JP3912556B2 - Fine void-containing polyester film - Google Patents

Description

【００５７】
【表２】

【００５８】
【表３】

[0001]
BACKGROUND OF THE INVENTION
The present invention includes various printing recording bodies, electrophotographic systems, ink jet systems, thermal systems, thermal transfer systems, sublimation transfer systems, and other various printer recording bodies, labels, cards, printing materials for building materials, heat insulating materials, light reflecting materials, Useful as flexible packaging materials, hard packaging materials, electrical insulation materials for livestock appliances, electrical insulation materials for motors, electrical insulation tapes, printed wiring boards, flexible printed wiring boards, prepreg base materials, and has sufficient strength as a film. In addition, the present invention relates to a homogeneous fine void-containing polyester film containing a large number of fine voids.
[0002]
[Prior art]
Synthetic paper, which is a substitute for paper based on synthetic resin, is water resistant, hygroscopic, dimensional stability, surface gloss, gloss and clarity of printed matter, and machine compared to paper based on natural pulp. Excellent in mechanical strength. For this reason, in recent years, various applications have been developed taking advantage of the characteristics.
The main raw materials for synthetic paper include polyolefin resins and polyester resins. Among them, polyester resins represented by polyethylene terephthalate are important requirements for synthetic paper because of their high heat resistance and stiffness. Due to its characteristics, it is still widely used in the present situation. In addition to paper that uses natural pulp as a main raw material, it is also used in fields where fabric and woven materials using synthetic resin fibers, glass fibers, metal fibers, natural fibers, etc. are used, such as heat insulating materials and electrical insulation. Excellent properties are also expected in the field of materials and the like.
[0003]
The most effective method for obtaining a film with a function similar to natural paper, using polyester resin as the main raw material, is to form countless fine cavities inside the polyester resin film. This is a method of giving moderate flexibility to the film itself, reducing the weight, and providing excellent writing properties and clear print / transfer properties. As a means for forming a cavity inside the film, a resin incompatible with the polyester resin is mixed with the raw material resin as a cavity forming agent, and after forming on the film, the cavity is formed at the resin interface by stretching it. It is a method to generate.
[0004]
Examples of the cavity forming agent used for the cavity formation include polyolefin resins typified by polypropylene resin and polymethylpentene resin (Japanese Patent Laid-Open No. 49-34755), and polystyrene resins (for example, Japanese Patent Publication No. 49-2016). No. Gazette, Japanese Patent Publication No. 54-29550, etc.) have been proposed.
[0005]
Among these, polyolefin resins, particularly polymethylpentene, have excellent cavity forming ability, and are very excellent in terms of reducing the weight of the film. On the other hand, due to the poor compatibility with polyester, the cavity forming agent is coarsely dispersed in the polyester, and the size of the cavity becomes very large. And there existed a problem that the flexibility of a film will fall remarkably, when the cavity was large and the waist strength of polyester itself was combined. This problem is manifested in the film manufacturing process, processing process, or handling of final products such as printed matter, and the film may be bent even if the film is bent slightly, or the film surface may be easily wrinkled. Cause problems. Flexibility is one of the very excellent characteristics of a synthetic resin film, and losing this characteristic is a fatal defect.
[0006]
On the other hand, the polystyrene-based resin can be finely dispersed in the cavity-forming agent due to a low degree of incompatibility with the polyester-based resin as compared with the polyolefin-based resin, and a large number of fine cavities can be formed. . Therefore, it is possible to minimize the decrease in flexibility of the film, which was a fatal defect of the polyolefin resin. However, when the film is stretched, the cavity forming agent tends to collapse in the thickness direction of the film, and the growth of the cavity is inhibited. As a result, the cavity forming ability is inferior and the film is not sufficient in terms of weight reduction. It was.
[0007]
On the other hand, several methods for maintaining the excellent cavity forming ability of the polyolefin-based resin and improving the dispersibility in the polyester have been proposed. For example, a method of adding a surfactant (Japanese Patent Publication No. 7-17779), polyethylene glycol (Japanese Patent Laid-Open No. 2-235942) or a polyether ester copolymer (Japanese Patent Laid-Open No. 4-264141) to polyester is used.
