Strain Engineering in Monolayer WS2 and WS2 Nanocomposites

There has been a massive growth in the study of transition metal dichalcogenides (TMDs) over the past decade, based upon their interesting and unusual electronic, optical and mechanical properties, such as tuneable and strain-dependent bandgaps. Tungsten disulfide (WS2), as a typical example of TMDs, has considerable potential in applications such as strain engineered devices and the next generation multifunctional polymer nanocomposites. However, controlling the strain, or more practically, monitoring the strain in WS2 and the associated micromechanics have not been so well studied. Both photoluminescence spectroscopy (PL) and Raman spectroscopy have been proved to be effective but PL cannot be employed to characterise multilayer TMDs while it is difficult for Raman spectroscopy to reveal the band structure. In this present study, photoluminescence and Raman spectroscopy have been combined to monitor the strain distribution and stress transfer of monolayer WS2 on a flexible polymer substrate and in polymer nanocomposites. It is demonstrated that WS2 still follows continuum mechanics on the microscale and that strain generates a non-uniform bandgap distribution even in a single WS2 flake through a simple strain engineering. It is shown that these flakes could be useful in optoelectonic applications as they become micron-sized PL emitters with a band gap that can be tuned by the application of external strain to the substrate. The analysis of strain distributions using Raman spectroscopy is further extended to thin-film few-layer WS2 polymer nanocomposites where it is demonstrated that the stress can be transferred effectively to WS2 flakes. The relationship between the mechanical behaviour of single monolayer WS2 flakes and that of few-layer flakes in bulk composites is investigated.


Introduction
The past two decades have witnessed an increasing interest in the application of 2D materials in various applications ranging from device fabrication to polymer composites. As one of the first 2D materials to be isolated graphene, in particular, has been studied extensively [1]. Graphene is, however, an electrical conductor that limits its wider applications where a bandgap is needed. Hence, TMDs have started to attract attention since their tunable bandgap offers more controllability [2,3]. As a typical example of TMDs, tungsten disulfide (WS2) has received significant interest for applications in transistors [4], photo-detectors [5], photovoltaic devices [6] and composites [7]. Of particular interest is that WS2 exhibits a transition from a direct-to indirect-bandgap semiconductor both as the number of layers increases [8] and when the WS2 flakes are subjected to strain [9]. Beyond the electronic applications, WS2 flakes have also been found to have a reasonable interfacial interaction with polymers as reflected by their good reinforcement of polymers, determined by their lateral size and its distribution [10], even at a low loadings [10,11]. This sets the foundation for making use of WS2 for the next generation multifunctional nanocomposites for a number of different applications, such as transistors, sensors, photo-detectors, photovoltaics and absorbers etc. [4][5][6]12].
In spite of the extensive work on the applications of WS2, however, relatively little is known about one fundamental core aspect of their behaviourthe deformation micromechanics of WS2 flakes for strain engineering in nanocomposites. For example, although the Young's modulus of WS2 has been measured by Liu et al. [13] to be ~270 GPa, its bilayer exhibits a lower stiffness indicating 25% reduction of stress transfer from one layer to the other due to weak interlayer interaction, similar to that observed in few-layer graphene [14]. The state of strain at the interface between TMDs and polymers, used either as coating in devices or as matrices in thin film composites [15], is also crucial. This is because it is a fundamental aspect of deformation mechanics to ensure that no mechanical failure occurs as well as the core aspect of bandgap control. This highlights the importance of strain engineering, particularly in the monitoring of the local strain in WS2 flakes and the state of stress at the interface.
Similar to earlier studies upon graphene and graphene oxide [16,17], Raman spectroscopy has been used to follow the deformation of WS2 and its Raman bands have been found to undergo a red-shift under tensile strain. In particular, the band widths and intensities are also found to correlate with the level of applied strain [18]. It is difficult, however, for Raman spectroscopy to be used to study the bandgap structure of WS2. Instead, photoluminescence (PL) has recently been employed for this purpose in TMDs and it has been found the PL peaks can be used to reveal the structural non-uniformity caused by strain and doping of MoS2, for which the energies of the PL peaks undergo a red shift with tensile strain [9,19,20]. This phenomenon was then applied to flexible WS2 devices to monitor structural uniformity and strain relaxation during cyclic loading [21]. This behaviour is advantageous as PL simultaneously senses the strain as well as monitoring the bandgap in WS2, suggesting a method of varying the bandgap in a single WS2 flake by simple strain engineering. A recent study [22] has shown that it is possible to tailor the PL of TMD monolayers by forming hydrocarbon-filled bubbles of monolayer TMDs on a substrate. These bubbles localise the PL by producing micrometre-sized strain gradients which form so-called "artificial atoms" that are well separated on the substrate. A downside of the PL approach, however, is also apparent since, unlike in Raman spectroscopy, the intensity of PL peaks decreases greatly as the number of layers increases so that most of the PL studies to date have so far concentrated mainly on TMD monolayers [23].
