Polymers in anisotropic environment with extended defects
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Polymers in anisotropic environment withextended defects
V. Blavatska and K. Haydukivska
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1,Svientsitskii Str., Lviv, 79011, Ukraine
Abstract.
The conformational properties of flexible polymers in d di-mensions in environments with extended defects are analyzed bothanalytically and numerically. We consider the case, when structuraldefects are correlated in ε d dimensions and randomly distributed inthe remaining d − ε d . Within the lattice model of self-avoiding ran-dom walks (SAW), we apply the pruned-enriched Rosenbluth method(PERM) and find the estimates for scaling exponents and universalshape parameters of polymers in environment with parallel rod-likedefects ( ε d = 1). An analytical description of the model is developedwithin the des Cloizeaux direct polymer renormalization scheme. Conformational properties of polymer macromolecules in solutions are the subject ofgreat interest in statistical polymer physics [1,2,3]. It is established, that long flexiblepolymers are characterized by a number of universal characteristics, independentof the chemical structure and architecture of the molecules. A typical example ofsuch a quantity is the averaged end-to-end distance of N -monomer polymer chain(characterizing the linear size measure of macromolecule), that scales according to: h R i ∼ N ν , (1)where ν is a universal exponent, depending on space dimension d only ( ν ( d = 3) =0 . ± . h A d i [5], whichequals 0 for sphere-like shapes and 1 for rod-like. For a polymer in a pure solvent in d = 3 it was found h A i = 0 . ± .
002 [6].For a long time, the computer experiment served as the best tool to investigatepolymer systems. In this approach, the most widely used model was and is until nowthe model of self-avoiding random walks (SAW) on a regular lattice. In spite of itssimplicity, it perfectly captures the universal conformational properties of polymers ina solvent. Among the analytical descriptions of polymer systems the most successfulare the field-theoretical approach [1], the direct polymer renormalization [3] and thetraditional Flory theory [1]. The values of critical exponents and universal shapeparameters, obtained within the frames of these theories, are in a good agreementwith results of computer experiments.
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Fig. 1.
Schematic presentation of polymer chain in an environment with structural defectsin form of parallel lines.
The study of polymers in solvents in the presence of structural impurities playsa considerable role due to the importance to understand their behavior in colloidalsolutions and near microporous membranes [7]. It also connects with the descriptionof proteins inside the living cells which can be treated as disordered environments[8]. The behavior of polymers in disordered environments encountered controversiesfor a long time. During more than fifteen years there has been a wide discussionwhether the presence of uncorrelated point-like defects leads to a change in universalproperties [9,10,11,12,13,14,15,16,17,18,19,20,21,22]. This problem was first posed byKremer [9] showing that there exists a new value of the size exponent (1) only whenthe concentration of defects is above the percolation threshold. Later, this result wasconfirmed both analytically [23] and numerically [24,25,26,27,28,29,30].The density fluctuations of obstacles create large spatial inhomogeneities and porespaces, which are often of fractal structure [31]. A special case arises when fractal ex-tended defects of parallel orientation are present in a system, which may cause theanisotropy of environment (see fig. 1). It is expected, that in this case there aretwo scaling exponents, governing the components of the size measure (1) in direc-tions parallel and perpendicular to the orientadion of the extended defects [32,33].The influence of such disorder on the critical behavior of magnetic systems with ε d -dimensional defects of parallel orientation within frames of the spin m − vector modelwas analyzed in refs. [32,34,35]. It was shown that there are two characteristic corre-lation lengths in the system, one parallel and other perpendicular to the defects. Asit is known [1], the scaling properties of polymers in solution can be described withinan m − vector model taking the de Gennes limit (polymer limit m → ε d -dimensional extendeddefects of parallel orientation (causing the anisotropy of environment), applying bothnumerical and analytical approaches. In the next Section, we study the special caseof defects in form of randomly placed lines of parallel orientation ( ε d = 1), applyingcomputer simulations. In the Section 3, we develop an analytical description of theproblem within the des Cloizeaux direct polymer renormalization scheme. We endgiving conclusions and an outlook in Section 4. ill be inserted by the editor 3 We study the conformational properties of self-avoiding random walks on a regularlattice with extended impurities in the form of parallel lines (see fig. 1), applying thepruned-enriched Rosenbluth method (PERM) [36]. The method combines the originalRosenbluth-Rosenbluth algorithm of growing chains [37] and population control [38].The n -th monomer is placed at a randomly chosen neighbor site of the previous( n − n ≤ N , where N is total length of the polymer). If this randomlychosen site is already visited by a chain trajectory or belongs to an impurity line,it is avoided without discarding the chain and the weight W n given to each sampleconfiguration at the n -th step is: W n = n Y l =1 m l , (2)where m l is the number of free lattice sites to place the l th monomer.The growth is stopped when the total length N of the chain is reached (or, at n < N , if a the “dead end” without possibility to make the next step is reached),then the next chain is started to grow from the starting point that is chosen randomlyevery time.To derive appropriate properties of the chain we apply a two-step averaging: firstaverage over configurations as usual and then averaging over different realizations ofdisorder. The configurational average for any quantity of interest then has the form: h ( . . . ) i = 1 Z N M X k =1 W kN ( . . . ) , Z N = M X k =1 W conf N , (3)where the summation is performed over the ensemble of all constructed N -step SAWs( M ∼ in our case). The disorder averaging is given in the form:( . . . ) = 1 p p X k =1 ( . . . ) , (4)where p is number of replicas (we take p = 400 in our study).The weight fluctuations of the growing chain are suppressed in PERM by prun-ing configurations with too small weights, and by enriching samples with copies ofhigh-weight configurations. These copies are made while the chain is growing, andcontinue to grow independently of each other. Pruning and enrichment are performedby choosing thresholds W
Fig. 2.
