Tetrahedral Amorphous Carbon Coatings Research Articles

Low cycle fatigue of doped tetrahedral amorphous carbon coatings by repetitive micro scratch tests

In contrast to the conventional progressive scratch testing where load is increased up to a critical damage level, the repetitive scratch testing is an assessment of coating behavior under repeatedly applied subcritical load. It often represents a better approximation to actual application conditions of well-designed coating systems where overcritical conditions typically do not occur. In this work, the cyclic micro scratch behavior of differently doped amorphous carbon coatings with similar coating thicknesses were studied using repetitive scratch tests at loads that correspond to 40 %, 60 % and 80 % of the critical loads from progressive scratch testing. Depending on the doping element, the structural and mechanical properties changed, in some cases considerably, which also had a clear effect on the test results. The damage progression was examined using optical microscopy, friction coefficient, and residual depth measurements. The progression of failure phenomena was categorized into different stages. Furthermore, the critical number of cycles for coating failure was recorded at the various load levels. For the identification of the critical cycles, the use of the deviation of the friction coefficient from its mean value has proven to be a valuable parameter. The approach of testing at different load levels allows the creation of low cycle fatigue curves to evaluate the coating behavior. It was found that the hardness and residual stresses of the coatings had different effects on the lifetime of the coatings at the various load levels, i.e. higher hardness and residual compressive stresses were only beneficial for moderate stress levels with low plastic substrate deformation. The results demonstrate that the cyclic scratch test is a powerful tool for comparative damage assessment at different load scenarios.

Enhanced high-temperature tribological performance of fluorinated tetrahedral amorphous carbon (ta-C:F) coatings in sliding applications

This study investigated the dry sliding behaviour of fluorinated tetrahedral amorphous carbon (ta-C:F) coatings against uncoated 52100 steel at temperatures ranging from 25 °C to 300 °C. The ta-C:F coatings demonstrated significant reductions in both the coefficient of friction(COF) during the running-in stage and at steady state, particularly within the temperature range of 25 °C to 200 °C, surpassing the performance of well-established a-C:H coatings. SEM analyses revealed the formation of transfer layers on the contact surface of 52100 steel when tested against ta-C:F coatings at temperatures up to 200 °C, while none were detected at temperatures ≥250 °C. Raman spectroscopy indicated a transition from sp3 to sp2 carbon structures in the carbonaceous transfer layers with increasing temperature, and XPS scans confirmed an increase in fluorine (F) concentration within these layers, correlating with reduced COF. The comparative analysis at 120 °C emphasized the intrinsic advantages of ta-C:F coatings in high-temperature applications, demonstrating a nearly 50% lower COF (0.08) when compared to traditional boundary-lubricated steel-to-steel sliding contacts. These findings have significant implications for enhancing the efficiency and durability of various mechanical systems, particularly in industries like automotive and manufacturing.

The deposition properties of tetrahedral amorphous carbon coatings deposited on piston ring: Molecular dynamics simulation

The tetrahedral amorphous carbon (ta-C) coatings are deposited on piston rings to improve the tribological property of the piston ring-cylinder liner system of the internal combustion engines. The deposition parameters are optimized by molecular dynamics simulation to reduce the cost of coatings’ fabrication. The ta-C coatings with higher sp3 fraction, lower friction coefficient, and superior anti-wear properties are achieved by optimizing the incident energy and substrate temperature of carbon atoms. The second nearest-neighbor modified embedded-atom method potential and Tersoff potential are used to describe the interatomic interactions. The effects of the incident energy of the carbon atoms and substrate temperature on the deposition properties of the ta-C coatings are discussed. The numerical results show that the ta-C coatings with high sp3 fraction, high density, and good interface mixing are obtained, and the deposition properties of the ta-C coatings are improved.

Effect of PAO-4 and MoDTC + ZnDTP oil additive lubricants on the tribological performance of novel tetrahedral amorphous carbon (ta-C) coatings

This study was carried out to assess the effect of PAO-4 and mineral oil mixed with MoDTC and ZnDTP on the tribological performance of novel tetrahedral-amorphous-carbon (ta-C) coatings namely P:ta-C, and M:ta-C, were analysed using the Pin-on-Disk Tribo-tester with a fixed normal load, temperature, sliding distance and speed of 20 N, 80 °C, 250 m and 0.068 ms−1 for 60 min, respectively. The C:ta-C, P:ta-C, and M:ta-C showed excellent wear resistance when lubricated with MoDTC + ZnDTP. The formation of weak Zn-derived tribofilm on the P:ta-C coating contributed to the strong wear resistance and reduced friction coefficient. Moreover, the passive transfer of ferrous compounds from the SUJ2 disk induced the development of weak Zn-derived tribofilm on the P:ta-C and M:ta-C.

