18374 DOI: 10.1021/la103229u Langmuir 2010, 26(23), 18374–18381Published on Web 11/01/2010

pubs.acs.org/Langmuir

© 2010 American Chemical Society

Molecular Dynamics Simulations of Grafted Layers of Bottle-Brush

Polyelectrolytes

Jan-Michael Y. Carrillo and Andrey V. Dobrynin*

Polymer Program, Institute of Materials Science and Department of Physics, University of Connecticut,Storrs, Connecticut 06269, United States

Received August 13, 2010. Revised Manuscript Received October 8, 2010

Usingmolecular dynamics simulations, we study the effect of the brush grafting density and degree of polymerizationof the side chains on conformations of brush layers made of charged bottle-brush macromolecules. The thickness of thebrush layer first decreases with increasing brush grafting density; then, it saturates and remains constant in the wideinterval of the brush grafting densities. The brush layers consisting of the bottle-brush macromolecules with longer sidechains have a larger layer thickness. The elongation of the side chains of the bottle-brushmacromolecules decreases withincreasing brush grafting density. This contraction of the side chains is due to counterion condensation inside the volumeoccupied by bottle-brushes. Our simulations showed that counterion condensation is a multiscale process reflectingdifferent symmetries of the bottle-brush layer.

1. Introduction

Bottle-brushes are macromolecules consisting of a flexiblebackbone with densely grafted side chains.1-4 In the case ofpolymer backbone being longer than the length of the side chains,

a bottle-brush molecule has a cylindrical shape with side chainspointing in the radial direction outward from the chain backbone.(See Figure 1.) The experimental,1-10 theoretical,11-13 andcomputational14-23 studies have shown that the structure of thesemacromolecules is the main reason for their unique response tothe external stimuli (temperature, pH, light, and solvent). This haspotential for applications of bottle-brush macromolecules formolecular engineering, for design of novel materials with pro-grammable properties, and for utilization as model systems inbiomedical research.1,3,4 In particular, the cylindrical bottle-brushmacromolecules have structure similar to aggrecanmolecules24,25

that are composed of polysaccharide chains tethered to a proteinbackbone. These aggrecan molecules play an important role incontrolling the lubricating properties of cartilage.26,27 The unique

*To whom correspondence should be addressed.(1) Ishizu, K.; Tsubaki, K.; Mori, A.; Uchida, S. Architecture of nanostructured

polymers. Prog. Polym. Sci. 2003, 28, 27–54.(2) Lee, H. I.; Pietrasik, J.; Sheiko, S. S.; Matyjaszewski, K. Stimuli-responsive

molecular brushes. Prog. Polym. Sci. 2010, 35, 24–44.(3) Matyjaszewski, K.; Tsarevsky, N. V. Nanostructured functional materials

prepared by atom transfer radical polymerization. Nature Chem. 2009, 1, 276–288.(4) Sheiko, S. S.; Sumerlin, B. S.; Matyjaszewski, K. Cylindrical molecular

brushes: synthesis, characterization, and properties. Prog. Polym. Sci. 2008, 33,759–785.(5) Hsu, H. P.; Paul, W.; Rathgeber, S.; Binder, K. Characteristic length scales

and radial monomer density profiles of molecular bottle-brushes: simulation andexperiment. Macromolecules 2010, 43, 1592–1601.(6) Polotsky, A.; Charlaganov, M.; Xu, Y. Y.; Leermakers, F. A. M.; Daoud,

M.;Muller, A.H. E.; Dotera, T.; Borisov, O. Pearl-necklace structures in core-shellmolecular brushes: experiments,Monte Carlo simulations, and self-consistent fieldmodeling. Macromolecules 2008, 41, 4020–4028.(7) Ruthard, C.; Maskos, M.; Kolb, U.; Grohn, F. Finite-size networks from

cylindrical polyelectrolyte brushes and porphyrins. Macromolecules 2009, 42,830–840.(8) Xu, Y. Y.; Bolisetty, S.; Drechsler, M.; Fang, B.; Yuan, J. Y.; Ballauff,

M.; Muller, A. H. E. pH and salt responsive poly(N,N-dimethylaminoethylmethacrylate) cylindrical brushes and their quaternized derivatives. Polymer 2008,49, 3957–3964.(9) Xu, Y. Y.; Bolisetty, S.; Drechsler, M.; Fang, B.; Yuan, J. Y.; Harnau, L.;

Ballauff, M.; Muller, A. H. E. Manipulating cylindrical polyelectrolyte brushes onthe nanoscale by counterions: collapse transition to helical structures. SoftMatter2009, 5, 379–384.(10) Zhang, B.; Grohn, F.; Pedersen, J. S.; Fischer, K.; Schmidt, M. Conforma-

tion of cylindrical brushes in solution: effect of side chain length. Macromolecules2006, 39, 8440–8450.(11) Connolly, R.; Bellesia, G.; Timoshenko, E. G.; Kuznetsov, Y. A.; Elli, S.;

Ganazzoli, F. “Intrinsic” and “topological” stiffness in branched polymers.Macromolecules 2005, 38, 5288–5299.(12) Denesyuk, N. A. Conformational properties of bottle-brush polymers.

