K. Haydukivska 1,2,V. Blavatska 1,31- Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine2- Institute of Physics, University of Silesia, 75 PuΕku Piechoty 1, 41-500 ChorzΓ³w, Poland3- Dioscuri Centre for Physics and Chemistry of Bacteria, Institute of Physical Chemistry, Polish Academy ofSciences, 01-224 Warsaw, PolandReceived July 26, 2023, in final form October 18, 2023We analyze the universal conformational properties of complex copolymer macromolecules based on twotopologies: the rosette structure containing ππ linear branches and ππ closed loops grafted to the central core, and the symmetric pom-pom structure, consisting of a backbone linear chain terminated by two branching points with functionalities π . We assume that the constituent strands (branches) of these structures can be of two different chemical species π and π. Depending on the solvent conditions, the inter- or intrachain interactions of some links may vanish, which corresponds to Ξ-state of the corresponding polymer species. Applying both the analytical approach within the frames of direct polymer renormalization and numerical simulations based on the lattice model of polymer, we evaluated the set of parameters characterizing the size properties of constituent parts of two complex topologies and estimated quantitatively the impact of interactions between constituent parts on these size characteristics.Key words: polymers, scaling, universal properties, renormalization group, numerical simulations
34) Scaling exponents Size characteristics exponents. As it was mentioned above, in the case of copolymer structure, we have three characteristic length scales, governed by two types of inter-chain interactions and and intra-chain 13301-8 Universal properties of branched copolymers in dilute solutions interaction , so that the segments of different species as well as correlations between segments are governed by different scaling exponents. Within the continuous chain model, the size exponents may be calculated using the expression : 2 1 = 2 + (cid:101) (cid:32) d ln 2 , d ln (cid:101) d ln 2 , d ln (cid:101) d ln 2 , d ln (cid:101) , (cid:101) + (cid:101) (cid:33) d ln 2 d ln (cid:101) + (cid:101) + (cid:101) . d ln 2 , d ln (cid:101) The estimates for the critical exponents governing correspondingly each of the three terms in (2.27) for the case of pom-pom structure read: = =
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= 1 2 1 2 1 2 ) , (1 + (cid:101) ) , + (cid:101) 2 (1 + (cid:101) (cid:18) 1 + (cid:101) (cid:19) , whereas for the case of rosette polymer we get: = = = 1 2 1 (1 + (cid:101) 2 (cid:18) 1 2 ) , (1 + (cid:101) ) , + 3(cid:101) 4 1 + (cid:101) (cid:19) . Similarly, like it was observed in the case of block copolymers in [43, 51, 52], scaling exponents and remain unchanged by the presence of the second species in the polymer structure and do not depend , one recovers the homopolymer behaviour. on the interaction between different species (cid:101) Though the universal exponent of a total gyration radius of the whole copolymer structure cannot by simply adding all the diagrams
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. With (cid:101) = (cid:101) be defined, we can derive an expression for the gyration radius 2 together. In general, it will read: 2 = 2 0 [1 + (. . .)] , being the Gaussian approximation and [1 + (. . .)] a swelling factor in one loop approximation with 2 in the coupling constants. According to the definition (2.35) we may evaluate an effective critical exponent for the cases of pom-pom and rosette structures, correspondingly: 0 =
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= + (cid:18) 12 2 + 6 2 (cid:101) + 2 (cid:101) 18 2 + 8 + 1 + 6 (cid:101) 1 2 1 2 1 + (3 + 3 2)(cid:101) 4 + 6 2 + 8 + 2 2 (2 + 2 1)(cid:101) 4 ) . 2(6 2 + 8 + 2 2 + (cid:101) (cid:101) (cid:19) , Again, for = , we recover an expression for an exponent of a homopolymer. These exponents are not properly defined. They may have a more practical application, since they can be related to the one observed in the experiment. Such calculations were previously performed in simulations in reference for star copolymers for which the topology dependent effective exponents were obtained. (2.35) (2.36) (2.37) (2.38) (2.39) (2.40) (2.41) (2.42) (2.43) (2.44) 13301-9 K. Haydukivska, V. Blavatska 1.3 2.0 1.1 1.5 wab=/4,wa=0 0.0 wab=3/16,wa=/8 3.5 4.0f 1.4 wab=0,wa=0 wab=0,wa=/8 0.5 1.2 1.0
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