A Heat Transfer Model for Graphene Deposition on Ni and Cu Foils in a Roll-to-Roll Plasma Chemical Vapor Deposition System

Roll-to-roll (R2R) manufacturing techniques have been implemented to mass-produce graphene, which is a two-dimensional carbon material [1–5]. Previous reports have demonstrated the successful application of R2R processes for graphene growth on Cu foil using thermal furnace chemical vapor deposition (CVD) [6–8], Joule heating CVD [9], and plasma CVD [10,11]. The extensive use of Cu substrates for large-scale deposition of graphene is due to the low cost of Cu, the self-limiting growth of graphene on Cu, and the ease of graphene transfer from Cu to other substrates [12]. Because of low carbon solubility in Cu, graphene growth on Cu is driven by surface catalytic-like reactions that occur at high temperatures (approximately 1200 K) [13]. Consequently, in R2R processes, the web speed controls the growth of graphene because the substrate temperature depends significantly on the duration of exposure to a high-temperature environment [7,11,14].

Alternatively, Ni substrates have been widely used for large-area deposition, from mono- to few-layer graphene and thin graphite [15–20]. Graphene growth on Ni is initiated by the decomposition of methane (CH₄) (or another hydrocarbon source) to intermediate carbon species at temperatures higher than 900 K. These carbon species dissolve in the Ni substrate to form metastable nickel carbides resulting from relatively high carbon solubility in Ni [21]. Upon cooling, elemental carbon precipitates and segregates on the Ni surface due to decreased carbon solubility as the substrate cools. As a result, graphene with different qualities and thicknesses emerge on the Ni surface, depending on the cooling rate [15]. However, large-scale production of graphene on Ni using R2R CVD methods has not been previously reported.

Following the work of Yu et al. [15], several studies have considered the effects of cooling rate on graphene deposition with Ni substrates, as summarized in Table S1 available in the Supplemental Materials on the ASME Digital Collection. Previous reports have categorized the cooling rates as either slow or fast, or according to an intermediate rate between the two. These cooling rates were primarily derived from thermocouple measurements of the substrate holder, and not directly from the substrate. Moreover, a constant cooling rate over time is unlikely due to the nonlinearity of convection and radiation processes associated with local gas flow and chamber walls. Consequently, the cooling rates associated with the successful growth of few-layer graphene are inconsistent in prior literature as evident from Table S1 available in the Supplemental Materials on the ASME Digital Collection. Therefore, the absence of a detailed heat transfer analysis of Ni substrate hinders the potential of its utilization for scalable graphene production.

In order to control large-scale R2R processes for graphene on Ni and Cu foils, a detailed understanding of the temperature distribution during the R2R process is necessary. The temperature profile of the substrate during R2R processing can be determined from analysis of a moving plate/web in a fluid medium. Heat transfer and fluid flow with a moving plate in a quiescent fluid medium have been modeled previously using boundary layer theory [22,23]. However, heat conduction within the moving plate itself was not included in these earlier studies because the plate was assumed to be isothermal. The temperature distribution of the moving plate/web was determined in later reports either by solving the coupled fluid and plate heat diffusion equations [24–26] or by using a prescribed heat transfer coefficient and solving the heat diffusion equation in the plate alone [27–31]. The temperature of the moving plate was found to depend on web speed, material properties, and flow conditions.

Here, we focus on the impact of heat transfer on the temperature distributions in Ni and Cu foils, and their relationship to graphene growth in a R2R plasma CVD system. The heat transfer model includes convection boundary conditions for quiescent flow and plasma gas, radiation exchange with the chamber wall and the plasma electrodes, and fin-type conduction and advection associated with the moving foil. The heat transfer model is informed by nonintrusive in situ temperature measurements of the substrate. Moreover, Raman spectroscopy and scanning electron microscopy (SEM) are utilized to assess graphene quality and uniformity as functions of web speed to determine optimal conditions for continuous R2R deposition of graphene on Ni and Cu foils. The effects of web speed on the quality and uniformity of graphene on Ni and Cu foils are correlated to the changes in the temperature profile of the substrate with web speed. The heat transfer analysis and graphene characterization methods reported in this work advance the possibility of mass production of graphene on Ni and Cu substrates.

The R2R plasma CVD system used here was custom-designed to deposit graphene on a variety of flexible substrates [11,14,32]. As-received nickel (201 annealed nickel, 76.2 μm (or 0.003 in.) thick and 25.4 mm (or 1.00 in.) wide purchased from All Foils Inc., Strongsville, OH) or copper (110 annealed copper, 76.2 μm (or 0.003 in.) thick and 25.4 mm (or 1.00 in.) wide purchased from Basic Copper Inc., Carbondale, IL) was placed in a top, free-moving winder, passed through a plasma region where carbon film was deposited because of the high gas temperature in the plasma, and finally collected at the bottom driving winder (Fig. 1). Initially, the system was evacuated to less than 0.1 mbar; then a gas mixture of 50:50% H₂:Ar by volume was introduced at 7 mbar to ignite the plasma at a power of 500 W. Thereafter, pressure and power were increased after a warm-up period to 15 mbar and 850 W, respectively, and CH₄ was added to the H₂/Ar mixture for graphene growth. The flow rates of Ar, H₂, and CH₄ during graphene growth were 150, 610, and 260 standard cubic centimeters per minute, respectively.

