties such as excellent mechanical performance, electrical and thermal conductivity, high specific surface area and good chemical resistance1. They have been incorporated in various applications including structural composites2, energy storage3, sensing4, catalysis5 and adsorption/separation6 to impart a wide range of functionalities. The combination of promising properties have also brought hopes for using CNFs as the building blocks of multifunctional components in weight and volume sensitive applications. One example to demonstrate the multifunctionality is structural energy storage which can greatly benefit from the incorporation of CNFs for instance in electrically powered vehicles. The electrically powered ground and air vehicles have the potential to be more environmentally-friendly with less sound and air pollution relative to hydrocarbon powered vehicles. However, at the current stage of development, batteries have much lower specific energy (energy per unit mass) than does hydrocarbon fuels. The weight 3



penalty associated with using batteries aboard can be decreased by developing multifunctional battery materials that both deliver electrical power and carry structural loads, i.e., structural batteries and supercapacitors7-12. Combining structural and energy storage functions is a natural step in the development of future vehicles systems because the propulsion system and load bearing frames are the two of the heaviest components of the vehicle. Interestingly, this solution is also motivated from a materials design perspective by the abundance of graphitic structures in structural carbon fibers and nanofibers and the electrodes of Li-ion batteries13. For instance, a structural energy storage material was made by using aramid nanofiber as reinforcement in graphene paper supercapacitor electrodes14. A trade-off between mechanical properties and energy storage capability was demonstrated. For example, the highest modulus and tensile strength of 13 GPa and 100 MPa corresponded to a 50% loss to energy storage capability. Another example with more relevance to modern aircraft is carbon fibers (CF). While CFs are primarily used as load bearing materials in structural components of aerospace and automotive applications due to their excellent specific strength and stiffness, lightweight and environmental resistance, there have recently been efforts to study the performance of this material as electrodes of batteries and supercapacitors. These studies often rely on inducing surface pores in CFs (i.e., activated CFs) to increase the specific surface area of the material to enhance the energy density of supercapacitors. For 4


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instance, commercial CFs were activated with potassium hydroxide (KOH), resulting in a 100-fold increase of specific area with no measurable reduction in mechanical performance15-16. Nevertheless, the specific surface area of activated CFs was limited to 23 m2/g, which is a small fraction of what can be achieved in other carbon materials such as porous carbon nanofibers (CNF)17-18, hollow CNF19-20 and carbon nanotubes21. Thus, although CFs remain an excellent choice for load bearing, they are yet to be demonstrated as promising materials for structural energy storage. One remedy to this problem could be the incorporation of porous CNFs instead of CFs, which due to the combination of their sub-micron diameter (surface area scales inversely with diameter) and interconnected network of internal and external pores can surpass the effective surface area of CFs by a large margin. In this regard, various forms of CNF, such as porous CNF20,

22-25,

hollow CNF19-20,

25

and activated CNF23, have been used as

electrodes in energy storage devices due to their excellent electrical conductivity, large specific surface area and good structural stability26. Electrospinning is commonly used to fabricate the precursors of CNFs out of polymers such as Polyacrylonitrile (PAN), Polyvinylidene fluoride (PVDF) and lignin. Among these polymers, PAN is the most widely used due to its excellent spinnability, low cost and high carbon yield 20. The porosity can be induced by incorporating sacrificial components in the precursor25, 27-31, which can be decomposed during pyrolysis to form pores. Examples of sacrificial component are 5



Poly(methyl methacrylate) PMMA17,

27, 30,

Polystyrene(PS)32 and SiO2

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22, 33.

