Surface Ripples of Polymeric Nanofibers under Tension: The Crucial

Article pubs.acs.org/Macromolecules

Surface Ripples of Polymeric Nanofibers under Tension: The Crucial Role of Poisson’s Ratio Shan Tang,*,†,⊥ Ying Li,‡,⊥ Wing Kam Liu,*,‡,§ and Xiao Xu Huang∥ †

Department of Engineering Mechanics, Chongqing University, Chongqing, China, 400017 Department of Mechanical Engineering, Northwestern University, Evanston, Illinois, United States § Distinguished Scientists Program Committee, King Abdulaziz University (KAU), Jeddah, Saudi Arabia ∥ College of Material Science and Engineering, Chongqing University, Chongqing, China, 400017 ‡

S Supporting Information *

ABSTRACT: Molecular dynamics and finite element simulations are performed to study the phenomenon of surface rippling in polymeric nanofibers under tension. Each nanofiber is modeled as a core−shell system that resembles most relevant features extracted from detailed molecular simulations and experiments. Accordingly, our model nanofiber consists of a dense glassy core embedded in a less dense, more flexible, rubbery shell. Poisson’s ratios of the core and shell layers are assumed close to that of compressible and incompressible materials, respectively. Surface rippling of the nanofiber is found, via combined finite element analysis and continuum theory, to be governed by a “polarization” mechanism at the core−shell interphase regime that is ultimately induced by the mismatch between Poisson’s ratios while a mismatch between Young’s moduli seems to play a secondary role. Plastic deformation is a prerequisite for the formation of rippled surfaces, that evolve from initial imperfections, and grow in the presence of uniaxial tension. For this reason, both strain rate and yield stress greatly influence the onset and modes of the observed rippled surface. Our findings are consistent with experimental observations on surface ripples of electrospun nanofibers and pave the way to design polymeric nanofibers with distinct surface morphologies.

1. INTRODUCTION Electrospun polymeric nanofibers, with diameters ranging from 10 to 600 nm, exhibit well-controlled diameters and surface morphologies, high surface area to volume ratios, as well as high stiffnesses and high tensile strengths.1−3 They can be massively produced by electrospinning at low processing cost.4,5 Constituting a novel class of one-dimensional nanomaterials, polymeric nanofibers qualify themselves for a wide range of applications, such as fine filtration,6 reinforcement for nanocomposites,2 templates for synthesizing carbon nanotubes and metallic nanowires,7,8 scaffolds for tissue growth and as drug delivery platforms,9,10 and nanoelectromechanical systems.11 Given the tremendous range of applications, it is important to understand the mechanical properties of polymeric nanofibers rooted in their microstructural features.2,12−18 Mechanical properties of polymeric nanofibers have been extensively studied through experiments and molecular dynamics (MD) simulations. Rutledge and co-workers have studied Young’s moduli and yield strengths of individual electrospun poly(trimethyl hexamethylene terephthalamide) (PA 6(3)T) polymeric fibers subjected to uniaxial tension.15 Both characteristics increase with decreasing diameter D as long as D < 500 nm.15 The enhanced stiffness and strength of thin nanofibers are found to be accompanied by enhanced molecular alignment.15 Buell et al. explored the mechanical properties of © XXXX American Chemical Society

glassy polyethylene (PE) nanofibers via MD simulations with D ranging from 3.7 to 17.7 nm.18 At fixed temperature, Young’s modulus slightly decreases with decreasing D. At the same time, Poisson’s ratio of the nanofibers reduces from 0.3 to 0.1.18 Both the polymer and entanglement densities at the surface of these fibers are found to be small compared with their counterparts at the nanofiber’s center.17,18 More importantly, the polymeric nanofiber is found to exhibit a core−shell structure. In previous experiments,13,16 the polymer density, amount of entanglement, and degree of crystallinity of the shell were found to exceed values obtained for the core in fibers of polystryene (PS) electrospun from dimethylformamide. The relatively hard shell is considered to be induced by the imbalance between slow radial solvent diffusion and rapid convective solvent loss at the surface during the electrospinning process.13,16 Very recently, Camposeo et al.19 correlated the internal structure of an electrospun functional nanofiber with its local stiffness. Their experiments directly unravel that the polymeric nanofiber is composed by a dense internal core embedded in a less dense polymeric shell.19 The dense core is found to be twice as stiff as the polymeric shell.19 Another interesting phenomenon has Received: June 17, 2014 Revised: August 4, 2014

