JP6429267B2 - Polymer - Google Patents

Description

  The present invention relates to a novel polymer.

Optical polymers are at the center of the optical technology field that is responsible for recording, displaying, and transmitting information, such as high-performance lenses, anti-reflective coatings, flat displays, optical discs, and optical fibers. A rate is required.
Conventionally, inorganic materials such as glass and ceramics have been widely used in optical components such as lenses. However, due to the recent increase in needs for weight reduction and price reduction, replacement of inorganic materials with resins in many applications Is progressing.

  Resin is difficult to control the refractive index compared to inorganic materials, but in order to obtain a high refractive index resin, introduction of an element or group having a high polarizability is effective, and halogen atoms other than fluorine, sulfur atoms, metal atoms It is considered that introduction of an aromatic ring or the like is effective. In particular, many high refractive index polymers containing sulfur have been reported (Non-Patent Documents 1 to 5).

It has also been reported that NN═C bonds and C—N═C bonds exhibit higher molecular refraction than C═C bonds (Non-patent Document 2).
As a polymer having such a bond, Patent Document 1 describes a triazine unit-containing poly (phenylene thioether). However, it is known that triazine units are isomerized by heat in the presence of a catalyst (Patent Document 2).
Patent Document 3 describes a fluorene unit-containing polymer.

JP 2011-219643 A Special table 2009-542755 JP 2012-38785 A

Matsuda, T .; Funae, Y .; Yoshida, M; Takaya, T., J. Macromol. Sci .., Pure Appl. Chem 1999, A36, 1271-1288. Choi, M .; Wakita, J .; Ha, C .; Ando, S., Macromolecules, 2009, 42, (14), 5112-5120. Liu, J.-g .; Nakamura, Y .; Shibasaki, Y .; Ando, S .; Ueda, M., Macromolecules, 2007, 40, (13), 4614-4620. Paquet, C .; Cyr, P. W .; Kumacheva, E .; Manners, I., Chem. Common. (Cambridge, U. K.) 2004, 234-235. Manners, I., J. Opt. A: Pure Appl. Opt. 2002, 4, S221-S223.

However, there is a further need for a polymer that is well balanced in heat resistance, high refractive properties, and moldability, and there is room for improvement in the polymer.
This invention is made | formed in view of the above, and makes it a subject to provide the polymer which is excellent in heat resistance, high refractive property, and a moldability with sufficient balance.

  The present inventors have intensively studied to solve the above problems. As a result, it has been found that the above problems can be solved by using a polymer having a specific structural unit, and the present invention has been completed. Embodiments of the present invention can be shown in [1] to [2] below.

［式（１）中、
Ｒ¹、Ｒ²及びＲ³はそれぞれ独立して、炭素数１〜１２の１価の有機基である。
ａは０〜２の整数であり、ａが２である場合、２つのＲ¹は同一でも異なっていてもよく、互いに結合して環構造の一部を形成してもよい。
ｂ及びｃはそれぞれ独立して、０〜４の整数である。ｂが２以上の場合、複数のＲ²は、同一でも異なっていてもよく、少なくとも２つのＲ²が互いに結合して環構造の一部を形成してもよい。ｃが２以上の場合、複数のＲ³は、同一でも異なっていてもよく、少なくとも２つのＲ³が互いに結合して環構造の一部を形成してもよい。
Ｙは独立して、−Ｏ−又は−Ｓ−である。
Ｚは、−Ｏ−、−Ｓ−、−ＣＯ−、−ＳＯ−、−ＳＯ₂−又は２価の有機基である。
ｄは０〜２の整数である。ｄが２の場合、複数のＲ³、Ｚ及びｃはそれぞれ、同一であっても異なっていてもよい。］ [1] A polymer having a structural unit represented by the following formula (1).

