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Versatile and scalable micropatterns on graphene oxide films based on laser induced fluorescence quenching effect

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Abstract

Here we report on the preparation of quasi-homogeneous fluorescence emission from graphene oxide (GO) film by modifying the local optical properties through the laser-induced fluorescence quenching effect, and the fabrication of single and multilayer micropatterns on quasi-homogeneous GO films. The modification is stemming from the photoreduction of GO, where the reduction degree and fluorescence intensity can be precisely tuned by changing the laser power and irradiation duration. This versatile approach with a mask-free feature can be readily used to fabricate various complex microstructures on quasi-homogeneous GO film from single layer to multilayer in vertical scale, as well as micrometers to centimeters in lateral scale. The micropatterns with varied optical properties are promising for applications in information storage, display technology, and optoelectronic devices.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Owing to unique physical properties, two-dimensional (2D) monolayer transition-metal dichalcogenides (TMD) with the common formula of MX2 (M = Mo, W, et al., X = S, Se, Te, respectively) have attracted significant research interest from the viewpoint of fundamental physics to the promising applications in various areas [1–3]. For example, atomic-layered TMD not only has been regarded as an ideal model to explore many-body correlated quasiparticles [4], but also has been used to design and fabricate photonics, electronics, and energy devices [5–7]. Very recently, the local modification on their optical properties have been achieved by direct laser patterning [8–10]. After laser irradiation, the local properties, such as layer thickness and fluorescence intensity, can be precisely controlled, and thus micropatterns can be created on the layered flake. This modification allows “micro-encryption” of information on the atomic-layered materials [8]. However, the practical applications of this significant and facile technique are hindered by either the small size of the TMD flakes (less than 100 μm in lateral scale), or the limitations in information storage capacity (only single layer can be used). These challenges can be solved by alternating TMD to graphene oxide, benefiting to its solution-processing compatibility and scalable production of high quality thin film in wafer scale [11].

Graphene oxide (GO), as one of the most important derivatives of graphene, is also a layered material featuring a variety of oxygen-containing functionalities and confined sp2 structures on the basal plane [12,13]. The abundant functional groups in GO not only result in good dispersion in aqueous, but also emerge strong and tunable fluorescence [14,15]. The good dispersion allows for the production of high-quality large-scale GO film in wafer scale [16,17]. This film is quasi-2D with the thickness of several to dozens nanometer, which has huge potential applications in the fabrication of microelectronic and optoelectronic devices [18–20]. For example, patterns of microcircuits and field-effect-transistor (FET) have been readily achieved using direct laser writing technology [21–24]. Even though many efforts have been made aiming to fabricate high quality microelectronics so for, very few microstructures based on the modification of GO’s optical properties have been developed. The main challenge is the intrinsic heterogeneous fluorescence emission from GO, which will strongly reduce the contrast of the patterns, and block further applications in high quality electronic and photonic devices.

Here we present the preparation of quasi-homogeneous fluorescence emission from GO film by pre-treatment processing with the irradiation of 405 nm continuous-wave (CW) laser, and then create single and multilayer micropatterns on the quasi-homogeneous GO films based on the laser-induced fluorescence quenching effect. The precise control on the fluorescence intensity is originating from the photoreduction of local GO film. In the experiment, the laser with high power is used as the writing pens which will effectively reduce local oxygen-containing functional groups of GO and result in the strong quenching of fluorescence intensity. While the same laser with extremely low power is applied as the reading tool to visualize the micropatterns. Various complex micropatterns, including alphabets, graphs, Quick Response (QR) code, have been successfully created on the single and multilayer GO films through this facile laser micro-writing pathway. Our approach is reliable and scalable, that can be operated in the ambient conditions without any special requirement. Micropatterns obtained with this approach can be used in information storage, micro-encryption, display, as well as optoelectronic devices.