[0008]
However, there is a limit to fine dispersion of polyolefin resin by these methods, and it is difficult to obtain fine dispersion equivalent to that of polystyrene resin. Further, it has been extremely difficult to obtain a stable fine dispersion effect without impairing other characteristics (for example, film strength and whiteness).
[0009]
That is, in the method of adding a surfactant, the surfactant is insufficiently heat-resistant, so that it is altered in the melt line and a stable fine dispersion effect cannot be obtained.
[0010]
In addition, in the method of adding a polyether-based resin, the polyether is extremely susceptible to thermal degradation, so that the polyether is decomposed in the raw material drying step, the resin melt line, or the step of collecting and recycling the waste film. . The degradation of the polyether not only causes variations in the fine dispersion effect of the polyolefin resin, but also causes a significant decrease in film strength by reducing the molecular weight of the polyester, and the problem of coloring the film yellow. This causes problems such as pungent odor due to the formation of aldehydes.
[0011]
Further, when the void content is increased by the above-described method, not only the excellent mechanical properties and heat resistance properties peculiar to the polyester film are lost, but also the productivity is lowered, and the specific gravity is particularly less than 1.0. In the region, and in the region below 0.7, not only the quality is lowered but also the industrial production is substantially impossible.
As described above, in the prior art, a fine cavity-containing polyester system having both excellent cavity forming ability of polyolefin resin, particularly polymethylpentene, and excellent flexibility of polystyrene resin, and having excellent heat resistance. Not only is a film not obtained, but industrial production is also impossible, particularly in the low specific gravity region.
[0012]
[Problems to be solved by the invention]
The present invention has been made in view of the circumstances as described above, and an object thereof is to provide a fine void-containing polyester film having both lightness and flexibility, and having excellent mechanical properties and heat resistance. There is.
[0013]
[Means for Solving the Problems]
  the aboveSectionContaining fine cavities of the present invention that could solve the problemLaminatedThe polyester film is at least one side of the fine void-containing layer (A layer)In addition,A polyester layer (B layer) that does not substantially contain fine cavities is bonded.RuContains fine cavitiesLaminatedA polyester film,
  The fine void-containing laminated polyester film has a laminated structure of A / B, A / B / A, or B / A / B,Apparent specific gravity is less than 0.3-1.0AndThe thickness of layer B is 5% or more and less than 30% of the total film thicknessThe
  The fine void-containing layer (A layer)It is obtained by biaxially stretching and heat treating a polymer mixture in which at least one thermoplastic resin incompatible with the polyester is mixed with the polyester.In addition, the thermoplastic resin incompatible with the polyester contained in the A layer is a polystyrene resin, a polymethylpentene resin, and a polypropylene resin, and the content (a wt%) of the polystyrene resin and polymethyl The content of pentene resin (b% by weight) and the content of polypropylene resin (c% by weight) satisfy the following relationship:It has a gist on the points.
    0.01 ≦ a / (b + c) ≦ 1
    c / b ≦ 1
    5 ≦ a + b + c ≦ 30
[0014]
Hereinafter, the constituent components of the present invention will be sequentially described below.
First, the polyester used in the present invention is an aromatic dicarboxylic acid or ester such as terephthalic acid, isophthalic acid or naphthalenedicarboxylic acid, and glycol such as ethylene glycol, diethylene glycol, 1,4-butanediol or neopentyl glycol. Polyester produced by polycondensation. In addition to the method of directly reacting aromatic dicarboxylic acid and glycol, these polyesters can be polycondensed by transesterification of alkyl ester of aromatic dicarboxylic acid and glycol, or diglycol ester of aromatic dicarboxylic acid. Can be produced by a method such as polycondensation. Typical examples of such polyester include polyethylene terephthalate, polyethylene butylene terephthalate, polyethylene-2, 6-naphthalate and the like. This polyester may be a homopolymer or a copolymer of a third component. In any case, in the present invention, a polyester having an ethylene terephthalate unit, a butylene terephthalate unit or an ethylene-2,6-naphthalate unit of 70 mol% or more, preferably 80 mol% or more, more preferably 90 mol% or more is preferable. .
[0015]
Next, as a thermoplastic resin incompatible with polyester, which is a component that characterizes the present invention (hereinafter sometimes abbreviated as a cavity forming agent), in addition to polystyrene resins, polymethylpentene resins and polypropylene resins may be used. Preferably used. Then, by using a mixture of a polystyrene resin and a specific polyolefin resin in a specific ratio, it is possible to obtain a fine void-containing polyester film having both lightness and flexibility and excellent heat resistance. Can do.