To further

increase the specific surface area, a sacrificial core can be added to the fibers by coaxial electrospinning to make hollow CNF20, 24-25. In this case, both internal and external surfaces of the hollow CNF can contribute to the specific surface area. While these efforts have effectively enhanced energy storage functionality in CNFs, from principles of solid mechanics and mechanics of materials, it is well established that pores can act as stress concentration sites and reduce the load bearing capability. Therefore, there is a clear tradeoff between load bearing and other functionalities that rely of porosity. However, the significance of this gap and ways to mitigate that has remained unclear at this point. To address this knowledge gap, in this article, for the first time we have addressed the tradeoffs between the induced surface area (porosity) and load bearing in porous CNFs. The outline of the present paper is as follows: In section 2 we discuss the fabrication and characterization of hollow CNFs with both solid shell and porous shell fabricated using a PMMA sacrificial material in PAN. In section 3 we present our experimental results on the load bearing capacity of porous CNFs as a function of pore geometry. The experimental results were interpreted and discussed in light of finite element models of porous CNFs to shed light on the effect of stress concentration around discontinuities (pores) on overall load bearing capacity of CNFs.

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3. Experimental 3.1. Fabrication of solid and porous shell CNF PAN(Mw~150kDa), PMMA with high and low molecular weight (~350kDa and ~15kDa) and DMF were obtained from Sigma-Aldrich and used as received. PMMA (350 kDa) was dissolved in DMF at 16 wt.% by stirring for 12 hrs and PAN was dissolved in DMF at 12 wt.% using a similar method. A PAN/PMMA blend solution was made by dissolving PAN and PMMA (15kDa) in DMF at 7.7 wt.% and 15.4 wt.%. Due to different solubility parameters of PAN and PMMA, the blend solution separates in two phases nearly spontaneously, one rich in PMMA and one rich in PAN. The blend solution was then stirred with a magnetic stir for 24 hr to form a stable emulsion with a continuous phase consisting of PAN solution and a dispersed phase of PMMA solution17. Polymer precursor fibers for hollow CNFs were electrospun using a custom designed coaxial electrospinning setup, the coaxial needle was comprised of a 12 gauge outer needle and a 21 gauge inner needle. To fabricate the precursor fibers for solid shell CNF, the 16 wt.% PMMA(350 kDa)/DMF solution was supplied to the core needle, and the 12 wt.% PAN/DMF solution was supplied to the shell needle with two separate syringe pumps (Harvard Apparatus Model 11). The concentration of both solutions was selected to obtain smooth and beadless fibers. To fabricate the precursor electrospun fibers for porous shell 7



CNFs, the PAN/PMMA emulsion was supplied to the shell needle and the 16 wt.% PMMA/DMF solution was supplied to the core needle. The total concentration of the polymers was selected to have similar viscosity with the 12 wt.% PAN/DMF solution used for the solid shell CNF. The ratio between PMMA and PAN concentration in the emulsion was selected to obtain high porosity17. A constant shell to core flow rate ratio of 1.4 was used in both cases to achieve similar shell thickness. The optimum shell and core flow rate to achieve steady Taylor cone size was found to be 0.7 ml/hr and 0.5 ml/hr for the solid shell CNF and 0.56 ml/hr and 0.4 ml/hr for the porous shell CNF. The electrospinning voltage and distance were set at 15 kV and 20 cm. Fibers were spun on a rotary drum at a take-up velocity of 3.9 m/s, corresponding to an angular velocity of 500 rpm. The temperature and relative humidity during electrospinning were controlled at 251°C and 402% respectively. The electrospun precursor fibers were peeled off from the drum collector and stabilized in a convection oven at 270°C for 2 hrs in an air atmosphere. Following the stabilization, the fibers were then treated at 1100°C in a tube furnace (MTI GSL-1700X) under nitrogen atmosphere for 1 hr with a ramp rate of 5°C/min. Details of the stabilization and carbonization process were obtained from literature 34-35.

