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# Towards conductive textiles: coating polymeric fibres with graphene

## Abstract

Conducting fibres are essential to the development of e-textiles. We demonstrate a method to make common insulating textile fibres conductive, by coating them with graphene. The resulting fibres display sheet resistance values as low as 600 Ωsq−1, demonstrating that the high conductivity of graphene is not lost when transferred to textile fibres. An extensive microscopic study of the surface of graphene-coated fibres is presented. We show that this method can be employed to textile fibres of different materials, sizes and shapes, and to different types of graphene. These graphene-based conductive fibres can be used as a platform to build integrated electronic devices directly in textiles.

## Introduction

The development of devices on textiles such as sensors1, photodetectors2, transistors3, electro-luminescent devices4, supercapacitors5 and solar cells6 is attracting great interest and has led to the emergence of the field of smart textiles. Smart textiles can find an enormous range of applications in several fields, including healthcare, military and fashion7. Since the concept of textiles is much wider than clothes and garments, the applications can extend to aviation, automotive and transport, construction, geo-textiles and packaging. While most commercial applications of smart textiles rely on conventional hardware simply mounted onto textiles, the integration of specific functionalities directly on textile fibres promises to revolutionize the field of wearable electronics. With the recent advances in nanotechnology and materials engineering, different functionalities can now be incorporated into textile fibres, such as antibacterial properties, static elimination, and electric conductivity8.

Conducting fibres are an important component of any e-textile, not only because they can be used as lightweight wiring for simple textile-based electronic components, but also because they can provide a platform for building electronic devices directly on textile fibres. For instance, such fibres can be used as gate electrodes for field-effect transistors, or bottom electrodes for light-emitting diodes and photovoltaic devices. Conductive textile fibres are currently used as a stronger and more flexible weight-saving material in the aviation sector, where the aircraft weight and fuel consumption are reduced by replacing metal wiring with electrically conductive cotton fibres like ARACON®. The most common approach to produce conductive fibres consists in mixing an insulating polymeric matrix with a conductive component, which can be a conducting polymer such as PEDOT:PSS9, silver nanowires10, nanocarbon fillers11, or their hybrids12,13,14,15. These composite fibres are usually produced by wet-spinning of the polymer with a suspension of the active conducting material, or via electrospinning16, techniques that require a large consumption of expensive materials and that can cause the loss of flexibility and transparency. A different approach relies on polymer-free fabrication of conducting fibres, but the methods include the use of strong acids and coagulants17, which greatly limits the potential for scaling up and commercialization. Another strategy is to impregnate fibres with conductive materials after they have been manufactured. This method has been used for fibres, yarns and fabrics that are highly porous with a complex structure of microfibrils, such as cellulosic fibres. These inks and dyes make use of several types of conductive materials, such as aluminium18, carbon nanotubes19, and graphene20, and have been in the base of demonstrations of wearable and stretchable electronics, including integration onto surfaces of live plants and insects21, 22. One of the limitations of this methodology is that such multifilament fibres and fabrics, compared to fibres coated prior or during manufacturing, tend to lose the conductive filling more easily if not completely encapsulated, and can pose end-of-life environmental concerns.

A more practicable emerging strategy to prepare conductive textile fibres is to coat insulating fibres with conductive atomically thin two dimensional layers such as graphene23. Graphene, a monoatomic carbon layer, is the strongest known material, the best electrical and thermal conductor which is also mechanically flexible and transparent24. Thus, it represents a radical alternative to conventional technologies as it can bend, stretch, compress, twist and deform into complex shapes while maintaining the same levels of performance and reliability25. There are already several examples of graphene-based textiles with different functionalities and for different applications26. The coating we propose is performed by electrostatic adhesion of graphene at the surface of monofilament fibres and does not involve impregnating an agglomerate of fibres. The adhesion of the graphene coating to the textile fibres is strong and durable, and a straightforward passivation can be achieved by encapsulation with an insulating polymeric layer. This method was developed for tape-shaped polypropylene (PP) and bio-based polylactic acid (PLA) fibres, two polymers with widespread use in the textile industry27. This approach depends on the size of the graphene sample, usually in the centimetre range, and although this might not be suitable for electronic wiring, it is appropriate to build electronic devices directly on textile fibres.

In this work, we demonstrate that graphene can be transferred to a large variety of thermoplastic monofilament textile fibres of different types and shapes. To further advance the development of this technique, it is important to understand the various factors that can influence the conductivity achieved by coating textile fibres with graphene. Surface topography and chemical nature seem to be determinant in the conductivity achieved. On the other hand, cracks and tears in the graphene coating will result in a decrease in conductivity, and therefore it is important to establish their origin. Thus, our present study aims at: (1) establishing how the above-mentioned factors actually influence the quality of the graphene coverage and how it correlates with the resulting conductivity; and (2) explore the suitability of our coating method for different materials, sizes and shapes.

## Results

Slight modifications in the fabrication of the fibres can result in significant changes in the fibre surface. To evaluate the influence of fibre topography in the graphene coating processed, we present a comparative study using PP fibres manufactured with different extrusion parameters. Different sources of polymer can also determine the topography and influence chemical nature of the surface, changing the quality of the graphene coating. To address this, a comparison between two formulations of PLA-based fibres was performed. Furthermore, the impact of ultraviolet-ozone (UVO) treatment on the surface of the fibres is discussed. Finally, we have employed the same methodology to different materials, such as polyethylene (PE) and nylon, and to different fibre size and shape, cylindrical (nylon) and tape-shaped (PP, PE, PLA).

Figure 1a shows photos of the textile fibres used in this work as reels (left) and cut to size (right). The fibres were coated with monolayer graphene following a method described previously23, which consists in etching the metal substrate in which the graphene was grown by CVD (Chemical Vapour Deposition), Cu in the case of monolayer graphene28 and Ni for few-layer graphene (FLG), and transferring the supported graphene onto the textile fibres immobilized on a rigid substrate. To evaluate the effect of fibre topography and different materials, extrusion parameters and precursor materials were varied. The difference between fibres PP1 and PP2 is only due to different extrusion parameters in their production, whereas PLA1 and PLA2 were extruded from different grade precursors. The difference in polymeric material used to prepare PLA1 and PLA2 is demonstrated by the Raman spectra of the two PLA fibres from both formulations, with PLA1 displaying two additional bands at 1615 and 1717 cm−1 compared to PLA2 (see Supplementary Fig. S1). As for both PP-based fibres, there is a good superposition between PP1 and PP2 Raman spectra, apart from minor differences in peak intensity, evidencing that both fibres are made from the same polymer.