[0016]
On the other hand, when polystyrene is used alone as the cavity forming agent, an excellent film is obtained in terms of the flexibility of the film as described above, but compared with the case where a polyolefin resin is used as the cavity forming agent. There is a limit in terms of light weight.
[0017]
  The content (a wt%) of the polystyrene resin in the present invention is 0.01 ≦ a with respect to the content (b wt%) of the polymethylpentene resin and the content (c wt%) of the polypropylene resin. / (B + c) ≦ 1To. And by making content of a polystyrene-type resin into the said range, it becomes possible to finely disperse | distribute a cavity formation agent in a film, and sufficient weight reduction is attained, ensuring the flexibility of a film. When the content of the polystyrene resin is less than the above range, the dispersion effect of the polystyrene resin on the polyolefin resin (polypropylene resin and polymethylpentene resin) becomes unstable, resulting in film unevenness and flexibility. Becomes defective. On the contrary, when the content of the polystyrene resin is larger than the above range, a sufficient weight reduction effect cannot be obtained.
[0018]
The content of the polystyrene resin can be arbitrarily adjusted within the above range, but a range of 0.1 ≦ a / (b + c) ≦ 0.5 is more preferable. This is because the film can be more easily reduced in weight by setting the content of the polystyrene resin within this range.
[0019]
In the fine void-containing polyester film of the present invention, a polymethylpentene resin and a polypropylene resin are particularly preferably used as the polyolefin resin in addition to the polystyrene resin. When a polypropylene resin is used alone as the polyolefin resin, the effect of reducing the weight of the film is hardly exhibited.
[0020]
  On the other hand, when a polymethylpentene resin is used alone as the polyolefin resin, it is relatively easy to reduce the weight of the film. However, polymethylpentene resin in the filmofDispersion unevenness appears, and a canvas-like unevenness is exhibited on the film surface. However, if the content of the polystyrene resin is made larger than the content of the polymethylpentene resin, the unevenness in the canvas ground shape is greatly reduced and falls within the practical range, but in this case, the lightening effect of the present invention cannot be obtained. .
[0021]
In the fine void-containing polyester film of the present invention, a polymethylpentene resin and a polypropylene resin are used in combination as a polyolefin resin, which is comparable to that when a polymethylpentene resin is used alone as a polyolefin resin. It is particularly preferable because excellent uniformity can be obtained while obtaining the lightening effect.
[0022]
The mixing ratio of the polymethylpentene-based resin and the polypropylene-based resin that can achieve the above effects needs to be in the range of c / b ≦ 1 when the content of each resin is b wt% and c wt%. Yes, preferably 0.01 ≦ c / b ≦ 0.5, particularly preferably 0.1 ≦ c / b ≦ 0.5. The desired purpose can be achieved by mixing a small amount of polypropylene resin. Conversely, when the mixing ratio exceeds 1, the effect of reducing the weight of the polymethylpentene resin is greatly impaired.
[0023]
The polystyrene-based resin of the present invention refers to a thermoplastic resin having a polystyrene structure as a basic component, and grafts or blocks other components in addition to homopolymers such as atactic polystyrene, syndiotactic polystyrene and isotactic polystyrene. A copolymerized modified resin, such as an impact-resistant polystyrene resin or a modified polyphenylene ether resin, or a mixture of a thermoplastic resin compatible with these polystyrene resins, such as polyphenylene ether, is included.
[0024]
The polymethylpentene resin in the present invention is a polymer having units derived from 4-methylpentene-1 at 80 mol% or more, preferably 90 mol% or more, and other components include ethylene units, Examples are derived units from propylene units, butene-1 units, 3-methylbutene-1, and the like.
[0025]
The melt flow rate of such polymethylpentene is preferably 200 g / 10 min or less, more preferably 30 g / 10 min or less, and particularly preferably 10 g / 10 min or less. This is because when the melt flow rate exceeds 200 g / 10 min, it is difficult to obtain the effect of reducing the weight of the film.
[0026]
The polypropylene resin in the present invention also includes modified resins obtained by grafting or block copolymerizing other components in addition to homopolymers such as isotactic polypropylene and syndiotactic polypropylene.