3.2. Material characterization The morphology of CNF was analyzed by scanning electron microscope (FEI Quanta 600 FE-SEM). To image the cross sections of CNF, they were cut with a razor blade and 8


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mounted with the cross sections normal to the electron beam. The fiber diameter and shell thickness were measured from SEM images with ImageJ software. Raman spectra of CNFs were obtained by Horiba Jobin-Yvon LabRam Raman confocal microscope with a He-Ne laser (633nm). The pore structure of the fibers was studied by collecting N2 adsorption isotherms at 77K with Quantachrome Autosorb iQ. Before the adsorption test, samples were degassed for 4 hrs at 250°C under vacuum. The specific surface area was calculated by Brunauer-Emmett-Teller (BET) theory. The pore size distribution was obtained by applying quenched solid density functional theory (QSDFT) assuming cylindrical pores. The total pore volume was obtained from total amount of N2 adsorption at relative pressure close to 1. The mechanical properties of the CNF were obtained by tensile tests on single fiber scale using an in-house designed MEMS device (fabricated by MEMSCAP Inc). Single CNF was manipulated under an Olympus optical microscope with a micro-manipulator and aligned with the MEMS device. To grip the CNF on the MEMS device, a 5 × 8 μm platinum block with thichness of 3 μm was deposited with Focused Ion Beam (FIB) (Tescan LYRA3). During the tensile test, the testing deviced was actuated with a picomotor actuator (Newport Picomotor Actuator 8303). Real time optical images were recorded with an Olympus optical microscope while the CNF were tested. The force and displacement were calculated by digital image correlation (DIC) analysis of the real time images using a 9



commercial DIC software (VIC-2D). The resolution of the stress and strain measurements using this method are better than 4 MPa and 0.1% respectively 34. The force applied to the CNFs in this method is calculated by multiplying the stiffness of the micromachined silicon compliant beams (i.e., load cell) by the deflection of the load cell. The former is calculated based on Euler–Bernoulli beam theory by knowing the elastic modulus of silicon (a well-documented value) and the dimensions of the load cell, which is designed and controlled with high precision via micromachining techniques. Moreover, the deflection of the beam and the elongation of the CNF are both calculated by tracking the motion of the micropads on which the CNF ends and the load cell ends are anchored to. This is carried out via using Digital image correlation applied to the optical images taken from the loaded CNFs, with a resolution better than ~30 nm. More details on the accuracy and validity of MEMS device and single fiber test can be found in the previous works 2, 36-37. The fracture surface of CNF was observed by SEM after each tensile test. The cross-sectional area of each CNF was measured on SEM images to determine the stress. For each type of sample, a minimum of 5 tests were performed and the average values and standard deviations were reported.

4. Results and discussion 4.1. Morphology of CNF with solid and porous shell As explained in the Experimental section, PMMA in DMF solution was used to form the core of the precursor fibers, while PAN/PMMA emulsion and PAN solution in DMF was 10


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used to form precursor shells of porous and solid shell CNFs, respectively. In the PAN/PMMA emulsion used for the porous shell fibers, PAN/DMF phase has lower surface tension than the PMMA/DMF phase, as a result, PAN/DMF formed the continuous phase and PMMA/DMF formed the dispersed phase

17-18, 25.

During the thermal treatment at

1100°C, PAN molecules undergo cyclization and form the carbon structure, PMMA in the both the shell and the core decomposes into gaseous phase and leave behind the hollow core and pores. SEM images of the CNF with both solid shell and porous shell are shown in Figure 1. As expected, hollow CNFs were fabricated with uniform shell thicknesses. The CNFs fabricated with no PMMA in the shell have solid cross-sections with no pores (Figure 1b). The CNFs fabricated with PAN/PMMA emulsion showed a highly porous cross-section (Figure 1d). The outer diameter of CNFs was measured on more than forty CNFs using SEM. The average and standard deviation of the outer diameter for solid and porous shell CNF were 1.50  0.23 μm and 1.61  0.29 μm, respectively. Similar method and sample size were used to measure the shell thicknesses. The average shell thickness for solid and porous shell CNF was nearly identical, with values of 0.23  0.06 μm and 0.24  0.06 μm respectively as previously reported 35.