2. Introduction

2.1 Sample preparation

GO dispersion, purchased from XFNANO materials Tech Co., Ltd. (Nanjing China), was synthesized by the modified Hummers method. The concentration was 2 mg/ml. The GO film was prepared by spin-coating the dispersion on the cleaned glass coverslip at 3000 rmp for 60 s, and then dried at 80°C in the Argon atmosphere. This approach was repeated for 10 times. Based on this approach, the GO film with relatively smooth surface and lager scale (in centimeters) can be readily prepared (See Appendix A for sample characterization). The multilayer GO structure was fabricated by spin-coating the transparent poly (vinyl alcohol) (PVA) polymer on the prepared first layer GO film. The concentration of the PVA solution was 40 mg/ml, the operation parameters of spin-coating process were 500 rmp for 5 s, 3000 rmp for 60 s, and 600 rmp for 10 s in turns. After dried in vacuum at 80°C, the second layer GO film was spin-coated on the PVA matrix, using the same parameters as that for the first layer. This procedure can be successive production of multilayer GO films, allowing to develop a three-dimensional device architecture. The final GO samples were placed on a three-dimensional nano-stage, used for further processing by focused laser beam.

2.2 Optical setup

To perform the micropatterns on the GO film, a home-built scanning confocal system based on an invert microscope (Nikon, TE2000-U) was used to irradiate the GO and take the fluorescence image. The detail descriptions of the experimental setup can be found in our pervious works [25,26]. Particularly, a CW laser with the wavelength of 405 nm was used to excite GO sample. Here the CW laser was not only used as the reduction laser with high power to directly write the information on the GO film, but also used as the probe laser with extremely weak power to read the final micropatterns. The laser beam was tightly focused by a 100 × oil immersion objective lens with a high numerical aperture (NA = 1.3). The focused laser spot was about 300 nm in diameter. The fluorescence from GO film was collected by the same objective. After passing through a dichroic mirror (Semrock, Di02-R405-25x36), and a long-pass filter (Semrock, BLP01-405R-25) to block the backscattered laser as well as background single, the fluorescence was further filtered spatially by a 100μm pinhole and detected by a single photon detector (PerkinElmer, SPCM-AQR-15). The micropatterns were created by moving the GO film with respect to the focused laser beam in a programmable and controlled way. The speed of the nano-stage movement was kept constant at 50 μm/s. The spatial step size of the stage was 100 nm.

3. Results and discussion

To pattern and visualize the microstructure on the GO film based on the modification of fluorescence intensity, the facile operation on the fluorescence emission with high spatial resolution is desired. Here the decay of GO’s fluorescence intensity after laser irradiation with high power, referred to laser induced fluorescence quenching (LIFQ) effect, is used as the manipulation to alter the fluorescence emission and directly write the information on the GO film. As presented in Fig. 1(a), a significant hole can be found after irradiation by the focused laser beam with the power of 10 mW and duration of 1 s. Figure 1(b) presents the detailed fluorescence decay as the function of irradiation duration. At the beginning, the intensity decays sharply, as shown in the inset of Fig. 1(b), then the fluorescence almost maintains at a steady intensity. As reported in our previous works [15], this LIFQ effect results from the photoreduction of oxygen-containing functional groups of GO. The origin of GO’s fluorescence can be distinguished into two aspects: the graphene-like confined sp2 clusters with weak fluorescence emission, and the sp3 domains consisting of oxygen-containing functional groups with strong fluorescence emission [15,27]. After laser irradiation, the strongly emissive sp3 domains will be photoreduced, either removed or turned into weakly emissive sp2 clusters. On the other hand, the high power density will also eliminate part of sp2 clusters. Consequently, the fluorescence intensity will be quenching [28].

 figure: Fig. 1

Fig. 1 (a) Confocal fluorescence image of GO film with a quenching hole, resulting from the laser irradiation with the power of 10 mW and duration of 1 s. Scale bar, 5 μm. (b) Fluorescence decay as a function of irradiation duration for the quenching area. The inset presents the decay in first 5 s. (c) and (d) are the fluorescence and Raman for original (GO) and reduction area (RGO), respectively. (e) FTIR spectra of GO and RGO obtained from a large area film (1.5 mm × 1 mm).

