Grafted Polymer Chains Interacting with Substrates: Computer Simulations and Scaling

Feature Article

Grafted Polymer Chains Interacting withSubstrates: Computer Simulations and Scaling

Radu Descas,* Jens-Uwe Sommer, Alexander Blumen

We review scaling methods and computer simulations used in the study of the static anddynamic properties of polymer chains tethered to adsorbing surfaces under good solventconditions. By varying both the grafting density and the monomer/surface interactions avariety of phases can form. In particular, for attrac-tive interactions between the chains and the sur-face the classical mushroom-brush transitionknown for repulsive substrates splits up into anoverlap transition and a saturation transitionwhich enclose a region of semidilute surfacestates. At high grafting densities oversaturationeffects and a transition to a brush state can occur.We emphasize the role of the critical adsorptionparameters for a correct description and under-standing of such polymer adsorption phenomena.

Introduction

Polymer chains grafted onto solid surfaces are funda-

mental in many applications in colloid and surface science,

as well as in biology.[1–4] In this paper, we focus on grafted

polymer chains whose monomers get adsorbed to the

surface. In a seminal work, using Monte Carlo simulations,

Eisenriegler et al.[5] have studied the adsorption behavior

of single chains grafted to the surface. It turned out that

polymer adsorption is related to a surface phase transition

and that in order to describe the situation correctly novel

surface exponents must be taken into account. Using these

results scaling arguments have been applied to explore the

R. Descas, A. BlumenTheoretische Polymerphysik, Universitat Freiburg,Hermann-Herder-Strasse 3, D-79104 Freiburg, GermanyFax: (þ49) 761 203 5906;E-mail: [email protected]. SommerLeibniz-Institut fur Polymerforschung Dresden,Hohe Strasse 6, D-01069 Dresden, Germany

Macromol. Theory Simul. 2008, 17, 429–453

� 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

behavior of polymer chains on adsorbing surfaces.[6] In

particular, extensions of the classical studies have focused

on systems formed from many chains, in which the

excluded volume interactions lead to a whole series of

features, such as the saturation of the adsorption on the

surface and the formation of blob-type structures.[7,8]

Some of these features also appear in polymer brushes

which form when the surface grafting density gets high.

However, the situation there is different due to the lack of

attractive monomer/surface interactions.[9,10]

A more formal, analytical approach to the problem of

polymer adsorption uses field theoretic methods, in

particular renormalization group techniques. Now, the

presence of the surface converts the infinite space

surrounding polymers in solution to a problem in semi-

infinite space: it turns out (as in the case of restricted

geometries) that the field theoretic methods work less well

in the semi-infinite than in the infinite geometry. For

instance, for very dilute systems one obtains for the cross-

over surface exponent very different values when different

methods to evaluate it are used.[11,12] Even more complex

is the problem at finite polymer densities; here most of the

DOI: 10.1002/mats.200800046 429

R. Descas, J.-U. Sommer, A. Blumen

RaduDescasstudied physics inCluj-Napoca,Romania,and obtained his Ph.D. in 2006 in the TheoreticalPolymer Physics group of A. Blumen in Freiburg. Inhis work, he has investigated the adsorption of poly-mers on surfaces using Monte-Carlo simulations andscaling analysis. During his Ph.D. studies he spentseveral months atthe ‘‘Institutde Chimie des Surfaceset Interfaces’’ in Mulhouse with J.-U. Sommer. Since2006 he is a postdoctoral researcher in the group of A.Blumen and he is collaborating with J.-U. Sommer atthe Leibniz-Institute of Polymer Research in Dresden.His research interest is the theory of polymeric sys-tems. Using both analytical and simulation methodshe focuses on adsorption of polymers on surfaces.

Jens-Uwe Sommer studied physics in Merseburg andJenaandobtainedhisPh.D.in1991workinginthegroupofG.Helmisondynamicalmodelsofpolymernetworks.After post-doctoral research in Regensburg and Saclay,he joined the group of A. Blumen in Freiburg were heobtained his Habilitation in 1998. In 2000, he became astaff scientist of the CNRS in France, where he workedat the ‘‘Institut de Chimie des Surfaces et Interfaces’’ inMulhouse. In 2006, he was appointed Full Professor forTheory of Polymers at the Technische Universitat Dres-den and is since then heading a research group at theLeibniz-Institute of Polymer Research in Dresden. Hisresearch is focused on the field of statistical physics ofsoft condensed matter using both analytical and simu-lation methods. His current research interests includepolymersatsurfacesandinterfaces,polymerdynamics,networks, and crystallization and structure formationfar from equilibrium.

AlexanderBlumenstudied physics at the University ofMunich and obtained his doctoral title and his habi-litation in Theoretical Chemistry in G. L. Hofacker’sgroup at the Technical University of Munich. He spentpart of his postdoctoral period at the MassachusettsInstitute of Technology in R. J. Silbey’s group and atExxon Research and Engineering Co., collaboratingwith J. Klafter. He was awarded a Heisenberg Stipendby the DFGin1984,whichheused in1985 fora researchstay at the Max-Planck Institute for Polymer Science inMainz. In 1986, he was appointed Professor for Theor-etical Physics at the University of Bayreuth, where hismain interest were the dynamics of and transportphenomena in molecular crystals. Since 1991 he is FullProfessor for Theoretical Polymer Physics at the Uni-versity of Freiburg. His current research centers onanalytical and numerical work on polymer systems,such as branched macromoleculesand networks, withparticular emphasis on their dynamics. Furthermore,he is interested in modeling stochastic processes insoft matter using continuous-time random walks(CTRW) and fractional integrodifferential equations.Recently, with O. Mulken, he started analyzing quan-tum-mechanical transport features in soft matterusing continuous-time quantum walks (CTQW), anextension of the classical CTRW.

430

field theoretic approaches use the analogy between

polymer chains and spin systems close to the critical

point.[13] Much attention has been focused on the chain

conformations and the profile of the adsorbed layer in the

saturated state.[14,15] Here, using scaling arguments, de

Gennes has predicted a power law behavior for the

concentration profile as a function of the distance from the

adsorbing surface, the so-called ‘‘self-similar layer.’’[14]

Densely adsorbed polymer layers have been studied using

variants of self-consistent field theories.[1,15] These meth-

ods have been extended to related problems, such as

polymer layers on curved and disordered surfaces and the

adsorption of polymers in slit-like geometries, where the

interaction between distinct polymer layers is of impor-

tance.[16–18]

Up to now, it is rather difficult to check these results

experimentally, since any convincing test requires very

smooth surfaces, given that one aims at determining (say

by scattering techniques) the density profiles on the

nanometer scale. When using AFM measurements one

must perform cuts of the polymer/substrate interface

without tampering with the polymer profile in the

nanometer range. Even more demanding is to determine

experimentally the behavior of individual chains adsorbed

to the surface.

