GRAPHENE POLYMER COMPOSITE

Abstract

The present invention relates to novel nanocomposite materials, methods of making nanocomposites and uses of nanocomposite materials.

Claims

1.-17. (canceled)

18. A method for the remote monitoring of the strain to which a nanocomposite is subjected, the nanocomposite comprising a substrate and graphene or functionalised graphene; the method comprising: taking Raman measurements of the graphene or functionalised graphene in the nanocomposite.

19. The method of claim 18, wherein the graphene or functionalised graphene takes the form of a plurality of discontinuous flakes.

20. The method of claim 19, wherein the graphene or functionalised graphene flakes are distributed throughout the substrate.

21. The method of claim 18, wherein the graphene or functionalized graphene is attached to the substrate and nanocomposite further comprises an adhesive component for adhering the graphene or functionalized graphene to the substrate.

22. The method of claim 21, wherein the nanocomposite further comprises a protective layer to cover the graphene or functionalized graphene nanocomposite material.

23. The method of claim 18, wherein the graphene or functionalised graphene is pristine graphene.

24. The method of claim 18, wherein the graphene or functionalised graphene is functionalised graphene.

25. The method of claim 24, wherein the functionalised graphene is graphene oxide.

26. The method of claim 18, wherein the substrate is a polymer selected from the group consisting of polyolefins, polyethylenes, polypropylenes, polyacrylates, polymethacrylates, polyacrylonitriles, polyamides, polyvinyl acetates, polyethyleneoxides, terphthalates, polyesters, polyurethanes, and polyvinylchlorides.

27. The method of claim 18, wherein the nanocomposite is a coating.

28. The method of claim 27, wherein the coating is on at least one surface of a structure.

29. The method of claim 28, wherein the structure is a bridge, a building, a ship, or an aircraft.

30. The method of claim 18, wherein the nanocomposite is a plastics product.

31. The method of claim 30, wherein the plastics product is selected from: a pipe; a component for use in the aerospace, defense or automotive industries, and a component of a civil structure.

32. The method of claim 18, wherein the step of taking Raman measurements of the graphene or functionalised graphene comprises measuring a wavelength of the graphene's or functionalised graphene's Raman G band.

33. The method of claim 32, wherein the method comprises determining a shift of the graphene's or functionalised graphene's Raman G band relative to a predetermined value and using said shift to determine an amount of strain to which the nanocomposite is being subjected.

Description

FIGURES

[0079] FIG. 1: The change in the band position of the G and D band in the GO-PVA films of example 1 as a function of exposure to the laser.

[0080] FIG. 2: The variation of the band position of the G and D band in the GO-PVA films of example 1 as a function of location on the film.

[0081] FIG. 3: Change in the G band of the GO-PVA films of example 1 as a function of strain. (Strain measured by the reference resistive gauge.)

[0082] FIG. 4: Change in the D band of the GO-PVA films of example 1 as a function of strain. (Strain measured by the reference resistive gauge.)

[0083] FIG. 5: The position of the G-band position as a function of strain as measured by the reference resistive strain gauge (for the strain sensitive coating of example 2).

[0084] FIG. 6: The position of the G-band position as a function of strain as measured by the reference resistive strain gauge (for the strain sensitive coating of example 2).

[0085] FIG. 7: The position of the G band of the graphene of example 3 as function of strain and time.

[0086] FIG. 8: A photograph of the coated PMMA beams used in example 4. Note the mounted strain gauge on the film.

[0087] FIG. 9: The deformation cycle applied to the PMMA beam of example 4.

[0088] FIG. 10: The peak position of the G band as it follows the strain shown in FIG. 9 of example 4.

[0089] FIGS. 11A and 11B: Contour maps of strain over the graphene flake of example 6 at different strains in the uncoated (FIG. 11A) and coated (FIG. 11B) states.

[0090] FIG. 12: Variation of the strain in the graphene of example 6 along the monolayer at a strain of 0.4% for both uncoated and coated with an SU-8 film.

[0091] FIGS. 13A and 13B: Single monolayer graphene composite.of example 7; FIG. 13A: Optical micrograph showing the monolayer graphene flake investigated; FIG. 13B: Schematic diagram (not to scale) of a section through the composite.