In the present study, we have combined Raman and photoluminescence spectroscopy to monitor the strain distribution in mechanically-cleaved monolayer WS2 flakes deformed on flexible substrates. This methodology is further extended to WS2/poly(vinyl alcohol) nanocomposites where the deformation micromechanics of multilayer WS2 flakes within the polymer is revealed.

Materials and processing
2.1.1 Substrate preparation. A poly(methyl methacrylate) (PMMA) beam was coated by spin coating a 600 nm layer of SU-8 (MicroChem SU-8 2000 Permanent Epoxy Negative Photoresist) as substrate in order to enhance the contrast with the substrate under the optical microscope. The coating was undertaken using a spin coater (WS-650Mz-23NPPB spin coater Laurell Technologies Corporation). The static coating procedure involved dispensing 0.5 ml of the SU-8 resin on the PMMA substrate, followed by 10 s spinning at 500 rpm with 100 rpm/s acceleration and 30 s at 2000 rpm with 300 rpm/s acceleration [24]. The specimens were then subjected to a 1 min soft bake at 95 C and 2 min post bake at 95 C [24].

Exfoliation and transfer. Monolayer WS2 flakes
were obtained by the exfoliation through the micromechanical cleavage [25] of bulk WS2 crystals with an average grain size of 200 μm supplied by HQ Graphene, Groningen, the Netherlands. The bulk WS2 crystals were peeled repeatedly with an adhesive tape until very thin flakes were obtained.
The flakes obtained were then transferred to the PMMA substrate with a lay of SU-8 on top by pressing the back of the adhesive film. After the transfer, monolayer flakes were located and identified using optical microscopy and a combination of Raman and photoluminescence (PL) spectroscopy.

Nanocomposite preparation. Liquid-phase
exfoliation was employed to produce WS2 dispersions, termed LPE WS2. A detailed procedure of the exfoliation was reported earlier by Bissett et al. [26]. Briefly, dispersions of LPE WS2 were produced by ultrasonication for 12 hours of commercially-available WS2 powder (Sigma-Aldrich) in a mixture of isopropanol and water (1:1 v/v) at a concentration of 10 mg/ml [26]. This was followed by centrifugation at three different speeds (1500, 3000 and 6000 rpm) so that a range of lateral dimensions could be obtained, namely LPE WS2-L, LPE WS2-M and LPE WS2-S, respectively. The WS2/PVA thin film nanocomposites were fabricated via a solutionmixing method. The poly(vinyl alcohol) (PVA, Sigma-Aldrich) was firstly dissolved in deionized water at ~90 C to form aqueous solutions (~50 mg/ml). The polymer solutions were mixed with the dispersions of WS2 in the appropriate ratio with WS2 to give loadings of 0.5, 0.8, 1.2, 2.0 and 5.0 wt%, respectively, followed by a short sonication (~5 min). Finally, the solutions were dried at room temperature for a few days (> 48 h) to obtain the composites. The LPE WS2 loading was converted to volume fraction Vf (vol%) from the weight fraction Wf (wt%) using the following equation: where ρp and ρW are density of the polymer and WS2 flakes, which were taken as 1.3 g/cm 3 [17] and 7.5 g/cm 3 [27], respectively. As a result, the 0.5, 0.8, 1.2, 2.0 and 5.0 wt% weight fraction loadings are converted to volume fractions of 0.09, 0.14, 0.21, 0.35 and 0.90 vol%, respectively.

