Coating method

Abstract

In a coating method according to a non-limiting aspect of the present disclosure, a surface of a tool is coated with a coating layer by means of physical vapor deposition, while the tool is revolved around a revolution axis while the tool is also rotated around a rotation axis parallel to the revolution axis, and a revolution table for revolving the tool and a rotation table for rotating the tool on the revolution table are used. When the rotational speed of the revolution table is defined as a (rpm) and the rotational speed of the rotation table is defined as b (rpm), the ratio b / a of a and b is larger than 2 and is not an integer.

Description

Coating Method

[0001] The present disclosure relates to coating methods.

[0002] Physical vapor deposition (PVD) is a known method for coating a surface of a tool (substrate) with a coating layer. PVD processing equipment that performs coating using the PVD method is also known.

[0003] For example, International Publication No. 2014 / 147979 (Patent Document 1) and Japanese Patent Application Laid-Open No. 2014-181385 (Patent Document 2) describe a PVD processing apparatus that includes a vacuum chamber that accommodates a plurality of substrates, a revolution table that is provided within the vacuum chamber and that supports the plurality of substrates while revolving these substrates around an axis of revolution, a plurality of rotation tables that support each of the plurality of substrates and rotate the substrate on the revolution table around an axis of rotation that is parallel to the axis of revolution, a plurality of targets formed from different types of film formation materials, and a table rotation mechanism that rotates each rotation table around its axis of rotation as the revolution table rotates.

[0004] A non-limiting aspect of the present disclosure is a coating method for coating a coating layer on a surface of a tool by physical vapor deposition while revolving the tool around a revolution axis and rotating the tool around a rotation axis parallel to the revolution axis, the coating method using a revolution table for revolving the tool and a rotation table for rotating the tool on the revolution table, where a rotation speed of the revolution table is a (rpm) and b rotation speed of the rotation table is b (rpm), and the ratio b / a of a and b is greater than 2 and is not an integer.

[0005] FIG. 2 is a plan view showing a tool rotating device (PVD processing device) used in the non-limiting method for coating one surface of the present disclosure; FIG. 3 is a side view of the tool rotating device shown in FIG. 1; FIG. 4 is a schematic view showing the positional relationship between a revolution table and a rotation table in the non-limiting method for coating one surface of the present disclosure; FIG. 5 is a schematic view showing the positional relationship between a revolution table and a rotation table in the non-limiting method for coating one surface of the present disclosure; FIG. 6 is a schematic view showing the positional relationship between a revolution table and a rotation table in the non-limiting method for coating one surface of the present disclosure; FIG. 7 is a schematic view showing the positional relationship between a revolution table and a rotation table in the non-limiting method for coating one surface of the present disclosure; FIG. 8 is a schematic view showing the positional relationship between a revolution table and a rotation table in the non-limiting method for coating one surface of the present disclosure; FIG. 9 is a schematic view showing the positional relationship between a revolution table and a rotation table in the non-limiting method for coating one surface of the present disclosure; FIG. 1 is a schematic diagram showing the positional relationship between a revolution table and a rotation table in a non-limiting one-sided coating method of the present disclosure. FIG. 2 is a schematic diagram showing the positional relationship between a revolution table and a rotation table in a non-limiting one-sided coating method of the present disclosure. FIG. 3 is a schematic diagram showing the positional relationship between a revolution table and a rotation table in a non-limiting one-sided coating method of the present disclosure. FIG. 4 is a schematic diagram showing the positional relationship between a revolution table and a rotation table in a non-limiting one-sided coating method of the present disclosure. FIG. 5 is a schematic diagram showing the positional relationship between a revolution table and a rotation table in a non-limiting one-sided coating method of the present disclosure.

[0006] The non-limiting one-surface coating method of the present disclosure will be described in detail below with reference to the drawings. However, for the sake of convenience, the figures referred to below show only the main components necessary for explaining the embodiment in a simplified form. Furthermore, the dimensions of the components in the figures do not faithfully represent the dimensions of the actual components or the dimensional ratios of each component. Note that, for ease of visual understanding, the rotating table and other components that overlap with the revolution axis when viewed from the side are omitted in Figure 2.