Experiments were conducted at a constant plasma power of 850 W and a fixed gas pressure of 15 mbar. Power was supplied to two parallel electrodes to generate a capacitively coupled rf plasma at a frequency of 80 kHz. The electrodes, which are made of graphite, had rectangular shapes with lengths of 12.5 cm and widths of 5.0 cm. The distance between the electrodes was fixed at 4.5 cm, and the distance between the right electrode and the foil was 1.5 cm (Fig. 1). The substrate moved in the plasma region at web speeds of 0, 50, and 150 mm/min which were assessed during the experiments with a speed uncertainty of 10%. The residence times for the null web speed were 21.3 and 17.0 min for Ni and Cu foils, respectively.

After the experiment, Raman spectroscopy was used to quantify the quality of graphene deposited on the foil without transfer of the carbon film. Raman spectra were measured using LabRAM Horiba Ltd. Raman spectrometer with a laser excitation of 532 nm and 100× magnification. Also, SEM using S-4800 from Hitachi Corp. was utilized to visualize carbon films on the foil with a 1 kV accelerating voltage at various magnifications.

The foil temperature in the plasma region was measured directly from its radiative emission using optical emission spectroscopy. The emission was collected and collimated using 50.8 mm diameter lenses, focused on a fiber optic held by a fixture that allowed motion to a precise location along horizontal and vertical axes using Zaber translation stages. The spatial resolution of the emission measurement was estimated to be approximately 2 mm. The fiber optic transmitted the emission to a spectrometer (Princeton Instruments, Acton SP-2756) which has a grating of 1800 G/mm and a CCD camera (Princeton Instruments, PIXIS 256E). The spectrometer was calibrated in the range of 670–850 nm using a mercury light source.

The measurement of emission was controlled by lightfield software (from Princeton Instruments) to acquire high-resolution spectral data over a broad wavelength range. Equally spaced spectra of about 15 nm increments were measured individually by rotating the grating to cover the 670–850 nm range. The discontinuity between each measured spectral band that arises from changes in grating diffraction efficiency with wavelength was corrected by calibrating using an intensity-calibration light source. The calibration of intensity plays an important role in accurately measuring the spectra from the substrate over the 670–850 nm range.

where k_B is Boltzmann's constant (1.381 × 10⁻²³ J/K), A is the linear offset of the spectrum, and B is a constant that represents Planck's constant (h = 6.626 × 10⁻³⁴ J s), the speed of light (c = 2.998 × 10⁸ m/s), the foil emissivity, and the contribution from the optical emission setup as described in Ref. [34]. Because the emissivity of Ni and Cu foils are nearly constant in the 670–850 nm wavelength range, foils are assumed to be gray surfaces with emissivity values of 0.14 and 0.04, respectively, at 1000 K [33]. The detector and grating responses as functions of wavelength were included in the intensity calibration. The temperature (T) and the constants (A and B) were found from a nonlinear least-squares fitting method using matlab as described in the Supplemental Materials on the ASME Digital Collection. The contributions from species emission were filtered out; thus, only the baseline of the spectrum was included in the fitting, as shown in Fig. S1 available in the Supplemental Materials.

The ability to measure the foil temperature at various vertical points along the plasma region allows for refinement and validation of a heat transfer model that considers a 1-m long foil starting from the top spool and ending at the bottom spool (Fig. 1(b)). During the R2R CVD process, the foil encounters three regions: preplasma, plasma, and postplasma as illustrated in Fig. 1(c). Convective flow, which is driven primarily by buoyancy, is assumed to be isothermal and uniform in each of the three regions. The gas temperatures in the pre- and postplasma regions (⁠$T∞1$ and $T∞3$⁠, respectively) are assumed to be 313 K, as measured by a thermocouple. On the other hand, the gas temperature in the plasma region (⁠$T∞2$⁠) is assumed to be 1100 K as estimated from the rotational temperature of H₂ using optical emission spectroscopy [35] (Fig. S2 available in the Supplemental Materials). Furthermore, the temperatures of the right and left electrodes (as indicated in Fig. 1(b)) are 892 and 870 K, respectively, as determined from their emission spectra in the same manner described for the foil temperature in Eq. (1).