Tetrahedral Amorphous Carbon Coatings with Al Incorporation Deposited by a Hybrid Technique of Sputtering and Arc Evaporation

In this paper, tetrahedral amorphous carbon (ta-C) coatings containing Al were deposited by a hybrid technique of sputtering and arc evaporation. The influence of Al incorporation in the structure and properties of the ta-C coatings were studied as a function of the Al concentration. It is found that Al tends to form a Al-O-C bond when the Al concentration is small. An Al-C bond was detected when the Al concentration is high. Al can facilitate the graphitization of the ta-C coatings and the graphite cluster size as well as the sp2/sp3 ratio of the coatings increase as the Al concentration increases. The decline of the sp3 fraction causes the drop in the hardness of the coatings. The incorporation of Al can effectively decrease the residual stress of the ta-C coatings. During friction tests, Al can facilitate the formation of the sp2-rich graphitic tribo-layer and decrease the friction coefficient. Nevertheless, the decline of the hardness due to the Al incorporation will result in the increase in the wear rate of the coating. It is believed that the ta-C coating with a proper concentration of Al appears to achieve a good comprehensive performance with high hardness, low residual stress, and a low friction coefficient and wear rate.

Diamond-like films of tetrahedral amorphous carbon deposited by anodic arc evaporation of graphite

A physical vapor deposition process using anodic arc evaporation in combination with a hollow cathode arc discharge was applied to the evaporation of graphite for deposition of hydrogen-free carbon layers. The diamond-like carbon (DLC) films deposited on 100Cr6 steel substrates were investigated by nanoindentation, Raman spectrometry, FE-SEM, AFM and spectroscopic ellipsometry. The relationships between the process parameters and the coating properties are discussed. Coatings deposited without bias voltage at substrate temperatures <200 °C are very hard (61–75 GPa) with also very high Young's modulus (588–685 GPa). The evaluation of the Raman spectra indicated a high proportion of tetrahedral sp3 bonds in the range of 70–88 %. Obviously, the vapor particle energies are high enough to achieve such high hardness values even without application of a bias voltage. The coatings proved to be completely droplet-free and have a very low surface roughness as confirmed by FE-SEM and AFM. The deposition rates in the range of 4–18 nm/s are exceptionally high for such tetrahedral amorphous carbon (ta-C) coatings, which is a good prerequisite for industrial applications. The ta-C layers with very high hardness and smooth surface are well suited as wear resistant layers.

Controlling adhesion and thermal stability of tetrahedral amorphous carbon coatings with intermediate sp3C fraction

Tetrahedral amorphous carbon (ta-C) coatings with an intermediate sp3C fraction were deposited on single-crystal silicon wafer substrates via intermittent filtered cathode vacuum arc (FCVA) deposition. The ta-C layer thickness and sp3C fraction in the coatings were adjusted using the deposition duration and bias voltage applied to the substrate, respectively. Some of the coated specimens were annealed at 300 and 600 °C for 4 h in a 5-Pa vacuum. The morphologies and microstructures of the specimens were characterized using scanning electron microscopy and transmission electron microscopy. The hybridization of carbon atoms was analyzed using X-ray photoelectron spectroscopy and micro-Raman spectroscopy. The mechanical characteristics of the coatings, including the critical adhesion load between the coatings and substrates, residual stress in the coatings, and hardness, were characterized. For the specimens deposited by the same bias voltage, even their top ta-C layer thickness ranged from approximately 500 nm to 1500 nm, their residual compressive stress in the coatings was maintained at a low value by the intermittent deposition of FCVA, which varied slightly with the coating thickness. The adhesion strength was improved by increasing the thickness of the as-deposited coating from approximately 12 to 40 N. The residual compressive stress of the post-annealing specimens with sp3C fraction of 45–53.8 % exhibited monotonously increasing or V-shaped variation with increasing annealing temperature. Herein, the underlying mechanisms are discussed in detail.

