Interactions between polymer brush-coated spherical nanoparticles: the good solvent case

THE JOURNAL OF CHEMICAL PHYSICS 135, 214902 (2011)

Interactions between polymer brush-coated spherical nanoparticles:The good solvent case

Federica Lo Verso,1,a) Leonid Yelash,1 Sergei A. Egorov,2 and Kurt Binder11Institut für Physik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Staudinger Weg 7, Germany2Department of Chemistry, University of Virginia, Charlottesville, Virginia 22901, USA

(Received 2 August 2011; accepted 2 November 2011; published online 2 December 2011)

The interaction between two spherical polymer brushes is studied by molecular dynamics simulationvarying both the radius of the spherical particles and their distance, as well as the grafting densityand the chain length of the end-grafted flexible polymer chains. A coarse-grained bead-spring modelis used to describe the macromolecules, and purely repulsive monomer-monomer interactions aretaken throughout, restricting the study to the good solvent limit. Both the potential of mean forcebetween the particles as a function of their distance is computed, for various choices of the parametersmentioned above, and the structural characteristics are discussed (density profiles, average end-to-end distance of the grafted chains, etc.). When the nanoparticles approach very closely, some chainsneed to be squeezed out into the tangent plane in between the particles, causing a very steep rise ofthe repulsive interaction energy between the particles. We consider as a complementary method thedensity functional theory approach. We find that the quantitative accuracy of the density functionaltheory is limited to large nanoparticle separation and short chain length. A brief comparison to Florytheory and related work on other models also is presented. © 2011 American Institute of Physics.[doi:10.1063/1.3663964]

I. INTRODUCTION

Spherical particles to which flexible macromoleculesare grafted occur in many physical contexts. For example,grafting relatively short polymer chains to colloidal particlesis a well-known recipe for preventing aggregation of col-loidal particles.1, 2 When these particles have radii in the mi-crometer range, the effective interaction between these parti-cles is often approximated as hard-sphere; however, recentlythere has been much interest in particles with radii below100 nm (see Refs. 3 and 4) and then the effective inter-action between such nanoparticles gives rise to nontrivialstructures, and requires a discussion in terms of the config-urational statistics of the polymers coating these particles.4

These systems are intermediate between standard colloidalparticles and star polymers (where the core has only a sizeabout 1 nm), for which the effective pair interaction has al-ready been discussed extensively.5–8 Brush-coated particlesare of potential interest as ingredients of composite materi-als to improve their mechanical properties. However, suchparticles embedded in a polymer matrix, (see Ref. 9 andreferences therein) or their possible use as building blocksfor nanocomposites10, 11 are out of consideration here. Re-lated systems are “smart nanoparticles,”12 i.e., microgels witha dense hydrophobic core to which hydrophilic chains aregrafted, and which are believed to be of interest for manymedical and biological applications,13 e.g., drug delivery sys-tems. Moreover, a successful incorporation of these macro-molecules into useful devices with specific mechanical prop-erties necessitates controlling of the specific strength andhardness of nanometer-scale objects interaction with their

a)Electronic mail: [email protected].

surroundings, including other molecules.13, 14 Nanoparticleswith grafted flexible polymers may also be considered asa model for spherical micelles,15 where the (insoluble) A-block of a binary AB-block copolymer in a selective sol-vent forms a dense core, resembling a nanoparticle to whichB-chains are grafted. Thus, there are many motivations tostudy the interplay between conformational entropy of thegrafted chains and the confining constraint provided by thedense grafting on a spherical surface by analytical theory,16–20

simulation,21–25 and experiment,4, 26–28 and understanding theinteractions between such spherical brushes is still a matter ofcurrent investigations.25, 27, 29–32

In the present paper, we extend our previous study,23

where the structure of the macromolecules grafted to anisolated spherical nanoparticle was considered, to the caseof the pair interaction between such spherical brushes, undergood solvent conditions (only repulsive effective interactionsbetween monomers being considered). The only previoussimulation30 of this problem that we are aware of used MonteCarlo methods and somewhat smaller values of chain lengthN and number of grafted chains f than were accessible in ourstudy. On the other hand, the inclusion of attractive dispersioninteractions between spherical brushes has been investigatedin Ref. 25. Another motivation to use molecular dynamicsmethods33 rather than Monte Carlo methods is that a more re-alistic description of chain dynamics is possible (but we defera detailed analysis of chain dynamics in spherical polymerbrushes to a forthcoming publication). In addition, we wishto perform a more extensive comparison with theoreticalapproaches. Cerda et al.30 noted disagreements between theirsimulations and the (then available) numerical results basedon self-consistent field theory (SCFT).29 Meanwhile moreelaborate SCFT treatments are available,31 and an alternative

0021-9606/2011/135(21)/214902/10/$30.00 © 2011 American Institute of Physics135, 214902-1

214902-2 Lo Verso et al. J. Chem. Phys. 135, 214902 (2011)

powerful approach is density functional theory (DFT). DFThas been shown to be an accurate tool to describe interactionsbetween planar polymer brushes,34 and an application tomodel an isolated spherical brush is found in our previouswork.23 However, an extension of DFT to the problem oftwo interacting spherical brushes has been discussed only forshort chain and moderate grafting density,35 and hence it willbe further analyzed in the present paper.

The outline of this work is as follows. In Sec. II, we de-scribe the model that is simulated and briefly comment on thesimulation method. In Sec. III, we present some pertinent re-sults for a single spherical brush that are needed in the laterdiscussion. Section IV contains the main part of our simula-tion results, discussing the interaction energy and potentialof mean force, density profiles, and chain radii. Section Vpresents then a comparison with some selected DFT resultsand discusses also other pertinent theoretical predictions. Wesummarize our work briefly in Sec. VI.

