Basic structural units in carbon fibers: Atomistic models and tensile behavior

C A R B O N 8 5 ( 2 0 1 5 ) 7 2 – 7 8

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Basic structural units in carbon fibers: Atomisticmodels and tensile behavior

http://dx.doi.org/10.1016/j.carbon.2014.12.0670008-6223/� 2015 Elsevier Ltd. All rights reserved.

* Corresponding author at: Department of Materials Science and NanoEngineering, Rice University, Houston, TX 77005, UniteE-mail addresses: [email protected] (E.S. Penev), [email protected] (V.I. Artyukhov), [email protected] (B.I. Yakobson), .

Evgeni S. Penev, Vasilii I. Artyukhov, Boris I. Yakobson *

Department of Materials Science and NanoEngineering, Department of Chemistry, The Richard E. Smalley Institute for Nanoscale Science

and Technology, Rice University, Houston, TX 77005, United States

A R T I C L E I N F O A B S T R A C T

Article history:

Received 8 October 2014

Accepted 18 December 2014

Available online 24 December 2014

Understanding the atomistic mechanisms of tensile failure in carbon fibers is important for

fiber manufacturing and applications. Here we design structural faults with atomistic

details, pertaining to polyacrylonitrile (PAN) derived fibers, and probe them using large-

scale molecular dynamics simulations to uncover trends and gain insight into the effect

of local structure on the strength of the basic structural units (BSUs) and the role of inter-

faces between regions with different degrees of graphitization. Besides capturing the

expected strength degrading with increasing misalignment, the designed basic structural

units reveal atomistic details of local structural failure upon tensile loading. Fracture initi-

ation is nearly always associated with the interface of the misoriented crystallite and its

environment.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Carbon fibers [1] have long been one of the technologically

most important carbon-based materials. More than half a

century of research and development, manufacturing, and

applications have resulted in a bulk of accumulated knowl-

edge on virtually any aspect of fiber structure and properties

[2,3]. As applications, from sporting goods to automobile parts

[4] to aircraft components etc., get progressively more sophis-

ticated, the demand for thorough understanding of fiber

structure–property relations [5,6] is steadily increasing and

becomes critical for further performance boost.

Despite the impressive specific tensile strength, the break-

ing stress of current fibers is well below the theoretical lim-

it � ðYc=dÞ1=2 [2,7], with Y being the Young’s modulus, c the

surface energy of the atomic planes normal to the loading,

and d the equilibrium distance between the latter. Under-

standing such behavior calls for detailed knowledge of fiber

structure. Although there is an overwhelming amount of

studies [1] the level of understanding of carbon fiber structure

still lacks the exquisite details known for carbon nanotubes

and graphene [8]. Ideally, one would like to design specific

microscopic structural features and probe their effect through

atomistic simulations [9] at a larger length scale. The utility of

such a bottom-up approach crucially depends on the physical

adequacy of the input atomic geometry and its computational

feasibility.

Among carbon fibers, those derived from a polyacryloni-

trile (PAN) precursor are of particular importance, presently

holding a major share of the overall fiber production/market

[10,11]. Unlike catalytically grown carbon nanotubes, the

PAN polymer chain, Fig. 1(a), is transformed into a planar aro-

matic structure via a complex multistage thermal processing

involving pyrolysis, stabilization, denitrogenation, carboniza-

tion, and eventually graphitization [12,1]. The sp2-carbon

planes stack up to further form graphitic crystallites, consid-

ered to be the basic structural units (BSUs) of PAN-based

fibers. However, these crystallites are usually imperfect. First,

d States.

C A R B O N 8 5 ( 2 0 1 5 ) 7 2 – 7 8 73

the arrangement of individual graphene planes relative to

each other is turbostratic [13], or their orientation relative to

the fiber axis may be suboptimal. Second, the degree of

graphitization is almost never complete, which means that

there is leftover (likely amorphous) sp3-carbon present, with

the graphitic–amorphous domain interfaces possibly forming

weak points [14,15]. Finally, it is plausible that the extensive

thermal treatment may result in defects within the sp2-car-

bon planes themselves that cannot be locally annealed,

which is a subject of its own right and deserves separate

attention [8,16].

