*e-mail : mschun@kist.re.kr1598-5032/10/393-05©2003 Polymer Society of Korea

393

Macromolecular Research, Vol. 11, No. 5, pp 393-397 (2003)

A Novel Simulation Architecture of Configurational-Bias Gibbs Ensemble Monte Carlo for the Conformation of Polyelectrolytes

Partitioned in Confined Spaces

Myung-Suk Chun*

Complex Fluids Research Team, Korea Institute of Science and Technology (KIST),PO Box 131, Cheongryang, Seoul 130-650, Korea

Received Mar. 31, 2003; Revised Sept. 18, 2003

Abstract: By applying a configurational-bias Gibbs ensemble Monte Carlo algorithm, priority simulation resultsregarding the conformation of non-dilute polyelectrolytes in solvents are obtained. Solutions of freely-jointed chainsare considered, and a new method termed strandwise configurational-bias sampling is developed so as to effectivelyovercome a difficulty on the transfer of polymer chains. The structure factors of polyelectrolytes in the bulk as wellas in the confined space are estimated with variations of the polymer charge density.

Keywords: polyelectrolyte, configurational-bias, Gibbs ensemble Monte Carlo, chain conformation, structure factor,Debye-Hückel interaction.

Introduction

Polyelectrolytes, one of the typical complex fluids, arepolymers bearing ionizable groups. They dissolve in water dueto the electrostatic repulsion between charged monomers,1

even though water is a poor solvent for most of syntheticpolymers. The conformational properties of polyelectrolytesare strongly affected by geometric confinement.2-4 The prob-lem of confined polyelectrolytes is relevant to numerousapplications such as in size-exclusion chromatography, vas-cularized spaces, and thin-film processing. Research trendhas become increasingly important in the design of eithermicro-biochips or microfluidic devices and the preparationof nanocomposites.

Even if the polyelectrolytes exist in an unbounded freespace, the explicit understanding of conformation is noteasy. The polyelectrolytes in confined spaces result in moredifficult problem. The difficulty of performing experimentsmakes rigorous simulations the more important. From themodeling the concentration partitioning of colloid particlesbetween the bulk and the slit-like pore, Chun and Phillips5

recognized the usefulness of Gibbs ensemble Monte Carlo(GEMC) simulations to study the static properties in con-fined spaces. It is obvious that the canonical ensemble MC

is not available to simulate each of the coexisting sub-systems. Recently, Park et al.3 examined the single chainproblem with the effects of the intramolecular interactionsusing a mean field approximation, where the interchaininteraction was safely neglected.

In contrast to the small molecules, when the GEMC isapplied to the polymer chains, the transfer of large moleculesbetween subsystems is rarely accepted always due to theoverlapping of the molecules. By employing the configura-tional-bias (CB) sampling, however, one can get efficacy insampling the trial conformations of polyelectrolytes. TheCB technique enhances significantly the acceptance ratecompared to the naive method that samples the whole chainconformation at each trial transfer step.4,6

Since the CBGEMC with sampling all monomers is thetime-consuming simulation, therefore, the strandwiseCBGEMC is newly developed. Here, a strand is defined bydividing a polyelectrolyte chain into several domains in itssequence. The ultimate examination of the present simulationaims the estimation of partition coefficient, however, thiscommunication addresses the results devoting the effect ofelectrostatic interactions on the conformation of polyelec-trolytes in both the bulk and the confined space. A chargedcylindrical pore is considered for the confined space, wherethe size of the pore is comparable to or larger than the radiusof gyration of the polymer.

Communications

Myung-Suk Chun

394 Macromol. Res., Vol. 11, No. 5, 2003

Strandwise Configurational-Bias Gibbs EnsembleMonte Carlo Method

GEMC Simulation without Volume Exchanges. TheGEMC method has been successfully applied to the studiesof phase equilibria ranging from Lennard-Jones fluids tomolecular solutions. In the original GEMC scheme, volumesare exchanged between subsystems to find the equilibriumstate, along with the canonical NVT moves of the particleswithin each subsystem. However, the volume exchangeprocess (i.e., NPT) is not necessary if the relevant system isguaranteed to be homogeneous and stable at any concent-ration,5 likewise this study.