II. MODEL AND SIMULATION METHOD

While in our previous work,23 the nanoparticle was de-scribed as a perfect sphere of radius Rc to which the firstmonomer of each chain is rigidly fixed (at regular posi-tions determined by a procedure of geodesic subdivision, seeRef. 23), we here rather describe the surface of the nanopar-ticle in a quasi-atomistic way providing a dense packing ofmonomer-like units. The arrangement of such units has beenobtained via the geodesic subdivision. Indeed, the same pro-cedure that was used previously23 to fix the positions of thegrafted monomers here is used to fix the positions of all theparticles forming the sphere surface.

As an example, Fig. 1(a) shows the internal core of thespherical brush with the grafting sites highlighted. In thiscase, the grafting density is ! = 0.118 (in Lennard-Jonesunits, see below), and f = 92 polymers are grafted to this coreparticle. Figure 1(b) then shows snapshot pictures where twosuch spherical brushes interact at two choices of distance r be-tween their centers. Here the number of effective monomersper chain is N = 40. The monomers of the chains (irrespectivewhether they are bonded to each other or not) interact with apurely repulsive Lennard-Jones potential,

VLJ(r)

=!

4"LJ{(!LJ/r)12 ! (!LL/r)6 + 1/4}, for r " 21/6!LJ,

0, for r > 21/6!LJ.

(1)

In the following "LJ = 1 and !LJ = 1 are chosen, set-ting the scales of energy and length, respectively. Note thatthe same potential VLJ(r) is also employed between themonomers and the particles in the surface layer of the core(the latter occurs at the rigidly fixed positions resulting fromthe geodesic subdivision procedure). We studied different val-ues of the core radius, namely Rc = 7 (corresponding tof = 42, ! = 0.068), Rc = 7.9 (corresponding to f = 92,! = 0.118), and Rc = 8.35 (corresponding to f = 162,! = 0.185), i.e., exactly as chosen in our previous work on

FIG. 1. (a) Internal core of the spherical brush with radius Rc = 7.9 and agrafting density ! = 0.118, i.e., a number f = 92 of grafted chains. Notethat the polymer chains are not shown in the picture, so that the regular ar-rangement of the grafting sites, highlighted by a different color, are easilydistinguished from the other particles forming the surface layer of the core.(b) Snapshot picture of two spherical brushes with f = 92, N = 40, and! = 0.118, at the separations r = 55 (upper part) and r = 20 (lower part)of their core centers. Grafting sites of the cores are highlighted by red color,other sites of the core are in gray color, while monomers of the flexible chainsare displayed in blue color.

single isolated spherical polymer brushes.23 As it is standardfor the simulation of polymer brushes36 bonded monomersinteract with the finitely extensible nonlinear elastic (FENE)potential,36–38

VFENE(r)

=!

0.5kFENE(R0/!LJ)2 ln[1 ! (r/R0)2] for r < R0,

#, for r > R0,

(2)

We fixed the constants kFENE = 30"LJ and R0 = 1.5!LJ asusual (then the total “spring potential” between two neighbor-ing monomers along the chain in our bead spring model hasits minimum at rmin $ 0.97!LJ).36–38 Note that VFENE(r) alsoacts between the first mobile effective monomer of a graftedchain and the rigidly fixed grafting site. The reduced temper-ature of our simulations was chosen kBT/#LJ = 1.2 throughoutand thus we consider the case of very good solvent conditionsonly; changing the solvent quality in our implicit solventmodel could be done by choosing a larger cutoff in Eq. (1), but

214902-3 Interactions between polymer brushes J. Chem. Phys. 135, 214902 (2011)

this is left to future work. Considering variable solvent qualitywould be desirable for a comparison to possible experiments,of course. Assigning to the monomer mass m, the value m = 1,

the characteristic time $ ="

m! 2LJ/"LJ is then unity too.

The monomers forming the core interact with the effec-tive monomers of the chains with the same potential, Eq. (1),with which the effective monomers interact with each other.The advantage of modelling the surface of the core particle asa regular arrangement of fixed particles is that the GROMACS

package39–41 is straightforwardly applicable to carry out themolecular dynamics simulations. Simulating grafted chainsof length N = 20, 40, 60, and 80 as in our previous work,23

where a code used for the simulation of star polymer42 wasadapted, we noted that the GROMACS code performs signifi-cantly faster.

To prepare starting configurations for our simulation, wegenerate a colloid with radially straight chains fixed to thecolloid surface at grafting points, which ensures no artificialmonomer overlaps at the beginning. This system was equi-librated using a Langevin thermostat with high friction pa-rameter until chains adopted their equilibrium distributions,which is controlled by energy, radial density profiles, andRe/Rg plots at different times. After equilibration, the produc-tion runs were started for t = 20000 using a Nose-Hooverthermostat and data on the trajectories of all particles weresaved after every dt = 10 time units to ensure uncorrelatedconfigurations in further statistical analysis.

To calculate the potential of mean force (PMF), we per-formed simulations of two colloid particles in the box. As astarting configuration, we take the configuration after the pro-duction run from the previous part, we put a copy of the par-ticle at distance r, which is bigger as a typical Re for a givensystem, increase the box size to enclose the second particlebeing in the box completely, and perform equilibration andproduction runs. This two-colloid system is used as a start-ing configuration for the next r, for which we move the sec-ond colloid particle by a distance %r closer to the first oneand repeat the equilibration procedure. Note that we choosethe shift %r depending on the distance r between the parti-cles. Whereas for large distances, one can choose %r = 1, onehas to decrease %r to %r = 0.1 due to monomer overlap andstrong increase of repulsion forces when particles are close toeach other. This excess energy, however, is removed from thesystem very fast using the Langevin thermostat during equi-libration. When equilibrated configurations for all distances rare generated, we start the production runs for t = 20000 asdescribed above.