A representative model for tensile failure that incorporates

the specifics of PAN-based fiber structure, Fig. 1(b), was pro-

posed by Reynolds and Sharp [17], highlighting the impor-

tance of crystallite environment and misorientation / as

limiting factors for mechanical performance. These misori-

ented crystallites are essentially shear stress concentrators.

It is however the specific structure/arrangement of the BSU

and the environment that ‘‘inhibits’’ the release of the

resolved shear stress on the BSU through shear between the

misoriented basal planes. The structural constraints on the

BSU due to the surrounding regions effectively convert the

tensile loading to local compressive stresses. Upon critical

tensile loading failure then occurs within the basal planes.

Within an isostress model the critical stress then can be

expressed [17] as rk > ð2c=sð/ÞplÞ1=2, where sð/Þ is the effective

basal–plane compliance of the misoriented BSU, and l is the

critical crack length.

The purpose of this article is thus twofold. We first intro-

duce and discuss atomistic models representing structural

flaws in carbon fiber—misoriented BSUs and interfaces

between graphitic and amorphous carbon domains—and

then explore their mechanical behavior upon tensile loading,

(b)(a)

Fig. 1 – (a) A fragment of a polyacrylonitrile chain –[CH2-CHCN]n–

highlighted); (b) schematic of a misoriented basic structural unit

The size along the fiber axis (dash-dotted line) is Lak, in the in-pla

planes. / indicates the BSU misorientation angle; (c) Atomistic

rendered. The misoriented BSU (’ 1:5� 4:5� 9 nm3) is highligh

thick arrow, by displacing thin slabs (‘‘handles’’) at the top and

(A colour version of this figure can be viewed online.)

using large-scale molecular dynamics simulations as a theo-

retical testing platform [8], Fig. 1(c). The modeling was carried

out using classical molecular dynamics simulations. Atomic

interactions are described by the ReaxFF force field [18,19]

which gives satisfactory description of graphene ripping [20]

and has also been used to simulate thermal transformation

of PAN [21]. All calculations were performed with the LAMMPS

[22] package. Detailed description of the computational

framework is given in the next Section 2 and results of the

simulations are presented in Section 3.

2. Computational protocol

Large-scale atomistic models [23] have proven crucial in sim-

ulating various nanocarbon forms, from graphite [24] to nano-

porous carbon [25], pyrocarbons [26], coal [27] and soot [28]. In

the context of ex-PAN fibers and composites, effort has been

limited to some aspects of the carbonization process [21,29]

and mechanical behavior [30,31]. In designing an atomistic

model we single out and focus on a structural feature, the

misoriented BSU, that is crucial for understanding tensile fail-

ure [2,7,17]. We note, however, that a detailed structural rep-

resentation of the BSU itself is hardly achievable given its

complexity. Instead, here we shall pay particular attention

to the immediate environment of the BSUs.

The initial system for all BSU models considered in Section

3.1 is an orthorhombic graphitic block with dimensions

Lak � La? � Lc ’ 9� 4:5� 1:5 nm3, with c along the [0001]

direction in graphite [z, Fig. 1(c)] and k being the zigzag direc-

tion ðyÞ. The block is then ‘‘misoriented’’ by transforming the

atomic coordinates of each graphene layer only within the

zigzag plane f10 �10g using a sigmoid function fðyÞ. Different

misorientations are achieved by varying the displacement

x

z

y

(c)

in isotactic helical (TG)3 conformation (a single monomer is

in carbon fiber. Gray sheets represent the sp2-carbon planes.

ne orthogonal direction La?, and Lc in direction normal to the

representation of (b); for clarity part of the geometry is not

ted. Load is applied along the fiber axis, as indicated by the

bottom of the system, schematically represented as plates.