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dx.doi.org/10.1021/ma5012599 | Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. (a) Molecular model of the polymeric nanofiber at T = 0.2. Periodic boundary conditions are applied only in the axial (z) direction of the nanofiber. The blue lines represent the simulation box. (b) Number density of polymer beads within different layers at temperatures T = 1.0 and T = 0.2. Note that the layer thickness depends on T. The number density of polymer beads versus temperature for (c) layer 1 and (d) layer 23. In part b, the polymeric nanofiber is divided into 25 layers along the radial direction from the center to surface of nanofiber. Layers 1 and 25 represent the inner- and outermost layer, respectively. In parts c and d, the solid lines are fitted straight lines from the low temperatures (T = 0.2, 0.28, 0.36) or high temperatures (T = 0.84, 0.92, 1.0). The slope change of fitted lines signals the glass transition temperature Tg, as given in part c.

been observed during the deformation process of electrospun polyacrylonitrile (PAN) nanofibers. Naraghi et al.20 investigated the deformation behaviors of 12 μm long PAN nanofibers with diameters D = 300−600 nm at strain rates between 10−2 and 10−4 s−1. A cascade of periodical ripples (necks) are found on the surface of these nanofibers at the two largest rates (see Figures 4 and 5 in ref 20), while at lower strain rates the nanofiber deforms homogeneously without showing surface ripples. In the field of soft materials, the ripple instability problem has attracted tremendous attention,13,16,21,22 as it is very sensitive to the environmental stimuli due to the relatively low stiffness of materials belonging to this class. Compared with conventional hard materials such as metal alloys, soft materials lose geometric stability easily, leading to ordered or disordered surface patterns. On one hand, surface instabilities can greatly affect the properties and performances of soft materials and their devices in real applications, and induce malfunction or failures. To help avoiding them, it is important to theoretically study the critical conditions for their appearance. On the other hand, instabilities can be controlled to form selected surface patterns during processing, such as required for stretchable electronics and biomedical applications.23−25 Patterns can signal specific material properties, e.g., surface optical properties, hydrophobicity, and hydrophilicity, as well as surface adhesion. In summary, understanding the relationship between the surface instability of soft materials and their geometries, mechanical properties and external loadings is critical for resolving both issues. A main part of this task is given by the

need to understand the physical origin of the experimental observations on the surface ripples of polymeric nanofibers under tension. In classical continuum mechanics, the instability of linear elastic materials under compression is well-understood as being induced by a bifurcation in the solution to the equations of static equilibrium.26 As polymeric nanofibers are usually considered to be homogeneous, incompressible, and hyperelastic materials,2,14 it is difficult to understand why the surface rippling occurs under uniaxial tension, as pointed out by Bignoni.27 Upon focusing on surface tension effects, Wu et al.28 made the first attempt to interpret the surface rippling of elongated polymeric nanofibers. In their model, the nanofiber is simplified as a homogeneous, isotropically hyperelastic Mooney−Rivlin solid with a constant surface tension. On the basis of linear perturbation analysis, they found that axisymmetric buckling can occur for infinitely long, homogeneous, and hyperelastic cylinders.28 This observation contradicts many previous studies where asymmetric Euler-buckling is usually preferred.27,29 Moreover, recent studies suggest that mechanical property differences (mismatch of Young’s moduli), rather than the surface tension, play the dominant role in the surface instabilities of core−shell structures,13,21,30−34 and thin film/ substrate structures.23,35−38 Therefore, we hypothesize that the surface ripples of nanofibers following uniaxial tension originate from their core−shell microstructure. In this work, we adopt both MD and finite element analysis (FEA) to study surface ripples of polymeric nanofibers under uniaxial tension. In section 2, we demonstrate by means of our B


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Figure 2. (a) Viral stress σz along the axial z-direction of the polymeric nanofiber versus the engineering strain εz. (b) Representative snapshots of the polymeric nanofiber during the uniaxial tension at various engineering strains εz. Periodic boundary conditions are applied in the axial direction. The arrows point at the surface instability (necking) sites.

diameter D is determined by the Gibbs dividing surface method, as suggested by Rutledge and co-workers.17,18 To accurately quantify the density change of the forming nanofiber, we divide the space occupied by the nanofiber into 25 cylindrical layers of identical thickness and calculate the number density of each layer, as given in Figure 1b. The layer numbers 1 and 25 represent the inner- (core) and outermost (shell) layers of the nanofiber, respectively. Clearly, the number density of the inner layers is larger than that of outer layers, due to the existing surface tension of the polymeric nanoobject, while the number density renders almost constant up to layer 20. At temperature T = 1, this plateau density is about 0.95, and thus it is very close to the number density of a bulk finite extensible nonlinear elastic (FENE) polymer under melt conditions.40 Beyond the plateau, the number density is gradually decreasing and approaches zero at the surface of the nanofiber. According to the number density difference, we can consider that the polymeric nanofiber has a core−shell structure, where the core of nanofiber has a constant number density as opposed to the shell. The volume fractions of the core and shell in the nanofiber are about 64% and 36%, respectively, for T = 1. When the nanofiber is annealed from T = 1.0 to T = 0.2, the number density will be increased, and the volume fraction of the core will also be slightly enlarged. These observations are consistent with the density distribution of glassy PE nanofiber predicted by MD simulations.17 Note that the volume fractions of the core (and shell) are dependent on the diameter D of the nanofiber; an issue to be discussed in the following section. From the calculated number densities (Figure 1b), we can further estimate the glass transition temperature Tg of the model nanofiber. As given in Figure 1c, the number density of layer 1 representing the constant number density within the core region is plotted against temperature T. Glassy and rubbery polymers exhibit different thermal expansion coefficients. This is confirmed by a change of slope of the number density (and thus also the specific volume) upon crossing Tg (Figure 1c), and we can extract Tg by linear extrapolation from the glassy and rubbery sides of the transition. The same method