III. ISOLATED SPHERICAL BRUSHES: REVIEWOF SELECTED RESULTS

Since in our previous study,23 we have already presenteda detailed analysis of the structure of single isolated sphericalbrushes, we keep this section very brief. However, the presentwork studies a slightly different model than was used inRef. 23, and thus it is important to test whether the differencesof the models leads to (undesirable) essential differences intheir properties.

8 10 12 14 18 20 22 2426r

0.001

0.01

0.1

1

10

100

!(r)

y~x4/3

f=92 N=20 DFT GROMACS MD Ref.[20] N=40 DFT GROMACS MD Ref.[20]

FIG. 2. Density profile of a spherical polymer brush with Rc = 7.9,! = 0.118, and two choices of N, N = 60 and N = 80. Simulation resultsfor the model of Ref. 23 and DFT results23 are compared to the present MDsimulations. The scaling prediction & % r!4/3 resulting in the star polymerlimit also is included.

Figure 2 presents a comparison of the monomer densityprofiles of isolated single spherical polymer brushes from theprevious work23 with the present simulation. Although themodels of the spherical core particle slightly differ, as dis-cussed above, the density profiles differ only very slightlyand only very close to the grafting surface (near r $ 9 inFig. 2 such small differences can be detected). However, wedid not focus here on local details on the scale of a sin-gle monomer, which are non-universal. In the outer parts ofthe profile, the present and previous simulations are indis-tinguishable. As noted already in Ref. 23, the DFT predic-tions for the profile &(r) differ slightly but systematicallyfrom the simulation results, despite a striking qualitative sim-ilarity, and the discrepancy increases with increasing chainlength.

Similar data have been obtained also for the other cases,Rc = 7(f = 42) and Rc = 8.35(f = 162). In all cases, the previ-ous model (with a non-corrugated surface of the nanoscopicsphere onto which the chains are grafted) and the presentmodel with a corrugated surface (Fig. 1(a)) differ at mostslightly very close to the grafting surface, where the layeringof the density profile occurs. At larger distances, the detailedstructure of the grafting surface does not have any specificeffects.

IV. PROPERTIES OF TWO INTERACTINGSPHERICAL BRUSHES

We started out with two spherical brushes at such a largeseparation r between the centers of their cores that the chainsinteract only very weakly with each other, Fig. 1(b). It isstraightforward to sample the total interaction energy fromthe pairwise interaction between monomers of chains frombrush 1 with brush 2. Then we decrease r by a small step%r, re-equilibrate the system again and then sample the inter-brush potential energy again. This procedure is repeated manytimes, until the cores of the two brushes almost touch eachother (see Sec. II for details).

214902-4 Lo Verso et al. J. Chem. Phys. 135, 214902 (2011)

0 1000 2000 3000r2

0.1

1

10

100

1000

W(r)

N=20N=40N=60N=80

0 1000 2000 3000 4000r2

0.010.11101001000

W(r)

N=20N=40N=60N=80

0 1000 2000 3000 4000 5000r20.010.11101001000

W(r)

N=20N=40N=60N=80

f=42

f=92

f=162

FIG. 3. Potential of mean force W(r) plotted versus the distance r betweenthe centers of the two spherical polymer brushes, for the case Rc = 7, f = 42(upper part), Rc = 7.9, f = 92 (middle part), and Rc = 8.35, f = 162 (lowerpart). The chain lengths of the grafted polymers were N = 20, 40, 60, and 80as indicated.

We begin by discussing the PMF between the two spheri-cal polymer brushes. It can be expressed in terms of the excessmean force F(r) as

W (r) =# #

r

dr &F (r &). (3)

Figure 3 shows the simulation results for the potentialof mean force W(r) for the three core radii Rc = 7, 7.9, and8.35, respectively, including also four choices of the chainlength N in each case. One can see that there are two dis-tinct regimes for each Rc: in a regime of close contact, 2Rc

< r < 2Rc + !LJ, there occurs a very steep increase by sev-eral orders of magnitude, and this increase is almost indepen-dent of N. Due to the fact that a close contact of the nanopar-ticles requires that chains grafted in the contact region needto be squeezed out in the xy directions perpendicular to thez-axis (connecting the two nanoparticle centers), cf. the snap-shot picture, Fig. 4, a rather large number of monomers iswithin the range of the particles in the surface region of thecore particles. Thus it is plausible that a very large repulsiveenergy as well as potential of mean force result. It is also plau-sible that the energy resulting from the monomers squeezed inthe contact region is almost independent of the chain length ofthe grafted chains. Indeed we observed that the scale for theenergy at larger distances (where then also the N-dependencematters) is only of the order 0.1kBT to 10kBT at most, implyingthat the typical configuration of two weakly interpenetratingspherical polymer brushes have only very few intermolecularcontacts. A more detailed study in the regime of close contactbetween the two nanoparticles has shown that the behaviordepends on the orientations of the two spherical brushes rela-

FIG. 4. Snapshot of the contact region between two spherical polymerbrushes with Rc = 7.9 at a distance r = 16.5 (exceeding 2Rc = 15.8 onlyslightly). Each nanoparticle has f = 92 chains of length N = 60, but only afew chains that are squeezed out from the contact region are shown, the re-maining chains being suppressed for the sake of clarity. Chains belonging tonanoparticle 1 are shown in red color, chains belonging to nanoparticle 2 areshown in blue color.