[0027]
In the present invention, the above components are particularly preferably used, but other resins can be used in combination as the cavity forming agent. Examples of resins that can be added in this case include polyphenylene ether resins, polyolefin resins, polyacrylic resins, polycarbonate resins, polysulfone resins, cellulose resins, polyphenylene sulfide resins, cyclic olefin-ethylene copolymer resins, and cyclic resins. Examples thereof include, but are not limited to, olefin polymers or ring-opened polymers thereof.
[0028]
The mixing amount of these void forming agents, that is, the thermoplastic resin incompatible with the polyester, with respect to the polyester varies depending on the amount of the target void, but may be in the range of 5 to 30% by weight with respect to the entire film. it can. If it is less than 5% by weight, there is a limit to increasing the amount of generated cavities. On the other hand, if it is 20% by weight or more, the stretchability of the film is remarkably impaired, and the phenomenon that the heat resistance, the strength, and the waist strength are impaired is easily developed.
[0029]
In order to effectively prevent this phenomenon and achieve both the lightness, mechanical properties, and heat resistance of the film, it is effective to make the film into a multilayer structure of two or more layers, especially the amount of addition of the cavity forming agent is 15 wt. % Or more, or an apparent specific gravity of 0.8 or less, it is possible to achieve both weight reduction, mechanical properties, and heat resistance only by a multilayer structure. The above objective is accomplished by joining a polyester layer that is substantially free of void forming agents and therefore substantially free of fine voids to the fine void containing layer. As the structure of bonding, the polyester layer is bonded to one side of the film or disposed in the center of the film, and the structure containing the fine cavity-containing layers on both sides, or the fine cavity-containing layer is disposed in the center, The method of joining this polyester layer to the both surfaces is mentioned.
[0030]
As a method for forming a multilayer structure, a coextrusion method is preferably used. This is because only the coextrusion method in which a multilayer structure is formed at the stage of an unstretched film is effective in improving the stretchability of the film forming process.
[0031]
The thickness of the polyester layer substantially free of fine cavities should be 5% or more of the total film thickness from the viewpoint of maintaining mechanical strength and heat resistance and ensuring stretchability during film formation. preferable. On the other hand, the upper limit of the thickness of the skin layer is preferably 30% or less of the total film thickness in order to ensure the light weight of the entire film.
[0032]
The lightweight property of the fine void-containing polyester film of the present invention is arbitrary, but the apparent specific gravity is preferably in the range of 0.3 to 1.0 as a range in which the effects of the present invention can be effectively exhibited. This is because when the apparent specific gravity is outside this range, the film becomes insufficient in strength, the flexibility becomes insufficient, the cushioning properties and the flexibility, etc., the characteristics given by cavity formation, that is, weight reduction, are effectively exhibited. Because it will not be done.
[0033]
Polyesters that form substantially void-free layers are aromatic dicarboxylic acids or esters such as terephthalic acid, isophthalic acid, naphthalene dicarboxylic acid and glycols such as ethylene glycol, diethylene glycol, 1,4-butanediol, neopentyl glycol. Is a polyester produced by polycondensation. In addition to the method of directly reacting aromatic dicarboxylic acid and glycol, these polyesters can be polycondensed after transesterification of alkyl ester of aromatic dicarboxylic acid and glycol, or diglycol ester of aromatic dicarboxylic acid Can be produced by a method such as polycondensation. Typical examples of such polyester include polyethylene terephthalate, polyethylene butylene terephthalate, polyethylene-2, 6-naphthalate and the like. This polyester may be a homopolymer or a copolymer of a third component. In any case, in the present invention, a polyester having an ethylene terephthalate unit or an ethylene-2,6-naphthalate unit of 70 mol% or more, preferably 80 mol% or more, more preferably 90 mol% or more is most preferably used. .
[0034]
In addition, in the film, inorganic or organic particles may be added to the polyester or the cavity forming agent as necessary in order to improve the concealability and the like. Examples of particles that can be added include silica, kaolinite, talc, calcium carbonate, zelite, alumina, barium sulfate, carbon black, zinc oxide, titanium oxide, zinc sulfide, organic white pigments, and the like. is not.
[0035]
The method for producing the fine void-containing polyester film of the present invention is arbitrary and is not particularly limited, but after forming the mixture having the above composition into a film shape to obtain an unstretched film, A general method of stretching can be used.