Phys. Rev. E 2003, 67, 051803.(13) Subbotin, A.; Saariaho, M.; Ikkala, O.; ten Brinke, G. Elasticity of comb

copolymer cylindrical brushes. Macromolecules 2000, 33, 3447–3452.(14) Elli, S.; Ganazzoli, F.; Timoshenko, E. G.; Kuznetsov, Y. A.; Connolly, R.

Size and persistence length of molecular bottle-brushes by Monte Carlo simula-tions. J. Chem. Phys. 2004, 120, 6257–6267.(15) Feuz, L.; Leermakers, F. A. M.; Textor, M.; Borisov, O. Bending rigidity

and induced persistence length of molecular bottle brushes: a self-consistent-fieldtheory. Macromolecules 2005, 38, 8891–8901.

(16) Kosovan, P.; Kuldova, J.; Limpouchova, Z.; Prochazka, K.; Zhulina, E. B.;Borisov, O. V. Amphiphilic graft copolymers in selective solvents: moleculardynamics simulations and scaling theory. Macromolecules 2009, 42, 6748–6760.

(17) Nap, R. J.; Szleifer, I. Structure and interactions of aggrecans: statisticalthermodynamic approach. Biophys. J. 2008, 95, 4570–4583.

(18) Saariaho, M.; Ikkala, O.; Szleifer, I.; Erukhimovich, I.; ten Brinke, G. Onlyotropic behavior of molecular bottle-brushes: A Monte Carlo computer simula-tion study. J. Chem. Phys. 1997, 107, 3267–3276.

(19) Saariaho, M.; Subbotin, A.; Szleifer, I.; Ikkala, O.; ten Brinke, G. Effect ofside chain rigidity on the elasticity of comb copolymer cylindrical brushes: aMonteCarlo simulation study. Macromolecules 1999, 32, 4439–4443.

(20) Saariaho, M.; Szleifer, I.; Ikkala, O.; ten Brinke, G. Extended conforma-tions of isolated molecular bottle-brushes: influence of side-chain topology.Macromol. Theory Simul. 1998, 7, 211–216.

(21) Yethiraj, A. A Monte Carlo simulation study of branched polymers.J. Chem. Phys. 2006, 125, 204901.

(22) Zhulina, E. B.; Leermakers, F. A.M. On the polyelectrolyte brushmodel ofneurofilaments. Soft Matter 2009, 5, 2836–2840.

(23) Zhulina, E. B.; Leermakers, F. A. M. The polymer brush model ofneurofilament projections: effect of protein composition. Biophys. J. 2010, 98,462–469.

(24) Ng, L.; Grodzinsky, A. J.; Patwari, P.; Sandy, J.; Plaas, A.; Ortiz, C.Individual cartilage aggrecan macromolecules and their constituent glycosamino-glycans visualized via atomic force microscopy. J. Struct. Biol. 2003, 143, 242–257.

(25) Ng, L.; Grodzinsky, L. J.; Sandy, J.; Plaas, A.; Ortiz, C. Persistence lengthof cartilage aggrecan macromolecules measured via atomic force microscopy.Macromol. Symp. 2004, 214, 1–4.

(26) Klein, J. Molecular mechanisms of synovial joint lubrication. Proc. Inst.Mech. Eng., Part J 2006, 220, 691–710.

(27) Klein, J. Chemistry repair or replacement-a joint perspective. Science 2009,323, 47–48.

DOI: 10.1021/la103229u 18375Langmuir 2010, 26(23), 18374–18381

Carrillo and Dobrynin Article

lubricating properties of a brush layermadeof cartilage aggrecansare supported by nanomechanical studies.28-33

In this Article, we use molecular dynamics simulations to studythe effect of the electrostatic interactions and structure of bottle-brush macromolecules on the properties of brush layer. Bychanging the brush grafting density and the side chain degree ofpolymerization of the bottle-brush macromolecules, we establishwhat effects these parameters have on counterion distributioninside and outside the brush layer, conformations of individualbottle-brush molecules, and layer thickness.

2. Simulation Details

We performed molecular dynamics simulations34,35 of graftedlayers of the charged bottle-brushes with explicit counterions.Polyelectrolyte bottle-brush macromolecules were modeled bychains of charged Lennard-Jones (LJ) particles (beads) withdiameter σ. The bottle-brushmacromolecules consisted of a mainchain with the degree of polymerization N = 97 and sidechains with the degree of polymerization m. (See Figure 1.) We

performed simulations of the bottle-brush macromolecules withthe side chain degree of polymerizationsm=9, 15, and 21. Therewas a total 31 side chains per each bottle-brush macromoleculethat corresponds to every third monomer of the main chain tohave a side chain attached to it. Only side chains were chargedwith the fraction of the chargedmonomers f=1/3 correspondingto every third monomer carrying a negative electrical charge,-e.The number NB of the bottle-brush macromolecules was graftedto a substrate. The substratewasmodeled by a periodic hexagonalpacked lattice of beads composed of 80� 70 beads with diameterσ located at z = 0. The grafting density of the substrate, Fg wasvaried between 2.06� 10-4 σ-2 and 8.04� 10-3 σ-2 by changingthe number of grafted chainsNB per simulation box from 1 to 39.The grafting points were randomly distributed over the substratesurface. A similar nonselective surface was located at the oppositeside of the simulation box to prevent counterions from escapingand, hence, maintaining 2-D periodicity in the lateral (x and y)directions. The system dimensions in the xy plane were 70.0σ �69.28σ. The height of the simulation box Lz was equal to 150.0σfor all simulations. The system configuration is shown inFigure 1,and the system parameters are listed in Table 1.

