dissipative particle dynamics simulations of polymer brushes: comparison with molecular dynamics...

668 DOI: 10.1002/mats.200600048 Full Paper

Summary: The structure of polymer brushes is investigatedby dissipative particle dynamics (DPD) simulations thatinclude explicit solvent particles. With an appropriate choiceof the DPD interaction parametersaij, we obtain goodagreement with previous molecular dynamics (MD) resultswhere the good solvent behavior has been modeled byan effective Lennard–Jones potential. The present resultsconfirm that DPD simulation techniques can be applied forlarge length scale simulations of polymer brushes. A relationbetween the different length scalesrc andσ is established.

Polymer brush at a solid–liquid interface.

Dissipative Particle Dynamics Simulations ofPolymer Brushes: Comparison with MolecularDynamics Simulations

Sandeep Pal,a Christian Seidel∗

Max Plank Institute of Colloids and Interfaces, Theory Department, Science Park Golm, D-14424 Potsdam, GermanyE-mail: [email protected]

Received: July 10, 2006; Revised: September 11, 2006; Accepted: September 12, 2006; DOI: 10.1002/mats.200600048

Keywords: dissipative particle dynamics; polymer brush; scaling; simulations; structure

Introduction

The study of polymer chains grafted to an interface(polymer brushes) is an area which has received increasingattention in recent years.[1–4] Such systems have importanttechnological applications which range from colloidalstabilization and lubrication[5,6] to nanoparticle formation atthe polymer brush/air interface.[7–9] In biological sciences,there is a growing interest to understand the importanceof the polymer matrix (grafted or otherwise) in celladhesion.[10–12]

a Present address: Biosystems Informatics Institute, MarlboroughHouse, Marlborough Crescent, Newcastle upon Tyne NE1 4EE,UK.

Compared to polymers in solution, there appearsa new length scale in grafted systems: the distancebetween grafting pointsd = √

A with A being theaverage area available for each polymer at the interface.When the grafting densityρa = 1/A is high, nearbychains repel each other, forcing the polymer to stretchout away from the grafting plane in order to gaininteraction energy. The stretching, however, gives riseto a decreasing configurational entropy which resultsin a penalty in free energy. In a simple Flory-likemean-field theory, the equilibrium height is obtained byminimizing the corresponding total free energy whichresults in

h0 ∼ Nb(υ2ρa/b)1/3 (1)

Macromol. Theory Simul. 2006, 15, 668–673 © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Dissipative Particle Dynamics Simulations of Polymer Brushes:. . . 669

with N being the number of monomers of sizeb andυ2 is thesecond virial or the so-called excluded-volume parameter.Thus, the brush height scales linearly with the contourlengthNb as was originally obtained by Alexander,[13] i.e.,the chains are strongly stretched away from the graftingsurface. The simple scaling approach predicts the brushheight h correctly in the limit of long chains and strongoverlap. It has been confirmed by experiments[14–19] andcomputer simulations.[20–25]

The scaling result mentioned above uses a step-likedensity profile, i.e., all chains are assumed to be stretchedexactly in the same way. Numerical studies, however,exhibit a more rounded monomer density profile with amaximum near the grafting surface and a continuous decayto zero at the rim of the brush.[26] Due to Semenov,[27]

classical polymer paths which minimize the free energyfor given start and end positions, play an important rolein stretched polymer systems. Based upon this observationMilner et al.[28] and Skvortsov et al.[29] proposed ananalytical theory that predicts a parabolic profile. Theparticular profile has subsequently been confirmed incomputer simulations[20,21,23] and experiments.[18,19] LaterNetz and Schick[30,31] pointed out, however, that thestretching of the brush is not correctly described for short orweakly interacting chains. A full mean-field theory has beendeveloped in terms of a single (dimensionless) interactionparameterβ, defined as[30]

β = N

(3υ2

2ρ2a

2b2

)1/3

= 3

2

(h0

R0

)2

(2)

