Star copolymers in porous environments: scaling and its manifestations
aa r X i v : . [ c ond - m a t . s o f t ] O c t Star copolymers in porous environments: scaling and its manifestations
V. Blavatska, ∗ C. von Ferber,
2, 3 and Yu. Holovatch
1, 4 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,79011 Lviv, Ukraine Applied Mathematics Research Centre, Coventry University, Coventry, UK Institute of Physics, Universit¨at Freiburg, D-79104 Freiburg, Germany Institut f¨ur Theoretische Physik, Johannes Kepler Universit¨at Linz, A-4040, Linz, Austria
We consider star polymers, consisting of two different polymer species, in a solvent subject toquenched correlated structural obstacles. We assume that the disorder is correlated with a power-law decay of the pair correlation function g ( x ) ∼ x − a . Applying the field-theoretical renormalizationgroup approach in d dimensions, we analyze different scenarios of scaling behavior working to firstorder of a double ε = 4 − d , δ = 4 − a expansion. We discuss the influence of the correlated disorderon the resulting scaling laws and possible manifestations such as diffusion controlled reactions inthe vicinity of absorbing traps placed on polymers as well as the effective short-distance interactionbetween star copolymers. PACS numbers: 82.35.Jk, 36.20.Fz, 64.60.ae, 64.60.F-
I. INTRODUCTION
Understanding the behavior of polymer macromolecules in solutions in the presence of structural obstacles is ofgreat interest in polymer physics. The presence of defects often leads to a large spatial inhomogeneity and may createpore spaces of fractal structure [1]. Such situations can be encountered in studying, e.g., polymer diffusion throughmicroporous membranes [2] or within colloidal solutions [3].Solutions of polymer macromolecules in disordered environment are subject to intensive studies. Numerous sim-ulations [4–9] and analytical studies [7, 10–16] have focussed on the case of uncorrelated structural defects at thepercolation threshold of the remaining accessible sites, this situation is shown to alter significantly the universal be-havior of polymer macromolecules. Recently, another special type of disorder has been brought to attention, whichdisplay correlations in mesoscopic scale. This case can be described within the frames of a model with long-range-correlated quenched defects, considered in Refs. [17–19] in the context of magnetic phase transitions. Here, structuraldefects are characterized by a pair correlation function g ( x ), which in d dimensions falls off at large distance x according to a power law: g ( x ) ∼ x − a . (1)In general, any value of 0 ≤ a ≤ d can be realized by defects that form clusters of fractal dimension d f = d − a .For integer dimension d f these include the following special cases: uncorrelated point-like defects ( d f = 0), mutuallyuncorrelated straight lines of random orientation ( d f = 1), mutually uncorrelated planes of random orientation( d f = 2). The influence of such long-range correlated defects on the universal properties of single polymer has beenanalyzed within the renormalization group approach in Refs. [20, 21].To describe the universal properties of polymer chains in good solvents, one may due to universality in the longchain limit consider the model of self-avoiding walks (SAWs) on a regular lattice [22, 23]. In particular, the averagesquare end-to-end distance h R e i and the number of configurations Z N of SAWs with N steps obey in the asymptoticlimit N → ∞ the following scaling laws: h R e i ∼ N ν , Z N ∼ R − η − /ν . (2)Here, the second equation shows the power law in terms of the effective polymer size R ≡ p h R e i ∼ N ν , ν and η areuniversal exponents that only depend on the space dimensionality d . For d = 3, high order renormalization groupestimates are [24] ν = 0 . ± . η = 0 . ± . f linear polymer chains or SAWs, linkedtogether at their end-points. The study of star polymers is of great interest since they serve as building blocks of ∗ [email protected] polymer networks [25, 26] and can be used to model complex polymer micellar systems and gels [27–29]. For a singlestar with f arms of N steps (monomers) each, the number of possible configurations obeys a power law in terms ofthe size R of the isolated chain of N monomers [25, 26]: Z N,f ∼ R η f − fη . (3)Here, the exponents η f are universal star exponents, depending on the number of arms f ( η = 0, η = 1 /ν − − η ).Scaling properties of star polymers are well studied both numerically [30–35] and analytically [26, 36–42]. It has beenshown that the presence of long-range-correlated disorder may have interesting consequences for the scaling propertiesof polymer stars such as entropic separation of polymers according to their architecture [43].Linking together polymers of different species, we receive non-homogeneous star polymers with a much richer scalingbehavior [44–47]. A particular case is the star copolymer, consisting of polymer chains of two different species. It hasbeen shown [45], that the number of configurations Z f f of a copolymer star with f arms of species 1 and f armsof species 2 scales as: Z f f ∼ ( R ) η f f − f η − f η , (4)where η f f constitutes a family of copolymer star exponents. These exponents are universal and depend only onspace dimension d and the number of chains f , f , as well as three different types of fixed points that govern the richscaling behavior [44].Depending on the temperature, a situation may occur, where one or more of the inter- or intrachain interactionsvanishes. Indeed, for each polymer system one finds a so-called Θ-temperature, at which attractive and repulsiveinteractions between monomers compensate each other (see, e.g., [22, 23]). Such a polymer chain can effectively bedescribed by a simple random walk (RW). In this case, scaling laws (Eq. (2)) hold with exponents: ν = 1 / η = 0.As a result, for example, there may be only mutual excluded-volume interactions between chains of different species,while chains of the same species can freely intersect. That is, some species behave effectively like RWs. Withina copolymer star, the subset of chains of such species builds up a substar of random walks, possibly avoiding thesecond part of the star, which can be either of random walks or self-avoiding random walks (see Fig. 1). Cates andWitten [48] have shown, that this situation can also be interpreted as describing the absorption of diffusive particleson polymers.Another example, where star exponents govern physical behavior concerns the short-range interaction betweencores of star polymers in a good solvent [25, 49, 50]. The mean force F fg ( r ) acting on the centers of two stars withfunctionalities f and g is inversely proportional to the distance r between their cores: F fg ( r ) = k B T Θ fg r , (5)with k B T denoting the thermal energy, Θ fg the universal contact exponent, related to the family of exponents of starpolymers by scaling relations: Θ fg = η f + η g − η f + g . (6) FIG. 1: (color online) Schematic representation of copolymer stars consisting of two polymer species (denoted as red and blue).Solid lines present species behaving like SAWs, dashed lines present RWs. The two different sets in each example may furtherbe either mutually avoiding or mutually “transparent”. (c)(a) (b)
We are interested to generalize this relation to the case of copolymers, and to analyze the impact of disorder onmutual interactions between two star copolymers.The questions of the influence of correlated disorder in the environment on the scaling behavior of star copolymersand the resulting consequences remain so far unresolved and are the subject of the present study. We will also analyzethe spectrum of scaling exponents in particular for the above mentioned process of trapping diffusive particles in thevicinity of absorbing polymers in disordered environments.The paper is organized as follows: in the next section we will give a field-theoretical representation of the modelLagrangean. The field-theoretical renormalization group method, which we use to find the qualitative characteristicsof scaling behavior, is shortly described in Section 3. In Section 4 we discuss the results obtained. We finish by givingconclusions and an outlook.
II. THE MODEL