Characterisation
An atomic force microscope (AFM, Bruker dimension 3100) was used in the tapping mode to provide morphological information about the WS2 flakes deposited onto a Si/SiO2 substrate. For transmission electron microscopy (TEM) analysis, the WS2 flakes were tape exfoliated onto a Si/SiO2 substrate. They were then transferred to a lacey-carbon Cu TEM grid via a polymer-free technique. The TEM phase contrast images were obtained using A JEOL 2100 field emission TEM with an accelerating voltage of 200 kV in a bright field detector. The acquisition time was between 0.5 s and 2.5 s.
The concentrations of WS2 dispersions were determined using a UV-Visible spectrophotometer (Thermal Scientific Evolution 201) with a Beer's Law Coefficient of 2756 ml/mg/m (for peak at λ=629 nm) [28]. Raman and PL spectroscopy were conducted using a Horiba LabRam system using λ=488 nm laser excitation at room temperature. The laser power for both spectroscopy measurements was set at a low level to avoid laser heating of specimens. The orientation of the LPE WS2 filler was examined using polarised Raman spectroscopy [29]. The in-situ deformation testing of the LPE WS2/PVA nanocomposites was undertaken using a Renishaw system 1000 Raman spectrometer with a λ=514.5 nm laser excitation with low power (<0.2 mW). The tensile properties of the neat PVA and LPE WS2/PVA composites with different WS2 loadings were determined based on the ASTM D3039 method using an Instron-1122 universal testing machine. Prior to mechanical testing, the specimens were left for 24h in an air-conditioned laboratory where the temperature was set as 23.0 ± 0.1 C with a relative humidity of ~50 ± 5%. The specimens were deformed using a crosshead speed of 1 mm/min and four or five specimens for the neat PVA sample and each of the LPE WS2/PVA composites sample were tested.

In-situ deformation with PL and Raman spectroscopy
For deformation of the monolayer WS2 flakes, the PMMA substrate was mounted on a 4-point bending rig and placed under the microscope stage in Raman spectrometer. A resistance strain gauge was fixed to the PMMA beam close to the WS2 flakes to monitor the strain. The specimen was then deformed stepwise with the Raman or PL spectra collected for each strain step. Mapping was undertaken using a grid size of 0.5 m  0.5 m or 1 m  1 m depending upon sizes of the flakes and the degree of precision needed. The laser was polarised parallel to the direction of tensile strain. For the deformation test of thin film nanocomposites, the films were mounted and stretched in a tensile rig designed in house and Raman spectra were obtained for the LPE WS2/PVA nanocomposites. The details of the experimental method have been described in detail in our previous report [7]. Figure 1a shows an optical image of a typical WS2 flake prepared via micromechanical cleavage consisting of a thick multilayer and a monolayer adjacent to it as confirmed by PL. Figure 1b shows the PL spectrum of the monolayer consisting of an extremely strong peak with two features, namely: a dominant peak centered at ~2.00 eV corresponding to a neutral A exciton and a weaker peak with slightly lower energy known as the Trion exciton (Aexciton) caused by undesired doping during exfoliation [30,31]. In contrast, Figure 1c also shows that the intensity of the PL peaks reduces significantly as the number of layers of WS2 increases, and the spectrum from the bulk crystal shows nearly no peak at all [23,32]. This is a result of the WS2 undergoing the transition from indirect band gap (Γ-Κ) to direct band gap (Κ-Κ) semiconductor as the number of layer decreases towards monolayer [23,32]. This can therefore be used to identify monolayer WS2 since only the monolayer region shows a strong intensity of the A exciton peak whereas there is no detectable signal from thicker counterparts (inset in Figure 1a).