The parameterβ is proportional to the square of theratio of the Alexander prediction for the brush height,h0,and the unperturbed Gaussian chain radius,R0 = bN1/2.Therefore, it is a measure of the stretching of the brush basedon the scaling prediction. In the infinite-stretching limit(ISL) β → ∞ the classical theories are correct. However,deviations from the parabolic profile become progressivelyimportant as the degree of polymerizationN or the graftingdensityρa decreases. For finiteβ the full mean-field theoryand the classical approximation lead to different resultsand both show deviations from the parabolic profile. Theresults of the full mean-field theory were confirmed byextensive molecular dynamics (MD) simulations where alsoindividual polymer paths have been traced.[32]

In this paper, we use dissipative particle dynamics(DPD) simulations to study grafted polymers. DPD is astochastic dynamics technique where each bead representsa volume of a fluid that is large compared to molecularscales. As a consequence of the coarse-graining, theconservative interactions are soft and a long time-step canbe employed. Therefore, DPD can handle larger length andtime scales compared to atomistic or even coarse-grainedMD simulations.[33–39]

For the first time, Malfreyt and Tildesley[40] used DPDsimulations to study equilibrium properties of polymerbrushes. The density distribution under good solventconditions are shown to be in good agreement with thetheoretically predicted parabolic shape.[28,29] The DPDtechnique was also applied to simulate polymer brushesunder shear.[41,42] Recently, the interface between a brushand a melt under shear was investigated by MD simulationsusing a DPD thermostat.[43]

In this paper, we compare DPD simulation results withprevious MD results and SCF predictions to investigatemonomer density and chain stretching. A mapping betweenDPD and MD length scales is found by comparing theuniversal scaling behavior obtained both in DPD and MD.

The organization of the paper is as follows: In the nextsection we give a brief summary of computational details. Inthe main part, results are discussed and compared with MDsimulations and theoretical predictions. Finally, we givesome conclusions and an outlook for further studies.

Computational Details

The model used in this study consists of solvent beads,polymer beads connected along the chains, and twoboundary walls on opposite sides of the simulation box.Thus we have a 2D+ 1 slab geometry and periodicboundary conditions are applied only inx, y directions,while perpendicular to the grafting surface the system isrestricted to one layer.

The DPD “particles” represent a coarse-grainedmesoscopic picture of the underlying microscopic model.A polymer bead represents a few atomic repeat units of thechain; a solvent particle represents an element of the fluidcontaining hundreds of solvent molecules. In our model,all the soft beads have the same massm0 = 1 and radiusrc. Time and length scales are set by these parameters andshould be extracted from the simulations. Forces in DPDsimulations are pairwise additive, conserve momentum,have no hard core, and are short ranged, i.e., the range of theforce defines the size of the soft beads. The effective forcesare chosen to reproduce correct hydrodynamic behavior,i.e., the overall movement of the particles obeys the Navier–Stokes equation.

The DPD method solves the Newtonian equations ofmotion for particles (massm0) subject to conservative,dissipative, and random forces. Thus

dri

dt= vi,

dvi

dt= fi/m0, (3)

wherefi is a pairwise additive force

fi =∑j �= i

(FC

ij + FDij + FR

ij

). (4)

Macromol. Theory Simul. 2006, 15, 668–673 www.mts-journal.de © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

670 S. Pal, C. Seidel

The conservative forceFCij acting between beadi and j

separated by a distancerij = |rij| = |ri − rj| is a soft-repulsive interaction which is linear up to a cutoff distancerc that sets the size of the DPD particles

FCij =

{aij(1 − rij/rc)rij (rij < rc)

0 (rij ≥ rc), (5)

where aij is the repulsion parameter for ani–j typeinteraction andrij = rij/rij gives the direction of the force.The dissipative forceFD

ij between two beads is linear in therelative momentumvij = vi − vj and can be expressed as

FDij = −γij ωD(rij)(rij · vij)rij, (6)

where γij is the strength of the dissipation (frictioncoefficient) between beadsi and j. The random forcebetween a bead pair is given by

FRij = σij ωR(rij)

ζij√�t

rij, (7)

whereσij is the noise amplitude,ωD(rij) andωR(rij) areweight functions which become zero forrij ≥ rc, andζij is arandom variable sampled from a Gaussian distribution withzero mean and unit variance. Espanol and Warren[34] haveshown that the system will sample a canonical ensembleand obey the fluctuation–dissipation theorem if

