Size, shape and diffusivity of a single Debye-Hückel polyelectrolyte chain in solution
aa r X i v : . [ c ond - m a t . s o f t ] A ug Size, shape and diffusivity of a single Debye-H¨uckel polyelectrolyte chain insolution
W Chamath Soysa, B. D¨unweg, and J. Ravi Prakash a) Department of Chemical Engineering, Monash University, Melbourne, VIC 3800,Australia Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz,Germany Condensed matter physics, TU Darmstadt, Karolinenplatz 5, 64289 Darmstadt,Germany (Dated: 14 October 2018)
Brownian dynamics simulations of a coarse-grained bead-spring chain model, with Debye-H¨uckel electrostaticinteractions between the beads, are used to determine the root-mean-square end-to-end vector, the radiusof gyration, and various shape functions (defined in terms of eigenvalues of the radius of gyration tensor)of a weakly-charged polyelectrolyte chain in solution, in the limit of low polymer concentration. The long-time diffusivity is calculated from the mean square displacement of the centre of mass of the chain, withhydrodynamic interactions taken into account through the incorporation of the Rotne-Prager-Yamakawatensor. Simulation results are interpreted in the light of the OSFKK blob scaling theory (R. Everaers,A. Milchev, and V. Yamakov, Eur. Phys. J. E 8, 3 (2002)) which predicts that all solution propertiesare determined by just two scaling variables—the number of electrostatic blobs X , and the reduced Debyescreening length, Y . We identify three broad regimes, the ideal chain regime at small values of Y , the blob-pole regime at large values of Y , and the crossover regime at intermediate values of Y , within which themean size, shape, and diffusivity exhibit characteristic behaviours. In particular, when simulation results arerecast in terms of blob scaling variables, universal behaviour independent of the choice of bead-spring chainparameters, and the number of blobs X , is observed in the ideal chain regime and in much of the crossoverregime, while the existence of logarithmic corrections to scaling in the blob-pole regime leads to non-universalbehaviour.PACS numbers: 05.10.-a, 05.40.Jc, 82.35.Rs, 61.25.he, 36.20.-r, 66.30.hkKeywords: dilute polyelectrolyte solutions; blob scaling theory; Brownian dynamics; shapes of polyelec-trolytes; translational diffusivity I. INTRODUCTION
Many synthetic polymers and most biopolymers arepolyelectrolytes. Their use in a range of industrial andbiological applications makes a thorough understandingof their behaviour highly desirable. While the scaling ofmany important observable properties of neutral poly-mers in solution has been successfully predicted by blobtheories, much is yet to be understood with this ap-proach even for the simple case of dilute polyelectrolytesolutions at equilibrium. The electrostatic blob, whichsets the length scale at which the energy of electrostaticinteractions is of order k B T (where k B is the Boltzmannconstant and T is the temperature), provides the ba-sis for scaling theories of such a system. A commonlyused scaling theory for polyelectrolyte solutions is theOSFKK scaling picture (after Odjik, Skolnick, Fix-man, Khokhlov, and Khachaturian), which predicts thescaling of the mean chain size in different scaling regimesbased on the dominating physics. Even though simula-tions have examined the predictions of OSFKK scalingtheory in detail, many questions remain unanswered and a) Electronic mail: [email protected] not many studies have applied the blob picture to proper-ties other than the root-mean-square end-to-end vector.The aim of this work is to carry out Brownian dynam-ics (BD) simulations in order to investigate the depen-dence of several static and dynamic properties of dilutepolyelectrolyte solutions on the intrinsic parameters thatgovern their behaviour. In particular, in addition to themean size, we use the OSFKK scaling picture to inter-pret simulation predictions of the shape and diffusivity ofpolyelectrolyte chains. A variety of shape functions, de-fined in terms of the eigenvalues of the radius of gyrationtensor, are used to examine changes in polymer shape inthe different regimes of conformational phase space. Fur-ther, hydrodynamic interactions have been incorporatedvia the Rotne-Prager-Yamakawa tensor, in order to ob-tain an accurate prediction of the chain diffusivity in var-ious scaling regimes. The OSFKK scaling picture ignoresthe presence of logarithmic corrections in the blob-poleregime of the conformational phase diagram, which arepredicted to be important by more refined scaling theo-ries. For all the properties examined here, we explorewhether logarithmic corrections can account for depar-tures from OSFKK scaling in the blob-pole regime.In most scaling theories, polyelectrolyte molecules arerepresented as freely-jointed chains with N K Kuhn steps,each of length b K . The number of charges per Kuhn step, !" I II V III IV VI <=> ’0;8 ?<=@ ’ ! % = ξ %$ ’;’8 <=> ’ ! % = ξ %$ ’;’8’ FIG. 1. The phase diagram in the OSFKK scaling picture. Various regimes are marked I-VI, with scaling predictions forthe end-to-end vector r e , in each regime, listed in Table I.Equations governing the boundaries between regimes are in-dicated in the figure, and the abbreviation “RW” implies ran-dom walk statistics are obeyed. f = αN m,K , where, N m,K is the number of monomers perKuhn step, and α is the degree of ionization per chain,is used to characterize the extent of ionic group dissocia-tion. Counterions and salt ions are not modelled explic-itly, and the screening of monomer charges due to thepresence of free ions is quantified via the Debye screen-ing length, l D . Another important length scale is theBjerrum length, l B , which is the distance at which theCoulomb energy between two unit elementary charges inthe solvent is equal to the thermal energy k B T , and isdefined by, l B = e πε ε r k B T (1)where e is the charge of an electron, ε is the vacuum per-mittivity, and ε r is the relative dielectric constant of thesolvent. Such a representation of a dilute polyelectrolytesolution is referred to here as the bare-model, and in whatfollows, it is always assumed that the macromolecules aredissolved in a theta solvent (which is simulated by switch-ing off excluded volume interactions).The electrostatic blob denotes the length scale belowwhich the conformation of a polyelectrolyte chain is prac-tically unaffected by electrostatic interactions, since theelectrostatic energy of the sub-chain within a blob is lessthan the thermal energy. This length scale can be usedto divide the chain into a number blobs, X , of diam-eter, ξ el , with random walk statistics followed withinthe blob, while the conformation of the chain of blobsas a whole depends on the electrostatic interactions be-tween the blobs. In addition, within the OSFKK scalingpicture, the screening of electrostatic interactions is ac-counted for by the scaled variable Y , which is the ratioof the Debye length to the blob size. In terms of thebare-model parameters, the OSFKK expressions for the blob scaling variables are, X = N K ˆ ξ (2)and, Y = ˆ l D ˆ ξ el (3)where, ˆ l D ≡ l D /b K , and the scaled blob size ˆ ξ el ≡ ξ el /b K ,is given by, ˆ ξ el = (cid:16) f ˆ l B (cid:17) − (4)with, ˆ l B ≡ l B /b K . Note, for future reference, that it issufficient to prescribe the reduced set of bare-model pa-rameters { N K , f, ˆ l B , ˆ l D } in order to determine the scal-ing variables X and Y . The OSFKK scaling picture isbriefly summarized in Fig. 1, which is a phase diagram in( X, Y ) space, highlighting the different scaling regimes.Table I lists the scaling expressions for the chain size, interms of both blob and bare parameters, in each of theregimes. In regime I, when X ≪ r e /ξ el ∝ X / , where r e is theroot-mean-square end-to-end vector. The Debye lengthis irrelevant in this regime as thermal energy completelydominates the electrostatic interactions between the seg-ments. When X ≫ XY plane is divided intodifferent regimes according to how large Y is comparedto X . The blob-pole conformation occurs in regime II,when Y → ∞ , and electrostatic interactions betweenblobs are unscreened. In this regime, r e /ξ el ∝ X . When Y decreases below X / , electrostatic interactions act asshort ranged repulsive forces and a crossover regime fol-lows, which, according to Khokhlov and Khachaturian is decomposed into regimes III-V. Here the confirmationsare governed by an intricate interplay of chain stiffnessand excluded volume interactions (both of electrostaticorigin). Since our numerical investigations lack the res-olution to study the details of these regimes we collec-tively refer to them as the crossover regime. Finally when Y ≪ X − / the Debye length is sufficiently small for thescaling to return to that of an ideal chain.Everaers, Milchev, and Yamakov have carried out ex-tensive single chain Monte Carlo simulations with Debye-H¨uckel electrostatic interactions and their results supportthe OSFKK scaling picture for the end-to-end vector.Pattanayek and Prakash used single chain Browniandynamics simulations with Debye-H¨uckel electrostatic in-teractions in order to investigate the scaling of the end-to-end vector and viscometric functions in simple shearflow. They showed that when Brownian dynamics sim-ulation data were represented in terms of blob scalingvariables, predictions independent of the choice of pa-rameters in the bead-spring chain model were obtained. TABLE I. Scaling expressions for the end-to-end vector in the various regimes shown schematically in Fig. 1, in terms of bareand blob parameters (the value of the Flory excluded volume exponent has been approximated to be 3 / X and Y are defined in Eqs. (2) and (3), respectively.Regime r e /ξ el (blob parameters) r e /b K (bare parameters)I X / N / II X f / ˆ l / N K III