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The reduction and fluorescence quenching have also been proved by their fluorescence, Raman and Fourier transform infrared (FTIR) spectra. Fluorescence and Raman spectra were carried out by the same confocal microscope system. FTIR was performed with a commercial spectrometer (Thermo Scientific Nicolet iS50), where the signal was obtained from a large area film (~1.5 mm × 1 mm). For reduced GO (RGO), the detection area has been irradiated spot by spot with the power of 10 mW and duration of 1s for each spot. The spatial step size of the stage in this process was 2 μm. As shown in Fig. 1(c), two features can be found from the fluorescence spectra of GO and RGO. One is that the fluorescence intensity of RGO is weaker than GO, coinciding with LIFQ effect. Well fitted Gauss peaks is another feature. We attributed the low frequency peak (~780 nm) to the emission from the confined sp2 clusters, and the high frequency peak (~675 nm) to that from sp3 domains. The decrease in both peaks indicates the elimination of sp2 clusters and sp3 domains. This result is further illuminated by the decrease of Raman intensity for RGO, as presented in Fig. 1(d). Moreover, the D and G bands can be clearly determined in GO and RGO area, respectively. The increased ID/IG ratio from 0.86 of GO to 1.25 of RGO, hints the formation of defects and disorder structures during photoreduction [29]. According to the previous lectures [30,31], the bands at 3600-2800, 1735, 1621, and 1067 cm−1 in FTIR spectra can be attributed to the stretching modes of O-H, C = O, C = C, and C-O groups, respectively. While the peak located at 1258 cm−1 is assigned to the C-O-C asymmetric stretching vibration of epoxy group. The decrease or even disappear of hydroxyl and epoxy groups, as well as carboxyl strongly indicates the effective reduction during laser irradiation.

The results of LIFQ effect strongly rely upon the laser power and irradiation duration in the processing. In order to optimize reduction parameters, an array of experiments were performed on the GO film, in which the laser power was varied from 0.2 μW to 10 mW, and the irradiation duration was varied from 10 ms to 210 s, respectively. Figure 2(a) presents the normalized fluorescence decay as functions of the laser power and irradiation duration. It can be determined that when the laser is extremely weak, such as weaker to 0.2 μW, the fluorescence shows a slow decay. However, when the power is larger than 0.5 mW, a sharp drop appears, leading the fluorescence quenching to the minimum intensity within 10 s. On the other hand, as the laser power fixed, the fluorescence intensity I(t) decays with the irradiation duration t in the exponential manner, expressed as:

I(t)=I0+A.et/τ
where I0 is the rest intensity after fully photoreduction, τ and A are the fluorescence decay time and exponential factors. The decay time τ can be used to characterize the fluorescence behaviour under different powers. As presented in Fig. 2(b), the τ is larger than 200 s for the power of 0.2 μW, while that is less than 1 s for the power of 10 mW. These results strongly suggest that not only can we use the high power laser to directly write the patterns on the GO film, which will induce the dramatic quenching in the fluorescence intensity, but also can we use the same laser with extremely weak power to read the patterns, which has no significant influence on the optical properties of GO film. Here, the power of 10 mW with duration of 100 ms for each pixel was used as the writing laser, and that of 0.2 μW with the integral time of 10 ms for each pixel was used as the reading laser. The fluorescence decays under the irradiation of the writing and reading laser have been shown in Fig. 2(c). After irradiation by the writing laser, the fluorescence almost decays to the minimum value (but not zero). The rest fluorescence is mainly originating from the sp2 clusters [15]. Conversely, the fluorescence intensity still maintains 90% after irradiation by the reading laser with the duration of 10 s. Considered the integral time for each pixel was only 10 ms during confocal fluorescence imaging to visualize the patterns, the information is still available after 1000 cycles reading. To further illustrate the reading effect, Figs. 2(d) and 2(e) present the pattern of the 1st and 50th reading, respectively. Note that the patterns can still be clearly distinguished at the 50th reading, without significant changes. As shown in Fig. 2(f), the fluorescence variations across the original and reduction areas (highlighted by the yellow dashed lines) during reading approaches further elucidate this conclusion.

 figure: Fig. 2

Fig. 2 (a) Normalized fluorescence decay as functions of laser power and irradiation duration. (b) Fluorescence decay time as a function of laser power. (c) Normalized fluorescence decay under the irradiation of writing and reading laser as a function of irradiation duration. The writing and reading laser are both 405 nm CW laser with the power of 10 mW and 0.2 μW, respectively. (d) and (e) are the results of the 1st and 50th reading, visualized by confocal fluorescence imaging. Scale bar, 10 μm. (f) Fluorescence variation of the GO and RGO (highlighted by the yellow dashed lines in d and e) during reading processes.