tive to the z-axis (by “orientation” we refer to the axis from aparticle center to the grafting site which is closest to the axisconnecting the two particle centers which defines the z-axis: ifthese two axes coincide, the two spheres upon contact wouldtouch precisely at a grafting site, while otherwise the nearestgrafting site occurs somewhat away from the contact point).We have not studied the details of this behavior; however, be-cause our model is not atomistically realistic, so such shortscale details are of minor interest. Note also that it may hap-pen that monomers of the chain get “arrested” in the minimaof the effective corrugation potential provided by the particlesurfaces (Fig. 1) when two particles approach each other up toalmost direct contact. Such “arrested” monomers imply thatthe system is not fully in equilibrium, of course. The detailsof the strongly rising part of W(r) at the smallest possible dis-tances in Fig. 3 may be affected by such problems, and hencethis region will not be discussed further here. Of course, themain interest in W(r) is in the region of the larger r, wherethe chains of the two spherical brushes weakly interpenetrate.We defer a discussion of this region to Sec. V, where W(r) iscompared to DFT results and other theoretical treatments.

Figure 5 gives a kind of density profile as a function ofthe z-coordinate (remember that the z-axis is oriented alongthe straight line connecting the centers of the two nanoparti-cles), by simply counting the numbers of Nxy(z) of monomersfollowing into a slice of width 'z = 0.45, 0.5, and 0.75 inthe simulation box, for N = 20 and f = 42 (upper part),f = 92 (middle part), and f = 162 (lower part), respectively.The left particle has its center at zL = 30, the right particle hasits center at zR = zL + r, and various choices of r are shown ineach part of these figures. Figure 6 is the analog of Fig. 5, butfor N = 60. Note that zL was chosen as zL = 45 in this case

214902-5 Interactions between polymer brushes J. Chem. Phys. 135, 214902 (2011)

10

20

30

40

50

Nxy

r=15r=18r=20r=25r=30r=40

20406080100

Nxy

r=16.5

0 20 40 60 80 100z

050100150200250

Nxy

r=17r=18

f=42

f=92

f=162

FIG. 5. Number of monomers Nxy(z) in a slice normal to the z-axis rang-ing from z ! 'z, z + 'z with 'z = 0.45, 0.5, and 0.75 in the simulationbox, for N = 20 and f = 42 (upper part), f = 92 (middle part), and f = 162(lower part), respectively. The center of the left spherical brush always is atzL = 30, and of the right particle at zR = zL + r, and the different choices ofr are indicated.

and 'z = 0.75, 0.8, and 0.75 for N = 20 and f = 42 (upperpart), f = 92 (middle part), and f = 162 (lower part), respec-tively. One sees that for larger r, there occurs a deep minimumin between the two brushes, but as r gets smaller, this mini-mum is progressively filled up, and when r exceeds 2Rc onlyslightly, one rather finds a sharp maximum at z = zL + r/2,due to the chains that are squeezed out in between two parti-cles into x and y directions (cf. Fig. 4). Of course, even moreinformation is contained in an analysis of a two-dimensionaldensity distribution, &(z, r||), where r2

|| = x2 + y2, see Figs. 7and 8. We note that these data are qualitatively rather similar

50

100

150

Nxy

r=15r=20r=30r=40

0 20 40 60 80 100 120 140z

0

100

200

300

400

Nxy

r=17

100

200

300

Nxy

r=16.5

f=42

f=92

f=162

FIG. 6. Same as Fig. 5, but for N = 60. Note that zL was chosen as zL = 45 inthis case and 'z = 0.75, 0.8, and 0.75 for N = 20 and f = 42 (upper part), f= 92 (middle part), and f = 162 (lower part), respectively.

to corresponding recent calculations with the self-consistentfield theory.31

One can also ask whether the reorientation of the graftedpolymer chains leads to a change in the linear dimensions ofthe grafted chains. For large separation r the end-to-end vec-tor 'Re of each of the grafted chains on average is oriented per-pendicular to the tangent plane to the surface of the nanopar-ticle in the grafting point; for small r, some of the chainsare rather oriented in the xy plane, since they are squeezedout from the contact region between the two nanoparticles.Figures 9 and 10 show that indeed for small r, the chains onaverage are more stretched, as one should expect. This effectis a consequence of a crowding of monomers in the surround-ings of the tangent plane between the two nanoparticles, attheir point of closest approach: unfavorable contacts can onlybe avoided if the chains strongly stretch in the radial directionin this plane. Of course, chains with grafting sites that are notclose to this contact region are not affected by the approach ofthe two spherical brushes. Thus, when one averages over allof the grafted chains (as done in Figs. 9 and 10), the overallr-dependence is only minor despite the fact that a few chainsare much more strongly stretched for small r.

V. COMPARISON WITH THEORETICAL PREDICTIONS

The quantity of major interest for us is the PMF be-tween two brush-coated spherical nanoparticles. Given thespherically symmetric nature of the brushes, the PMF de-pends only on the separation r between the core centers ofthe two nanoparticles. Given that the solvent in our approachis treated implicitly, the total number of segments in the prob-lem is fixed (via the values of the grafting density and thechain length), and the PMF can be written as the Helmholtzfree energy of the system when the two brushes are separatedby r relative to its value at infinite separation,35, 43

W (r) = F [&p(Rp); r] ! F [&p(Rp); #], (4)

where &p(Rp) is the polymer density profile of the twobrushes.