For the CBGEMC, each simulation boxes are constructedas shown in Figure 1. The total probability distribution ofsubsystems is

(1)

where N (= Nb + Ncs) is the total number of polyelectrolytechains, and Nb and Ncs are the numbers of chains existing inthe bulk and the confined space, respectively. For Nb chainsin the bulk of volume Vb, the probability distribution isgiven by

(2)

where kBT is the Boltzmann thermal energy, r i the positionvector of i-th monomer bead, and a the radius of bead. Thelong-range energy of Coulomb interaction E(r i,r j) between apair of monomers is given by7

(3)

where Qi corresponds to the charge of i-th monomer. Theproduct of Dirac delta functions in Eq. 2 represents the con-straint of the freely-jointed bead-chain with neighboringsegments separated by twice the bead radius.3 Similarly, theprobability distribution for Ncs chains in the pore of volumeVcs is

(4)

where φ(r) is the electrostatic potential at the position r dueto the charged wall. This study employs the Coulomb inter-actions treated at the Debye-Hückel (or Yukawa) app-roxi-mation level.

Charged beads are located at both ends of the chain andevenly distributed at every fixed number of monomers. Forthe NVT moves, a monomer is chosen with equal probabilityand a random trial is generated under a constraint of thefreely-jointed chain. The acceptance criteria for the move inthe specified subsystem is given by

(5)

where Rand is a uniform random number. The randomexchange of grand canonical µVT ensemble is then per-formed for the condition of chemical potential equilibrium.The transfer is accepted if

(6)

where the subscripts d and r indicate the donating andreceiving subsystems, respectively.

Strandwise Configurational-Bias Sampling. The isolatedpolymer chain can be modeled by a self-avoiding walk of Mmonomers.1,2 The chain is divided into a number of pieceswith a strand as shown in Figure 2. The number of monomersin a strand may be varying depending upon the charge dis-tribution of the chain and the monomer concentration of the

P N( ) Pb Nb( )Pcs Ncs( )=

Pb Nb( )Vb

Nb

Nb!------- 1

kBT--------– E r i r j,( )

i j<∑ δ

i j,⟨ ⟩∏ r i r j– 2a–( )exp=

E r i r j,( )∞, if r i r j– 2a<

QiQj

κ r i r j––( )expr i r j–

--------------------------------- otherwise,

=

Pcs Ncs( )Vcs

Ncs

Ncs!--------- 1

kBT--------– E r i r j,( )

i j<∑exp=

1kBT-------- Qiφ r i( )

i∑ δ

i j,⟨ ⟩∏ r i r j– 2a–( )–

Rand min< 1E∆

kBT--------–

exp,

Rand min< 1NdVr

Nr 1+( )Vd

------------------------E∆ d E∆ r+( )

kBT---------------------------–

exp,

Figure 1. A schematic of polyelectrolytes in the bulk as well asin the confined space of narrow pore, where the periodic boxesare illustrated.

Figure 2. Schematic drawing of the polyelectrolyte chaindivided into strands. Dark beads denote the charged monomers.

Configurational-Bias Gibbs Ensemble Monte Carlo for Polyelectrolytes

Macromol. Res., Vol. 11, No. 5, 2003 395

subsystem. For a neutral polymer that has only the excludedvolume interaction, the optimal strand length may be appro-ximately given as the ν-th power of an average size of thevoids in the solution. For charged polymers, the chargedmonomers have strong preferences in their positions to lowerthe potential energy. An optimal strand size may be deter-mined in such a way that each strand has a charged monomerat the end of each strand.

Let us describe a brief algorithm for the strandwise CBsampling for transfer of a chain from the bulk to the pore asan example. The first segment is assigned as a strand since itis always charged in the present model. We find N1st differentpositions randomly for the insertion of the first monomer intothe pore, and then calculate the correct Boltzmann weightw1(j) = exp[-E(j)/kBT], where E(j) is the interaction energyof the first monomer at r j (j = 1, 2,Î , N1st). According tothe probability distribution Pr(j) = w1(j)/Σjw1(j), we selectone among N1st trial positions, and Nstr strands are generatedwith an appropriate length starting from the end position ofthe previous strand. Both the calculation of w2(j) = exp[-E(j)/kBT] for j-th strand with j = 1, 2,Î , Nstr and selecting oneamong Nstr trial strands based on the probability distributionare repeating until the end of the selected chain. We calculatethe Rosenbluth factor of the new conformation,6,8