σij = √2kBTγij and ωR(rij) =

√ωD(rij). (8)

For convenience, we use

ωD(rij) ={

(1 − rij/rc)2 (rij < rc)

0 (rij ≥ rc). (9)

The random forces have the symmetry propertyζij(t) =ζji(t) that ensures the local momentum conservation andhence the correct hydrodynamic behavior of the simulatedfluid on long length scales. Polymer chains are constructedby using a conservative spring force between neighboringDPD particles

FSij = ks(rij − b0)rij, (10)

whereks is the spring constant andb0 the equilibrium springlength. Here we use a parameter set reported by Malfreytand Tildesley:[40] ks = −225kBT/r2

c andb0 = 0.85rc. Notethat due to the soft-repulsive forces the equilibrium bondlengths are larger by a factor of about two.

A wall at z = 0 has to make sure that no particle crossesthe grafting plane. In MD simulations, the wall is modeledby a short-ranged purely repulsive potential, the exact form

Table 1. DPD simulation parameters. B, S, and W representbrush, solvent and wall beads, respectively (in units ofkBT/rc).

Symbol Interaction parameter (aij)

aBS 5aBB 25aSS 25aWW 0aWS 25aWB 25

of which is arbitrary. However, such a potential is not slowlyvarying in the spirit of DPD and would cause restrictions tothe choice of time step. Therefore, in this study, the wall ismodeled by several layers of densely packed and fixed DPDbeads (their equations of motions are not integrated). Threelayers were found to be sufficient to prevent penetration bysolvent and polymer beads. Obviously, such a wall is softerthan a potential of the range of about half a bead size. Thedifference might have some effect on the particle densityclose to the wall. However, on large length scales of theorder ofR0, we are actually interested in, it has no effect.

The simulations were performed in a periodic cubic boxof constant volumeV = Lx × Ly × Lz, whereLx, Ly, Lz

are the simulation box side lengths in units of the beaddiameterrc. In all the cases, 50 polymer chains were graftedon the wall to form a polymer brush. The box lengths inx and y directions are varied according to the particulargrafting density. Inz direction the length is kept fixed at40rc. The total number of beads (polymer + solvent) inthe simulation box is chosen to fix the overall reduceddensity of the simulation boxρr3

c = 3.0. In this paper, westudy the structural properties of polymer brushes undergood solvent conditions. The interaction parametersaij

are chosen to model that case, see Table 1. Note thataBS < aBB and aBS < aSS, which represents a polymerbrush under good solvent conditions. The friction parameteris set asγij = γDPD = 2.25

√m0kBT/r2

c. The equation ofmotion [Equation (3)] is integrated by means of a modifiedvelocity-Verlet algorithm[35] with a time step�t = 0.02t0with t0 = √

m0r2c/kBT being the unit of time. Due to

the slowly varying DPD potentials, the time step can bechosen one order of magnitude larger than in previousMD simulations.[32] Configurations are saved every 1 000time steps. The first 50 000 configurations are discarded;production runs were done for the next 50 000 configura-tions. For simplicity, mass, and temperature are set tom0 =1, kBT = 1. Note that previous MD simulations were donewith kBT = 1.2ε,[32] where the Lennard–Jones parameterε sets the strength of the effective interaction betweenmonomers. A mapping of the interaction strengths in thedifferent models is beyond the scope of this communication.Thus there remains an uncertainty in the exact value of thetemperature which is expected to play no role as long as thesystem remains in the good solvent regime.



Figure 1. Monomer number density per chain,ρ(z)/Nρa, as afunction of the scaled distance from the grafting surfacez/RF.DPD simulation results are obtained for 50 anchored chains oflengthN = 50 at varying grafting densityρa (in units ofr−2

c ).