[0007] The coating method may be a method of coating a coating layer on the surface of the tool 301 (see FIG. 2). The tool 301 coated with a coating layer on its surface may also be called a coated tool. Note that coating may also be called film formation. The coating method may also be called film formation method. The coating layer may also be called a coating. The coating layer may also be called simply a film.

[0008] The tool 301 to be coated may be plate-shaped. For example, the tool 301 may be in the shape of a square plate. However, the shape of the tool 301 is not limited to a square plate. For example, the top surface of the tool 301 may be triangular, pentagonal, hexagonal, or circular.

[0009] The tool 301 may be a cutting tool. The tool 301 may also have a through hole. When the tool 301 is a cutting tool, the through hole can function as a portion to which a fixing screw, a clamp member, or the like is attached when the cutting tool is held in a holder.

[0010] The tool 301 is not limited to a specific size. For example, the length of one side of the upper surface may be set to about 3 to 20 mm. The height from the upper surface to the lower surface may be set to about 5 to 20 mm. The tool 301 is not limited to being plate-shaped. The tool 301 may be rod-shaped or the like.

[0011] Examples of materials for the tool 301 include cemented carbide and cermet. Examples of cemented carbide compositions include WC-Co, WC-TiC-Co, and WC-TiC-TaC-Co. WC, TiC, and TaC may be hard particles, and Co may be a binder phase. Cermet may be a sintered composite material in which a ceramic component is combined with a metal. An example of a cermet is a titanium compound mainly composed of TiC or TiN.

[0012] The number of tools 301 to be coated may be, for example, about 5 to 80. The number of tools 301 exemplified may be the number per support pillar, which will be described later.

[0013] The composition of the coating layer coated on the surface of the tool 301 may include, for example, titanium carbide (TiC), titanium nitride (TiN), titanium carbonitride (TiCN), and alumina (Al2O3).

[0014] The coating layer is not limited to a specific thickness. For example, the average thickness of the coating layer may be set to about 0.1 to 10 μm. The thickness of the coating layer may be measured by cross-sectional observation using an electron microscope. For example, the thickness may be measured at 10 or more measurement points at any position on the coating layer, and the average value may be calculated. Examples of electron microscopes include a scanning electron microscope (SEM) and a transmission electron microscope (TEM).

[0015] 1 and 2, the coating method may be a method in which the tool 301 is revolved around the revolution axis O1 while being rotated around the rotation axis O2 parallel to the revolution axis O1, and a coating layer is coated on the surface of the tool 301 by a PVD method. Examples of the PVD method include an ion plating method and a sputtering method.

[0016] In the coating method, a revolution table 1 and a rotation table 3 may be used.

[0017] The revolution table 1 may be a member that revolves the tool 301 around the revolution axis O1. Furthermore, the central axis of the revolution table 1 may coincide with the revolution axis O1. The revolution table 1 is rotatable around the revolution axis O1. The arrow Y1 in FIG. 1 and other figures may indicate the direction of rotation of the revolution table 1. Note that the revolution table 1 may rotate in the opposite direction to the arrow Y1.

[0018] A motor and gears may be used to rotate the revolution table 1. That is, the driving force of the motor may be transmitted to the revolution table 1 via gears to rotate the revolution table 1. This also applies to other rotatable members such as the rotation table 3 described below.

[0019] The revolution table 1 may be disk-shaped, and the outer diameter of the revolution table 1 may be set to, for example, about 300 to 600 mm.

[0020] The turning table 3 may be a member that rotates the tool 301 on the revolution table 1 about a rotation axis O2 that is parallel to the revolution axis O1. The turning table 3 is also rotatable about a central axis O3 of the turning table 3. The central axis O3 of the turning table 3 may be parallel to the revolution axis O1. The arrow Y2 in FIG. 1 and other figures may indicate the rotation direction of the turning table 3. The turning table 3 may also rotate in the direction opposite to the arrow Y2.

[0021] 2, the rotation table 3 may have a plate-shaped main body 5 and a top plate 7 facing the main body 5 across an interval in the direction along the central axis O3. The main body 5 and the top plate 7 may be disk-shaped with an outer diameter smaller than that of the revolution table 1. The outer diameters of the main body 5 and the top plate 7 may be set to, for example, about 100 to 150 mm.