Quite generally, the Helmholtz free energy can be writ-ten as a sum of three terms: the ideal term, the excess term,and the contribution due to the external field created by thenanoparticle cores,35, 43

F [&p(Rp); r] = Fid [&p(Rp); r] + Fex[&(r); r]

+#

Vext (r)&(r)dr, (5)

where the last two contributions are expressed in terms ofmonomer density distribution &(r).

Regarding the ideal term, while its calculation is straight-forward for simple liquids, it is extremely involved forpolymers.44 However, in a recent DFT study of interactionsbetween dendrimers,44 it was argued that the contribution ofFid to the PMF is negligible compared to Fex. Here we followthe same approach and neglect the first term in Eq. (5).

Within the quasi-atomistic model employed in the MDsimulations, the surface of the nanoparticles is formed bya dense packing of monomer-like units. By contrast, in

214902-6 Lo Verso et al. J. Chem. Phys. 135, 214902 (2011)

FIG. 7. Two-dimensional density distribution &(z, r||) shown in the form of a contour plot where curves &(z, r||) = constant are shown for six representativecases for N = 20, namely, r = 30 (left side), f = 42(a), f = 92(b) and f = 162(c), and r = 20 (right side), f = 42(d), f = 92(e) and f = 162(f).

DFT calculations, the brush-coated nanoparticles are mod-eled as spherical shells of thickness ! uniformly composedof Lennard-Jones particles, on which the polymer chains aregrafted uniformly, i.e., without specifying the individual graft-ing points. At short separations between the nanoparticlesurfaces (up to a few monomer diameters), where the contri-bution from Vext(r) to the PMF is significant and model depen-dent, it does not make sense to compare directly the results forPMF. Accordingly, in what follows, we focus on comparisonbetween MD and DFT results at intermediate and long sepa-

rations between the nanoparticles. We emphasize here that theexternal potential does enter DFT calculations but its preciseform is different from the one used in the simulations.

Turning next to the calculation of the intrinsic Helmholtzfree energy, we note that it is customary to divide F into idealand excess components. In a recent DFT study of interactionsbetween dendrimers,44 it was argued that the contribution ofFid to the PMF is negligible compared to Fex and therefore thePMF can be approximated as follows:

W (r) $ Fex[&(r), r] ! Fex[&(r),#]. (6)

FIG. 8. Same as Fig. 7, but for N = 80.

214902-7 Interactions between polymer brushes J. Chem. Phys. 135, 214902 (2011)

20 30 40 50 60 70r

1.2

1.3

1.4

1.5

1.6R

e/N"

N=20N=40N=60N=80

20 30 40 50 60 70r

0.48

0.5

0.52

0.54

Rg/N

"

FIG. 9. Rescaled radius of gyration (R2g)1/2/N( (bottom) and end-to-end

distance (R2e )1/2/N( (top) plotted for f = 42 as a function of the separation

between the particles.

In our study of interactions between sterically stabilizednanoparticles,35 we have assessed the accuracy of the aboveapproximation by comparing the corresponding DFT resultsfor the PMF with the simulation data, and found Eq. (6) tobe quite accurate for relatively short grafted chains (up toN = 20) and moderate grafting densities.

In the present work, we are studying considerably longerchains (up to N = 80) and higher grafting densities. Hence,the accuracy of the DFT approach needs to be reassessed.As discussed above, in our earlier work the polymer seg-ment density profiles were obtained within the DFT frame-work iteratively and self-consistently, i.e., by minimizing thegrand potential.35 For longer chains and more densely graftedbrushes considered in the present study, we found that this

1.4

1.6

1.8

2

Re/N

"

N=20N=40N=60N=80

0.5

0.55

0.6

0.65

Rg/N

"

20 30 40 50 60 70 80 90r

0.3

0.35

0.4

0.45

Re/N

FIG. 10. Rescaled radius of gyration (R2g)1/2/N( (middle) and end-to-end

distance (R2e )1/2/N( (top) plotted for f = 162 as a function of the separation

between the particles. For comparison in the same figure, we show also therescaled end-to-end distance (R2

e )1/2/N (bottom) plotted for f = 162 as afunction of the separation between the particle.

iterative calculation of PMF is plagued by convergence prob-lems and is numerically demanding. Therefore, we take analternative approach by substituting the polymer segmentdensity profiles as given by MD simulations directly intoEq. (6); a similar hybrid MC-DFT method has been recentlyemployed for studying multidimensional entropic forces.45

In order to compute the excess Helmholtz free energy, wefollow our earlier DFT studies23, 35, 43, 46 and employ a simpleweighted density approximation due to Tarazona,47

Fex[&(r)] =#

dr&(r)fex (&̄(r)), (7)

where fex(&) is the excess free energy density per site of thesystem with site density & and &̄(r) is the weighted densitygiven by47

&̄(r) = 34)! 3

#dr&&(r&)*(|r ! r&|), (8)

where *(r) is the Heaviside step function.The excess free energy density per site we write as a sum

of two terms: the first term accounts for the excess free en-ergy of the homogeneous fluid of unconnected hard spheresas given by the Carnahan-Starling equation of state,48 andthe second term accounts for the connectivity of chain of Nmonomers and is given by the Wertheim’s expression whichwas obtained on the basis of the first-order thermodynamicperturbation theory,49

fex(&) = 4+ ! 3+2

(1 ! +)2!

$1 ! 1

N

%ln

1 ! +/2(1 ! +)3

(9)

where + = )! 3&/6 is the packing fraction. We note herethat more sophisticated free energy functionals are cur-rently available, such as, based on the Fundamental MeasureTheory.45, 50, 51 Their accuracy will be investigated in our fu-ture work.