Let us consider a polymer star with f arms of different species in a solvent. We are working within the Edwardscontinuous chain model [51, 52], representing each chain by a path r i ( s ), parameterized by 0 ≤ s ≤ S i , i = 1 , , . . . , f .The central branching point of the star is fixed at r (0). The partition function of the system is then defined by thepath integral [26]: Z f ( S i ) = Z D [ r , . . . , r f ] × exp [ −H f ] f Y i =2 δ d ( r i (0) − r (0)) . (7)Here, a multiple path integral is performed for the paths r , . . . , r f , the product of δ -functions reflects the star-like configuration of f chains, each starting at the point r (0), H f is the Hamiltonian, describing the system of f disconnected polymer chains: H f = 12 f X i =1 Z S i d s (cid:18) d r i ( s )d s (cid:19) + 16 f X i ≤ j =1 u ij Z d rρ i ( r ) ρ j ( r ) , (8)where ρ i ( r ) = R S i d s δ d ( r − r i ( s )) and u ij is a symmetric matrix of bare excluded-volume interactions between chains i and j .The continuous chain model (7) can be mapped onto a corresponding field theory by a Laplace transform in theGaussian surface S i to the conjugated chemical potential variable (mass) ˆ µ i [22, 44]: b Z f (ˆ µ i ) = Z Y b d S j exp[ − ˆ µ j S j ] Z f ( S i ) . (9)One may then show that the Hamiltonian H is related to an m -component field theory with a Lagrangean L in thelimit m → L{ ϕ j , µ j } = 12 f X i =1 Z d d x (cid:0) µ i | ~ϕ i ( x ) | + |∇ ~ϕ i ( x ) | (cid:1) + 14! f X i ≤ j =1 u ij Z d d x ϕ i ( x ) ϕ j ( x ) , (10)where ϕ mi = { ϕ i , . . . , ϕ mi } and µ i are bare critical masses. On the base of the Lagrangean (10) the one-particleirreducible vertex functions Γ ( L ) of the theory can be obtained: δ ( X q i )Γ Li ,...,i L ( q i ) = Z e iq i r i d r · · · d r L h ϕ i ( r ) . . . ϕ i L ( r L ) i L P I , (11)where only those contributions, that have non-vanishing tensor factors in the limit m → b Z f (ˆ µ i ) has a vertex part, which is defined by the insertion of thecomposite operator Q i ϕ i : δ ( p + X j q j )Γ ∗ f ( p, q , . . . , q f ) = (12) Z e i ( pr + q j r j ) d r d r . . . d r f h ϕ ( r ) · · · ϕ f ( r ) ϕ ( r ) · · · ϕ f ( r f ) i L P I . Let us note that we are interested in the case of a copolymer star, having f chains of one species and f of another,so that f + f = f . To keep notations simple we will consider in the following discussion only two fields ϕ and ϕ ,corresponding to two different “species”. Thus, in (10) we have interactions u , u between the fields of the same“species” and u between different fields. The composite operator in (12) has the form of a product ( ϕ ) f ( ϕ ) f .We introduce disorder into the model (10), by redefining ˆ µ i → ˆ µ i + δ ˆ µ i ( x ), where the local fluctuations δµ i ( x )obey: hh δµ i ( x ) ii = 0 , hh δµ i ( x ) δµ j ( y ) ii = g ij ( | x − y | ) . Here, hh· · ·ii denotes the average over spatially homogeneous and isotropic quenched disorder. The form of the paircorrelation function g ( x ) is chosen to decay with distance according to the power law (1).In order to average the free energy over different configurations of the quenched disorder we apply the replicamethod to construct an effective Lagrangean: L eff = Z d x X i =1 n X α =1 h ( ~ ∇ ~ϕ αi ) + µ i ( ~ϕ αi ) i + X i ≤ j =1 n X α =1 u ij
4! ( ~ϕ αi ) ( ~ϕ αj ) (13) − Z d x d y X i ≤ j =1 n X α,β =1 g ij ( | x − y | )( ~ϕ αi ) ( ~ϕ βj ) . Here, the coupling of the replicas is given by the correlation function g ( x ) of Eq. (1), Greek indices denote replicasand the replica limit n → k , the Fourier-transform ˜ g ij ( k ) of g ij ( x ) (1) reads:˜ g ij ( k ) ∼ v ij + w ij | k | a − d . (14)Thus, rewriting Eq. (13) in momentum space, one obtains an effective Lagrangean with 9 bare couplings: u , u , u , v , v , v , w , w , w . As it was pointed out in Ref. [14], once the limit m, n → u ij and v ij terms acquire the same symmetry, and an effective Lagrangean with couplings ( u ij − v ij ≡ u ij ) of O ( mn = 0)symmetry results. This leads to the conclusion that weak quenched uncorrelated disorder i.e. the case a = d isirrelevant for polymers. Taking this into account, we end up with only 6 couplings in an effective Lagrangean: u , u , u , w , w , w . For a < d , the momentum-dependent coupling w ij k a − d has to be taken into account. Notethat ˜ g ij ( k ) must be positively definite being the Fourier image of the correlation function. Thus, we have w ij > k . Note, that the couplings u ij should be positive, otherwise the pure system would undergo a 1st ordertransition.The resulting Lagrangean in momentum space then reads: L eff = 12 n X α =1 2 X i =1 X k (cid:2) k + µ i (cid:3) ( ϕ αi ( k )) + X i ≤ j =1 X k k k k u ij n X α =1 δ ( k + k + k + k ) ~ϕ αi ( k ) ~ϕ αi ( k ) ~ϕ αj ( k ) ~ϕ αj ( k ) (15) − w ij n X α,β =1 | k + k | a − d δ ( k + k − k − k ) ~ϕ αi ( k ) ~ϕ αi ( k ) ~ϕ βj ( k ) ~ϕ βj ( k ) . In the next section, we apply the field-theoretical renormalization group approach in order to extract the scalingbehavior of the model (15).