[0092] FIGS. 14A and 14B: Shifts of the Raman G band during loading and unloading of the monolayer graphene composite.of example 7; FIG. 14A: Change in the position of the G band with deformation; FIG. 14B: Shift of the G band peak position as a function of strain. (The blue circles indicate where the loading was halted to map the strain across the flake).

[0093] FIGS. 15A and 15B: Distribution of strain in the graphene composite of example 7 in the direction of the tensile axis (x) across a single monolayer at 0.4% strain; FIG. 15A: Variation of axial strain with position across the monolayer in the x-direction (The curve fitted to the data is Equation (1)); FIG. 15B: Variation of axial strain with position across the monolayer in the vertical direction (The curve is calculated from Equation (1) using the value of ns=20 determined from a) and taking into account the change in width of the graphene sheet with position, y).

[0094] FIG. 16: Distribution of graphene strain of the composite of example 7 in the direction of the tensile axis (x) across a single monolayer at 0.6% strain; variation of axial strain with position across the monolayer mapped in the x-direction. The solid lines are fitted to the data to guide the eye.

[0095] FIG. 17: Raman spectra for different layer flakes of graphene

[0096] FIGS. 18A and 18B: Deformation patterns for a discontinuous flake in a polymer matrix.

[0097] FIG. 19: Balance of stresses acting on an element of length, dx, of the flake of thickness, t, in the composite.

[0098] FIG. 20: Model of a flake within a resin used in shear-lag theory. The shear stress, , acts at a distance z from the flake centre.

[0099] FIGS. 21A and 21B: FIG. 21A: Distribution of strain in the graphene in direction of the tensile axis across a single monolayer at 0.4% strain. The curves are fits of Equ. SI.12 using different values of parameter ns. FIG. 21B: Variation of interfacial shear stress with position determined from Equ. SI.13 for the values of ns used in FIG. 21A.

[0100] FIG. 22: Distribution of strain in the graphene in direction of the tensile axis across a single monolayer at 0.4% strain showing the variation of fibre strain with position across the monolayer in the vertical direction. The curves were calculated from Equ SI.12 using different values of ns.

[0101] FIGS. 23A and 23B: FIG. 23A: G band shift for a nanocomposite according to example 11 after being subjected to a load; FIG. 23B: G band shift for a nanocomposite according to example 11 after being subjected to a load.

[0102] FIGS. 24A and 24B: FIG. 24A: G band shift for a nanocomposite according to example 12 after being subjected to a load; FIG. 24B: G band shift for a nanocomposite according to example 12 after being subjected to a load.

EXAMPLES

Example 1: Strain Sensitive Coating Comprising Graphene Oxide (GO)-Polyvinyl Alcohol (PVA) which was Deposited onto a Polymethyl Methacrylate (PMMA) Beam Specimen

[0103] This example serves to illustrate that graphene oxide (a highly substituted and widely commercially available graphene material) can be used as a strain sensitive coating despite having a modulus of 20% of the modulus of pristine graphene (and therefore a smaller Raman peak shift as compared with pristine graphene).

[0104] A graphene oxide (GO)-polyvinyl alcohol (PVA) coating was deposited on a PMMA beam following the method of Xin Zhao et al. (Macromolecules, 2010, 43, 9411-9416) and as described in detail in the following paragraphs.

[0105] 10 ml of 1 wt % PVA solution was prepared and a separate beaker of 10 ml of 0.1 mg/ml GO solution was also prepared. (The GO solution was made using a method as described in (i) Eda, G.; Fanchini, G.; Chhowalla, M., Large-Area Ultrathin Films of Reduced Graphene Oxide as a Transparent and Flexible Electronic Material. Nat Nano 2008, 3, 270-274; or (ii) Hummers, W. S.; Offeman, R. E., Preparation of Graphitic Oxide. JACS 1958, 80, 1339-1339.) The beam was then coated using the following procedure:

[0106] (i) the PMMA beam was placed in the PVA solution for 10 minutes;

[0107] (ii) the beam was dried in air;

[0108] (iii) the beam then washed by placing it in deionised water for 2 minutes;

[0109] (iv) the beam was dried in air;

[0110] (v) the beam was placed into the GO solution for 10 minutes;

[0111] (vi) the beam then washed by placing it in deionised water for 2 minutes;

[0112] (vii) the beam was dried in air.