To assess the electrical conductivity of the fibres we measured the sheet resistance for a variety of conditions. Figure 1b shows the sheet resistance values for different types of fibres coated with graphene, untreated (lines and circles) and previously submitted to UVO treatment (lines and squares), as well as their behaviour upon bending, starting in a flat position, then down to a radius of 4.6 mm, and back to a flat position. This study was performed to assess if the failure observed was due to the fibre itself failing or from damage to the graphene coating, and does not intend to be a systematic study. In general, the sheet resistance of the fibres coated with graphene is lower for fibres produced to have an improved surface (i.e. PP2, PLA2 and PE, represented in full lines in Fig. 1b), all falling below ca. 4 kΩsq−1, with UVO treatment bringing values even lower than 1 kΩsq−1. However, there seems to be a more pronounced increase in sheet resistance upon bending as compared to the PP1 and PLA1 fibres (PP1 and PLA1, dotted lines in Fig. 1b). This may be because both PP2 and PLA2 fibres are thinner and more fragile compared with PP1 and PLA1. The effect of UVO treatment in increasing the conductivity of the graphene-coated fibre samples is evident, but we found that there is a larger dispersion in sheet resistance values as the number of samples increases (Fig. 1c). This apparent irreproducibility of the UVO effect on sheet resistance is related to the fact that this treatment visibly contributes to the degradation of the fibres to some extent, particularly in the case of the more fragile and biodegradable PLA-based fibres. On the other hand, very often the sheet resistance of a fibre decreases around the bending radii of 9.0 and 7.4 mm. Although the fibres are being subjected to strain in these bending measurements, and sometimes fail completely, they are also recovering to their natural position, since they are kept in reels after the production, and therefore they have a natural curvature to them.

Transmittance measurements were performed for both types of transparent fibres, PP and PE, to evaluate the effect of graphene coating on their transparency, and are shown in Fig. 1d. A loss of only 2% and 3% in mean transmittance was observed for graphene-coated PP2 (Fig. 1d, in pink, right inset) and PE (Fig. 1d, in green, left inset), respectively, over the wavelength range between 430–900 nm, which covers the visible and near infra-red portions of the spectrum. These values are in accordance with a 2.3% absorbance for monolayer graphene25. The Raman spectra of fibres of the three materials (PP2, PE and PLA2), before and after the graphene transfer, are shown in Fig. 1e. For all graphene-coated fibres it was possible to identify the graphene G band at 1585 cm−1. The 2D peak was clearly visible for graphene-coated PP1 and PLA2, at 2685 cm−1, as well as a small D peak for PE and PLA2. These values match those found for G and 2D bands of the same type of graphene transferred to SiO2 (see Supplementary Fig. S2; more extended Raman spectra of the graphene-coated fibres is shown in Supplementary Fig. S3, along with details of the G and 2D peaks and corresponding integrations for PP2, PE and PLA2).

To study the factors that lead to the observed differences in sheet resistance, it is important to understand the influence of the topography of the fibres on the effectiveness of the graphene coating. A non-contact optical method was used to determine the macroscopic surface parameters of the untreated fibres. The images obtained are shown in Supplementary Figs S4, S5 and S6, and the parameters are listed in Supplementary Table S1. Figure 2 shows the Atomic Force Microscopy (AFM) amplitude and topography for 5 × 5μm images of the PP fibres before coating and corresponding height profiles taken at the highlighted lines parallel to the extrusion axis. Compared to PP1, PP2 fibres show considerable differences in terms of AFM topography. PP2 does indeed have a smoother height profile (Fig. 2a, right) with less pronounced height differences than PP1 (Fig. 2a, left). The UVO treatment created a fine roughness throughout the whole surface, which seems to create more points where the graphene sheet can effectively adhere to the surface of the fibre (Fig. 2a, middle). The same conclusions are valid for larger and smaller AFM scanning areas (30 × 30 μm and 1 × 1 μm images, see Supplementary Fig. S7). Although in terms of overall thickness and surface features, PP1 and PP2 are very similar (see Supplementary Table S1), in a smaller scale AFM shows that PP2 has areas with less pronounced features than PP1, which is also in accordance with the smaller Kurtosis value in PP2. A similar study was performed for the PLA-based fibres PLA1 and PLA2 (Fig. 2b and Supplementary Fig. S8), showing that the changes in polymer source grade do not have a substantial impact on the surface morphology in the AFM scale. However, the difference in roughness is much more significant macroscopically, with PLA2 is rougher than PLA1, showing visible ridges perpendicular to the extrusion lines (Supplementary Fig. S5). On the other hand, we found that the UVO treatment does change the surface of PLA-fibres significantly (Fig. 2b, middle). All three monofilament tape-shaped samples, PP2, PLA2 and PE were also subjected to UVO treatment. Both PLA1 and PLA2 fibres often appeared to be damaged after UVO-treatment, particularly towards the edges, where propagating cracks and microfibrils tearing from the sample were clearly visible to the naked eye (see Supplementary Fig. S8, top middle, where a loose microfibril is clearly noticeable). The damage induced by the UVO treatment is even more severe in PLA2 fibres. The protuberances that are observed on the fibres as a consequence of the UVO treatment are probably a sign of that degradation, and biodegradable polymers like those based on PLA are prone to be damaged even more rapidly. These protuberances can be thermally caused or appear due to the chemical reactions of oxygen radicals at the surface of the polymer. Attempted coating of these UVO-treated PLA2 fibres with graphene was unsuccessful, as all the samples remained insulating after the graphene transfer. The immersion in warm acetone during the PMMA ((poly(methyl methacrylate)) cleaning step seemed to damage the fibres even further, causing them to curl and shrink. It should be noted that the UVO treatment was performed under the same conditions for all fibres, and even though the bio-based PLA1 fibres also showed some signs of quicker degradation when subjected to the UVO treatment, it was still possible to achieve considerably low sheet resistance. The fact that PLA2 fibres are slightly smaller than PLA1, 1.0 mm wide and 0.07 mm thick, compared to 1.2 mm wide and 0.1 mm thick, may also account for the fact that the formers are more fragile and easily degradable.

