JP6852747B2 - A method for producing a positive electrode composition for a non-aqueous electrolyte secondary battery, a non-aqueous electrolyte secondary battery, and a positive electrode composition for a non-aqueous electrolyte secondary battery. - Google Patents

Description

The present invention relates to a positive electrode composition for a non-aqueous electrolyte secondary battery such as a lithium ion secondary battery. In particular, the present invention relates to a positive electrode composition in which the output characteristics of a battery are improved, and the cycle characteristics and the viscosity stability of a positive electrode slurry are also improved.

In recent years, mobile devices such as VTRs, mobile phones, and notebook computers have become widespread and miniaturized, and non-aqueous electrolyte secondary batteries such as lithium ion secondary batteries have come to be used as their power sources. Furthermore, it is also attracting attention as a power battery for electric vehicles and the like in response to recent environmental problems.

As a positive electrode active material for a lithium secondary battery, LiCoO ₂ (lithium cobalt oxide) is generally widely used as a material capable of forming a 4V class secondary battery. However, since _{cobalt, which is a raw material of LiCoO 2} , is a rare resource and is unevenly distributed, it is costly and causes anxiety about the supply of raw materials.

In response to these circumstances, a lithium transition metal composite oxide having a layered structure such as lithium nickel cobalt manganate in which Co of _{LiCoO 2 is replaced with an element such as Ni or Mn has been developed.} Generally, among layered lithium transition metal composite oxides, those having a large proportion of nickel tend to have an unstable crystal structure, and a lithium compound tends to be easily precipitated in a positive electrode slurry during production for a positive electrode. Further, when the ratio of cobalt becomes small, the output characteristics tend to decrease.

On the other hand, there is also a technique of mixing a boron compound such as boric acid with a lithium transition metal composite oxide or allowing the boron compound to be present on the surface of the lithium transition metal composite oxide, depending on various purposes.

Patent Document 1 describes that a positive electrode using lithium manganate contains a boron compound soluble in an electrolytic solution such as boron oxide, orthoboric acid, metaboric acid, and tetraboric acid to form a spinel structure with lithium manganate and a hydrogen halide. A technique for suppressing the reaction with hydroacid and improving the cycle characteristics is disclosed.

In Patent Document 2, a surface treatment layer having excellent ionic conductivity containing a coating element such as boron, hydroxide, oxyhydroxydo, etc. is formed on the surface of a lithium transition metal composite oxide to increase the discharge potential and have a lifetime. Techniques for improving properties are disclosed. The specifically disclosed coating method is obtained by precipitating a coating element dissolved in a solvent on the surface of a lithium transition metal composite oxide and removing the solvent.

Patent Document 3 discloses a technique for preventing gelation of an electrode paste by containing boric acid or the like as an inorganic acid in an electrode using a lithium transition metal composite oxide or the like. The only lithium transition metal composite oxide specifically disclosed is lithium nickelate.

In Patent Document 4, a boric acid compound such as ammonium borate or lithium borate is adhered to the surface of lithium transition metal composite oxide particles in which nickel or cobalt is essential, and heat-treated in an oxidizing atmosphere. As a result, a technique for increasing the capacity of the secondary battery and improving the charge / discharge efficiency of the secondary battery is disclosed. The specifically disclosed lithium transition metal composite oxide is lithium nickelate in which a part of nickel is replaced with cobalt and aluminum.

Japanese Unexamined Patent Publication No. 2001-257003 JP-A-2002-124262 Japanese Unexamined Patent Publication No. 10-079244 JP-A-2009-146739

Lithium nickel-cobalt manganate, which uses nickel, cobalt, and manganese as transition metals, has a relatively good balance between its cost and various battery characteristics, but cycle characteristics and / or output characteristics have recently been improved in a specific composition range. I can no longer meet the demand.

The present invention has been made in view of these circumstances. An object of the present invention is to provide a positive electrode material for a non-aqueous electrolyte secondary battery in which cycle characteristics and output characteristics are still improved by using a lithium transition metal composite oxide based on nickel-cobalt-manganate lithium.

In order to achieve the above object, the present inventors have made extensive studies and have completed the present invention. The present inventors have found that the output characteristics and the cycle characteristics are improved by preparing a positive electrode composition using a lithium transition metal composite oxide having a specific composition and a boron compound in a specific state. It was also found that the positive electrode slurry using such a positive electrode composition has little change in slurry viscosity with time.