[0022] There may be multiple rotation tables 3. The multiple rotation tables 3 may be located on the outer periphery of the revolution axis O1 on the top surface 9 of the revolution table 1, as in the non-limiting example shown in Figure 1. The multiple rotation tables 3 may be located at equal intervals in the circumferential direction of the revolution axis O1. When there are multiple rotation tables 3, the number of rotation tables 3 may be, for example, about 2 to 10.

[0023] Here, the rotation speed of the revolution table 1 may be a (rpm), and the rotation speed of the rotation table 3 may be b (rpm). The rotation speed may also be referred to as the number of rotations.

[0024] In this case, if the ratio b / a of a and b is not an integer, in other words, if the values ​​of a and b are not an integer ratio, the uniformity of the coated film is likely to be improved. This is because when the rotation of the revolution table 1 brings the tool 301 close to the target in the PVD processing apparatus described below, it is easy to avoid the same surface of the tool 301 facing the target.

[0025] In order to avoid the same surface of the tool 301 facing the target, b = m a + n + δ (m, nεN, m≠0, 0<δ<1), and theoretically δ may be an irrational number. Note that, since it is practically difficult to set δ to an irrational number in the strict sense in the coating process, δ may be set to a number with two or more decimal places.

[0026] The ratio b / a of a and b may be 0<b / a<1, or may be 1<b / a. Since the driving force required to rotate the revolution table 1 is greater than the driving force required to rotate the rotation table 3, it is easy to suppress the power consumption required for the coating process when 1<b / a. From the perspective of improving the uniformity of the film coated during one rotation of the revolution table 1, it is desirable that b is greater than twice a, that is, 2<b / a. Therefore, when the ratio b / a of a and b is greater than 2 and is not an integer, excellent coatability is achieved.

[0027] The uniformity of the coated film may be evaluated as follows. First, the film thickness of the film coated on the upper surface of the tool 301 is measured. The location for measuring the film thickness is set as follows, depending on the shape of the upper surface of the tool 301. If the upper surface of the tool 301 is polygonal, measurements are taken at locations near at least three vertices. If the upper surface of the tool 301 is circular, measurements are taken at locations near the tangent points on the circumference obtained by dividing the circumference into thirds. The nearby locations may be locations 0.5 mm away from the vertices or tangent points. Next, the maximum value of the measured film thickness is defined as X, the minimum value as Y, and the average value as Z. Then, the film may be evaluated as uniform if the value calculated by applying X, Y, and Z to the formula: [[(X-Y) / Z] / 2] x 100 does not exceed 25%.

[0028] The relationship between a and b may be such that b<5a, in which case the uniformity of the coated film is likely to be improved.

[0029] For example, a may be set to 0.5 to 10 rpm, and b may be set to 8 to 20 rpm. The settings of a and b may be made by adjusting the gear ratio, etc.

[0030] As shown in a non-limiting example in Figure 3, a reference position S1 may be set for the positions of the revolution table 1 and the rotation table 3 at the start of rotation. As shown in a non-limiting example in Figure 4, the deviation of the rotation table 3 from the reference position S1 when the revolution table 1 has made one rotation may be set to α°. α° may be in the range of 10°≦α≦350°. In this case, the uniformity of the coated film is likely to be improved. Specifically, it is likely to be avoided that the thickness of the film coated on the outward-facing surface of the tool 301 at the reference position S1 is excessively thick compared to other parts.

[0031] 4 shows a case where α=30° as a non-limiting example. Furthermore, the reference position S1 may be a position facing a target in a PVD processing apparatus, which will be described later.

[0032] The coating method may use a tool rotation device 101. The tool rotation device 101 may include a revolution table 1 and a rotation table 3, as shown in a non-limiting example in FIGS.

[0033] As described above, the turning table 3 is rotatable around the central axis O3 of the turning table 3. The turning table 3 may also have a support 11, as shown in a non-limiting example in FIG. 2 . The support 11 is capable of holding the tool 301. If the tool 301 has the above-described through-hole, the support 11 may be inserted into the through-hole to hold the tool 301 on the support 11. If the tool 301 is rod-shaped, the tool 301 may be held on the support 11 using a jig or the like.

[0034] The support pillar 11 may be located between the main body 5 and the top plate 7. The support pillar 11 may also be cylindrical. For example, the diameter of the support pillar 11 may be set to about 1 to 10 mm. The length of the support pillar 11 may be set to about 200 to 700 mm.