Photoluminescence spectroscopy of monolayer WS2
The high sensitivity of PL spectroscopy to detect features at the submicron level also reveals some non-uniformity around the centre of flake where wrinkles and contaminant are present [20]. This can also be seen by the AFM (Figure 1d) The AFM height profile also shows some contamination on the flake, perhaps due to bubbles trapped by the flake, commonly seen for micromechanically-exfoliated flakes of 2D materials [22,33]. It should be noted that the height profile along the line in Figure 1d is around 10 nm, significantly greater than the thickness of WS2 monolayer. This may arise from high degree of roughness and viscoelastic nature of polymer substrate. The lattice structure of the WS2 flakes can also been clearly seen by using atomic resolution TEM as shown in Figure 1e 34]. It should be noted that the size of WS2 flake (~10 µm) is three orders of magnitude smaller than the length of substrate (~70 mm) so that the flake can be assumed to be strained uniaxially [16]. A clear red-shift of the PL peak can be seen as the strain level increases, with the peak fitted with two Gaussian peaks ( Figure 2b) [7,35]. Since the band gap in monolayer WS2 is mostly determined by the 3p orbital of the S atoms and the 5d orbital of the W atoms [36,37], the red-shift occurs as the strain alters the W-W and W-S bond lengths, giving rise to a reduction in the orbital hybridization and d-band width [38]. In more detail, as the strain increases, the red-shift of both the A exciton peak and Aexciton peak is in the order of tens of meV as shown in Figure 2c, which agrees with other studies on TMDs [19,23,38,39]. The shift rate is found to be -58.7 ± 1.4 meV/% strain for the A exciton peak, and -89.9 ± 4.9 meV/% strain for the Aexciton peak, respectively. The apparent tension at 0% strain is likely to be due to the residual strain induced during specimen preparation. These shift rate values are similar to those predicted by theoretical simulation [40], and amongst the highest experimental values reported [35,41]. The PL spectra x-y mapped on a 0.5  0.5 m grid across a monolayer WS2 flake, in a region showing no contamination, ( Figure 2d) were collected at both 0% and 0.35% strain, A reduction of the energy (bandgap) can be clearly seen towards the centre of the flake for 0.35% strain (Figure 2e). This implies that the simple uniaxial tension applied to the monolayer flake on the substrate generates a non-uniform bandgap distribution across an individual WS2 flake. This is of particular relevance for local fine tuning of the bandgap in a TMD flake through strain engineering [20]. The PL energy varies by some 40 meV over a region of a few microns in the WS2 monolayer crystals at 0.35% strain. Moreover, this variation of PL energy can be controlled by the application of external strain to the substrate as long as the WS2 monolayer crystals remain intact and do not debond from the substrate. It is interesting to compare and contrast this present study with that of Tyurina and coworkers [22] who demonstrated that it was possible tailor the PL using hydrocarbon-filled bubbles under monolayer TMDs on a substrate. Although they were able to localise the PL variation in submicron regions ("artificial atoms") they were only able to tune the behaviour though the choice of different TMDs rather than by external strain as demonstrated in our study.
The measurements of the positions of the A exciton peaks in Figure 2e were converted to the strain through the use of the correlation found in Figure 2c, leading to maps of the monolayer WS2 strain distributions at different applied strain (the strain that was applied to the substrate) levels (Figure 3a  & b). It can be seen that at 0% applied strain the flake has an average strain about 0% but with some variation, which could be attributed to the formation of wrinkles in the flake from the specimen preparation by pressing the flake onto the substrate [42,43], that can be flattened by the tensile strain applied afterwards. When the applied strain is increased to 0.35%, the strain in the flake is found to increase from the edges and eventually plateau in the middle of the flake. This behaviour is shown more clearly in Figure 3c

0%
The behaviour shown in Figure 3 is characteristic of that seen during the deformation of monolayer graphene on a substrate for which it has been demonstrated that continuum mechanics [44] can be employed to model strain distributions using the well-established 'shear-lag' theory [16,45]. In this theory, the strain in the WS2 flake, εf, at a position x along the strain direction is given by an equation of the form [16,46]: where εm is the level of strain applied to the matrix, Ef is the Young's modulus of the WS2 flake, Gm is the shear modulus of the polymer, l is the length of WS2 flake along with strain direction, t and T are the thickness of the WS2 flake and the representative volume, respectively and s is the aspect ratio of the WS2 flake (l/t). A 'shear-lag' curve drawn using Eq. 2 with ns = 10 ( Figure  3c) at an applied strain ~0.35% shows the shear-lag theory to be an effective model to predict the behaviour of WS2 flake on a polymer substrate, and suggests an intact interface between monolayer and polymer. The critical length, lc, which is defined as twice the distance from the flake edge (0% strain) to where 90% of the maximum strain occurs [46], has also been estimated to be in the order of ~4 µm at a strain of 0.35%, which provides an important guideline regarding the minimum lateral dimension of WS2 flakes reinforcing nanocomposites. Similarly, the variation of the shear stress, τ, along the WS2 flake can be determined by developing Eq. 2 using a force balance [16] to give: By using ns = 10, E f = 272 GPa [13] and t = 0.65 nm [13], the shear stress at the edges of the monolayer WS2 flake is found to be ~1.1 MPa. This value is similar to that found for monolayer graphene [16,47] but is an order of magnitude lower than that of the carbon fibres in composites (~20-40 MPa) [46].