All particles in the system interacted through truncated-shiftedLennard-Jones (LJ) potential

ULJðrijÞ ¼

4εLJσ

rij

!12

-σ

rij

!6

-σ

rcut

� �12

þ σ

rcut

� �624

35 re rcut

0 r > rcut

8><>:

ð1Þ

Figure 1. Snapshot of the simulation box with grafted polyelec-trolyte bottle-brushes and counterions. The red, blue, and blackbeads correspond to positively charged counterions, negativelycharged, and neutral monomers belonging to bottle-brush macro-molecules, respectively. The surface beads are colored in green.

Table 1. System Parametersa

N m NB Fg [σ -2] Npart Nc

1 97 9 1 2.06� 10-4 11669 932 97 9 3 6.19� 10-4 12607 2793 97 9 6 1.24� 10-3 14014 5584 97 9 9 1.86� 10-3 15421 8375 97 9 12 2.47� 10-3 16828 11166 97 9 15 3.09� 10-3 18235 13957 97 9 18 3.71� 10-3 19642 16748 97 9 21 4.33� 10-3 21049 19539 97 9 27 5.57� 10-3 23863 2511

10 97 9 33 6.80� 10-3 26677 306911 97 9 39 8.04� 10-3 29491 362712 97 15 1 2.06� 10-4 11917 15513 97 15 3 6.19� 10-4 13351 46514 97 15 6 1.24� 10-3 15502 93015 97 15 9 1.86� 10-3 17653 139516 97 15 12 2.47� 10-3 19804 186017 97 15 15 3.09� 10-3 21955 232518 97 15 18 3.71� 10-3 24106 279019 97 15 21 4.33� 10-3 26257 325520 97 15 27 5.57� 10-3 30559 418521 97 15 33 6.80� 10-3 34861 511522 97 15 39 8.04� 10-3 39163 604523 97 21 1 2.06� 10-4 12165 21724 97 21 3 6.19� 10-4 14095 65125 97 21 6 1.24� 10-3 16990 130226 97 21 9 1.86� 10-3 19885 195327 97 21 12 2.47� 10-3 22780 260428 97 21 15 3.09� 10-3 25675 325529 97 21 18 3.71� 10-3 28570 390630 97 21 21 4.33� 10-3 31465 455731 97 21 27 5.57� 10-3 37255 585932 97 21 33 6.80� 10-3 43045 716133 97 21 39 8.04� 10-3 48835 8463

aNB, number of brushmolecules per simulation box;Npart, number ofparticles in a system including substrates; and Nc, number of counter-ions.

(28) Dean, D.; Han, L.; Grodzinsky, A. J.; Ortiz, C. Compressive nanomecha-nics of opposing aggrecan macromolecules. J. Biomech. 2006, 39, 2555–2565.(29) Dean, D.; Han, L.; Ortiz, C.; Grodzinsky, A. J. Nanoscale conformation

and compressibility of cartilage aggrecan using microcontact printing and atomicforce microscopy. Macromolecules 2005, 38, 4047–4049.(30) Dean, D.; Seog, J.; Ortiz, C.; Grodzinsky, A. J. Molecular-level theoretical

model for electrostatic interactions within polyelectrolyte brushes: Applications tocharged glycosaminoglycans. Langmuir 2003, 19, 5526–5539.(31) Han, L.; Dean, D.; Mao, P.; Ortiz, C.; Grodzinsky, A. J. Nanoscale shear

deformation mechanisms of opposing cartilage aggrecan macromolecules. Bio-phys. J. 2007, 93, L23–L25.(32) Han, L.; Dean, D.; Ortiz, C.; Grodzinsky, A. J. Lateral nanomechanics of

cartilage aggrecan macromolecules. Biophys. J. 2007, 92, 1384–1398.(33) Seog, J.; Dean, D.; Rolauffs, B.; Wu, T.; Genzer, J.; Plaas, A. H. K.;

Grodzinsky, A. J.; Ortiz, C. Nanomechanics of opposing glycosaminoglycanmacromolecules. J. Biomech. 2005, 38, 1789–1797.(34) Plimpton, S. Fast parallel algorithms for short-range molecular dynamics.

J. Comput. Phys. 1995, 117, 1–19.(35) Frenkel, D.; Smit, B.UnderstandingMolecular Simulations; Academic Press:

New York, 2002.