In such situations computer simulations are an alter-

native method, which is able to explore the properties of

polymeric systems on the nanoscale and hence can

provide a test for the theory (see ref.[19] for a recent

review on the last developments in Monte Carlo simula-

tions of polymer systems on lattices). However, similar to

the field theoretic calculations, computer simulations face

particular problems when used to estimate the critical

exponents of the adsorption transition. A very important

issue here is the possibility to determine the critical

adsorption strength (or, equivalently, the transition

temperature) using calculations based on chains of finite

length.[20–22] As has been shown in ref.,[23] even slight

variations in the estimation of the critical point substan-

tially influence the determination of the surface expo-

nents. In this work, we will compare and discuss various

methods used in the literature in order to obtain such

critical parameters.

Scaling at the critical point of adsorption (CPA) (which is

a special point in the context of field theory) is the

precondition which permits, using simple concepts, to

explore the behavior of polymer layers formed by many

chains. Bouchaud and Daoud[7] have used the results of

ref.[5] to develop a scaling theory for the semidilute surface

state. In ref.,[7] the properties of single chains and the

density profiles above the surface have been determined as

a function of the adsorption strength and of the monomer

concentration on the surface. These properties of the

quasi-2D surface layer have been calculated using argu-

DOI: 10.1002/mats.200800046

Grafted Polymer Chains Interacting with Substrates . . .

ments developed for polymers in the bulk. Of particular

interest for systems formed from many chains is the so-

called semidilute surface state, which is intermediate

between a state in which a single chain is adsorbed and

states in which the surface is saturated with chains. In a

recent work, using computer simulations, the semidilute

surface state has been considered in detail.[24]

Using grafted chains the surface concentration can be

increased beyond the values achievable by only adsorbing

free chains from the solution. For grafted chains one

observes a transition from configurations in which most of

the polymer segments are close to the surface to a brush-

like regime, in which after the formation of a dense surface

layer most polymers end by having an orientation

perpendicular to the surface. The possible arrangements

of grafted polymers on adsorbing substrates can be studied

using scaling arguments: this leads to a phase diagram in a

parameter space spanned by the interaction strength and

by the grafting density, as we will show in the next section

of this work. We mention here that for end-anchored

chains without attraction to the surface there are other

interesting features which appear.[25] In this case, much

recent attention was devoted to the compression of the

end-grafted layers by different mechanisms;[10,26,27] one

also studied entanglements in polymer brushes.[28,29] The

influence of the end-grafted chains on the adsorption of

simple fluids and of spherical molecules on surfaces has

been investigated using density functional approaches

and computer simulations.[30,31]

Another open problem concerns the dynamics of

polymer chains both at the CPA and in the adsorbed

state. Here, simulation studies are rare. Using the results

for single chains in good solvents and in the absence of

hydrodynamic interactions Milchev and Binder[32] have

monitored the diffusion of single monomers and the

correlation functions of the chain ends and have found a

transition from 3D to 2D dynamics. The dynamics of single

chains at the CPA has been investigated previously, and it

has been found that the dynamics of a 3D chain does not

change qualitatively at the adsorption point.[23] This

conclusion was inferred by focusing on the dynamical

exponent which relates the longest relaxation time of the

chain to its length. The dynamics in the adsorbed state can

be understood using dynamic scaling arguments,[33]

results which confirm and extend observations first made

in ref.[32] In the adsorbed state the excluded volume

interaction influences the dynamics of the monomer

fluctuations, so that the dynamics change in the semi-

dilute surface state. Using scaling arguments, the simula-

tion results in this region can be again satisfactorily

explained as a dynamic cross-over from a 2D excluded

volume situation to a Gaussian behavior at long times.

When the grafting density exceeds the saturation point,

another dynamical regime appears, in which brushes form



and the fluctuations become independent of the strength

of the adsorption. We will discuss these features in the

subsequent sections of this work.

Static Properties

In this section, we review the static equilibrium properties

of adsorbed polymers. In the limit of infinite chain length,

N, the adsorption of a single chain can be viewed as being a

surface phase transition, as first pointed out by Eisenrieg-

ler et al.[5] There exists a CPA, denoted as special point in

the context of surface phase transitions, which separates

the adsorbed surface state from the non-adsorbed state. To

see that there is a critical point in the polymer adsorption

problem one has to take into account that an impenetrable

wall suppresses many conformational degrees of freedom

if a chain is located close to it. This results in an effective

(entropic) repulsion of the chain which, in turn, will avoid

the surface region. On the other hand, if the surface

provides a strong enthalpic attraction, e, for the monomers,

the chain will form a flat conformation on the wall. Here,

the number of monomers in contact with the wall is

proportional to the number of monomers in the chain. In

between the repulsive and attractive regimes there exists a

critical strength of adsorption ec where the chain is neither

rejected from wall nor completely adsorbed. This point can

be understood as the CPA. However, the transition from

the bulk to the adsorbed state is only a true (infinitely

sharp) phase transition if the chain length tends to infinity.

In the formal language of field theory the double limit

" ! "c and N ! 1 is called the ‘‘special transition.’’ Many

concepts of the theory of critical phenomena at surfaces

can be used in polymer physics. For finite chains the phase

transition displays a smooth cross-over, which is con-

trolled by a single scaling variable U given by

� ¼ kNf (1)

Now the constant k (which characterizes the strength of

adsorption) can be expressed as a dimensionless quantity:

k � D"="c � ð"� "cÞ="c (2)

where we denoted by e the attraction strength and by ec its

value at the CPA. In Equation (1), f is the cross-over

exponent;[5,11,34] f relates the number of surface contacts

Mc at CPA to the length of the chain N:

Mc � Nf (3)

When the surface is not impenetrable the exponent f

equals f¼ 1� n, a result which has been obtained by de

Gennes[35] using scaling arguments only. The calculation

of the cross-over exponent for impenetrable surfaces via

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432

analytical approaches (such as field theoretic methods) is

much more difficult; in fact reliable results have been

obtained only recently.[12,36] Also computer simulations do

not yield the cross-over exponent in a simple and

straightforward manner. Here a major problem is to

determine from simulations performed with finite chains,

at the same time both the CPA and also the cross-over

exponent. In ref.[5] a value of about f¼ 0.6 is reported,

value which is much larger than the one found for

penetrable surfaces. Analyses based on chain growth

algorithms[37] lead to f-values which are close to 0.5 and

which lie even below 0.5; such values, however, were

found to lead to rather poor overall scaling results.[38] We

have shown in recent work[23] that even a quite small

change in the estimation of ec leads to a strong change in

the value of f and that both values depend crucially on the

analysis used to determine them. Since f is essential for

the scaling behavior of chains at adsorbing surfaces, we

review in Section ‘‘Single Chains, Critical Parameters, and

the Cross-over Exponent’’ various methods to determine

the set of critical parameters (ec, f).