An AFM study of the graphene-coated fibres also provides relevant information concerning the quality of graphene coverage at the surface. In particular, given that AFM phase imaging is capable of sensing variations in stiffness, we can distinguish between parts of the surface of the fibre that are exposed from those that are fully covered with graphene. Figure 3a shows the AFM amplitude images of the fibres before and after coating, as well as the AFM phase images of the graphene-coated fibres. All AFM phase images show a uniform graphene coverage of samples coated with no pre-treatment (Fig. 3a7, 8 and 9). These results demonstrate that a continuous surface coverage is also achievable without UVO treatment. Moreover, a detailed analysis of the topography of the graphene-coated fibres does not support the hypothesis that the tears and cracks in the graphene layer covering the surface are caused by significant differences in height. In fact, they show that several samples of PP1 and PLA1 displayed large differences in phase AFM, which correspond to tears and cracks, in areas where the topography is rather smooth (Supplementary Fig. S9, top left and middle), and areas with large differences in height that do not coincide with the discontinuities revealed by the phase images (Supplementary Fig. S9, bottom left and middle). Additionally, PP2 and PLA2 show continuous graphene coverage in areas where topography shows abrupt variations (see Supplementary Fig. S9 top and bottom right). Nonetheless, whereas continuous graphene coverage was only observed for UVO-treated PLA1 fibres, for PLA2 no pre-treatment was needed, and that might be due to the differences between both PLA sources used in the production of the fibres and how they influence the chemistry at the surface. These differences can play an important role during the graphene transfer, namely in terms of wettability, since the transfer is performed in an aqueous medium. There are essentially two ways of improving wettability in polymers: by changing surface energy and/or topography29. It is important to note that the UVO treatment not only changes the topography of the textile fibres, but it also causes the chemistry of the surface of the fibre to change. We observed that the wettability of all polymers improved after UVO treatment, resulting in a decrease in repulsion of the water droplets that separate the floating PMMA-supported graphene sheet, and the surface of the fibre during the transfer process, facilitating adhesion. Common methods of improving wettability, and consequently dyeability, of polymers include cold oxygen-based plasma treatment and the use of siloxane-bearing coatings30. As well as UVO, oxygen plasma treatment of polymers also induces ageing and degradation, namely due to formation of free radicals at the surface, which in turn can create a cross-linked polymeric layer. This layer is both rougher and has significantly different surface energy than that of the untreated polymer, and that is accountable for the increase in wettability, much like what happens during the UVO treatment. Nevertheless, UVO treatment is gentler than plasma treatment, as it has been documented in the literature, mainly as methods commonly used to clean organic residues on silicon-based surfaces31.

Besides AFM phase imaging, Scanning Thermal Microscopy (SThM) can also be used to assess the extent and quality of graphene coverage at the surface of the textile fibres. Surface topography and thermal conductivity contrast (TCC) often show a direct correlation in SThM. Surface roughness induces changes in the tip-sample contact area which alters the head flux between tip and sample. Abrupt depressions (and elevations) in the sample surface will translate in an apparent increase (and decrease) in thermal conductivity contrast. Although these textile fibres have been designed and produced as to have smoother surfaces, they are still inherently rough due to the extrusion process in which they were manufactured. This means that, for most samples, particularly with the PLA and PE-based textile fibres, TCC is mostly a feature of topography and actually not related with changes in the thermal conductivity of the surface. This is the reason why the SThM study is limited to PP fibres, as this new set of PP2 fibres displayed visibly smoother surfaces than all the other fibres used in this work. We have indeed found some cracks and holes in some graphene-coated PP samples which are undoubtedly non topography-induced (Fig. 3b). The same type of cracks was also visible in SEM (Scanning Electron Microscopy) images, as depicted in Fig. 3c. A comparative optical microscopy study on the effect of bending on the surface of the sample, subjecting coated PP fibres to severe and moderate bending, did not show a direct correlation between the bending stress and the appearance of cracks. For this reason, we believe that these cracks are not due to the handling and bending of the samples after the graphene transfer, but due to the manipulation of graphene sheets during the transfer process itself. The actual stress to which graphene is subjected during the wet transfer procedure, particularly as the floating PMMA-supported graphene sheet is lifted with the target sample, is most probably what is causing these cracks. Using a thicker layer of supporting PMMA would minimize the stress applied to the graphene sheet, with the drawback of generating yet more PMMA residue at the surface of the coated fibres. In fact, these PMMA residues are often visible in our studies, both in the AFM and SEM imaging (see Supplementary Fig. S10).

To demonstrate the flexibility of the methodology we have developed to coat textile fibres with graphene, we performed a similar study on cylindrical shaped fibres using commercially available nylon fishing lines. As mentioned above, these fibres pose another challenge, they are cylindrical and considerably smaller than the tape-shaped fibres, with diameters ranging from 0.35 to 0.6 mm (Fig. 4a). We have assumed a graphene coverage of half a cylinder, which we believe is the maximum coverage that can be achieved with this method in cylindrical samples. This means that the sheet resistance is most likely underestimated, since the values are normalized using the aspect ratio (width/length) of the graphene sheet that is actually covering the sample, and this area is likely to be smaller than a full half cylinder coverage. Figure 4b shows the dispersion of sheet resistance of the graphene-coated nylon fibres, with values between ca. 2 and 15 kΩsq−1, which is of the same order of magnitude as the sheet resistance of the tape-shaped fibres. The conductivity of the nylon fibres increases with diameter because this increases the aspect ratio of the graphene-covered surface, which is in line with what we observed for the tape shaped fibres as well. For larger areas the electro-static adhesion of graphene is facilitated and furthermore, the effect of eventual cracks and discontinuities in the coating is minimized, which contributes to increased electrical conductivity. In any case, we found continuous graphene coverage for all samples (AFM images of N1 and N4 depicted in Supplementary Fig. S11), with both amplitude and phase AFM images of the coated fibres showing only a few drop-shaped contaminations which are probably due to PMMA residue, particularly visible in the case of P1 (see Supplementary Fig. S9c and e). As with the tape-shaped fibres, the G peak region of the Raman spectra was used to identify the presence of graphene at the surface of the nylon fibres, as show in Fig. 4c. The G peak is clearly visible in the Raman spectra of N1 and N2 (top and bottom left in Fig. 4c) at ca. 1580 cm−1. Fibres N3 and N4 are dyed (third and fourth fibres from the right in Fig. 4a, dyed in yellow and green) and exhibit a significant degree of fluorescence in the Raman spectra, as can be seen on the right-hand side of Fig. 4c. A small G peak is still visible in the Raman spectra of N3 (top right in Fig. 4c), but slightly shifted at 1586 cm−1, indicating doping of graphene. This doping effect may be due to the dye, and may also be accountable for the lower sheet resistance measured in both N3 and N4 samples. This will be addressed in further studies.