Y X / f / ˆ l / ˆ l D N / IV Y / X / f / ˆ l / ˆ l / N / V Y / X / f / ˆ l / ˆ l / N / VI X / N / As in the present study, these Brownian dynamics simu-lations were unable to distinguish between the differentregimes in the crossover region due to the computationalcost of simulating the extremely long chains required toverify OSFKK scaling. By including a characteristic non-dimensional shear rate as an additional scaling variablein shear flow, Pattanayek and Prakash were able toshow that the equilibrium electrostatic blob model pro-vides a framework to obtain universal scaling even fornon-equilibrium properties. Both these simulation stud-ies, however, did not attempt to describe the scaling ofother static properties, or the diffusivity in the differentregimes, nor did they examine the occurrence of logarith-mic corrections in the blob-pole regime.The appropriateness of using a Debye-H¨uckel poten-tial to represent electrostatic interactions in Brown-ian dynamics simulations has been examined by Stoltz,de Pablo, and Graham . They compared the predic-tions of chain size as a function of Bjerrum length (atvarious concentrations in the dilute regime) of a bead-spring chain model with explicit counterions and pair-wise Coulomb interactions between charges, with thoseof a model with pair-wise Debye-H¨uckel electrostatic in-teractions between beads on chains. They found that theresults of both models are nearly identical for values ofBjerrum length roughly equal to or less than the distancebetween the beads on a chain (all of which were assumedcharged). For larger values of Bjerrum length, in ac-cord with Manning’s theory, they observed the onsetof counterion condensation. We have adopted a Debye-H¨uckel potential in this work, and as discussed in greaterdetail subsequently, chosen parameter values in the bead-spring chain model to ensure that the simulations alwaysremain in the regime where this approximation is valid,and there is no counterion condensation.Property predictions from Brownian dynamics simu-lations of bead-spring chains depend on several modelparameters, such as the number of beads, the chargeon the beads, the finite extensibility parameter for thesprings, and so on. A key issue is a rational choice of val-ues for these parameters. The earlier BD simulations ofmean size and viscometric functions by Pattanayek andPrakash has shown the advantage of using blob scal- ing variables to interpret results of simulations since thisleads to a description independent of the level of coarse-graining, which is extremely useful for comparing simu-lation results with experiments. In this study, we inves-tigate the shape and diffusivity of polyelectrolyte chainsin the various regimes of the phase diagram, by recastingBD simulation results in terms of blob scaling variables,and examine their independence from the specific choiceof bead-spring chain model parameters.Many studies have shown that the shape of a neutralpolymer chain, even at equilibrium, is not spherical aboutthe centre of mass of the chain. The nature of theasymmetry in chain shape has been examined in termsof a number of different quantities, such as the degree ofprolateness , the asphericity , the acylindricity , the shapeanisotropy , and so on, which are functions of the eigen-values of the radius of gyration tensor, since the breakingof symmetry is reflected in the three eigenvalues differingfrom each other. The symmetry or otherwise of polyelec-trolyte chain shapes, particularly in the different scal-ing regimes, has not yet been systematically investigated.Here, we examine if a universal description of polyelec-trolyte chain shapes can be obtained, when BD simula-tion results are represented in terms of electrostatic blobscaling variables.While the OSFKK scaling picture ignores the presenceof logarithmic corrections to the scaling of chain sizewith degree of polymerisation in the blob-pole regime,their existence has been derived in a number of differ-ent ways, ranging from Flory type energy minimisationarguments, to refined scaling theories that account forthe nonuniform stretching of polyelectrolyte chains alongthe elongation axis. In Appendix A (for the sake ofcompleteness), we have used a Flory type argument toshow how logarithmic corrections arise in regime II. Inparticular, it can be shown that r e obeys the followingscaling expression in terms of the blob scaling variable X , r e ξ el ∼ X [ln X ] (5)Liao, Dobrynin, and Rubinstein have carried outmolecular dynamic simulations of bead-spring chainswith explicit counterions, and have shown in terms ofbare-model parameters, that for sufficiently long chains,the scaling of the end-to-end vector does indeed exhibitlogarithmic corrections in the blob-pole regime. Here, weexamine whether results of BD simulations in the blob-pole regime exhibit logarithmic corrections as describedby Eq. (5).A common assumption in the various theoretical de-scriptions of the blob-pole regime is that electrostatic in-teractions lead to chain stretching along one direction,while leaving the chain conformation unperturbed in di-rections perpendicular to the stretching direction. Interms of blob scaling variables, this implies that chain di-mensions lateral to the stretching direction are expectedto scale as ξ el X . By examining the eigenvalues of thegyration tensor, we verify if chain dimensions perpendic-ular to the stretching direction indeed obey ideal chainscaling laws.The concept of the Zimm diffusivity of a blob has beensuccessfully used to develop scaling relations for the diffu-sivity of neutral polymer chains, both in the dilute con-centration regime (in terms of thermal blobs ), and inthe semidilute regime (in terms of correlation blobs ).Here, we examine if the scaling of the diffusivity of poly-electrolyte chains in the various regimes of the phase dia-gram, becomes independent of bead-spring chain param-eters, when the Zimm diffusivity of an electrostatic blobis used to interpret simulation results. In particular, sincethe blob-pole is reminiscent of the shish-kebab model forrodlike polymers (with blobs taking the place of beads),we examine if the diffusivity of a polyelectrolyte chain inregime II can be understood in terms of the translationaldiffusivity of rodlike polymers.The paper is structured as follows. In section II wedescribe the bead-spring chain model used to representpolyelectrolyte chains, and the governing equations forthe time evolution of the position vectors for the beads.Section III, which summarises our results and the rele-vant discussions, is subdivided into four sections; the firstlooks at the scaling of the chain size and the extent of log-arithmic corrections in the blob-pole regime, the secondand third consider the scaling of various functions thatdescribe the shape of the chain, while the fourth exam-ines the scaling of chain diffusivity and relaxation time.Finally, the key findings of this work are summarised insection IV. II. THE BEAD-SPRING CHAIN MODELA. Governing equations and Brownian dynamicssimulations
A polyelectrolyte chain is modelled in the BD sim-ulations by a coarse-grained version of the bare-model,i.e., by a bead-spring chain consisting of N b beads of ra-dius a , connected linearly by ( N b −
1) finitely extensiblenon-linear elastic (FENE) springs, with spring constant H and maximum stretch Q . The beads act as centresof frictional resistance, with a Stokes friction coefficient, ζ = 6 πη s a (where η s is the solvent viscosity). The totalcharge on the bead-spring chain is set equal to that of achain in the bare-model, distributed uniformly along thelength of the chain with each bead having an identicalcharge q , given by q = f N K N b (6)The time evolution of the position vector r µ ( t ) of bead µ , is described by the non-dimensional stochastic differ-ential equation d r µ = 14 X ν D µν · F ν d t + 1 √ X ν B µν · d W ν (7)where, the length scale l H = p k B T /H and time scale λ H = ζ/ H have been used for non-dimensionalization.The dimensionless diffusion tensor D µν is a 3 × µ and ν . It is related tothe hydrodynamic interaction tensor, as discussed fur-ther subsequently. The sum of all the non-hydrodynamicforces on bead ν due to all the other beads is representedby F ν , the quantity W ν is a Wiener process, and B µν is a non-dimensional tensor whose presence leads to mul-tiplicative noise. Its evaluation requires the decompo-sition of the diffusion tensor. Defining the matrices D and B as block matrices consisting of N × N blocks eachhaving dimensions of 3 ×
3, with the ( µ, ν )-th block of D containing the components of the diffusion tensor D µν ,and the corresponding block of B being equal to B µν ,the decomposition rule for obtaining B can be expressedas B · B t = D (8)The non-hydrodynamic forces on a bead µ are com-prised of the non-dimensional spring forces F s µ andnon-dimensional electrostatic interaction forces F es µ , i.e., F µ = F s µ + F es µ . The entropic spring force on bead µ due to adjacent beads can be expressed as F s µ = F c ( Q µ ) − F c ( Q µ − ) where F c ( Q µ − ) is the force betweenthe beads µ − µ , acting in the direction of the con-nector vector between the two beads Q µ − = r µ − r µ − .Specifically, the spring force in the FENE springs usedhere is given by, F c ( Q µ ) = Q µ − ( Q µ /b ) (9)where b = HQ /k B T is the dimensionless finite exten-sibility parameter. The vector F es µ is given in terms ofthe dimensionless electrostatic potential U es µν between thebeads µ and ν of the chain, F es µ = − N b X ν =1 ν = µ ∂∂ r µ U es µν (10)The OSFKK scaling theory assumes Debye-H¨uckel elec-trostatic interactions between charges, which is justi-fied for weakly charged polyelectrolyte chains in the ab-sence of Manning counterion condensation. Weadopt a Debye-H¨uckel potential in this work, U es µν = l ∗ B q r µν exp (cid:18) − r µν l ∗ D (cid:19) (11)with r µν = | r µν | , where r µν = r µ − r ν , is the vectorbetween beads ν and µ , and l ∗ B and l ∗ D are the nondimen-sional Bjerrum and Debye lengths, respectively.The non-dimensional diffusion tensor D νµ is related tothe non-dimensional hydrodynamic interaction tensor Ω through D µν = δ µν δ + (1 − δ µν ) Ω ( r ν − r µ ) (12)where δ and δ µν represent a unit tensor and a Kroneckerdelta, respectively, while Ω represents the effect of themotion of a bead µ on another bead ν through the dis-turbances carried by the surrounding fluid. The hydro-dynamic interaction tensor Ω is assumed to be given bythe Rotne-Prager-Yamakawa (RPY) regularisation of theOseen function Ω ( r ) = Ω δ + Ω rr r (13)where for r ≡ | r | ≥ a ∗ , Ω = 34 a ∗ r (cid:18) a ∗ r (cid:19) and Ω = 34 a ∗ r (cid:18) − a ∗ r (cid:19) (14)while for 0 < r ≤ a ∗ , Ω = 1 − ra ∗ and Ω = 332 ra ∗ (15)Here, a ∗ is the non-dimensional bead radius, which isrelated to the conventionally defined hydrodynamic in-teraction parameter, h ∗ , by a ∗ = √ πh ∗ . We have set a ∗ = 0 . h ∗ close to 0.25ensures that the non-draining limit is reached (and con-sequently universal predictions obtained), at relativelysmall values of N b . The spatial configuration of the chain at any time t ,i.e., r µ ( t ) for all beads µ = 1 , . . . , N b , is obtained by inte-grating Eq. (7) using a semi-implicit predictor-correctorscheme proposed by Prabhakar and Prakash. In thepresence of fluctuating HI, the problem of the computa-tional intensity of calculating the Brownian term is re-duced by the use of a Chebyshev polynomial represen-tation for the Brownian term.