[0113] These steps were repeated 20 times so that the coating on the PMMA beam comprised of 20 alternating GO-PVA layers in a laminate-form. It is thought that each polymer layer will partially infiltrate the underlying layer. The number of layers is not important; in this case 20 layers are being used to build up thickness of GO on the substrate (although it is likely that these steps only need to be repeated two or three times). A reference resistive strain gauge was then mounted onto the coating.

[0114] Raman spectra was then collected from the coating using a 514 nm laser at 2.5 mW power at the laser head (Renishaw 1000 system). The positions of the G and D Raman bands were found to be sensitive to the time the laser spent on region of the film being studied (FIG. 1).

[0115] However, it was found that the peak position was repeatable for a given exposure period, such that there was a variation in the position of the bands <0.5 cm.sup.1 across the sample (FIG. 2) as measured over a collection time of 50 seconds.

[0116] The coated PMMA beam was then deformed with the strain increased stepwise (in increments of 0.04%). For each strain step, the average band position was taken across 5 locations on the beam (FIGS. 3 and 4). A peak shift of 3 cm.sup.1 per % was recorded, showing that the GO was a viable strain gauge. (A peak shift of 3 cm.sup.1 per % corresponds to an accuracy of 0.17% for the 0.5 cm.sup.1.)

Example 2: Strain Sensitive Coating Comprising Graphene Oxide (GO)-Polyvinyl Alcohol (PVA) which was Deposited onto a Steel Sample

[0117] This example also serves to illustrate that graphene oxide (a highly substituted graphene material) can be used as a strain sensitive coating. This example provides an alternative substrate to that used in example 1 and an alternative method of applying the PVA-GO coating to that employed in example 1.

[0118] A GO-PVA coating was solution cast onto the steel sample. 0.12 g GO solution (1 mg GO per ml) was mixed with 1.2 g aqueous PVA solution (0.05 wt %) and stirred for 30 minutes. The method for making the GO solution is described in (i) Eda, G.; Fanchini, G.; Chhowalla, M., Large-Area Ultrathin Films of Reduced Graphene Oxide as a Transparent and Flexible Electronic Material. Nat Nano 2008, 3, 270-274; or (ii) Hummers, W. S.; Offeman, R. E., Preparation of Graphitic Oxide. JACS 1958, 80, 1339-1339. The mixture was then dispersed using a sonic bath for another 30 minutes. A drop of the GO-PVA solution was then casted onto 0.4572 mm (0.5 mm) thick spring steel beams and left to dry. The concentration of GO in the final PVA/GO composites was 20 wt %. The resulting GO-PVA coating is a homogeneous mixture of GO and PVA. The reference resistive strain gauge was mounted onto the steel next to the coated area.

[0119] The virtual absence of the G band from the GO meant that that this band could not be used for strain measurements. Likewise, the shift of the G band with strain was found to be within scatter of the homogeneity of the samples (FIG. 5). However, the D peak was found to have a shift rate of 14 cm.sup.1 per % strain, up to a maximum strain of 0.18% at which the interface failed (FIG. 6).

Example 3: The Stability of a Epoxy-Mechanically Exfoliated-Graphene-PMMA Coating on a PMMA Beam: Stability and Interface Failure

[0120] This example serves to illustrate that pristine, mechanically exfoliated graphene (i.e. an unsubstituted graphene material) can be used as a strain sensitive coating. In this example, an epoxy film is being used as an adhesive layer rather than the PVA adhesive of examples 1 and 2.

[0121] A thin epoxy film (300 nm) was spin coated onto a PMMA beam (5 mm thick). Mechanical exfoliated graphene flakes were then deposited on this epoxy film and a PMMA film (50 nm) coated onto the graphene flakes. A reference resistive strain gauge was then mounted onto the top of the PMMA.

[0122] The PMMA beam was deformed stepwise and the peak position was recorded as a function of time at each strain. The Raman G band position was found to decrease with increasing strain up to a strain of 0.3%, at which point the interface between the graphene and surrounding polymer failed. It is noted that the interface of the GO-PVA composites of examples 1 and 2 do not fail at this level of strain. Without meaning to be bound by theory, it is thought that the presence of oxygen in GO provides a better interface with the PVA than the interface between the pristine graphene and expoxy as in this example. This shows the that the present invention can be easily tuned to meet any specific needs relating to accuracy and interface strength. At a given strain, the strain readings were found to be constant within 1.36 cm.sup.1 up to strains of 0.3%. It should be noted that 0.3% strain is useful for most mechanical applications of the present invention.