III. RENORMALIZATION GROUP APPROACH
We apply the renormalization group (RG) method [53] in the massive scheme renormalizing the one-particle ir-reducible vertex functions, in particular Γ (2) , Γ (4) and Γ , , as well as the vertex function Γ ∗ ( f ,f ) , with a single( ϕ ) f ( ϕ ) f insertion. Note that the polymer limit of a zero component field leads to an essential simplification: eachfield ϕ i , mass µ i and coupling u ii renormalizes as if the other fields were absent. The renormalized couplings u ij , w ij are given by: u ii = µ ε Z − ϕ i Z ii u ii , i = 1 , w ii = µ δ Z − ϕ i Z ii w ii , i = 1 , u = µ ε Z − ϕ Z − ϕ Z u , (18) w = µ δ Z − ϕ Z − ϕ Z w . (19)Here, µ is a scale parameter, equal to the renormalized mass, and parameters ε = 4 − d , δ = 4 − a . The renormalizationfactors Z have the form of power series, the coefficients of which are calculated perturbatively order by order.The star vertex function Γ ∗ ( f ,f ) is renormalized by a factor Z ∗ f ,f : Z f / ϕ Z f / ϕ Z ∗ f ,f Γ ( ∗ f f ) = µ ( f + f )( ε/ − − ε . (20)The variation of the coupling constants under renormalization defines a flow in parametric space, governed bycorresponding β -functions: β u ij ( u ij , w ij ) = µ dd µ u ij , β w ij ( u ij , w ij ) = µ dd µ w ij , i, j = 1 , . (21)The fixed points (FPs) of the RG transformation are given by the solution of the system of equations: β u ij ( u ∗ ij , w ∗ ij ) = 0 , β w ij ( u ∗ ij , w ∗ ij ) = 0 , i, j = 1 , . (22)The stable FP, corresponding to the critical point of the system, is defined as the fixed point where the stabilitymatrix possesses eigenvalues { λ i } with positive real parts.The flow of the renormalizing factors Z in turn gives rise to RG functions η ϕ i and η ∗ f f as follows: µ dd µ ln Z ϕ i = η ϕ i ( u ij , w ij ) , (23) µ dd µ ln Z ∗ f f = η ∗ f f ( u ij , w ij ) . (24)At the FP of the renormalization group transformation, the function η ϕ i describes the pair correlation function criticalexponent, while the functions η ∗ f f define the set of exponents for copolymer stars: η = η ϕ i ( u ∗ ij , w ∗ ij ) (25) η f f = η ∗ f f ( u ∗ ij , w ∗ ij ) . (26)In the next section, we will present expressions for the β and η functions, together with a study of the RG flow andthe fixed points of the theory. IV. THE RESULTSA. Fixed points and scaling exponents
According to the renormalisation group prescriptions, we obtain the RG functions of the model (15) within amassive scheme up to the one-loop approximation: β u ii = − ε (cid:20) u ii − u ii I (cid:21) − δ u ii w ii (cid:20) I + 13 I (cid:21) + (2 δ − ε ) w ii I , (27) β w ii = − δ (cid:20) w ii + 23 w ii I + 23 w ii I (cid:21) + ε w ii u ii I , i = 1 ,
2; (28) β u = − ε (cid:20) u − u I − u ( u + u ) I (cid:21) − δ (cid:20) u w I + 12 u ( w + w ) I + 12 u ( w + w ) I ) (cid:21) +(2 δ − ε ) (cid:20) w I + 16 w ( w + w ) (cid:21) , (29) β w = − δ (cid:20) w + 13 w I + 13 w ii I (cid:21) + ε (cid:20) w u I + 16 w ( u + u ) I + 16 w ( w + w ) I ) (cid:21) . (30)Note, that expressions for β u ii , β w ii restore the corresponding RG functions for a single polymer chain in long-rangecorrelated disorder [20, 21]. Here, I i are the loop-integrals: I = Z d ~q ( q + 1) ,I = Z d ~q q a − d ( q + 1) ,I = Z d ~q q a − d ) ( q + 1) ,I = ∂∂k (cid:20)Z d ~q q a − d [ q + k ] + 1) (cid:21) k =0 . (31)We make the couplings dimensionless by redefining u ij = u ij µ d − and w ij = w ij µ a − , therefore, the loop integralsdo not explicitly contain the mass. Besides, we absorb geometrical factors S d , resulting from angular integration intothe couplings.Additionally, we need the RG function η ∗ f f ( u ij , w ij ), which we find in the form: η ∗ f f = − ε (cid:18) u f ( f − I + u f ( f − I + u f f I (cid:19) ++ δ (cid:18) w f ( f − I + w f ( f − I + w f f I (cid:19) . (32)The perturbative expansions for RG functions may be analyzed by two complementary approaches: either byexploiting a double expansion in the parameters ε = 4 − d, δ = 4 − a [17, 20, 21], or by fixing the values of theparameters d, a [20]. Let us note, that within the one-loop approximation the latter method can not give reliableresults [20], and we exploit the double expansion in ε = 4 − d , δ = 4 − a for a qualitative analysis. The resultingexpressions for β - and η -functions read: β u ii = − εu ii + 43 u ii − u ii w ii + 23 w ii (33) β w ii = − δw ii − w ii + 23 u ii w ii , i = 1 ,