18376 DOI: 10.1021/la103229u Langmuir 2010, 26(23), 18374–18381

Article Carrillo and Dobrynin

where rij is the distance between ith and jth beads and σ is the beaddiameter chosen to be the same regardless of the bead type. Thecutoff distance, rcut = 2.5σ, was chosen for polymer-polymerinteractions, and rcut = 21/6σ was chosen for all other pairwiseinteractions. The interaction parameter εLJ was equal to kBTfor polymer-counterion, counterion-counterion, polymer-substrate, and counterion-substrate interactions, where kB is theBoltzmann constant andT is the absolute temperature. The valueof the Lennard-Jones interaction parameter for the polymer-polymer pair was set to 0.3 kBT, which is close to a θ solventcondition for the polymer. By selecting the strength of thepolymer-polymer interactions close to the θ point, weminimizedthe effect of the short-range interactions on the bottle-brushproperties.

The connectivity ofmonomers to bottle-brushmacromoleculeswas maintained by the finite extension nonlinear elastic (FENE)potential

UFENEðrÞ ¼ -1

2kspringR

2max ln 1-

r2

R2max

!ð2Þ

with the spring constant kspring=30kBT/σ2, whereRmax=1.5σ is

the maximum bond length. The repulsive part of the bondpotential was represented by the truncated-shifted LJ potentialswith εLJ = kBT and rcut = 21/6σ.

Interaction between any two charged particles with chargevalences qi and qj and separated by a distance rij, was given by theCoulomb potential

UCoulðrijÞ ¼ kBTlBqiqj

rijð3Þ

where lB = e2/εkBT is the Bjerrum length, defined as the lengthscale on which the Coulomb interaction between two elementarycharges, e, in a dielectric mediumwith the dielectric constant, ε, isequal to the thermal energy, kBT. In our simulations, the value ofthe Bjerrum length, lB, was equal to 1σ.

The particle-particle particle-mesh (PPPM) method imple-mented in LAMMPS34 with the sixth-order charge interpolationscheme and estimated accuracy of 10-5 was used for calculationsof the electrostatic interactions between all charges in the system.The 2D periodic images of the systemwere periodically replicatedalong the z direction with distance L = 3Lz between theirboundaries. This reduced the problem of calculation of theelectrostatic interactions in a 2D periodic system to those in a3D system.

Simulations were carried out in a constant number of particles,volume, and temperature ensemble (NVT) with periodic bound-ary conditions. The constant temperature was maintained bycoupling the system to a Langevin thermostat. In this case, theequation of motion of the ith particle is

mdvBiðtÞdt

¼ FBiðtÞ- ξvBiðtÞþFBR

i ðtÞ ð4Þ

where vBi(t) is the bead velocity and FBi(t) is the net deterministicforce acting on the ith bead ofmassm.FBi

R(t) is the stochastic forcewith zero average value ÆFBi

R(t)æ=0and δ-functional correlationsÆFBi

R(t)FBiR(t0)æ= 6ξkBTδ(t- t0). The friction coefficient ξ was set

to ξ=m/τLJ, where τLJ is the standard LJ time τLJ= σ(m/εLJ)1/2.

The velocity-Verlet algorithm with a time step of Δt = 0.01τLJwas used for integration of the equations of motion (eq 4).

Simulations were performed using the following procedure.At the beginning of each simulation run: the main and side chainsof the bottle-brush macromolecules were in fully extended

conformation with the main chains pointing along the z directionand side chains were randomly oriented in the x-y plane.Neutralizing monovalent counterions were uniformly distributedover the volume of the simulation box. The system was pre-equilibrated for 2 � 106 MD steps. This was followed by theproduction run lasting 106MD steps. During the production run,we averaged the brush thickness, bottle-brush gyration tensor,and counterion and monomer distribution functions.

We have also performed simulations of neutral bottle-brushlayers consisting of the bottle-brush macromolecules with thedegree of polymerization of the side chains m = 21 in the samerange as the brush grafting densities. The parameters for bondpotential and Lennard-Jones interaction potentials were the sameas those in the case of the charged bottle-brush macromolecules.The total duration of the simulation runs for these systems was4 � 106 MD steps, and the last 106 MD steps were used for thedata analysis.

3. Simulation Results

3.1. Layer Structure.We begin discussion of our simulationresults by describing the dependence of the bottle-brush layerstructure on the brush grafting density. In Figure 2a, we plot thelayer thickness as a function of the bottle-brush grafting density.The average layer thickness ÆHæ was obtained from the brushdensity profile F(z) along the normal to the surface direction z

ÆHæ ¼ 2

Z Lz

0

zFðzÞ dz=Z Lz

0

FðzÞ dz ð5Þ

The average layer thickness first decreases with increasing thebrush grafting density, Fg; then, it saturates. For bottle-brusheswith longer side chains, this decease in the layer thickness is lesspronounced. This behavior is qualitatively different from the oneobserved for the brush layer made of linear polyelectrolyte chainsfor which the thickness of the brush layer usually increases withthe brush grafting density.36-38 Note that increase in the brushlayer thickness with increasing the brush grafting density was alsoobserved for charged bottle-brushmacromolecules with rigid sidechains.39 The decrease and saturation of the layer thickness is dueto high linear charge density that is achieved in the bottle-brush bygrafting charged side chains to the polymer backbone. Therefore,the bottle-brush macromolecules are strongly stretched becauseof electrostatic repulsion between charged side chains. It isimportant to point out that the decrease in the brush layerthickness can be explained by counterion condensation withinthe brush layer with increasing the brush grafting density. (See thediscussion below.) The layer thickness increases with increasinglength of the side chains. This increase in the brush layer thicknessis due to the orientation of the side chains located close to the topof the grafted bottle-brush macromolecules along the brushbackbone. (See the discussion below.)