Example 4: Cyclic Loading of a Epoxy-Mechanical Exfoliated-Graphene-PMMA Coating on a PMMA Beam

[0123] This example serves to illustrate that pristine graphene (i.e. an unsubstituted graphene material) coated onto a PMMA substrate via an epoxy film can be used as a strain gauge. The example also demonstrates the principle of the strain hardening effect.

[0124] A graphene composite coating was deposited onto a PMMA beam, in the same manner as described in previous examples (example 3 above and example 7 below). A reference strain gauge (denoted as reference numeral 3) was mounted on the film (FIG. 8). The remaining reference numerals of FIG. 8 relate to the substrate (1), mechanical exfoliated graphene (2) and the electrodes (4) that at attached to the strain sensor (3). The strain was increased stepwise, but with an increasing peak strain level in each successive cycle, and then decreased as shown in FIG. 9. It can be seen that, as with example 3 above, the interface fails at 0.3% strain. The strain was increased beyond 0.3% to investigate the effects after interface failure. The Raman peak shift with the strain is shown in FIG. 10. As can be seen, the peak position of the G band followed the deformation of the PMMA beam. As table 1, shows, some strain hardening of the composite was observed, with the modulus increasing by a factor 3.

TABLE-US-00001 TABLE 1 Shift rate and effective Young's modulus of graphene subjecting to cyclic deformation with increased strain steps. Shift Rate Effective Maximum (cm.sup.1/ modulus Strain (%) Cycle % strain) (TPa) 0.1% loading 25.10 0.50 unloading 32.40 0.65 0.2% loading 59.49 1.19 unloading 59.05 1.18 0.3% loading 65.63 1.31 unloading 67.59 1.35 0.4% loading 79.52 1.59 unloading 84.84 1.70 0.5% loading 86.91 1.74 unloading 89.19 1.78 (Note that 50 cm.sup.1/% strain = ~1 TPa)

Example 5: Straining Hardening of Graphene Composite Compared to a Single-Walled Nanotubes (SWNT) Composite

[0125] This example serves to illustrate the advantageous differences between graphene composites compared with SWNT composites.

[0126] A graphene composite coating was deposited onto a PMMA beam, as previously described in examples 3 and 4 with a reference strain gauge also mounted on the film (see FIG. 8 which illustrates a strain gauge mounted onto a film). A comparable single walled nanotubes composite (SWNT) was produced by mixing 0.1 wt % HiPco SWNTs (see http://www.nanointegris.com/en/hipco) in epoxy and depositing a layer of this mixture on a epoxy beam.

[0127] The beams were deformed to a strain just beneath that at which the carbon interface failed (0.3% for the graphene and 0.8% for the SWNTs) and then unloaded. This loading cycle was repeated for a total of 4 times. The effective modulus of the SWNTs and graphene in the samples was calculated using a calibration of 1 TPa is equivalent to 50 cm.sup.1 per %. Table 2 summarises the results of the experiment.

[0128] The first conclusion to note is that the shift rate is approximately 3 times higher for the graphene samples as compared to the SWNT samples. This means that a graphene based strain sensor is 3 times more sensitive than a nanotube based strain sensor. Secondly, the effective modulus of the SWNTs remained approximately constant with each cyclic loading, where as the modulus for the graphene samples increases from 1.07 to 1.35 GPa on loading from the 1.sup.st and 4.sup.th loading cycles. This shows the benefit of pre-treatment of the graphene composites to increase their modulus.

TABLE-US-00002 TABLE 2 A summary of the SWNT and graphene cyclic deformation up to same strain level (Graphene-0.3% and SWNT-0.8%) SWNT Graphene (max strain of 0.8%) (max strain of 0.3%) Shift rate Effective Shift rate Effective (cm.sup.1/ modulus (cm.sup.1/ modulus Cycle % strain) (TPa) % strain) (TPa) 1 Loading 17.48 0.35 53.68 1.07 Unloading 16.10 0.32 47.53 0.95 2 Loading 16.72 0.33 48.61 1.10 Unloading 13.94 0.28 48.81 0.98 3 Loading 15.95 0.32 58.11 1.16 Unloading 12.43 0.25 53.80 1.08 4 Loading 15.72 0.31 67.33 1.35 Unloading 11.69 0.23 48.21 0.96

Example 6: Graphene Vs Graphene Sandwich

[0129] This example serves to illustrate that a sandwiched graphene composite works as well as a non-sandwiched graphene composite as a strain sensor given sufficiently large graphene flakes and good interface between the graphene and the underlying polymer. This is important as a sandwiched graphene composite will be harder wearing than a non-sandwiched graphene composite and therefore the real-life utility of a strain sensor comprising graphene is improved.