2; (34) β u = − εu + 23 u + 13 u ( u + u ) − u ( w + w ) − u w + 13 w + 16 w ( w + w ) , (35) β w = − δw − w + 13 u w + 16 w ( u + u ) − w ( w + w ) , (36) η ∗ f f = − f ( f − u − w ) − f ( f − u − w ) − f ( f − u + f ( f − w . (37)Substituting Eqs. (33)-(36) into (22), we find a number of fixed points, corresponding to different scenarios of thescaling behavior of the model. Pure solution
First, let us consider the case, when disorder is absent ( w = w = w = 0) and we recover theproblem of the so-called ternary solution of two polymer species in a good solvent [44]. Solving the equations β u ij = 0, i, j = 1 ,
2, we find eight fixed points in correspondence with Refs. [45–47]. The trivial FPs: G ( u ∗ = u ∗ = u ∗ = 0), U ( u ∗ = 0 , u ∗ = u ∗ = 0), U ′ ( u ∗ = 0 , u ∗ = u ∗ = 0) and S ( u ∗ = u ∗ = 0 , u ∗ = 0) describe sets of two mutuallynon-interacting polymer species. More interesting are the FPs denoted as G , U , U ′ , S , describing two mutuallyinteracting species, their coordinates are given in the upper part of Table I. Corresponding values of the exponents η f f read: η Gf f = − ( f f ) ε ,η Uf f = η U ′ f ,f = − f ( f + 3 f − ε ,η Sf f = − ( f + f )( f + f − ε . (38)Note, that η Sf f just recovers the exponent of a homogeneous polymer star with f = f + f arms. The values ofthese exponents are known up to 4th order of the ε -expansion [26, 54] and in the fixed d approach [45]. Solution in the presence of long-range correlated disorder . Next, let us turn on the disorder. Apart from the eightFPs listed above, now we have a whole set of new FPs describing two polymer species in the case, when one or bothof the species feel the presence of long-range correlated disorder. Indeed, to find these FPs one has to solve thesystem of 6 second-order equations (22) with the β -functions given by (33) − (36). In principle, this may lead to 2 solutions [55]. In what follows we consider only four nontrivial points, corresponding to copolymer stars of mutuallyinteracting species, both feeling the presence of disorder, which are of foremost interest (see Table I). These FPsdescribe particular situations of two mutually interacting sets of RWs ( G L ), SAWs ( S L ) and two interacting sets ofRWs and SAWs ( U L , U ′ L ). Note that due to the special form of the β -functions the fixed points with u ∗ ii = 0, w ∗ ii = 0do not exist, i.e. one cannot describe simple random walks in the media with long-range-correlated disorder.We are interested in the points, which are stable in all coordinate directions. After analyzing the stability andphysical accessibility of all the points, we come to the conclusion, that only the FPs S and S L are stable in alldirections and their stabilities are determined by the conditions: • fixed point S is stable for ε > δ , • fixed point S L is stable for δ < ε < δ .Although the remaining FPs ( G L , U L and U ′ L from the Table 1) are unstable, they can be reached for δ < ε < δ under specific initial conditions. In particular, G L is reachable from the initial condition u = u = w = w = 0, U L is reachable for u = w = 0 and U ′ L for u = w = 0. Substitution of these FPs values into the expansion(37) results in the following estimates for η f f : η G L f f = − ( f f ) δ,η U L f f = η U ′ L f ,f = − f ( f + 3 f − δ , TABLE I: Non-trivial fixed points of the model (15). u ∗ u ∗ u ∗ w ∗ w ∗ w ∗ G ε U ε ε U ′ ε ε S ε ε ε G L δ ( ε − δ ) δ ( ε − δ )( δ − ε ) U L δ ε − δ ) δ ε − δ ) 3 δ ( ε − δ )2( δ − ε ) δ (2 δ − ε )4( ε − δ ) U ′ L δ ε − δ ) 9 δ ε − δ ) δ ( ε − δ )2( δ − ε ) 9 δ (2 δ − ε )4( ε − δ ) S L δ ε − δ ) 3 δ ε − δ ) 3 δ ε − δ ) 3 δ ( ε − δ )2( δ − ε ) 3 δ ( ε − δ )2( δ − ε ) 3 δ ( ε − δ )2( δ − ε ) η S L f f = − ( f + f )( f + f − δ . (39)Here, η S L f f gives the exponent for the homogeneous star with f + f arms in solution in long-range-correlated disorder, η G L f f and η U L f f describe f random walks, interacting with f RWs and with f SAWs respectively, in long-range-correlated disorder. All this leads to a variety of new scaling behavior for copolymer stars in a disordered medium.
