The transition from highly to fully stretched polymer brushes in good solvent
TThe transition from highly to fully stretched polymer brushes in goodsolvent
Ivan Coluzza and Jean-Pierre Hansen
Department of Chemistry, University of Cambridge,Lensfield Road, Cambridge CB2 1EW, UK. (Dated: November 9, 2018)
Abstract
The stretching of brushes of long polymers grafted to a planar surface is investigated by Monte Carlo sim-ulations in the limit of very high grafting densities, as achieved in recent experiments. The monomer densityprofiles are shown to deviate considerably from the parabolic limiting form predicted by self-consistent fieldtheory. A rapid transition is observed from parabolic to fully stretched polymers, characterized by a dra-matic change in the end-monomer height distribution and by a clear cross-over in the slope of the brushheight versus scaled grafting density.
PACS numbers: 68.47.Mn, 68.47.Pe, 61.25.Hq, 82.35.Lr a r X i v : . [ c ond - m a t . s o f t ] S e p here have been numerous experimental [1], theoretical [2, 3, 4] and simulation [5, 6] studiesof polymer “brushes” grafted to substrates, confirming the Alexander scaling of the brush heightwith grafting density [7]. They show that the monomer density profiles go over to a universalparabolic profile as a function of the distance from the surface, for sufficiently strong stretch-ing, as predicted by self-consistent field theory (SCF) [3, 4, 8]. Recent experimental techniquesbased on surface-initiated polymerization [9, 10] allow higher grafting densities than previouslyachieved. In this letter we analyse the break-down of SCF theory for long chains and high graftingdensities by advanced Monte Carlo simulations of polymers in good solvent. Upon increasingthe grafting density, we observe a transition from the parabolic regime to fully stretched brusheswhich correlates with a dramatic change in the end-monomer density profile. The exposure of theend monomers of very dense brushes opens new possibilities for the development of chemicallyactive soft surfaces.Consider a layer of N identical flexible polymers of M monomers and contour length L =( M − ) b (where b is the segment length), in good solvent. The first monomer of each chain ischemically grafted to a grafting site on a planar substrate of area A; the dimensionless graftingdensity is defined as σ = Nb / A . For convenience b will henceforth be the unit length. The keylength scales are the mean distance between grafting sites, d ∼ σ − / , the radius of gyration of anisolated polymer, R g ∼ L ν (where the Flory exponent ν = .
588 under good solvent conditions)and the brush height h which, according to the Alexander-de Gennes mean field argument [2, 7],scales like h ∼ L σ − ν ν ∼ L σ . ∼ Ld − . (1)i.e. the chains stretch upon increasing the grafting density due to their mutual repulsion. Thisscaling has been confirmed by neutron scattering diffraction experiments [1]. An important di-mensionless parameter is σ g = σ R g ∼ R g / d . When σ g <
1, polymers grafted to neighbouringsites do not overlap and the height h of quasi-independent coils scales like h ∼ R g ∼ L ν (“mush-room regime”). When σ g >
1, neighbouring coils interact and begin to stretch, i.e. h ∼ L . Thisregime has been widely studied within SCF theory [2, 3, 8] and by MC simulations of variousmodels, involving in general rather short chains ( L < P ( z ) , where z is the vertical distancefrom the substrate, normalized such that R L P ( z ) dz = N . Within SCF, the dimensionless controlparameter is β = ( h / R go ) (where R go ∼ L / is the radius of gyration of the non-interacting, Gaus-sian chain), and in the strong stretching regime β >>
1, SCF reduces to its “classical” limit, which2 P [ z ]/ σ / z/(L σ ) σ =0.0020.0040.0200.0400.0800.1200.2500.500 P [ z ] z/L Figure 1: (Color on-line)Scaled monomer distribution function versus scaled altitude for brushes of SAWpolymers of length L =
50, and several grafting densities 0 . ≤ σ ≤ .
