Process for the preparation of block copolymers for tire tread compositions and these copolymers

A process for the preparation of a functionalized, coupled or starred block copolymer which is usable in a sulphur-cross-linkable rubber composition comprising carbon black and of reduced hysteresis in the cross-linked state, one at least of said blocks consisting of a polyisoprene and at least one other block consisting of a diene elastomer other than a polyisoprene the molar ratio of units originating from one or more conjugated dienes of which is greater than 15%, wherein said process comprises:

• (i) copolymerization of one or more monomers comprising at least one conjugated diene other than isoprene using a catalytic system comprising a hydrocarbon solvent, a compound A of a metal of group IIIA, a compound B of an alkaline-earth metal and a polymer initiator C comprising a C—Li bond which is formed of a monolithiated non-functionalized polyisoprene intended to form said or each polyisoprene block, said polyisoprene block(s) having a number-average molecular weight Mn1 of between 2,500 and 20,000 g/mol, so that said block(s) formed of a diene elastomer other than a polyisoprene has/have a content of trans-1,4 linkages equal to or greater than 70% and a number-average molecular weight Mn2 of between 65,000 and 350,000 g/mol, and
• (ii) addition to the product of said copolymerization of a functionalizing, coupling or starring agent comprising one, two or at least three epoxy groups respectively, so that said block formed of a diene elastomer other than a polyisoprene is functionalized, coupled or starred.