To illustrate the difference in behavior of charged and neutralbottle-brush layers, in Figure 2b, we show dependence of theaverage layer thickness of the charged and neutral bottle-brushlayers consisting of the bottle-brush macromolecules with the

(36) Netz, R. R.; Andelman, D. Neutral and charged polymers at interfaces.Phys. Rep. 2003, 380, 1–95.

(37) Ballauff, M.; Borisov, O. Polyelectrolyte brushes. Curr. Opin. ColloidInterface Sci. 2006, 11, 316–323.

(38) Carrillo, J.-M. Y.; Dobrynin, A. V.Morphologies of planar polyelectrolytebrushes in a poor solvent: molecular dynamics simulations and scaling analysis.Langmuir 2009, 25, 13158–13168.

(39) Cao, Q. Q.; Zhu, C. C.; Li, L. J. Molecular dynamics simulations of end-grafted centipede-like polymers with stiff charged side chains. Eur. Phys. J. E 2010,32, 1–12.

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Carrillo and Dobrynin Article

degree of polymerization of the side chainsm=21.As one can see,the electrostatic interactions between charges lead to much stron-ger elongation of the charged bottle-brush macromolecules incomparison with elongation of the neutral bottle-brushes inducedby the short-range (Lennard-Jones) interactions. The backbonesof the polyelectrolyte brushes are strongly stretched and have rod-like conformations, whereas the backbones of the neutral bottle-brushes have coil-like conformations. (See the insets in Figure 2b.)The end-to-end vector of the polyelectrolyte brush backboneshows much stronger alignment with the z-axis in comparisonwith that for the end-to-end vector of the neutral bottle-brushes.The strong orientation of the backbone of the charged bottle-brushes is due to long-range interbrush electrostatic repulsion. Thethickness of the neutral brush layer starts to increase with increas-ing brush grafting density when the bottle-brush macromoleculesbegin to overlap. The increase in the layer thickness is due tointerbrush short-range repulsion. The thickness of the chargedbottle-brush macromolecules stays almost constant throughoutthe entire interval of the studied brush grafting densities.

The side chains of the charged bottle-brush macromoleculesshow larger separation than those of the neutral bottle-brush.

(See the insets in Figure 2b.) This is due to strong electrostaticrepulsion between the side chains. The electrostatic repulsionbetween side chains of the charged bottle-brush also leads to theirorientation with respect to the brush backbone. In Figure 3, weshow variation along the z direction of the average value of thecosine of the angle between the end-to-end vector of the brushbackbone and that of the side chains. It follows from this plot thatin the middle of the bottle-brush the side chains are orientedperpendicular to the bottle-brush backbone. This chain orienta-tion minimizes the electrostatic repulsion between chains. How-ever, for the side chains located close to the top and bottom of thebottle-brush macromolecules, the side chains are aligned alongthe bottle-brush backbone. At the top of the bottle-brush, the sidechains show a stronger orientational correlation with the brushbackbone than chains at the bottom of the bottle-brush. This isdue to the different origin of the side-chain orientation. At the topof the bottle-brush, the side-chain orientation is due to electro-static repulsion of the side chains located on the top portion of thebottle-brush from the chains located underneath of them. How-ever, at the bottom of the bottle-brush, the side-chain orientationis caused by the short-range repulsion between side chains andsubstrate.

The size of the side chains decreases with increasing brushgrafting density. (See Figure 4.) For this plot, we used only theside chains located in the middle of the bottle-brush covering 1/3of the bottle-brush layer thickness. This allowed us to eliminatethe end effects in obtaining the size of the side chains. The solidline in this Figure describes the average distance between graftedbottle-brush backbones

RB � ðπFgÞ- 1=2 ð6Þ

The data points on this line represent the average value of thedistance between grafting points obtained by averaging the actualdistances between neighboring grafting points of the bottle-brush

Figure 2. (a) Dependence of the brush layer thickness on thebrush grafting density for charged bottle-brush macromoleculeswith different degrees of polymerization of the side chains: m= 9(red squares), m = 15 (blue rhombs), and m = 21 (black circles).(b) Dependence of the brush layer thickness on the brush graftingdensity for charged (filled circles) and neutral (open circles) bottle-brushmacromolecules with the degree of polymerization of the sidechains m= 21. Insets show typical bottle-brush configurations.

Figure 3. Orientation of side chains of the charged bottle-brushalong the backbone of the bottle-brush macromolecules withm= 21 at different brush grafting densities: Fg = 6.19 � 10-4 σ-2

(gray hexagons), Fg = 1.86� 10-3 σ-2 (blue rhombs), Fg = 3.09�10-3 σ-2 (red squares), and Fg = 4.33 � 10-3 σ-2 (black circles).Inset shows a typical configuration of the charged bottle-brushmacromolecule.