The heat transfer coefficients in the three regions are derived from fitting the model to the measured Cu foil temperatures (Fig. S3 available in the Supplemental Materials). The coefficients h₁ and h₃ are similar and fixed at 5 W/m² K, and we found in preliminary analyses that the results are relatively insensitive to these values as long as the coefficient is relatively small. By corollary, the present model and measurements clearly lack the fidelity to capture the complex fluid-thermal behavior that would be required to refine this assumption. The heat transfer coefficient in the plasma region (h₂) is used as a fitting parameter of the model to match the experimentally measured foil temperatures as explained later in this section.

In addition, a steady-state flow condition is assumed in the analysis because chamber wall temperatures reached a steady-state, and the foil moved at a constant prescribed web speed. Because of the small foil thickness (δ) and width (w), which are 76.2 μm and 25.4 mm, respectively, the Biot number is below 0.01. Therefore, a one-dimensional fin model is developed to solve for the substrate temperature along the foil length using analytical and numerical approaches as explained below.

where $θi=(T−T∞,i), a=ρUcP/k$⁠, and $mi=(2hi/kδ)+((hrad,Li+hrad,Ri)/kδ)$⁠. The regions $i=1,2,$ and 3 correspond to the preplasma, plasma, and postplasma locations, respectively. Also, ρ is the density of the foil (8933 and 8900 kg/m³ for Cu and Ni, respectively [33]), and U is the web speed.

where C₁ and C₂ are constants that are determined with appropriate boundary conditions. Note that $y*$ is the shifted y-coordinate written as $y*=y−0.4375$ m to simplify the solutions of C₁ and C₂.

The energy balance in the plasma region encompasses heating from the plasma constituents. For instance, heat transfer from the plasma accounts for substrate heating from electrons, neutrals, and ions, as well as surface reactions [36]. These heat fluxes depend on electron temperature, electron number density, reaction rates, and gas temperature, which are unknown and challenging to measure (except for the plasma gas temperature, $T∞2$⁠). Nevertheless, the heat fluxes from plasma processes can be lumped into an effective heat transfer coefficient (h₂) multiplied by the temperature difference between the foil and plasma gas (⁠$T−T∞2$⁠). Hence, h₂ is used as a fitting parameter of the model to match foil temperature measurements, resulting in values of 55, 30, and 29 W/m² K for Cu and 36, 30, and 28 W/m² K for Ni at 0, 50, and 150 mm/min, respectively. For both Cu and Ni, h₂ decreases with increased web speed indicating lessened interactions of the plasma gas with the foil.

The general solutions of the temperature distributions (Eqs. (3)–(5)) have six constants that can be evaluated by applying the following boundary conditions:

• Infinite fin boundary condition: $θ3(y*→∞)=0$⁠, leading to $C1=0$ in Eq. (3).

• Infinite fin boundary condition: $θ1(y*→−∞)=0$⁠, leading to $C4=0$ in Eq. (4).

• Temperature continuity at the interface of the preplasma and plasma regions: $θ1(y*=0)+T∞1=θ2(y*=0)+T∞2$⁠.

• Heat conduction flux continuity at the interface of the preplasma and plasma regions: $k1 ((dθ1(y*=0))/dy*)=k2 ((dθ2(y*=0))/dy*)$⁠.

• Temperature continuity at the interface of the plasma and postplasma regions: $θ2(y*=L)+T∞2=θ3(y*=L)+T∞3$⁠.

• Heat conduction flux continuity at the interface of the plasma and postplasma regions: $k2 ((dθ2(y*=L))/dy*)=k3 ((dθ3(y*=L))/dy*)$⁠.

where L is the length of the plasma region (i.e., the length of the electrodes, L = 0.125 m). A coordinate transformation simplifies the form of the constants C₂, C₃, C₅, and C₆, which are found by solving the set of linear equations from boundary conditions (c)–(f). Final expressions for the constants C₂, C₃, C₅, and C₆ are included in the Supplemental Materials.

The temperature profile is assumed to vary linearly between cell centroids [37]. Hence, heat fluxes from the top and bottom cells (i.e., q_N and q_S, respectively) are found by $qN=(kwδ/(Δy))(Tj−1−Tj)$ and $qS=(kwδ/(Δy))(Tj+1−Tj)$⁠, where $Δy$ is the cell size. Also, T_j, $Tj+1$⁠, and $Tj−1$ are the central cell (j), its south (j + 1), and north (j − 1) neighbor cell temperatures, respectively (Fig. S4 available in the Supplemental Materials). Temperatures T_N and T_S of the neighboring cells, using the central differencing discretization scheme, are given by $TN=(Tj−1+Tj)/2$ and $TS=(Tj+1+Tj)/2$⁠. The convection term is written as $qconv=hiw(Δy)(Tj−T∞i)$ for i = 1, 2, and 3, representing the preplasma, plasma, and postplasma regions, respectively. The factor of 2 in front of q_conv in Eq. (7) accounts for convection from both sides of the foil, which is assumed to be symmetric. Radiation from the foil to the electrode and to the chamber wall on the right and left sides (i.e., $qrad,R$ and $qrad,L$⁠, respectively) are derived in the Supplemental Materials.