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The optimized parameters of laser irradiation allow for direct micropatterns on GO film based on the LIFQ effect. However, the fabrication of distinguished micropatterns is still challenging, because the fluorescence from GO film is heterogeneity in most cases. This heterogeneity will result in the low contrast of the patterns, and even unreadable. Generally, this heterogeneity results from two aspects: one is the inhomogeneous thickness of GO film, as shown in Fig. 3(a). The other is the wide variability in the type and coverage of the oxygen containing functional groups [13], and also the broad distribution in the lateral sizes and amounts of sp2 confined clusters. Figure 3(b) presents the typical fluorescence image of the inhomogeneous GO film. The heterogeneous fluorescence intensity will hinder the patterning with high contrast, and also block further applications in high quality electronic and photonic devices. Here we solve this challenge by a pre-treatment processing. Firstly, the intensity distribution will be fitted by multiple peaks Gauss function, as shown in Fig. 3(c). Secondly, a background intensity, Ib, will be defined as the central value of the first Gauss peak, i.e. Ib = 33.9 kcps. Thirdly, the GO film will be pre-treatment by laser irradiation with calculated duration for each pixel under certain laser power. The duration of the pixel with the intensity weaker than or equal to Ib will be set as zero. While that with the intensity stronger than Ib will be calculated according to the formula of

ti,j=τP.1n(1(Ii,j(t=0)Ib)/Ai,j)
where ti,j is the duration for the pixel of (i, j), τP and Ai,j are the decay time and exponential factor under the power of P, Ii,j(t = 0) is the initial intensity for the pixel. (See Appendix B for the derivation of Eq. (2) and details of the calculation.) The calculated results under the power of 10 mW has been presented in Fig. 3(d). After pre-treatment, the fluorescence shows a reasonable homogeneity, as presented in Figs. 3(e) and 3(f), respectively. Note that the central value of the intensity distribution is a little lower than Ib. And more importantly, the full width at half maximum (FWHM) after pre-treatment is much smaller.

 figure: Fig. 3

Fig. 3 (a) Optical image of the prepared GO film with some thick flakes on the surface. (b) and (c) are the fluorescence image and intensity distribution of the inhomogeneous GO film, respectively. (d) The calculated irradiation duration for the pre-treatment processing by laser irradiation with the power of 10 mW. (e) and (f) are the corresponding fluorescence image and intensity distribution after pre-treatment processing. Scale bar: 10 μm.

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The high quality in fluorescence intensity provides a good platform for further micropatterns on GO film. As shown in Figs. 4(a) and 4(b), alphabets (“GO”) and graph (“Panda”) with high contrast have been successfully fabricated. We also produce a pattern of barcode on the GO film, as shown in Fig. 4(c). As we known that the barcodes represent the information by varying the widths and spaces of parallel lines, thus readable barcodes require sharp and clear edges. To improve the quality of the barcode pattern, the fluorescence intensity distribution has been carried out, as shown in Fig. 4(d). Two distinct peaks can be clearly determined, which represents the fluorescence emission from RGO and GO, respectively. Here, a threshold value Ith, defined as Ith = I1 + 3σ1, is used to distinguish the areas of RGO and GO, where I1 and σ1 are the central value and standard deviation of the first Gauss peak. As the fluorescence emission stronger than the Ith, the areas are set as white color. On the other hand, as the fluorescence weaker than or equal to Ith, the areas are set as black. After processing, a new barcode with sharp edges and high resolution is determined, as shown in Fig. 4(e). Even through a few errors can be found, the information of “GO” can be readily read from the barcode with a smart phone. According to the same processing, a quick response code (QR code) is also successfully achieved (Fig. 4(f)).

 figure: Fig. 4

Fig. 4 Micropatterning of RGO structures on the GO film. (a)-(c) are the confocal fluorescence image of the alphabets (“GO”), graph (“Panda”) and barcode, respectively. Scale bar, 10 μm. (d) Fluorescence intensity distribution of the barcode. (e) and (f) are the patterns after processing with the fluorescence thresholds. The barcode and QR code represent the information of “GO” and “http://laserspec.sxu.edu.cn/”, respectively. The lateral scales for them are 50 μm × 50 μm.