In Fig. 11, we show simulation and theoretical results forthe PMF between the two brush-coated spherical nanoparti-cles where the system parameters are as follows: Rc = 7,f = 42, and N = 20. Symbols give the MD simulation re-sults, while the blue line presents the DFT result obtainedwith MD density profiles as input. One sees that at interme-diate and long separations (r > 2Rc + 6!LJ), the theory is ingood agreement with the simulation, while at short separa-tions, DFT underestimates the PMF. As discussed above, thisdiscrepancy most likely stems from the differences in the mi-croscopic models of the nanoparticles employed in the twoapproaches.

In Fig. 12, we show simulation and theoretical results forthe PMF between the two brush-coated spherical nanoparti-cles where the system parameters are as follows: Rc = 8.35,f = 162, and N = 80. Symbols give the MD simulation re-sults, while the blue line presents the DFT result obtained withMD density profiles as input. Once again, theory underesti-mates the PMF at short separations, as expected due to differ-ences in microscopic models. However, in this case of largerand denser brushes, one also observes discrepancies at inter-mediate separations, where the theory on the Tarazona leveloverestimates the PMF somewhat. It is also clear that for thislonger value of N, the DFT prediction is not perfectly accurate

214902-8 Lo Verso et al. J. Chem. Phys. 135, 214902 (2011)

16 20 24 28r

0

10

20

30

40

50

60

70

W(r

)

MDMD-DFT

FIG. 11. Simulation and theoretical results for the PMF between the twobrush-coated spherical nanoparticles; the system parameters are as follows:Rc = 7, f = 42, and N = 20. Symbols: simulation results; line: DFT resultobtained with MD density profiles as input. Recall that DFT cannot describeW(r) for distances r close to the value where the two spheres touch (r = 2Rc= 14).

also for large r. As noted in our earlier work on single spher-ical polymer brushes,23 DFT does not yield the star polymer-like scaling for N * #. This discrepancy is expected to affectthe PMF for large N and large r as well. We plan to study thisproblem in future work.

As discussed in Ref. 30, the attempts to obtain the effec-tive brush-brush interaction has been restricted so far mostlyto specific limiting cases namely star polymers, i.e., high cur-vature regime, and almost flat brushes, low curvature regime.In Ref. 30, the authors computed by means of off-latticeMonte Carlo simulations, the force acting between sphericalsurfaces, grafted by polymer chains. Each polymer is repre-

20 40 60r

0

500

1000

1500

2000

W(r

)

MDMD-DFT

FIG. 12. Simulation and theoretical results for the PMF between the twobrush-coated spherical nanoparticles; the system parameters are as follows:Rc = 8.35, f = 162, and N = 80. Symbols: simulation results; line: DFTresult obtained with MD density profiles as input. Recall that DFT cannotdescribe W(r) for distances r close to the value where the two spheres touch(r = 2Rc = 16.7).

sented by the pearl-necklace model. The authors found thattheir force profile can be divided into two regimes: a WPregime, at short separation between brushes described by theWitten and Pincus (WP) theory16 and a Flory regime, wherean extension of the Flory theory30 for dilute polymer solutionsreproduce the results at larger separation. Here we follow thestudy presented by Cerda et al.30 and we compare our datawith the Flory theory. Note that we do not have a WP regime,for several reasons. First of all in the region where our curvesbecome independent from the number of monomers per chain,the core is strongly repulsive. Additionally, the ratio betweensize of the core and number of monomers per chain is in gen-eral bigger than in the case of Ref. 16.

According to the generalization of the Flory theory de-scribed in Ref. 30, the potential of mean forces between spher-ical brushes at larger core-core separation r has a Gaussianprofile,

WG(r) = c · exp(!d · (r ! e)2). (10)

Here the c and d coefficients depend on the length ofthe chains and on the number of grafting sites. In Fig. 13,we plotted the potential of mean force obtained by simula-tions and the fitting curves of our data points with the formulaEq. (10). The system parameters are as follows: Rc = 7,f = 42, and N = 20, 40, and 80 (top) and Rc = 8.35, f = 162,and N = 20, 40, and 80 (bottom). Here we shifted the centerof the Gaussian curves by a constant e (proportional to Rc).The height and the width of the curve’s peak are comparablewith the coefficients obtained by Cerda et al.30 The differ-ences between our results in Fig. 13 and the correspondingresults of Ref. 30 can be attributed to the use of somewhatdifferent models.

VI. FINAL DISCUSSION AND SOME CONCLUSIONS

In this paper, we have presented results from moleculardynamics simulation for the problem of two interactingspherical polymer brushes, varying the distance betweentheir centers, for several representative choices of graftingdensity and chain length of the grafted polymers. The latterwere described by a coarse-grained bead-spring model, andgood solvent conditions were assumed throughout. Our studycan be viewed as a reference case for planned extensions tovariable solvent conditions, and to the more difficult case thatthe two brush-coated spherical nanoparticles are embeddedin a polymer melt. As it is well-known, embedding nanopar-ticles into polymer melts is of major interest for improvingapplication properties of composite materials. The graftingof the chains to the surface of the nanoparticles is done withthe intention to prevent that in such materials such nanopar-ticles cluster together and phase separate from the polymermelt, forming large aggregates, which is undesirable. Fora discussion of this tendency of the particles to aggregate,understanding of the effective forces between them isindispensable.