In Section ‘‘Many Chains and their Phase Diagram,’’ we

turn to the many chain problem, namely to the investigation

of the semidilute surface state. This regime has been

theoretically addressed using scaling arguments by Bou-

chaud and Daoud,[7] but is difficult to reach experimentally.

The reason is that here already small concentrations of

polymers in the bulk are sufficient to saturate the surface,

since in this regime the free energy gain per adsorbed chain

is much larger than the thermal energykT. Using simulation

results, we discuss the consequences of semidilute surface

scaling, an approach which turns out to faithfully reproduce

several features found when the surface concentration

varies. In Section ‘‘Deviations from Semidilute Scaling and

Saturation Scaling,’’ we discuss for higher surface concen-

trations the influence of the excluded volume effect in more

detail. In particular, we show that the excluded volume

interactions influence the stability of the adsorption blobs

and hence lead to corrections of the semidilute surface

scaling. In this section, we demonstrate that introducing a

new scaling variable related to the saturation concentration

helps in understanding the behavior found at high surface

concentrations. Finally, we address in Section ‘‘Oversatura-

tion Effects’’ the problem of oversaturation and the

transition of grafted polymer chains to brush-like config-

urations.

Single Chains, Critical Parameters, and the Cross-OverExponent

For a flexible chain close to a surface the geometrical

restrictions introduced by the presence of the surface (the

fact that a part of the space is no more accessible to the

monomers of the chain) lead to a lower conformational



entropy. Despite this entropic effect, in many cases

polymers still adsorb on the surface because of an

enthalpic gain, usually due to a van der Waals-type

interaction. We will denote by �e the attractive energy

gained by each monomer from getting adsorbed on the

surface. Above the value e¼ ec, i.e., above the CPA, the

polymers change their conformation from a three-dimen-

sional form (a mushroom shape) to a two-dimensional

form (a pancake).[5,13,35] Based on the formal analogy of the

polymer adsorption problem to the problem of critical

phenomena in the n-vector model of a magnet with a free

surface, it was shown[5,34] that for e> ec the adsorbed phase

of the polymer chain corresponds to a situation in which

the surface of the magnet gets ordered before the bulk does

so. Moreover, then ec corresponds to a multi-critical

(special) point of simultaneous bulk and surface order

(see ref.[5,34]).

Order Parameter for Single Chain Adsorption and Scaling

The observable which plays the role of order parameter in

the single chain adsorption transition is the fraction of

monomers in contact with the surface. This quantity is

given by

m ¼ M

N(4)

In Equation (4), M is the number of adsorbed monomers

and N is the chain length. For e> ec (i.e., in the adsorbed

state) m is finite for N ! 1. For e< ec (in the non-adsorbed

state) M grows less than linearly with N and therefore

m ! 0 in the limit N ! 1.

Directly at the CPA, for e¼ ec, the number of adsorbed

monomers is related to the chain length by Equation (3).

The cross-over exponent f has to be considered as a new

(surface) critical exponent which cannot be a priori

obtained from bulk exponents in the case of an

impenetrable surface. Since f plays a key role in the

analysis of polymer adsorption,[6,7,34] its estimation is a

very important task.

According to the theory of critical phenomena, the

partition function Z (Z counts the number of chain

conformations) can be expressed, close to the CPA, in

scaling form.[5] The relevant scaling variable is given by

Equation (1). This implies that also other statistical

variables directly related to Z can be written in scaling

form.[5,7] Alternatively, one can view the scaling variable

as the ratio of two characteristic length scales: one of these

is given by the extension of the free chain, R � Nn, and the

second is controlled by the adsorption strength, L � k�n=f

(see further below).

The scaling behavior of M close to the CPA was

determined theoretically in ref.[5,7,13] Based on these

works, other scaling relations for M follow readily. The

DOI: 10.1002/mats.200800046


number M of adsorbed monomers on the surface can be

written in scaling form as

Macrom

� 2008

M ¼ McfMð�Þ � NffMð�Þ; with fMð0Þ ¼ 1 (5)

where fM is a function of the scaling variable U [see

Equation (1)]. For � � 1, the whole chain is adsorbed on

the surface and the number of surface contacts is

proportional to N. With this remark it follows that

N � M � NffMð�Þ (6)

This proportionality holds only if fM(U) is a power law,

fMð�Þ � ðkNfÞn1 (7)

Equation (6) becomes now

N � M � NfðkNfÞn1 � Nfkn1Nn1f (8)

Comparing the powers of N one finds n1 ¼ (1�f)/f and

from Equation (6) one infers that

M=Nf � ðkNfÞð1�fÞ=f ðfor kNf � 1Þ (9)

In the other limit, when � � 0, the chain is in

mushroom form and M � N0. Following the same line of

thought we obtain

M=Nf � ðkNfÞ�1 ðfor kNf � 1Þ (10)

By considering length scales in the direction perpendi-

cular to the surface it is useful to introduce a new

dimensionless variable z through

z ¼ z=Nn (11)

where z is the distance from the surface.

Scaling Regions of Polymer Adsorption

As a function of the scaling variable U one can distinguish

four cases:

Non-Adsorbed Phase

This phase corresponds to large negative values of U (i.e.,

the case when the wall is repulsive). Then, the tethered

chain is in a mushroom form. The chain extension is

proportional to that of a free chain with the same N, whose

characteristic length, here the radius of gyration Rg0, obeys

Rg0 � Nn; furthermore, the monomer density profile has a

smooth maximum at Rg0.

ol. Theory Simul. 2008, 17, 429–453

WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Critical Point of Adsorption and Adsorption Cross-Over

For U¼ 0, the surface appears effectively neutral to the

polymer. For this to happen a non-zero adsorption strength

ec is necessary, since the non-penetrable surface reduces

the conformational entropy of the chain. Now, the chain

conformation, in particular the monomer density profile in

the direction perpendicular to the wall, differs both from

the mushroom state and from the free state. As discussed

by Eisenriegler et al.[5] and by de Gennes and Pincus,[6] in

the presence of a non-penetrable but effectively neutral

surface a singular monomer profile � z�n evolves, where n

is the proximal exponent. A further analysis given by

Eisenriegler reveals a non-trivial cross-over for the shape of

the density profile at the CPA[13] (see also Section ‘‘Analysis

of the Monomer Density Profile’’).