The positive electrode composition for a non-aqueous electrolyte secondary battery of the present invention has the general formula Li _a Ni _1-xy Co _x Mn _y M _z O ₂ (1.00 ≦ a ≦ 1.50, 0 <x ≦ 0). .50, 0 <y≤0.50, 0.00≤z≤0.02, 0.40≤x + y≤0.70, M is selected from the group consisting of Zr, Ti, Mg, Ta, Nb and Mo. It is characterized by containing a lithium transition metal composite oxide represented by (at least one kind) and a boron compound containing at least oxygen.

The method for producing the positive electrode composition for a non-aqueous electrolyte secondary battery of the present invention is described in the general formula Li _a Ni _1-xy Co _x Mn _y M _z O ₂ (1.00 ≦ a ≦ 1.50, 0 <. x ≦ 0.50, 0 <y ≦ 0.50, 0.00 ≦ z ≦ 0.02, 0.40 ≦ x + y ≦ 0.70, M is a group consisting of Zr, Ti, Mg, Ta, Nb and Mo. The synthetic step of synthesizing the lithium transition metal composite oxide represented by (at least one selected from the above), the lithium transition metal composite oxide obtained in the synthetic step, and the raw material compound of the boron compound are mixed and used as a raw material. It is characterized by including a mixing step of obtaining a mixture and a firing step of firing the raw material mixture obtained in the mixing step.

Since the positive electrode composition of the present invention has the above-mentioned characteristics, the cycle characteristics and the output characteristics can be improved without impairing the characteristics of lithium nickel-cobalt manganate. Further, it is possible to suppress an increase in slurry viscosity when the positive electrode slurry is used.

Since the method for producing a positive electrode composition of the present invention has the above-mentioned characteristics, it is possible to obtain a positive electrode composition having improved cycle characteristics and output characteristics and good productivity.

FIG. 1 shows a change in viscosity of a positive electrode slurry using the positive electrode composition of the present invention and a positive electrode composition for comparison.

Hereinafter, the positive electrode composition of the present invention will be described in detail with reference to embodiments and examples. However, the present invention is not limited to these embodiments and examples.

[Lithium transition metal composite oxide]
The lithium transition metal composite oxide is a lithium-based nickel-cobalt manganate containing nickel, cobalt and manganese (combined as main components) in the transition metal. If necessary, zirconium, tantalum, niobium, molybdenum and the like may be further contained. For example, zirconium is suitable for improving storage properties, and titanium and magnesium are suitable for further improving cycle properties. If these additional elements are up to about 2 mol%, various purposes can be achieved without impairing the characteristics of lithium nickel cobalt manganate.

If the ratio of nickel in the main component is too large, the stability of the crystal structure is difficult, and the viscosity of the positive electrode slurry is likely to increase. On the other hand, if the amount is too small, the charge / discharge capacity will decrease. Considering these balances, it is set to 40 mol% or more and 60 mol% or less. On the other hand, the ratio of cobalt and manganese shall be 50 mol% or less, respectively, after satisfying the ratio of nickel. Too much cobalt leads to increased costs and reduced output characteristics, and too much manganese causes reduced output characteristics and charge / discharge capacity. When the ratio of cobalt and manganese is 5 mol% or more and 35 mol% or less, respectively, the balance of various properties is good, which is more preferable.

If the amount of lithium in the lithium transition metal composite oxide is large, the output characteristics tend to be improved, but if the amount is too large, it is difficult to synthesize. Further, even if it can be synthesized, sintering proceeds, and subsequent handling tends to be difficult. Based on these, the amount of lithium is set to 100 mol% or more and 150 mol% or less with respect to the element of nickel site. Considering the balance of characteristics, ease of synthesis, etc., 105 mol% 125 mol% or less is preferable.

Based on these, the composition of the lithium transition metal composite oxide is the general formula Li _a Ni _1-xy Co _x Mn _y M _z O ₂ (1.00 ≦ a ≦ 1.50, 0 <x ≦ 0.50). , 0 <y ≦ 0.50, 0.00 ≦ z ≦ 0.02, 0.40 ≦ x + y ≦ 0.70, M is at least selected from the group consisting of Zr, Ti, Mg, Ta, Nb and Mo. It is represented by (a kind). However, this alone is insufficient for at least one of the output characteristics and the cycle characteristics. Further, since the viscosity of the positive electrode slurry also increases, it is combined with a boron compound described later to form a positive electrode composition.