Although found previously for graphene [16], graphene oxide [45] and MoS2 [20], it is shown previously [44] and in this work that monolayer WS2 also follows continuum mechanics at the microscale. Hence it seems that this behaviour may be typical for all 2D layered materials. In particular, it has been demonstrated that the non-uniformity of strain determines the distribution of the bandgap, as monitored by the energy of PL peak. This suggests a way of tuning the bandgap of WS2 even within a single flake by making use of the non-uniformity in strain-engineered TMDs.
A recent study [48] has reviewed the experimentallydetermined PL shift rates for a number of TMDs, including WS2. Values of shift rate ranging from as low as -1.3 meV/% [49] to -61.8 ± 3.8 meV/% [50] have been reported for the A exciton peak of WS2. The shear lag analysis described above enables this discrepancy to be explained. The stress transfer efficiency is controlled by the parameter ns and the interaction of the WS2 flake with the substrate [51]. Better stress transfer is given by high values of ns, i.e. using stiffer substrates (with higher Gm values leading to a higher n) and long flakes with high aspect ratios s. For example the low value of shift rate was found using a very flexible polydimethylsiloxane (PDMS) substrate [49]. This present study used long monolayer flakes on a relatively-stiff PMMA substrate with a layer of SU-8 top coating.

Raman spectroscopy of WS2 flakes
As shown above, the intensity of the PL peaks in WS2 drops significantly as the number of layers increases. This limits its wider application in fields where few-layer and multilayer TMDs are employed, such as in nanocomposites [15]. In contrast, the intensity of the Raman bands in WS2 tends to increase as the number of layers increases and so Raman Monolayer and few-layer WS2 flakes were located on a substrate ( Figure 4a) and their Raman spectra are shown in Figure 4b. The monolayer was confirmed by the characteristic strong PL emission (inset in Figure 4b) and, in the Raman spectra, the increase of number of layers results in the increasing separation between E 1 2g and A1g bands (~62 cm -1 for monolayer) [52]. As the tensile strain is increased up to ~0.55%, both the E 1 2g and A1g bands of the monolayer and few-layer WS2 not only undergo a detectable red-shift, but also show a splitting of the E 1 2g band into two bands, namely the E 1-2g and E 1+ 2g mode, respectively ( Figure S1). This arises from the removal of the degeneracy of the E 1 2g mode as a result of strain [35,39]. As the E 1+ 2g band stays nearly unshifted, we will now demonstrate how the use of the Raman E 1 2g mode in a similar way to our previous report [7] enables the strain in WS2 flake to be determined. (The band will be referred to as the E 1 2g mode in the following discussion for simplicity). In order to investigate the micromechanics of monolayer WS2 flake using Raman spectroscopy, the PMMA beam was deformed using the procedure summarised in Figure S2. Briefly, the beam with monolayer WS2 on top was deformed initially to a strain ~0.55%. The specimen was then released and top-coated with a thin layer of SU-8 photoresist polymer. In the second cycle the specimen was then loaded to the same strain of ~0.55%. Using the calibration between the Raman E 1 2g band and strain established in our previous study [7], the strain distributions of the uncoated monolayer WS2 at 0% and 0.55% applied strain were mapped using a 1 m  1 m grid as shown in Figure 5a and b, respectively.

0.55%-Uncoated
It can be seen that the strain distribution in Figure 5 is quite uniform at 0% strain, apart from the compression at the edge of flake perhaps due to sample preparation. The similarity between the strain distributions shown in Figure 5 obtained using Raman spectroscopy and those obtained for a similar WS2 monolayer specimen using PL in Figure 3 is striking.