The potential of our method to cover textile fibres with a conductive graphene coating was further demonstrated by using a different type of graphene. Whereas all previous samples were coated with monolayer graphene, we have successfully coated PP2 fibres with few-layer graphene (FLG). The Raman spectrum of FLG-coated PP2 fibres is shown in Supplementary Fig. S12. These samples show 77% transparency, accounting for the increased number of graphene layers, as shown in Supplementary Fig. S13. Sheet resistance behaviour upon bending is shown in Fig. 1b (in black), with values lower than the most conducting fibres coated with monolayer graphene. Thus, using FLG we have achieved resistance values as low as 600 Ωsq−1, which is desirable for wiring textile-based electronic components. Furthermore, the increase in sheet resistance upon bending is even less significant with FLG-coated PP2 than with fibres coated with monolayer graphene, with a full recovery of the resistance after bending. This may be due to the fact that, even if the upper graphene layers are damaged with the bending stress, the remaining layers will still be continuous. AFM imaging of FLG-coated PP2 samples is presented in Supplementary Fig. S14, showing two different FLG-coated samples. One of the samples (a and b) shows what seem to be portions of the upper graphene sheets that are cracked and lifted up towards the edges, whereas the other sample (c and d) seems to have a much smoother and homogeneous surface. Both samples have similar values of sheet resistance, ca. 600 Ωsq−1. These values are slightly higher than those reported in the literature, of the order of a few hundreds of Ωsq−132,33,34, and those of the same type of FLG transferred onto glass, 200–300 Ωsq−1.

To demonstrate the mechanical resilience of our samples, and to directly compare the effect of bending on monolayer of few-layer graphene coating, we monitored the resistance of PP2 fibres while subjected to 1000 bending cycles, with a fixed bending radius of 5 mm. Figure 5a shows the change in sheet resistance, normalized to that of the device before bending for PP2 coated with monolayer graphene (dark yellow) and FLG (black). These measurements demonstrated that the resistance of either fibres does not change significantly during a large number of bending cycles.

The high conductivity observed in textile fibres coated with monolayer graphene and FLG make these materials valuable candidates for future applications requiring conductive textile fibres. Figure 5b compares the conductivity of our graphene-coated fibres with other types of conductive fibres reported in the literature taking into account the thickness of the conducting layer. The samples prepared in this work display considerably higher conductivity as compared to most of the fibres in the literature. These high values are due to the fact that conductivity is essentially two-dimensional, given the atomically thin nature of graphene films (further details in Supporting Information), whereas the other mentioned fibres have the conductive layer embedded in their structure. Furthermore, the conductivity of the fibres prepared by this method does not depend on the thickness of the coated fibre. Therefore, the outstanding electrical conductivity and optical transparency make graphene-coated fibres promising transparent conductors for future smart textile applications.

## Conclusions

We have demonstrated a versatile method to coat a wide range of textile fibres with graphene, including materials such as pure PP, PLA, PE and nylon. This method is applicable to tape-shaped fibres as well as cylindrical fibres. In addition to monolayer graphene, this method has the potential to transfer other types of graphene or 2D materials to textile fibres, as demonstrated by the FLG coating. Comparing the quality and continuity of the surface coverage of all the fibres, continuous graphene surface coverage is easily achieved without the need of a surface pre-treatment. Nevertheless, a pre-treatment of ultraviolet ozone can still improve the conductivity of the corresponding coated fibres, but often damaging them and making them more fragile. We have also demonstrated that, although the UVO pre-treatment of the fibres can lower their sheet resistance after the coating with graphene, it is possible to achieve low values with untreated fibres, especially if the surface is uniformly rough. In general, the sheet resistance of the fibres coated with graphene is lower for fibres produced with smoother surface, of the order of the kΩsq−1, and can withstand up to 1000 bending cycles down to a bending radius of 5 mm with no significant change in resistance. An extensive microscopic study of the surface of the coated fibres, by means of AFM, SEM and SThM, helped ruling out that the cracks often observed in conducting samples is caused by their manipulation or by significant height differences. The study also suggests that it is the stress to which the sheet is subjected during the transfer process that causes these small cracks, with minor effects on the sheet resistance. The same method to coat textile fibres with graphene was successfully employed to cylindrical nylon fibres, which were considerably smaller than the tape-shaped textile fibres. Sheet resistance values of the same order of magnitude were obtained, increasing with decreasing fibre diameter and good surface coverage was observed also without surface pre-treatment needed to achieve continuous coverage.

## Methods

Tape-shaped textile fibres were fabricated by Centexbel using a monofilament extrusion line from polypropylene (PP), polylactic acid (PLA) and polyethylene (PE). By varying the extrusion parameters, two types of PP fibres were produced: PP1 (0.04 mm thick, 1.6 mm wide); and PP2 (0.03 mm thick, 2.4 mm wide). The precursor was changed and compared PP and PE fibres with similar dimensions (0.03 mm thick, 2.4 mm wide). PLA-based fibres obtained from different grade precursors, i.e. PLA1 Bio-based polylactic acid polymer (0.1 mm thick x 1.2 mm wide) and PLA2 from PLA Naturework 6400D (0.07 mm thick, 1.0 mm wide) were produced. The fibres were rolled onto a bobbin and cut to the desired length. Commercial reels of monofilament nylon fishing lines were used: Mitchell® Marine Performance with 0.35 mm diameter (N1); Blue Pacific Expert Line Monofilament, 0.40 mm (N2); Climax® Monofilament Perlon, 0.55 mm (N3); Climax® Monofilament Perlon, 0.60 mm (N4).

### UVO Treatment

Ultra-Violet Ozone (UVO) treatment was performed using a UVO Cleaner® 144AX-220 (Jelight Company Inc.) for 11 minutes (6 minutes exposure time and 5 minutes exhaust time).

### Graphene Coating

The untreated fibres were coated with high quality monolayer graphene produced by cold-wall chemical vapour deposition35 and few-layer graphene (purchased from Graphene Supermarket) using the transfer method previously reported23.

### Profilometry

The surface profile parameters of the fibres were determined using a Taylor Hobson TalyScan 150 non-contact scanning instrument, at a speed of 30 µm/s and a spacing of 5 µm. To allow a direct comparison, PP1 and PP2 were placed side by side and measured in a single scan. A rectangular section with the same area was selected from each fibre to perform the calculations of the surface parameters. The same was done for PLA1 and PLA2, and using two sections of a PE fibre.