[0035] There may be a plurality of support columns 11. That is, the turntable 3 may have a plurality of support columns 11. The plurality of support columns 11 may be positioned at equal intervals in the circumferential direction of the central axis O3.

[0036] The rotation axis O2 may be located closer to the outer periphery of the rotation table 3 than the central axis O3. The support column 11 may extend along the rotation axis O2. The support column 11 is rotatable around the rotation axis O2.

[0037] For example, the main body 5 and the top plate 7 may have a through-hole through which the end of the support column 11 can be inserted, or a recess into which the end of the support column 11 can be inserted, and the end of the support column 11 may be positioned in this through-hole or recess to rotate the support column 11. Alternatively, the drive force of a motor may be transmitted to the support column 11 via a gear to rotate the support column 11. Alternatively, the drive force may be transmitted to the support column 11 via a gear and a kicker to rotate the support column 11. The arrow Y3 in FIG. 2 may indicate the rotation direction of the support column 11. The support column 11 may also rotate in the direction opposite to the arrow Y3.

[0038] In the coating method, when the tool rotation device 101 is used, it is easy to rotate and revolve the multiple tools 301 stably.

[0039] Here, when the number of the support columns 11 is n1, α° and n1 may be expressed as α°=360 / n1. In this case, the uniformity of the coated film is likely to be improved.

[0040] The number of support columns 11, n1, may be set to 4 to 36. In the case where there are multiple turntables 3, n1 may be the number of support columns 11 on one turntable 3.

[0041] The coating time may be t (minutes). The value calculated by substituting a and t into the formula: a × t may be an integer multiple of the value calculated by substituting α° into the formula: 360° / α°.

[0042] t may be set to 4 to 240 minutes. Furthermore, the integral multiple does not necessarily have to be an integral multiple in the strict sense. The integral multiple may be an approximate integral multiple. If the value calculated by applying α° to the formula: 360° / α° is within the range of ±0.2 of the integer N1, it may be considered to be an integral multiple.

[0043] The above relationship will be explained using an example where α = 30°. First, the revolution table 1 and the rotation table 3 may start rotating from the non-limiting example state shown in Figure 3. Then, the revolution table 1 and the rotation table 3 may sequentially change from the state shown in Figure 3 to the states shown in Figures 4 to 15.

[0044] The non-limiting example shown in Figure 4 may show a state in which the revolution table 1 has made one rotation and returned to its original position (the position shown in Figure 3). At this time, the rotation table 3 may rotate m1. m1 may be set to 4 to 60 rotations.

[0045] The rotation table 3 may also be located at a position 30° ahead of its original position. In other words, when the revolution table 1 rotates 360°, the rotation table 3 may rotate by (360°×m1)+30°.

[0046] The non-limiting example shown in Figure 5 may show a state in which the revolution table 1 has made one more rotation and returned to its original position (a total of two rotations). At this time, the rotation table 3 may make another m1 rotation. The rotation table 3 may also be at a position 60° forward from its original position. The rotation table 3 may also be at a position 30° forward from the position of the rotation table 3 when the revolution table 1 has made one rotation (the position shown in Figure 4).

[0047] The non-limiting example shown in Figure 6 may show a state in which the revolution table 1 has made one more rotation and returned to its original position (a total of three rotations). At this time, the rotation table 3 may make another m1 rotation. The rotation table 3 may also be at a position 90° forward from its original position. The rotation table 3 may also be at a position 30° forward from the position of the rotation table 3 when the revolution table 1 has made two rotations (the position shown in Figure 5).

[0048] The non-limiting example shown in Figure 7 may show a state in which the revolution table 1 has made one more rotation and returned to its original position (a total of four rotations). At this time, the rotation table 3 may make another m1 rotation. The rotation table 3 may also be at a position 120° forward from its original position. The rotation table 3 may also be at a position 30° forward from the position of the rotation table 3 when the revolution table 1 has made three rotations (the position shown in Figure 6).

[0049] The non-limiting example shown in Figure 8 may show a state in which the revolution table 1 has made one more rotation and returned to its original position (a total of five rotations). At this time, the rotation table 3 may rotate an additional m1. The rotation table 3 may also be at a position 150° forward from its original position. The rotation table 3 may also be at a position 30° forward from the position of the rotation table 3 when the revolution table 1 has made four rotations (the position shown in Figure 7).