[0130] The specimen was prepared following the general procedure of examples 3 and 4 above and employed a 5 mm thick poly(methyl methacrylate) beam spin-coated with 300 nm of SU-8 epoxy resin. The graphene was produced by mechanical cleaving of graphite and deposited on the surface of the SU-8. This method produced graphene with a range of different numbers of layers and the monolayers were identified both optically and by using Raman spectroscopy. The PMMA beam was deformed in 4-point bending up to 0.4% strain with the strain monitored using a strain gage attached to the beam surface. Well-defined Raman spectra could be obtained from the graphene monolayer using a low-power HeNe laser (1.96 eV and <1 mW at the sample in a Renishaw 2000 spectrometer) and the deformation of the graphene in the composite was followed from the shift of the 2D (or G) band. The laser beam polarization was always parallel to the tensile axis and the spot size of the laser beam on the sample was approximately 2 m using a 50 objective lens.

[0131] Raman spectra were obtained at different strain levels through mapping over the graphene monolayer in steps of between 2 m and 5 m by moving the x-y stage of the microscope manually and checking the position of the laser spot on the specimen relative to the image of the monolayer on the screen of the microscope. The strain at each measurement point was determined from the position of the 2D Raman band using the calibration in FIG. 2 and strain maps of the monolayer were produced in the form of colored x-y contour maps using the OriginPro 8.1 graph-plotting software package, which interpolates the strain between the measurement points (see FIGS. 11A and 11B).

[0132] The beam was then unloaded and another thin 300 nm layer of SU-8 epoxy resin was then spin-coated on top so that the graphene remained visible when sandwiched between the two coated polymer layers. The beam was then reloaded initially up to 0.4% strain, unloaded and then reloaded to various other levels of strain. The strain in the graphene monolayer was mapped fully at each strain level as well as in the unloaded state (see FIGS. 11A and 11B).

[0133] As can be seen from comparing the coated and uncoated contour maps of FIGS. 11A and 11B and the strain plots of FIG. 12, the presence of a coating on the top of the graphene has no deleterious effect on the sensitivity of the material.

Example 7

[0134] A graphene polymer composite was prepared using a 5 mm thick poly(methyl methacrylate) beam spin-coated with 300 nm of SU-8 epoxy resin. The graphene, produced by the mechanical cleaving of graphite, was deposited on the surface of the SU-8. This method produced graphene with a range of different numbers of layers and the monolayers were identified both optically [26] and using Raman spectroscopy. A thin 50 nm layer of PMMA was then spin-coated on top of the beam so that the graphene remained visible when sandwiched between the two coated polymer layers as shown in FIG. 13A. FIG. 13B illustrates a schematic diagram (not to scale) of a section through the composite.

[0135] The PMMA beam was deformed in 4-point bending and the strain monitored using a strain gauge attached to the beam surface. A well-defined Raman spectrum could be obtained through the PMMA coating using a low-power HeNe laser (1.96 eV and <1 mW at the sample in a Renishaw 2000 spectrometer) and the deformation of the graphene in the composite was followed from the shift of the G band [22-25] (see FIGS. 14A and 14B). The laser beam polarization was always parallel to the tensile axis.

Example 8Characterisation of the Graphene Using Raman Spectroscopy

[0136] Raman spectroscopy has been employed to follow the deformation of the graphene in the polymer composite. FIG. 17 shows that the technique can also be used to differentiate between flakes of graphene with different numbers of layers.

Example 9Shear Lag Analysis for a Graphene Single Monolayer.SUP.S2, S3

[0137] In the case of discontinuous graphene flakes reinforcing a composite matrix, stress transfer from the matrix to the flake is assumed to take place through a shear stress at the flake/matrix interface as shown in FIGS. 18A and 18B. Before deformation parallel lines perpendicular to the flake can be drawn before deformation from the matrix through the flake. When the system is subjected to axial stress, .sub.1, parallel to the flake axis, the lines become distorted since the Young's modulus of the matrix is much less than that of the flake. This induces a shear stress at the flake/matrix interface. The axial stress in the flake will build up from zero at the flake ends to a maximum value in the middle of the flake. The uniform strain assumption means that, if the flake is long enough, in the middle of the flake the strain in the flake equals that in the matrix. Since the flakes have a much higher Young's modulus it means that the flakes carry most of the stress in the composite.