### Scanning Thermal Microscopy

All measurements were performed in ambient conditions using the PARK SYSTEMS XE7 Scanning Thermal Microscopy (SThM) equipment, with standard thermal probes from KELVIN Nanotechnology, thermoresistive probes where a Palladium resistive element is lithographically patterned on the AFM tip apex mounted on a Silicon nitride cantilever (400 nm thick). The electrical connection is made to two Au pads on the base of the probes. The thermal tip, with a radius of approximately 100 nm, has a sensitivity of ~1 Ω/°C and provides a 50 nm spatial resolution. The study was performed operating in the thermal conductivity contrast mode, with a bridge circuit to keep the probe at a constant temperature (determined by its resistance). During this type of measurement the tip is kept in contact with the sample. By electrically feeding the tip and monitoring the electrical current (output of the system in this mode) needed to keep its temperature constant during a scan, the thermal response of the material is detected in images of sub-micrometric resolution.

### Atomic Force Microscopy

AFM was performed with a Digital Instruments (Veeco/Bruker) Dimension 3100 with a Nanoscope IIIa controller in tapping mode using commercial Olympus Si tips (OTESP).

### Raman Spectroscopy

Raman spectra were measured in air and at room temperature with a Renishaw spectrometer using a 532 nm laser wavelength with a 1.5 μm spot size and a 1 mW of incident power.

### Electrical Measurements

Electric conductivity was measured by a two-probe method using tungsten probes and a Keithley 237 source-measure unit. The contacts were drawn with graphite paste to facilitate the measurements. Sheet resistance was calculated by subtracting the contact resistance and taking into account the aspect ratio of the graphene section on each sample. Contact resistance was estimated using a common method for graphene devices36, 37, based on measuring the two-probe resistance in devices with different contact separation while keeping the width of the channel constant. Since the two-probe resistance of a given sample is $${R}_{2p}={R}_{G}+2{R}_{C}$$, where $${R}_{G}=\rho L/W$$ is the graphene resistance ($$\rho$$ is graphene resistivity, $$L$$ is channel length and $$W$$ is channel width) and $${R}_{C}$$ is the contact resistance, the contact resistance can be estimated from samples with different lengths $${L}_{1}$$ and $${L}_{2}$$ ($$W$$ is the same). In this case the contact resistance is given by $${R}_{2p}=({R}_{2p}^{Dev1}-{\rho }_{G}({L}^{Dev1}/{W}^{dev1}))/2$$ and $${\rho }_{G}=({R}_{2p}^{Dev1}-{R}_{2p}^{Dev2}){({L}^{Dev1}/{W}^{Dev1}-{L}^{Dev2}/{W}^{Dev2})}^{-1}$$. A contact resistance of the order of 1 kΩ was found for our devices, which is agreement with previous experiments37.

### Optical Measurements

A custom-built UV-Vis micro-spectrometer was used to acquire the transmission spectra. Light from an incandescent bulb was focused onto the samples through a condenser lens and collected by a X20 microscope objective, then delivered to the entrance slit of a spectrometer (Princeton Instruments Acton SP2500) equipped with a 1200 g/mm grating and CCD camera (Princeton Instruments PIXIS400). Transmittance as a function of wavelength λ was calculated as $$T=I/{I}_{0}\,$$where $$I$$ is the measured spectral intensity through the sample and $${I}_{0}$$ is the incident light intensity. Corrections to account for the optics and gratings efficiencies at different wavelengths were applied using Princeton Instruments IntelliCal system.

### Scanning Electron Microscopy

Scanning electron microscopy (SEM) was performed using a Hitachi SU-70 scanning electron microscope at an accelerating voltage of 4 kV, working distance of 6200 µm, emission current of 44000 nA and magnifications of 100x, 200x, 500x, 1000x and 10000x.

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## Acknowledgements

The authors would like to thank Dr Yat-Tarng (Tommy) Shyng for the non-contact scanning measurements and would like to acknowledge financial support from the UK Engineering and Physical Sciences Research Council (EPSRC grants EP/J000396/1, EP/K017160, EP/K010050/1, EP/G036101/1, EP/M002438/1, EP/M001024/1), the Royal Society Travel Exchange Grants 2012 and 2013, the European Commission FP7-ICT-2013-613024-GRASP and H2020-MSCA-IF-2015-704963, and the Portuguese Foundation for Science and Technology (FCT), co-financed by FEDER (PT2020 Partnership Agreement), under contracts PTDC/QEQ-SUP/1413/2012, RECI/CTM-CER/0336/2012, IF/01088/2014, BI/UI89/2015, POCI-01-0145-FEDER-007679 (UID/CTM/50011/2013) and COMPETE:FCOMP-01-0124-FEDER-027465.

## Author information

Authors

### Contributions

A.I.S.N. developed the method and performed the experiments, I.S. produced the textile fibres, D.P.R. performed SEM imaging and studied the influence of bending, A.D.S. executed the transmittance measurements, E.T.A. prepared the fibres coated with few-layer graphene and performed the bending cycles test, M.S.P. and V.S.A. performed the SThM studies, and L.V.M. and A.I.S.N. performed the A.F.M. imaging. H.A. and M.F.C. conceived the work, A.I.S.N., H.A., S.R. and M.F.C. discussed the results and A.I.S.N. wrote the manuscript and Supplementary Information.

### Corresponding author

Correspondence to Ana I. S. Neves.

## Ethics declarations

### Competing Interests

The authors declare that they have no competing interests.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Neves, A.I.S., Rodrigues, D.P., De Sanctis, A. et al. Towards conductive textiles: coating polymeric fibres with graphene. Sci Rep 7, 4250 (2017). https://doi.org/10.1038/s41598-017-04453-7

• Accepted:

• Published:

• ### The Influence of Substrate Functionalization for Enhancing the Interfacial Bonding between Graphene Oxide and Nonwoven Polyester

• Junaid Khan
•  & M. Mariatti

Fibers and Polymers (2021)

• ### Electrical Heating Performance of Electro-Conductive Para-aramid Knit Manufactured by Dip-Coating in a Graphene/Waterborne Polyurethane Composite

• Hyelim Kim
• , Sunhee Lee
•  & Hanseong Kim

Scientific Reports (2019)

• ### Graphene electronic fibres with touch-sensing and light-emitting functionalities for smart textiles

• Elias Torres Alonso
• , Daniela P. Rodrigues
• , Mukond Khetani
• , Dong-Wook Shin
• , Hugo Joulie
• , Isabel de Schrijver
• , Anna Baldycheva
• , Helena Alves
• , Ana I. S. Neves
• , Saverio Russo
•  & Monica F. Craciun

npj Flexible Electronics (2018)


text:
# Basic Magnifying Glasses

The idea behind a magnifying glass is to make something look bigger, but how does this work? The principle is so simple you already know it! The farther away something is, the smaller it looks, right? So to make something look bigger, just move it closer to your eye.