[In Formula (1),
R ¹ , R ² and R ³ are each independently a monovalent organic group having 1 to 12 carbon atoms.
a is an integer of 0 to 2, and when a is 2, two R ^{1 s} may be the same or different and may be bonded to each other to form a part of a ring structure.
b and c are each independently an integer of 0 to 4. When b is 2 or more, the plurality of R ² may be the same or different, and at least two R ² may be bonded to each other to form a part of the ring structure. When c is 2 or more, the plurality of R ³ may be the same or different, and at least two R ³ may be bonded to each other to form a part of the ring structure.
Y is independently —O— or —S—.
Z is —O—, —S—, —CO—, —SO—, —SO ₂ — or a divalent organic group.
d is an integer of 0-2. When d is 2, the plurality of R ³ , Z and c may be the same or different. ]

［式（２）中、
Ｒ⁴及びＲ⁵はそれぞれ独立して、炭素数１〜１２の１価の有機基である。
ｇ及びｈはそれぞれ独立して、０〜６の整数である。ｇが２以上の場合、複数のＲ⁴は、同一でも異なっていてもよく、少なくとも２つのＲ⁴が互いに結合して環構造の一部を形成してもよい。ｈが２以上の場合、複数のＲ⁵は、同一でも異なっていてもよく、少なくとも２つのＲ⁵が互いに結合して環構造の一部を形成してもよい。
ｅ及びｆはそれぞれ独立して、０又は１である。］ [2] The polymer according to [1], wherein Z in the formula (1) is -S- or a group represented by the following formula (2).

[In Formula (2),
R ⁴ and R ⁵ are each independently a monovalent organic group having 1 to 12 carbon atoms.
g and h are each independently an integer of 0-6. When g is 2 or more, the plurality of R ⁴ may be the same or different, and at least two R ⁴ may be bonded to each other to form a part of the ring structure. When h is 2 or more, the plurality of R ⁵ may be the same or different, and at least two R ⁵ may be bonded to each other to form a part of the ring structure.
e and f are each independently 0 or 1. ]

According to the present invention, it is possible to provide a polymer having excellent balance between heat resistance, high refractive property and moldability.
Furthermore, according to the present invention, a polymer having a high refractive index and light transmittance and excellent heat resistance can be provided.

FIG. 1 is a ¹ H NMR spectrum of polymer P1 obtained in Example 1. FIG. 2 is a ¹³ C NMR spectrum of the polymer P1 obtained in Example 1. FIG. 3 is a DSC curve of the polymer P1 obtained in Example 1. 4 is a UV-vis spectrum of the polymer P1 obtained in Example 1. FIG. FIG. 5 is a ¹ H NMR spectrum of the polymer P2 obtained in Example 2.

≪Polymer≫
The polymer according to the present invention is a polymer having a structural unit represented by the following formula (1) (hereinafter also referred to as “polymer (1)”). The polymer (1) is excellent in heat resistance, high refractive property and moldability with a good balance.

［式（１）中、
Ｒ¹、Ｒ²及びＲ³はそれぞれ独立して、炭素数１〜１２の１価の有機基である。
ａは０〜２の整数であり、ａが２である場合、２つのＲ¹は同一でも異なっていてもよく、互いに結合して環構造の一部を形成してもよい。
ｂ及びｃはそれぞれ独立して、０〜４の整数である。ｂが２以上の場合、複数のＲ²は、同一でも異なっていてもよく、少なくとも２つのＲ²が互いに結合して環構造の一部を形成してもよい。ｃが２以上の場合、複数のＲ³は、同一でも異なっていてもよく、少なくとも２つのＲ³が互いに結合して環構造の一部を形成してもよい。
Ｙは独立して、−Ｏ−又は−Ｓ−である。
Ｚは、−Ｏ−、−Ｓ−、−ＣＯ−、−ＳＯ−、−ＳＯ₂−又は２価の有機基である。
ｄは０〜２の整数である。ｄが２の場合、複数のＲ³、Ｚ及びｃはそれぞれ、同一であっても異なっていてもよい。］

[In Formula (1),
R ¹ , R ² and R ³ are each independently a monovalent organic group having 1 to 12 carbon atoms.
a is an integer of 0 to 2, and when a is 2, two R ^{1 s} may be the same or different and may be bonded to each other to form a part of a ring structure.
b and c are each independently an integer of 0 to 4. When b is 2 or more, the plurality of R ² may be the same or different, and at least two R ² may be bonded to each other to form a part of the ring structure. When c is 2 or more, the plurality of R ³ may be the same or different, and at least two R ³ may be bonded to each other to form a part of the ring structure.
Y is independently —O— or —S—.
Z is —O—, —S—, —CO—, —SO—, —SO ₂ — or a divalent organic group.
d is an integer of 0-2. When d is 2, the plurality of R ³ , Z and c may be the same or different. ]

Examples of the monovalent organic group having 1 to 12 carbon atoms in R ¹ , R ² and R ³ include a monovalent hydrocarbon group having 1 to 12 carbon atoms and at least one carbon atom or hydrogen of the hydrocarbon group And a group in which the atom is substituted with at least one atom selected from the group consisting of an oxygen atom, a sulfur atom and a nitrogen atom.