B. Diffusion-limited reaction rates
Let us consider the f -arm star polymer with arms of linear size R s and absorbing sites all along these arms. At thecenter of the star a particular absorbing trap is placed. Free particles A which diffuse in solution are trapped or reactat these sites. We are interested in the reaction rate k f of simultaneously trapping f randomly walking particles A .This rate is proportional to the averaged moments of the concentration ρ of the particles near this trap and scales as[45–48]: k f ∼ h ρ f i ∼ R − λ f f s . (40)This process is an example of a so-called diffusion-limited reaction [56, 57], with the rate depending on the sum of thediffusion coefficients of the reactants [58]. As far as the presence of disorder lowers the diffusion coefficients [59, 60],it is predicted to lower rates of association in diffusion-limited circumstances. It is interesting to check this predictionanalytically, analyzing the behavior of star copolymers in long-range correlated disorder. In terms of the path integralsolution of the diffusion equation, one finds that to calculate the rate of a reaction at the absorber that involves f particles simultaneously one needs to consider f RWs that end at this point. The moments of concentration in Eq.(40) are thus defined by a partition function of a star comprising f RWs [25, 26]. Finally, introducing the mutualavoidance conditions between the absorbing star and a “star” of diffusive particles one ends up with the problem ofcalculating the partition function of a copolymer star with two species f , f . By means of the short-chain expansion[49] the set of exponents η f f in (4) can be related to the exponents λ f f in (40) [45–47, 61]: λ RWf f = − η Gf f ,λ SAWf f = − η Uf f + η Uf . (41)Based on these relations, the resulting values for the pure solution read [61]: λ RW pure f f = ε f f ,λ SAW pure f f = 3 ε f f . (42)Let us note, that the case f = 2 corresponds to a trap located on the chain polymer, whereas f = 1 corresponds toa trap attached at the polymer extremity.Corresponding values for the exponents defining these processes in an environment with long-range correlateddisorder can be obtained by substituting Eqs. (39) into (41): λ RW L f f = − η G L f f = δf f ,λ SAW L f f = − η U L f f + η Uf = 3 δ f f . (43)Comparing relations (42) and (43) at fixed values ε = 1 ( d = 3) and varying the parameter δ , one notes that thepresence of correlated disorder results in an increase of the exponents λ . Moreover, the stronger the correlation ofdefects, the larger is λ . Recalling the definition (40), we immediately conclude that, as expected, the presence oflong-range correlated disorder results in lowering the rates of diffusion-limited reactions. The crucial point is thatwhile long-range correlated disorder apparently does not influence the RW itself (there is no new fixed point with u ii = 0, w ii = 0), the fact that the absorbing polymer changes its conformation and fractal dimension in the LRbackground leads to a change of the diffusive behavior of particles being absorbed (or catalyzed) on the polymer.Let us analyze several particular cases: • For a given f -star absorber i.e. a reactive site placed at one end of an otherwise absorbing polymer increasingthe size R s by a factor of l changes the reaction rate to k ′ f f ∼ ( lR s ) − λ f f , so that: k ′ f f /k f f ∼ l − λ f f . (44)Increasing the size of the polymer thus leads to a reaction rate decrease by a factor of l − λ f f . Since λ Lf f is larger than λ puref f , we conclude, that the presence of long-range correlated defects makes the reaction ratedecreases more slowly as compared to the pure solution case. • For a fixed number f of particles to be trapped simultaneously the effect of attaching f ′ additional arms to an f -arm star absorber decreases the reaction rate: k f + f ′ f /k f f ∼ R − (cid:18) λ f f ′ f − λ f f (cid:19) s , (45)as far as λ f + f ′ f > λ f f . This decrease is suppressed to some extent in the presence of long-range correlateddefects. • For a given f -star absorber an increase of the number of particles to be trapped simultaneously results in adecrease of the reaction rate: k f f + f ′ /k f f ∼ R − (cid:18) λ f f f ′ − λ f f (cid:19) s , (46)since λ f f + f ′ > λ f f . Again, presence of disorder makes the reaction rate decrease more slowly as comparedto pure case. C. Interaction between star copolymers