where i = 1, 2, and 3 for the preplasma, plasma, and postplasma regions, respectively. Also, ε is the emissivity of the foil, and σ is the Stefan–Boltzmann constant (5.67 × 10⁻⁸ W/m² K⁴). A mesh-independence study (Fig. S5 available in the Supplemental Materials) led to a cell size (⁠$Δy$⁠) of $10−4$ m (i.e., the number of cells is 10,000). $Tj*$ denotes the central cell temperature from the previous iteration. Similarly, $Jfoil−R*$ and $Jfoil−L*$ are the radiosities of the right and left foil surfaces, respectively, from the previous iteration.

An iterative scheme solution is implemented in matlab to update temperature-dependent properties (i.e., c_P and k), radiosity (⁠$Jfoil−R*$ and $Jfoil−L*$⁠), and temperature from the previous iteration (⁠$Tj*$⁠) in Eq. (8). The solution is assumed to be converged when the relative temperature change between current and previous iterations is less than 10⁻⁴.

Both analytical and numerical models have identical temperature profiles when the radiation heat transfer is neglected (Fig. S6 available in the Supplemental Materials on the ASME Digital Collection). However, the analytical model with the inclusion of radiation heat transfer overpredicts the foil temperature in the plasma region due to the linearization of the radiation terms. Thus, the results from the analytical solution provide qualitative insights into the heat transfer processes in the foil, but the numerical model, which treats the radiation exchange and temperature-dependent properties more thoroughly, is used to find the Ni and Cu foil temperature distributions in the remainder of this work.

Figure 2(a) shows emission spectra from the plasma with and without Cu foil. The emission spectrum without the inclusion of foil exhibits only narrow spectral lines originating from active plasma species (such as Ar, Ar⁺, H₂, and H) [11,14]. The plasma emission with the presence of Cu foil has similar narrow spectral lines, but with a broad baseline that represents blackbody emission from the Cu substrate. Thus, the temperature of the foil can be determined from its emission (using Eq. (1)) which was measured at different vertical positions along the foil to validate the heat transfer model at various web speeds for Ni and Cu foils.

Figures 2(b) and 2(c) show the emission spectra of Cu and Ni foils, respectively. At a web speed of 0 mm/min, the Cu (Ni) foil temperature decreases slightly from its peak of 1100 K (985 K) at the plasma centerline, around y = 0.49 m, to 1045 K (962 K) at the plasma edge, around y = 0.46 m (as indicated in Fig. 1(c)). This temperature drop occurs because of heat loss by conduction to the lower temperature preplasma region. In addition, the temperature at the plasma edge, around y = 0.46 m, reduces further to 960 K and 938 K for Cu and Ni, respectively, when the foil is moving downward at a web speed of 50 mm/min. Advection of the moving foil lowers the substrate temperature, especially near the plasma edge.

Figure 3 presents the temperature distributions of Cu and Ni foils based on experiment and model at web speeds of 0, 50, and 150 mm/min. The temperature of both Ni and Cu foils is high in the plasma region due to the elevated gas temperature in the plasma $T∞2$⁠. Due to the lower thermal conductivity of Ni (72 W/m K at 1000 K [33]), its temperature at 0 mm/min remains more uniform in the plasma region in comparison to the Cu foil, which exhibits a strong decaying profile because of its higher thermal conductivity (352 W/m K at 1000 K [33]). Consequently, Ni foil conducts less heat to the pre- and postplasma regions, and the substrate temperature reaches the gas temperatures $T∞1$ and $T∞3$ more readily. For Cu, because of its higher thermal conductivity, the temperature field penetrates to longer distances in the pre- and postplasma regions.

At the given web speeds, Ni foil has a lower peak temperature in the plasma region than Cu foil. This temperature difference is related to the higher emissivity of Ni, which is 0.14, compared to 0.04 for Cu at 1000 K [33]. Hence, Ni foil experiences higher radiation exchange with the low-temperature chamber wall and plasma electrodes, resulting in a higher heat loss from Ni foil than from Cu foil in the plasma region [38].

As illustrated in Fig. 3, both Ni and Cu foil temperature profiles evolve as a result of heat advection via foil movement. As web speed increases from 0 to 150 mm/min, the temperature profile becomes asymmetric with lower and higher temperatures in the pre- and postplasma regions, respectively. The peak temperature in the plasma region decreases with increased web speed due to the cooling effect provided by advection of the moving foil. Furthermore, the position of peak temperature shifts downstream as web speed increases. Hence, the temperature of the foil has less uniformity in the plasma region with increased web speed.