Examples of such a group include a methyl group, an ethyl group, an n-propyl group, an isopropyl group, an n-butyl group, a sec-butyl group, a tert-butyl group, a methylthio group (—SCH ₃ ), and an ethylthio group (— SC ₂ H ₅ ), methylsulfonyl group (—SO ₂ CH ₃ ), cyclohexyl group, morpholino group, and phenyl group.

a is preferably 0 or 1.
The bond that binds to the pyrimidine ring (Y that binds to the pyrimidine ring) is preferably the meta position, and the 4th and 6th positions of the pyrimidine ring are preferable. When a is 1, R ¹ is preferably bonded to the 2-position of the pyrimidine ring.
When a is 2, examples of the ring structure that can be formed by bonding two R ^{1 to} each other include a cyclopentane ring, a cyclohexane ring, a cyclopentadiene ring, and a benzene ring. Between the carbon-carbon bonds in the ring structure, -O-, -CO-, -COO-, -OCOO-, -NR ^* -(R ^* is a hydrogen atom or a monovalent hydrocarbon group. ), - CS -, - S -, - SO -, - SO 2 - and the like may be introduced.

b and c are each independently preferably 0.
When b or c is 2 or more, the ring structure formed by bonding at least two R ² or R ³ to each other is the same ring structure as the ring structure formed by bonding the two R ¹ to each other. Examples include the structure.

d is preferably 1.
When d is 1 or 2, the substitution positions of Z and Y on the benzene ring are preferably the 1 and 4 positions of the benzene ring to which the respective groups are bonded.

Z is preferably -S- or a group represented by the following formula (2).
When Z is -S-, the polymer has a high refractive index and light transmittance. Moreover, since this polymer is excellent in injection moldability, a molded body can be easily formed by injection molding.
On the other hand, when Z is a group represented by the following formula (2), the polymer is more excellent in heat resistance.

［式（２）中、
Ｒ⁴及びＲ⁵はそれぞれ独立して、炭素数１〜１２の１価の有機基である。
ｇ及びｈはそれぞれ独立して、０〜６の整数である。ｇが２以上の場合、複数のＲ⁴は、同一でも異なっていてもよく、少なくとも２つのＲ⁴が互いに結合して環構造の一部を形成してもよい。ｈが２以上の場合、複数のＲ⁵は、同一でも異なっていてもよく、少なくとも２つのＲ⁵が互いに結合して環構造の一部を形成してもよい。
ｅ及びｆはそれぞれ独立して、０又は１である。］

[In Formula (2),
R ⁴ and R ⁵ are each independently a monovalent organic group having 1 to 12 carbon atoms.
g and h are each independently an integer of 0-6. When g is 2 or more, the plurality of R ⁴ may be the same or different, and at least two R ⁴ may be bonded to each other to form a part of the ring structure. When h is 2 or more, the plurality of R ⁵ may be the same or different, and at least two R ⁵ may be bonded to each other to form a part of the ring structure.
e and f are each independently 0 or 1. ]

Examples of the monovalent organic group having 1 to 12 carbon atoms in R ⁴ and R ⁵ are the same groups as those exemplified for the monovalent organic group having 1 to 12 carbon atoms in R ¹ , R ² and R ³ . Etc.
e and f are each independently preferably 0.

g and h are each independently preferably 0.
When g or h is 2 or more, the ring structure formed by bonding at least two R ⁴ or R ⁵ to each other is the same ring structure as the ring structure formed by bonding the two R ¹ to each other. Etc.
The case where g or h is 5 or 6 means that when e or f is 1, respectively, 5 or 6 R ⁴ or R ⁵ are bonded to the naphthalene ring.

  The polymer (1) may contain a structural unit other than the structural unit represented by the formula (1). Another structural unit will not be specifically limited if it does not impair the effect of this invention.