18378 DOI: 10.1021/la103229u Langmuir 2010, 26(23), 18374–18381

Article Carrillo and Dobrynin

backbones. Therefore, for most of the brush grafting densitiesused in our simulations, the bottle-brushmolecules donot overlapbecause the distancebetween bottle-brush chains is larger than thesize of the side chains. This corresponds to the so-called “mush-room regime”.37 It is interesting to point out the similaritybetween contraction of the side chains with increasing bottle-brush grafting density and contraction of the polyelectrolytechains in dilute polyelectrolyte solutions.40-42 In both cases, thecontraction is caused by accumulation of the counterions withinthe volume occupied by macromolecules, which reduces theintrachain electrostatic repulsion resulting in chain contraction.A more detailed discussion of the effect of the counterions onthe structure of the bottle-brush is given below in Section 3.2.Note that only bottle-brush macromolecules with the longestside chains crossover to the overlap bottle-brush regime. It isinteresting to point out that neutral bottle-brushes crossover tothe overlap brush regime at lower brush grafting densitiesthan polyelectrolyte bottle-brushes. This is due to the larger sizeof the neutral bottle-brushes in the xy plane. (See insets inFigure 2b.)

Further analysis of data shown in Figure 4 indicates that thedata sets obtained for bottle-brushes with different degrees ofpolymerization of the side chains are shifted with respect to eachother. To illustrate universality in the deformation of the sidechains of the bottle-brushmacromolecules in Figure 5, we plottedthe dependence of the reduced side chain size ÆRm

2 æ1/2/mb (whereb = 0.971σ is the average bond length of the side chains) on thereduced value of the brush grafting density. The data pointscorresponding to bottle-brushmacromolecules with the degree ofpolymerization of the side chains m= 15 and 21 show a reason-ably good overlap. The data obtained for the bottle-brush with

the shortest side chainsm=9demonstrateweaker dependenceonthe reduced brush grafting density. The difference between twodata sets can be explained by a finite size effect, which is strongerfor shorter side chains.3.2. Counterion Distribution. A counterion distribution

plays an important role in determining a structure of a brushlayer. There are different length scales that are involved in acounterion distribution process. First, the counterion localizationoccurs within a brush layer. With increasing the brush graftingdensity, the Gouy-Chapman length, λGC=(2πlBFg fmNm)

-1,determining a length scale within which half of the brushcounterions is localized, becomes on the order of the brushthickness. This corresponds to a crossover to the so-called“osmotic brush” regime when the stretching of the bottle-brushbackbone is caused by a translational entropy of localized withinbrush counterions.37 For bottle-brush systems, this occurs at verylow brush grafting densities because of the large value of the netcharge efmNm on each bottle-brush macromolecule in compar-ison with that for the brush made of the linear chains. In addition

Figure 4. Dependence of the square root of the mean squareaverage value of the end-to-end distance of the charged bottle-brush side chains on the brush grafting density for different degreesof polymerizations of the side chains:m=9 (red squares),m=15(blue rhombs), and m = 21 (black circles). The solid line corre-sponds to eq 6, and open symbols represent the average distancebetween grafting points of bottle-brushmacromolecules withm=9 (squares), m = 15 (rhombs), and m = 21 (circles). Insets showsnapshots of the top view of the charged bottle-brush layer.

Figure 5. Dependence of the normalized square root of the meansquare average value of the end-to-end distance of the bottle-brushside chains on the normalized brush grafting density for differentdegrees of polymerizations of the side chains:m=9 (red squares),m= 15 (blue rhombs), and m= 21 (black circles).

Figure 6. Density distribution of the charged monomers (O) andcounterions (b) along the z direction for brush of bottle-brushmacromoleculeswithm=21andbrush grafting density Fg=4.33�10-3 σ-2. The solid line corresponds to a counterion distributiongiven by eq 7.

(40) Liao, Q.; Dobrynin, A. V.; Rubinstein,M.Molecular dynamics simulationsof polyelectrolyte solutions: nonuniform stretching of chains and scaling behavior.Macromolecules 2003, 36, 3386–3398.(41) Dobrynin, A. V.; Rubinstein, M. Theory of polyelectrolytes in solutions

and at surfaces. Prog. Polym. Sci. 2005, 30, 1049–1118.(42) Dobrynin, A. V. Theory and simulations of charged polymers: from

solution properties to polymeric nanomaterials. Curr. Opin. Colloid InterfaceSci. 2008, 13, 376–388.

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Carrillo and Dobrynin Article

to counterion localization within a brush layer, we have counter-ion localization within the volume occupied by the bottle-brushmacromolecule. Localization of the counterions within bottle-brush volume reduces the effective linear charge density alongbottle-brushes to the critical Manning-Oosawa value.41,43 Be-cause of the internal structure of the bottle-brush macromole-cules, the counterions can be localized in the vicinity of the sidechains and prevent an increase in the linear charge density of theside chains above the critical value.Note that this process can alsobe described in the framework of theManning-Oosawa counter-ion condensation (localization) model. Below we will illustratethis multiscale picture of counterion distribution within a brush.

