Binder, electrode and lithium battery including the same, and method of preparing the binder

1. A binder comprising a cross-linked product of at least a first polymer, a second polymer, and a third polymer, wherein

the cross-linked product is cross-linked by at least two ester bonds or at least one ester bond and at least one amide bond;

the first polymer comprises polyimide, polyamic acid, a copolymer thereof, or a combination thereof, wherein the first polymer comprises a structural unit comprising an alkali metal and at least one hydroxyl functional group in the same or different structural units;

wherein the alkali metal is coordinated to a carbonyl group of the polyimide or the copolymer thereof, the alkali metal is bonded to a carboxyl group of the polyamic acid or the copolymer thereof, the alkali metal is coordinated to an amide group of the polyamic acid or the copolymer thereof, or a combination thereof;

the second polymer comprises poly(acrylic acid), poly(methacrylic acid), a copolymer thereof, or a combination thereof; and

the third polymer comprises polyvinyl alcohol, polyacrylamide, polymethacrylamide, a copolymer thereof, or a combination thereof.

2. The binder of claim 1 , wherein an amount of the alkali metal is about 0.2 equivalents to about 1 equivalent with respect to a total content of carboxyl groups, amide groups, and carbonyl groups of the first polymer.

3. The binder of claim 1 , wherein the first polymer comprises a structural unit represented by Formula 1, Formula 2, or a combination thereof:

wherein in Formulae 1 and 2,

Ar 1 is an aromatic ring group of a substituted or unsubstituted tetravalent C6-C24 arylene group, a substituted or unsubstituted tetravalent C6-C24 heteroarylene group, or a combination thereof,

Ar 2 is an aromatic ring group of a substituted or unsubstituted C6-C24 arylene group, a substituted or unsubstituted C6-C24 heteroarylene group, or a combination thereof,

each of the aromatic ring groups is independently a single aromatic ring, a group in which two or more aromatic rings are condensed, or a group in which the two or more aromatic rings are linked to each other via a single bond, —O—, —S—, —C(═O)—, —S(═O) 2 —, —Si(R a )(R b )— (wherein R a and R b are each independently a C1-C10 alkyl group), a C1-C10 alkylene group that is unsubstituted or substituted with a halogen atom, or —C(═O)═NH—,

M is an alkali metal,

Y is —OH, and

* is a binding site to an adjacent atom.

4. The binder of claim 3 , wherein Ar 1 of Formulae 1 and 2 comprises a residue having a structural unit of a formula of Group 1 below:

wherein in the formulae of Group 1,

each residue is substituted or unsubstituted, and

* is a binding site to an adjacent atom.

5. The binder of claim 3 , wherein Ar 2 of Formulae 1 and 2 comprises a residue having a structural unit of a formula of Group 2:

wherein in the formulae of Group 2,

each residue is substituted or unsubstituted,

Y a is —OH and

* is a binding site to an adjacent atom.

6. The binder of claim 3 , wherein an amount of the structural unit is 20% by weight or greater based on a total weight of the first polymer.

7. The binder of claim 1 , wherein the first polymer is represented by Formula 3, Formula 4, or a combination thereof:

wherein in the Formulae 3 and 4,

each Ar 3 and Ar 4 are the same or different, and are each independently an aromatic ring group of a substituted or unsubstituted C6-C24 arylene group, a substituted or unsubstituted C6-C24 heteroarylene group, or a combination thereof, the aromatic ring group is a single aromatic ring, a group in which two or more aromatic rings are condensed, or a group in which the two or more aromatic rings are linked to each other via a single bond, —O—, —S—, —C(═O)—, —S(═O) 2 —, —Si(R a )(R b )— where R a and R b are each independently a C1-C10 alkyl group, a C1-C10 alkylene group that is unsubstituted or substituted with a halogen atom, or —C(═O)═NH—,

each M is the same or different, and is lithium or sodium,

each Y 1 is the same or different, and is a hydrogen atom, a halogen atom, —COOH, —OH, —CHO, —C(═O)NH 2 , a substituted or unsubstituted C1-C10 alkyl group, a C6-C20 aryl group that is unsubstituted or substituted with a halogen atom, or a C2-C20 heteroaryl group that is unsubstituted or substituted with a halogen atom, or an combination thereof,

each Y 2 is the same or different, and is —OH, and

n and m are mole fractions in repeating units, wherein n and m for Formulae 3 and 4 are each independently 0<n<1, 0<m<1, and n+m=1.