[Boron compound]
In the positive electrode composition in the present invention, the boron compound contains oxygen. Although the details are unknown, it is presumed that at least a part of the boron compound forms a composite oxide with lithium or the like. Further, it is considered that at least a part of the boron compound is present on the surface of the lithium transition metal composite oxide particles in the form of coating or the like. A form such as coating is preferable because it has a high effect of preventing the elution of lithium positive electrode slurry into the binder.

The above-mentioned boron compound is obtained as a result of sufficiently mixing the lithium transition metal composite oxide and the raw material compound of the boron compound and firing them. It is considered that at least a part of the raw material compound finally reacts with some elements such as lithium in the lithium transition metal composite oxide to form a composite oxide. The form of the composite oxide thus obtained is different from the form obtained only as a result of physically mixing the lithium transition metal composite oxide and the raw material compound of the boron compound. This difference can be confirmed, for example, in the spectrum of XPS (X-ray photoelectron spectroscopy).

The content of the boron compound in the positive electrode composition is preferably 2.0 mol% or less as boron with respect to the lithium transition metal composite oxide. If it is too large, the charge / discharge capacity of the entire positive electrode composition will decrease. Further, if the amount is too small, the above-mentioned effect of preventing the elution of lithium is weak. The preferred range is 0.5 mol% or more and 1.5 mol% or less.

When the raw material compound of the above-mentioned boron compound is at least one selected from the group consisting of boron oxide, oxo acid of boron and oxo acid salt of boron, the final boron compound is in a form suitable for the object of the present invention. It is preferable because it becomes. The oxo acid (salt) of boron includes polyboric acid (salt) such as orthoborate (salt), metaboric acid (salt), diboric acid (salt), and triboric acid (salt). When the oxo acid salt is used as a raw material compound, a lithium salt or an ammonium salt is preferable. Specific examples include lithium borate (Li ₂ B ₄ O ₇ ), ammonium pentaborate (NH ₄ B ₅ O ₈ ) and the like. In addition, these raw material compounds may have hydration water. The raw material compound is preferably an oxo acid of boron in terms of ease of handling and the final form of the boron compound. In particular, orthoboric acid (so-called ordinary boric acid) is preferable.

[Manufacturing of positive electrode composition]
Next, a method for producing the positive electrode composition will be described. The positive electrode composition is produced by a synthesis step of synthesizing a lithium transition metal composite oxide, a mixing step of mixing a lithium transition metal composite oxide with a raw material compound of a boron compound to obtain a raw material mixture, and firing the raw material mixture. Includes firing step.

[Synthesis process]
The lithium transition metal composite oxide is synthesized by appropriately using a known method. The raw material compound that decomposes into oxides at high temperature is mixed according to the target composition. The raw material compound that is soluble in the solvent is dissolved in the solvent, and the precursor is precipitated by temperature adjustment, pH adjustment, addition of a complexing agent, etc. , Etc. The mixed raw material may be appropriately adjusted, and the obtained mixed raw material may be fired at about 700 ° C. to 1100 ° C.

[Mixing process]
The lithium transition metal composite oxide obtained in the synthesis step and the raw material compound of the boron compound are sufficiently mixed. It is sufficient if the two can be mixed using an existing stirrer or the like to the extent that there is no bias between the two, but if the boron compound is present on the surface of the lithium transition metal composite oxide particles due to the mechanochemical effect, etc. More preferred. It is presumed that at least a part of the raw material of the boron compound forms a composite oxide with lithium or the like in this mixing step. As described above, at least one selected from the group consisting of boron oxide, boron oxo acid and boron oxo acid salt is preferably used as the starting compound. When an oxo acid salt of boron is used, a lithium salt or an ammonium salt is preferable. As described above, boron oxoacid is more preferable as the raw material compound, and orthoboric acid is particularly preferable. In this way, the positive electrode composition of the present invention can be obtained.