Tribological behavior of different doped ta-C coatings for slip-rolling contacts with high hertzian contact pressure

PurposeThe curved and tribologically highly stressed surfaces of bearing components pose a major challenge for steel alloys or tribological resistant coatings like tetrahedral amorphous carbon (ta-C) coatings which in particular have an increased risk of delamination due to the significantly increased residual stresses. A possibility to prevent coating failure is the use of dopants while maintaining or even increasing tribological properties. This study aims to compare the tribological behavior of several doped diamond-like-carbon coatings with an undoped ta-C coating under varying slip conditions and Hertzian pressure up to 1800 MPa.Design/methodology/approachFor this purpose, the tribological behavior was studied using of a ball-on-disc tribometer and a two-disc test rig under mixed/boundary conditions. The tests were conducted with coated specimens against uncoated 100Cr6 steel. Additionally, the influence of lubrication additives was studied due to the use of two fully formulated PAO-based oils, one without and one with molybdenum containing additives. The friction was measured in situ, and the wear was analyzed trough laser scanning microscopy and tactile measurement.FindingsIt was shown that the use of doped ta-C coatings exhibited a tendency for a more favorable tribological behavior compared to undoped ta-C coatings, with no general dependence on the lubricants used. The use of the most suitable coatings reduced the wear of the steel counter-body considerably.Originality/valueTo the best of the authors’ knowledge, this is the first approach of testing the tribological behavior of these doped ta-C coatings, developed for friction efficiency, in dependency on lubrication additives under the given load collective. The approach is relevant to determine whether the friction reduction and the wear inhibition of these coatings are suitable for higher contact pressures and load cycles.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-11-2022-0336/

Microstructure evolution and mechanical performance of tetrahedral amorphous carbon coatings with dense droplets during annealing

Tetrahedral amorphous carbon (ta-C) coatings with approximately 50% sp3C were deposited onto cemented carbide alloy YG6 and Si wafers via direct vacuum cathode arc (DVCA) evaporation at different substrates temperatures. Coated specimens were annealed in a 5 Pa argon atmosphere. The micromorphology and chemical bonding structure of these coatings with different treatments were characterized using field emission scanning electron microscopy (FE-SEM), Raman spectroscopy, and X-ray photoelectron spectroscopy (XPS). The mechanical performance was tested using a nanoindenter and film stress instrument. The obtained ta-C coatings contained dense droplets of various sizes (“abnormal particles” (An-Ps)). The densities of the An-Ps were controlled by the graphite cathode target arc rather than the substrate temperature, and their ID/IG ratios were close to their counterpart coating matrix. Increasing the substrate temperature was beneficial for increasing the sp2C fraction and the number of aromatic ring sp2C sites. Growth, coalescence, and leveling-diffusion of aggregates in the coating matrix and An-Ps occurred during annealing, which was related to the existing state of the sp2C sites, including the olefinic chain and aromatic ring. The residual stress variation with annealing temperature also depended on the sp2C state. The residual stress in the coating increased during annealing when the number of aromatic ring sp2C sites exceeded a certain critical value.

Wear Mechanism of Superhard Tetrahedral Amorphous Carbon (ta‐C) Coatings for Biomedical Applications

AbstractTetrahedral amorphous carbon (ta‐C) coatings have the potential to protect biomedical implants from wear and increase their service life. This study elucidates the biocompatibility, mechanical properties, adhesion, and wear resistance of ta‐C coatings fabricated by physical vapor deposition on cobalt‐chromium‐molybdenum (CoCr) and titanium (Ti64) alloys as well as ultrahigh molecular weight polyethylene (UHMWPE). Satisfactory cytocompatibility is verified using contact angle and surface tension measurements as well as indirect and direct cell testing. Scratch testing demonstrates excellent adhesion to the substrates and as confirmed by nanoindentation, the coatings represent an up to 13‐fold and 182‐fold increase in hardness on the hard and soft materials. In metal pin‐on‐UHMWPE disk sliding experiments under simulated body fluid lubrication, the wear rates of the disk are reduced by 48% (against CoCr) and 73% (against Ti64) while the pin wear rates are reduced by factors of 20 (CoCr) and 116 (Ti64) compared to uncoated pairings. From optical and laser scanning microscopy, Raman measurements, and particle analyses, it is shown that the underlying substrates remain well protected. Nonetheless, focused ion beam scanning electron microscopy revealed coating process‐related and thermally driven subductions as well as tribologically induced near‐surface fatigue, which can potentially constitute critical wear mechanisms.