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Beyond single layer micropatterns, this procedure also allows the fabrication of multilayer micropatterns. Figure 5(a) shows a schematic of the multilayer patterns in this work. The thickness of the GO film and transparent PVA matrix can be readily controlled by the concentration, spin-coated parameters and repeated times. Figure 5(b) presents the fluorescence intensity as the laser focus scanning from the bottom to the top. Two distinct peaks emerge with FWHM ~2 μm, revealing that the laser focal along the axial direction is about 2 μm. The thickness of the PVA matrix can be estimated to be ~21 μm. The tightly focused laser beam in the axial direction allows good separation of the fluorescence emission from the two GO films. Multilayer micropatterns can be created by the location of the microscope’s focal plane on each GO film. Figures 5(c) to 5(e) present the confocal fluorescence image on the bottom GO film, the middle of PVA matrix and the top GO film, respectively. It can be determined that the micropatterns have been successfully created on the two GO layers without any interference.

 figure: Fig. 5

Fig. 5 (a) Schematic diagram of the experiment for the multilayer micropatterns. (b) Fluorescence intensity as a function of the height of the prepared GO sample. (c)-(e) are the confocal fluorescence image with the focal plane on the 1st GO film, the middle of PVA matrix and the 2nd GO film, respectively. Scale bar, 10 μm.

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4. Conclusions

In this work, the fluorescence quenching effect under laser irradiation was determined, which results from the photoreduction of GO and the elimination of both strongly emissive sp3 domains and weakly emissive sp2 clusters. By changing the laser power and irradiation duration, the reduction degree and fluorescence intensity of GO can be precisely tuned. Consequently, the quasi-homogeneous fluorescence emission was achieved. Many complex microstructures with high contrast have been created on the quasi-homogeneous GO film based on the fluorescence quenching effect. This versatile approach is reliable and scalable, which can be used to create multilayer micropatterns in wafer scale. These microstructures with varied optical properties are needed for applications in optoelectronic devices, display technology and information storage.

Appendix A sample characterization

In principle, the significant LIFQ effect can be observed in monolayer, with the thickness about 1.5 nm, as reported in our previous work [15]. However, the lateral size of monolayer GO flakes is only 0.5-3 μm, which is not enough to create micropatterns. Thus, in our scheme the GO samples were prepared by spin-coating for 10 times to make sure that the GO film is continuum. Figure 6(a) presents a large scale (60 mm×24 mm) GO film prepared on the cleaned glass substrate. A slightly yellow-brown colour can be found on the substrate, resulting from the original colour of GO materials. Figure 6(b) presents the optical image of the GO film. It can be found that for some area the surface was relative smooth, while thick flakes overload on the smooth surface in most case, resulting in the heterogeneous thickness. To further investigate the surface property, atomic force microscopy (AFM) was used to estimate the thickness of the GO film. Film thickness and surface roughness were analysed using AFM in contact-mode. As shown in Fig. 6(c), many folded wrinkles can be visualized, which was formed during flakes collapsed atop of each other from dispersion to the film. As presented in Fig. 6(d), the thickness of the selected lines are in the region of 12 to 22 nm, with some wrinkles close to 32 nm.

 figure: Fig. 6

Fig. 6 (a) The photograph GO film with the lateral scale of 60 mm × 24 mm on the cleaned glass coverslip. (b) The optical image of the prepared sample. Scale bar: 10 μm. (c) and (d) are AFM topographic image and the height profiles of the selected lines, respectively. Scale bar: 5 μm.

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Appendix B derivation of the duration for the pre-treatment processing

To obtain the quasi-homogeneous fluorescence emission from GO film, the film was processing by pre-treatment through laser irradiation with calculated duration for each pixel under certain laser power. The derivation of this formula can be outlined as follows.

Here, the fluorescence intensity decays with the irradiation duration in a single exponential manner, as given in Eq. (1), that is

Ii,j(t)=Ii.j0+Ai,jexp(tτP)
where Ii,j(t) is the fluorescence intensity at time t for the pixel of (i, j), and Ii.j0 is the rest fluorescence intensity after fully photoreduction, which is mainly originating from the sp2 clusters. τP is the decay time under the lase power of P, and Ai,j is the corresponding exponential factor.

Thus, at t=0,

Ii,j(t=0)=Ii,j0+Ai,j

At t=ti,j,

Ii,j(t=ti,j)=Ii,j0+Ai,j.exp(ti,jτP)
Subtract Eq. (A3) from Eq. (A2), we obtain
Ii,j(t=0)Ii,j(t=ti,j)=Ai,j(1exp(ti,jτP))
The Eq. (2) can be easily determined by transferring Eq. (A4).

Now, Ii,j(t= ti,j) is setting as the central value of the first Gauss peak, i.e. Ii,j(t)=33900 cps, and Ii,j(t=0) is the initial intensity for each pixel. Once τP and Ai,j were determined, the duration ti,j can be obtained.