We have adopted thisstrategy, and the details of the exact algorithm followedhere are given in Ref. 45.While the majority of results reported here have beenobtained with the “single-chain” BD algorithm describedin Ref. 45, the predictions of eigenvalues and shapefunctions have been obtained with the “multi-chain” BD algorithm described in Ref. 48. The most sig-nificant difference between the two algorithms is thatthe latter (in which multiple chains are simulated in abox with periodic boundary conditions), accounts for inter-particle hydrodynamic and electrostatic interac-tions in addition to intra-particle interactions. Essen-tially, N c bead-spring chains are simulated in a cubicsimulation box of length L , such that the concentration c = N b N c /L . With the overlap concentration c ∗ definedby, c ∗ = N b / [(4 / πr ], where r g,eq is the radius of gy-ration of a chain at equilibrium, we maintain a scaledconcentration c/c ∗ = 10 − , to ensure that the system isin the dilute limit. Since the typical distance betweenchains at these concentrations is much greater than theDebye screening lengths considered here, the strength ofshort-ranged inter-molecular Debye-H¨uckel electrostaticinteractions is effectively zero. The equivalence of theuse of either of the algorithms has been verified by com-parison of predictions of the end-to-end vector and theradius of gyration, and ensuring that no difference wasobserved. Plots which contain data from the multi-chainBD algorithm are identified as such in the figure captions,and unless explicitly stated, most plots report data ob-tained using the single chain BD algorithm. B. Size, shape and diffusivity
The two static properties examined here are: (i) theend-to-end distance, r e ≡ p h r i , with, h r i = h ( r N b − r ) · ( r N b − r ) i (16)where, h ( . ) i represents an ensemble average, and r N b and r are the positions of the two beads at either end ofthe chain, and, (ii) the radius of gyration of the chain, r g ≡ q h r i , with, h r i = h λ i + h λ i + h λ i (17)where, λ , λ , and λ are eigenvalues of the gyrationtensor G (arranged in ascending order), with, G = 12 N N b X µ =1 N b X ν =1 r µν r µν (18)Note that, G , λ , λ , and λ are calculated for eachtrajectory in the simulation before the ensemble averagesare evaluated.The asymmetry in equilibrium chain shape has beenstudied previously in terms of various functions definedin terms of the eigenvalues of the gyration tensor. Apart from λ , λ , and λ , themselves, we have exam-ined the following shape functions : the asphericity ( B ),the acylindricity ( C ), the degree of prolateness ( S and S ∗ ), and the shape anisotropy ( κ and κ ∗ ), as definedin Table II. As is evident from the Table, the latter two TABLE II. Definitions of shape functions in terms of eigenvalues of the gyration tensor, G . Note that, I = λ + λ + λ , and, I = λ λ + λ λ + λ λ , are invariants of G .Shape function DefinitionAsphericity B = h λ i − (cid:2) h λ i + h λ i (cid:3) (19)Acylindricity C = h λ i − h λ i (20)Degree of prolateness S ∗ = (cid:28) (3 λ − I )(3 λ − I )(3 λ − I )( I ) (cid:29) (21) S = (cid:10) (3 λ − I )(3 λ − I )(3 λ − I ) (cid:11)D ( I ) E (22)Relative shape anisotropy κ ∗ = 1 − (cid:28) I I (cid:29) (23) κ = 1 − h I ih I i (24) quantities are evaluated in terms of two different defi-nitions. Typically, it is easier to evaluate S and κ foranalytical calculations, rather than S ∗ and κ ∗ , whichrequire the evaluation of averages of ratios of fluctuatingquantities. The only dynamic property that is directly calculatedhere is the long-time self-diffusion coefficient D , definedby, D = lim t →∞ (cid:28) | r c ( t ) − r c (0) | t (cid:29) (25)where, r c = 1 N b P N b µ =1 r µ , is the position vector of thecenter of mass of the chain.The calculation of the radius of gyration and the dif-fusivity of a chain, enables an estimation of the longestrelaxation time from the expression, τ = h r i D (26)An ensemble averaging method has been used to es-timate all the properties, details of which are given inAppendix A of Ref. 49. We find that trajectories leveloff after roughly one relaxation time, but a true station-ary state from which equilibrium properties can be esti-mated, generally occurs after tens of relaxation times.Typically, more than O (10 ) data points are used inthe estimation of averages and standard errors of mean,where a data point is a saved property value at some time after the beginning of the stationary state. For proper-ties evaluated with the single chain code, a time step sizeof 0.005 was used, and data was saved roughly after every200 time steps. Averages were carried out over 1000 tra-jectories, with 1000 data points in each trajectory. Forproperties evaluated with the multi-chain code, a timestep size of 0.005 was used, and data was saved roughlyafter every 10 to 20 time steps. Averages were carried outover 64 trajectories, each with 15 chains in the simulationbox, and 2000 data points in each trajectory. C. Mapping between bead-spring chain and blob variables
The concept of an electrostatic blob enables the rep-resentation of experimental and simulation data fordilute polyelectrolyte solutions in terms of a signifi-cantly reduced number of scaling variables ( { X, Y } ),when compared to the number of bare-model parame-ters ( { N K , f, ˆ l B , ˆ l D } ). Further, the universal nature ofproperty predictions when expressed in terms of blobscaling variables has been established previously by sim-ulations, which have shown that various combinations ofbare-model parameters lead to identical results, providedvalues of the scaling variables are kept fixed. In orderto achieve a similar demonstration for BD simulationsof bead-spring chains, it is necessary to map the bead-spring chain parameters { N b , b, q, l ∗ B , l ∗ D } onto the blobscaling variables { X, Y } . This issue has been discussedpreviously by Pattanayek and Prakash . For the sake of TABLE III. Values for the bead-spring chain parameters { N b , b, q, l ∗ B } , the resultant scaling variables X and ξ ∗ el , and thecorresponding values for the bare model parameters { N K , f, ˆ l B } , used in Brownian dynamics simulations. Note that the listedvalues of the scaling variable Y are specified independently of other parameters by suitably choosing l ∗ D (not tabulated here). L ∗ is the non-dimensional contour length of the bead-spring chain, and γ is the Manning parameter.Blob model Bead-spring chain model Bare-model X Y ξ ∗ el { N b , b, q, l ∗ B } L ∗ { N K , f, ˆ l B } γ { , . , . , . } { , . , . } { , . , . , . } { , . , . } { , . , . , . } { , . , . } { , . , . , . } { , . , . } { , . , . , . } { , . , . } { , . , . , . } { , . , . } { , . , . , . } { , . , . } { , . , . , . } { , . , . } { , , . , . } { , . , . } { , . , . , . } { , . , . } { , , . , . } { , . , . } { , . , . , . } { , . , . } completeness, we summarise their arguments here.The key first step is to map bead-spring chain pa-rameters onto bare-model parameters, after which themapping onto blob scaling variables is straight-forward.This is achieved by equating the end-to-end vectors andthe contour lengths of the bare-model and bead-springchains. If L is the contour length, then, L = N K b K = ( N b − Q = ( N b − √ b l H (27)Further, using the analytical result for the end-to-endvector of FENE chains, we get, h r i = N K b = ( N b −
1) 3 bb + 5 l (28)Equations (27) and (28) can be solved for N K , and b ∗ K = b K /l H , to give, N K = ( N b − b + 53 ; b ∗ K = 3 √ bb + 5 (29)Since ˆ l D = l ∗ D /b ∗ K , ˆ l B = l ∗ B /b ∗ K , and ˆ ξ el = ξ ∗ el /b ∗ K , substi-tuting from Eq. (29) for N K and b ∗ K into Eqs (2) and (3),and using Eq. (6), leads to the following equations forthe blob scaling variables in terms of bead-spring chainparameters, X = ( N b − (cid:20) bb + 5 (cid:21) ξ ∗ el2 (30) Y = l ∗ D ξ ∗ el (31)where, ξ ∗ el = (cid:20) N b − b N q l ∗ B ( b + 5) (cid:21) / (32) It is clear from the expressions above that various com-binations of bead-spring chain parameters can result inidentical values for X and Y . Table III displays the var-ious sets of values of bead-spring chain parameters, andthe resultant sets of scaling variables X and Y , that havebeen used in the simulations reported here, and the cor-responding bare-model parameter values. Two to threedifferent values of N b are typically used for each valueof X . Note that Y can be varied independently of X by varying l ∗ D (and keeping all other bead-spring chainparameters fixed). As a result, the dependence of prop-erties of dilute polyelectrolyte solutions on X and Y canbe explored by carrying out BD simulations, and theirextent of universality assessed.In terms of bare-model parameters, the Manning pa-rameter, γ , which is defined as the number of elementarycharges per Bjerrum length, is given by, γ = f ˆ l B . TheOSFKK scaling theory assumes that γ ≪
1, since coun-terion condensation, which occurs for γ &
1, is not takeninto account.
The last column in Table III shows thatthe Manning parameter is significantly smaller than onein all the simulations reported here.