[0138] The relationship between the interfacial shear stress, .sub.i, near the flake ends and the flake stress, .sub.f, can be determined by using a force balance of the shear forces at the interface and the tensile forces in a flake element as shown in FIG. 19.

[0139] The main assumption is that the forces due to the shear stress at the interface, .sub.i, is balanced by the force due to the variation of axial stress in the flake, d .sub.f, such that if the element shown in FIG. 19 is of unit width

.sub.idx=td.sub.f(SI.1)

and so

[00004]$\begin{array}{cc}\frac{d.Math..Math.{}_f}{\mathrm{dx}}=-\frac{{}_i}{t}& \left(\mathrm{SI}.Math.\mathrm{.2}\right)\end{array}$

[0140] The behaviour of a discontinuous flake in a matrix can be modelled using shear lag theory in which it is assumed that the flake is surrounded by a layer of resin at a distance, z, from the flake centre as show in FIG. 20. The resin has an overall thickness of T. It is assumed that both the flake and matrix deform elastically and the flake-matrix interface remains intact. If u is the displacement of the matrix in the flake axial direction at a distance, z, then the shear strain, , at that position is be given by

[00005]$\begin{array}{cc}=\frac{\mathrm{du}}{\mathrm{dz}}& \left(\mathrm{SI}.Math.\mathrm{.3}\right)\end{array}$

The shear modulus of the matrix is defined as G.sub.m=/ hence

[00006]$\begin{array}{cc}\frac{\mathrm{du}}{\mathrm{dz}}=\frac{}{G_m}& \left(\mathrm{SI}.Math.\mathrm{.4}\right)\end{array}$

The shear force per unit length carried by the matrix is transmitted to the flake surface though the layers of resin and so the shear strain at any distance z is given by

[00007]$\begin{array}{cc}\frac{\mathrm{du}}{\mathrm{dz}}=\frac{{}_i}{G_m}& \left(\mathrm{SI}.Math.\mathrm{.5}\right)\end{array}$

This equation can be integrated using the limits of the displacement at the flake surface (z=t/2) of u=u.sub.f and the displacement at z=T/2 of u=u.sub.T

[00008]$\begin{array}{cc}\underset{u_t}{\overset{u_T}{}}.Math.\mathrm{du}=\left(\frac{{}_i}{G_m}\right).Math.\underset{t.Math.\text{/}.Math.2}{\overset{T.Math.\text{/}.Math.2}{}}.Math.\mathrm{dz}& \left(\mathrm{SI}.Math.\mathrm{.6}\right)\\ \mathrm{hence}.Math..Math.u_T-u_f=\left(\frac{{}_i}{2.Math.G_m}\right).Math.\left(T-t\right)& \left(\mathrm{SI}.Math.\mathrm{.7}\right)\end{array}$

It is possible to convert these displacements into strain since the flake strain, e.sub.f and matrix strain, e.sub.m, can be approximated as e.sub.fdu.sub.f/dx and e.sub.mdu.sub.T/dx. It should be noted that this shear-lag analysis is not rigorous but it serves as a simple illustration of the process of stress transfer from the matrix to a flake in a graphene-flake composite. In addition, is given by Equation (SI.2) and so differentiating Equation (SI.7) with respect to x leads to

[00009]$\begin{array}{cc}{e}_f-e_m=\frac{\mathrm{tT}}{2.Math.G_m}.Math.\left(\frac{d^2.Math.{}_f}{{\mathrm{dx}}^2}\right)& \left(\mathrm{SI}.Math.\mathrm{.8}\right)\end{array}$

since T>>t. Multiplying through by E.sub.f gives

[00010]$\begin{array}{cc}\frac{d^2.Math.{}_f}{{\mathrm{dx}}^2}=\frac{n^2}{t^2}.Math.\left({}_f-e_m.Math.E_f\right).Math..Math.\mathrm{where}.Math..Math.n=\sqrt{\frac{2.Math.G_m}{E_f}.Math.\left(\frac{t}{T}\right)}& \left(\mathrm{SI}.Math.\mathrm{.9}\right)\end{array}$