[0036]
The conditions for stretching / orienting the unstretched sheet are closely related to the formation of cavities. In the following, the stretching / orientation conditions will be described by taking as an example a sequential biaxial stretching method used most preferably, particularly a method of stretching an unstretched sheet in the longitudinal direction and then in the width direction. First, in the first-stage longitudinal stretching step, stretching is performed between two or many rolls having different peripheral speeds. As a heating means at this time, a method using a heating roll or a method using a non-contact heating method may be used, or they may be used in combination. However, in order to develop a large number of cavities at the incompatible resin interface, the stretching temperature is 3 to 5 times at a polyester secondary transition temperature Tg + 50 ° C. or lower. Next, the uniaxially stretched film is introduced into a tenter, and stretched 2.5 to 5 times in the width direction at a temperature of the melting point Tm-10 ° C. or less of the polyester.
[0037]
The biaxially stretched film thus obtained is subjected to heat treatment as necessary. The heat treatment is preferably performed in a tenter, and is preferably performed in the range of the melting point Tm-50 ° C. to Tm of the polyester.
[0038]
The fine void-containing polyester film of the present invention may have a coating layer on at least one surface thereof. Then, by providing the coating layer, it is possible to improve the paintability and adhesiveness of inks and coating agents. The compound constituting the coating layer is preferably a polyester resin, but in addition to this, a compound disclosed as a means for improving the adhesion of a normal polyester film such as a polyurethane resin, a polyester urethane resin, or an acrylic resin. Etc. are applicable.
[0039]
In addition, as a method for providing the coating layer, conventionally used methods such as a gravure coating method, a kiss coating method, a dip method, a spray coating method, a curtain coating method, an air knife coating method, a blade coating method, and a reverse roll coating method can be applied. As a step of applying, any method such as a method of applying before stretching of the film, a method of applying after longitudinal stretching, and a method of applying to the film surface after the orientation treatment is possible.
[0040]
The fine void-containing polyester film thus obtained achieves both high flexibility and light weight at a high level. In addition, it does not require a surfactant or polyether resin as a dispersant for polyolefin resin, so it has excellent heat resistance, small change in color tone even when reused self-recovery materials, and stable film production. Also excellent in properties. A preferable use ratio when reusing the self-recovered raw material is 5 to 50% by weight.
[0041]
【Example】
Next, examples and comparative examples of the present invention will be shown.
The measurement / evaluation method used in the present invention is shown below.
[0042]
1) Apparent specific gravity
The film is accurately cut into a 10 cm × 10 cm square, and its thickness is measured at 50 points to obtain an average thickness t (unit: μm). Next, the weight of the sample is measured to 0.1 mg and is set to w (unit: g). And the apparent specific gravity was calculated by the following formula.
Apparent specific gravity (−) = (w / t) × 100
[0043]
2) Flexibility of film
The film is cut into strips of 5 cm in length and 1 cm in width, wound around a stainless steel rod having a diameter of 2 mm, and then squeezed. Thereafter, the sample was stretched again, and the state of wrinkles generated on the surface was observed using a stereomicroscope.
[0044]
3) Breaking strength of the film
Evaluation was performed according to JIS C-2318.
[0045]
Example 1
As raw materials, 70% by weight of polyethylene terephthalate resin with an intrinsic viscosity of 0.64, 3% by weight of polystyrene resin with melt flow index 2.0 (Topolex 570-57U manufactured by Mitsui Toatsu Co., Ltd.), polypropylene resin with melt flow index 1.7 (Mitsui Toatsu Co., Ltd.) Noblene FO-50F made by company, 6% by weight, 21% by weight of polymethylpentene resin (TPI, DX-845 made by Mitsui Petrochemical Co., Ltd.) with a melt flow index of 8 is mixed with pellets and supplied to a twin screw extruder. After the strand was cast in water and cooled, it was cut with a strand cutter to produce a master pellet (A) containing a cavity forming agent.
[0046]
The obtained master pellets were dried with hot air (170 ° C. × 3 hours), then 60% by weight of the master pellets, 38% by weight of polyethylene terephthalate resin having an intrinsic viscosity of 0.62 and dried with hot air, and an average particle size of 0.3 μm (electron microscopic method). 2% by weight of anatase type titanium dioxide (TA-300 manufactured by Fuji Titanium Co., Ltd.) was mixed and used as a raw material for the film.