D. Mapping from experiments to blob variables
While the results of the simulations reported here havenot been compared directly with experimental observa-tions, it is worth outlining how experimental data on di-lute polyelectrolyte solutions can be mapped onto blobvariables, to facilitate future comparisons. In this con-text, the various equations for evaluating the blob vari-ables X and Y for an arbitrary polyelectrolyte solutionare derived in Appendix B.In the current simulation, the behaviour of variousstatic properties and the diffusivity of dilute polyelec-trolyte solutions has been explored for a range of blob h r e i / ξ e l YX = 24 , N b = 28 X = 24 , N b = 32 X = 24 , N b = 40 X = 48 , N b = 52 X = 48 , N b = 60 X = 72 , N b = 75 X = 72 , N b = 85 (a) h r g i / ξ e l YX = 24 , N b = 28 X = 24 , N b = 32 X = 24 , N b = 40 X = 48 , N b = 52 X = 48 , N b = 60 X = 72 , N b = 75 X = 72 , N b = 85 (b)FIG. 2. (Color online) Measures of mean chain size versus thereduced screening length, Y , at various values of the numberof blobs, X : (a) scaled end-to-end vector, h r i /ξ , (b) scaledradius of gyration, h r i /ξ . Simulation data acquired at dif-ferent values of chain length, N b , are displayed at each valueof X , to demonstrate parameter-free data collapse. The vari-ous combinations of bead-spring chain parameters that corre-spond to the displayed values of N b , are shown in Table III.Error bars are of the order of symbol size. variable values, 20 < X <
80 and 0 . ≤ Y ≤ γ / III. RESULTSA. Mean chain size as a function of X and Y From the OSFKK scaling picture depicted schemat-ically in Fig. 1, we expect the chain size to be inde- h r e i / ξ e l X Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 Y = 30 , X = 48 . , N b = 52 (a) r e / ( ξ e l X ) ln X . ± . r e / ( ξ e l X ) ln X . ± . Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (b)FIG. 3. (Color online) Demonstration of the existence of log-arithmic corrections to scaling at Y = 30 (in the blob-poleregime) for the scaled end-to-end vector. (a) The dotted lineshows that h r i /ξ departs from the X scaling predicted byOSFKK theory for X &
30. (b) A power law fit through thedata for r e / ( ξ el X ) versus ln X (solid line) shows that the loga-rithmic corrections have the expected exponent of 1 / N b are shown in Table III. Errorbars are of the order of symbol size. pendent of Y in the limits of Y → Y → ∞ (regime II) for X ≫
1, while at intermediatevalues of Y (in the crossover regimes III, IV, and V),we expect the chain size to depend on Y . This expecta-tion has been verified by several prior simulations, whichhave examined the scaling of the end-to-end vector. Figs. 2 (a) and 2 (b) show that in the current simulationsas well, both the scaled end-to-end vector, h r i /ξ , andthe scaled radius of gyration, h r i /ξ , asymptotically ap-proach constant values at the two extremes of Y . Notethat the asterisk in ξ ∗ el2 has been dropped for simplicity ofnotation, since the non-dimensional character of the blobsize is clear from the context. In the crossover regime, thechain size increases monotonically with Y as the chainstarts to swell due to increasing electrostatic repulsionwith increasing Debye screening length. As mentionedpreviously, our BD simulations are unable to distinguish h r g i / ξ e l X Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 Y = 30 , X = 48 . , N b = 52 (a) r g / ( ξ e l X ) ln X . ± . r g / ( ξ e l X ) ln X . ± . Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (b)FIG. 4. (Color online) Demonstration of the existence of weaklogarithmic corrections to scaling at Y = 30 (in the blob-poleregime) for the scaled radius of gyration. (a) The dotted lineshows the onset of departure from X scaling for h r i /ξ atlarge values of X . (b) A power law fit through the last fourdata points for r g / ( ξ el X ) versus ln X (solid line) shows thatthe logarithmic corrections have an exponent smaller than1 /
3. The various combinations of bare parameters corre-sponding to the value of N b are shown in Table III. Errorbars are of the order of symbol size. between the different regimes in the crossover region dueto the computational cost of simulating the extremelylong chains that would be required for such a purpose.Notably, as long as the value of X is the same, thepredicted values of both h r i /ξ , and h r i /ξ , are inde-pendent of the specific choice of bead-spring chain pa-rameters, at all values of Y . For instance at X = 48,both N b = 52 and N b = 60, lead to identical results.Such a parameter-free data collapse was previously shownto occur for BD simulation predictions of h r i /ξ byPattanayek and Prakash , who also noted that the onlyconstraint to the simulations appeared to be to ensurethat there were more beads than blobs in a chain, i.e., N b ≥ X .The OSFKK scaling picture ignores the presence of log-arithmic corrections in the blob-pole regime. However, asseen in Appendix A, and as shown previously by several different analytical studies, such corrections are ex-pected to arise in this regime, and previous molecularsimulations have shown that such is indeed the case. Inthe absence of logarithmic corrections, the scaled end-to-end vector, h r i /ξ , is expected to scale as X inregime II (see Table I). It is clear from Fig. 3 (a) thatwhile this scaling is obeyed for values of X .
30, thereis a departure at larger values of X . A similar devia-tion from the expected scaling in regime II was observedpreviously in the BD simulations reported by Pattanayekand Prakash . However, they did not verify if the sourceof the deviation was due to logarithmic corrections. Interms of the blob scaling variable X , Eq. 5 shows thatthe logarithmic correction term has an exponent of 1 / r e / ( ξ el X ) versus ln X would be a straight line with a slope of 1 /
3. Fig. 3 (b)shows that the logarithmic corrections to OSFKK scal-ing, picked up by the BD simulations, do in fact have theexpected dependence on X .While the radius of gyration is also a measure of meanchain size, Figs. 4 reveal that the logarithmic correctionsto scaling for h r i /ξ in the blob-pole regime are muchweaker than for h r i /ξ . This could be attributed to thefact that the radius of gyration not only depends on thestretched axis of the chain (which is expected to be in thedirection of the end-to-end vector), but also depends onthe dimensions of the chain perpendicular to the elon-gation axis. The scaling of the chain in these lateraldimensions is generally considered to obey random walkstatistics. We examine the validity of this expectationbelow, when we consider the eigenvalues of the gyrationtensor.In the limit of long chains, the ratio of the mean squareend-to-end vector to the mean square radius of gyra-tion, h r i / h r i , has a universal value of 6 for ideal chains,while it is equal to 12 for rigid rods. We anticipate,therefore, that for the polyelectrolyte solutions consid-ered here, the value of the ratio will increase from beingclose to 6 in the limit Y ≪ Y → ∞ , where thepolyelectrolyte chain adopts a stretched rodlike config-uration. Fig. 5 (a) shows that the ratio does approacha constant value close to 6 as Y approaches zero, andincreases monotonically with increasing Y . However,though the ratio levels off as the chain enters the blob-pole regime in the limit of large Y , the asymptotic valueis less than 12, for the values of X considered here. Thedeparture from the limiting value for rigid rods can beconsidered to reflect the degree of flexibility remaining inthe chain in this regime. Interestingly, the ratio appearsto be independent of X for a range of values of Y , start-ing with the ideal regime, and progressing well into thecrossover regime (i.e., the red and blue symbols overlapin Fig. 5 (a)). The ratio starts to become X dependentfor values of Y approaching the blob-pole regime. As wasseen in Figs. 3 (b) and 4 (b) above, though both h r i and h r i depart from the OSFKK scaling theory predictionsin this regime due to the presence of logarithmic correc-0 h r e i / h r g i YX = 24 , N b = 28 X = 24 , N b = 32 X = 24 , N b = 40 X = 48 , N b = 52 X = 48 , N b = 60 (a) h r e i / h r g i X h r e i / h r g i XY = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (b)FIG. 5. (Color online) The variation of the ratio h r i / h r i with: (a) the reduced screening length Y , for two differentvalues of the number of blobs X , and (b) the number of elec-trostatic blobs X , at Y = 30. Error bars are of the order ofsymbol size in (a), and are explicitly displayed in (b). tions, the strength of these corrections is different in thetwo cases. This difference is responsible for a persistentdependence of h r i / h r i on X in the blob-pole regime, ascan be seen in Fig. 5 (b), where the value of the ratiois plotted as a function of X , at Y = 30. We can an-ticipate, however, that with increasing values of X (i.e.,chain length), the appearance of logarithmic corrections,and the consequent onset of non-universal behaviour, ispostponed to larger and larger values of Y , due to a delayin the inception of the blob-pole regime (see Fig. 1). B. Eigenvalues of the radius of gyration tensor