We will first consider distribution of counterions outside thebrush layer. Counterions localized within the bottle-brush layerreduce an effective surface charge density of the brush layer toΣ = (1-xH)Fg fmNm, where xH is the fraction of counterionslocalized within the brush layer. For the remaining counterionsoutside the brush, their density distribution is given by44

FcðzÞ ¼ 1

2πlBh2s2

cos2ðsð1- ðz-HÞ=hÞÞ ð7Þ

where we defined h=Lz-H, and the parameter s is a solution ofthe equation s tan(s) = 2πlBΣh. Equation 7 is a solution of thenonlinear Poisson-Boltzmann equation, which couples the dis-tribution of counterions to the electrostatic potential. In Figure 6,we plotted the distribution of the charged monomer and counter-ion densities as a function of the distance from the substratesurface. The solid line corresponds to eq 7, with the effectivesurface charge density of the brush layer evaluated at the averageheight of the brush layer, H = 79σ (shown by an arrow on the

plot). The selection of the average brush thicknessH as a dividingplane for the diffusive counterion layer was based on the follow-ing arguments. At the distances larger than the average brushthickness z>79σ, the density of the charged monomers becomessmaller than the counterion density (Figure 6), and counterionsbegin to provide a dominant contribution to the local chargedistribution. This choice of the dividing plane results in a verygoodagreement between theoretical expression for the counteriondensity distribution and simulation results. It is interesting topoint out that inside the brush layer counterions almost comple-tely neutralize chargeon thebottle-brush.The excess chargeof thebottle-brush layer is located close to the edge of the brush layerwhere the side chains demonstrate alignment with the brushbackbone. This excess charge is compensated by the diffusivelayer of counterions outside the brush.Note that neutralization ofthe bottle-brush bulk charge minimizes the electrostatic energystored within brush layer.

A qualitatively similar counterion density profile outside thebrush layerwas also observed in simulations of the polyelectrolytebrushes under good45 and poor38 solvent conditions for thepolymer backbone. This supports the idea that the distributionof counterions outside the brush does not depend on the brushstructure and is only controlled by the effective surface chargedensity of the brush layer.

A strong electrostatic attraction between bottle-brush andcounterions localized with the brush layer leads to localizationof the counterions within the bottle-brush volume. The localiza-tion of counterions within the bottle-brush volume is a result ofoptimization of the counterion translation entropy, electrostaticrepulsion between uncompensated charges of the bottle-brush,and elastic free energy of the side chains. Note that the elastic freeenergy of the bottle-brush backbone does not play a significant

Figure 7. (a) Snapshot of a charged bottle-brushmoleculewith counterions. The bottle-brush backbone is covered by spheres with size beingequal to that of side chains. The volume covered by these spheres was considered to be the bottle-brush volume used for calculations offraction xB of localized within bottle-brush counterions. (b) Dependence of the reduced linear number charge density of the bottle-brush onthebrushgraftingdensity forbottle-brushmoleculeswithdifferent degreesofpolymerizationof side chains:m=9(red squares),m=15(bluerhombs), andm= 21 (black circles). Logarithmic scales. Inset shows dependence of the fraction of condensed counterions xB on the brushgrafting density.

(43) Manning, G. S. The critical onset of counterion condensation: a survey ofits experimental and theoretical basis. Ber. Bunsen-Ges. 1996, 100, 909–922.(44) Evans, D. F.; Wennerstrom, H. The Colloidal Domain; Wiley-VCH:

New York, 1999.

(45) Hehmeyer, O. J.; Arya,G.; Panagiotopoulos, A. Z.Monte Carlo simulationand molecular theory of tethered polyelectrolytes. J. Chem. Phys. 2007, 126,244902/1–244902/11.

18380 DOI: 10.1021/la103229u Langmuir 2010, 26(23), 18374–18381

Article Carrillo and Dobrynin

role in this process because the brush layer thickness remainsconstant in the wide interval of the brush grafting density. Tocalculate the fraction of counterions localized within the volumeoccupied by the bottle-brush macromolecules, we have sur-rounded each monomer of the brush backbone by a sphere withradius equal to that of the square root of themean-square averagevalueof the end-to-enddistance of the side chains ÆRm

2 æ1/2 (Figure 7a)and counted the number of counterions belonging to this volume.It is important to point out that for overlapping bottle-brushes weassigned the counterion to the nearest bottle-brush macromole-cule. This prevented double counting of counterions.

Figure 7b shows the dependence of the reduced number chargedensity of the bottle-brush on the reduced brush grafting density.The fraction of the condensed counterions xB increases withincreasing the brush grafting density (inset in Figure 7b), resultingin a decrease in the net charge of the bottle-brush. This isconsistent with the reduction in counterion localization penaltyas the brush grafting density increases. For the bottle-brushmacromolecules with the side chain degree of polymerizationm = 9, we see a saturation of the reduced linear number chargedensity at 1.2. For the other side chain degrees of polymerization,even for our lowest grafting densities, the systems are far from thesaturation limit. Note that lines corresponding to different degreeof polymerizations of the side chains converge with increasingbrush grafting density. This could be an indication of thecrossover to the regime with universal counterion distribution.Unfortunately, one has to consider bottle-brush macromoleculeswith longer side chains to eliminate any doubts about theexistence of this regime.