74 C A R B O N 8 5 ( 2 0 1 5 ) 7 2 – 7 8

Dz for fðLakÞ ¼ 1, Fig. 1(c). This procedure also results in a loss

of the initial Bernal stacking thus partially mimicking a tur-

bostratic arrangement [13].

Here we consider four sets of geometries f/ng generated by

Dz ¼ 2nd0; n ¼ 0; 1; 2;3; ð1Þ

with d0 being the interplanar distance in graphite. This choice

results in initial geometries with maximal misorientation

angles

/max � tan�1 f 0��Lak=2

� �� 0–30�; ð2Þ

the prime denoting first derivative in y. The unit is then

‘‘sandwiched’’ by a block of graphite with size ’ La? along x

to ensure that the misoriented unit is in contact with an

aligned graphitic block for any /.

As a next step, the voids formed by the misoriented gra-

phitic block are filled out to represent different structural con-

straints on the BSU, realizing also three distinct basal

interfaces: (i) aligned graphitic block with unpassivated

edges, (ii) disordered, amorphous carbon (a-C) [32,33] (gener-

ated through a procedure described below), and (iii) folded,

graphene hairpin-like inclusions [34]. The initial geometries

of the three models are shown in Fig. 2; the final size of the

designed model is ’ 41,000 carbon atoms. All structure gener-

ation and manipulation procedures are performed using VMD

[35]. As transmission electron microscopy (TEM) is a major

source of structural information about carbon fibers, we pro-

vide also the corresponding simulated [36] TEM images. Sim-

ilar motifs can be identified in the experimental TEM imaging

of various carbon fibers: the models in Fig. 2(a) and (c), for

instance, qualitatively match longitudinal-section TEM

images in Ref. [37] as well as those of the T700 fibers in Ref.

[38], while that in Fig. 2(b) may be seen in the high-resolution

TEM images of PAN-based commercial C/C composites

reported in Ref. [39].

Mechanical behavior of all structural models is analyzed at

T = 300 K. Tensile deformation is realized by displacing thin

slabs of C atoms in opposite directions along y, Fig. 1(c), so

(a) (

Fig. 2 – Structural models of misoriented BSUs (left subpanels;

interfaced with (a) active-edge graphene planes, (b) disordered,

like inclusions. For clarity, the aligned graphitic part is rendered

are shown in the corresponding right subpanels. (A colour vers

that the initial separation between them DLkðt ¼ 0Þ increases

in time

DLkðtÞ ¼ DLkð0Þð1þ 2_etÞ; ð3Þ

with constant (engineering) strain rate _e ¼ const ¼ 10�3 ps�1,

such that L_e� u;u being the speed of sound in graphite [40].

Periodic boundary condition is applied along x and Nose–Hoo-

ver barostat is used to maintain the corresponding pressure

component Pxx ¼ 0.

In order to evaluate the possible importance of interfaces

between graphitized and amorphous regions in the fiber (Sec-

tion 3.2), the graphiticjamorphous interfaces were generated

according to the following procedure. First, a graphite slab

of size about 2:5� 2:5� 11 nm3, comprising 7680 atoms, was

generated with periodic boundary conditions applied in all

dimensions, with the long dimension running along the

½11 �20� zigzag direction of the layers. A region of half the

length of the structure was then transformed into amorphous

carbon [see Fig. 6(a) below] via simulated annealing using the

AIREBO potential [41] to accelerate the calculations, while the

atoms in the remaining parts were held fixed. In this way, we

were able to make sure that the carbon atoms near the amor-

phization region boundary would be affected by their gra-

phitic neighbors and thereby form a smooth continuous

interface between the regions to be explored in more accurate

ReaxFF computational nanomechanical tests.