5. The two lowest σ correspond tothe “mushroom” regime, while deviations from SCF scaling occur for σ ≥ .
25. The inset shows the endmonomer altitude profiles versus z / L for the same values of σ predicts a universal parabolic curve when P ( z ) σ ( − ν ) / ν is plotted versus z / (cid:16) L σ ( − ν ) / ν ) (cid:17) [3, 4].This scaling is confirmed by MC simulations, with deviations from the parabolic shape at smalland large z in the case of moderate stretching [5, 6, 8, 11, 12]. Some deviations from a universalmaster curve, due to a flattening of the profiles at intermediate heights, have been observed in sim-ulations at the highest grafting densities σ g explored so far [8, 13, 14]. Meanwhile experimentalgrafting techniques based on surface-initiated polymerization [9] have achieved extremely highgrafting densities, up to nearly one site per nm [9, 10].To explore the structure of polymer brushes in the regime of such high grafting densities, cor-responding to the regime of σ g >>
1, which may be achieved by increasing the grafting density σ or considering very long polymers, we have carried out extensive MC simulations of the simplest3attice model of flexible polymers in good solvent, namely self (and mutually) avoiding walks(SAW) on a cubic lattice. The grafting sites are on a square lattice of identical spacing b . We used“annealed” grafting conditions whereby the first monomer of each chain can jump between near-est neighbor grafting sites in order to speed up equilibration [12]. Three types of MC moves wereused: discrete translations of entire chains on the square lattice in the x − y grafting plane; chainre-growth moves based on the Configurational Bias Monte Carlo (CBMC) algorithm [15] ; seg-ment re-growth moves based on CBMC with fixed end-points [15]. This combination of moves isexpected to guarantee good equilibration of the simulated brush up to very high grafting densities(corresponding to moderately high monomer volume fractions in the brush). The initial conditionswere generated with fully stretched polymer conformations attached to randomly chosen graftingcentres on the square lattice. The grafting centres were allowed to diffuse on the square lattice(annealed sampling), which is equivalent to a random sampling of fixed grafting points, followedby a statistical average over the randomly chosen initial configuration [12]. After initial equili-bration, statistical averages of the monomer and end-monomer profiles were taken over typically2 · MC moves.We have systematically computed the monomer and end-monomer distribution functions P ( z ) and P M ( z ) over a wide range of σ , and for lengths L =
50, 200, 400 and 800. Scaled profiles forshort chains ( L =
50) and several σ are shown in Fig. 1. The universal scaling regime is reachedfor σ > .
04 ( σ g > . σ = .
25 ( σ g (cid:39) σ (cid:39) .
04 ( σ g < . P M ( z ) moves to higher z as σ increases, as one might expect. The probability of the end monomer“returning” towards the grafting surface decreases with increasing σ [8]. Fig. 2 shows similarresults, but for longer chains ( L = σ < . σ g < . . ≤ σ ≤ .
05 (0 . ≤ σ g ≤ σ > .
05, the profiles deviate increasingly from the scaling prediction, and for σ > . σ g > P ( z ) flattens and stretches considerably. Concomitantly the end-monomer profiles P M ( z ) move furtherand further towards their upper limit, z = L . These trends are considerably enhanced in the case L = σ < .
02 ( σ g <
10) whilefor higher σ , P ( z ) flattens and stretches towards the rectangular profile ( P ( z ) = z < L ; P ( z ) = z > L ) which is the exact limit for the lattice model, when σ = L because of the excluded volume constraint). This strong stretching behavior4orrelates well with the behavior of P M ( z ) , which undergoes a dramatic transition around σ = . σ g = z = L (in the limit σ → P M ( z ) → δ ( z − L ) ).The mean height h of the brush is the first moment of the probability density P ( z ) ; the MCresults for the ratio h / R g are plotted in Fig. 4 vs σ ( − ν ) / ν g for L = σ g (cid:39)
10. Thelinearity agrees with the prediction of Alexander-de Gennes scaling [1], but the change in slopesuggests a cross-over from the SCF regime to the regime dominated by strong excluded volumecorrelations which enhance stretching of the chains. Although the slope ( (cid:39) .