The problem is that your eye can only focus on something that is far enough for your eye’s lens to “accommodate.” Bringing something closer makes it bigger, but fuzzy, and that doesn’t help. Typically, this distance is about 10″ or 25 cm. Children can often focus closer than 25 cm, and people over 50 can’t normally focus even this close, so take the 10″/25cm as a rough approximation. If you’re as nearsighted as I am, you can focus 4″/10cm from your eye. It’s great for inspecting things, but not so good for driving.

The question is now, “How can I bring something closer to my eye, but still focus on it?” And the answer is, “a magnifying glass.”

A magnifying glass has two essential properties: its focal length and its diameter. The diameter, or how big the lens is doesn’t really matter that much. If you want to hold the lens far from your eye, then bigger is better, but it doesn’t change the magnifying power. The magnifying power is determined by the focal length.

Focal length is basically the distance from the magnifying glass to where it forms an image of a distant object. For example, you can make a small image of the moon by holding a magnifying glass a few inches (6-12 cm) from a piece of paper.

The magnifying power of a magnifying glass is 10″ (25 cm) divided by its focal lenth. For example, if you have a lens with a focal length of 4″ (10 cm) it will have a magnifying power of 10 divided by 4 (or 25 divided by 10), which is 2.5X.

The optical design of a magnifying glass is simple. Both sides must have the same curvature (so it doesn’t matter which way you hold the lens), and the glass is almost always simple soda-lime glass, so the focal length determines the design. The radius is given by

$R = 2(n-1)F$

where R is the radius of the surfaces, n is the refractive index of the glass (roughly 1.52) and F is the desired focal length. So for a 2.5X magnifying glass (focal length = 10 cm), R = 2*(1.52 – 1) * 10 = 10.4 cm.

Simple lenses are pretty good magnifiers up to about 5X. If you want more magnification, go on to Advanced Magnifying Glasses.

format: plain


text:
# zbMATH — the first resource for mathematics

Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue. (English) Zbl 1195.20041
Summary: This article surveys many standard results about the braid group, with emphasis on simplifying the usual algebraic proofs.
We use van der Waerden’s trick to illuminate the Artin-Magnus proof of the classic presentation of the braid group considered as the algebraic mapping-class group of a disc with punctures.
We give a simple, new proof of the $$\sigma_1$$-trichotomy for the braid group, and, hence, recover the Dehornoy right-ordering of the braid group.
We give three proofs of the Birman-Hilden theorem concerning the fidelity of braid-group actions on free products of finite cyclic groups, and discuss the consequences derived by Perron-Vannier and the connections with Artin groups and the Wada representations.
The first, very direct, proof, is due to Crisp-Paris and uses the $$\sigma_1$$-trichotomy and the Larue-Shpilrain technique. The second proof arises by studying ends of free groups, and gives interesting extra information. The third proof arises from Larue’s study of polygonal curves in discs with punctures, and gives extremely detailed information.

##### MSC:
 20F36 Braid groups; Artin groups 20E36 Automorphisms of infinite groups 20F05 Generators, relations, and presentations of groups 20E05 Free nonabelian groups 20F60 Ordered groups (group-theoretic aspects) 57M07 Topological methods in group theory
Full Text:
##### References:
 [1] A’Campo N., Math. Ann. 213 pp 1– (1975) · Zbl 0316.14011 [2] DOI: 10.1007/BF02950718 · JFM 51.0450.01 [3] DOI: 10.2307/1969218 · Zbl 0030.17703 [4] DOI: 10.2307/1970830 · Zbl 0237.57001 [5] Bohnenblust F., Ann. Math. 48 pp 127– (1947) · Zbl 0030.17801 [6] Chow W.-L., Ann. Math. 49 pp 654– (1948) · Zbl 0033.01002 [7] DOI: 10.1016/0021-8693(87)90229-8 · Zbl 0628.20029 [8] DOI: 10.1515/advg.2005.5.4.607 · Zbl 1156.20033 [9] DOI: 10.2140/pjm.2005.221.1 · Zbl 1147.20033 [10] Dehornoy P., Sér. I 309 pp 143– (1989) [11] Dehornoy P., J. Pure Appl. Algebra 61 pp 123– (1989) · Zbl 0686.20041 [12] DOI: 10.2307/2154598 · Zbl 0837.20048 [13] DOI: 10.2140/pjm.1999.191.49 · Zbl 1009.20042 [14] Funk J., Theory Appl. Categ. 9 pp 121– (2001) [15] Larue D. M., Algebra Universalis 31 pp 104– (1994) · Zbl 0793.08007 [16] Magnus W., Math. Ann. 109 pp 617– (1934) · Zbl 0009.03901 [17] Manfredini S., Topology Appl. 78 pp 123– (1997) · Zbl 0965.20016 [18] Perron B., Math. Ann. 306 pp 231– (1996) · Zbl 0863.32013 [19] Short H., Enseign. Math. 46 pp 279– (2000) [20] Shpilrain V., Internat. J. Algebra Comput. 11 pp 773– (2001) · Zbl 1024.20036 [21] DOI: 10.1016/0040-9383(92)90029-H · Zbl 0758.57008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Vistat

a reproducible gallery of statistical graphics

# Brownian Motion with R

We can use the animation package to produce animations in R. This short article shows you how to create a Brownian motion with the brownian.motion() function.

Simply speaking, a Brownian motion shows the trace of the coordinates

where $\epsilon_i$ is i.i.d from a standard Normal distribution. That is fairly easy to program in R – it is nothing but cumsum(rnorm(n)), and that is what brownian.motion() does internally.

library(animation)
ani.options(nmax = 50)  # create 50 image frames
set.seed(20121106)
brownian.motion(n = 20, pch = 21, cex = 4, col = "red", bg = "yellow",
xlim = c(-10, 10), ylim = c(-15, 15))

Done.