The dependence of the three scaled eigenvalues of theradius of gyration tensor, h λ i /ξ , h λ i /ξ , and h λ i /ξ ,on the reduced screening length Y , is shown in Figs. 6.As was observed for h r i /ξ and h r i /ξ , all the eigenval-ues increase in the crossover regime from constant valuesat small values of Y , to constant values in the blob-poleregime at large values of Y . The collapse of data for dif- h λ i / ξ e l Y h λ ∗ i l ∗ D X = 24 , N b = 28 X = 24 , N b = 32 X = 48 , N b = 52 X = 48 , N b = 60 (a) h λ i / ξ e l Y h λ ∗ i l ∗ D X = 24 , N b = 28 X = 24 , N b = 32 X = 48 , N b = 52 X = 48 , N b = 60 (b) h λ i / ξ e l YX = 24 , N b = 28 X = 24 , N b = 32 X = 48 , N b = 52 X = 48 , N b = 60 (c)FIG. 6. (Color online) Scaled eigenvalues of the gyration ten-sor as a function of Y , at two values of X : (a) the smallesteigenvalue h λ i /ξ , (b) the intermediate eigenvalue h λ i /ξ ,and (c) the largest eigenvalue h λ i /ξ . The insets in (a)and (b) display the dependence of the unscaled eigenvalueson the non-dimensional Debye length l ∗ D close to the over-shoot, for two different chain lengths, N b , at the same valueof X . Data obtained with the multi-chain algorithm, with c/c ∗ = 10 − . Error bars are of the order of symbol size. ferent bead-spring chain model parameters, when repre-sented in terms of blob scaling variables, is also observedfor these properties. Clearly, the chain size increases withincreasing electrostatic repulsion between blobs, not onlyin the direction of maximum chain stretching ( h λ i ), but1 h λ i / ξ e l X Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (a) p h λ i / ( ξ e l X ) ln X . ± . p h λ i / ( ξ e l X ) ln X . ± . Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (b)FIG. 7. (Color online) Demonstration of the existence of log-arithmic corrections to scaling at Y = 30 (in the blob-poleregime) for the largest scaled eigenvalue of the gyration ten-sor. (a) The dotted line shows that h λ i /ξ departs fromthe X scaling predicted by OSFKK theory for X &
30. (b)A power law fit through the the last four data points for p h λ i /ξ el X versus ln X (solid line) shows that the logarith-mic corrections have an exponent close to the expected valueof 1 /
3. Data was obtained using the multi-chain algorithmwith c/c ∗ = 10 − . Error bars are of the order of symbol sizein (a), and are explicitly displayed in (b). also in the lateral directions ( h λ i and h λ i ). The sig-nificant differences in the magnitude of the chain dimen-sions in the three different principal directions clearlyillustrates the highly anisotropic shape of the chain in allregions of the phase diagram. This is examined in moredetail in terms of shape functions in the section below. Acurious observation, for which we do not have an obviousexplanation, is the presence of an overshoot in h λ i /ξ and h λ i /ξ for values of Y at the threshold of the blob-pole regime, i.e. just before the chain enters the finalscreening-length-independent regime from the crossoverregime. The continued stretching of h λ i with increas-ing Y at the verge of the blob-pole regime appears tobe accommodated by a shrinkage in h λ i and h λ i . Theovershoot also becomes more pronounced with increas-ing X . The insets in Figs. 6 (a) and (b) demonstratethat even the occurrence of an overshoot, which is dis- h λ i / ξ e l X
21 1 . ± . Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (a) h λ i / ξ e l X
21 1 . ± . Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (b)FIG. 8. (Color online) Dependence on the number of blobs X , of (a) the smallest scaled eigenvalue of the gyration tensor, h λ i /ξ , and (b) the intermediate scaled eigenvalue of thegyration tensor, h λ i /ξ , at a reduced screening length, Y =30. Data was obtained using the multi-chain algorithm with c/c ∗ = 10 − . Error bars are of the order of symbol size. tinct for different bead-spring chain parameters, collapsesonto a single curve when represented in terms of scalingvariables—signifying that blob variables accurately cap-ture the essential physics, even for phenomena that gobeyond the simple picture upon which they were origi-nally based. The use of finitely extensible springs in thesimulations is probably not responsible for the overshoot,and it is more likely to be an electrostatic phenomenon,since the springs are weakly stretched even in the blob-pole regime. For instance, from the value of h r i /ξ inFig. 2 (a) ( h r i /ξ ≈ N b = 60 and X = 48,in the limit of large Y ), and the values of ξ ∗ el and L ∗ corresponding to this value of N b and X in Table III( ξ ∗ el = 1 .
85 and L ∗ = 471), we can see that the ratio ofthe end-to-end vector to the contour length is less than16%.The scaling of r e with X in the blob-pole regime wasseen in Fig. 3 (b) to exhibit logarithmic corrections in linewith the prediction of Eq. (5). Since the largest eigen-value of the radius of gyration tensor is expected to be2 TABLE IV. Normalized eigenvalues of the radius of gyration tensor for neutral random walk polymers compared with thosefor a polyelectrolyte chain (PE) in the ideal chain regime of the phase diagram. Predictions by Koyama and Wei wereobtained from approximate analytical calculations for infinite Gaussian chains, while those by ˇSolc , ˇSolc and Stockmayer and Kranbuehl and Verdier were obtained with lattice Monte Carlo simulations. Values reported from Theodorou andSuter are for atactic polypropylene, obtained from Monte Carlo simulations based on a rotational isomeric state model,while those from Zifferer are from lattice Monte Carlo simulations of “nonreversal random walks”, which are considered torepresent θ -solvents. Data for the polyelectrolyte solution was obtained using the multi-chain algorithm, with c/c ∗ = 10 − .Solution Chain length h λ i / h r i h λ i / h r i h λ i / h r i h λ i / h λ i h λ i / h λ i Neutral N b → ∞ . .
175 0 .
754 2 . . N b → ∞ .
071 0 .
179 0 . · · · . N b = 100 0 .
065 0 .
175 0 .
75 2 . . N b = 63 0 .
065 0 .
176 0 .
76 2 .
70 11 . N b = 999 0 . ± .
004 0 . ± .
01 0 . ± .
06 2 . ± . . ± . N b → ∞ . . . · · · · · · PE ( Y = 0 . X = 24 0 . ± . . ± . . ± . . ± .
02 12 . ± . X , at a reducedscreening length, Y = 30. Data was obtained using the multi-chain algorithm, with c/c ∗ = 10 − . X h λ i / h r i h λ i / h r i h λ i / h r i h λ i / h λ i h λ i / h λ i − h λ i / h λ i . ± . . ± . . ± . . ± .
02 54 . ± .
22 0 . ± . . ± . . ± . . ± . . ± .
03 62 . ± .
30 0 . ± . . ± . . ± . . ± . . ± .
09 68 . ± .
88 0 . ± . . ± . . ± . . ± . . ± .
04 74 . ± .
43 0 . ± . . ± . . ± . . ± . . ± .
11 81 . ± .
12 0 . ± . . ± . . ± . . ± . . ± .
04 86 . ± .
40 0 . ± . . ± . . ± . . ± . . ± .
16 100 . ± . . ± . in the direction of maximum chain stretching, we expect p h λ i /ξ el to also exhibit logarithmic corrections accord-ing to Eq. (5). Figure 7 (a) displays the departure fromthe OSFKK scaling prediction in the blob-pole regime,while Fig. 7 (b) shows that the logarithmic correctionshave an exponent close to the expected value of 1 / h λ i /ξ and h λ i /ξ are plotted as functions of X . For the limited range ofvalues of X examined here, the smallest and intermedi-ate eigenvalues appear to obey power law scaling withexponents that lie somewhere between ideal chain andblob-pole scaling.Eigenvalues of the radius of gyration tensor for neutralpolymer chains are usually reported in terms of ratios,either between individual eigenvalues, or with the meansquare radius of gyration, since they are expected to at-tain universal values in the limit of long chains. For a chain with a spherically symmetric shape about the cen-tre of mass, we expect h λ i i / h r i = 1 /
3, for i = 1 , , h λ i i / h λ j i = 1 for all combinations i and j . Pre-dictions reported previously in the literature for flexibleneutral chains in θ -solutions, from a variety of differ-ent approaches, are displayed in Table IV. The strongasymmetry in the shapes of chains is clearly apparentfrom the extent of departure from the expected values.Steinhauser has carried out molecular dynamics sim-ulations to examine the influence of solvent quality onthe ratios of eigenvalues, and noted that while their val-ues are relatively similar in good and theta solvents,they change significantly in poor solvents, with valuesthat imply that chains become much more spherical asthey collapse into globules. From OSFKK theory, we ex-pect polyelectrolyte chains to behave like ideal chains for Y ≪
1. As can be seen from the last row of Table IV,this is indeed the case for the results obtained with thecurrent simulations, with all the different ratios being ingood agreement with results for neutral chains.The dependence of the eigenvalues scaled with themean square radius of gyration, on the reduced screen-ing length Y , is displayed in Figs. 9. Starting with thelimiting values reported in Table IV for polyelectrolytechains in the ideal chain regime, both h λ i / h r i and h λ i / h r i (not shown here) decrease with increasing Y in the crossover regime until they reach their asymp-3 h λ i / h r g i Y X = 24 , N b = 28 X = 24 , N b = 32 X = 48 , N b = 52 X = 48 , N b = 60 (a) h λ i / h r g i YX = 24 , N b = 28 X = 24 , N b = 32 X = 48 , N b = 52 X = 48 , N b = 60 (b)FIG. 9. (Color online) Dependence of the eigenvalues of theradius of gyration tensor, normalised by the radius of gyra-tion, on the reduced screening length Y , at two values of thenumber of blobs, X : (a) the smallest eigenvalue h λ i / h r i ,and (b) the largest eigenvalue h λ i / h r i . Data was obtainedusing the multi-chain algorithm, with c/c ∗ = 10 − . Errorbars are of the order of symbol size. totic values in the blob-pole regime. The reason forthe decrease in the magnitude of these ratios is because h r i = h λ i + h λ i + h λ i is dominated by the behaviourof h λ i , which can be seen from Figs. 6 to increase muchmore rapidly with Y than both h λ i and h λ i . For thesame reason, the ratio h λ i / h r i increases with Y , nearlyapproaching a value of 1 in the limit of large Y . The over-shoot in h λ i and h λ i that was observed in Figs. 6 (a)and (b) at the threshold of the blob-pole regime, is notnoticeable when the eigenvalues are scaled with h r i inplace of ξ (as displayed in Fig. 9 (a) for h λ i / h r i ).The extent of asymmetry in chain shape in the planeperpendicular to the elongation axis can be gauged bythe quantity, 1 − h λ i / h λ i , which would be zero if theshape was symmetric. For neutral chains, and for poly-electrolyte chains in the ideal chain regime, using thevalues of h λ i / h r i and h λ i / h r i reported in Table IV,we can calculate 1 − h λ i / h λ i ≈ .