Moreover, the temperatures in the preplasma region decrease with increased web speed due to the higher value of the exponent in the temperature distribution of Eq. (6) (i.e., (⁠$a2+4m1+a$⁠), where $a=(ρUcP)/k$ and U is the web speed). Thus, as web speed increases, more heat is carried downward by the moving foil. On the other hand, the exponent in the temperature distribution for the postplasma region of Eq. (6) (i.e., (⁠$a2+4m3−a$⁠)) decreases with increased web speed, resulting in a higher substrate temperature. In other words, the foil needs a longer distance to cool to the ambient gas temperature (⁠$T∞3$⁠) as web speed increases [31,38,39].

Furthermore, Ni has a higher advective Pe number than Cu because of the former's low thermal diffusivity, resulting in larger heat transfer by advection than diffusion processes in Ni foil [33]. The advective Pe number is defined as $Pe=LU/α$⁠, where α is the thermal diffusivity of the foil, L is the plasma length, and U is the web speed. For example, Pe number for Ni increases from 7.3 at 50 mm/min to 21.8 at 150 mm/min, whereas Pe number rises slightly from 1.2 to 3.6 for Cu for the same web speeds. Hence, the temperature distribution of Ni foil changes more considerably than Cu foil with web speed.

The temperature distribution from the model is fitted with experimental measurements to find the best fit of h₂ at various web speeds for Cu and Ni foils as shown in the inset of Fig. 3. The best-fit values of h₂ are 55, 30, and 29 W/m² K for Cu and 36, 30, and 28 W/m² K for Ni at 0, 50, and 150 mm/min, respectively. The difference in h₂ between Cu and Ni is low with increased web speed indicating lessened interactions of the plasma gas with the foil. However, h₂ is higher for Cu than Ni at 0 mm/min which could be related to the difference in their temperatures with the plasma gas temperature. The fluid flow characteristics adjacent to the foil, which have not been assessed in this work because of their complexity, could be altered leading to the difference in h₂ for Cu and Ni at 0 mm/min. The uncertainty bands on h₂ are derived by finding the best fit of the model to the minimum and maximum values of the foil temperature measurements which are 990 and 1210 K, respectively (which is 10% of the plasma gas temperature, 1100 K). For example, the ranges of h₂ for Cu foil are 29–80 W/m² K at 0 mm/min, 17–60 W/m² K at 50 mm/min, and 17–50 W/m² K at 150 mm/min. Similarly, the ranges of h₂ for Ni foil are 20–65 W/m² K at 0 mm/min, 17–55 W/m² K at 50 mm/min, and 17–50 W/m² K at 150 mm/min.

Furthermore, a sensitivity analysis is made to determine the influence of the main input parameters (including h₂) on the model as shown in Table S2 and Figs. S7 and S8 available in the Supplemental Materials on the ASME Digital Collection. The model is more sensitive to change in the plasma gas temperature (⁠$T∞2$⁠), followed by the electrode temperatures (T_RE and T_LE for the right and left electrodes, respectively), the heat transfer coefficient in the plasma region (h₂), and the foil emissivity (ε). The gas temperature in the preplasma region (⁠$T∞1$⁠), heat transfer coefficient in the preplasma region (h₁), and velocity (U) have negligible sensitivity coefficients on model. Also, $T∞2$⁠, h₂, T_RE, and T_LE have positive sensitivity coefficients, whereas ε has a negative sensitivity coefficient that indicates a reduced foil temperature with increased emissivity.

Graphene can be deposited on Ni foil during the R2R process with different qualities and thicknesses, depending on the web speed. Figure 4 shows SEM images of deposited films on Ni with substantial differences in contrast and morphology for different web speeds. The graphene thickness varies significantly across the Ni surface at 0 mm/min, as shown in Figs. 4(a) and 4(b). The darker regions represent thicker carbon films, whereas the lighter regions suggest the presence of graphene with a reduced number of layers. This thickness variation within the sample likely arises from the various orientations in the polycrystalline Ni foil that have distinct carbon precipitation rates during the substrate cooling period [40–42].

Conversely, as web speed increases, the uniformity of graphene is considerably enhanced, as shown in the SEM images of Figs. 4(c) and 4(e), with web speeds of 50 and 150 mm/min, respectively. Furthermore, the contrast between the images in Figs. 4(d) and 4(f) decreases with higher web speed (or lower residence time), indicating decreased graphene thickness. With increased residence time in the plasma, the thickness increases because of the larger concentration of dissolved carbon atoms in Ni [43].