Effect of doping elements to hydrogen-free amorphous carbon coatings on structure and mechanical properties with special focus on crack resistance

The doping of amorphous carbon is one way of specifically modifying its properties. The present work is a contribution to understand the effect of dopant selection on structure, mechanical properties and crack resistance of hydrogen-free tetrahedral amorphous carbon (ta-C) coatings. For this purpose, nonmetallic (B, Si) and metallic (Mo, Fe, Cu) dopants were selected. The coatings were produced by the Laser-Arc technique and investigated by Raman spectroscopy, transmission electron microscopy, nanoindentation, bending, and scratch tests. It was found that doping generally affected the sp2 content and cluster size of the sp2 carbon phase, and thus hardness, Young's modulus, and intrinsic compressive stresses, but also the defect content and the crack resistance. The addition of metallic dopants led to an increase and clustering of the sp2-bonded carbon and, consequently, to a significant reduction in hardness and Young's modulus. This effect was much less pronounced when doping with non-metallic elements. The failure behavior changed toward higher ductility and crack resistance when metals were added. The crack resistance was found to be directly dependent on the H3/E2 value, with H3/E2 > 0.2 leading to a more brittle behavior with unstable crack propagation, while at H3/E2 < 0.2 the material was more ductile and exhibited stable crack propagation and thus higher crack resistance.

Friction Response of Piston Rings for Application-like Starvation and Benefit of Amorphous Carbon Coatings

The oil supply at the interface between the top ring and the cylinder liner (TRCL) plays a major role in an internal combustion engines efficiency. In particular, the interface forms a trade-off between the serving of enough lubricant for sufficient lubrication conditions and emissions through subsequent combustion. This can lead to deficient top ring lubrication conditions. In this study, a new developed reciprocating long-stroke tribometer, enabling the variation of oil supply, is used to investigate such application-like starved lubrication conditions of the TRCL interface. With the simulative investigations, a comparison with the fired engine is possible. The performance of diamond-like carbon coatings is compared to standard nitrided piston rings. It was found that the tetrahedral amorphous carbon (ta-C) coatings exhibit up to 31% reduced friction as well as a lower wear under starved lubrication conditions. Simulative investigations show a good correlation between engine friction and tribometer measurements for selected oil supply conditions.

Low-Friction of ta-C Coatings Paired with Brass and Other Materials under Vacuum and Atmospheric Conditions.

Vacuum environments provide challenging conditions for tribological systems. MoS2 is one of the materials commonly known to provide low friction for both ambient and vacuum conditions. However, it also exhibits poor wear resistance and low ability to withstand higher contact pressures. In search of wear-resistant alternatives, superhard hydrogen-free tetrahedral amorphous carbon coatings (ta-C) are explored in this study. Although known to have excellent friction and wear properties in ambient atmospheres, their vacuum performance is limited when self-paired and with steel. In this study, the influence of the paired material on the friction behavior of ta-C is studied using counterbodies made from brass, bronze, copper, silicon carbide, and aluminum oxide, as well as from steel and ta-C coatings as reference materials. Brass was found to be the most promising counterbody material and was further tested in direct comparison to steel, as well as in long-term performance experiments. It was shown that the brass/ta-C friction pair exhibits low friction (µ < 0.1) and high wear in the short term, irrespective of ambient pressure, whereas in the long term, the friction coefficient increases due to a change in the wear mechanism. Al2O3 was identified as another promising sliding partner against ta-C, with a higher friction coefficient than that of brass (µ = 0.3), but considerably lower wear. All other pairings exhibited high friction, high wear, or both.