For a finite chain a slight excess of adsorption energy is

necessary in order to localize the chain at the interface (and

to change the density profile). Here, therefore adsorption is

a cross-over phenomenon: in order to adsorb a floating

chain from the solvent and to localize it at the surface the

energy of adsorption, namely Mk, must balance the loss kT

in entropy. Hence, a finite chain is localized at the surface

when k fulfills

Nfk � 1 (12)

Adsorbed State

For � � 1, but small k many of the monomers of the chain

are adsorbed, but the binding energy of each monomer is

low. It is important to note that the adsorption of a single

long chain does not require a strong monomer-interface

attraction in units of ec.

Therefore, although the chain is adsorbed as a whole, a

large number of monomers are located at distances much

larger than a single segment length from the surface. This

situation corresponds to the so-called weak coupling

limit.[14] The adsorbed chain can be viewed as being a

string of adsorption blobs. This string of blobs forms a

quasi-two-dimensional layer on the surface (see Figure 1).

An adsorption blob is on the verge of adsorption (still

dominated by free conformations).[7] Its characteristic

parameters are the number g of monomers which it

contains and its characteristic length L. Thus, the criteria of

being at adsorption implies

gfk � 1 (13)

whereas L and g are connected through L � gn. It follows

that

L � gn � k�n=f (14)

Each blob carries an adsorption energy of about kT.



Figure 2. Scaling plot of M/Nf as a function of jkjNf for e> ec(upper part) and e< ec (lower part).[33] To include both e> ec ande< ec in the same graph we plot the scaling variable as jkjNf, sincefor e< ec, k becomes negative. The different symbols representsimulation results, with N lying between 20 and 200. Scaling isfound for ec¼ 1.01 and f ’ 0:59.

Figure 1. Sketch of an adsorbed chain which can be viewed asbeing a two-dimensional chain composed of adsorption blobs.[24]

The view from the side indicates the localization length L and theclassification of the regions above the surface. The view from thetop shows the two-dimensional self-avoiding chain of adsorptionblobs on the surface.

434

Strong Coupling Limit

For k � 1 each monomer is strongly attracted by the

surface. The density profile in this regime displays a quasi-

exponential decay beyond the length scale of a single

segment. One might consider this case as the limit g ! 1.

Here, scaling analysis breaks down and the chain can be

simply envisaged as being a 2D object. For values larger

than k ’ 10 the adsorption becomes irreversible (see

ref.[39–44]).

The analysis of the first three (scaling) regions requires

exact values for f and for ec (via k). In the following, we

discuss various methods to obtain and assess the

correctness of the set of critical parameters ec and f.

Best Scaling Analysis

Based on the idea that self-similar behavior should be

visible close to the CPA, several observables can be tested

for scaling by varying the two parameters ec and f and by

examining the results.

As an example we present in Figure 2 an analysis[23] of

the scaling behavior of M based on Equation (9) and (10).

Here, the simulation data which correspond to different

values of N and e lie on common curves (master curves)

when one plots M/Nf as a function of jkjNf for

Macrom

� 2008

f ’ 0:59 and "c ¼ 1:01 (15)

To find this optimal pair (f, ec) of values, ec was

systematically varied from 0.8 to 1.1 and f from 0.4 to 0.6

(see ref.[23]).

ol. Theory Simul. 2008, 17, 429–453


Intersection Method

An alternative way to determine ec was proposed by

Metzger et al.[38] and makes use of the chain’s extension.

Since the presence of the surface breaks the isotropy of the

problem, the parallel and perpendicular components of

position-dependent quantities have to be considered

separately. In the following, we denote by Rgk and by

Rg? the radius of gyration of the polymer chain parallel and

perpendicular to the surface. One has then

Rgkð�Þ ¼ Rgkð0Þfkð�Þ (16)

and

Rg?ð�Þ ¼ Rg?ð0Þf?ð�Þ (17)

with fkð0Þ ¼ 1 and f?ð0Þ ¼ 1. One has at U¼ 0

Rgkð0Þ ¼ rgkNn (18)

and

Rg?ð0Þ ¼ rg?Nn (19)

with rgk and rg? being constants. For � � 1, the chain is

completely adsorbed on the surface and Rg?ð�Þ is

independent of N, thus Rg?ð�Þ � N0. The situation is

different parallel to the surface; here the chain behaves as

a two-dimensional self-avoiding walk; hence, Rgkð�Þ � Nn2

(with n2 being the Flory exponent in two dimensions).

DOI: 10.1002/mats.200800046


Considering now the ratio wð�Þ ¼ Rg?ð�Þ=Rgkð�Þ and

using Equation (18) and (19) together with Equation (16)

and (17) one obtains

Figsymsymatpatarecurestthis

Macrom

� 2008

wð�Þ ¼ wgfrð�Þ (20)

Figure 4. Display of m¼M/N as a function of N in doublelogarithmic scales. This plot renders evident the correlationbetween ec and f, see text for details.

with wg ¼ rgk=rg? and frð�Þ ¼ fkð�Þ=f?ð�Þ. Hence, for U¼ 0

the ratio between both components should be a constant,

independent of N. For different values of N, these curves

must intersect at a single point, which gives ec. This

method to obtain ec was also used in ref.[20–22] The plot

w2 ¼ R2g?=R

2gk versus e is given in Figure 3. In Figure 3, the

curves intersect close to the value e¼ 0.98� 0.03. However,

a zoom of the region around e¼ 0.98 (see the inset of

Figure 3) shows that the curves do not intersect in a single

point. This fact was also reported in ref.[20–22,38] The inset

in Figure 3 shows the full region in which the curves

intersect. We note that the intersection method tacitly

assumes simple scaling at the CPA (it does not consider

corrections to scaling at finite N).

Analysis of the Order Parameter

This method was first used by Eisenriegler et al. in ref.[5] and

is based on an analysis of the order parameterm¼M/N [see

Equation (4)]. For large values ofN,m is expected to obey the

following asymptotic power laws (see also Section ‘‘Order

Parameter for Single Chain Adsorption and Scaling’’):

m �N�1 for k < 0Nf�1 for k ¼ 0N0 for k > 0

8<: (21)

ure 3. Ratio R2g?=R2

gk, plotted as a function of e. Differentbols belong to different values of N, as given in the list ofbols. Since in the simulation box the reflecting wall is located

z¼ 100, for N¼ 200 and small e deviations from the generaltern appear. Therefore, for N¼ 200 only the results for e>0.75included in the plot. The location of the intersection of the

ves (whose variation is given by the width of the bar) allows toimate ec as being ec¼0.98�0.03. The inset presents a zoom of

region.

ol. Theory Simul. 2008, 17, 429–453


The principle of the method is illustrated in Figure 4.