### Meta

Keywords: Categories: Reviewer: You can find the R Markdown source document in the vistat repository on GitHub.


text:
# Directed Grid Graphs; All Possible Paths Through Nodes

I have a problem in which I am interested in taking a matrix of positive integer weights, including zero, where the matrix has dimensions nrow x ncol and the columns always sum to the same arbitrary number. I want to search for a list of paths (sets of edges essentially) that traverse through the grid space of the same dimension & create the paths such that the # of edges in a path going through a node is equal to the nodes weight (from the matrix). Ie: if a particular index in my matrix was "3", there would be 3 edges that would run into (and out of) this node.

Some important restrictions. Firstly, The only direction the edges can move is rightward (so vertical edges are disallowed) but only one column distance at a time. I do allow for the edges to go from any row_j (in column_i) to any other possible row_j in column_i+1, so diagonal edges (either upwards or downwards are allowed). Here's an example of such a solution (it is non singular which is why Im interested in ALL possible paths)

Most importantly, I am interested in two things. First I want all possible paths from this process, and even more critically, I want to minimise the number of possible diagonal edges that my resulting paths will contain.

Any sort of algorithm here to solve this would be hugely helpful.

I have managed to solve the case when I don't care about the number of diagonal edges, and just want a set of paths that match with the weights. To do that, I used the weights at adjacent columns to generate a Markov transition process. This gave me a series of transition matrices (of length ncol-1) which from there I was able to construct probabilistically what my paths through my weights were.

Let me know if anyone needs any more details/explanations here.

Update & Further Questions

I've looked into the mincost flow algo's & although they are definitely in the right wheelhouse for this problem here, they aren't 100% exactly what I need for a few reasons. First one important thing to note is that in my real world scenario, these weights are actually more of a probability mass, so they can take any value (0,1) non inclusive. But for a toy example, the integer weights work ok to derive a framework.

I think the way I had solved the problem (prior to posting this question) essentially was in a network flow like manner already without realizing it, ie: looking at fluxes coming in & out of the various nodes across my lattice. However, the main problem with this manner of solving the problem is how this solution affects my inference of downstream results. Essentially behind this 'weight' matrix (in the example, it is a 3x3 grid) is a grid that encodes with a binary alphabet {0,1} a letter for each weight. Ie one possible reference grid (as I call it) could be

1 0 0
0 1 1
0 0 0

By using this reference grid & weighted panel together, we end up with a series of weighted references which has two important properties we can calculate (given some weights & some ref panel): the column means from this weighted reference panel, and the expected covariance (how often do 1's stay 1's essentially) of the weighted reference encodings.

Ie. in the example outlined above, we have on column A a weight of (4 * 1 + 0 * 1 +0 * 1)/(4+1+1) = 0.67.

The covariance terms are slightly more difficult to derive but essentially we look at what proportion of the total flow (that we have already derived) comes from elements encoded '1' in column i and stays encoded '1' column in i+1. Here that contribution would only come from {row_1,column_1}->{row_2,column_2}.

To get the longer distance covariance what I do is perform matrix multiplication on the flow matrices. You can think of the flow matrices as a series (of length ncol--1) of matrices (each with dimensions nrow x nrow) which look from each successive column (i->i+1), how much % flow is going from one row to itself, or to other rows.

When I perform this inference, I am able to properly calculate the correct expected value for the covariance based on the flow given an underlying reference matrix & weight matrix. However, I don't really want the expected value, but actually an expected value conditional on us having not used as high a proportion of diagonal edges as we would have in the naive case. The reason for this is when we perform this operation on larger matrices, the covariance terms (the proportion of times 1's flow to other 1's along the columns) is weakened over long distances (this is because this is a memoryless markov process if that helps) since with this diagonal flow we don't get the same set of encodings as the reference panel. (eg: in our example we would looking at the reference have only ({1,0,0},{0,1,1},{0,0,0}) as the possible paths, but with diagonal flow we could have some element such as {0,1,0}.

So in essence I guess what I'm wondering is if there is some sort of expected covariance I can derive, based on the graph traversal (or the framework I have developed here) that would allow me to get the expected covariance based on different # of diagonal switches possible, or more broadly to look at ways how to minimise this long-distance loss phenomena I am facing here.

One idea I had was to sort my reference panel based on the positional burrow wheels transform (http://alimanfoo.github.io/2016/06/21/understanding-pbwt.html) which would allow me to look at series of blocks in my reference panel that were more similar to one another. Unsure exactly how I would use that however?

• I am unclear what you mean by "all possible paths". If my matrix is two columns of $n$ rows consisting entirely of $1$s, do you expect a single solution ($n$ horizontal edges, since that minimizes the number of diagonal edges), or $n!$ solutions (all permutations)? – Aaron Rotenberg Jul 7 '20 at 23:49
• Hi Aaron, in that trivial example, I would want the former solution to be the case (single solution, n horizontal edges). I actually havent come across these sets of algorithms before, so this should be helpful thank you! – Eulers Jul 7 '20 at 23:52

[*] Mincost flow is usually presented with capacity constraints on the arcs rather than on the vertices. This is easily worked around by splitting each vertex in two, with a capacity-limited arc in-between. That is, if a vertex $$v$$ has a capcity limit $$c$$, replace $$v$$ with two vertices $$v_1$$ and $$v_2$$ such that all the arcs that entered $$v$$ instead enter $$v_1$$, all the arcs that left $$v$$ instead leave $$v_2$$, and there is an arc with capacity limit $$c$$ from $$v_1$$ to $$v_2$$.


text:
# Binary Tree

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 65536/65536 K (Java/Others)
Total Submission(s): 42    Accepted Submission(s): 25
Special Judge

Problem Description The Old Frog King lives on the root of an infinite tree. According to the law, each node should connect to exactly two nodes on the next level, forming a full binary tree.

Since the king is professional in math, he sets a number to each node. Specifically, the root of the tree, where the King lives, is   1. Say   froot=1.

And for each node   u, labels as   fu, the left child is   fu×2  and right child is   fu×2+1. The king looks at his tree kingdom, and feels satisfied.

Time flies, and the frog king gets sick. According to the old dark magic, there is a way for the king to live for another   N  years, only if he could collect exactly   Nsoul gems.

Initially the king has zero soul gems, and he is now at the root. He will walk down, choosing left or right child to continue. Each time at node   x, the number at the node is   fx  (remember   froot=1), he can choose to increase his number of soul gem by   fx, or decrease it by   fx.