MD results that the polymeric nanofiber consists of an amorphous shell surrounding a glassy core. The density and stiffness of the core are much higher than those of the shell. According to this observation, we develop a FEA model to quantitatively study the surface ripples and morphological transition of the nanofibers with core−shell structure under tension in section 3. The “polarization” mechanism,39 induced by the mismatch between the Poisson’s ratios of the core and shell layers, is identified as the key element that drives the surface instability. With the help of plastic deformation mechanisms, the effect of strain rate on the appearance of surface instabilities, observed by the experiments,20 can be reproduced by our FEA simulations. In section 4, a simplified theoretical model is proposed to illustrate the underlying physical mechanisms of the surface ripples and to predict the critical strain, found to be consistent with our FEA analysis. Conclusions are drawn in section 5.

2. RESULTS FROM MOLECULAR DYNAMICS SIMULATIONS 2.1. Annealing and Uniaxial Tension of Polymeric Nanofibers. Applying a method to generate a nanofiber composed of multibead anharmonic spring-connected chains by the procedure outlined in detail in the appendix, a cylindrically shaped, stable polymeric nanofibre can be formed in our MD simulation, as demonstrated in Figure 1a. The stability is due to sufficiently strong attractive nonbonded Lennard-Jones (LJ) interactions and chain connectivity, in the presence of periodic boundaries along the fiber’s main axis. After the nanofiber has been fully equilibrated at a typical melt temperature (T = 1 in LJ units) for the corresponding bulk system, it has been further annealed to the glassy temperature (T = 0.2) through canonical ensemble (NVT) MD. The temperature is controlled via conventional Nosé−Hoover thermostat. The annealing process takes about 10 million time steps in the simulation, corresponding to 100 000 LJ units. During this process, the effective diameter D of the model nanofiber gradually shrinks from the initial value 58 down to 50 LJ units and the density increases accordingly. Here the C


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properties, we adopted the same methodology given in above section to construct polymeric nanofibers with diameters D ranging from 19.78 to 98.36. Note that the diameters D of these nanofibers are determined by the Gibbs dividing surface method, as suggested by Rutledge and co-workers.17,18 Figure 3 shows the number density of polymer beads along the radial direction of the nanofiber, from its center to surface.

has been widely adopted in experiments and simulations to determine the Tg of polymers.17,41,42 As demonstrated in Figure 1c, Tg of the core is located at about 0.42, which is very close to the known Tg of bulk FENE polymers.42 We tried to adopt the same methodology to obtain a Tg of the shell (see Figure 1d). However, there is no distinct change of slope of the number density curve. Therefore, we must conclude that the shell (surface layer) of the model nanofiber does not undergo a transformation from a rubbery to a glassy state. In a recent experimental study performed by Camposeo et al.,19 authors have measured the local mechanical properties of electrospun nanofibers, made by prototype conjugated polymers. The electrospun fiber was found to be formed by a dense internal core embedded in a less dense polymeric shell. Moreover, Young’s modulus of the core was found to exceed the one of the polymeric, less dense, shell by a factor of 2. All these findings are in qualitative agreement with our observations and previous MD simulation results on glassy PE nanofibers.17 According to the markedly different variations of number density within core and shell, we can further see that the core undergoes a transformation from a rubbery to a glassy state, while the shell does not. At T = 0.2, the core is formed by the dense glassy polymer with higher Young’s modulus and the shell mainly consists of a less dense rubbery polymer with lower Young’s modulus. It is well-known that Poisson’s ratios for glassy and rubbery polymers are about 0.3 (compressible material) and 0.5 (incompressible material),43,44 respectively. Therefore, it is reasonable to assume that the Poisson’s ratios of the core and shell are about 0.3 and 0.5, respectively, which will be conformed by further MD simulations given in following section. After the nanofiber has been annealed, it has been further equilibrated under the isothermal−isobaric ensemble (NPT) to remove its internal stress along the axial (z) direction. After the equilibration, the length of the nanofiber further shrinks from 312 to 309; while its diameter approaches D = 50. Under the canonical ensemble (NVT), the simulation box as well as the polymeric nanofiber is then stretched along the axial (z) direction at fixed strain rate ε̇z = 5 × 10−5, which is low enough that the stress can equilibrate along the length of the fiber.45 The viral stress σz along the axial direction has been plotted against the engineering strain εz, as given in Figure 2a. When the deformation is small (εz < 0.04), σz rapidly increases with deformation and reaches its peak value at εz = 0.04. Hereafter, σz decreases with increasing deformation, while corrugation instabilities (neckings) are formed on the surface of the nanofiber. Up to εz = 0.6 the surface instabilities propagate over a long distance on the surface of the nanofiber. The backbone of the polymer chains start to become stretched under the deformation. This latter phenomenon marks the onset of strain hardening behavior during which σz monotonically increases with the deformation. The corresponding snapshots of the nanofiber during the uniaxial tension are shown in Figure 2b. When the strain is large enough (εz = 0.36), the surface instabilities (neckings) occur on the surface of the nanofiber. With the deformation increasing beyond εz = 0.6, the surface instabilities do not disappear but keep propagating on the surface of the nanofiber. 2.2. Size Effect of Polymeric Nanofibers. As has been reported by Rutledge and co-workers,15,17,18 the mechanical and thermal properties of polymeric nanofibers are highly dependent on their sizes/radii. To explore the influence of nanofiber size on their core−shell structures and mechanical