Raman spectra of graphene on Ni foil are shown in Fig. 5 as a function of web speed. Raman peaks of graphene include the D peak at 1350 cm⁻¹, G peak at 1580 cm⁻¹, D$′$ peak at 1620 cm⁻¹, and 2D peak at 2700 cm⁻¹ [44,45]. The D and D′ peaks occur because of defects in the graphene lattice, whereas the G and 2D peaks are signatures of graphitic films. The intensity ratio of the 2D and G peaks (I_2D/I_G) at 0 mm/min is 0.31. However, as web speed increases to 50 and 150 mm/min, the I_2D/I_G ratio drops to 0.20 and 0.12, respectively. The 2D peak decreases with increased web speed because of the decreased foil residence time in the plasma and the lower substrate temperature that minimizes carbon diffusion [42].

The I_D/I_G peak ratio, which represents relative defect density, suggests that the deposited graphene is negatively affected by ion bombardment from plasma species such as Ar⁺ [11]. The I_D/I_G ratio values on Ni remain similar at around 1.4 as a function of web speed. The D′ peak is present at 0 mm/min, whereas this peak disappears as web speed increases (Fig. 5). In addition, the G-peak position shifts from 1586 cm⁻¹ at 0 mm/min to 1602 cm⁻¹ at 50 mm/min and finally to 1606 cm⁻¹ at 150 mm/min. The roll-to-roll process could induce strain in the carbon films that results in the shift of G-peak position from its original position of 1580 cm⁻¹ to 1586 cm⁻¹ at 0 mm/min [10]. However, the shift of the G peak and the disappearance of the D′ peak with increased web speed could be related to the existence of higher strain and larger disorder in the graphene lattice, leading to the broadening of the G peak and its possible overlap with D′ peak [46,47].

Graphene deposition on Cu foil exhibits better uniformity than on Ni foil but with a lower thickness of deposited film as web speed increases, as also reported in our previous work [11]. However, the quality of graphene on Cu foil varies significantly with web speed [7,11,14]. Table 1 compares results from Raman spectroscopy of graphene on Ni and Cu foils as a function of web speed. Unlike graphene on Ni foil, the I_D/I_G ratio of graphene on Cu foil decreases steeply with increased web speed. For example, the I_D/I_G ratio of graphene on Cu is higher than graphene on Ni at 0 mm/min. However, the I_D/I_G ratio on Cu drops significantly to 0.76 and 0.66 with increased web speeds of 50 and 150 mm/min, respectively. This decline of the I_D/I_G ratio of graphene on Cu with web speed is opposite to previous results using a thermal CVD system [7]. Decreasing the residence time in the plasma minimizes the defects created by ion bombardment, because graphene growth on Cu occurs during its exposure to the plasma where ample supply of intermediate carbon species exists to accelerate graphene growth on Cu by catalytic reactions on its surface [48]. On the other hand, the I_D/I_G ratio of graphene on Ni varies little with the reduction of residence time since the growth process is primarily driven by carbon diffusion and the segregation/precipitation process in the postplasma region.

Furthermore, the I_2D/I_G ratio for graphene on Cu declines sharply from 0.53 at 0 mm/min to 0.10 at 50 mm/min (Table 1). The 2D peak is negligible when the web speed is raised to 150 mm/min. The I_2D/I_G ratio is larger for Cu than Ni at the null speed, but the I_2D/I_G ratio of graphene on Cu becomes lower than on Ni at 50 and 150 mm/min. The variation of the I_2D/I_G ratio with web speed found here is similar to previous results in a R2R thermal CVD system [7]. Decreased residence time of Cu foil in the plasma region with increased web speed could affect growth kinetics, especially for lower Cu foil temperatures at which the graphene growth rate is very low [49]. Thus, the graphitization processes are hindered at lower substrate temperatures and shorter residence times because of less active C atoms at the Cu surface.

Indeed, the changes of the temperature distributions of Ni and Cu foils with web speed strongly affect the quality of graphene growth in a R2R process. Increasing the web speed to 150 mm/min minimizes the foil temperature in the preplasma region and lowers the temperature uniformity in the plasma region in which graphene growth occurs. These temperature changes, combined with reduced residence time with increased web speed, affect the growth kinetics of graphene by altering catalytic reactions of methane on Cu foil, carbon diffusion in Ni foil, and the heating and cooling rates of the Cu and Ni foils as discussed below.

The variations of graphene thickness and uniformity on Ni foil with web speed in Fig. 4 can be clarified first by estimating the diffusion length of carbon atoms in the Ni substrate during the growth process. The diffusion length (⁠$LD$⁠) is defined as $LD=2DTt$⁠, where t is the Ni foil residence time in the plasma region: 1278, 150, and 50 s at 0, 50, and 150 mm/min web speeds, respectively. Carbon diffusivity, $DT$⁠, depends on Ni foil temperature, T, as $DT=Do exp(−(ED/kBT))$⁠, where k_B is Boltzmann's constant which is 8.617 × 10⁻⁵eV/K [21,50,51]. Also, D_o equals 2.4818 cm²/s and E_D equals 1.74 eV representing the entropic prefactor and the diffusion activation energy, respectively [21,52].