The effective interaction between two star copolymers at short distance r between their cores can be estimatedfollowing the scheme of Refs. [25, 49, 50], based on short distance expansion. The partition sum Z f f g g (r) of thetwo stars with f = f + f and g = g + g arms of species 1 and 2 at small center-to-center distances r factorizes intoa function C f f g g ( r ) and the partition function Z f + g f + g of the star with f + g arms of species 1 and f + g arms of species 2 which is formed when the cores of the two stars coincide: Z f f g g ( r ) ≃ C f f g g ( r ) Z f + g f + g . (47)For the function C f f g g ( r ) it was shown [25, 49] that power-law scaling for small r holds in the form: C f f g g ( r ) ≃ r Θ f f g g . (48)To find the scaling relation for this power law, we take into account (4) and change the length scale in an invariantway by: r → ℓr , R → ℓR . Eq. (47) then can be written as: ℓ η f f − f η − f η ℓ η g g − g η − g η Z f f g g ( r ) = ℓ Θ f f g g ℓ η f g f g − ( f + g ) η − ( f + g ) η Z f + g f + g . (49)Collecting powers of ℓ provides the scaling relation for the contact exponent:Θ f f g g = η f f − f η − f η + η g g − g η − g η − (50)( η f + g f + g − ( f + g ) η − ( f + g ) η ) = η f f + η g g − η f + g f + g . FIG. 2: (color online) Three non-trivial examples of copolymer stars where the interaction is governed by contact exponentsΘ
S Sf f g g (a), Θ U Uf f g g (b) and Θ G Gf f g g (c). (a) (b) (c) r between their centers the mean force F f f g g ( r ) acting on the centers canbe derived as the gradient of the effective potential V eff ( r ) = − k J T log[ Z f f g g ( r ) / ( Z f g Z f g )]. For the force atshort distances r this results in [62]: F f f g g ( r ) = k J T Θ f f g g r . (51)For two mutually interacting star copolymers we have the three following nontrivial situations. First one may havetwo stars, each consisting of two species (with numbers of arms f , f and g , g respectively) all behaving as mutuallyavoiding SAWs (see Fig. 2a). This situation is equivalent to two SAW star polymers of the same species. A secondpossible situation is the interaction between two star copolymers, the first containing f SAWs and f RWs, another g SAWs and g RWs (Fig. 2b). Thirdly, one may have two stars, each consisting of two species (with f , f and g , g arms respectively) all behaving like RWs but with mutual avoidance between the species (Fig. 2c). It is easy tocheck, that any other case can be represented in terms of these three nontrivial examples. E.g., putting f = 0 in thecase corresponding to Fig. 2b, we obtain a homogeneous f -arm star polymer interacting with a star copolymer, etc.Taking into account Eqs. (38), (39) we find the following contact exponents corresponding to the three nontrivialsituations described above.
1) Pure Solution Θ S Sf f g g = η Sf + f + η Sg + g − η Sf + f + g + g = ε f + f )( g + g ) (52)Θ U Uf f g g = η Uf f + η Ug g − η Uf + g f + g = ε f g + 3 f g + 3 g f ) (53)Θ G Gf f g g = η Gf f + η Gg g − η Gf + g f + g = ε f g + f g ) . (54)
2) Presence of LR disorder Θ ( S S ) L f f g g = η S L f + f + η S L g + g − η S L f + f + g + g = δ f + f )( g + g ) (55)Θ ( U U ) L f f g g = η U L f f + η U L g g − η U L f + g f + g = δ f g + 3 f g + 3 g f ) (56)Θ ( G G ) L f f g g = η G L f f + η G L g g − η G L f + g f + g = δ ( f g + f g ) . (57)Qualitative estimates for the contact exponents in d = 3 can be found by direct substitution of ε = 1 in the aboverelations. To discuss the physical interpretation of these results, let us consider Fig. 3, comparing the cases of purelattice and LR disorder with a = 2 . a = 2 .