[Baking process]
The positive electrode composition obtained in the mixing step is used as a raw material mixture and further fired. Most of the boron compounds in the obtained positive electrode composition are present in the form of coating the surface of the lithium transition metal composite oxide particles. The boron compound obtained through the firing step and coating the surface of the lithium transition metal composite oxide particles forms a chemical or physical bond with the elements constituting the lithium transition metal composite oxide and is firmly integrated. It is thought that it is. As a result, it is considered that the elution of lithium from the lithium transition metal composite oxide is suppressed and the increase in viscosity of the positive electrode slurry is suppressed. At the same time, it enhances the lithium ion conductivity of the entire positive electrode composition and contributes to the improvement of output characteristics.

Note that if the firing temperature is too high, the reaction between the lithium transition metal composite oxide and the boron compound (or its raw material compound) proceeds too much, and the lithium transition metal composite oxide cannot exhibit its original characteristics. If it is too low, the effect of the firing process cannot be expected. The preferred range is 450 ° C. or lower, and the more preferable range is 200 ° C. or higher and 400 ° C. or lower.

Pure water in a stirred state was adjusted in the reaction vessel, and aqueous solutions of nickel sulfate, cobalt sulfate, and manganese sulfate were added dropwise at a flow rate ratio of Ni: Co: Mn = 4: 3: 3. After completion of the dropping, the liquid temperature was adjusted to 50 ° C., and a certain amount of an aqueous sodium hydroxide solution was dropped to obtain a precipitate of nickel-cobalt-manganese composite hydroxide. The obtained precipitate was washed with water, filtered, and separated, and mixed with lithium carbonate and zirconium oxide (IV) so that Li: (Ni + Co + Mn): Zr = 1.07: 1: 0.005, and the mixed raw material was mixed. Obtained. The obtained mixed raw material was fired at 885 ° C. for 15 hours in an air atmosphere to obtain a sintered body. The obtained sintered body was pulverized and subjected to a dry sieve to obtain a lithium transition metal composite oxide represented by _{the composition formula Li 1.07} Ni _0.4 Co _0.3 Mn _0.3 Zr _0.005 O _2. It was.

0.5 mol% boric acid as a raw material for the boron compound was mixed with the obtained lithium transition metal composite oxide with a high-speed shear type mixer to obtain a raw material mixture. The obtained raw material mixture was calcined in the air at 250 ° C. for 10 hours to obtain the desired positive electrode composition.

The desired positive electrode composition was obtained in the same manner as in Example 1 except that the boric acid to be mixed was 1.0 mol% with respect to the lithium composite oxide.

A nickel-cobalt-manganese composite hydroxide precipitate was obtained in the same manner as in Example 1 except that Ni: Co: Mn = 5: 2: 3. The obtained precipitate was washed with water, filtered, and separated in the same manner as in Example 1 except that Li: (Ni + Co + Mn): Zr = 1.14: 1: 0.005, and the composition formula Li _1.14 Ni _0.5. A lithium transition metal composite oxide represented by Co _0.2 Mn _0.3 Zr _0.005 O _{2 was obtained.}

0.3 mol% boric acid as a raw material for the boron compound is mixed with the obtained lithium transition metal composite oxide with a high-speed shear type mixer to obtain a raw material mixture. The obtained raw material mixture was calcined in the air at 250 ° C. for 10 hours to obtain the desired positive electrode composition.

The desired positive electrode composition was obtained in the same manner as in Example 3 except that the boric acid to be mixed was 0.5 mol% with respect to the lithium composite oxide.

The desired positive electrode composition was obtained in the same manner as in Example 3 except that the boric acid to be mixed was 1.3 mol% with respect to the lithium composite oxide.

The desired positive electrode composition was obtained in the same manner as in Example 3 except that the raw material mixture was calcined in the air at 350 ° C. for 10 hours.

The desired positive electrode composition was obtained in the same manner as in Example 3 except that the raw material mixture was calcined in the air at 500 ° C. for 10 hours.

The desired positive electrode composition was obtained in the same manner as in Example 3 except that 0.5 mol% lithium metaborate was used as a raw material for the boron compound.

[Comparative Example 1]
The lithium transition metal composite oxide of Example 1 was used for comparative examples.

[Comparative Example 2]
The lithium transition metal composite oxide of Example 3 was used for comparative examples.

[Evaluation of output characteristics]
For Examples 1 to 8 and Comparative Examples 1 and 2, DC-IR (DC internal resistance) was measured as follows.