63. Fig. 10 shows thatwith increasing electrostatic repulsion between blobs, thealready highly asymmetric shape becomes even more so, − h λ i / h λ i Y − h λ i / h λ i YX = 24 , N b = 28 X = 24 , N b = 32 X = 48 , N b = 52 X = 48 , N b = 60 FIG. 10. (Color online) Asymmetry in chain shape in theplane perpendicular to the stretching axis as a function of thereduced screening length, Y . Data was obtained using themulti-chain algorithm, with c/c ∗ = 10 − . levelling off to a constant value in the limit of large Y .A notable aspect of the normalised eigenvalue ratiosdisplayed in Figs. 9 and 10 is that their values are inde-pendent of the number of blobs X in the chain, for a widevariety of values of Y , ranging from the ideal chain regimeand well into the crossover regime. The ratios become X dependent only as the chains approach the blob-poleregime with increasing screening length. This suggeststhat for sufficiently long chains, as in the case of neutralpolymer chains, the normalised eigenvalues of polyelec-trolyte chains attain universal values in those parts ofthe phase diagram that lie outside the blob-pole regime.Within the blob-pole regime, we expect that ratios ofquantities that scale nearly identically with X will attainroughly constant values, such as h λ i / h r i and h λ i / h r i (which follows from the dominance of the growth of h λ i with X , compared to h λ i and h λ i ), while ratios of quan-tities that do not scale identically, would retain a de-pendence on the number of blobs. This is examined inTable V, where the dependence of a variety of ratios onthe number of blobs X , at the particular value, Y = 30,which is in the blob-pole regime, is tabulated. At anyfixed value of Y , however, we expect chains to move outof the blob-pole regime as X → ∞ , and for all the ratiosto become universal. C. Shape functions
For chain shapes with tetrahedral or greater symme-try, the asphericity B = 0, otherwise B >
0. Forchain shapes with cylindrical symmetry, the acylindric-ity C = 0, otherwise C >
0. With regard to the degreeof prolateness, its sign determines whether chain shapesare preponderantly oblate (
S, S ∗ ∈ [ − . , S, S ∗ ∈ [0 , κ and κ ∗ ),on the other hand, lies between 0 (for spheres) and 14 B / h r g i YX = 24 , N b = 28 X = 24 , N b = 32 X = 48 , N b = 52 X = 48 , N b = 60 C / h r g i YX = 24 , N b = 28 X = 24 , N b = 32 X = 48 , N b = 52 X = 48 , N b = 60 (a) (b) S ∗ YX = 24 , N b = 28 X = 24 , N b = 32 X = 48 , N b = 52 X = 48 , N b = 60 κ ∗ YX = 24 , N b = 28 X = 24 , N b = 32 X = 48 , N b = 52 X = 48 , N b = 60 (c) (d)FIG. 11. (Color online) The dependence of shape functions on the reduced screen length Y , at two values of X : (a) thenormalised asphericity B/ h r i (see Eq. (19)), (b) the normalised acylindricity C/ h r i (see Eq. (20)), (c) the degree of prolateness, S ∗ (see Eq. (21)), and, (d) the shape anisotropy, κ ∗ (see Eq. (23)). When not displayed explicitly, error bars are of the orderof symbol size. Data obtained with the multi-chain code, with c/c ∗ = 10 − .TABLE VI. Shape functions for neutral random walk polymers compared with those for a polyelectrolyte chain (PE) in theideal chain regime of the phase diagram. Predictions by Steinhauser were obtained with molecular dynamics simulations,while those by Bishop and Michels and Bishop et al. were obtained with Brownian dynamics, and off-lattice Monte Carlosimulations, respectively. Methods used to evaluate shape functions in Refs. 20, 28 and 27 are given in the caption to Table IV.Data for the polyelectrolyte solution was obtained using the multi-chain algorithm, with c/c ∗ = 10 − . Solution Chain length B/ h r i C/ h r i S ∗ S κ ∗ κ Neutral N b → ∞ · · · · · · . . . . N b = 999 0 . ± .
04 0 . ± . · · · · · · . ± . · · · Neutral N b = 48 · · · · · · . ± .
021 0 . ± .
245 0 . ± .
011 0 . ± . N b → ∞ · · · · · · · · · · · · . ± .
001 0 . ± . N b → ∞ . ± . · · · · · · · · · . ± .
003 0 . ± . N b → ∞ · · · · · · . . . . Y = 0 . X = 24 0 . ± . . ± . . ± . . ± . . ± . . ± . (for rods). In this context, the values of the normalisedshape functions B/ h r i and C/ h r i , S and S ∗ , and κ and κ ∗ , predicted by various techniques for neutral polymerchains are compared with the predictions of the currentsimulations for polyelectrolyte chains in the ideal chainregime, in Table VI. Clearly, in this regime, polyelec-trolyte chains behave identically to neutral random walk chains, and share the same high degree of anisotropy.The dependence of the shape functions B/ h r i , C/ h r i , S ∗ and κ ∗ on Y , for two different values of X , is dis-played in Figs. 11. As we might expect, chains appear tobecome more aspherical, more cylindrical, more prolate,and more rodlike, with increasing electrostatic repulsionbetween the blobs. The functions S and κ behave simi-5 B / ξ e l X Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 Y = 30 , X = 48 . , N b = 60 C / ξ e l X Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 Y = 30 , X = 48 . , N b = 60 (a) (b) S X S X Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 Y = 30 , X = 48 . , N b = 60 κ X κ X Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 Y = 30 , X = 48 . , N b = 60 (c) (d)FIG. 12. (Color online) The dependence of shape functions on the number of blobs X , in the blob-pole regime, at a particularvalue of the reduced screening length Y = 30: (a) the scaled asphericity B/ξ , (b) the scaled acylindricity
C/ξ , (c) the degreeof prolateness S , and, (d) the shape anisotropy κ . Error bars are of the order of symbol size in (a) and (b). Data obtainedwith the multi-chain code, with c/c ∗ = 10 − . larly to S ∗ and κ ∗ .The independence of the values of the shape functionsfrom the number of blobs X , in all the regimes exceptthe blob-pole regime, suggests that for sufficiently longchains, at any given value of the reduced screening length Y , the shapes of polyelectrolyte chains are universal. Inthe blob-pole regime however, as noted earlier, the ap-pearance of corrections to scaling (to varying extents inthe three principal directions), leads to a dependence ofchain shapes on the chain length. The nature of thisdependence on X , at a particular value of Y = 30, isdisplayed in Figs. 12 for B/ξ , C/ξ , S and κ . Thefunctions S ∗ and κ ∗ behave similarly to S and κ . D. Translational diffusivity interpreted with the OSFKKscaling picture
The dependence on the reduced screening length Y ,of all the static properties examined here so far, has fol-lowed a similar pattern: they exhibit Y independent be-haviour in the ideal chain and blob-pole regimes, while they are functions of Y in the crossover regime betweenthese two limits. By defining the Zimm diffusivity of anelectrostatic blob through the expression, D ξ = k B T πη s ( ξ el /
2) (33)we find that a similar pattern is obeyed by the ratio
D/D ξ , where D is the translational diffusivity of thepolyelectrolyte chain as a whole.In the ideal chain regime, since D ∼ k B T / (6 πη s r e ), weexpect D/D ξ ∼ X − / . In the blob-pole regime, an ex-pression for the diffusivity can be derived by drawing ananalogy with the shish-kebab model for rodlike polymers.In the latter, a rodlike polymer of length L is modelled as N = ( L/d ) beads of diameter d , placed along a straightline. The translational diffusivity of such a shish-kebabcan be shown to be, D = ln( L/d )3 πη s L k B T (34)By mapping d → ξ el , and L → r e , it follows that the6 D / D ξ Y X = 24 , N b = 28 X = 24 , N b = 32 X = 24 , N b = 40 X = 72 , N b = 75 X = 72 , N b = 85 FIG. 13. (Color online) Dependence of the ratio of the trans-lational diffusivity of the chain D , to the diffusivity of theelectrostatic blob D ξ , on the reduced screening length Y , attwo values of the number of blobs, X . Error bars are of theorder of symbol size. diffusivity of a blob-pole is given by, D = ln( X )3 πη s r e k B T (35)Within the OSFKK scaling ansatz, since ( r e /ξ el ) ∼ X inthe blob-pole regime, we see that, DD ξ = ln( X ) X (36)If logarithmic corrections to the scaling of r e (accordingto Eq. (5)) are taken into account, we expect, DD ξ = [ln( X )] / X (37)In either case, we see that the ratio D/D ξ is independentof Y in the both the limiting regimes of small and large Y .In between these two limits, we expect D/D ξ to dependon Y .These arguments are substantiated in the plot of D/D ξ versus Y , at two values of X , displayed in Fig. 13. We seethat for each value of X , the ratio D/D ξ crosses over froma constant value in the ideal chain regime to a constantvalue in the blob-pole regime. As has been observed inthe case of all the properties examined so far, the useof blob scaling variables to interpret the results of BDsimulations leads to behaviour that is independent of thespecific choice of bead-spring chain parameters.The importance of accounting for logarithmic correc-tions to the scaling of r e with X in Eq. (35), is stud-ied in Figs. 14. The plot of D/ ( D ξ ln X ) versus 1 /X in Fig. 14 (a) shows that while the scaling described byEq. (36) is obeyed at relatively small values of X , thereis a departure from OSFKK scaling for X &
30. On theother hand, the plot of D/ ( D ξ X − ) as a function of ln X D / ( D ξ l n X ) /X Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (a) D / ( D ξ X − ) ln X . ± . D / ( D ξ X − ) ln X . ± . Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (b)FIG. 14. (Color online) Demonstration of the importanceof accounting for logarithmic corrections to the scaling of r e with X in Eq. (35) for the translational diffusivity D in theblob-pole regime: (a) Departure from OSFKK scaling at largevalues of X (see Eq. (36)) (b) Scaling behaviour in agreementwith Eq. (37), with the expected exponent of ln X ≈ / displayed in Fig. 14 (b) shows that Eq. (37) is indeedobeyed, with the exponent of ln X close to the expectedvalue of 2 / Y →
0) diffuse more than twiceas fast as nearly unscreened chains ( Y ≫ X . This behaviour is examinedin more detail in Fig. 15, which displays the ratio of thediffusivity of a chain with Y = 0 .