Figure 6 illustrates the diffusion length of carbon in Ni as a function of web speed. At 0 mm/min, the diffusion length is about 41.1 μm, which is slightly greater than half of the Ni foil thickness which is 38.1 μm. Hence, carbon atoms can diffuse throughout the Ni foil since its both sides are exposed to plasma. Diffused carbon atoms are expected to precipitate and segregate on the Ni surface during the cooling stage, yielding a thick carbon film as seen in the corresponding optical image at 0 mm/min in Fig. 6 and the SEM images of Figs. 4(a) and 4(b).

Because of the decline in both the residence time and substrate temperature with increased web speed, the diffusion length drops to 12.5 and 6.5 μm at 50 and 150 mm/min, respectively. Therefore, less carbon can dissolve into the Ni foil as web speed increases, leading to the growth of graphene with lower thickness as evident from the SEM images at 150 mm/min (Figs. 4(e) and 4(f)) and the optical image of graphene on Ni at 150 mm/min (Fig. 6). On the other hand, because carbon solubility is negligible in Cu, the growth of graphene on Cu is controlled not by diffusion processes as in Ni, but rather by catalytic reactions of carbon species on Cu at high temperatures [13].

In addition, the rate of temperature change (⁠$dT/dt$⁠) is derived by taking the first derivative of the temperature distribution (⁠$dT/dy$⁠) and applying the chain rule: $dT/dt=(dT/dy)(dy/dt)=U (dT/dy)$⁠, where U is the web speed. Here, the derivative of the temperature distribution from the numerical model (Eq. (8)) is used to find the rate of temperature change as a function of position (⁠$dT/dy$⁠). The heating and cooling rates of the foil occur in the pre- and postplasma regions, respectively.

Figure 7 presents the heating and cooling rates of Cu and Ni foils at 50 and 150 mm/min. The heating rate of the foil is higher than the cooling rate of the foil. Both rates are nonlinear because of variations in the foil temperature in the three regions due to convection, radiation, and advection processes. Also, Ni has higher heating and cooling rates than Cu, which exhibits more gradual changes. These differences arise from the distinct temperature distributions of Ni and Cu, primarily due to the high thermal conductivity of the latter (Fig. 3).

Web speed significantly influences both the heating and cooling rates because of the substantial contribution from the advection term to the temperature distribution profiles (Fig. 3). For instance, at the interface of the preplasma and plasma regions, the heating rate increases sharply since the foil enters the plasma region rapidly. Similarly, the maximum cooling rates in the postplasma region rise with increased web speed, with values of 12.3 and 19.2 K/s at 50 mm/min and 150 mm/min, respectively, for Ni foil. On the other hand, Cu has lower maximum cooling rates of 7.9 and 18.7 K/s at 50 and 150 mm/min, respectively.

The cooling rate decreases exponentially as the foil moves to the bottom spool until the substrate temperature equilibrates with the temperature of the spool (i.e., the gas temperature in the postplasma region, (⁠$T∞3$⁠)). However, at 150 mm/min, the foil requires a longer distance to cool to the postplasma gas temperature, which explains the sharp rise in the cooling rate near the bottom spool, as shown in Fig. 7.

Consequently, uniform deposition of few-layer graphene on Ni foil can be achieved at high web speeds of 50 and 150 mm/min. However, due to the decreased Cu foil temperature with increased web speed (Fig. 3), the catalytic reactions of CH₄ (and other carbon species) limit the production rate of high-quality graphene on Cu [53]. For example, the I_2D/I_G ratio for graphene on Cu is negligible at 50 and 150 mm/min as presented in Table 1. Hence, the improved I_2D/I_G ratio for graphene on Ni is likely due to diffusion/segregation processes that drive the growth on Ni rather than by surface catalysis reactions, which are strongly temperature-dependent. Furthermore, because of the lower diffusion length and higher cooling rate with increased web speed, graphene uniformity on Ni foil is improved at higher web speeds, allowing for higher throughput of graphene deposition than would be possible on Cu foil.

The difference of cooling rates associated with the successful growth of few-layer graphene on Ni listed in Table S1 available in the Supplemental Materials on the ASME Digital Collection can be correlated with substrate thickness as demonstrated in Fig. 8(a). A lower cooling rate (less than 1 K/s) yields graphene films on Ni substrate with thicknesses of less than 10 μm. However, the successful deposition of graphene on thicker Ni (greater than 10 μm) is reported at cooling rates higher than 1 K/s.