Figure 3. Number density of polymer beads within different layers at temperature T = 1.0 (solid lines) and T = 0.2 (dashed lines) for nanofibers with different diameters: 19.78σ, 29.46σ, 59.04σ, and 98.36σ. The polymeric nanofiber is divided into 25 layers along the radial direction from the center to surface. The layers 1 and 25 represent the most inner and outer layers, respectively.

Similar to the results given in Figure 1b, the core−shell structure is also found in these different sized nanofibers. The number density of the core is very close to that of the bulk value; while, the number density of the shell is generally much smaller. We also observe the rubbery-to-glassy transition for the core of all these nanofibers during the annealing process while the shells do not go through the glass transition in these processes (cf. Figure 1c,d). More interestingly, the core volume fractions of these nanofibers vary with their diameters. For example, the volume fractions of core and shell are about 57.76% and 42.24%, respectively, for a nanofiber with D = 19.78. However, when the diameter of the nanofiber is as large as D = 98.36, the fractions are 84.64% and 15.36% for core and shell regions, respectively.Table 1 summarizes the core and Table 1. Fiber Diameter, Shell Thickness, and Volume Fractions of Core and Shell Layers for Simulated Polymeric Nanofibers with Different Diameters diameter D [LJ units] core volume fraction shell volume fraction shell thickness [LJ units]

19.78 57.76% 42.24% 4.75

29.46 64.00% 36.00% 5.89

59.04 77.44% 22.56% 7.08

98.36 84.64% 15.36% 7.87

shell volume fractions for differently sized polymeric nanofibers. The core and shell volume fractions are monotonically increasing and decreasing with D, respectively. Curgul et al. found that the shell layer thickness of model PE nanofibers slightly enlarged from 1.03 to 1.39 nm with their diameters D increasing from 2.908 to 22.95 nm.17 The volume fraction of shell layers is thus confirmed by our findings to be monotonically decreasing with D increasing, while the trend is more pronounced for the model nanofiber. The mechanical properties, i.e., Young’s modulus and Poisson’s ratio, of our polymeric nanofibers were determined D


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Figure 4. Mechanical properties of polymeric nanofibers with different diameters at temperature T = 0.2: (a) Young’s modulus and (b) Poisson’s ratio. In part a, the Young moduli of nanofibers are normalized by the bulk modulus, 40. In part b, the Poisson ratios of bulk rubbery and glassy polymers are determined by the uniaxial tension tests.