Indentation modulus extrapolation and thickness estimation of ta-C coatings from nanoindentation

Coatings used in tribological applications often exhibit high hardness and stiffness to achieve high wear resistance. One coating characterization method frequently used is nanoindentation which allows the determination of indentation hardness and indentation modulus among other material properties. The indentation modulus describes the elastic surface behavior during indentation and is, among hardness, a direct indicator for wear resistance. To obtain the true indentation modulus of a coating, it must be measured with varying loads and then extrapolated to zero load. Current recommendation of the standard ISO 14577-4:2016 is a linear extrapolation which fits poorly for nonlinear curves. Such nonlinear curves are commonly found for high hardness mismatches between coating and substrate, for example, superhard tetrahedral amorphous carbon coatings (ta-C) on a steel substrate. In this study, we present a new empirical fit model, henceforth named sigmoid. This fit model is compared to several existing fit models described in the literature using a large number of nanoindentation measurements on ta-C coatings with wide ranges of indentation modulus and coating thickness. This is done by employing a user-independent and model agnostic fitting methodology. It is shown that the sigmoid model outperforms all other models in the combination of goodness of fit and stability of fit. Furthermore, we demonstrate that the sigmoid model’s fit parameter directly correlates with coating thickness and thus allows for a new approach of determining ta-C coating thickness from nanoindentation.

Effect of Energy and Temperature on Tetrahedral Amorphous Carbon Coatings Deposited by Filtered Laser-Arc.

In this study, both the plasma process of filtered laser-arc evaporation and the resulting properties of tetrahedral amorphous carbon coatings are investigated. The energy distribution of the plasma species and the arc spot dynamics during the arc evaporation are described. Different ta-C coatings are synthesized by varying the bias pulse time and temperature during deposition. An increase in hardness was observed with the increased overlapping of the bias and arc pulse times. External heating resulted in a significant loss of hardness. A strong discrepancy between the in-plane properties and the properties in the film normal direction was detected specifically for a medium temperature of 120 °C during deposition. Investigations using electron microscopy revealed that this strong anisotropy can be explained by the formation of nanocrystalline graphite areas and their orientation toward the film’s normal direction. This novel coating type differs from standard amorphous a-C and ta-C coatings and offers new possibilities for superior mechanical behavior due to its combination of a high hardness and low in-plane Young’s Modulus.

Boride-Carbon Hybrid Technology for Ultra-Wear and Corrosive Conditions

This work discusses a study on a surface treatment for creating extremely durable low-friction, wear and corrosion-resistant surfaces for tribological components in harsh conditions. A duplex surface treatment was developed that combines the advantages of ultra-fast electrochemical boriding with those of hard tetrahedral amorphous carbon coatings. The friction and wear properties of the duplex treatment are compared to the boride-only treatment of AISI 1045 steel, while corrosion and contact fatigue behaviors of the duplex layer are compared to that of the single-layer carbon coating on low carbon steel. The duplex treatment yields wear rates as low as 6 × 10−8 mm3·N−1·m−1 and a coefficient of friction of 0.14 when tested against a steel counter face. The contact fatigue impact tests reveal that the high hardness of 1200 HV0.05 of the borided layer in the duplex treatment leads to higher resistance against indentation but is accompanied by a higher incidence of crack initiation, being in good agreement with the finite-element modeling of nanoindentation results. The duplex coatings exhibit resistance to pinhole corrosion as evidenced by a 3 h exposure to 15% HCl at room temperature.

Nanotextured Mold Surface with DLC Coating for Reduction in Residual Ceramic Particles.

In the ceramic industry, ceramic particles remain on a mold surface due to which the mold requires frequent cleaning during press molding, reducing productivity. Surface texturing and tetrahedral amorphous carbon (ta-C) coatings are well-known surface-energy controllable treatments developed for low adhesion, low friction, and high wear resistance. In the present paper, we demonstrate the effect of reducing ceramic residues using nanotexturing, ta-C coatings, and their combination. We compare two surface morphologies (i.e., 770 nm pitch nanotexturing and flat) and five materials (i.e., nonhardened steel, hardened steel, ta-C, and two types of nitrogen-doped ta-C (ta-CNx). Molding test results show that the ta-C coating on flat surfaces with the highest hardness of 30 GPa shows the lowest residual amount of 5.9 μg for Al2O3 ceramic particles. The amount is 82% less than that of the nonhardened steel. The ta-CNx20, made with a nitrogen flow rate of 20 sccm, shows the lowest residual amount of 234 μg for SiO2 ceramic particles, which is 81% less than that of the nontextured ta-CNx20. In conclusion, we provide design guidelines for nanotextured mold surfaces including the texturing pitch should be small enough for ceramic particles; the mold surface should be sufficiently hard; the lower the surface energy per unit area, the less residues of ceramic particles.