[Physical properties of polymer (1)]
The number average molecular weight (Mn) in terms of polystyrene is preferably 1,000 to 400,000, more preferably 10,000 to 200, because the polymer (1) is a polymer having excellent heat resistance. The molecular weight distribution (Mw / Mn) is preferably 1.5 to 12.0, and more preferably 1.7 to 6.0.
In addition, the said molecular weight can be measured by the method as described in the following Example.

The polymer (1) has a refractive index (nD) of preferably 1.63 or more, more preferably 1.65 or more, with respect to light of D line (wavelength 589 nm). A polymer having a refractive index in the above range can be said to be a polymer having a high refractive index.
In addition, the said refractive index can be measured by the method as described in the following Example.

The polymer (1) preferably has a thermal decomposition temperature T _d5 (5% weight loss temperature) measured by thermogravimetric analysis (TGA) of 200 to 600 ° C., more preferably 250 to 600 ° C.
A polymer having T _d5 in the above range can be said to be a polymer having excellent heat resistance.
T _d5 can be specifically measured by the method described in the following examples.

The polymer (1) preferably has a glass transition temperature Tg measured by differential scanning calorimetry (DSC) of 100 to 250 ° C, more preferably 120 to 230 ° C.
A polymer having Tg in the above range is excellent in moldability.
In addition, Tg can be specifically measured by the method described in the following examples.

≪Method of polymer synthesis≫
In the polymer (1), for example, a compound represented by the following formula (A) and a compound represented by the following formula (B) are mixed with N, N-dimethylacetamide in the presence of an alkali metal compound such as potassium carbonate. It can be obtained by heating and polymerizing in a suitable organic solvent such as (DMAc).

［式（Ａ）中、Ｒ¹及びａはそれぞれ、式（１）中のＲ¹及びａと同義であり、Ｘはハロゲン原子である。］

Wherein (A), R ¹ and a are the same as R ¹ and a in the formula (1), X is a halogen atom. ]

Wherein ^{(B), R 2, R} 3, Z, b, respectively c and d, the same meanings as R ² in formula ^{(1), R 3, Z} , b, c and d.
In formula (B), each R ^a independently represents a hydrogen atom, a methyl group, an ethyl group, an acetyl group, a methanesulfonyl group or a trifluoromethylsulfonyl group, and among them, a hydrogen atom is preferable.

The heating conditions are not particularly limited as long as the polymerization reaction proceeds, but the heating temperature is preferably 60 to 250 ° C, more preferably 80 to 200 ° C, and the heating time is preferably 0.5 to 100. Time, more preferably 1 to 24 hours.
In particular, the heating temperature is preferably about 60 to 110 ° C., the heating time is 4 to 20 hours, and a high molecular weight polymer with a small amount of low molecular weight components from the viewpoint of obtaining a polymer without coloring. In view of obtaining (1), the heating temperature is more preferably about 100 ° C.

  In the polymer (1), the use ratio of the compound represented by the formula (A) and the compound represented by the formula (B) is the compound represented by the formula (A) and the compound represented by the formula (B). When the total amount of the compounds is 100 mol%, the compound represented by the formula (A) is preferably 45 mol% to 55 mol%, more preferably 50 mol% to 52 mol%, still more preferably 50 mol%. The compound represented by the formula (B) is preferably 45 mol% or more and 55 mol% or less, more preferably 48 mol% or more and 50 mol% or less, and further preferably 48 mol% or more. It is less than 50 mol%.

  Examples of the alkali metal compound used in the reaction include alkali metals such as lithium, potassium and sodium; alkali hydrides such as lithium hydride, potassium hydride and sodium hydride; lithium hydroxide, potassium hydroxide and sodium hydroxide And alkali metal carbonates such as lithium carbonate, potassium carbonate and sodium carbonate; alkali metal hydrogen carbonates such as lithium hydrogen carbonate, potassium hydrogen carbonate and sodium hydrogen carbonate. These can be used alone or in combination of two or more.

With respect to the amount of the alkali metal compound used, the amount of metal atoms in the alkali metal compound is usually 1 to 3 times equivalent, preferably 1 —O—R ^a in the compound represented by the formula (B). Is used in an amount of 1.1 to 2 times equivalent, more preferably 1.2 to 1.5 times equivalent.

  EXAMPLES Hereinafter, although this invention is demonstrated more concretely based on an Example, this invention is not limited to a following example at all.

  Analytical conditions for the polymers obtained in the following examples are as shown below.