End-to-end distance component perpendicular (left) and parallel (right) to theimpurity lines as functions of chain length in double logarithmic scale at several values ofdefects concentrations.
Let us introduce the position vectors r n = { x n , x n , x n } of each n -th monomer(1 ≤ n ≤ N ). In contrary to the the purely isotropic case (1), space anisotropycauses different scaling behavior for the end-to-end distance components in directionsparallel h R k i ≡ h ( x N − x ) i and perpendicular h R k i ≡ h ( x N − x ) + ( x N − x ) i to the lines of defects (we assume, that the defects are extended in the x direction).Namely: h R k i ∼ N ν k , h R ⊥ i ∼ N ν ⊥ , (5)with ν k = ν ⊥ .Analyzing our results for the end-to-end distance components the parallel andperpendicular to the lines of defects (see Fig. 2), we can see that when concentrationof impurity lines is small, there is a crossover between two types of behavior: one forshort length ( ∼
40 monomers) and the other for longer chains. We can interpret this bythe fact that at small concentration of defects the short chain does not feel the presenceof impurities (when averaged distances between impurity lines approximately equalthe length of the chain). In such situations, to calculate the value of size exponentswe have to take into account only the second part as it is closer to the asymptoticalregime.As it was suggested, there are two different size exponents ν k and ν ⊥ , which canbe evaluated applying the least-square fitting of results presented in Fig. 2 to the form Fig. 3.
Size exponents (5) at various concentration of defects.ill be inserted by the editor 5
Fig. 4.
Aspheriity as function of the chain length at several different concentrations ofdefects (left). The change of aspheriity with increasing the defect concentration at fixed N = 100 (right). (5). For example, in the case of concentration 20 % we have ν ⊥ = 0 . ± .
005 and ν k = 0 . ± .
07 which should be compared with corresponding pure value. Resultsfor ν ⊥ , ν k at different concentrations of impurity lines are given in Fig. 3. Whereasexponent ν k gradually tends to the value of 1 as concentration of defects grows (thepolymer chain extends in direction parallel to defects), the exponent ν ⊥ is alwayssmaller and gradually tends to zero.To analyze the shape of polymers we calculate the components of gyration tensorgiven by: Q ij = 1 N N X m =1 N X n =1 ( x in − x im )( x jn − x jm ) (6)where i, j = 1 , ,
3. The shape of a typical polymer chain conformation can be de-scribed by a quantity: A = 32 T r ˆ Q ( T rQ ) ˆ Q = Q − ˆ I T rQ . (7)Our results for h A i are given in Figure 4. One can see, that at small concentrationsof impurity lines the polymer chain does not change its shape in spite of the existenceof two different characteristic lengths. Increasing the defect concentration to mediumconcentrations the shape become more prolate and tends to a rod-like state.As a result, polymers in an anisotropic environment with defects aligned in a givendirection are elongated in this direction. In our case this is caused by the impuritiesthat are extended along this direction and which repel the polymer. Let us start with a continuous model, where the polymer chain is presented as a pathparametrized by r ( s ), with s = 0 . . . S . An effective Hamiltonian of the system isgiven by: H = 12 Z S (cid:18) d r ( s ) ds (cid:19) ds + u Z S ds ′ Z s ′ ds ′′ δ ( r ( s ′ ) − r ( s ′′ )) ds ++ Z S V ( r ( s )) ds. (8) Will be inserted by the editor
Note that r ( s ) is a d -dimentional vector with unit [length] while the parameters S , s have units [length] . S is also called the Gaussian surface. Here, the first termdescribes the chain connectivity, the second term reflects the excluded volume effectwith coupling constant u , and the last term arises due to the interaction between themonomers of the polymer chain and the structural defects in the environment givenby potential V .We are interested in the case, when defects are correlated in ε d dimensions andrandomly distributed in the remaining d − ε d . The correlation function of the densitiesof defects is thus assumed to be given by [33,32]: V ( r ( s )) V ( r ( s ′ )) = vδ d − ǫ d ( r ( s ) − r ( s ′ )) . (9)Averaging the partition sum of the model over different realizations of disorderwith (9) results in: Z dis = Z D{ r } exp " − Z S (cid:18) d r ( s ) ds (cid:19) ds − u Z S ds ′ Z s ′ ds ′′ δ ( r ( s ′ ) − r ( s ′′ )) ds −− v Z S