In the experiment, the irradiation power in this step was 10 mW. Hence, τP can be obtained from Fig. 2(b), which is about 0.086 s, i.e. τP=86 ms. By comparing the fluorescence intensity after fully photoreduction with the initial intensity, the rest fluorescence intensity is about 0.18 of the initial intensity, that is

Ii,j00.18×Ii,j(t=0)

Thus,

Ai,j0.82×Ii,j(t=0)

Consequently, the duration ti,j can be calculated by

ti,j(ms)=86×1n(1Ii,j(t=0)339000.82Ii,j(t=0))

The results have been presented in Fig. (4c).

Funding

The project is sponsored by the National Key Research and Development Program of China (Grant No. 2017YFA0304203), the National Natural Science Foundation of China (Grant No. 11434007), the Natural Science Foundation of China (Nos. 11404200, 61527824, 11374196, 61605104 and U1510133), the PCSIRT (No. IRT13076).

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Figures (6)

Fig. 1
Fig. 1 (a) Confocal fluorescence image of GO film with a quenching hole, resulting from the laser irradiation with the power of 10 mW and duration of 1 s. Scale bar, 5 μm. (b) Fluorescence decay as a function of irradiation duration for the quenching area. The inset presents the decay in first 5 s. (c) and (d) are the fluorescence and Raman for original (GO) and reduction area (RGO), respectively. (e) FTIR spectra of GO and RGO obtained from a large area film (1.5 mm × 1 mm).
Fig. 2
Fig. 2 (a) Normalized fluorescence decay as functions of laser power and irradiation duration. (b) Fluorescence decay time as a function of laser power. (c) Normalized fluorescence decay under the irradiation of writing and reading laser as a function of irradiation duration. The writing and reading laser are both 405 nm CW laser with the power of 10 mW and 0.2 μW, respectively. (d) and (e) are the results of the 1st and 50th reading, visualized by confocal fluorescence imaging. Scale bar, 10 μm. (f) Fluorescence variation of the GO and RGO (highlighted by the yellow dashed lines in d and e) during reading processes.
Fig. 3
Fig. 3 (a) Optical image of the prepared GO film with some thick flakes on the surface. (b) and (c) are the fluorescence image and intensity distribution of the inhomogeneous GO film, respectively. (d) The calculated irradiation duration for the pre-treatment processing by laser irradiation with the power of 10 mW. (e) and (f) are the corresponding fluorescence image and intensity distribution after pre-treatment processing. Scale bar: 10 μm.
Fig. 4
Fig. 4 Micropatterning of RGO structures on the GO film. (a)-(c) are the confocal fluorescence image of the alphabets (“GO”), graph (“Panda”) and barcode, respectively. Scale bar, 10 μm. (d) Fluorescence intensity distribution of the barcode. (e) and (f) are the patterns after processing with the fluorescence thresholds. The barcode and QR code represent the information of “GO” and “http://laserspec.sxu.edu.cn/”, respectively. The lateral scales for them are 50 μm × 50 μm.
Fig. 5
Fig. 5 (a) Schematic diagram of the experiment for the multilayer micropatterns. (b) Fluorescence intensity as a function of the height of the prepared GO sample. (c)-(e) are the confocal fluorescence image with the focal plane on the 1st GO film, the middle of PVA matrix and the 2nd GO film, respectively. Scale bar, 10 μm.
Fig. 6
Fig. 6 (a) The photograph GO film with the lateral scale of 60 mm × 24 mm on the cleaned glass coverslip. (b) The optical image of the prepared sample. Scale bar: 10 μm. (c) and (d) are AFM topographic image and the height profiles of the selected lines, respectively. Scale bar: 5 μm.

Equations (9)

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I ( t ) = I 0 + A . e t / τ
t i , j = τ P .1 n ( 1 ( I i , j ( t = 0 ) I b ) / A i , j )
I i , j ( t ) = I i . j 0 + A i , j exp ( t τ P )
I i , j ( t = 0 ) = I i , j 0 + A i , j
I i , j ( t = t i , j ) = I i , j 0 + A i , j . exp ( t i , j τ P )
I i , j ( t = 0 ) I i , j ( t = t i , j ) = A i , j ( 1 exp ( t i , j τ P ) )
I i , j 0 0.18 × I i , j ( t = 0 )
A i , j 0.82 × I i , j ( t = 0 )
t i , j ( m s ) = 86 × 1 n ( 1 I i , j ( t = 0 ) 33900 0.82 I i , j ( t = 0 ) )
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