<NMR spectrum>
The NMR (Nuclear Magnetic Resonance) spectra of the polymers obtained in Examples 1 and 2 below were measured as follows.
¹ H NMR spectrum and ¹³ C NMR were measured using JEM-EX400 FT NMR manufactured by JEOL Ltd. The solvent was deuterated chloroform, and tetramethylsilane (0 ppm) was used as a reference substance. The analysis program Delta version. 4.3.4 manufactured by JEOL Ltd. was used for the analysis. For the chemical shift, the value analyzed by the program was used as it was.

<Measurement of number average molecular weight and molecular weight distribution>
Number average molecular weight (Mn) and molecular weight distribution (Mw / Mn) were measured using size exclusion chromatography (manufactured by Tosoh Corporation, HLC-8320GPC, UV-8320).
In addition, the liquid feeding speed was 0.35 mL / min, and the column oven was set to 40 ° C. for measurement. Two analytical columns, TSKgel Super Multipore HZ-M (4.6 mm I.D. × 15 cm), were connected in series, and chloroform was used as the solvent.
The calibration curve was prepared using polystyrene standard samples (Mn = 427000, 190000, 96400, 37900, 18100, 10200, 5970, 2630, 1050, 500).

<Thermogravimetric analysis>
Thermogravimetric analysis (TGA) was performed by using a TGA / 6200 manufactured by Seiko Instruments Inc., and using an aluminum pan under a nitrogen atmosphere at a heating rate of 10 ° C./min to measure a 5% weight loss temperature. .

<Measurement of refractive index>
A polymer dissolved in an appropriate amount of methylene chloride was coated on a glass plate and dried overnight at room temperature and normal pressure to form a coating film. Next, residual methylene chloride was removed from the obtained coating film with a vacuum dryer to obtain a polymer film. The refractive index of this film was measured using a prism coupler (model PC-2010) manufactured by Metricon. The refractive index was measured using light of three wavelengths of 408, 633, and 828 nm, and the refractive index (nD) for the D line (589 nm) was determined using the Cauchy equation.

<DSC measurement>
Differential scanning calorimetry (DSC) measurement was performed using a DSC6200 manufactured by Seiko Instruments Inc. and using an aluminum pan in a nitrogen atmosphere. Specifically, the temperature was raised from 100 ° C. to 220 ° C. at 20 ° C./min (1st Heating). Thereafter, the temperature was lowered to 100 ° C. at 20 ° C./min, and then raised to 220 ° C. at 20 ° C./min (2nd Heating).

[Example 1]
Under a nitrogen atmosphere, add 4,6-Dichloropyrimidine (0.75g, 5mmol), Bis (4-hydroxypenyl) Sulfide (1.09g, 5mmol), Potassium carbonate (2.07g, 15mmol) and DMAc (11mL) to a 2-neck eggplant flask. The reaction was carried out at 100 ° C. for 6 hours. After completion of the reaction, isolation and purification were performed by reprecipitation using distilled water and methanol. Thereafter, the precipitate was collected and dried under reduced pressure at 80 ° C. to obtain a white solid (polymer P1). The yield was 91%.

The structure of the obtained polymer P1 was confirmed by ¹ H NMR and ¹³ C NMR. The results are shown in FIGS. 1 and 2, respectively.
The Mn of the polymer P1 was 108000, Mw / Mn was 1.76, the refractive index (nD) was 1.682, and the 5% weight loss temperature was 436 ° C.

The result of DSC measurement of the obtained polymer P1 is shown in FIG.
From the result of DSC measurement, it was found that the glass transition temperature of the polymer P1 was 132 ° C. In 1st Heating, a melting point was observed at 186 ° C. This suggests that crystallization may have occurred.

Of the obtained polymer P1, cyclohexanone (CHO), CHCl ₃ , tetrahydrofuran (THF), dimethyl sulfoxide (DMSO), CH ₂ Cl ₂ , N, N-dimethylacetamide (DMAc), N, N-dimethylformamide (DMF) ), N-methyl-2-pyrrolidone (NMP) and tetrachloroethane (TCE) were evaluated.
When the polymer P1 (2 mg) was placed in 1 mL of each of the above solvents at room temperature and stirred for 10 minutes, the polymer dissolved was defined as (++), and was not dissolved at room temperature for 10 minutes, but when heated to 80 ° C. The case where it melt | dissolved was set to (+), and the case where it did not melt | dissolve even by the heating by 80 degreeC was evaluated as (-). The results are shown in Table 1.