from the uniaxial tension test at a small strain rate 10−5, following the method described in the previous section. During this process, the nanofiber is deformed along its axial direction and its surface is allowed to freely fluctuate (unconstrained) at temperature T = 0.2. The stress−strain curve in the linear elastic regime (εz < 0.04, as shown in Figure 2a), is used to calculate Young’s modulus of a nanofiber. As already mentioned, the change of the nanofiber radius is determined by the Gibbs dividing surface method, following previous works.17,18 The Poisson’s ratio of the nanofibers is then directly obtained as the radial strain divided by the axial strain. To determine the bulk properties of the model polymer, we have built another model with 212 chains, each carrying N = 500 monomers. For this setup, periodic boundary conditions are applied in all directions. The bulk structure has been gradually cooled from T = 1.0 down to 0.2 within 10 million time steps. Two configurations at different temperatures (T = 0.5 and 0.2) were saved to determine bulk properties. As it is known that the glass transition temperature Tg = 0.42 for the model system,42 the configurations at T = 0.5 and T = 0.2 are utilized to calculate the bulk mechanical properties of rubbery and glassy polymers, respectively. Similar to polymeric nanofibers, the uniaxial tension is applied on one dimension of the bulk system, while the other two orthogonal dimensions are allowed to fluctuate under the barostat (pressure p = 0), following the work done by Capaldi et al.46 The chosen strain rate is the same as that of nanofibers, ie. about ε̇z = 10−5. Similarly, Young’s modulus and Poisson’s ratio of the bulk systems can be determined from data collected during the uniaxial tensions. When T = 0.2, the bulk polymer is in the glassy state. Its Young’s modulus and Poisson’s ratio were calculated to be about 40 LJ units and 0.36, respectively. These obtained values agree well with the results reported for the same polymer model in refs 47−49. At T = 0.5, on the other hand, the bulk polymer is in the rubbery state. Young’s modulus and Poisson’s ratio are found to be 0.2 LJ units and 0.49, respectively. Again, these values are in good agreement with the results given by Makke.49 Figure 4a denotes the Young moduli of differently sized nanofibers. Upon increasing diameter D, the Young moduli of these nanofibers are monotonically increasing and approaching the bulk modulus of the model polymer. Workum and de Pablo50 have adopted the same polymer model with only N = 16 monomers per chain to study the mechanical properties of

polymer thin films. The thickness of their films varied from 6 to 24, while the aspect ratio (length/thickness) of these films was held constant at approximately 4. The in-plane moduli of these films were determined by compression deformation simulations at temperature T = 0.2. With the film thickness increasing from 6 to 24, the in-plane Young’s modulus of these films gradually increases and approaches the bulk value, similar to our results given in Figure 4a. Poisson’s ratio of model polymeric nanofibers is presented in Figure 4b. Interestingly, when the diameter of the nanofiber is as small as 19.78, its Poisson’s ratio is very close to that of the bulk polymer in its rubbery state. However, when the nanofiber’s diameter is large enough, i.e. 98.36σ, its Poisson’s ratio is about the same as that of the glassy polymer. We recall that when the diameter of the nanofiber is very small, the shell layer (rubbery layer) occupies about 42.24% of the fiber volume (cf. Table 1). Therefore, Young’s modulus of the nanofiber is only about 68% of Young’s modulus of bulk glassy polymer, while its Poisson’s ratio is large and comparable to the rubbery polymer (0.5). However, when the diameter of nanofiber is sufficiently large, most of its polymer chains are in the glassy state. In this case, Young’s modulus is almost 90% of Young’s modulus of the bulk glassy polymer. Its Poisson’s ratio is also small and similar to the glassy polymer (0.36). Buell et al. have studied the mechanical properties of PE nanofibers with diameters D ranging from 3.7 to 17.7 nm through atomistic MD simulations.18 Young’s moduli of these PE nanofibers were found to be increasing with increasing diameter D at different temperatures (100, 150, and 200 K), which is consistent with our above findings (Figure 4a). However, Poisson’s ratio of these PE nanofibers increases from 0.1 to 0.3 as the diameter increases from 3.7 to 17.7 nm at 100 and 150 K,18 which is different from the current results on the Poisson’s ratio (Figure 4b). Apparently, as a highly coarsegrained model has been adopted in the present study, each polymer bead represents several monomers. Thus, the diameter of the model fiber is much larger than that of the glassy PE fibers. Above discrepancy could be induced by the different sizes of model fibers, as well as different potentials being used. From the above discussion, the polymeric nanofiber consists of a hard core and a soft shell. The number density of the core is close to that of the bulk material, while the density of the shell is much smaller. Therefore, we can hypothesize that Young’s modulus of the core equals the one of the bulk E