Results and Discussion

In Figure 1, we report the reduced monomer densitydistribution ρ(z)/Nρa at different grafting densities as afunction of the normalized distance from the grafting planez/RF, where RF is the Flory radius of a free polymerwith identical contour length. The value ofRF calculatedfrom a separate DPD simulation of a free polymer chainin good solvent conditions is found to beRF ∼ 12.36rc.Note that in contrast to the ISL[28,29] as well as the full SCFcalculations,[31] at higher grafting densities the monomerdensity profiles shown in Figure 1 exhibit no parabolicbehavior but a region where the monomer profiles becomerather flat. Such a behavior, is in agreement with previousMD[21,32] and DPD[40–42] simulation results. As pointedout by Laradji et al.,[25] the discrepancy is expected tooccur due to higher order virial terms being present in thesimulation but being neglected in the SCF approach. Forlower grafting densities, the monomer profiles rise sharplyfrom the grafting plane and decay smoothly to zero atlarge distances. The stretching of the chains with increasinggrafting densities is evident. Note that at weak grafting themonomer density of the first layer close to the grafting planeis indeed considerably larger than that previously obtainedwith a rather hard wall.[32]

Several simulation studies have shown that the brushheight follows the theoretical predictionh ∼ Nρ

1/3a

provided the grafting density is above the critical oneρ∗a ∼

N−6/5.[21,32,40–42] In Figure 2, we plot the scaling behaviorof the brush height measured both as the first moment ofthe density distributionh = ∫

zρ(z)dz/∫ρ(z)dz and as thez

component of the radius of gyrationRG,z = 1/(N + 1) ×〈∑N

i = 0(zi − z2cm)〉1/2, wherezcm is thez coordinate of the

Figure 2. DPD simulation results for rescaled average monomerheighth/N(ρab

2)1/3 (squares) andz-contribution to the radius ofgyration of the polymer brushRG,z/N(ρab

2)1/3 (circles) versusN(ρab

2)1/3/RF.

center of mass. The scaling variable for thex axis is chosenas N(ρab

2)1/3/RF ∼ β1/2, whereβ is the SCF stretchingparameter introduced in Equation (2). At medium to higherstretching, both rescaled measures of the brush heightexhibit a nearly constant value. The point where the scalingrelationship holds is termed as the universal crossoverpoint between the mushroom regime and brushes.[21,32]

Seidel and Netz[32] have reported the universal crossoverpoint from mushroom to brush regime asN(ρ∗

ab2)1/3/RF ∼

1.4σ−1LJ , whereσLJ is the Lennard–Jones length chosen as a

typical Kuhn length of a polymer. Similar discussion of thecrossover points are present in the literature from other MDsimulations.[21] In our DPD simulation the crossover pointis achieved atN(ρ∗

ab2)1/3/RF ∼ 2.75r−1

c . The difference inthe numerical values between MD and DPD occurs dueto the different length scales used in the two simulations.Comparing the result for the crossover point obtained hereby DPD simulations with that found in MD simulationsby Seidel and Netz, we get the relationrc = 1.9σ. Withinthe error bars, the same relation holds, if we equate thegrafting densities from DPD and MD simulations for similarrescaled brush monomer density profiles (see Figure 1,Figure 5 in ref.,[32] and Table 2). The characteristic Kuhn

Table 2. Mapping of DPD and MD grafting densities. Relatedare grafting densities which give more or less identical monomerdensity profilesρ(z).

ρDPDa (r−2

c ) ρMDa (σ−2) X, relation (rc = Xσ)

0.62 0.17 1.910.35 0.09 1.970.22 0.06 1.910.15 0.04 1.940.08 0.02 1.98


672 S. Pal, C. Seidel

Figure 3. Effective chain stretching factorγ as a function ofthe rescaled anchoring density [N(ρab

2)1/3/RF]2. The dashed linebeing a guide to the eyes.

length of a flexible polymer isσ ≈ 1 nm. Thus, in the DPDsimulations we haverc ≈ 1.9 nm. This relationship gives anapproximate mapping between DPD and MD length scales.Note that similar mapping relations can be derived also fromother quantities calculated by means of both simulationapproaches.