When the specimen was subjected to a strain of 0.55%, the strain in most of the WS2 flake increased to about 0.55%, demonstrating effective stress transfer from the substrate to the flake. However, a slight strain concentration, higher than the applied strain ~0.55%, can be found in the bottom part perhaps due to defects or edge effects [53]. The data points extracted along the vertical dashed white lines in tensile strain direction (Figure 5c) clearly show a strain plateau in the middle of the flake at the strain of 0.55%, suggesting that monolayer WS2 flake also follows the shear lag behaviour shown in Figure 3. Similarly, by using ns = 12 in Eq. 2, the 'shear-lag' curve is drawn for an applied strain ~0.55% (Figure 5c). Similar to the behaviour shown in Figure 3d, the curve matches the data points very well, further confirming the validity of shear-lag theory. The shear stress at the edges of the monolayer WS2 flake is found to be ~1.5 MPa (Eq.3), similar to that found by using PL (Figure 3).
The specimen shown in Figure 5 was unloaded and a top coat of SU-8 applied so that the deformation of the WS2 monolayer sandwiched between two polymer films, as in a nanocomposite, could be studied as shown in Figure 6. Although the strain in the monolayer is relatively uniformly distributed at 0% strain ( Figure 6a) it is, however, reduced to ~-0.35% which means that there is a uniform 0.35% compressive strain in the flake as a result of the shrinkage of the top-coating SU-8 polymer during curing [54]. Subsequently, the coated flake was subjected to a strain of 0.55% (Figure 6b) and a strain concentration at top part of the flake is observed. The 'shear-lag' curve drawn using Eq. 2 (ns=12) for an applied tensile strain of 0.55% is offset by -0.35% to take into account the 0.35% compressive strain, hence the maximum strain is only about 0.20% [53]. In addition the strain drops sharply in the middle of the flake although the strain distribution of other parts still approximately follows the shear lag curve (Figure 6c). The reason for this is attributed to the occurrence of two cracks in the middle of the WS2 flake perpendicular to the tensile axis as highlighted with dashed white frames and the inset in Figure 6b, similar to the fracture behaviour observed for monolayer graphene [55]. This implies that the failure strain of this WS2 flake is no more than 0.55%, due probably to the presence of defects. This strain to failure, in combination with its Young's modulus ~272 GPa [13], implies the strength of this monolayer WS2 flake was ~1.5 GPa. The interfacial adhesion has not been improved by the SU-8 top coating. This is because for monolayer WS2 sandwiched and stretched between SU-8 layers, the adhesion with one side interface is not significantly different from that with double sides. This is in similarity with graphene, but if few-layer graphene is stretched the SU-8 top coating can provide extra adhesion to compensate for the interlayer sliding due to the weak interlayer bonding [14].  Figure 7c that the nanocomposites reinforced with LPE WS2-L flakes exhibit a significant red-shift at a rate of -0.66 ± 0.15 cm -1 /% strain (Raman spectra shown in Figure S3), while the shift rate decreases as the lateral dimension decreases because of lower level of reinforcement provided by the smaller flakes [34]. It was found that the rate of band shift decreased WS2-L/PVA above 0.3% strain due possible to failure of the interface between the PVA and WS2 flakes above this strain [57]. This is further evidenced by similar behaviour of the A1g band ( Figure S4) where the LPE WS2-L flakes show significantly larger shift than the other two. The results were found to be very reproducible, in that only the LPE WS2-L flakes with large lateral dimensions of 8.1 ± 4.7 µm give rise to good stress transfer and so can be used to reinforce nanocomposites effectively.
The critical length for the monolayer WS2 flakes was found to be 4 m and this is indicated on Figure 7a. It should be noted that the lateral sizes of most of the LPE WS2-L flakes are higher than 4 m whereas they are lower than 4 m for the LPE WS2-M and LPE WS2-S flakes. This is clear indication of the importance of flake size in having good stress transfer in nanocomposites [58]. It is likely that the larger flakes may also be thicker and it might be better to consider the flake aspects ratios, s, rather than just their lateral dimensions for a complete analysis in the future. The Raman band shift rate reveals the level of strain in the materials as a result of stress transfer from matrix and the stress/strain-induced Raman band shift and has been used to estimate the effective modulus of graphene [59], graphene oxide [17] and boron nitride [60] in nanocomposites. We have made an attempt to extend this methodology to WS2 flakes [45]. The effective modulus of the reinforcement is given by where dω(E 1 2g )/dε is the shift rate of the Raman E 1 2g band as the function of strain ε. The reference band shift rate value, dω(E 1 2g )/dε(ref) for the deformation of isolated flakes can be taken as -2.05 cm -1 /% as found previously [7]. If the Young's modulus of a monolayer WS2 flake Ef is taken as 272 GPa [13], then the value of dω(E 1 2g )/dε measured -0.66 ± 0.15 cm -1 /% for LPE WS2-L flakes in this work (Figure 7c) leads to an effective modulus of LPE WS2-L (Eeff) of ~87 ± 20 GPa, i.e. around 30% of the value of 272 GPa determined from the direct measurement on monolayer flakes [13]. No apparent slipping in either model devices or nanocomposites has been observed in PL and Raman experiments. This is for two reasons: (1) the level of strain achieved is not sufficiently high to induce an interfacial damage and (2) the WS2 flake cracks first prior to the interfacial failure, probably due to the fact that the strength of WS2 is not as high as graphene, especially with the presence of defects [55].