[0050] The non-limiting example shown in Figure 9 may show a state in which the revolution table 1 has made one more rotation and returned to its original position (a total of six rotations). At this time, the rotation table 3 may rotate an additional m1. The rotation table 3 may also be at a position 180° forward from its original position. The rotation table 3 may also be at a position 30° forward from the position of the rotation table 3 when the revolution table 1 has made five rotations (the position shown in Figure 8).

[0051] The non-limiting example shown in Figure 10 may show a state in which the revolution table 1 has made one more rotation and returned to its original position (a total of seven rotations). At this time, the rotation table 3 may rotate an additional m1. The rotation table 3 may also be at a position 210° forward from its original position. The rotation table 3 may also be at a position 30° forward from the position of the rotation table 3 when the revolution table 1 has made six rotations (the position shown in Figure 9).

[0052] The non-limiting example shown in Figure 11 may show a state in which the revolution table 1 has made one more rotation and returned to its original position (a total of eight rotations). At this time, the rotation table 3 may rotate an additional m1. The rotation table 3 may also be at a position 240° forward from its original position. The rotation table 3 may also be at a position 30° forward from the position (the position shown in Figure 10) of the rotation table 3 when the revolution table 1 has made seven rotations.

[0053] The non-limiting example shown in Figure 12 may show a state in which the revolution table 1 has made one more rotation and returned to its original position (a total of nine rotations). At this time, the rotation table 3 may rotate an additional m1. The rotation table 3 may also be at a position 270° forward from its original position. The rotation table 3 may also be at a position 30° forward from the position of the rotation table 3 when the revolution table 1 has made eight rotations (the position shown in Figure 11).

[0054] The non-limiting example shown in Figure 13 may show a state in which the revolution table 1 has made one more rotation and returned to its original position (a total of 10 rotations). At this time, the rotation table 3 may make another m1 rotation. The rotation table 3 may also be at a position 300° forward from its original position. The rotation table 3 may also be at a position 30° forward from the position (the position shown in Figure 12) of the rotation table 3 when the revolution table 1 has made nine rotations.

[0055] The non-limiting example shown in Figure 14 may show a state in which the revolution table 1 has made one more rotation and returned to its original position (a total of 11 rotations). At this time, the rotation table 3 may make another m1 rotation. The rotation table 3 may also be at a position 330° forward from its original position. The rotation table 3 may also be at a position 30° forward from the position (position shown in Figure 13) of the rotation table 3 when the revolution table 1 has made 10 rotations.

[0056] The non-limiting example shown in Figure 15 may show a state in which the revolution table 1 has made one more rotation and returned to its original position (a total of 12 rotations). At this time, the rotation table 3 may make another m1 rotation. The rotation table 3 may also be in a position 360° forward from its original position. The rotation table 3 may also be in a position 30° forward from the position (position shown in Figure 14) of the rotation table 3 when the revolution table 1 has made 11 rotations.

[0057] The non-limiting example shown in Figure 15 may show the positional relationship between the revolution table 1 and the rotation table 3 at the end of coating. If the value calculated by substituting a and t into the formula: a x t is an integer multiple of the value calculated by substituting α° into the formula: 360° / α°, the revolution table 1 and the rotation table 3 are likely to be in the state shown in Figure 15 at the end of coating. If the revolution table 1 and the rotation table 3 are in the state shown in Figure 15 at the end of coating, the uniformity of the coated film is likely to improve. Note that the state shown in Figure 15 may be repeated several times during coating.

[0058] The rotation speed of the support 11 may be c (rpm). b and c may be b>c. In this case, the uniformity of the coated film is likely to be improved.

[0059] For example, c may be set to 0.5 to 4 rpm. Also, c may be set by adjusting the gear ratio.

[0060] The coating method may use a PVD processing apparatus 201 that performs coating using a PVD method. The PVD processing apparatus 201 may include a tool rotation device 101, a target 203, and a vacuum chamber 205, as shown in FIG. 1 as a non-limiting example.

[0061] The target 203 may be located outside the revolution table 1. The target 203 may also be located outside in the radial direction of the revolution table 1. The target 203 may also be located on the inner wall surface of the vacuum chamber 205.

[0062] The target 203 may be formed from a film-forming material that is a raw material for the coating layer. The target 203 may also be called an evaporation source or a vapor deposition source. The target 203 may be in the form of a plate.