The model is used to understand the effects of Ni thickness on the temperature distribution and cooling rate at three thickness values of Ni: 5, 50, and 500 μm in accordance with the previous reports. The other input parameters are the same as in Fig. 3(b) for Ni foil at 50 mm/min. Heat transfer by advection and conduction within the foil is enhanced with increasing thickness (through increasing the cross-sectional area), leading to a higher temperature in the postplasma region with increased thickness (Fig. 8(b)). The foil requires a longer distance to cool to $T∞3$ for the 500 μm-thick Ni foil, whereas the 5 μm-thick foil equilibrates thermally with the ambient gas in the postplasma region (⁠$T∞3$⁠) over a short distance (less than 20 cm after leaving the plasma region).

The cooling rate at various thicknesses is shown in Fig. 8(c). The 5 μm-thick foil cools quickly with a maximum rate of 59.9 K/s at 0.56 m, which then decreases sharply with position until it becomes 0.1 K/s at 0.65 m. Therefore, the reported low cooling rates for thin Ni substrates in Fig. 8(a) could be due to the fact that the substrate temperature decreases sharply after the growth period, as illustrated in Fig. 8(c) for 5 μm-thick foil. The additional slow cooling rate of less than 0.1 K/s for the 5 μm-thick foil is not expected to contribute to graphene synthesis since the initial fast cooling rate can impede carbon from diffusing from the Ni surface to the bulk gas (Fig. 8(c)). Hence, the additional slow cooling rate, which lasts for a longer distance (a surrogate of time in Fig. 8(c)), can minimize the averaged cooling rate of the 5 μm-thick Ni foil.

On the other hand, the cooling rate for the substrate at 500 μm is initially lower than the substrates at 5 and 50 μm. The maximum cooling rate for the 500 μm foil is 2.7 K/s at 0.56 m, gradually decreasing to 0.4 K/s at 0.93 m. The cooling rate becomes higher for the 500 μm foil than the other foils starting at crossover positions of 0.60 and 0.64 m for 5 and 50 μm foils, respectively, as shown in the inset of Fig. 8(c). On average, thicker Ni substrates are cooled at a faster rate to allow graphene growth to avoid the diffusion of carbon back to the bulk gas [15,43]. For instance, graphene sheets were not observed when the cooling rate was 0.1 K/s for Ni substrate of 500 μm thickness [15], whereas graphene films were detected in the same cooling rate range using thinner Ni substrates as summarized in Fig. 8(a). Nevertheless, graphene can be deposited on thin Ni substrates using a higher cooling rate of 10 K/s [17].

The effects of other process conditions such as the gas mixture (particularly the carbon source and its concentration), growth temperature, gas pressure, and growth time play important roles in graphene deposition [16,18,20,40,43,54–58]. These parameters affect the solubility of carbon atoms upon methane (or another carbon source) addition for graphene deposition. For instance, Reina et al. reported the influence of methane concentration on graphene deposition with Ni substrates at various cooling rates [16]. They found that graphene was deposited using methane fractions of 0.5 vol % and 0.6 vol %, whereas a lower methane concentration of 0.4 vol % did not yield graphene even though the same cooling rate and substrate thickness was used for these experiments [16]. Hence, these parameters should be optimized to control the amount of carbon dissolved in Ni to allow successful graphene deposition.

Temperature distributions of Ni and Cu foils during roll-to-roll deposition of graphene have been determined through a combined experimental, analytical, and numerical study. The maximum substrate temperature, which occurs in the high-temperature plasma region, decreases with increasing web speed caused by higher heat advection of the moving foil. Consequently, graphene deposition on Cu foil declines with increased web speed since the growth mechanism is catalytically driven with less contribution from carbon diffusion. The throughput of graphene production can be increased by utilizing Ni as a substrate because carbon solubility in Ni is higher than in Cu. Accordingly, carbon films can be deposited on Ni foil within a variety of thicknesses, from few-layer graphene to thin graphite films, depending on web speed.

The cooling rate of the substrate during the R2R process is derived from basic heat transfer principles, correlating the cooling rate with web speed. Thus, due to high carbon solubility in Ni, graphene can be deposited at higher web speeds because of enhanced cooling rates. Thin Ni foils offered higher temperature with better uniformity and faster cooling rate, allowing for controlled deposition of few-layer graphene films. These findings reveal the importance of using heat transfer analysis to optimize graphene deposition on Ni and other substrates with high carbon solubility in a R2R process. Even though uniform graphene was successfully deposited on Ni at high web speed, the temperature distribution, which strongly depends on web speed, affects the quality of the deposited graphene. Hence, to further improve the throughput deposition of graphene on Ni or Cu, separate temperature-controlled pre- and postgrowth regions could be utilized to regulate heating and cooling rates independently.

We thank Guy Telesnicki, Dr. Anurag Kumar, and the staff of Birck Nanotechnology Center for the support during the experimental setup of the R2R plasma CVD system.

• National Science Foundation, Scalable Nanomanufacturing program (Grant No. CMMI-1344654; Funder ID: 10.13039/100000001).