We also investigated the reinforcement of the WS2/PVA nanocomposites further by undertaking tensile testing upon the nanocomposites reinforced with different loadings of the LPE WS2-L flakes. The stress-strain curves in Figure 8a show that the incorporation of WS2 at low loadings (up to ~0.21 vol%) there is a linear increase in the Young's modulus of the nanocomposites (Ec) (Figure 8). Above this loading, the reinforcement efficiency from the WS2 decreases, probably due to aggregation and percolation effects [61], although the maximum Ec is found to be ~940 ± 38 MPa for the highest loading, corresponding to nearly a 60% increase in modulus with respect to the neat PVA. In contrast, the strain to failure and ultimate strength remain unchanged, regardless of the WS2 loading (Figure 8a). These findings are consistent with the Raman band shifts shown in Figure 7c. We have demonstrated previously that the Young's modulus of the nanocomposites, Ec, can be analysed using the modified rule of mixtures [46,58]: where Eeff is the effective Young's modulus of LPE-WS2, Ef and Em are the Young's modulus of the LPE-WS2 and PVA, respectively. Vf is the volume fraction of WS2 in the nanocomposites. Figure 8b shows a linear increase of Ec as the function of Vf, up to 0.21 vol% loading that enables the Eeff to be determined as ~93 GPa, similar to that obtained from Raman band shifts using Eq. 4, validating the use of Raman spectroscopy to predict the value of Eeff for the TMDs.
The effective modulus of WS2 flakes in the nanocomposites is given by [58] E eff =ηlηoEf (6) The length factor, ηl, which has a value of between 0 and 1, reflects the dependence of reinforcement on flake length and increases with the flake aspect ratio s [62]. The Krenchel orientation factor ηo enables the effect of filler orientation upon the reinforcement efficiency to be determined and it ranges from 8/15 for randomly oriented to 1 for well-aligned  Figure  8b) shows that the intensity of the A1g band is independent of the angle of direction of laser polarisation. This corresponds to a random orientation of the WS2 flakes in the nanocomposites, based on the methodology established earlier [29]. The length factor ηl is given by [64] 2 / ) 2 / tanh( -1 = l ns ns η (7) The parameter ns was found to be of the order of 10 for the monolayer flakes which would give value of ηl  0.8 (Eqs. 2 and 7) as the average number of WS2 layers is about 1-3 [26]. Considering the above, the values of Ef can be calculated to be up to 220 GPa according to Eq. 6, in broad agreement with the experimental theoretical value ~270 GPa. It should be noted that the value of Ef may also be lower for few-layer materials than the monolayer, as has been found for graphene [14], due to interlayer sliding.

Conclusions
In this work, photoluminescence and Raman spectroscopy have been combined to successfully monitor the strain distribution and stress transfer of monolayer WS2 on a flexible polymer substrate and also in thin film bulk nanocomposites. It is demonstrated that monolayer WS2 still follows continuum mechanics. Particularly, a non-uniform bandgap distribution has been achieved by strain engineering even within a single WS2 flake due to the non-uniform strain distribution through stress transfer from the substrate. It has been demonstrated that this could have useful applications in optoelectonics in producing tuneable micron-sized PL emitters on a substrate. The micromechanics developed for monolayer WS2 flake has been extended to thin film nanocomposites, where it has been found that the stress can be transferred effectively to thicker WS2 flakes. Their effective Young's modulus is around 30 % of their theoretical modulus which means that their reinforcement efficiency is comparable to that of few-layer graphene-reinforced nanocomposites.