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polymer Ebulk. The stiffness of the shell Eshell is more compliant, with Eshell < Ebulk. Subjected to uniaxial tension along its axis, Young’s modulus E of the nanofiber can be estimated according to the Voigt rule of mixing by assuming uniform strain throughout the nanofiber. Then, E = Ebulk f bulk + Eshell fshell. Here f bulk and fshell are the volume fractions of the bulklike core and shell layers, respectively, while fshell = 1−f bulk. As the bulk modulus Ebulk had already been determined, there are only two unknown parameters, Eshell and f bulk, which can be fitted using the relationship between Young’s moduli of nanofibers and their diameters (cf. Figure 4a). The core volume fraction f bulk can be further evaluated by f bulk = (1−ξ/Rfiber)2, where ξ and Rfiber are the mechanically effective thickness of the shell and radius of the nanofiber, respectively; ξ also characterizes the effective length scale over which the stiffness of the nanofiber varies. If the nanofiber radius Rfiber is smaller than ξ, we set ξ = Rfiber, and the value of ξ is assumed to be a unique function of temperature. The above model has been successfully applied to study the radius dependent moduli of PE nanofibers (see Figure 6 in ref 18). At temperature 100 K, the bulk modulus of PE is about 2758 MPa. The fitted shell modulus Eshell and shell thickness ξ are 1050 MPa and 3.4 nm,18 respectively. The ratio Ebulk/Eshell is about 2.62, which agrees with the experimental observation that Young’s modulus of the core is twice the stiffness of the shell.19 We also applied the above model to fit the curve given in Figure 4a. However, the results from present study cannot be well-fitted by the above method. We suspect that this could be induced by the variation of shell thickness ξ with the nanofiber radius Rfiber (cf. Table 1). According to the previous experimental observations19 and MD simulations,18 as well as present simulation results, we set Ebulk/Eshell = 2.0 in the following FEA and theoretical study. Since the core and shell layers are glassy and rubbery, respectively, it is reasonable to assume that their Poisson’s ratio vcore ∼ 0.3 and vshell ∼ 0.5 to simplify the finite element simulation and theoretical analysis in the following parts.

Figure 5. Schematic of a polymeric nanofiber with core−shell structure. The core and shell layers, represented by different phases, are assigned by different mechanical properties. At the interphase between core and shell layers, the “polarization” of the materials can occur. dp is the interphase thickness, which is introduced in the theoretical model to account for the polarization mechanism.

by Young’s modulus Ei and Poisson ratio νi through a classical relationship.63 The subscripts i ∈ {core, shell} represent the core and shell layers, respectively. The material and geometric parameters in the FEA model (cf. Figure 5) are defined as Eshell/Ecore = 0.5, νcore = 0.3, νshell = 0.499, L0/R0 ∈ {2,5,10,20,30,40}, Ro/Ri = 1.5, and Ro/dp = 6, unless otherwise stated. The stiffness of the core Ecore is taken to be twice of that of the shell Eshell, according to the experimental observation19 and above MD simulations. The Poisson’s ratios of the core and shell are taken to be close to that of glassy and rubbery polymers, as discussed in the previous section. Here we should emphasize that all the FEA results are not sensitive to the volume fractions of the core and shell layers, i.e., values of Ro/Ri. 3.1. Instability Modes of Polymeric Nanofibers. To obtain the instability modes of polymeric nanofibers under uniaxial tension, a linear perturbation analysis has been performed on the above FEA model. A uniform displacement UZ was applied on one end of the nanofiber, along its axial direction while the opposite end was held fixed. The analysis was done by solving the incremental equilibrium equations div(Ṡ) = 0,26 where Ṡ is the incremental nominal stress. The eigenvalue (critical deformation) and eigenvector (critical instability mode) are obtained upon solving the rate form of the equilibrium equations, in the absence of body forces. This way, a critical strain UZ/L0 for the surface instability to occur can be easily obtained. For the case of uniaxial stretching, the critical strain is almost insensitive to the length of the nanofiber, and assumes a value between 0.58 and 0.59 (Figure 6a), while is is known to be affected by fiber length in the case of uniaxial compression.26 Figure 6b shows the surface instability modes for nanofibers with different lengths. Under unaxial tension deformation, elastic instabilities can occur, resulting in rippled surfaces of these polymeric nanofibers. In a number of previous studies,13,18,19,58 the difference in Young’s moduli between the core and shell layers and the related size effect on mechanical properties of nanofibers had been considered. However, the difference in Poisson’s ratio between the core and shell layers and its effect on surface instabilities remained unexplored. During close inspection of the interphase region between the core and shell layers, we observed a periodic displacement of the shell, relative to the core (also see the inset of Figure 5). Such a periodic displacement seems to happen because the interphase can relieve some of the contraction or

3. RESULTS FROM FINITE ELEMENT ANALYSIS Although the above MD simulations shed light on the microstrcuture of polymeric nanofibers as well as their mechanical properties, the approachable spatial and temporal scales are still limited. To efficiently study the surface rippling phenomenon observed in the experiments,20 we adopted the FEA method, which has been widely used to study the mechanical behaviors of polymers and polymer composites.51−61 As informed by above molecular and experimental observations, a FEA model of the polymeric nanofiber representing the core−shell structure is described in Figure 5. The lengths of the fiber in the undeformed (initial) and deformed (current) configurations are L0 and L, respectively. The thickness of the shell layer is δ. Its inner and outer radii are Ri and Ro, respectively. We use a cylindrical coordinate system, where X = (R,Θ,Z) and x = (r,θ,z) denote undeformed (initial) and deformed (current) positions, respectively. Thus, the deformation gradient is F = ∂x/∂X. To efficiently implement the simulation, we define a modified deformation gradient as F̅ = J−1/3F with volume change eliminated (J = det(F)). The compressible Neo-Hookean model62 with strain energy function W = 1/2 μi(I1̅ − 3) + K(J − 1)2 is used to characterize the nonlinear elastic behavior of the polymeric nanofiber in which I1̅ = trace(FF ̅ ̅ T) and μi and Ki are the initial shear and bulk moduli, respectively. Note that μi and Ki can be expressed F