ds ′ Z s ′ ds ′′ δ d − ǫ d ( r d − ǫ d ( s ′ ) − r d − ǫ d ( s ′′ )) ds , (10)Here, r d − ǫ d ( s ) denotes the component of position vector r ( s ) perpendicular to theorientation of the extended defects and the functional integration R D{ r } is to beperformed over different chain configurations. Note that the coupling u must be pos-itive, which corresponds to an effective mutual repulsion of the monomers due to theexcluded volume effect, whereas the coupling v should have negative sign, weakeningthe covolume effect.To evaluate quantitative estimates for the conformational characteristics of sucha model, we apply the direct polymer renormalization scheme [3]. All properties ofinterest can be found in the form of a perturbation theory series in an couplingconstants. In particular, for the components of the end-to-end distances (5) we found: R d − ǫ d = S ( d − ǫ d ) z u (2 − d )(3 − d ) + z v (2 − d − ǫ d )(3 − d − ǫ d ) ! , (11) R ǫ d = Sǫ d z u (2 − d )(3 − d ) ! , (12)here z u = u (2 π ) − d/ S − d/ and z v = v (2 π ) − ( d − ε d ) / S − ( d − ε d ) / are dimentionlesscouplinds. Note, that ε d indicates the dimentionality of the subspace parallel to thedefects and d − ε d the dimentionality of the corresponding perpendicular subspace.In the asymptotic limit S → ∞ the physical quantities, presented in the formof series expansions in the coupling constants, are, however, divergent. To obtainasymptotical values of the corresponding physical parameters a renormalization ofthe coupling constants needs to be performed [3]. The scaling exponents attain finitevalues when evaluated at the stable fixed point of the renormalization group trans-formation. The fixed points are defined as the common zeros of the flow equations,which in our problem read: β u = ǫz u − z u − z u z v ,β v = δz v − z v − z u z v , (13) ill be inserted by the editor 7 here ǫ = 4 − d , δ = ǫ + ε d . We find four fixed points: z ∗ u = 0 , z ∗ v = 0 , (14) z ∗ u = ǫ/ , z ∗ v = 0 , (15) z ∗ u = 0 , z ∗ v = δ/ , (16) z ∗ u = ǫ/ − δ/ , z ∗ v = δ/ − ǫ/ . (17)The first of them is the well known Gaussian fixed point, corresponding to an ideal-ized polymer chain (random walk) without excluded volume interactions. The seconddescribes another well known problem namely the polymer in a good solvent (disorderis absent). The third point describes the Gaussian polymer in an anisotropic environ-ment and the last (most interesting one) describes SAW in anisotropic environment.Note, that at ε d = 0 we restore the situation with uncorrelated point-like defects,studied previously [23,41,10,17]. To correspond to a physical critical point of the sys-tem, the given fixed point should be stable and physically accessible. Unfortunately,in our problem the two last points, which are of main interest, are non-physical (e.g.attain appropriate values z u > z v < ε d < m − vector model with extended defects [35],when one tries to take the m → We analyzed the conformational properties of flexible polymer chains in disorderedanisotropic environments. The anisotropy is introduced in the form of ε d -dimensionalextended impurities of parallel orientation. Both numerical and analytical approachesare developed.In numerical part of our work, we studied the scaling properties of a SAW modelon a lattice with defects in a form of lines ( ε d = 1), extended along a fixed coordinatedirection. We conclude that there exist two critical exponents that describe the sizemeasure of the polymer chain in directions parallel and perpendicular to the defects.The parallel exponent ν || is always larger than the corresponding value for polymersin a pure solvent and gradually approaches the limit of 1 when increasing the de-fect concentration. The perpendicular exponent ν ⊥ is always smaller and reaches thelimit of 0. We also found, that increasing the concentration of defects, the shape ofthe polymers becomes more anisotropic, elongated in the direction parallel to theextended defects and gradually reaches the limit of a rod-like shape.Our analytical study is developed on the basis of a continuous chain model withapplying the direct polymer renormalization scheme. However, we encounter somecontroversies when analyzing stability and physical accessibility of fixed points, cor-responding to a critical point of the system. Clarification of this problem will be thesubject of our further work. Will be inserted by the editor
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