Here, the order parameter m is displayed in a double

logarithmic plot as a function of the chain length N. As can

be seen from Equation (21), for k 6¼ 0 and N ! 1 only two

types of asymptotic behavior appear, namely M=N ! N0

for k> 0 and M=N ! N�1 for k< 0. Therefore, all these

curves should bend toward their corresponding, asympto-

tic fixed points. A straight line with a slope of f� 1, see

Equation (21), should appear only at k¼ 0 (i.e., at e¼ ec).

However, in the range of 0.98< ec < 1.05 we do not observe

that the curves bend, at least not for the N-values which

we consider here. We can only infer that ec obeys

0.98< ec < 1.05. Here, Figure 4 provides additional infor-

mation about the correlation between ec and f. As can be

learned from Figure 4, smaller values of e are related to

smaller slopes in Figure 4 and thus to smaller values of f.

Analyzing Figure 4, we infer that for ec¼ 1.01 the best

choice is f ’ 0:59, and that for ec ¼ 0.98 it is f ’ 0:5. We

will thus compare in the following the results based on the

two parameter sets (ec, f)¼ (1.01, 0.59) and (ec, f)¼ (0.98,

0.5).

We now turn to study the scaling behavior of M

according to Equation (9) and (10). The expected asympto-

tic power laws correspond in a double logarithmic

representation to straight lines, as indicated in Figure 2

and 5. Figure 2 shows the data obtained for the set (1.01,

0.59), whereas Figure 5 gives the data for the set (0.98, 0.5).

The simulation data follow nicely the expected power laws

for the two regions, e> ec and e< ec. A critical value ec¼ 0.98

is suggested by Figure 3; according to the discussion of

Figure 4 the corresponding cross-over exponent is f ’ 0:5.

The value ec¼ 1.01 is slightly larger, but it is still in the

range of the bar of Figure 3. A closer look at Figure 2 and 5

shows, however, that for the set of parameters (0.98, 0.5)

the scaling behavior is worse than for the set (1.01, 0.59).



Figure 5. Same as Figure 2. Here the parameters are ec¼0.98 andf¼0.5. The straight lines indicate the asymptotic behavior of thescaling function[23] according to Equation (9) and (10).

Figure 6. Double logarithmic plot of the normalized free-enddistribution,[23] Z(z), as a function of z/Nn at fixed N¼ 160. Thee-values are as listed. The inset reproduces the plot for e¼0.9,closely below the CPA.

436

Moreover, in Figure 5 the asymptotic slope is not yet

reached. This is similar to observations made in ref.[38]

An analysis of the scaling behavior of the chain’s

extension indicates that this quantity is less sensitive to a

change in the set of critical parameters than the order

parameter. This fact is in line with the results of ref.[23]

Analysis of the Monomer Density Profile

Yet another possibility to find the correct set of critical

parameters is to analyze the distribution of distances of

the end-monomers from the surface and the density

profile of all the monomers. According to ref.,[5] the

number of grafted chains whose free end is at the distance

z from the surface obeys in the vicinity of the CPA a scaling

form which depends on two variables. One has

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ZðzÞ ¼ N�1þnð1�h?ÞFðz;�Þ (22)

whereU is given by Equation (1) and zby Equation (11). Here,

we introduce the surface critical exponentsh? and hk, which

characterize the singularity of the monomer distribution

functions for N ! 1 (see ref.[5]). Namely, at the CPA the

variable� ¼ kNf vanishes andFdepends only onz. For small

z, i.e., for distances smaller than Rg(0), a sharp power-law

behavior near the wall is expected. This power law can be

written (dispensing with N) in the form:[5,36]

Z0ðzÞ � z�Dh ðfor z � 1Þ (23)

where

Figure 7. Double logarithmic plot of the normalized free-enddistribution,[23] Z0(z)Nn, for ec¼ 1.01, given as a function of

Dh ¼ h? � hk (24)

z/Nn. The same data, but in linear scales, are displayed in theinset. The straight line indicates the best fit to the simulationpoints in the region z � 1.

In order to estimate ec one can now analyze the shape of

the free-end monomer distributions as a function of e. At ec

ol. Theory Simul. 2008, 17, 429–453


the distributions change from mushroom forms with

pronounced peaks at about Rg0 to distributions having

their maxima at the surface. This is illustrated in Figure 6,

where we display the normalized distribution function Z(z)

for N¼ 160. Here, e varies between 0.1 and 1.8.

Note that in the region " ’ "c, but e still below the CPA,

the density profile gets large in the proximal region and

exhibits two peaks (see ref.[5,13,35]). In the inset of Figure 6,

this is visible at the value e¼ 0.9, below ec. According to

Equation (23), at e¼ ec and for small values of z the

distribution function of the free-end monomer should

follow a power law.

By comparing results for different chain lengths it is

possible to check the scaling with respect to N. In Figure 7,

DOI: 10.1002/mats.200800046


we display the results for ec ¼ 1.01 and for three values of

N. From the figure we can determine the exponent Dh, see

Equation (23) and (24), using the singular part of the

profile. By taking a fit to the points in the region z � 1 we

estimate the exponent to be Dh ’ 0:25 [see Equation (23)].

This value is intermediate between the result Dh ¼ 0:11 of

ref.[11,36,45] and the result Dh ¼ 0:47 of ref.[5]

Alternatively, one may use the density profile of

polymer chains to gain information about ec. As has been

discussed by de Gennes and Pincus,[6] see also Section

‘‘Scaling Regions of Polymer Adsorption,’’ at k¼ 0 and

z � 1 the density profile is expected to show scaling,

Figmeec¼z/N

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r0ðzÞ � z�n (25)

where n denotes the proximal exponent,[6] given by

n ¼ 1 þ ðf� 1Þ=n (26)

One should note that the power law of the density

profile is controlled by the cross-over exponent f (see ref.[6]

for details). Hence, this relation can be used to estimate (or

test) the set of critical parameters. An example is given in

Figure 8 for the case ec ¼ 1.01.

A fit to the points in the region z � 1 allows to estimate

the proximal exponent given by Equation (26). We obtain

n¼�0.33, which again agrees very well with a cross-over

exponent of f¼ 0.59.