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Figure 1. SEM images of (a), (b) solid shell and (c), (d) porous shell CNFs. (e) Raman spectra of solid and porous shell CNFs Despite differences in the morphology of the shell, the Raman spectra for porous shell and solid shell CNFs, shown in Figure 1e, suggests that both types of fibers have similar partially graphitic structures. The two peaks appeared at ∼1336 cm-1 and ∼1580 cm-1 12


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corresponds to defects (D peak) and graphitic structures (G peak) of carbon materials. The intensity ratio of the D peak to G peak indicates the defect density and quality of graphitic domain, and it is related to the graphitic domain size and defect density within the graphitic domains

38-40.

The ID/IG ratio for the porous shell and solid shell is very close,

1.21 and 1.16 respectively. The similar ID/IG ratio suggests that the existence of PMMA in the shell did not affect the molecular structure of carbonized PAN.

Figure 2. (a) N2 adsorption isotherm of solid and porous shell CNF. (b) Mesopore size distribution of solid and porous shell CNF, calculated using QSDFT method. The porous structure was studied by N2 adsorption. The N2 adsorption isotherm of CNFs with solid and porous shells are both presented in Figure 2a. Adsorption at low relative pressure 0~0.2 is attributed to micropores (pore width smaller than ~3 nm). In both types of fibers, the low adsorption amount indicates that the micropores constitute a small portion of all the pores. These micropores are likely surface pores and roughness on the inner and outer surfaces of the shell. Adsorption at higher relative pressure 0.8~1 13



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is related to mesopores (pores width between 3 nm to 50 nm). In the case of solid shell CNFs, there is small amount of mesopores, as can be seen from the low adsorption amount as well as on the pore size distribution plot in Figure 2b. In contrast, the porous shell CNF have significantly more mesopores, which can be seen from the high adsorption and the pore size distribution, consistent with SEM images (Figure 1d). The specific surface area (SSA) of the CNF with solid shell and porous shell is respectively 36.1 m2/g and 87.2 m2/g, and the total pore volume is 0.070 cm3/g and 0.243 cm3/g respectively, as shown in Table 1. Therefore, the increase in SSA (to ~3x) and pore volume (to ~3.5x) is a result of the mesoporous structure created by decomposition and departure of PMMA during the thermal treatment which leave pores behind. Table 1. Porous structure properties of hollow CNF

Type of CNF Solid shell Porous shell

BET specific surface area (SSA) m2/g 36.1 87.2

Specific pore volume (Vp) cm3/g

Porosity(P)

Normalized porosity (Pn)

0.070 0.243

11.2% 30.4%

19.2%

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4.2. Mechanical properties of CNFs with solid and porous shells Representative apparent stress-strain curves of CNFs with solid and porous shells are shown in Figure 3a. The apparent stress was calculated by  apparent  F / Ashell where F is the measured force applied to the fiber by the microdevices, and Ashell is the total crosssectional shell area (as shown in Figure 4c), which is measured from the SEM images of the fractured surfaces. In both cases, CNF behaved in a linear elastic fashion until fracture and experienced brittle fracture. The mechanical properties are summarized in Table 2 and Figure 3b. SEM images of the fracture surface are shown in Figure 4.

Figure 3. (a) Tensile test stress strain curve of solid and porous shell CNFs. (b) average apparent modulus, apparent strength and stain to failure of CNF.

The average apparent modulus, apparent strength and strain to failure for the solid shell CNF was 75.69.2GPa, 1.990.18GPa and 2.80.2%. The average apparent modulus, strength and strain to failure for the porous shell CNF was 65.06.2GPa, 1.280.14GPa and 15



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2.10.3%. Surprisingly, the mechanical properties of the porous shell CNF were not significantly lower than for the solid shell CNF: the apparent modulus, apparent strength and the strain to failure of the porous shell CNF reduced by ~14%, 36% and 26%, respectively. Table 2. Mechanical properties of hollow CNF