2) of the low σ g regime is close to that of the earlier estimates obtained for different polymer models [5, 6], aclear-cut cross-over to the strong stretching regime has not been reported earlier. Indications inthat direction are contained in the early work of Grest et al. [16, 17]. By using a continuousmodel of polymer brushes, they observed deviations from the SCF scaling regime for graftingdensities of 0.15 and chains 200 monomers long, resulting in enhanced stretching of the brush.However there was no clear evidence of a sharp transition between the SCF and the fully stretchedscaling regimes, similar to to that shown in the end-monomer distribution function (Fig. 3), or inthe scaling plot (Fig. 4) of the mean brush height. This was probably due to insufficient polymerlength and to the presence of an attractive potential between the monomers.Let us know consider the “roughness” of the brush/solvent interface may be characterized bythe relative fluctuation f M of the end-monomer altitude, (cid:104)(cid:10) z M (cid:11) − (cid:104) z M (cid:105) (cid:105) / / (cid:104) z M (cid:105) which is readilyderived from P M ( z ) . The MC results for f M are plotted in Fig. 4 versus σ for L=50 and 800. Thedifference is dramatic: while f M saturates rapidly at a value close to 0.4 for the shortest chains, thefluctuation is seen to decrease steadily for the longer chains, pointing to a relatively smooth upperinterface of the brush. It is worthwhile to note that the corresponding fluctuation of the center ofmass (CM) of the chains follows very similar curves, slightly below those for f M .Sampling of stretched polymer conformations at high grafting densities becomes increasinglydifficult for long polymers, such that σ g > . A way out is to switch to a multi-blob represen-tation of the grafted chains, as recently proposed for homopolymers and block copolymers in thebulk [18, 19]. Within this coarse-graining procedure each chain is divided into n blobs of length l = L / n and radius of gyration r g ∼ l ν . The average blob density within the volume of the brush ofheight h is ρ b = Nn / ( Ah ) ∼ NLh ( − ν ) / ν , while the overlap density of blobs is ρ ∗ b ∼ / r g ∼ / l ν so that ρ b / ρ ∗ b ∼ ( L / n σ / ν ) ν − . The minimum number of blobs required to ensure that blobs5 P [ z ]/ σ / z/(L σ ) σ =0.0020.0040.0160.0320.0480.0960.1600.2000.250 P [ z ] z/L Figure 2: (Color on-line)Same as in Fig. 1, but for L =
200 and 0 . ≤ σ ≤ .
25. Increasing deviationsfrom SCF scaling regime are observed for σ > . do not, on average, overlap is hence n ∼ σ / ν g . Effective interactions between the CM’s of thebonded and non-bonded blobs and between a blob and the substrate, are determined by averag-ing over monomer degrees of freedom for given CM-CM distances. The coarse-grained multi-blob model requires effective pair potentials v ( r ) and φ ( r ) between the CM’s of non-bonded andbonded blobs on the same or different (in the case of v ( r ) ) chains, as well as an effective wall-blobpotential ψ ( z ) , and an effective tethering potential φ ( r ) between the CM of the first blob and thegrafting centre. All these effective interactions are determined by inverting MC results for the pairdistribution functions between the CM’s of a single grafted polymer made up of a small numberof blobs [18, 19, 20], and are assumed to be transferable to finite grafting density conditions, aslong as the system is in the weak overlap ( ρ b < ρ ∗ b ) regime. The form of the repulsive potential6 P [ z ] z/L σ =0.0020.0040.0160.0320.0480.0960.1600.2000.250 P [ z ]/ σ / z/(L σ ) Figure 3: (Color on-line)End-monomer height distribution function of grafted SAW polymers of length L =
800 versus z / L for grafting densities 0 . ≤ σ ≤ .