Of course, there are many other instances where relatedproblems occur. For example, if nanoparticles are used asdrug carriers in the blood, also their aggregation needs to beavoided. Related problems also occur in micellar solutions

214902-9 Interactions between polymer brushes J. Chem. Phys. 135, 214902 (2011)

15 20 25 30 35 40r

1

10

100W

(r)

N=20N=40N=80

17 21 25 29 33 37 41 45 49 53 57r

10

100

1000

W(r

)

N=20N=40N=80

f=42

f=162

FIG. 13. Simulation results for the PMF between the two brush-coatedspherical nanoparticles; the system parameters are as follows: Rc = 7,f = 42, and N = 20, 40, and 80 (top) and Rc = 8.35, f = 162, andN = 20, 40, and 80 (bottom). The full lines correspond to the fit ofthe MD data according to formula described in Eq. (10). The fit pa-rameters for f = 42 are c = 41.925, d = 0.017, and e = 11.614(N = 20), c = 73.143, d = 0.006, e = 9.255 (N = 40), and c = 118.957,d = 0.002, e = 3.86 (N = 80). The fit parameters for f = 162 are c = 795.38,d = 0.017, and e = 11.614 (N = 20), c = 1134.75, d = 0.010, and e = 10.277(N = 40), and c = 1249.11, d = 0.001, and e = 2.381 (N = 80).

(if the polymer forming the core of a block copolymer mi-celle due to poor solvent conditions is collapsed into a densestructure, while the polymer in the corona is under good sol-vent conditions, the situation is very similar to the presentproblem).

In our study, we have found that for weak interpenetrationof the two polymer brushes, the potential of mean force canbe described (Fig. 13) by a simple formula deriving from aFlory-like theory (Eq. (10)), if the phenomenological param-eters of that theory are treated as adjustable constants. How-ever, we did not see an obvious route for a prior prediction ofthese constants. In contrast, the DFT can make a predictionfor both the radial monomer density profiles and the potentialof mean force, if one restricts attention to very short chains(Fig. 11). However, for long chains, the theory was foundto suffer from numerical problems, and a way out neededto use simulated profiles as an input to the DFT calculation(Fig. 12). Thus, at the present stage, the predictive power ofDFT for this problem is rather limited, and clearly more re-search is desirable to resolve the situation.

It is also obvious from our simulations that for smallerseparation between the two spherical brushes, the PMF risesmore steeply than either the Flory theory or the DFT wouldpredict. We find that in this regime where ultimately thetwo nanoparticles are close to direct contact, the chainsare “squeezed out” in the directions perpendicular to the

z-direction (which is the axis connecting the centers of thetwo nanoparticles). This is not only obvious from the directinspection of chain conformations in simulation snapshots,but is also evident from the monomer density distributions(Figs. 5–9). These chains that are squeezed out become some-what more stretched than would be normal for chains ofan isolated spherical polymer brush, and thus lead to anenhancement even of the average chain linear dimensions(Figs. 9 and 10) at small separations.

Of course, a direct comparison of our results to corre-sponding experiments would be very desirable. We note thatmeasurements of forces between spherical polymer brushescan be done by holding such particles by optical tweezers.52, 53

Such experiments have been carried out for particles with2 µm diameter, on which DNA chains were grafted, andforces were measured varying the separation between the par-ticles (on the scale of about 10 nm to 300 nm, with a resolutionof ±3 nm).

Unfortunately, a quantitative comparison to such exper-iments is not possible for us for several reasons: (1) Esti-mating that a Lennard-Jones diameter of our course-grainedmodel corresponds to a distance of 1 nm or smaller, the dis-tances available in our simulation are on the lower end of theexperimental range. (2) The experimental system is a poly-electrolyte brush, while we consider a neutral brush. Onlythe strongly salted brushes are similar to the case studiedhere (for this case in the limit where the substrate surfacesare essentially flat, the brush height is expected to scale ash $ N! 1/3c

!1/3s , cs being the salt concentration, similar to

ordinary neutral brushes for which h $ N! 1/3 in this limit).(3) The focus of our work is on spherical polymer brushesfor which the brush height and nanoparticle radius Rc arecomparable, while the experiments known to us use Rc +h. Nevertheless, the existence of such experiments is veryencouraging, and thus we hope that our work is not onlystimulating for further theoretical developments, but also forthe refinement of experimental techniques making studiesof spherical polymer brushes in the considered size rangefeasible.

ACKNOWLEDGMENTS

S.A.E. acknowledges partial support from the Alexandervon Humboldt foundation and from the Deutsche Forschungs-gemeinschaft (DFG) under Grant No. SFB625/A3. L.Y. ac-knowledges partial support from the Deutsche Forschungs-gemeinschaft (DFG) under Grant No. PA473/8. We are verygrateful to Martin Oettel for his advice on the density func-tional approach. A grant of computer time at the JUROPA ma-chine at the John von Neumann Center for Computing (NICJülich) is gratefully acknowledged.

1D. H. Napper, Polymeric Stabilization of Colloidal Dispersions (Academic,London, 1983).

2Polymer Adsorption and Dispersion Stability, ACS Symposium Series,Vol. 240, edited by E. Goddard and B. Vincent (ACS, Washington, 1984).

3K. Ohno, T. Morinaga, S. Takeno, Y. Tsujii, and T. Fukuda, Macro-molecules 39, 1245 (2006).

4K. Ohno, T. Morinaga, S. Takeno, Y. Tsujii, and T. Fukuda, Macro-molecules 40, 9143 (2007).

214902-10 Lo Verso et al. J. Chem. Phys. 135, 214902 (2011)

5C. N. Likos, H. Löwen, M. Watzlawek, B. Abbas, O. Jucknischke, J.Allgaier, and D. Richter, Phys. Rev. Lett. 80, 4450 (1998).

6C. N. Likos and H. M. Harreis, Condens. Matter Phys. 5, 173 (2002).7F. Lo Verso, C. N. Likos, and L. Reatto, Prog. Colloid Polym. Sci. 133, 78(2006).