Solid shell CNF Porous shell CNF Reduction

Apparent Modulus (GPa) 75.69.2 65.06.2 13.92.1%

True Modulus (GPa) 80.57.2 -6.51.0%

Apparent Modulus (GPa) 1.990.18 1.280.14 35.54.9%

True strength (GPa) 1.590.17 20.12.8%

Strain to failure 2.80.2% 2.10.3 % 25.84.5%

4.3. Strength compromising parameters and design of resilient porous nanofibers As discussed in section 3.1, the similar ID/IG ratio in Raman spectra indicates similar graphitic domain quality for two cases. Thus, the loss of strength and modulus should be attributed to the differences in stress distribution between porous and solid shell CNFs. The pores can cause stress concentrations and also reduce the effective load bearing area which can in principle lower strength. The former will lead to first local and then propagating failure by generating nonuniform stress fields, while the strength loss in the latter is proportional to the porosity of the fibers. 16


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Figure 4. SEM images of fracture surfaces of (a) porous shell and (b) solid shell CNFs (c) schematic of hollow CNF. The total shell area in the porous CNFs is hatched and includes the area of the pores. On the other hand, the magnitude of stress concentration is strongly dependent on the curvature of pore cap along the loading direction. For instance, consider an elongated ellipsoidal pore with major axis of a and a minor axis of r (a>r), and another spherical pore with radius of r. If the loading direction is along the major axis of the ellipsoidal pore, the ellipsoidal pore has lower curvature (r/a2) than the spherical pore (1/r), resulting in a lower stress concentration and consequently lower loss in strength 41. Therefore, to relate strength loss to porosity in CNFs, the shape of the pores was examined by SEM. To this end, internal pores were exposed by making longitudinal cuts in porous shell CNFs via FIB. As shown in Figure 5 b & c, the pores have high aspect ratio 17



and are elongated along the fiber direction. This pore shape is due to the stretching of the dispersed PMMA in the emulsion during electrospinning, consistent with other findings in literature17. The formation of elongated pores may explain why the mechanical properties of porous shell CNT did not reduce significantly comparing the solid shell CNF. In addition to stress concentration, the reduction in load bearing portions of the crosssection area (porosity) can also lower the strength. The porosity can be calculated using the total specific pore volume (i.e., pore volume per unit mass, Vp) from adsorption and the density of PAN based carbon fiber (ρ = 1.7~1.9 g/cm3)42 as P=Vp/(Vp+1/ρ). The porosity of CNF with solid shell and porous shell turns out to be 11.20.6% and 30.41.2%, respectively. As shown previously, the CNF with solid shell has no observable pores inside the shell, therefore the porosity is mainly due to surface pores. To relate the mechanical properties to porosity in porous shell CNFs, the porosity in porous shell CNFs must be adjusted with respect to the porosity of solid shell CNFs. This adjustment assumes that the CNF with solid shell and porous shell have similar surface pore volume with similar contribution to the mechanical properties. Thus, a normalized porosity (Pn) for the porous shell CNF can be calculated as the porosity difference between the porous and solid shell CNF. The normalized porosity (Pn) of the porous shell CNF is 19.21.3% (calculated by subtracting the porosity of solid shells from that of porous shell CNFs).

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Figure 5. (a) SEM image of fracture surface after mechanical test on porous shell CNF. (b), (c) Longitudinal cross-section of porous shell CNF obtained through FIB etching, (d) Schematic for the shape of the pore and the RVE used in the finite element analysis. Considering the measured apparent mechanical properties, the loss in apparent modulus, 13.92.1%, is comparable to the normalized porosity especially when the uncertainties in measurements are taken into account, whereas the reduction in apparent strength of 35.54.9% is much higher than the normalized porosity. Thus, the loss in apparent modulus is mainly caused by the reduction in the load bearing area of the CNFs when pores are introduced. In other words, if pores are fully aligned with the fiber axis 19



and have very high aspect ratio, the apparent modulus can be predicted by the rule of mixtures, and the reduction in modulus will be equal to the reduction in area. On the other hand, the reduction in apparent strength outweighs the reduction in load bearing area of CNFs by a large margin (loss in strength is ~36%), and therefore requires the consideration of stress concentrations. To separate the effect of area loss and stress concentration on mechanical properties, the true modulus and true strength for the porous shell CNF can be calculated by excluding the pore area from the total cross-sectional area when calculating stress, i.e.  true  F

A

shell



 Apore . Assuming the adjusted porosity (Pn) is only a result of the

mesoporous structure, the area porosity is equal to the normalized porosity, the true modulus and strength will be equal to their corresponding apparent values divided by 1Pn. Thus, the true modulus and true strength of porous CNFs are 80.57.2 GPa and 1.590.17 GPa. Compared to solid shell CNF, the true strength decreased by 20.1% and the strain to failure decreased by 25.8%. The losses in true strength and strain to failure are very close, within the uncertainty. As such, the loss in strain to failure is also mainly attributed to stress concentration as discussed earlier in this section.