01 to that of a chainwith Y = 30, for various values of X . When logarithmicscaling corrections are ignored, we expect this ratio toscale as X . As can be seen from Fig. 15, this scaling isapproximately satisfied. As in other examples consideredhere, the inclusion of logarithmic corrections in the blob-pole regime leads to a better fit of the data, as shown inthe inset to Fig. 15.From the scaling of r e /ξ el (see Eq. (5)) and D/D ξ (seeEq. (37)) with X in the blob-pole regime, it can be seen7
20 40 60 75 D ( Y = . ) / D ( Y = ) X . D ( Y = . ) / D ( Y = ) X . Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 Y = 30 , X = 72 . , N b = 85 ( D ( Y = ) / D ( Y = . ) ) X / ln X / ( D ( Y = ) / D ( Y = . ) ) X / ln X / Y =30 , X =20 . , N b =32 Y =30 , X =24 . , N b =28 Y =30 , X =27 . , N b =32 Y =30 , X =34 . , N b =38 Y =30 , X =37 . , N b =40 Y =30 , X =72 . , N b =85 FIG. 15. (Color online) Dependence on the number of blobs X , of the ratio of the translational diffusivity of the chain inthe ideal chain regime ( Y = 0 .
01) to the diffusivity of a chainin the blob-pole regime ( Y = 30). Inset shows the importanceof accounting for logarithmic corrections to scaling. Errorbars are of the order of symbol size. that the ratio, h r i /ξ D/D ξ = h r i /Dτ ξ ∼ X where, τ ξ , the blob relaxation time, is defined by, τ ξ = ξ /D ξ . Notably, this ratio is free from logarithmic cor-rections even in the blob-pole regime. Figure 16 (a) dis-plays the dependence on the number of blobs X , of theratio of the relaxation time in the blob-pole regime to therelaxation time of an electrostatic blob, τ /τ ξ . Thoughthe relaxation time τ has been defined in terms of h r i instead of h r i (see Eq. (26)), we see from Fig. 16 (a) that(in the range of X values that have been considered), anyresidual logarithmic corrections are extremely weak, andthat the ratio scales with X with the expected power lawexponent.The difference in the conformations of the chain in thetwo regimes, leads to the long-time relaxation in the blob-pole regime being nearly an order of magnitude slowerthan in the ideal chain regime, as displayed in Fig. 16 (b).As can be seen from the figure, the ratio of the two lim-iting relaxation times, τ ( Y =30) /τ ( Y =0 . , increases withthe number of blobs X with the expected power law, X .The ratio U RD , defined by, U RD = r g r h (38)where, r h is the hydrodynamic radius, r h = k B T / (6 πη s D ), has an universal value for neutral poly-mer solutions in the long chain limit, since both r g and r h scale identically with chain length. In partic-ular, by extrapolating finite chain data acquired fromhighly accurate BD simulations, to the long chain limit,Sunthar and Prakash have shown that for θ -solutions, τ / τ ξ X Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (a) τ ( Y = ) / τ ( Y = . ) X . Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (b)FIG. 16. (Color online) Dependence on the number of blobs X , of (a) the ratio of the relaxation time of the chain in theblob-pole regime to the relaxation time of an electrostaticblob, and (b) the ratio of the relaxation time of the chain inthe blob-pole regime (at Y = 30) to the relaxation time inthe ideal chain regime (at Y = 0 . U θ RD = 1 . ± .
01. We anticipate that for the polyelec-trolyte solutions considered here, U RD will have a similarvalue in the ideal chain regime.In the blob-pole regime, by substituting for D fromEq. (35) into Eq. (38), we see that, U RD ∼ (cid:18) r g r e (cid:19) ln X (39)As a result, U RD does not have a universal value in thisregime, but rather depends on the number of blobs X .In the context of the current simulations, Figs. 3 (b)and 4 (b) show that the scaling expressions, r e / ( ξ el X ) ∼ (ln X ) . , and r g / ( ξ el X ) ∼ (ln X ) . , respectively, pro-vide good fits to the simulation data. As a result, weexpect from Eq. (39) that, U RD ∼ (ln X ) . .Figure 17 (a) displays the dependence of U RD on thereduced screening length Y , for two different values of X . We see that for small values of Y , the ratio has aconstant universal value which is close to that for neutral8 U RD YX = 24 , N b = 28 X = 24 , N b = 32 X = 24 , N b = 40 X = 72 , N b = 75 X = 72 , N b = 85 (a) U RD ln X . ± . Y = 30 , X = 20 . , N b = 32 Y = 30 , X = 24 . , N b = 28 Y = 30 , X = 27 . , N b = 32 Y = 30 , X = 30 . , N b = 32 Y = 30 , X = 34 . , N b = 38 Y = 30 , X = 37 . , N b = 40 (b)FIG. 17. (Color online) Ratio of the radius of gyration to thehydrodynamic radius, U RD (see Eq. (38)), as a function of(a) the reduced screening length Y , at two different values ofthe number of blobs X , and (b) demonstration of logarithmiccorrections to scaling in the blob-pole regime, at a constantvalue of the reduced screening length, Y = 30. Error bars areof the order of symbol size. chains. The ratio increases, while remaining universal(i.e., independent of X ) for increasing values of Y in thecrossover regime, before levelling off to a non-universalvalue in the blob-pole regime. The dependence of U RD on ln X in the blob-pole regime, at the particular value Y = 30, is displayed in Fig. 17 (b). We see that theexponent of ln X is indeed close to the expected value of0 . IV. CONCLUSIONS
The size, shape and diffusivity of a weakly-chargedpolyelectrolyte chain in solution have been examined inthe limit of low polymer concentration using Brown-ian dynamics simulations of a coarse-grained bead-springchain model, with Debye–H¨uckel electrostatic interac-tions between the beads. Simulation results have beenrecast in terms of the scaling variables X (the num- ber of electrostatic blobs), and Y (the reduced screeninglength), which are defined within the framework of theOSFKK blob scaling theory. While the root-mean-squareend-to-end vector and radius of gyration have been usedto represent the mean size of the chain, various func-tions defined in terms of the eigenvalues of the radiusof gyration tensor have been used to characterise chainshape in all parts of the { X, Y } phase-space. The trans-lational diffusion coefficient of the chain, which is a dy-namic property, has been determined accurately from thedisplacement of the chain centre of mass, by taking hy-drodynamic interactions into account through incorpora-tion of the Rotne-Prager-Yamakawa tensor into the BDalgorithm.The key results of the present work are summarised inthe list below:1. Interpretation of simulation results in terms of blobscaling variables X and Y leads to a descriptionof solution behaviour independent of the level ofcoarse-graining, i.e., of the choice of the number ofbeads N b , and other parameters in the bead-springchain model. This should prove extremely usefulfor comparing simulation results with experiments.The procedure for mapping from experimental vari-ables to blob scaling variables, in order to facilitatethis comparison, has been provided in Appendix B.2. Three broad domains of behaviour can be clearlyidentified: (i) the ideal chain regime correspondingto small values of Y , (ii) the crossover regime cor-responding to intermediate values of Y , and (iii)the blob-pole regime corresponding to large valuesof Y .3. In the blob-pole regime, BD simulations appear topick up the existence of logarithmic corrections forfairly short chains, which are in good agreementwith predictions of refined scaling theories.4. Various universal ratios of eigenvalues, the as-phericity, the acylindricity, the degree of prolate-ness and the shape anisotropy of polyelectrolytechains in the ideal chain regime, have been com-pared with published results in the literature forflexible neutral random walk polymers.5. When suitably scaled, the size, shape and diffu-sivity of a chain are independent of the reducedscreening length Y in the ideal chain and blob-poleregimes, and depend on Y only in the crossoverregime.6. The translational diffusivity of the chain in theblob-pole regime can be described by drawing ananalogy with the translational diffusivity of a rod-like polymer modelled as a shish-kebab. The nor-malisation parameter that enables the collapse ofBD data for the translational diffusivity is theZimm diffusivity of the electrostatic blob.97. All properties of polyelectrolyte solutions, whensuitably normalised, appear to exhibit universal be-haviour, independent of the number of electrostaticblobs X in the chain, in all regimes of the { X, Y } phase-space, except in the blob-pole regime, wherethe occurrence of logarithmic corrections to scalingleads to non-universal behaviour. V. ACKNOWLEDGEMENTS
The authors gratefully acknowledge CPU time grantsfrom the National Computational Infrastructure (NCI)facility hosted by the Australian National University, andVictorian Life Sciences Computation Initiative (VLSCI)hosted by the University of Melbourne.
