Interfacial and Internal Stress Transfer in Carbon Nanotube Based Nanocomposites
Robert J. Young1*, Libo Deng1,2, Tamer Z. Wafy1,3 andIan A Kinloch1
1School of Materials, University of Manchester, Manchester, M13 9PL, United Kingdom
2Chinese Academy of Sciences, Shenzhen Institute of Advanced Technology, Shenzhen 518055, China
3Military Technical College, Kobry Elkobbah, Egypt
*Corresponding Author - Tel: +44-161-306-3550, Email:
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]
(1)
where I() is the resulting intensity of a forced damped harmonic oscillator model of phonon excitation, 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 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]:
(2)
where is the frequency G`-peak in graphene (cm-1) that is laser energy dependent and 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:
(3)
where is the stress transfer efficiency parameter, 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:
(4)
The relationship between the transmittance of light (%), 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, of Equation (1) may be given, between the limits 2500 cm-1 2800 cm-1, as
(5)
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:
(7)
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 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 , 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.
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 (=0.9770.977), were calculated for each strain in the range 0.0-1.5% is given for n= 10 layers by:
(8)
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, for the experimental shift of the Raman G' bands, the value of the stress transfer efficiency parameters, were varied between zero and one in steps of 0.1.
The different values of the stress transfer efficiency parameters, 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 ≈ 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 = 1.0 and = 0, respectively as shown in 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
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