[0063] There may be multiple targets 203. The multiple targets 203 may be formed from different types of film formation materials or may be formed from the same type of film formation material. The multiple targets 203 may be positioned at intervals in the direction along the revolution axis O1.

[0064] Furthermore, the multiple targets 203 may be positioned at intervals along the circumferential direction of the revolution table 1. For example, when there are two targets 203, the two targets 203 may be positioned so as to face each other with the revolution table 1 in between. When there are multiple targets 203, the number of targets 203 may be, for example, about 2 to 16.

[0065] The vacuum chamber 205 may accommodate the tool rotation device 101 and the target 203 therein. The vacuum chamber 205 is capable of depressurizing its interior. For example, an exhaust pipe may be connected to the vacuum chamber 205, and a vacuum pump or the like may be connected to the exhaust pipe to evacuate the interior to a vacuum or extremely low pressure. A gas supply pipe may be connected to the vacuum chamber 205 to supply an inert gas, a reactive gas, or the like into the vacuum chamber 205.

[0066] The above provides examples of the non-limiting one-side coating method and the other-side coating method of the present disclosure, but it goes without saying that the present disclosure is not limited to the above-described embodiments and can be any method as long as it does not deviate from the gist of the present disclosure.

[0067] The present disclosure will be described in detail below with reference to examples, but the present disclosure is not limited to the following examples.

[0068] The surface of the tool was coated with a coating layer. The coating conditions were as follows: PVD method: arc ion plating method Tool shape: square plate Number of tools to be coated: 36 (number per support) Coating layer composition: TiN Average coating layer thickness: 2.5 μm Number of rotating tables: 4 a: 4 rpm b: 13.2 rpm c: 1.6 rpm α: 30° n1: 12 (number of supports per rotating table) t: 60 minutes Other than the above, the conditions were as shown in FIGS. 1 to 15.

[0069] As a result of the coating, the uniformity of the coated film was good. Specifically, the value calculated from the above formula: [[(X-Y) / Z] / 2] x 100 was not more than 25%. Therefore, it can be said that the above coating method has excellent coatability.

[0070] DESCRIPTION OF SYMBOLS 1: revolution table 3: rotation table 5: main body 7: top plate 9: upper surface 11: support 101: tool rotation device 201: PVD processing device 203: target 205: vacuum chamber 301: tool O1: revolution axis O2: rotation axis O3: central axis S1: reference position

Claims

1. 1. A coating method for coating a surface of a tool with a coating layer by physical vapor deposition while revolving the tool around an axis of revolution and rotating the tool around an axis of rotation parallel to the axis of revolution, comprising: A revolution table that revolves the tool and a rotation table that rotates the tool on the revolution table are used, A coating method in which, when the rotation speed of the revolution table is a (rpm) and the rotation speed of the rotation table is b (rpm), the ratio of a to b, b / a, is greater than 2 and is not an integer.

2. 2. The coating method according to claim 1, wherein a and b satisfy b<5a.

3. 2. The coating method according to claim 1, wherein reference positions for the position of the revolution table and the position of the rotation table at the start of rotation are set, and when the deviation of the rotation table from the reference position when the revolution table has made one rotation is defined as α°, α° satisfies the range of 10°≦α≦350°.

4. the revolution table and the rotation table, the rotating table is rotatable around a central axis of the rotating table, the rotation axis is located closer to the outer periphery of the rotation table than the central axis, the rotating table has a plurality of support columns capable of holding the tool; The support post extends along the rotation axis and is rotatable around the rotation axis; and 4. The coating method according to claim 3, wherein when the number of the supports is n1, the α° and the n1 are such that α°=360 / n1.

5. 4. The coating method according to claim 3, wherein, when the coating time is t (minutes), the value calculated by substituting a and t into the formula: a × t is an integer multiple of the value calculated by substituting α° into the formula: 360° / α°.

6. the revolution table and the rotation table, the rotating table is rotatable around a central axis of the rotating table, the rotation axis is located closer to the outer periphery of the rotation table than the central axis, the rotating table has a plurality of support columns capable of holding the tool; The support post extends along the rotation axis and is rotatable around the rotation axis; and 6. The coating method according to claim 1, wherein when the rotation speed of the support is c (rpm), b and c satisfy b>c.