Counterions localized within bottle-brush volume are prefer-entially located in the vicinity of the side chains. This reduces thecharge on the side chains. To calculate the effective charge of theside chains, we have covered each monomer belonging to a sidechain by a sphere with radius Rcut = 2σ and calculated thefraction xA of counterion located within this volume. In Figure 8,we plotted the dependence of the reduced linear number chargedensity along the side chains of the bottle-brush on the brushgrafting density. (The inset shows the variation of the fraction ofthe condensed counterions xA with the brush grafting density.)Note that for this plot we used only side chains located within

thickness ÆHæ/3 from the brush center of mass. This allowed us tominimize the end effects on the counterion distribution. It followsfrom Figure 8 that the linear charge density along the side chainsdoes not depend on the side chain degree of polymerization. Thishas a very simple explanation. The side chains are stronglystretched by the electrostatic repulsion between uncompensatedcharges such that ÆRm

2 æ1/2 � m and the effective charge of the sidechains is proportional to (1-xA)fm, resulting in the ratio (1 -xA)fm/ÆRm

2 æ1/2 �(1-xA)f to be independent of the degree ofpolymerization m of the side chains. The effective linear chargedensity of the side chains decreases with increasing brush graftingdensity. This reduction of the side chain charge is consistent withreduction of the configurational entropy penalty for the counter-ion localizationwithin the bottle-brush volume. The lines begin tospread as systems approach overlap grafting densities and sidechains begin to shrink.

4. Conclusions

We have studied the effect of the side chain degree of poly-merization and brush grafting density on conformations of graftedbottle-brush polyelectrolytes. Our molecular dynamics simula-tions showed that the thickness of the brush layer first decreaseswith increasing brush grafting density; then, it saturates andremains constant in the wide interval of the brush graftingdensities. The saturation of the brush thickness indicates thatthe stretching of the brush backbone does not change in the wideinterval of the brush grafting density. The side chains of thebottle-brush contract with increasing layer grafting density. Thecontraction of the side chains was explained by the counterioncondensation within the bottle-brush volume. This counterioncondensation reduces the charge of the bottle-brush, weakeningthe electrostatic repulsion between and within side chains.

Counterion condensation in the bottle-brush layer is a multi-scale process. This reflects different symmetries of the problems.At the distances from the grafting surface larger than the brushlayer thickness, a brush layer can be considered as a chargesurface with effective surface charge density, the magnitude ofwhich depends on the fractionof the condensedwithin brush layercounterions. On these length scales, one can use a solution of theone-dimensional nonlinear Poisson-Boltzmann equation to de-scribe counterion distribution outside the brush layer. Inside thebrush layer on the length scales larger than the thickness of thebottle-brush molecules, the electrostatic potential and distribu-tion of counterion density have a cylindrical symmetry. On theselength scales, the counterion distribution around the bottle-brush molecules can be viewed as counterion condensation on acylindrical polyion and can be described in the framework of theManning-Oosawa model of counterion condensation. In thiscase, the counterion condensation precludes the linear chargenumber density of the bottle-brush macromolecule to be largerthan a critical value. Finally, on the length scales smaller than thebottle-brush thickness, there is a counterion condensation on theside chains. This multiscale counterion condensation is a uniquefeature of the brushes made from bottle-brush polyelectrolytes.

Flexibility of the side chains plays an important role indetermining bottle-brush properties. Comparison of our simula-tions with simulations of the bottle-brush chains with rigid sidechains shows that the thickness of the brush layer made of thebottle-brushes with flexible chains does not depend on the brushgraftingdensity in thewide interval of the brush grafting densities,whereas simulations of the bottle-brushes with rigid side chains39

demonstrate an increase in the layer thickness with increasingbrush grafting density. This qualitatively different behavior can

Figure 8. Dependence of the reduced linear number charge den-sity of the side chains on the brush grafting density for bottle-brushmacromolecules with different degrees of polymerization of sidechains: m = 9 (red squares), m = 15 (blue rhombs), and m = 21(black circles). Logarithmic scales. Insets show dependence of thefraction of condensed counterions xA on the brush grafting densityand snapshots of the top view of the brush layer.

DOI: 10.1021/la103229u 18381Langmuir 2010, 26(23), 18374–18381

Carrillo and Dobrynin Article

be explained by contraction of the side chains with increasingbrush graftingdensity. The rigid side chains lead to stronger brushoverlap at lower brush grafting densities resulting in strongerelectrostatic repulsion between chains and stretching of the brushbackbone. The detailed analysis of the effect of the side chain

rigidity as well as orientational correlations between brush back-bone and side chains will be the subject of future studies.

Acknowledgment. This work was supported by the NationalScience Foundation under the grant DMR-1004576.