To compare theoretical results with that obtained bysimulations, in ref.[32] the effective stretching parameterγ has been defined as the relation between the averageend-point height at a certain grafting density and that ofa single chain end-grafted at an impenetrable wallγ(ρa) =〈ze(ρa)〉/〈ze, single chain〉. In Figure 3, we plotγ as a functionof the rescaled grafting density [N(ρab

2)1/3/RF]2 ∼ β. Atlow grafting densities, one hasγ ≈ 1, i.e., basically thechains exhibit the behavior of grafted single chains andare relatively unaffected by changes in grafting density.However at largerρa the stretching parameter rises sharplyas a function of the increasing grafting density. The behavioris reasonable and quite similar to that predicted by theoryand found in MD simulations.[32]

Considering the crossover point between mushroom andbrush regime atN(ρ∗

ab2)1/3/RF ≈ 2.75 discussed above,

and using Figure 3 as a calibration curve for the DPDsimulation, we find that the polymer brush regime is reachedat an effective chain stretchingγ ≥ 1.6, which is in goodagreement with the value previously obtained by MDsimulations.[32]

Figure 4 shows the distribution of free endsρe(z)normalized to unity as a function of the rescaled distancefrom grafting planez/〈ze〉 where 〈ze〉 is the averageendpoint height. The profiles are in very good agreementwith those obtained by Murat and Grest.[21] In agreement

Figure 4. Normalized density of free ends of polymer chainsversus scaled end point positionz/〈ze〉 at varying grafting densityρa (in units ofr−2

c ).

with Murat and Grest at lower grafting densities,ρe(z) isconvex forz < zm, with zm being thez-value whereρe(z)reaches its maximum. Forz > zm, ρe(z) decays gradually.At higher grafting densities,ρe(z) increases rapidly to themaximum value atz = zm, and then drops to zero.

Conclusion

In this article, we have reported structural properties ofpolymer brushes in a good solvent, obtained by DPDsimulations with explicit solvent particles and comparedthem with previous results found by MD simulations withan effective Lennard–Jones potential that models the goodsolvent behavior of polymer chains. In the first place, ouraim was to show that equilibrium properties of brushescan be properly studied by the DPD technique. That iswhy we omit here a discussion of solvent profiles alsoprovided by the DPD simulations. With an appropriatechoice of the DPD interaction parametersaij, we obtainedgood agreement between the MD results by Seidel andNetz[32] and the present DPD data. The relation betweenLennard–Jones length scaleσ and DPD scalerc is foundto be rc = 1.9σ. This contribution is a preliminary workto benchmark DPD simulations of polymer brushes forfollowing large length scale simulations. DPD simulationsto study nanoparticle aggregation inside a polymer brushare currently under progress.

Acknowledgements: We thank Julian Shillcock for makingavailable his DPD code to us as well as for useful commentsand discussions. S.P. acknowledgesAndrea Grafmuller for helpfulcomments on the DPD code.



[1] A. Halperin, M. Tirrell, T. P. Lodge,Adv. Polym. Sci. 1991,100, 31.

[2] S. T. Milner,Science 1991, 251, 905.[3] J. Ruhe, M. Ballauf, M. Biesalski, P. Dziezok, F. Grohn, D.

Johannsmann, N. Houbenov, N. Hugenberg, R. Konradi, S.Minko, M. Motornov, R. R. Netz, M. Schmidt, C. Seidel,M. Stamm, T. Stephan, D. Usov, H. N. ZhangAdv. Polym.Sci. 2004, 165, 79.

[4] A. Naji, C. Seidel, R. R. Netz,Adv. Polym. Sci. 2006, 198,149.

[5] D. H. Napper, “Polymeric Stabilization of ColloidalDispersions”, Academic Press, London 1983.

[6] A. Elgsaeter, B. T. Stokke, A. Mikkelsen, D. Branton,Science 1986, 234, 1217.

[7] V. Z.-H. Chan, J. Hoffman, V. Y. Lee, H. Iatrou, A.Averopoulos, N. Hadjichristidis, R. D. Miller, E. L.Thomas,Science 1999, 286, 1716.

[8] R. B. Thompson, V. V. Ginzburg, M. W. Matsen, A. C.Balazs,Science 2001, 292, 2469.

[9] Z. Liu, K. Pappacena, J. Cerise, J. Kim, C. J. Durning, B.O’Shaughnessy, R. Levicky,Nanoletters 2002, 2, 219.