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Figure 6. (a) Critical strain vs length of the nanofiber obtained from linear perturbation analysis. (b) Snapshots of the surface instability modes of polymeric nanofibers with different lengths. Horizontal plots show the surface at θ = π/2. Vertical plots show the cylinder surface at R = Ro. Blue and red colors represent the smallest and largest perturbation displacement, respectively.

regimes, with a strain rate dependent yield strength. The effect of strain rate on Young’s modulus and Poisson’s ratio are however negligible.20,64 The elasticity parameters, i.e., Ei and μi, are therefore taken as constants, regardless of the imposed strain rate. To perform the calculations for plastic deformations, we adopted the simple classical J2 plasticity model.63 For this model, the plastic deformation behavior is governed by the potential Φ = σe−σy = 0 where σy = σi0(1 + ep/e0)n. The effective stress is σe and the initial yielding stress is σi0 for i ∈ {core,shell}. The so-called hardening component is denoted by n. The reference strain is taken as e0 = 0.025 and the accumulated plastic strain is ep. We set σcore 0 /Ecore = 0.05, 0.1, 0.025, σshell 0 /Ecore = 0.025, 0.05, 0.0125, and n = 0.4, unless otherwise stated. The elastic behavior of the nanofibers is still described by the compressible Neo-Hookean model as in the previous linear perturbation analysis. Following this procedure, most of the experimental observations can be predicted and reproduced. As aforementioned, the yield strength of nanofibers is dependent on the imposed strain rate.20 Accordingly, the different deformation behaviors induced by strain rates can be well represented by the different yield strengths. We thus consider three different initial yielding stresses: Case 1) σcore 0 / core shell Ecore = 0.05, σshell 0 /Ecore = 0.025; Case 2) σ0 /Ecore = 0.1, σ0 / shell Ecore = 0.05; Case 3) σcore 0 /Ecore = 0.025, σ0 /Ecore = 0.0125, denoting the fastest, slowest and medium strain rates, respectively. All these cases are investigated at L0/Ro = 40.

dilation energy by moving from the compressed regions toward stretched regions, when the interphase is alternately compressed and stretched along the axial direction. Such a mechanism had been discussed by Read et al.,39 and denoted as “polarization”, in analogy to the separation of charges in a polarizable medium. We should emphasize that such a polarization mechanism is induced by the difference between Poisson’s ratios of core and shell layers. If they have the same Poisson’s ratio, the nanofiber will deform uniformly under tension without the appearance of surface ripples. If νcore is larger than νshell, the polarization still cannot occur, as observed from our FEA results. The polarization phenomenon is triggered by νcore rc ⎩ 0,

(9)

where rij denotes the distance between bead i and j, and rc = 21/6σ is a cutoff radius. The connectivity of spring-connected beads is ensured by the additional attractive FENE potential:60,71,75 UijFENE

⎧ 1 2 2 ⎪− kR 0 ln[1 − (rij/R 0) ], rij ≤ R 0 =⎨ 2 ⎪ 0, rij > R 0 ⎩

(10)

71,75

Following the previous works, the FENE spring coefficient is set to be k = 30ε and the maximum bond length is R0 = 1.5σ. Such a spring is strong enough to avoid bondcrossing between different chains, while it is also weak enough to allow for a large time step to be adopted in the simulation. The combination of the attractive potential UFENE and the ij repulsive one UWCA gives rise to an equilibrium bond length ij ≈0.97 at T = 1.60 All beads, except those interconnected by FENE+WCA springs, interact with each other through a smoothed version of a LJ pair potential that vanishes for r ≥ rout ≥ rc:

5. CONCLUSION In this work, the physical mechanism for surface ripples of polymeric nanofiber under tension has been investigated through large scale MD simulations and FEA calculations. The nanofiber is found to be composed by a hard core embedded in a soft shell. More importantly, the Poisson’s ratios of the core and shell layers are close to that of the glassy (compressible) and rubbery (incompressible) polymers, respectively. Informed by these findings, a FEA model has been developed to study the instability condition of these surface ripples. The polarization mechanism, occurred in the interphase regime between the core and shell layers and induced by the mismatch of their Poisson’s ratios, plays the key role. The importance of this mechanism has been further validated and verified through our newly developed theoretical model. By considering the plastic deformation in 3D FEA calculations, the surface ripples can be tuned through applied strain rate and yield strength of nanofibers, which are consistent with experimental observations. In previous studies,16,21,22,30−32 the surface instabilities of core−shell structures had been attributed to the mismatch of Young’s moduli between core and shell layers. However, in light of the present FEA results and accompanying theoretical analysis, the surface instability of polymeric nanofibers under tension we find to be primarily induced by the mismatch of Poisson’s ratios, furthermore evidenced by MD simulation and a simplified theory. The present findings seem to open a new way to understand the surface instability of polymeric nanofibers under tension. In biological materials, such as bones, the mismatch of Poisson’s ratios between different components was found to play an important role in their mechanical properties.66−70 The present study may thus also