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4.4. Prediction of strength reduction in porous shell CNF via continuum mechanics Although carbon fibers (CFs) have achieved great success in structural applications, the failure mechanics of CFs is still not well understood. Even less is known about the failure mechanics of CNFs. The failure of CFs initiates mostly at two types of defects: (1) misoriented turbostratic (i.e. crystalline) regions surrounded by oriented turbostratic regions, similar to a kink in a crystal, and (2) the interface between turbostratic and amorphous regions

43.

Misoriented crystalline regions are essentially shear stress

concentrators. Penev et al

44

used molecular dynamics to study these defects and found

that both the tensile strength and the strain to failure decreased with increasing angle of misorientation. A misorientation angle of 12 degrees can lower the strength by almost 50%. Continuum models have also been used to study the mechanics of carbonized structures with misoriented regions. A seminal study by Reynolds and Sharp

45

simply

resolved the components of the applied stress to obtain the shearing stress acting on the misoriented plane. The analysis is independent of the size of the defect, only targeting orientation. Tagawa and Miyata

46

extended this method to include Weibull strength

distributions that account for the volume (i.e. size) of the fiber, but not the defect size. Naraghi and Chawla

43

developed an experimentally verified model for the failure of the

crystalline/amorphous interface by adapting failure models for short-fiber composites. 21



Each CNF is composed of amorphous carbon (matrix), filled with turbostratic domains43. According to the model, the strength of a portion of CNF as a composite depends on the following variables: the angle between the direction of loading and the axis of the reinforcement, the strength of the matrix (amorphous carbon), the strength of the filler in the planar direction (in plane strength of turbostratic domains), the shear strength of the interface between the turbostratic domains and the matrix, the volume fraction of fillers, their length and diameter, and the critical length of the reinforcements. It was determined that the likely cause of the failure of CNFs is the interface failure between the turbostratic domains and the amorphous carbon, leading to domain pull-out. The presence of pores can trigger additional failure mechanisms, further complicating strength predictions. To the best knowledge of the authors, the failure mechanics of porous CNFs is not studied in the scientific literature. However, given the fact that the failure of CNFs is brittle, prior studies on failure of porous brittle materials, such as porous ceramics, can be elucidating, as discussed below. Several models have been reported in the literature to study the relationship of porosity and strength of porous materials

47.

Balshin

48

suggested a model to capture the

relationship of the tensile strength (often apparent strength) of porous metal and ceramics:

   0 1 P 

b

where

 0 is the strength with no porosity P is the porosity and b

is an empirical constant. Ryshkewitch 49 proposed another relationship between strength 22


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of porous material

   0 e kP where k is an empirical constant. Another relationship

between porosity and strength was developed by Hasselman

50:

   0  cP , where c is

empirical constant. A common theme in all these models is that the apparent strength of porous materials drops more than linearly with porosity, i.e., the true strength of the material drops when porosity is introduced into the material. An explanation for this drop is the development of gradient of stress fields in the material, which together with the requirement to satisfy overall equilibrium with the applied load, results in locally higher than average stress fields, promoting failure. Another common feature in all the aforementioned models is that they are empirical, requiring experimental calibration and validation, to identify the scope of applicability. Less empirical models have also been developed. An example of that is the simple relationship suggested by Haynes porosity:

   0 1 P  K

51

which relates tensile strength of brittle material to

. In this model, the stress concentration factor for spherical

pores is determined by elastic theory. A non-spherical pore shape was not considered. The effect of porosity and the development of stress gradients in the material which can promote failure has also been developed by considering the details of the microstructure of hierarchical structures 52. However, these models are mainly applicable to highly porous foams in which the structure can be simplified as a combination of simpler structural components such as beams and columns. 23



To address the above limitations in understanding the effect of pores on mechanics of CNFs, we studied the mechanics of the building blocks of porous CNFs via Finite Element Analysis (FEA). The material properties used in the model was obtained from studies on PAN based carbon fiber

53.