All the methods described above yield a very good

scaling of various observable quantities at the CPA when

choosing the set of critical parameters according to

Equation (15), a caveat being that the results are based

on moderately long chains. Since infinitely long chains are

neither available in experiments nor in computer simula-

ure 8. Double logarithmic plot of the normalized total mono-r density[23] r0(z), namely r0(z)Nn, as a function of z/Nn for1.01. The straight line is a fit to the simulation points for smalln. The inset presents the same data, but in linear scales.

ol. Theory Simul. 2008, 17, 429–453


tions, we continue here by using the parameters of

Equation (15), which allows us to further explore the

adsorption of grafted polymer chains.

Many Chains and their Phase Diagram

When grafting many polymer chains on the surface, the

interplay between the grafting density, the adsorption

strength and the excluded volume interaction leads to the

appearance of different regimes. The grafting density is

defined to be

s ¼ 4n

A(27)

where n is the total number of chains in the system and A is

the surface area. In Equation (27), the factor four appears

because in the simulation method which we use (see

Appendix A) a monomer occupies four lattice sites on the

adsorbing surface. Since all polymers are grafted to the

surface, all monomers contribute to the surface layer.

Therefore, the surface excess concentration G (given by the

integral over the surface excess concentration density

profile[7]) is

G ¼ 4nN

A¼ Ns (28)

There exist two characteristic monomer surface con-

centrations. First, the surface concentration, G�, at which

the adsorbed chains start to overlap is

G� ¼ N

R2gk0

(29)

In Equation (29), N represents the number of monomers

per chain and Rgk0 is the extension parallel to the surface of

the radius of gyration of isolated chains ðG ! 0Þ. Now, Rgk0

depends on N and k in the following way:[7,35]

Rgk0 � Nn2kDn=f (30)

with Dn ¼ n2 � n being the difference between the 2D and

the 3D Flory exponent. This expression represents the

extension of a 2D self-avoiding chain composed of

adsorption blobs [see Figure 1 and Equation (13) and

(14)]. Thus, G� can be written as

G� � N1�2n2k�2Dn=f (31)

A second characteristic monomer surface concentration

arises in situations when under an increase in the total

concentration the surface and its vicinity (the proximal

layer) cannot be enriched anymore, because of the

excluded volume interactions between the adsorption



438

blobs. In the case of non-grafted chains this characteristic

concentration is called saturation concentration;[7] in our

case we will call it surface saturation concentration and

denote it by G��. Using Equation (14) we obtain

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� 2008

G�� g

L2� kð2n�1Þ=f (32)

Figure 9. Phase diagram for grafted chains on adsorbing surfaces.As a function of k and s five different regions could be found.Region M: the tethered chains are in a mushroom-like state.

The Phase Diagram

In order to construct the phase diagram for grafted

polymer chains on adsorbing surfaces we use the grafting

density s¼G/N, see Equation (28), instead of G, since this

quantity can be related more directly to experiments.

Using Equation (31) we can write
Region AD: The adsorbed chains are in a dilute state. Region ASD:The adsorbed chains form a 2D semidilute surface state. RegionOSB: the chains are in an oversaturated brush state. For finitechains a narrow cross-over region OSA appears between ASD and
s�ðkÞ ¼ G�

N� N�2n2k�2Dn=f (33)

OSB, which we call adsorbed oversaturated region. Region B: thetethered chains are in a brush state. The small filled circles (red inthe online version) denote adsorption blobs, whereas the largeopen circles represent Alexander-de Gennes blobs.

Let us recall the fact that the surface phase-transition is

very sharp only for infinite chains at k¼ 0 (i.e., at ec ¼ e). As

already discussed in Section ‘‘Scaling Regions of Polymer

Adsorption’’, see Equation (12), for finite chains adsorption

is a cross-over effect with a characteristic value of

kc � N�f (34)

Below kc the chains are only weakly perturbed by the

surface. Inserting now Equation (34) into Equation (33) we

obtain

s� ¼ s�ðkcÞ � N�2n2N2Dn � N�2n (35)

In the case of non-adsorbing surfaces this corresponds to

the cross-over from the mushroom to the brush state, as

first discussed by Alexander and de Gennes.[46,47] We note

that changes in the conformational properties might be

observable only for values of s/s�> 10, as has been

reported recently by Chen et al.[48,49] This implies a rather

broad transition zone. In a similar way, for the case of

adsorbing surfaces and above kc, we have also observed in

our simulations a cross-over region between the adsorbed

semidilute and oversaturated brush regions (see Figure 9

and the explanations below).

Using Equation (32) we obtain as corresponding grafting

density at the saturation limit

s��ðkÞ � N�1kð2n�1Þ=f (36)

At k¼ kc, Equation (36) becomes

s��ðkcÞ � N�1N�ð2n�1Þ � N�2n (37)

which again corresponds to the mushroom-brush transi-

tion. Thus, an adsorbing surface splits s� into two cross-

over lines which open the region of a semidilute surface

state. This is illustrated in Figure 9, where we show the

ol. Theory Simul. 2008, 17, 429–453


phase diagram whose parameters are the adsorption

strength k and the grafting density s. Note that k is directly

related to the temperature by the relation k¼ (Tc � T)/T,

where Tc corresponds to the temperature at the CPA.

Next, we discuss the various regions which can be

defined in this phase diagram:

For k< kc the polymer chains are not adsorbed. At s� a

transition from a mushroom state (M) to a brush-like state

(B) takes place.

For k> kc the chains are adsorbed and three new regions

can be identified. The adsorbed chains are diluted in the

(AD) region below the ‘‘overlap-line’’ s�(k) [see Equation

(35)]. Above the ‘‘overlap line’’ the adsorbed chains form a

2D semidilute surface state (ASD), which extends up to the

‘‘saturation-line’’ s��(k) [see Equation (36)]. The ‘‘overlap-

line’’ and the ‘‘saturation-line’’ intersect at k¼ kc which

gives s�, the grafting density which triggers the mush-

room-brush transition. If the grafting density is increased

beyond the ‘‘saturation line’’ a brush-like state forms on

the top of the adsorbed layer. We call this region the

oversaturated brush regime OSB, to be explained below.