[1. Preparation of cathode]
85 parts by weight of the positive electrode composition, 10 parts by weight of acetylene black, and 5.0 parts by weight of PVDF (polyvinylidene fluoride) were dispersed in NMP (normal methyl-2-pyrrolidone) to prepare a positive electrode slurry. The obtained positive electrode slurry was applied to an aluminum foil, dried, compression-molded with a roll press, and cut into a predetermined size to obtain a positive electrode.

[2. Fabrication of negative electrode]
A negative electrode slurry was prepared by dispersing 97.5 parts by weight of artificial graphite, 1.5 parts by weight of CMC (carboxymethyl cellulose), and 1.0 part by weight of SBR (styrene butadiene rubber) in water. The obtained negative electrode slurry was applied to a copper foil, dried, compression-molded with a roll press, and cut into a predetermined size to obtain a negative electrode.

[3. Preparation of non-aqueous electrolyte solution]
EC (ethylene carbonate) and MEC (methyl ethyl carbonate) were mixed at a volume ratio of 3: 7 to prepare a solvent. Lithium hexafluorophosphate (LiPF ₆ ) was dissolved in the obtained mixed solvent so that its concentration was 1 mol / l to obtain a non-aqueous electrolytic solution.

[4. Assembly of evaluation battery]
After attaching lead electrodes to the current collectors of the positive electrode and the negative electrode, vacuum drying was performed at 120 ° C. Next, a separator made of porous polyethylene was arranged between the positive electrode and the negative electrode, and they were stored in a bag-shaped laminate pack. After storage, it was vacuum dried at 60 ° C. to remove the moisture adsorbed on each member. After vacuum drying, the above-mentioned non-aqueous electrolyte solution was injected and sealed in the laminate pack to obtain a laminate-type non-aqueous electrolyte secondary battery for evaluation.

[5. DC-IR measurement]
A weak current was passed through the obtained battery for aging, and the positive electrode and the negative electrode were sufficiently blended with the electrolyte. After that, discharging with a high current and charging with a weak current were repeated. The charge capacity in the 10th charge was defined as the total charge capacity of the battery, and after the 10th discharge, the battery was charged to 40% of the total charge capacity. After charging, the battery was placed in a constant temperature bath set at −25 ° C., left for 6 hours, discharged at 0.02A, 0.04A, and 0.06A, and the voltage was measured. The intersections were plotted with current on the horizontal axis and voltage on the vertical axis, and the slope of the straight line connecting the intersections was defined as DC-IR. A low DC-IR means that the output characteristics are good.

[Viscosity measurement of positive electrode slurry]
For Examples 1 and 2 and Comparative Example 1, the viscosity of the positive electrode slurry was measured as follows.

[1. Initial viscosity measurement]
30 g of the positive electrode composition, 1.57 g of PVDF, and 12.48 g of NMP were placed in a 150 ml polyethylene container and kneaded at room temperature (about 25 ° C.) for 5 minutes. After kneading, the obtained slurry was immediately measured with an E-type viscometer. A cone plate type was used as the blade type, and the rotation speed of the rotor was 5 rpm. In this way, the measured value of the initial viscosity was obtained.

[2. Evaluation of viscosity change]
Next, the slurry in the polyethylene container was left to stand in a constant temperature bath at 60 ° C., and the viscosity was measured after 24 hours and 48 hours. Before the measurement, the mixture was kneaded at room temperature for 2 minutes.

[Cycle characterization]
The cycle characteristics of Examples 1 to 8 and Comparative Examples 1 and 2 were measured as follows.

The secondary battery for evaluation similar to that for evaluation of output characteristics was aged with a weak current, and the positive electrode and the negative electrode were sufficiently blended with the electrolyte. After aging, the battery is placed in a constant temperature bath set at 20 ° C., and charged with a charging potential of 4.2 V and a charging current of 1.0 C (current that discharges in 1 C≡1 hour), and a discharge potential of 2.75 V. Discharge at a discharge current of 1.0 C was set as one cycle, and charging and discharging were repeated. The value obtained by dividing the discharge capacity of the nth cycle by the discharge capacity of the first cycle was defined as the discharge capacity retention rate of the nth cycle. A high discharge capacity retention rate means that the cycle characteristics are good.