To this end, a representative volume element (RVE) of the

porous fiber was constructed, Figure 5d. The pores were modeled as cylinders with semiellipsoidal caps (ellipsoid of revolution), Figure 5d. The pore shape was defined by three parameters: length (l), minor axis of the ellipsoidal cap (r) and major axis of ellipsoidal cap (a). The RVE contains two identical pores with each one being one-eighth of a complete pore. The inclusion of two pores with finite length was guided by experimental observation of discontinuous pores. Moreover, the consideration of two pores allows us to take into account the effects each pore can have on the stress fields around the other. Based on SEM images of porous shell CNF, the ratio of between the major and minor axis, a/r, was chosen to be 1-5, and the aspect ratio of the pores (l/r) was chosen to be 5-40. In this RVE, the porosity is defined as P ~

r2 , note this is the area porosity, which is equal 4hw

to the adjusted volumetric porosity stated earlier in section 3.3. In the FEA model (Figure 5d), symmetric boundary conditions were applied to planes at x=0, y=w, z=h. A planar constraint was applied to y=0, i.e. the flat plane remains planer and parallel to the initial undeformed plane after the deformation. A displacement boundary condition was applied at x=l, where the displacement u=l*1% (equivalent to a 24


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1% strain). The resultant force Fd and the maximum principal stress σmax was obtained from the output of the FEA analysis. Due to the existence of the pore, the stress field is nonuniform. In this model, the apparent stress is equal to the resultant force divided by the total cross section area,

 apparent  Fd (hw)

, and the true stress is equal to the ratio of the

resultant force over the net cross section area of the fiber excluding the area of the pores,



 true  Fd hw   r 2 4

and true stress:

then defined as

 . Thus, the following relationship can be obtained between apparent

 true   apparent

K apparent 

concentration factors is

1 P  . The apparent and true stress concentration factors are

 max  apparent

and

K true 

 max  true . The relationship between two stress

K true  1 P  K apparent

. It is also to be noted that since the material

is linear elastic, both the maximum true stress and apparent stress scale linearly with strain, and as such, their ratio is independent of the applied strain. Therefore, the values of and

K true

K apparent

can be calculated regardless of the value of the applied strain and strength of

the CNF. As a result of the applied displacement, the resultant force Fd and the maximum principal stress σmax was obtained from the output of the FEA analysis. It is to be noted that due to the existence of the pore, Fd is a result of a nonuniform stress field. In this model, the apparent stress is equal to the resultant force divided by the total cross section area,  apparent  Fd (hw) , and the true stress is equal to the ratio of the resultant force over 25



the net cross section area of the fiber excluding the area of the pores,  true  Fd

 hw   r 4 . A relationship between apparent and true stress can be obtained: 2

 true   apparent 1 P  . The apparent and true stress concentration factors are then defined as K apparent 

 max  max and K true  . The relationship between two stress concentration  true  apparent

factors is Ktrue  1 P  K apparent . Since the CNFs experienced brittle fracture, the maximum principal stress was chosen as the failure criteria. Hence, the relative strength reduction of the porous shell CNF can be calculated as



0



K 1 , where  is the strength loss and  0 is the strength of a K

fiber with no pores. When Kapparent is used, the strength reduction is due to the combination of both area loss and stress concentration, shown as black line in the results, whereas when Ktrue is used, the strength reduction is only a result of stress concentration effect, shown as blue line in the results. The real pore geometry in the porous CNFs is more complex than the model. Hence, the reliability of the model needs to be examined. In the model, with a porosity of 20%, a/r=3 and l/r=20, Kapparent is 1.89 and the strength reduction is 47.1%. This result is relatively close to the 35.54.9% strength reduction with a porosity of 19.21.3% in the experimental results (gray box in Figure 6b,c). Therefore, the model is fairly accurate and can give good qualitative prediction for the strength of the porous CNF. 26


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Figure 6. Influence of (a) pore aspect ratio (l/r) (b) pore neck curcature (r/a2) and (c) porosity on strength of porous shell CNF.