[10] E. Zimmerman, B. Geiger, L. Addadi,Biophys. J. 2002, 82,1848.

[11] M. Cohen, E. Klein, B. Geiger, L. Addadi,Biophys. J. 2003,85, 1996.

[12] M. Cohen, D. Joester, B. Geiger, L. Addadi,Chem. Phys.Chem. 2004, 5, 1393.

[13] S. Alexander,J. Phys. (Paris) 1977, 38, 983.[14] G. Hadziioannou, S. Patel, S. Granick, M. Tirrell,J. Am.

Chem. Soc. 1986, 108, 2869.[15] H. J. Taunton, C. Toprahcioglu, L. J. Fetters, J. Klein,

Nature 1988, 332, 712.[16] H. J. Taunton, C. Toprahcioglu, L. J. Fetters, J. Klein,

Macromolecules 1990, 23, 571.[17] T. Cosgrove,J. Chem. Soc., Faraday Trans. 1990, 86, 1323.[18] P. Auroy, L. Auvray, L. Leger,Phys. Rev. Lett. 1991, 66,

719.[19] [19a] P. Auroy, L. Auvray, L. Leger,Macromolecules

1991, 24, 2523; [19b] P. Auroy, L. Auvray, L. Leger,Macromolecules 1991, 24, 5158.

[20] T. Cosgrove, T. Heath, B. van Lent F. Leermakers, J.Scheutjens,Macromolecules 1987, 20, 1692.

[21] M. Murat, G. S. Grest,Macromolecules 1989, 22, 4054.[22] A. Chakrabarti, R. Toral,Macromolecules 1990, 23, 2016.[23] P.-Y. Lai, K. Binder,J. Chem. Phys. 1991, 95, 9288.[24] G. S. Grest,Macromolecules 1994, 27, 418.[25] M. Laradji, H. Guo, M. Zuckermann,J. Phys. Rev. E 1994,

49, 3199.[26] A. M. Skvortsov, I. V. Pavlushkov, A. A. Gorbunov,Polym.

Sci. USSR 1988, 30, 487.[27] A. N. Semenov,Sov. Phys. JETP 1985, 61, 733.[28] S. T. Milner, T. A. Witten, M. E. Cates,Europhys. Lett.

1988, 5, 413;Macromolecules 1988, 21, 610.[29] A. M. Skvortsov, I. V. Pavlushkov, A. A. Gorbunov, Y. B.

Zhulina, O. V. Borisov, V. A. Pryamitsyn,Polym. Sci. USSR1988, 30, 1706.

[30] R. R. Netz, M. Schick,Europhys. Lett. 1997, 38, 37.[31] R. R. Netz, M. Schick,Macromolecules 1998, 31, 5105.[32] C. Seidel, R. R. Netz,Macromolecules 2000, 33, 634.[33] P. J. Hoogerbrugge, J. M. V. A. Koelman,Europhys. Lett.

1992, 19, 155.[34] P. Espanol, P. Warren,Europhys. Lett. 1995, 30, 191.[35] R. D. Groot, P. B. Warren,J. Chem. Phys. 1997, 107,

4423.[36] R. D. Groot, T. J. Madden,J. Chem. Phys. 1998, 108,

8713.[37] P. B. Warren,Curr. Opin. Colloid Interface Sci. 1998, 3,

620.[38] J. B. Gibson, K. Zhang, K. Chen, S. Chynoweth, C. W.

Manke,Mol. Simul. 1999, 23, 1.[39] J. C. Shillcock, R. Lipowsky,J. Chem. Phys. 2002, 117,

5048.[40] P. Malfreyt, D. J. Tildesley,Langmuir 2000, 16, 4732.[41] D. Irfachsyad, D. Tildesley, P. Malfreyt,Phys. Chem. Chem.

Phys. 2002, 4, 3008.[42] C. M. Wijmans, B. Smit,Macromolecules 2002, 35,

7138.[43] C. Pastorino, K. Binder, T. Kreer, M. Muller,J. Chem. Phys.

2006, 124, 064902.


dissipative particle dynamics simulations of polymer brushes: comparison with molecular dynamics...

Documents