[0047]
This raw material was used as a raw material for the film core layer, and a polyethylene terephthalate resin having an intrinsic viscosity of 0.64 was separately used as a raw material for the skin layer. Then, the core layer raw material was supplied to the twin screw extruder, and the skin layer raw material was supplied to the vent type twin screw extruder and supplied to the feed block to bond the skin layers to both sides of the core layer. At this time, the discharge amount was controlled using a gear pump so that the volume ratio of the skin layer material and the core layer material was 1: 4. Next, the sheet was extruded onto a cooling drum adjusted to 30 ° C. using a T die to prepare an unstretched sheet having a thickness of about 630 μm. Subsequently, the obtained unstretched sheet was heated to 86 ° C. using a heated roll, and longitudinally stretched 3.4 times between the rolls. Subsequently, it is heated to 120 ° C with a tenter, stretched 3.7 times, fixed in width, heat-treated at 230 ° C for 5 seconds, and further relaxed by 4% in the width direction at 220 ° C. A polyester film was obtained.
[0048]
Comparative Example 1
A fine void-containing polyester film was obtained in the same manner as in Example 1 except that a polyethylene terephthalate skin layer was not laminated in the preparation of the unstretched sheet.
[0049]
Example 2
A fine void-containing polyester film was obtained in the same manner as in Example 1 except that the raw material for the skin layer was polyethylene naphthoate in the production of the unstretched sheet.
[0050]
Example 3
Same as Example 1 except that the film raw material used for the core layer in Example 1 is used for the skin layer, polyethylene terephthalate is used for the core layer, and the volume ratio of the skin layer raw material to the core layer raw material is 9: 1. Thus, a fine void-containing polyester film was obtained.
[0051]
Example 4, Comparative Examples 2, 3
Implemented except that the composition of the master pellet and the composition of the film were changed as shown in Table 1 and Table 2, respectively, and the thickness of the unstretched sheet was adjusted so that the film thickness after biaxial stretching was 75 μm. A film was produced in the same manner as in Example 1.
[0052]
The properties of the film thus obtained are shown in Table 1.
[0053]
From Table 1, it can be considered as follows. First, it can be seen that Example 1 and Example 2 satisfying the configuration requirements of the present invention provide sufficient flexibility and strength despite the low specific gravity, and can be produced stably. In addition, it can be seen that in Examples 3 and 4 in which the specific gravity is further lowered, the film can be stably formed.
[0054]
On the other hand, when it does not have the layer which does not contain the fine cavity which is an essential component of this invention, it is inferior to flexibility by the low specific gravity of a film, and it also becomes easy to fracture | rupture. (Comparative Example 1) In addition, when the mixing ratio of the cavity forming agent deviates from the appropriate range of the present invention, it is presumed that sufficient low specific gravity cannot be achieved, or (Comparative Example 2) due to poor dispersion of the cavity forming agent. As a result, it is found that satisfactory quality cannot be obtained as a fine void-containing polyester film.
[0055]
【The invention's effect】
According to the present invention, it can be seen that a film having good flexibility, strength and surface condition can be stably obtained, and the film is suitable for various applications.
[0056]
[Table 1]

[0057]
[Table 2]

[0058]
[Table 3]

Claims (1)

Fine on at least one side of the void-containing layer (A layer), a substantially polyester layer (B layer) is a fine void-containing laminated polyester ing been joined film containing no fine voids,
The fine void-containing laminated polyester film has a laminated structure of A / B, A / B / A, or B / A / B, an apparent specific gravity is less than 0.3 to 1.0 , and the thickness of the B layer There Ri than 30% der least 5% of the film total thickness,
The fine void-containing layer (A layer) is obtained by biaxially stretching and heat-treating a polymer mixture obtained by mixing at least one thermoplastic resin incompatible with the polyester into the polyester , and included in the A layer. Polyester resin, polymethylpentene resin, and polypropylene resin are thermoplastic resins that are incompatible with polyester, and the content of polystyrene resin (a wt%) and the content of polymethylpentene resin ( b wt%) and the content of the polypropylene resin (c wt%) of fine void-containing laminated polyester film characterized that you satisfy the following relationship.
0.01 ≦ a / (b + c) ≦ 1
c / b ≦ 1
5 ≦ a + b + c ≦ 30