With the verified model, we conducted a few simple parametric studies on the influence of pore geometry on the strength of the porous CNF. The influence of aspect ratio on the strength was first studied by changing l, with r=50 nm, a=150 nm and P=20%. The result is shown in Figure 6a. As shown in the result, the aspect ratio only affects the strength when the aspect ratio is smaller than 10 and it has very small influence on the strength. In the following study, an aspect ratio of 20 was used for all cases. In order to study the influence of pore neck curvature (r/a2) (curvature along the loading direction), r was kept constant at r=50 nm and a was changed to from 50 nm to 250 nm, P=20% was used. The results are shown in Figure 6b. There seems to be a strong dependence of strength on the pore neck curvature (r/a2). With lower curvature, the stress concentration reduces significantly which leads to lower strength reduction. Furthermore, the influence of porosity on strength was also studied, r=50 nm and a=150 nm was used in this case, w was changed from 75 nm to 400 nm to have a porosity of 5% to 25%. The results are 27



shown in Figure 6c. Thus the total porosity has very small influence on the stress concentration (blue line) and loss in strength. This insignificant reduction is mainly caused by a reduction in load bearing area. In Figure 6b & c, we have also presented the experimental results which clearly shows the reliability of the model. The results presented here can be used to design porous CNFs for multifunctional applications in which load bearing is one of the main functions.

5. Conclusion Hollow carbon nanofibers (CNFs) with porous and nonporous (solid) shells were fabricated by coaxial and emulsion electrospinning using PAN as a precursor and PMMA as sacrificial component. The microstructural characterization showed a rather uniform pore distribution in the porous CNFs. The BET specific area of 87.2 m2/g was achieved on the porous shell CNF, ~200% improvement with respect to solid shell CNFs. The mechanical properties of the hollow CNF with porous and solid shell were characterized by single fiber MEMS test. The average values of modulus, strength, and strain to failure of the solid shell CNF and porous shell CNF was 75.69.2GPa and 65.06.2GPa, 1.990.18GPa and 1.280.14GPa and 2.80.2% and 2.10.3% respectively. The modulus, strength, and the strain to failure decreased by 13.92.1%, 35.54.9% and 25.8%4.5% when pores were introduced in CNFs. The loss in true mechanical properties, such as true strength and modulus, was mainly attributed to stress concentrations around pores. 28


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Continuum mechanics models of the porous shell CNF were built to study the influence of the pore geometry on the strength of porous CNF. The model predicts a 40%~70% strength reduction at 20% porosity, comparable to the experimentally measured loss in properties. Porosity and pore aspect ratio have limited influence on strength. Pore curvature along the loading direction was identified to have the most significant influence on the strength. It is important to note that since the intrinsic material properties was not affected by the pore generating process, a better designed pore geometry can be developed in the future to mitigate stress concentration and achieve even less strength loss.

6. Author Information 6.1. Corresponding Author * E-mail: [email protected]

6.2. Notes The authors declare no competing financial interest.

29



7. Acknowledgments We would like to thank Air Force Office of Scientific Research for supporting this work under the award number FA9550-15-1-0170. We are also grateful to Materials Characterization Facility (MCF) and Microscopy and Imaging Center (MIC) at Texas A&M University and Shared Equipment Authority (SEA) at Rice University for the use of their characterization equipment. We also appreciate the support from the seed grant funding from Texas A&M Energy Institute.

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TOC figure

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Mechanics of Emulsion Electrospun Porous Carbon Fibers as Building

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