Appendix A: Scaling of the end-to-end vector in theblob-pole regime
The scaling of the end-to-end vector in the blob-poleregime can be derived from a Flory type energy min-imisation argument. The essential assumption is thatchains adopt an ellipsoidal shape, with electrostatic inter-actions causing stretching in the direction of the major-axis, while leaving chain conformations in the directionof the minor-axes unaffected. As a result, the long half-axis of the ellipsoid scales as r e /
2, while the aspect ratioscales as r e / ( b K N / ). The total energy of such a chaincan be obtained by combining the electrostatic energy ofan ellipsoid with the elastic energy of a stretched chain.Ignoring prefactors of order unity, the electrostatic en-ergy of an ellipsoid is, k B T l B ( f N K ) r e ln r e b K N K ! while the elastic energy of a stretched chain is, k B T (cid:18) r b N K (cid:19) From Flory theory, it follows that the equilibrium end-to-end vector can be derived by minimising the total energy, Uk B T ∼ (cid:18) r b N K (cid:19) + l B ( f N K ) r e ln r e b K N K ! (A1)It is convenient to proceed in two steps—first deriving theequilibrium end-to-end vector by neglecting the logarith-mic term in the electrostatic energy, and then accountingfor it in the next step. Minimising the sum, (cid:18) r b N K (cid:19) + l B ( f N K ) r e with respect to r e , leads to, r e = f ˆ l B N K b K (A2)This is identical to the expression derived by OSFKK the-ory for the end-to-end vector in the blob-pole regime (seeTable I). We now assume that including the logarithmicterm in the electrostatic energy leads to a modificationof the equilibrium end-to-end vector, r e = f ˆ l B N K b K x (A3)where, the factor x remains to be determined. Substi-tuting the modified expression for r e from Eq. (A3) intoEq. (A1) (and absorbing the common factor N K ( f ˆ l B ) into the energy), leads to the following sum, x + 1 x ln h N K f ˆ l B x i which must be minimised with respect to x in order todetermine x . This leads to, x = ln h N K f ˆ l B i + ln x − x ≫
1, implies, x = (cid:26)
12 ln h N K f ˆ l B i(cid:27) (A5)where, the factor (1 /
2) in front of the logarithmic termhas been introduced for future convenience. The assump-tion that x is large can be seen to be justified for N K ≫ x from Eq. (A5) intoEq. (A3), shows that this two-step procedure leads to aprediction of the equilibrium end-to-end vector scalingwith chain size according to, r e b K ∼ f ˆ l B N K n ln h N K f ˆ l B io (A6)Clearly, the logarithmic correction to the OSFKK scalingexpression in the blob-pole regime comes from taking thelogarithmic term in the expression for the electrostaticenergy of an ellipsoid into account. Equation (A6) is sim-ilar to the expression derived previously by Dobrynin andRubinstein . This expression takes a particularly sim-ple form in terms of blob scaling variables (see Eqs. (4)and (2)), r e ξ el ∼ X [ln X ] (A7) Appendix B: Mapping experiments to blob variables forNaPSS
Consider a polyelectrolyte chain of molecular weight M , dissolved in a solvent at temperature T , with a tem-perature dependent relative permittivity ε r ( T ). If the0 TABLE VII. Mapping experimental data for sodium poly(styrene sulfonate) to the blob model M = 20600 ( N m = 100 , N K = 10)Solution properties Blob variables α ( f ) c ms (I) (mol/ltr) T ◦ C l B (nm) l D (nm) γ ξ el (nm) X Y × − ) 15 0.7077 4321.0 0.0283 17.67 0.2001 244.508425 0.7158 4297.0 0.0286 17.61 0.2016 244.046335 0.7247 4270.0 0.0290 17.53 0.2033 243.54220.01 (0.01) 15 0.7077 3.056 0.0283 17.67 0.2001 0.172925 0.7158 3.038 0.0286 17.61 0.2016 0.172635 0.7247 3.019 0.0290 17.53 0.2033 0.17220.5 (0.5) 15 0.7077 0.4321 0.0283 17.67 0.2001 0.024525 0.7158 0.4297 0.0286 17.61 0.2016 0.024435 0.7247 0.4270 0.0290 17.53 0.2033 0.02440.3 (3.0) 0 (1 . × − ) 15 0.7077 788.9 0.8492 1.830 18.6534 431.003325 0.7158 784.5 0.8589 1.824 18.7951 430.188935 0.7247 779.6 0.8696 1.816 18.9512 429.30020.01 (0.01) 15 0.7077 3.056 0.8492 1.830 18.6534 1.669325 0.7158 3.038 0.8589 1.824 18.7951 1.666135 0.7247 3.019 0.8696 1.816 18.9512 1.66270.5 (0.5) 15 0.7077 0.4321 0.8492 1.830 18.6534 0.236125 0.7158 0.4297 0.8589 1.824 18.7951 0.235635 0.7247 0.4270 0.8696 1.816 18.9512 0.2351 M = 206000 ( N m = 1000 , N K = 100) α ( f ) c ms (I) (mol/ltr) T ◦ C l B (nm) l D (nm) γ ξ el (nm) X Y × − (4 . × − ) 15 0.7077 151.8 0.2831 3.808 43.1119 39.876625 0.7158 151.0 0.2863 3.793 43.4393 39.801235 0.7247 150.0 0.2899 3.777 43.8001 39.71900.1 (0.1) 15 0.7077 0.9662 0.2831 3.808 43.1119 0.253825 0.7158 0.9608 0.2863 3.793 43.4393 0.253335 0.7247 0.9548 0.2899 3.777 43.8001 0.25280.8 (0.8) 15 0.7077 0.3416 0.2831 3.808 43.1119 0.089725 0.7158 0.3397 0.2863 3.793 43.4393 0.089635 0.7247 0.3376 0.2899 3.777 43.8001 0.08940.35 (3.5) 4 × − (4 . × − ) 15 0.7077 149.5 0.9907 1.652 229.0977 90.537625 0.7158 148.7 1.0021 1.645 230.8376 90.366635 0.7247 147.8 1.0146 1.639 232.7550 90.17990.1 (0.1) 15 0.7077 0.9662 0.9907 1.652 229.0977 0.585025 0.7158 0.9608 1.0021 1.645 230.8376 0.583935 0.7247 0.9548 1.0146 1.639 232.7550 0.58270.8 (0.8) 15 0.7077 0.3416 0.9907 1.652 229.0977 0.206825 0.7158 0.3397 1.0021 1.645 230.8376 0.206435 0.7247 0.3376 1.0146 1.639 232.7550 0.2060 monomer concentration in molar units is c mp , and thedegree of ionization per chain is α , then the number ofcounterions in solution per unit volume is, α c mp ( N A N A is Avagadro’s number. If the added salt A m B n ,with molar concentration c ms , dissociates in solution ac-cording to, A m B n −→ mA z+ + nB z − where, z + and z − are the cation and anion valences, re-spectively, then the number of cations A per unit volumein solution is c + = m c ms ( N A c − = n c ms ( N A I of the solution is given by, I = 12 (cid:0) mz + nz − (cid:1) c ms ( N A z p α c mp ( N A z p is the counterion valence.The Bjerrum length l B can be calculated from Eq. (1), while the Debye length is given by, l D = (8 π l B I ) − (B2)Mapping the experimental system onto blob variablesis straightforward once it is mapped onto the bare-modelparameters { N K , f, ˆ l B , ˆ l D } . In order to do so, it is neces-sary to know the intrinsic persistence length of the poly-electrolyte chain, ℓ , the molecular weight of a monomer, M m , and the monomer length b m . With this informa-tion, we can find, (i) the Kuhn step length, b K = 2 ℓ ,(ii) the number of monomers in a chain, N m = M/M m ,(iii) the number of Kuhn steps, N K = ( N m b m ) /b K , and(iv) the number of charges per Kuhn step, f = αN m,K ,where, N m,K = b K /b m , is the number of monomers perKuhn step. It is also straight forward to find ˆ l B = l B /b K ,and ˆ l D = l D /b K , at the given temperature and ionicstrength. Once the bare-model parameters are known,the blob scaling variables, { X, Y, ˆ ξ el } can be determined1from Eqs. (2)–(4).Table VII displays the blob scaling variables calculatedwith this procedure for sodium poly(styrene sulfonate)(NaPSS) at two different molecular weights, M = 20600Dalton and M = 206000 Dalton. For NaPSS, the molec-ular weight of a monomer, M m = 206 Dalton, and themonomer length b m = 2 . The intrinsic persistencelength of NaPSS is variously estimated as lying inthe range, 9˚A ≤ ℓ ≤ ℓ = 12 . c mp = 10 − mol/ltr is chosen asthe monomer concentration, since at this concentration,according to the phase diagram in Ref. 52, the solu-tion remains in the dilute regime for all the values of N m considered here. The degree of ionization (sulfona-tion) per chain for NaPSS is commonly assumed to be α = 0 . Here, we choose a range of values between0.01 and 0.35 in order to generate a range of values ofthe blob scaling variables X and Y . For water, ε r ( t ) =87 . − . t +9 . × − t − . × − t , where,0 ◦ C ≤ t ≤ ◦ C. We assume monovalent cations andanions for the added salt, with composition stochiometry m = 1, and n = 1. The salt concentration c ms is varied inorder to change the ionic strength, and consequently, theDebye length. Finally, three values of temperature arechosen that bracket typical room temperature values.As mentioned earlier, in the current simulation, blobvariable values in the range, 20 < X <
80 and 0 . ≤ Y ≤ X and Y , while maintaining the Manning parameter, γ / P.-G. de Gennes,
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