7. Fig. 3a presents the contact exponent Θ
U Uf f g governing theinteraction between a star copolymer and a homogeneous star with g arms of RWs. We fix g = 8 and change f and f in such a way that f + f = 8. The case f = 0, describing two stars of RWs, results in zero value contactexponents and thus the absence of interaction. Increasing the parameter f (SAW component) leads to the gradualincrease of the strength of the interaction. For f = 8, we have a star of SAWs interacting with a star of RWs withmaximal interaction strength. Fig. 3b depicts the situation of a star copolymer interacting with a star of g = 8SAWs. Again, we change f and f as above. The case f = 0 describes a star of SAW interacting with a star of RWsand is a particular case of Fig. 3a described above. Increasing f leads to a gradual decrease of the strength of theinteraction. For f = 8, we have two interacting stars of SAWs with minimal interaction strength. Fig. 3c depicts asituation of two interacting star copolymers with f , f and g , g arms correspondingly. We fixed g = g = 4 andagain change f and f as described above. The case f = 0 describes a copolymer star interacting with a star ofRWs. Increasing the parameter f leads to a gradual increase of the strength of the interaction, until it reaches itsmaximal value at f = 8, corresponding to the interaction between a star copolymer and a star of SAWs. The case f = f = 4 describes the interaction between two identical copolymer stars.Finally, we conclude, that in all situations considered above, the presence of correlated disorder leads to an increaseof the contact exponent. The stronger the correlation (the smaller the value of correlation parameter a ), the strongeris the interaction between polymers in such an environment. Let us recall, that the exponent Θ ( S S ) L f f g g correspondsto the situations of two interacting homogeneous polymer stars of f = f + f and g = g + g arms in solution in thepresence of long-range-correlated disorder. This problem has previously been analyzed [43] using a two-loop expansionseries for Θ ( S S ) L fg in d = 3. The quantitative estimates obtained predict a decrease of the contact exponents with thestrength of the disorder correlations in contrast to our present ε, δ -expansion results. Revising the resummation asperformed in [43] we conclude that the number of terms in the two-loop expansion is probably too small to rely onthose quantitative results.1 FIG. 3: Contact exponents of interaction between a copolymer star with f SAWs and f RWs and: an 8-armed star of RWs(a);an 8-armed star of SAWs(b); a copolymer star with 4 arms of SAWs and 4 arms of RWs in d = 3. Boxes: pure case ( a = 3),circles: a = 2 .
7, triangles: a = 2 . f U U f f f (a) U U f f f (b) f U U f f f f (c) V. CONCLUSIONS
In the present paper, we have studied the scaling properties of copolymer stars, consisting of f arms of polymerspecies 1 and f arms of species 2 in a solution in which one or more of the intra- and interspecies interactions arefound to be at their Θ-point with the further complication of a disordered environment with correlated structuraldefects. We assume that the disorder is correlated with a power-law decay of the pair correlation function g ( x ) ∼ x − a at large distance x . This type of disorder is known to be relevant for simple polymer chains [20, 21] and homogeneous2polymer stars [43], and we address the question of the scaling of copolymer stars in this situation.Considering the f -arm absorbing star polymer with a special trap placed at the center of the star where f freeparticles (RWs) are to be trapped simultaneously, the reaction rate of this diffusion-limited reaction is found to scalewith exponents, connected to the spectrum of critical exponents η f f of star copolymers [48]. Such a process isan example of a so-called diffusion-limited reaction, with the rate depending on the sum of diffusion coefficients ofthe reactants. Another example, where star exponents govern physical behavior concerns the short-range interactionbetween the cores of star polymers in a good solvent. The present study aims to analyze the impact of structuraldisorder in the environment on these processes.In the frames of the field-theoretical renormalization group approach, we obtain estimates for the critical exponents η f f up to the first order of an ε = 4 − d, δ = 4 − a -expansion, which belong to a new universality class. In particular,this enables us to conclude that the rates of diffusion-limited reactions are slowed down by the presence of long-range-correlated disorder. The crucial point is that while long-range correlated disorder apparently does not influence theRWs and thus the universal behavior of diffusion itself, the fact that the absorbing polymer changes its conformationin the LR background leads to a change of the rate with which particles are absorbed (or catalyzed) on specific sitesof the polymer.The contact exponents, governing the repulsive interaction between two star copolymers in correlated disorder, arefound to be larger than in the pure solution case. The stronger the correlation of the defects, the stronger is theinteraction between polymers in such a disordered environment. Acknowledgment
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