The annealing was performed in a two-step protocol. In

the first step, the structure was annealed under NVT condi-

tions using the Berendsen thermostat [42] at 10,000 K and a

s ¼ 10 fs time constant over 5 ps (time step dt ¼ 0:1 fs). The

structure was then relaxed under NPT conditions for 50 ps

(dt ¼ 0:25 fs) with a Nose–Hoover thermostat (quenching from

T = 5000–1000 K, using s ¼ 0:1 ns time constant) and Parri-

nello–Rahman barostat [43] (P ¼ 0, s ¼ 0:5 ns). During the sec-

ond stage, the mass of the atoms was scaled by a factor of ten

to reduce the atomic velocities while maintaining the high

kinetic energy needed to overcome the potential energy barri-

ers for relaxation.

b) (c)

perspective views along x, cf. Fig. 1, for the case of /3)

amorphous carbon region, and (c) folded, graphene hairpin-

transparent. Simulated TEM images (200 kV electron beam)

ion of this figure can be viewed online.)

C A R B O N 8 5 ( 2 0 1 5 ) 7 2 – 7 8 75

3. Results and discussion

3.1. Tensile failure of a misoriented basic structural unit

To probe the mechanical behavior of the three structural mod-

els, Fig. 2, we compute the axial stress rk vs. strain e for differ-

ent misorientation angles /, cf. Fig. 1. The stress is calculated

from the volume average of the superposition of the per-atom

virial stress tensor S [44] as obtained from LAMMPS [22],

r ¼ hSiV �1

VðeÞXi2V

Si; Si ¼ �mivi vi �X

j

rji Fji; ð4Þ

for the constituent atoms excluding those utilized as ‘‘han-

dles’’, Fig. 1(c). In Eq. (4), mi and vi in the kinetic term are,

Fig. 3 – Tensile stress–strain curves for the structural models

in Fig. 2 (schematic icons, lower-left corners) for four

misorientation angles /n. For the model in Fig. 2(c), the

reference, aligned system (dotted curve) is the same as in

(a). The insets show the corresponding tensile stress to

failure rtk values relative to that for a perfectly aligned BSU,

rt0 � rt

kð/ ¼ 0�Þ. The end-points of the /-arrows are those

determined from Eq. (5) while the start-points give the

actual initial misorientation angles resulting from geometry

pre-relaxation. The direction of the arrows indicates the

gradual decrease of / in the course of the tensile loading. (A

colour version of this figure can be viewed online.)

respectively, the mass and velocity of atom i, and Fji is the

force due to atom j at distance rji.

Results are summarized in Fig. 3. In relating mechanical

behavior to designed structural details, a few provisions need

to be made regarding the misorientation angle /. Quantita-

tively, we will associate it with its initial (maximal) value as

it can be rigorously defined for the constructed initial geome-

tries, through Eqs. (1) and (2). Upon pre-relaxation and subse-

quent tensile loading, / is a fluctuating and decreasing

quantity. For given extension its geometrically ‘‘minimal’’

value is limited by

tan�1ðDz=DLakÞK / ð5Þ

and depends on the structural constraints due to the BSU

environment, Fig. 2.

The overall trend in Fig. 3 corroborates the expected

degrading strength with increasing the BSU misalignment.

For virtually all configurations tensile failure occurs in the

range e � 0.1–0.15, whereas the stress at failure can be low-

ered to about half of its value for perfectly aligned BSU, see,

e.g., the inset in Fig. 3(a). Differences are clearly detectable

both within the same model, or the same misorientation /n

across different models.

The arrangement in the /0-geometry for the sample type

shown in Fig. 2(a) accommodates essentially two aligned

neighboring graphitic blocks and its stress–strain curve is rep-

resentative of a perfectly graphitized structure within the

present theoretical framework (ReaxFF). Even a small mis-

alignment of the BSU reduces considerably its strength. The

edges of the terminated graphene planes may bind with the

outermost layers of the BSU to form topological features

which are locally similar to the Y-type dislocations that occur,

e.g., in anthracite [45]. This rebonding can be viewed as an

effective reduction of Lc. The /3-geometry, for the sample type

in Fig. 2(a), appears to be the weakest of all studied. The fail-

ure is initiated by rupture of one of the Y-junctions, Fig. 4(a),

associated with the first drop of rk at et ’ 0:09;rtk � rkðetÞ. Fur-

ther extension leads to breaking of a second Y-junction [right

panel in Fig. 4(a)], and complete failure of the entire BSU.