25. Note the rather sharp transition to full stretchingwhen σ (cid:39) .
05, which contrasts with the gradual stretching observed for shorter chains in Fig. 1-2. Inset:scaled monomer distribution function versus scaled altitude for the same values of σ . Deviations from theSCF scaling regime set in for σ (cid:38) .
01, and the profiles become nearly rectangular (full stretching) at thehighest grafting densities. v ( r ) is Gaussian to a good approximation v ( r ) k B T (cid:39) Ae − α ( r / r g ) (2)with A (cid:39) . α (cid:39) .
8. The tethering potential φ ( r ) is well represented by the sum of v ( r ) and a harmonic spring potential, while φ ( r ) is a similar harmonic spring potential, and the wall-blob potential ψ ( z ) is an exponential repulsion of range (cid:39) r g [20]. MC simulations of the multi-blob representation of grafted polymers sample single blob and total chain displacements with astandard Metropolis acceptance criterion, which are sufficient because of the softness of the above7 h / R g σ g(1- ν )/(2 ν ) L=50200400800 0.2 0.4 0.6 0.8 0 1 2 3 4 5 6 f M σ g(1- ν )/(2 ν ) L=50800
Figure 4: (Color on-line)Plot of the mean height h = < z M > of a brush of SAW polymers (divided by R g ), versus σ ( − ν ) / ν g for L = σ g and high σ g data indicate a cross-over from the SCF regime to the fully stretched regime around σ g (cid:39) (cid:39) .
2) of the low σ g regime (dashed line) is close to earlier estimates obtained for differentpolymer models [5, 6]. Inset: relative fluctuation f M = (cid:104)(cid:10) z M (cid:11) − (cid:104) z M (cid:105) (cid:105) / / (cid:104) z M (cid:105) of the brush height versus σ ( − ν ) / ν g for L =
50 and L = effective potentials. If n is chosen such as to satisfy the weak overlap condition above, the effectiveinteractions can be extracted from low density MC simulations. This coarse-graining leads to anenhancement of sampling efficiency by several orders of magnitude, because of the reduction of thetotal number of degrees of freedom by a factor l , and the softness of the effective interactions. MCresults for the CM probability densities, as determined for coarse-grained polymer chains of n = P [ z ]/ σ b2 / z/(n σ b1/3 ) σ b =0.6170.7711.2331.850 0 0.002 0 20 40 60 P ( z / r g ) z/r g Figure 5: (Color on-line)Scaled monomer distribution function versus scaled altitude for brushes of multi-blob polymers of length n =
20 blobs, and 1 . ≥ σ b ≥ . σ b = σ r g . Increasing deviations fromSCF scaling regime are observed for σ b > .
2. Inset: comparison between the scaled monomer distributionfunction of a SAW polymer of length L =
800 at a grafting density σ = .
008 (triangles) and a multi-blobpolymer under the corresponding physical condition of σ b = .
617 (continuous curve). The plot shows verygood agreement between the two representations. agreement with the full monomer-level MC results of Fig. 1-4.In summary, we have shown by MC simulations of long grafted polymer chains that the brushprofile and height undergo a rapid transition when the grafting density σ g = σ R g >
10. Beyondthat value, the probability density P ( z ) switches rapidly from universal, quasi-parabolic profileto a quasi-rectangular shape, while the fluctuations of the end-monomer altitude drops sharply,pointing to a rather flat brush/solvent interface. These changes are induced by correlations betweenmonomers on neighbouring chains, which are neglected in SCF theory. We conjecture that thesecorrelations will lead to an effective, entropic repulsion between stretched polymers, similar to the9elfrich interaction [21] between stacked membranes, which in turn may induce an ordering of thegrafting centres on the substrate if these centres are mobile, as for copolymers anchored at a liquid-liquid interface, just as the Helfrich repulsion leads to a lamellar phase of stacked membranes. Ourresults point to new possibilities in experimental realization of chemically active surfaces, becausethe height distribution of the end monomer controls the chemical activity of the brush. Moreoverthe activity of such soft surfaces could be controlled by the length of the grafted polymers. Acknowledgments
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