INTERFACIAL AND INTERNAL STRESS TRANSFER IN CARBON NANOTUBE
BASED NANOCOMPOSITES
Robert J. Young1*, Libo Deng1,2, Tamer Z. Wafy1,3 and Ian A Kinloch1
1
School of Materials, University of Manchester, Manchester, M13 9PL, United Kingdom
2
Chinese Academy of Sciences, Shenzhen Institute of Advanced Technology, Shenzhen
518055, China
3
Military Technical College, Kobry Elkobbah, Egypt
*Corresponding Author - Tel: +44-161-306-3550, Email: robert.young@manchester.ac.uk
Keywords: Carbon nanotubes, Nanocomposites, Micromechanics, Raman spectroscopy
SUPPLEMENTARY MATERIAL
Modelling internal stress transfer in MWNT/epoxy nanocomposites
The G'-peak (or 2D-peak) for CNTs can be described by a Lorentzian function, given by
[1]
𝐼
0
𝐼(𝜔) = 𝜋𝛤
[
1
2
(𝜔–𝜔𝐺′ ) +(𝛤)2
]
(1)
where I() is the resulting intensity of a forced damped harmonic oscillator model of phonon
excitation, 𝐼0 is the maximum intensity of the G'-peak, (arbitrary units),  is the half width at
half maximum intensity, HWHM (cm-1), while FWHM (cm-1) is the full width at half
maximum intensity 2 (cm-1),  is the oscillating frequency of the external field, (cm-1), and
 G'CNT
is the centre of G'-peak, the natural vibration frequency of carbon nanotubes in the
composites, (cm-1).
It has been demonstrated that the frequency of the G'-peak depends on the tube diameter of
the isolated SWNTs through [2]:
35.4
𝜔𝐺` = 𝜔𝐺′0 − (
𝐷
)
(2)
S1
where 𝜔𝐺′0 is the frequency G`-peak in graphene (cm-1) that is laser energy dependent and
𝐷 is the diameter (nm) of the individual tubes in MWNTs and SWNTs. The stress sensitive
Raman bands (G and G' band) can be employed to follow the stress transfer behaviour in
CNT/polymer composites and monitor interfacial adhesion.
The dependence of the frequency G'-peak in graphene upon strain has been adapted to
carbon nanotubes by incorporation of the stress transfer efficiency parameter for the CNTs
(outer wall)/polymer interface and for the inner walls interface within the CNTs, as well as
the applied strain such that the stress-induced shift rate of the G'-band in carbon nanotubes
can be defined as:
𝜔𝐺′0 = 𝜔𝐺′𝐶𝑁𝑇 – (𝑘𝑖 ∗ 𝑆(0) ∗ )
(3)
where 𝑘i is the stress transfer efficiency parameter, 𝑆(0) is the stress-induced shift rate of the
G'-band, (cm-1) and  is the applied strain, (%). After substitution Equation (4) in Equation
(3), the frequency of the G'-peak is written as:
35.4
𝜔𝐺′ = (𝜔𝐺′𝐶𝑁𝑇 − ( 𝑘𝑖 ∗ 𝑆(0) ∗ )) − (
𝐷
)
(4)
The relationship between the transmittance of light 𝑇 (%), and the number of graphene
layers has been widely investigated and monolayer graphene is found to transmit about
97.7% of visible white light [3]. The most extreme case (T = 100%) is when the light is fully
transmitted through the outmost wall of the SWNT/polymer composites. Hence the intensity
of the frequency of the G'-peak is affected by the absorption the light. Based on the frequency
of the G'-peak of Equation (4), the intensity, 𝐼(𝜔) of Equation (1) may be given, between the
limits 2500 cm-1  𝜔  2800 cm-1, as
1
𝐼(𝜔) = [
π
1
∗T
𝛤
(𝜔−𝜔𝐺`𝐶𝑁𝑇 +(𝑘𝑖 ∗𝑆(0) ∗)+(
2
35.4
))
𝐷
]
(5)
+(𝛤)2
Finally the line shapes of the G'-peak for SWNTs/epoxy composites were calculated for
each strain level in the range 0.0 - 1.5%, in which the laser light is being transmitted through
the SWCNTs, (5, 5) layer is given by:
1
𝐼(𝜔) = π [
1
∗(0.954)0
25
2
35.4
(𝜔−2614+(1∗14∗)+(
))
𝐷(5,5)
]
(7)
+(25)2
In Figure 4 the experimental shift of the Raman G' band positions and widths (FWHM) as
a function of strain from 0 to 1.5 % for the hot-cured 0.1 wt% SWNT/epoxy nanocomposites
and their corresponding model simulation calculated by Equation (6) using a symmetrical
Lorentzian are shown. As seen in Figure 4(a) there is a very good agreement with the
S2
measured and the line simulation shapes of the G'-peak for SWNTs in SWNT/epoxy
composites upon tensile loading as a function of strain from 0 to 1.5 %. It is known, that
whenever there is good adhesion between nanotubes and a polymeric matrix a linear
relationship is obtained, but when there is a gradual loss of adhesion the line begins to deviate
from linearity. The G'-band shift is linear up to about 0.7% strain followed by a deviation
from linearity where debonding occurs, giving a change in stress transfer efficiency between
the nanotubes and polymeric matrix.