When the BSU is constrained by a-C domains, Fig. 2(b), ini-

tially strength rk is slightly affected by misorientation as

observed for the /0;1;2 samples in Fig. 3(b). Qualitatively, this

behaviors resembles the ‘‘fault tolerance’’ with respect to

the crack size seen in atomistic simulations of fracture of a-

C nano-samples [46]. For the largest misalignment /3 failure

is initiated again at the outer layer in contact with the tip

region of the a-C wedge, as indicated in the left panel of

Fig. 4(b). It is shortly followed by nearly simultaneous fracture

of pair of layers, and subsequent collapse of the misoriented

BSU. It should be noted also that this model represents a

basal–plane type interface between the BSU and poorly

graphitized domain as represented by the a-C. As designed

here, however, there exist boundaries of these mostly sp3-C

domains in contact with the zigzag face of the aligned sp2-C

graphitic block. The mechanical response of such an interface

is of sufficient interest and will be discussed in greater detail

in the next subsection.

The tensile loading of the model in Fig. 2(c) was performed

only for misaligned samples /1;2;3 as for the perfectly aligned

geometry /0 we approximately adopt the one for the model in

Fig. 5 – Simulation snapshots capturing the sequential

failure (from left to right) of the misoriented graphene-like

planes for the BSU model in Figs. 2 and 4(c). Corresponding

strain values ek (cf. also the curve for the /3-geometry in

Fig. 3(c)) are indicated in the lower-right corners and color

coding [47] corresponds to the ryz shear stress component.

(A colour version of this figure can be viewed online.)

(a) (b) (c)

Fig. 4 – Representative snapshots (views along x) of subsequent stages in the tensile failure of the initially most misoriented

samples (/3) for the three models studied; see corresponding panels in Fig. 2. Left subpanels show the failure initiation

(circled area). The outer BSU layer along with narrow slab of the surrounding material are highlighted in red. For clarity, the

aligned graphitic part is rendered transparent, similar to Fig. 2. (A colour version of this figure can be viewed online.)

76 C A R B O N 8 5 ( 2 0 1 5 ) 7 2 – 7 8

Fig. 2(a). The initial misorientation angle for the /3 sample is

the largest realized here [cf. inset in Fig. 3(c)] and the corre-

sponding stress–strain curve, Fig. 3(c), appears distinct from

those of all other models. Note that at ek ¼ 0:1 a slight kink

in the curve occurs. We find it to be a signature of an effect

related to the high curvature of the constraining hairpin-like

inclusion. At that extension, the outer plane of the BSU

‘‘unzips’’ along x and slips normally to bind with the tip of

the narrow hairpin-like inclusion, as indicated in Fig. 4(c).

This restructuring survives further loading but ultimately ini-

tiates the overall failure of the BSU, right subpanel in Fig. 4(c).

Structural failure for all models in Fig. 4, having largest ini-

tial misorientations / � 20–25�, occurs earliest in the course

of loading as compared to all other less misoriented geome-

tries. In real fibers / displays certain (experimentally accessi-

ble) distribution and therefore large local tiltings of BSUs

which are twice or higher than the mean value for the fiber

are probable. Within the Reynolds–Sharp model [17], the mis-

oriented BSU represents a critical shear stress concentrator,

owing to the specifics of local fiber structure. The atomistic

details discussed above may be considered as possible micro-

scopic mechanisms through which local structure can initiate

fracture of the misoriented BSU. In Fig. 5, the distribution of

the ryz shear stress component is illustrated for selected

sequential states of the BSU model of Fig. 2(c) undergoing a

tensile failure. Clearly, ryz is mainly concentrated within the

misaligned BSU.