8. The binder of claim 1 , wherein the first polymer is represented by Formula 5, Formula 6, or a combination thereof:

wherein in Formulae 5 and 6,

each R 1 , R 2 , R 3 , R 4 , R 5 , R 6 , R 7 , and R 8 is the same or different, and are each independently a hydrogen atom, a halogen atom, —COOH, —OH, —CHO, —C(═O)NH 2 , a substituted or unsubstituted C1-C10 alkyl group, a C6-C20 aryl group that is unsubstituted or substituted with a halogen atom, a C2-C20 heteroaryl group that is unsubstituted or substituted with a halogen atom, or a combination thereof,

each Ar 5 is the same or different, and is an aromatic ring group of a substituted or unsubstituted C6-C24 arylene group, a substituted or unsubstituted C6-C24 heteroarylene group, or a combination thereof,

the aromatic ring group is a single aromatic ring, a group in which two or more aromatic rings are condensed, or a group in which the two or more aromatic rings are linked to each other via a single bond, —O—, —S—, —C(═O)—, —S(═O) 2 —, —Si(R a )(R b )— where R a and R b are each independently a C1-C10 alkyl group, a C1-C10 alkylene group that is unsubstituted or substituted with a halogen atom, or C(═O)—NH—,

each Y 3 is the same or different, and is —OH, and

n and m are mole fractions in repeating units, wherein n and m for Formulae 5 and 6 are each independently 0<n<1, 0<m<1, and n+m=1.

9. The binder of claim 8 , wherein Ar 5 of Formulae 5 and 6 comprises a residue having a structural unit of a formula of Group 3:

wherein in the formulae of Group 3,

each residue is substituted or unsubstituted,

Y b is —OH and

* is a binding site to an adjacent atom.

10. The binder of claim 8 , wherein n and m for Formulae 5 and 6 are each independently 0.2<n<0.8, 0.2<m<0.8, and n+m=1.

11. The binder of claim 1 , wherein the first polymer is represented by Formula 7, Formula 8, Formula 9, Formula 10, or a combination thereof:

wherein in Formulae 7, 8, 9, and 10,

each R 1 , R 2 , R 3 , R 4 , R 5 , R 6 , R 7 , and R 8 is the same or different, and are each independently a hydrogen atom, a halogen atom, —COOH, —OH, —CHO, —C(═O)NH 2 , a substituted or unsubstituted C1-C10 alkyl group, a C6-C20 aryl group that is unsubstituted or substituted with a halogen atom, or a C2-C20 heteroaryl group that is unsubstituted or substituted with a halogen atom, or a combination thereof, and

n and m are mole fractions in repeating units, wherein n and m for Formulae 7, 8, 9, and 10 are each independently 0<n<1, 0<m<1, and n+m=1.

12. The binder of claim 1 , wherein the second polymer further comprises a structural unit comprising an alkali metal.

13. The binder of claim 12 , wherein an amount of the alkali metal is about 0.2 equivalents to about 1 equivalent with respect to a total content of carboxyl groups and amide groups of the first polymer.

14. The binder of claim 1 , wherein the second polymer is poly(acrylic acid).

15. The binder of claim 14 , wherein poly(acrylic acid) has a weight average molecular weight of about 10,000 Daltons to about 1,000,000 Daltons.

16. The binder of claim 1 , wherein the third polymer is polyvinyl alcohol.

17. The binder of claim 16 , wherein the polyvinyl alcohol has a saponification ratio of 90% or greater.

18. The binder of claim 1 , wherein the polyvinyl alcohol has a weight average molecular weight of about 10,000 Daltons to about 500,000 Daltons.

19. The binder of claim 1 , wherein an amount of the first polymer is in a range of about 3% by weight to about 30% by weight based on a total weight of the cross-linked product.

20. The binder of claim 1 , wherein an amount of the third polymer is in a range of about 5% by weight to about 60% by weight based on a total weight of the cross-linked product.

21. The binder of claim 1 , further comprising an antifoaming agent.

22. The binder of claim 1 , wherein the cross-linked product has a weight average molecular weight of about 10,000 Daltons to about 2,000,000 Daltons.

23. An electrode comprising:

the binder of claim 1 ; and

a positive active material or a negative active material.

24. The electrode of claim 23 , wherein the negative active material comprises at least one of a silicon active material, a tin active material, a silicon-tin alloy active material, and a silicon-carbon active material.

25. The electrode of claim 23 , wherein a surface of the electrode has an average roughness Ra of 3 micrometers or less.

26. The electrode of claim 23 , wherein a cohesion between the positive active material and the binder, between the negative active material and the binder, or between adjacent binders is 100 grams force per centimeter or greater.

27. A lithium battery comprising:

the electrode of claim 23 , as a first electrode;

a second electrode; and

an electrolyte disposed between the first electrode and the second electrode.

28. A method of preparing a binder, the method comprising:

mixing a first composition comprising a non-aqueous solvent and a first polymer comprising polyimide, polyamic acid, a copolymer thereof, or a combination thereof, wherein the first polymer comprises a structural unit comprising an alkali metal and a structural unit comprising at least one hydroxyl functional group; a second composition comprising water and a second polymer comprising poly(acrylic acid), poly(methacrylic acid), a copolymer thereof, or a combination thereof; and a third composition comprising water and a third polymer comprising polyvinyl alcohol, polyacrylamide, polymethacrylamide, a copolymer thereof, or a combination thereof to prepare a mixture; and

heat-treating the mixture to prepare a cross-linked product and prepare the binder.

29. The method of claim 28 , wherein the heat treating is performed at 150° C. or greater.