8C. N. Likos, Soft Matter 2, 478 (2006).9I. Borukhov and L. Leibler, Macromolecules 35, 5171 (2002).

10M. K. Corbierre, N. S. Camron, M. Sutton, K. Laaziri, and R. B. Lennox,Langmuir 21, 6063 (2005).

11P. Akcora, H. Liu, S. K. Kumar, J. Moll, Y. Li, C. Benicewicz,L. S. Schadler, D. Acehan, A. Z. Panagiotopoulos, V. Pryamitsyn, V.Ganesan, J. Ilavsky, P. Thiyagarajan, R. H. Colby, and J. F. Douglas, NatureMater. 8, 354 (2009).

12M. Ballauf and Y. Lu, Polymer 48, 1815 (2007).13F. Lo Verso and C. N. Likos, Polymer 49, 1425 (2008).14M. E. McEwan, S. A. Egorov, J. Ilavsky, D. L. Green, and Y. Yang, Soft

Matter 7, 2725 (2011).15S. Foerster, E. Wenz, and E. Lindner, Phys. Rev. Lett. 77, 95 (1996).16T. A. Witten and P. A. Pincus, Macromolecules 19, 2509 (1986).17R. C. Ball, J. F. Morgo, S. T. Milner, and T. A. Witten, Macromolecules 24,

693 (1991).18E. B. Zhulina, O. Borisov, V. Pryamitsyn, and T. M. Birshtein, Macro-

molecules 24, 140 (1991).19N. Dan and M. Tirrell, Macromolecules 25, 2890 (1992).20C. M. Wijmans and E. B. Zhulina, Macromolecules 26, 7214 (1993).21R. Toral and A. Chakrabarti, Phys. Rev. E 47, 4240 (1993).22E. Lindberg and C. Elvigson, J. Chem. Phys. 114, 6343 (2001).23F. Lo Verso, S. A. Egorov, A. Milchev, and K. Binder, J. Chem. Phys. 133,

184901 (2010).24T. V. M. Ndoro, E. Voyiatzis, C. Ghanbari, D. N. Theodorou, M. C. Böhm,

and F. Müller-Plathe, Macromolecules 44, 2316 (2011).25K. T. Marla and J. C. Meredith, J. Chem. Theory Simul. 2, 1624 (2006).26D. A. Savin, J. Pyun, G. D. Patterson, T. Kowalewski, and K.

Matyjaszewski, J. Polym. Sci., Part B: Polym. Phys. 40, 2667 (2002).27Y. Mei and M. Ballauff, Eur. Phys. J. E 16, 341 (2005).28D. Dukes, Y. Li, S. Lewis, B. Benicewicz, L. Schadler, and S. K. Kumar,

Macromolecules 43, 1564 (2010).

29C. M. Wijmans, F. A. M. Leermakers, and G. J. Fleer, Langmuir 10, 4514(1994).

30J. J. Cerda, T. Sintes, and R. Toral, Macromolecules 36, 1407 (2003).31J. U. Kim and M. W. Matsen, Macromolecules 41, 4435 (2008).32V. Pryamitsyn, V. Ganesan, A. Z. Panagiotopoulos, J. Liu, and S. K. Kumar,

J. Chem. Phys. 131, 221102 (2009).33Monte Carlo and Molecular Dynamics Simulations in Polymer Science,

edited by K. Binder (Oxford University Press, New York, 1995).34S. A. Egorov, J. Chem. Phys. 129, 064901 (2008).35A. Striolo and S. A. Egorov, J. Chem. Phys. 126, 014902 (2007).36G. S. Grest and M. Murat, Monte Carlo and Molecular Dynamics Simu-

lations in Poylmer Science, edited by K. Binder (Oxford University Press,New York, 1995), pp. 476–578.

37G. S. Grest and K. Kremer, Phys. Rev. A 32, 3628 (1986).38K. Kremer and G. S. Grest, J. Chem. Phys. 92, 5057 (1990).39H. J. C. Berendsen, D. van der Spoel, and R. van Drunen, Comput. Phys.

Commun. 91, 43 (1995).40D. van der Spoel, E. Lindahl, B. Hess, G. Groenhof, A. E. Mark, and H. J.

C. Berendsen, Comput. Chem. 26, 1701 (2005).41B. Hess, C. Kutzner, D. van der Spoel, and E. Lindahl, J. Chem. Theory

Comput. 4, 435 (2008).42A. Jusufi, M. Watzlawek, and H. Löwen, Macromolecules 32, 4470 (1999).43N. Patel and S. A. Egorov, J. Chem. Phys. 121, 4987 (2004).44I. O. Götze, H. M. Harreis, and C. N. Likos, J. Chem. Phys. 120, 7761

(2004).45Z. Jin and J. Z. Wu, J. Phys. Chem. B 115, 1450 (2011).46N. Patel and S. A. Egorov, J. Chem. Phys. 123, 144916 (2005).47P. Tarazona, Mol. Phys. 52, 81 (1984).48N. F. Carnahan and K. E. Starling, J. Chem. Phys. 51, 635 (1969).49M. S. Wertheim, J. Chem. Phys. 87, 7323 (1987).50Y. X. Yu and J. Z. Wu, J. Chem. Phys. 116, 7094 (2002).51V. Botan, F. Pesth, T. Schilling, and M. Oettel, Phys. Rev. E 79, 061402

(2009).52K. Kegler, M. Salomo, and F. Kremer, Phys. Rev. Lett. 98(5), 8304

(2007).53K. Kegler, M. Konieczny, G. D. Espinosa, C. Gutsche, M. Salomo,

F. Kremer, and C. N. Likos, Phys. Rev. Lett. 100, 118302 (2002).