. (7)

The next stage calculates the Rosenbluth factor of the oldconformation in the bulk, by generating N1st-1 trial positionsfor the first monomer and Nstr-1 trial strands for each strand,

. (8)

The trial transfer is either accepted or rejected in accordancewith probability criterion,

. (9)

An analogous algorithm is used for the transfer from thepore to the bulk. An appropriate strand length is determineddynamically, depending upon the concentration of the objectsubsystem as well as upon the polyelectrolyte charge density.

Condition of Super Detailed Balance. This conditionrelevant to the chain transfer step can be expressed as

P(o)T(oç n)A(oç n) = P(n)T(nç o)A(nç o). (10)

Here, T(oçn) = exp[-E(n)/kBT]/W(n) and T(nço) = exp[-E(o)/kBT]/W(o) indicate transition probabilities with dummynotations of o and n standing for old and new configurations.From the super detailed balance, one can derive a conditionfor the acceptance probability,

. (11)

Eq. 9 satisfies Eq. 11, owing to the consequence of theacceptance criteria.

Simulation Results

A hypothetical cubic box made with Cartesian coordinatesis applied to the bulk, and the periodic boundary conditionsare imposed for all directions. For the confined space, theperiodic boundary conditions are embedded along the axialdirection of a cylindrical box. The wall of the space has uni-form surface charge density σw, and the freely-jointed chainhas contour length L = l(M - 1), where l = 2a is the segmentallength. All the variables are nondimensionalized by thesegmental length and the characteristic time at constant kBT.Charged beads are placed with equal spacing of inversepolyelectrolyte charge density 1/σp along the chain contour.Simulations are performed with variations of both N and L.About 1Ý 105 updates of the position of each monomer areperformed, and then chain transfer is attempted at 10 to 100updates per monomer. Suitable adjusting this value is neces-sary to guarantee the relaxed subsystem. Equilibrium statescan be determined by monitoring the change of total energyas a simulation proceeds, in which the simulation period isdivided into 50 blocks of configurations. The numerical codewas self-made in C++.

Chain Conformations. Snapshots of each simulationboxes are displayed in Figure 3. The moderate polyelectro-lyte charge density σp of 0.1 is considered, which means thatcharged monomers are located at every 10 beads. As theelementary length units, l is given 1.0. Debye screeninglength κ -1 (i.e., in dimensional, ) is pro-vided as 5.0, and Bjerrum length λB (i.e., in dimensional, e2/4πεkBT) is given 2.0. Here, e represents the elementarycharge, Zi the amount of charge on monomer i, ni the ionicconcentration, and ε the dielectric constant.

Although the other snapshots are not given, results showthat the chains have rod-like conformations in the diluteconcentration regime caused by the repulsion among thecharged monomers. In the moderate concentration regime,the intrachain electrostatic repulsions are screened and thechains are indeed overlapped. The dilute polyelectrolytechains reside near the center of confined space and stretchalong the axial direction once the polymer-wall interaction isrepulsive. When the chain concentration becomes moderate,however, such trends are generally disappeared. The estima-tion of radius of gyration RG provided in Table I supportsthis behavior. In the bulk, one takes ,and is introduced in the pore identi-fying the anisotropy. The radius of gyration is decreasedwith increasing the chain concentration, where its value inthe pore is higher than that in the bulk.

W n( ) wi j( )j

∑i

∏=

W o( ) w̃i j( )j

∑i

∏=

A o n→( ) min 1NbVcsW n( )

Ncs 1+( )VbW o( )--------------------------------------,=

A o n→( )A n o→( )----------------------

NbVcsW n( )Ncs 1+( )VbW o( )

--------------------------------------=

1 e2Σizi2ni εkB⁄ T⁄

RG2⟨ ⟩ r i r j–( )2⟨ ⟩ N⁄=

RG2⟨ ⟩ RG II,

2⟨ ⟩ RG ⊥,2⟨ ⟩+=

Myung-Suk Chun

396 Macromol. Res., Vol. 11, No. 5, 2003

Structure Factor. The structure factor represents themicrostructural information that gives all length scale data.The scattering wave vector q has a magnitude = , where k is the wave vector, λ the wave-length in the dispersion medium, and θ the scattering angle.7