The SWNTs (n = 1) were described by a single (5, 5) layer whereas the family of ten
coaxial commensurate tubes for MWNTs is shown in Figure 5 which it can be presented by
(5n, 5n), (5n-1, 5n-1), (5n-2, 5n-2),…….., (5n-9, 5n-9) for the outermost wall, the ninth wall,
the eighth wall,…….., and the innermost wall, respectively. The separation between
individual walls in MWCNTs is constant (~0.35 nm) for this family in tubes. The outermost
wall (5n, 5n) is in contact directly with the polymer epoxy matrix. The relationship between
the transmittance of light 𝑇, and the number of graphene layers has been investigated widely
and monolayer graphene is found to transmit about 97.7% of visible white light [3] as shown
in Figure S1.
Figure S1 A schematic diagram representing the light scattered from the layers of the
MWNTs.
S3
Recently, in graphene optical studies have shown that the transmittance of the light is
decreasies as the number of graphene layers increases. There was an argument, however,
about the dependence of the wavelength of the light and its transmission through the
graphene layers [4, 5]. This has not been taken into account in the present calculations. All
calculations were performed on computational software programs Mathematica 8 and
OriginPro 8.5.
For the line shapes of the G'-peak for MWNT/epoxy composites, the transmission values
follow the geometric sequence with common ratio 0.954 (=0.9770.977), were calculated for
each strain in the range 0.0 - 1.5% is given for n = 10 layers by:
1
∗(0.954)n−10
25
1
𝐼(𝜔) = π [
1
π
35.4
))
D(5n,5n)
1
∗(0.954)n−9
25
[
1
(ω−2678.53+((0.7)n−n ∗14∗)+(
(ω−2678.53+((0.7)n−1 ∗14∗)+(
[
π
35.4
))
D(5(n−1),5(n−1))
35.4
))
D(5(n−9),5(n−9))
+
+(25)2
Outermost tube
]
2
1
∗(0.954)n−8
25
(ω−2678.53+((0.7)n−9 ∗14∗)+(
]
2
+. . … +
+(25)2
ninth tube
]
2
(8)
+(25)2
Innermost tube
The individual components of the G'-band from the family of nanotubes and the resultant
overall G' band are presented in Figure 6 using the Equation (8). In order to know the
appropriate value of the stress transfer efficiency parameters, 𝑘i for the experimental shift of
the Raman G' bands, the value of the stress transfer efficiency parameters, 𝑘i were varied
between zero and one in steps of 0.1.
The different values of the stress transfer efficiency parameters, 𝑘i as a function of strain
from 0 to 1.5 % for hot cured 0.1wt % MWNTs /epoxy nanocomposites, along with the
experimental shift of the Raman G' bands positions and their corresponding model simulation
as a function of strain from 0 to 1.5 % for hot cured 0.1wt % MWNT/epoxy nanocomposites
are presented in Figure 6.
The calculations of the resulting line shapes of the G'-peak induced stresses, for 10 layers,
yield a best fit with a shear transfer efficiency parameter of 𝑘i ≈ 0.7. Some authors [6, 7] have
speculated that the limiting cases for the shear transfer efficiency parameter corresponding to
perfect and no shear transfer (full slip) are 𝑘i = 1.0 and 𝑘i = 0, respectively as shown in
S4
Figure 7. The reason for this could be attributed to the imperfect interface bonding between
layers, therefore the shear stress decreases steadily as the number of layers increases. In
addition, the load is taken mainly by the outside layer and the inner layers offer decreasing
reinforcement.
Comparisons between the experimental and simulated of the G`-bands downshifts are
presented in Figure 8(a)). It is observed that the G' band shifts approximately linearly and
suggests that there is good bonding between the MWNTs and epoxy matrix. However, on
increasing the tensile strain the shift becomes non-linear. For the MWNT/epoxy composites,
the most striking result to emerge from Figure 8(b) is that the narrowing of G`-band up to
1.5% strain is found in both the experimental and simulated data. This is probably due to the
outer walls being more highly strained than the inner walls, leading to larger band shifts for
the outer wall components of the G' band spectrum (Figure 6).
REFERENCES
[1]
A Jorio, MS Dresselhaus, R Saito, G Dresselhaus (2011) Raman spectroscopy in
graphene related systems. Wiley-VCH, Weunheim, Germany
[2]
R Saito, M Hofmann, G Dresselhaus, A Jorio, MS Dresselhaus (2011) Adv. Phys. 60:
413.
[3]
R Nair, P Blake, A Grigorenko, K Novoselov, T Booth, T Stauber, N Peres, A Geim
(2008) Science 320: 1308.
[4]
A Gray, M Balooch, S Allegret, S De Gendt, W-E Wang (2008) J. Appl. Phys. 104:
053109.
[5]
JW Suk, A Kitt, CW Magnuson, Y Hao, S Ahmed, J An, AK Swan, BB Goldberg, RS
Ruoff (2011) ACS Nano 5: 6916.
[6]
S Cui, IA Kinloch, RJ Young, L Noé, M Monthioux (2009) Adv. Mater. 21: 3591.
[7]
L Zalamea, H Kim, RB Pipes (2007) Comp. Sci. Tech. 67: 3425.
S5