This differential equation has the general solution

[00011]${}_f=E_f.Math.e_m+C.Math..Math.\sinh \left(\frac{\mathrm{nx}}{t}\right)+D.Math..Math.\cosh \left(\frac{\mathrm{nx}}{t}\right)$

where C and D are constants of integration. This equation can be simplified and solved if it is assumed that the boundary conditions are that there is no stress transmitted across the flake ends, i.e. if x=0 in the middle of the flake where .sub.f=E.sub.fe.sub.m then .sub.f=0 at x=l/2. This leads to C=0 and

[00012]$D=-\frac{E_f.Math.e_m}{\cosh \left(\mathrm{nl}.Math.\text{/}.Math.2.Math.t\right)}$

The final equation for the distribution of flake stress as a function of distance, x along the flake is then

[00013]$\begin{array}{cc}{}_f=E_f.Math.e_m\left[1-\frac{\cosh \left(\mathrm{nx}.Math.\text{/}.Math.t\right)}{\cosh \left(\mathrm{nl}.Math.\text{/}.Math.2.Math.t\right)}\right]& \left(\mathrm{SI}.Math.\mathrm{.10}\right)\end{array}$

Finally it is possible to determine the distribution of interfacial shear stress along the flake using Equation (SI.2) which leads to

[00014]${}_i={\mathrm{nE}}_f.Math.e_m.Math.\frac{\sinh \left(\mathrm{nx}.Math.\text{/}.Math.t\right)}{\cosh \left(\mathrm{nl}.Math.\text{/}.Math.2.Math.t\right)}$

[0141] It is convenient at this stage to introduce the concept of flake aspect ratio, s=l/t so that the two equations above can be rewritten as

[00015]$\begin{array}{cc}{}_f=E_f.Math.e_m\left[1-\frac{\cosh \left(\mathrm{ns}.Math.\frac{x}{l}\right)}{\cosh \left(\mathrm{ns}.Math.\text{/}.Math.2\right)}\right]& \left(\mathrm{SI}.Math.\mathrm{.12}\right)\end{array}$

for the axial flake stress and as

[00016]${}_i={\mathrm{nE}}_f.Math.e_m.Math.\frac{\sinh \left(\mathrm{ns}.Math.\frac{x}{l}\right)}{\cosh \left(\mathrm{ns}.Math.\text{/}.Math.2\right)}$

for the interfacial shear stress.

[0142] It can be seen that the flake is most highly stressed, i.e. the most efficient flake reinforcement is obtained, when the product ns is high. This implies that a high aspect ratio, s, is desirable along with a high value of n.

Example 10Fit of Experimental Data of the Graphene Monolayer to the Shear Lag Analysis

[0143] The experimental data on the variation of graphene strain across the monolayer flake are fitted to the shear lag analysis derived above in FIGS. 21A and 21B. It can be seen that the fits of the theoretical shear-lag curves to the strain distribution are sensitive to the value of ns chosen. Likewise the value of interfacial shear stress at the flake ends is very sensitive to the values of ns chosen.

[0144] FIG. 22 shows the fits of Equ. SI.12 to the vertical strain distribution across the graphene monolayer flake as it tapers to a point at y=0. It can be seen that the fits are very sensitive to the value of ns employed.

Example 11: SU-8/Mechanical Cleaved Graphene/SU-8/Steel

[0145] SU-8 epoxy was spin coated onto a steel substrate. Mechanically cleaved graphene was deposited on the SU-8 and a thin layer of SU-8 epoxy was laid on top of it. A bilayer of graphene was indentified and the shift of the G plotted as a function of strain as measured from a reference resistive gauge was recorded. The effective modulus of the graphene during loading was 0.28 TPa and unloading was 0.35 TPa.

Example 12: SU-8/Mechanically Cleaved Graphene/Steel

[0146] Mechanically cleaved graphene was deposited onto a steel substrate and SU-8 was spin coated on it. It was found that the mechanically cleaved graphene did not adhere well to the steel, without the epoxy adhesion layer. A graphene multilayer flake was indentified and the spectra collected as a function of strain, as measured by a reference resistive strain gauge. The poor adhesion between the graphene and the steel meant that the rate of the peak shift for the graphene was very low compared to when an adhesion layer is used.

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