It is interesting to consider the condition for the cross-

over from the ASD to the OSB region. The number of

monomers available for brush formation is given by

DN ¼ N � Nsat (38)

where Nsat is the number of adsorbed monomers per chain

at the threshold of saturation and is given by

Nsat ¼ðA=L2Þg

n� s�1kð2n�1Þ=f (39)

DOI: 10.1002/mats.200800046


where we have used Equation (14) and (27). Now, the

condition for a brush to be formed can be written as

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ðDNÞ2n � s�1 (40)

and therefore

N � cs�1kð2n�1Þ=f � s�1=2n (41)

where c is some constant. Hence, the condition to reach the

oversaturated brush state (OSB) is given by

N > s�1=2n þ cs�1kð2n�1Þ=f (42)

Thus, for finite N there exists an oversaturated adsorbed

state (OSA) between the ASD and the OSB. By considering

the relation s� � N�2n and s ffi s�� (close to the transition

line) we obtain

s�� < s < s�� 1 þ s�

s��

� �1=2n" #

(43)

Figure 10. Sketch of adsorbed chains in the ASD, OSA, and OSBregimes. In ASD a hierarchy of blobs is introduced, by which theadsorption blobs of size L are the effective monomers of theconcentration blobs of size j. In OSB 3D blobs form on top ofadsorbed ones. A view from the top displays the ASD state. Athigher monomer surface concentrations crowding effects showup and the extension of the chains perpendicular to the surfacebecomes much larger than the localization length L. This leads to

as the condition for the OSA region (see Figure 9).

Adsorbed Semidilute Surface State

Let us now turn to a more detailed analysis of the

adsorption controlled region k> kc. Here, we come back to

the parameter G which is the appropriate variable for

semidilute surface scaling.

For G>G� a two-dimensional semidilute regime devel-

ops on the surface. In this region, the chains overlap and

interactions between the adsorbed chains become impor-

tant. An indication of the overlap between different chains

is given by the scaling variable[7]

the appearance of an OSB regime through a cross-over regionOSA (for finite chains, see text for details). In the OSB, theadsorption of a single chain is restricted to an area which scales

~� ¼ G=G� (44)

with the inverse grafting density.
We can now extend the blob picture, see Figure 1, to
the semidilute surface regime. This has been considered

first by Bouchaud and Daoud.[7] Here, we follow closely

the arguments of ref.[7] As in the bulk case, the excluded

volume interactions are basically screened on length

scales larger than the correlation length j. The correla-

tion length defines the size of the 2D concentration

blobs, which cover the surface densely. A single chain

on the surface can now be viewed as a Gaussian chain

consisting of (uncorrelated) segments of size j. This is

illustrated on the left hand side of Figure 10. The 2D

concentration blobs are made of adsorption blobs; this

introduces a hierarchy of blobs as sketched in Figure 10.

The relation j> L is ensured in the ASD region of the

phase diagram (Figure 9). In this picture, it is tacitly

assumed that the concentration blobs of size j do not

ol. Theory Simul. 2008, 17, 429–453


interfere with the adsorption blobs of size L. We will

return to this point below. Under these conditions one

can write

j � gG

n2 (45)

where gG is the number of adsorbed blobs contained in a

concentration blob.

In the following, we apply the semidilute surface scaling

picture in order to evaluate the dimensions (extensions) of

the tethered chains, lengths which can be easily accessed

in computer simulations.



440

The Extension of the Chains Parallel to the Surface

We start by discussing the extension Rgk of a chain parallel

to the surface. According to the scaling assumption we

may express Rgk as

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RgkRgk0

¼ fkð~�Þ (46)

Here and in the following the index ‘‘0’’ indicates the

state of an isolated adsorbed chain ð~� ! 0Þ. The limit~� � 1 corresponds to the dilute limit and we obtain by

definition fkð~� � 1Þ ! 1. As discussed above, for ~� � 1

the screened chains behave as 2D Gaussian random walks.

In this limit one can write

Figure 11. Double logarithmic plot of R2gk=R

2gk0 as a function of the

scaling variable[24] ~� ¼ G=G�. The different symbols correspond�

Rgk � N1=2 (47)

to different values of e and G. Here, G is given by Equation (29).The solid arrow points to a new state (indicated by ‘‘’’) obtainedfor e¼ 1.4 after releasing the grafting constraints. The dashed

Introducing in Equation (46) the relation for Rgk0, given

by Equation (30) and (47), it follows that
arrow points to the initial state, in which the chains were stillgrafted.
N1=2 � Nn2kDn=ffkð~�Þ (48)

Now, the function fkð~�Þ has to be of the form

fkð~�Þ � ðG=G�Þh (49)

to satisfy Equation (48). Then, with Equation (31) and (49),

one can rewrite Equation (48) as

N1=2 � Nn2kDn=fðN1�2n2k�2Dn=fÞ�hGh (50)

Equating now the exponents of N in Equation (50), one

obtains h¼�1/2. With this value and taking into

consideration Equation (46) and (49), it follows that

RgkRgk0

� ~��1=2

and thus Rgk � N1=2G�1=2 (51)

The last equation can be tested by computer simula-

tions, as has been done in previous work.[24]

In Figure 11, the values of R2gk obtained from the

simulations are rescaled by the value of R2gk0 obtained from

simulations on single chains at the same value of e, and are

then plotted as a function of the scaling variable ~� given

by Equation (44). The expected asymptotic power law, see

Equation (51), is indicated in Figure 11 by a straight line. To

show that semidilute surface scaling also holds for chains

which are not grafted to the surface we have performed

additional simulations in which we removed the grafting

constraints. Thus, we consider a state which is originally

located in the scaling regime (e¼ 1.4, G¼ 0.44). We indicate

ol. Theory Simul. 2008, 17, 429–453


this state by a dashed arrow in Figure 11. After relaxing the

system by removing the grafting constraints a new state

emerges. Here, we take into account the effective surface

concentration by disregarding chains which do not touch

the surface. As can be seen from Figure 11, where the

position of the new state is indicated by a solid arrow,

there occurs a shift of the data toward smaller values of G

(and thus toward smaller values of ~�). The new state fits

again into the scaling region, in line with the data for the

grafted system.

Figure 11 shows that for large G (i.e., for large ~�) scaling

breaks down. We will discuss this issue in the following.

Also, the data for e¼ 1.1 do not follow the master curve.

This is explained by the fact that this value of e is very close

to the CPA (ec ¼ 1.01) and the chains are too short to

develop enough adsorption blobs necessary to display the

2D semidilute behavior.

The Extension of the Chains Perpendicular to the Surface

We turn now to the extension of the chains in the direction

perpendicular to the surface. In the case of ~� � 1, the

chains do not interact and their extensions in the direction

perpendicular to the surface are given by the correspond-

ing extension of an isolated adsorbed chain. As previously

discussed, in the adsorbed regime Rg?0 is given by the

localization length L, see Figure 1, and is independent of N.