Uijsmooth

⎧ 4ε[(σ /r )12 − (σ /r )6 + 1/4], r ≤ r ij ij ij in ⎪ ⎪ ⎪ 4 = ⎨∑ Cj(rij − rin) j , rin < rij ≤ rout ⎪ j=0 ⎪ ⎪ 0, rij > rout ⎩ (11)

Here rin denotes the maximum distance up to which the smoothed version exactly matches the LJ potential. The C’s are constants, calculated once to ensure that the potential, the force and its first derivative are continuous everywhere. Here we take rin = 2.4σ and rout = 2.5σ. Such values can maintain enough surface energy for the films and fibers formed by polymer chains.40−42,78 By setting σ = ε = m = 1 and the Boltzmann constant kB = 1, all molecular simulation results are given in the reduced LJ units. Here m is the mass of a single bead. The integration time step in the simulation is Δt = 0.01. It is well-known that the contour distance between entanglement points, so-called entanglement length is Ne = 48 ± 1 and Ne = 86 ± 2 for such a FENE polymer melt, as defined from either kinks or coils.79 Therefore, we choose N = 500 ≫ Ne to model highly entangled polymer chains; the number of entanglements per J


Macromolecules

Article

“Bairen” startup funding of Chongqing University for support. Y.L. warmly expresses thanks for the financial support provided by Ryan Fellowship and Royal E. Cabell Terminal Year Fellowship, as well as a supercomputing grant on Quest from Northwestern University High Performance Computing Center. W.K.L. expresses thanks for the support from AFOSR Grant No. FA9550-14-1-0032.

chain is then about Z = 10.75 There are M = 1200 chains and thus M × N = 600 000 beads contained in the simulation box whose size is about 100 × 100 × 312 in the x, y, and z directions, respectively. Periodic boundary conditions are applied only along the axial direction (z direction) of the simulation box. Nonperiodic boundary conditions (reflecting walls, whose properties will be irrelevant throughout the subsequent simulations) are applied along the x and y directions of the simulation box. The size of the simulation box in the x and y directions is kept unchanged during the simulation. During the equilibration process, the simulations are performed under the microcanonical ensemble (NVE) with temperature T = 1 controlled by a Langevin thermostat with friction coefficient Γ = 0.5 by using the LAMMPS package.80 The initial pre-equilibrated configuration for the nanofiber has been generated by using our hybrid Monte Carlo/ molecular dynamics method (see refs 75 and 81−83). Within this method, thin chains (with zero thickness) are placed randomly within the fiber. Subsequently, their volumes increase using an adaptive time step and artificial, time-dependent potentials. We have slightly adapted this algorithm, originally designed for the creation of polymer solutions, melts and brushes, to deal with the nonperiodic boundary conditions applied in lateral directions for the nanofiber. The resulting preequilibrated configuration has been further fully equilibrated by another hybrid method, the bond swap algorithm (BSA).75,84,85 The BSA attempts to swap bonds between different chains, effectively exchanging a part of one chain with that of another chain and then systematically reduces the system free energy, without changing the underlying polymer network (i.e., polymer chain length and number). This way, the polymer chains rapidly converge to relaxed conformations exhibiting stationary mean end-to-end distances Ree and radii of gyration Rg. During this process, due to the existing surface tension and the nonperiodic boundary conditions applied in the lateral directions of the nanofiber, the surface of the nanofiber shrinks and automatically forms a cylindric shape that renders stable due to the attractive nonbonded interactions and preserved connectivity within the polymers, in some analogy with the socalled “soft solid model” used for the description of filamentous networks and gels.86

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ASSOCIATED CONTENT

S Supporting Information *

Details of the theoretical model and derivations. This material is available free of charge via the Internet at http://pubs.acs. org/.

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AUTHOR INFORMATION

Corresponding Authors

*(S.T.) E-mail: [email protected]. *(W.K.L.) E-mail: [email protected]. Author Contributions ⊥

These authors contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Martin Kröger, John Moore, Jacob Smith, and Miguel Bessa for critical reading of the manuscript and helpful discussions. S.T. thanks the NSF of Chongqing (Project No. 0211002431039), the NSAF (Grant No. 11176035), and K


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Surface Ripples of Polymeric Nanofibers under Tension: The Crucial

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