3.2. Nanomechanical response of the sp2-C/a-C interface

The as-annealed structure, shown in Fig. 6(a) as a simulated

TEM image and atomistic structural model, was used for nano-

mechanical failure simulations with ReaxFF, preceded by a

short NPT re-equilibration from 3000 K to 300 K over 10 ps with

dt ¼ 0:25 fs. The structure was then pulled for 100 ps up to 15%

strain with zero-pressure boundary conditions in the trans-

verse direction. At a strain of 11.5%, the onset of breaking is

observed, producing a configuration shown in Fig. 6(b). This

is soon followed by complete failure at 13.5% strain, Fig. 6(c),

whereupon the two halves of the system remain connected

only through several dangling carbon chains [48].

The slope of the stress–strain curve in Fig. 6(d) is notably

shallower than of those in Fig. 3. This reflects the relative soft-

ness of a-C as described by ReaxFF. The different slope at very

small deformation may reflect the residual intrinsic strain

remaining from incomplete equilibration. However, the main

result of our simulation is that the fracture occurs near the

middle of the a-C region. Therefore, the boundary between

the graphitic and amorphous domains apparently does not

represent a natural weak spot in the structure as one might

expect from a two-phase interface [49]. In this regard, our

simulation predicts that a fixed amount of a-C in the sample

will be more beneficial to distribute between many small

(nanoscale, of size comparable to interface thickness) inclu-

sions, as opposed to few but larger ones. Thus, maximized

(a) (b) (c)

(d)

Fig. 6 – Simulations of graphiticjamorphous heterostructures

under tension. (a) Simulated TEM (left subpanel) atomistic

representation of the as-prepared structure. (b) Atomistic

structure during the initial failure event and (c) after the final

breaking. The failure region is highlighted. The plot in (d)

shows the corresponding stress–strain curve. (A colour

version of this figure can be viewed online.)

C A R B O N 8 5 ( 2 0 1 5 ) 7 2 – 7 8 77

surface-to-volume ratio of amorphous fraction can leverage a

greater relative strength of the graphiticjamorphous interface

compared to amorphous bulk. We also performed several sim-

ulations applying shear loads to the same system, with invari-

ably the same conclusion of failure in the middle of the

amorphous phase, rather than at the phase interfaces.

4. Conclusion

Carbon fiber structure is excessively complex and modeling

attempts necessarily rely on various approximations. Here

we have tested the nanomechanical response of idealized

structural models of BSUs in PAN-derived fibers. By design,

the models do not include any atomistic defects within the

basal planes (vacancies, dislocations, grain boundaries). This

allows to better isolate the effect of mesoscopic-scale flaws such

as the BSU misorientation and the omnipresent interfaces

between domains of different degrees of graphitization.

The designed BSU models appear to capture not only the

gist of the Reynolds–Sharp mechanism of tensile failure

[17,2], but also naturally reveal detailed atomistic processes

that may be responsible for the fracture initiation. In all cases

the interfaces of the misoriented BSU with the surrounding

environment are the culprit of BSU failure. In particular, it

appears that cracks are generated at the basal–plane interface

upon the compressive effect of the restraining surrounding

regions. Our analysis suggests however that the interface

between non-basal–plane graphitic carbon with amorphous

carbon domains may not be intrinsically weak and warrants

further exploration of the interface with other types of pre-

dominant sp2-type domains. All structural models considered

here are also believed to be suitable initial constructions for

introduction of more realistic atomistic details (e.g., better

account of the turbostratic nature of the BSUs; basal–plane

defects, interplanar cross linking leading to enhanced shear

modulus, etc.) in the attempts towards better models of indi-

vidual BSUs and thereby more informed approaches towards

improved PAN-fiber mechanical properties.

Acknowledgments

This work was funded by DARPA DSO and at later stage by the

US Air Force Office of Scientific Research grant FA9550-13-1-

0151. The authors acknowledge multiple discussions and col-

laboration with the Oak Ridge National Laboratory within this

project (S. Ozcan and C. Eberle), and with S. Kumar. Computer

resources were provided by XSEDE, which is supported by NSF

Grant No. OCI-1053575, under allocation TG-DMR100029 and

the DAVinCI cluster acquired with funds from NSF grant

OCI-0959097.

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