He will walk from the root, visit exactly   K  nodes (including the root), and do the increasement or decreasement as told. If at last the number is   N, then he will succeed.

Noting as the soul gem is some kind of magic, the number of soul gems the king has could be negative.

Given   N,   K, help the King find a way to collect exactly   N  soul gems by visiting exactly   K  nodes.
Input First line contains an integer   T, which indicates the number of test cases.

Every test case contains two integers   N  and   K, which indicates soul gems the frog king want to collect and number of nodes he can visit.

1T100.

1N109.

N2K260.
Output For every test case, you should output ” Case #x:” first, where   x  indicates the case number and counts from   1.

Then   K  lines follows, each line is formated as ‘a b’, where   a  is node label of the node the frog visited, and   b  is either ‘+’ or ‘-‘ which means he increases / decreases his number by   a.

It’s guaranteed that there are at least one solution and if there are more than one solutions, you can output any of them.

Sample Input

2 5 3 10 4
Sample Output

Case #1: 1 + 3 – 7 + Case #2: 1 + 3 + 6 – 12 +
Source 2015ACM/ICPC亚洲区上海站-重现赛（感谢华东理工）

Sample Input

2 5 3 10 4

Sample Output

Case #1: 1 + 3 – 7 + Case #2: 1 + 3 + 6 – 12 +

#include<stdio.h>
#include<string.h>
int casenum,casei;
int n,m;
long long b[64],s[64],num[64];
int main()
{
for(int i=1;i<=62;++i)b[i]=1ll<<i-1;
for(int i=0;i<=62;++i)s[i]=(1ll<<i)-1;
scanf("%d",&casenum);
for(casei=1;casei<=casenum;++casei)
{
scanf("%d%d",&n,&m);
printf("Case #%d:\n",casei);
memcpy(num,b,sizeof(b));
long long dif=s[m]-n;
if(dif&1){++dif;++num[m];}
dif>>=1;
for(int i=1;i<=m;++i)printf("%lld %c\n",num[i],dif&b[i]?'-':'+');
}
return 0;
}

#include<stdio.h>
#include<string.h>
#include<ctype.h>
#include<math.h>
#include<iostream>
#include<string>
#include<set>
#include<map>
#include<vector>
#include<queue>
#include<bitset>
#include<algorithm>
#include<time.h>
using namespace std;
void fre(){freopen("c://test//input.in","r",stdin);freopen("c://test//output.out","w",stdout);}
#define MS(x,y) memset(x,y,sizeof(x))
#define MC(x,y) memcpy(x,y,sizeof(x))
#define MP(x,y) make_pair(x,y)
#define ls o<<1
#define rs o<<1|1
typedef long long LL;
typedef unsigned long long UL;
typedef unsigned int UI;
template <class T1,class T2>inline void gmax(T1 &a,T2 b){if(b>a)a=b;}
template <class T1,class T2>inline void gmin(T1 &a,T2 b){if(b<a)a=b;}
const int N=0,M=0,Z=1e9+7,ms63=1061109567;
int casenum,casei;
int n,m;
LL b[64],s[64],num[64];
int main()
{
for(int i=1;i<=62;++i)b[i]=1ll<<i-1;
for(int i=0;i<=62;++i)s[i]=(1ll<<i)-1;
scanf("%d",&casenum);
for(casei=1;casei<=casenum;++casei)
{
scanf("%d%d",&n,&m);
printf("Case #%d:\n",casei);
MC(num,b);
LL dif=s[m]-n;
if(dif&1){++dif;++num[m];}
dif>>=1;
for(int i=1;i<=m;++i)printf("%lld %c\n",num[i],dif&b[i]?'-':'+');
}
return 0;
}
/*
【trick&&吐槽】

【题意】

【类型】

【分析】

【时间复杂度&&优化】
O(Tm)

*/

原文作者：B树
原文地址: https://blog.csdn.net/snowy_smile/article/details/50094987
本文转自网络文章，转载此文章仅为分享知识，如有侵权，请联系博主进行删除。


text:
## Solvability of quasilinear elliptic equations with strong dependence on the gradient.(English)Zbl 1005.35043

From the introduction: The aim of this paper is to study existence of weak and strong solutions of the following quasilinear elliptic problem: $-\Delta_pu= G\bigl(|x|,u,|\nabla u|\bigr) \text{ in }B\setminus\{0\},$
$u=0\text{ on }\partial B,\tag{1}$
$u(x)\text{ spherically symmetric and decreasing },$ where we assume strong dependence on both the unknown and the gradient. Here $$B=B_R(0)$$ is the ball of radius $$R$$ in $$\mathbb{R}^N$$, $$N\geq 1$$, $$1<p<\infty$$, $$\Delta_pv=\text{div} (|\nabla v|^{p-2}\nabla v)$$ is the $$p$$-Laplace operator. The elliptic problem (1) is studied by relating it to the corresponding singular ordinary integro-differential equation. Solvability range is obtained in the form of simple inequalities involving the coefficients describing the problem. We also study a posteriori regularity of solutions. An existence result is formulated for elliptic equations on arbitrary bounded domains in dependence of the outer radius of the domain.

### MSC:

 35J60 Nonlinear elliptic equations 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 45J05 Integro-ordinary differential equations
Full Text:


text:
# preg_match to .NET equivalent

I have the following code in PHP:

    if (preg_match('@^[a-z0-9/._-]+$@i',$script)
&& !preg_match('@([.][.])|([.]/)|(//)@', $script))  I'm making the assumption that the predicate for the if statement returns true for the string js/core.js. How would I translate this to C#? The dumb translation is as follows: if(Regex.IsMatch(script,"@^[a-z0-9/._-]+$@i")
&& !Regex.IsMatch(script,"@([.][.])|([.]/)|(//)@"))

but I suspect that the @ symbol has meaning associated with it that I can't get to the bottom of. A translation to .NET regex would be nice, but I'm entirely familiar with .NET regex, so an explanation of the relevant syntax differences would suffice.

Many thanks.

-

The @s are just delimiter. You don't need them in .NET
if(Regex.IsMatch(script,"^[a-z0-9/._-]+\$", RegexOptions.IgnoreCase)

The @ symbol in .net is syntactic sugar that denotes that the following string will be a literal and is located outside of the quotes. this allows for a string where you do not have to double-escape the backslash character (\). So "c:\\foo\\bar" becomes @"c:\foo\bar"