Table 1 shows the boron content (B content) in the lithium transition metal composite oxide (Structure A) of Examples 1 to 8 and Comparative Examples 1 and 2, the raw material compound of the boron compound (Structure B), and the positive electrode composition. , Battery characteristics are shown in Table 2. Further, FIG. 1 shows a change in viscosity of the positive electrode slurry using the positive electrode compositions of Examples 1 and 2 and Comparative Example 1.

From Tables 1 and 2, when boric acid was used as the raw material compound of the boron compound and the firing steps were performed, Examples 1 and 2 or Examples 3 to 8 had output characteristics and output characteristics as compared with Comparative Example 1 or 2 without the boron compound. It can be seen that both cycle characteristics are improved. Further, from FIG. 1, in Examples 1 and 2 in which boric acid was used as the raw material compound of the boron compound and the firing step was performed, there was almost no change in the viscosity of the positive electrode slurry, whereas in Comparative Example 1 without the boron compound, the positive electrode was rapidly formed. It can be seen that the viscosity of the slurry increases and reaches the measurement limit (40,000 mPa · s) after 24 hours.

By using the positive electrode composition of the present invention, it is possible to achieve both the features of lithium nickel cobalt manganate, high output characteristics and cycle characteristics in a non-aqueous electrolyte secondary battery. Further, since the increase in the viscosity of the positive electrode slurry is suppressed, the operability and the yield at the time of manufacturing the battery are improved. Based on these facts, the non-aqueous electrolyte secondary battery using the positive electrode composition of the present invention is used not only for mobile devices such as mobile phones, notebook computers and digital cameras, but also for powering large devices such as electric vehicles. Is also preferably available.

Claims (9)

General formula Li _a Ni _1-x-y Co _x Mn _y M _z O ₂ (1.00 ≦ a ≦ 1.50, 0 <x ≦ 0.50, 0 <y ≦ 0.50, 0.00 ≦ z ≦ 0.02,0.40 ≦ x + y ≦ 0.60 , M is Zr, Ti, Mg, and the lithium transition metal composite oxide represented by at least one) of Ta, is selected from the group consisting of Nb and Mo,
Contains at least oxygen-containing boron compounds
A positive electrode composition for a non-aqueous electrolyte secondary battery, which is a calcined product of a mixture of the lithium transition metal composite oxide and solid orthoboric acid. The positive electrode composition for a non-aqueous electrolyte secondary battery according to claim 1, wherein the content of the boron compound is 2.0 mol% or less as boron with respect to the lithium transition metal composite oxide. The positive electrode composition for a non-aqueous electrolytic solution secondary battery according to claim 1 or 2, wherein the fired body is obtained at a firing temperature of 450 ° C. or lower. The positive electrode composition for a non-aqueous electrolyte secondary battery according to any one of claims 1 to 3, wherein in the above general formula, 0.05 ≦ x ≦ 0.35 and 0.05 ≦ y ≦ 0.35. Stuff A non-aqueous electrolytic solution secondary battery using the positive electrode composition according to any one of claims 1 to 4 as a positive electrode. General formula Li _a Ni _1-x-y Co _x Mn _y M _z O ₂ (1.00 ≦ a ≦ 1.50, 0 <x ≦ 0.50, 0 <y ≦ 0.50, 0.00 ≦ z ≦ 0.02,0.40 ≦ x + y ≦ 0.60 , M is Zr, Ti, Mg, and the lithium transition metal composite oxide represented by at least one) of Ta, is selected from the group consisting of Nb and Mo,
A method for producing a positive electrode composition for a non-aqueous electrolyte secondary battery, which comprises at least an oxygen-containing boron compound.
The synthetic step of synthesizing the lithium transition metal composite oxide and
A mixing step of mixing the lithium transition metal composite oxide obtained in the synthesis step with solid orthoboric acid, which is a raw material compound of the boron compound, to obtain a raw material mixture.
A firing step of firing the raw material mixture obtained in the mixing step, and a firing step.
A method for producing a positive electrode composition for a non-aqueous electrolyte secondary battery, which comprises. The production method according to claim 6, wherein the firing temperature in the firing step is 450 ° C. or lower. The production method according to claim 6 or 7, wherein the mixing amount of the raw material compound in the mixing step is 2.0 mol% or less as boron with respect to the lithium transition metal composite oxide. The production method according to any one of claims 6 to 8, wherein the general formula is 0.05 ≦ x ≦ 0.35 and 0.05 ≦ y ≦ 0.35.

Images