Friction properties of thick tetrahedral amorphous carbon coating with different surface defects under dry contact conditions

Surface defects of tetrahedral amorphous carbon coatings are typically defined as spikes, droplets, and pores. In this study, a hybrid filtering method using a fixed physical and electrical filter (duct bias voltage) in a filtered cathodic vacuum arc system was used to control the surface defects. The introduction of the duct bias can control to trap the negatively charged macro-particles which is transported to the base material due to a floating potential and duct wall sheath that depend on the positive voltage. The applied positive duct bias voltage was 10, 15, 20, and 30 V. The results of this experiment revealed that as the duct bias voltage increased up to 30 V, the spikes and droplets were almost completely removed on the tetrahedral amorphous carbon surface and only the pores remained. The pores could not create a rough surface for the relative motion between interfaces due to which the friction coefficient increased as the duct bias voltage increased up to 30 V. It was concluded that the spike and droplet defects were positive elements and were required to introduce a carbon-based transfer interface and maintain a stable coefficient of friction.

Tribological properties of multilayer tetrahedral amorphous carbon coatings deposited by filtered cathodic vacuum arc deposition

Tetrahedral amorphous carbon (ta-C) has emerged as an excellent coating material for improving the reliability of application components under high normal loads. Herein, we present the results of our investigations regarding the mechanical and tribological properties of a 2-µm-thick multilayer ta-C coating on high-speed steel substrates. Multilayers composed of alternating soft and hard layers are fabricated using filtered a cathodic vacuum arc with alternating substrate bias voltages (0 and 100 V or 0 and 150 V). The thickness ratio is discovered to be 1:3 for the sp2-rich and sp3-rich layers. The results show that the hardness and elastic modulus of the multilayer ta-C coatings increase with the sp3 content of the hard layer. The hardness reached approximately 37 GPa, whereas an improved toughness and a higher adhesion strength (> 29 N) are obtained. The friction performance (µ = 0.07) of the multilayer coating is similar to that of the single layer ta-C thick coating, but the wear rate (0.13 × 10−6 mm3/(N·m)) improved under a high load of 30 N. We further demonstrate the importance of the multilayer structure in suppressing crack propagation and increasing the resistance to plastic deformation (H3/E2) ratio.

New Droplet Removal Polishing Method for Diamond-Like Carbon with Carbon Fiber Brush

In this paper, we propose a new polishing method for diamond-like carbon (DLC) coatings using a carbon fiber brush (CFB). Surface finishing is an important process for DLC coating applications. A lapping process is widely used for attaining tetrahedral amorphous carbon (ta-C) coatings, which are a type of DLC coating containing many droplets, to obtain fine flat surfaces. The lapping process removes protuberant parts of droplets rather than the entire droplet. In this paper, we propose a new polish brush material made of carbon fiber, called CFB. Carbon fiber has both mechanical strength due to its hard carbonaceous material and flexibility due to its fiber structure. In polishing tests, CFB removed droplets from ta-C coatings and the removal effect increased with the shortening of the brush length. The surface profiles of the polished surfaces indicated that a shorter brush length yielded deep scratch marks on ta-C surfaces. Consequently, the arithmetic average surface roughness of the polished ta-C surfaces, Sa, had almost the same value as that of a non-polished surface. Here, we show the ability of CFB to remove the droplets without an increase in the surface roughness. The CFB with the longest brush length in the present study (12 mm) showed a ten-point average roughness SZJIS= 75 nm and Sa= 4.7 nm, which were 59% and 22% lower than those of the non-polished surface, respectively. Furthermore, the longest CFB removed the entire droplets whereas a shorter CFB merely removed the protuberant part of the droplets. The result indicates that CFB polishing can remove entire droplets, which result in abrasive wear or deterioration. From other polishing tests, the optimum polishing distance was determined. Shorter polishing distances could not remove droplets sufficiently whereas longer polishing distances caused deep scratches on ta-C surfaces due to the material transferred to the CFB. Accordingly, the polishing distance of 600 m showed the best surface finishing with SZJIN= 25 nm and Ra= 0.43 nm, which were 86% lower than and similar to those of the non-polished ta-C surface, respectively.