  The polymer P1 showed good solubility in chloroform and tetrachloroethane. Moreover, a transparent self-supporting film could be produced from the tetrachloroethane solution by a solution coating method or the like.

Using a solution obtained by dissolving the obtained polymer P1 (2 mg) in 1 mL of tetrachloroethane, a thin film having a thickness of 10 μm made of the polymer P1 was prepared. The light transmittance of the polymer P1 was evaluated using this thin film and an ultraviolet-visible photometer V-630BIO (manufactured by JASCO). The results are shown in FIG.
The polymer P1 showed a high light transmittance of 90% or more with respect to light having a wavelength of 350 nm or more. Moreover, even after the said thin film was heat-processed (under nitrogen, 230 degreeC, 6 hours), it became clear that the light transmittance did not change and the polymer P1 showed the outstanding heat resistance.

[Example 2]
Under a nitrogen atmosphere, in a 2-neck eggplant flask, 4,6-Dichloropyrimidine (0.45g, 3mmol), Bis (4-hydroxypenyl) Sulfide (0.33g, 1.5mmol), 9,9-Bis (4-hydroxyphenyl) fluorene (0.53 g, 1.5 mmol), potassium carbonate (1.24 g, 9 mmol) and DMAc (7.5 mL) were charged and reacted at 100 ° C. for 18 hours. After completion of the reaction, isolation and purification were performed by reprecipitation using distilled water and methanol. Thereafter, the precipitate was collected and dried under reduced pressure at 80 ° C. to obtain a white solid (polymer P2). The yield was 93%.

The structure of the obtained polymer P2 was confirmed by ¹ H NMR. The results are shown in FIG.
The Mn of the polymer P2 was 104000, the Mw / Mn was 2.11. The refractive index (nD) was 1.680, and the 5% weight loss temperature was 375 ° C. The glass transition temperature determined by DSC measurement was 205 ° C.

Of the obtained polymer P2, cyclohexanone (CHO), CHCl ₃ , tetrahydrofuran (THF), dimethyl sulfoxide (DMSO), CH ₂ Cl ₂ , N, N-dimethylacetamide (DMAc), N, N-dimethylformamide (DMF) ), N-methyl-2-pyrrolidone (NMP) and tetrachloroethane (TCE) were evaluated.
When the polymer P2 (2 mg) was added to 1 mL of each of the above solvents at room temperature and stirred for 10 minutes, the case where the polymer was dissolved was defined as (++), and was not dissolved at room temperature for 10 minutes. The case where it melt | dissolved was set to (+), and the case where it did not melt | dissolve even by the heating by 80 degreeC was evaluated as (-). The results are shown in Table 2.

  The solubility was improved by introducing a fluorene skeleton having a cardo structure. The polymer P2 showed excellent solubility in many organic solvents such as chloroform, tetrachloroethane, and cyclohexanone. Further, by using a solution obtained by dissolving the polymer P2 in these organic solvents, a polymer film could be easily prepared by a solution coating method or the like.

  The polymer (1) of the present invention has a high refractive index and excellent heat resistance. Therefore, the polymer (1) of the present invention is suitably used in the optical technical field responsible for recording / displaying / transmitting information, such as high-performance lenses, antireflection coatings, flat displays, optical disks, optical fibers, etc. It is suitably used for optical parts such as optical films such as retardation plates, conical lenses, spherical lenses, special lenses such as cylindrical lenses, and lens arrays.

Claims (1)

Have a structural unit represented by the following formula (1),
The number average molecular weight (Mn) in terms of polystyrene is 1,000 to 400,000, and the molecular weight distribution (Mw / Mn) is 1.5 to 12.0.
Polymer.

[In Formula (1),
R ¹ , R ² and R ³ are each independently a monovalent organic group having 1 to 12 carbon atoms.
a is an integer of 0 to 2, and when a is 2, two R ^{1 s} may be the same or different and may be bonded to each other to form a part of a ring structure.
b and c are each independently 0.
Y is independently —O— or —S—.
Z is -S-.
d is 1 or 2. When d is 2, the plurality of R ³ , Z and c may be the same or different. ]

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