In the bulk, we can define the spherically-averaged staticstructure factor Ss(q) for all monomers of all chains, whichis the Fourier transform of the pair correlation function g(r)= . One obtains

(12)

U

In the pore, the static structure factor parallel to the wall is

evaluated via .As shown in Figures 4 and 5, the structure factor is con-

stant at lower value of q and then provides a decreasingtendency with increasing the q value. The Ss(q) is an oscillatoryfunction at intermediate values of q, but its σp dependency isrelatively decreased for larger value of q. The slope betweenq and Ss(q) is higher in the bulk compared to that in the

q 2 k θ 2⁄( )sin=4π λ⁄( ) θ 2⁄( )sin

Σm n, δ r r m–( )δ r r n–( )⟨ ⟩

Ss q( ) 1NM--------- iq– r i r j–( )⋅( )exp

i j<

N M,

∑2

⟨ ⟩≡

1NM--------- r iq r⋅( )expd∫ g r( )=

Ss ,|| q( ) Ss z, qz( )=

Figure 4. Static structure factor, Ss(q), in the bulk for variouspolyelectrolyte charge density, σp, at N = 20, L = 160, κ = 0.2,λB = 2.0, and σw = 0.1.

Table I. Estimations of Radius of Gyration in Each Sub-system for Different Polymer Charge Density, σp, with M = 100and σw = 0.1

σp

N = 20 N = 60 N = 20 N = 60

0.0 21.1 6.7 30.6 8.1

0.05 22.6 6.3 32.9 9.6

0.10 36.6 10.1 41.4 12.5

0.20 61.0 21.8 64.4 24.0

RG2⟨ ⟩bulk RG

2⟨ ⟩pore

Figure 3. Snapshots of polyelectrolytes (a) in the bulk and (b) in the confined space of charged pore by performing the CBGEMC sim-ulation with total configurations of 3Ý106. Total 60 chains are introduced, and each chain consists of 100 monomers.

Configurational-Bias Gibbs Ensemble Monte Carlo for Polyelectrolytes

Macromol. Res., Vol. 11, No. 5, 2003 397

pore. This means polyelectrolyte chains in the bulk experi-ence the collapsed globular conformation having strongcorrelations, whereas they become elongated with the self-avoiding walks conformation in the pore. As the σp

increases, the structure factor in the bulk is decreased fora given value of q. In the confined space, however, theopposite behavior is observed due to the confining walleffect.

Conclusions

Conventional MC simulations are restricted to examinethe conformation of non-dilute polyelectrolytes partitionedin confined spaces, for which a novel MC architecture withstrandwise CB Gibbs ensemble has successfully been devel-oped. This communication presented the distinct results ofstructure factor with variations of the polyelectrolyte chargedensity. In the future study, extensive results will be addressedby encouraging the present codes.

Acknowledgements. This work was supported by theBasic Research Fund (KOSEF: R01-2001-000-00411-0).The author also acknowledges helpful comments on the MCstudy of non-dilute polyelectrolytes provided by the TheoryGroup of Max-Planck Institut für Polymerforschung atMainz during the visiting research funded by the DFG.

References

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8850 (2000). (4) G. F. Hermsen, B. A. de Geeter, N. F. A. van der Vegt, and M.

Wessling, Macromolecules, 35, 5267 (2002).(5) M. -S. Chun and R. J. Phillips, AIChE J., 43, 1194 (1997).(6) D. Frenkel and B. Smit, Understanding Molecular Simula-

tion: From Algorithm to Applications, 2nd Ed., AcademicPress, New York, 2002.

(7) M. -S. Chun and H. S. Lee, Macromole. Res., 10, 297 (2002).(8) M. N. Rosenbluth and A. W. Rosenbluth, J. Chem. Phys., 23,

356 (1955).

Figure 5. Static structure factor, Ss,z(qz), in the confined space forvarious polyelectrolyte charge density, σp, at N = 20, L = 160,κ = 0.2, λB = 2.0, and σw = 0.1.