The situation changes when the chains start to be crowded

on the surface. Here, large loops and tails form, which leads

to an increase in the extension of the chains in the

DOI: 10.1002/mats.200800046


direction perpendicular to the surface. Finally, at high

surface coverage the loops and tails form an extended

layer, whose characteristic height is that of an unper-

turbed chain. Bouchaud and Daoud have assumed that this

limit is reached in the semidilute surface region. Thus, for~� � 1 one can write[7]

Figas aat anot

Macrom

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Rg? � Nn (52)

Now, according to the scaling assumption one can write

in general for the extension perpendicular to the surface

Rg?Rg?0

¼ f?ð~�Þ (53)

Making use of the fact that Rg?0 is given by the

localization length L for ~� � 1 and that Rg? is given by

Equation (52) for ~� � 1, one can rewrite Equation (53) as

Nn � k�1f?ð~�Þ (54)

Here, the approximate relation[5,12,23] n ’ f was used.

Now Equation (54) can be satisfied only when the function

f?ð~�Þ obeys a power law. Hence, using the exact result

n2 ¼ 3/4 we obtain

Rg?Rg?0

� ~�2n

and Rg? � k�2nfð2n�1Þ

G2nNn (55)

Figure 12 displays in a double logarithmic plot the

simulation results for the radius of gyration of the chains

in the direction perpendicular to the surface,[24] R2g?,

plotted as a function of the scaling variable ~�. In this figure

ure 12. Same as in Figure 11, only that here R2g?=R2

g?0 is displayedfunction of G/G�. For isolated chains R2

g?0 becomes very smallttraction strengths larger than e> 1.6, so that such results areshown here.

ol. Theory Simul. 2008, 17, 429–453


R2g? is rescaled by the simulation results obtained for single

chains, R2g?0, at the same value of e. The expected slope is

indicated in the figure by a straight line.

In contrast to the results for the chain’s extension

parallel to the surface Figure 12 displays rather poor

scaling. One can notice only an accumulation of data

points around the asymptotic region given by Equation

(55).

It is important to note that the ASD surface scaling for

Rg? is based on additional assumptions. While the scaling

of Rgk follows the same arguments as for semidilute

solutions in 3D (SD), one has to assume here, a priori, that

Rg? � Nn at G ffi G�. Since G� is difficult to monitor (by

contrast to G��, see ref.[7]) there are no experimental

observations available to prove the relation Rg?ðG�Þ � Nn.

Deviations from Semidilute Scaling and Saturation Scaling

From the analysis of Figure 11 and 12 it turns out that for

large values of G the simulation points do not follow the

master curve according to the power laws obtained from

the scaling analysis in the semidilute surface regime.

As mentioned above, the scaling analysis of the ASD

surface state assumes that the adsorption blobs are not

perturbed by the excluded volume interaction. Now,

excluded volume interactions increase the free energy of

the polymer layer. Let us estimate at the mean-field level

and as a function of concentration the excess free energy

per adsorption blob. The probability that two adsorption

blobs overlap is given by

pover �G

g

� �L2 (56)

In analogy to the situation of concentration blobs in

semidilute solutions,[35] for a mutual penetration of blobs

one needs an energy of at least kT; in fact the free energy

penalty is limited to kT by the self-avoidance of the

adsorption blobs. Hence, we can express the free energy

per blob due to repulsion as

Frep � kTpover � G=G�� (57)

Inserting Equation (56) in Equation (57) and using

Equation (14) the free energy per blob due to repulsion is

Frep � kTGk�ð2n�1Þ=f (58)

where we obtained the last expression by comparing Frep

to Equation (32). The free energy increase due to the dense

packing of the adsorption blobs has to be confronted to the

free energy decrease due to adsorption, which is again

around kT per blob. The dimensionless scaling variable



442

which measures this free energy excess is then given by

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~V ¼ G

G�� (59)

Figure 13. Scaling plot of the chain order parameter assumingsaturation scaling[24] according to Equation (66).

This relation shows that the saturation concentration

G�� controls the balance between concentration blobs and

adsorption blobs.

Thus, semidilute surface scaling is fulfilled in the regime

G� � G � G��, because saturation leads to only weak

perturbations, as already remarked by Bouchaud and

Daoud.[7]

To check the saturation scaling we use the single chain

order parameter m (see Section ‘‘Order Parameter for Single

Chain Adsorption and Scaling’’), which is only weakly

influenced by chain overlap. For G>G�, the formation of

large loops and tails slightly decreases the number of

adsorbed monomers per chain, an effect which hardly

affects the relation M � N. Therefore, only saturation

scaling is expected to change the order parameter and we

assume that

m

m0¼ fmð~VÞ (60)

where m0 is the chain order parameter for G ! 0 and is

given by:[7,23]

m0 � kð1�fÞ=f (61)

In the case of (non-grafted) chains adsorbed from

solution, the limit ~V � 1 is unphysical because G is

limited by G��. In the case of tethered chains, however,

G>G�� can be reached using higher grafting densities, thus

attaining the OSD and OSB regions in the phase diagram of

Figure 9.

Simulation results for the chain order parameter using

saturation scaling are displayed in Figure 13. In this figure

m is rescaled by the m0 value obtained from the simulation

of single chains and is then plotted against the saturation

scaling variable given by Equation (59). One can see that

the different symbols corresponding to different values of eand G follow a common curve. Saturation scaling does not

hold anymore when the size of the adsorption blobs is

comparable to the distance between grafting points. Such

a situation occurs for chains close to the adsorption

threshold, see the data for e¼ 1.1. In this particular case,

the chains are not long enough to form many adsorption

blobs.

Close to G�� the free energy excess due to the repulsion

between adsorption blobs is essential and changes the

scaling behavior also for such observables which follow

semidilute surface scaling (i.e., the chain extensions

parallel to the surface). This was already visible in

ol. Theory Simul. 2008, 17, 429–453


Figure 11, where the data points for higher values of the

surface concentration deviate from the scaling predictions.

In order to analyze this effect we consider an observable

O in the semidilute surface state. Then, we assume that the

deviations with respect to this reference state are

controlled by saturation scaling, i.e., by the parameter ~V

defined in Equation (59):

O ¼ OASDfOð~VÞ (62)

where the index ‘‘ASD’’ indicates the value of the

observable in the semidilute surface state and fO is a

function of the saturation scaling variable ~V.

A suitable example is the chain extension in the

direction parallel to the surface:

Rgk ¼ RgkASDfkð~VÞ ¼ G�1=2N1=2fkð~VÞ (63)

where for Rgk Equation (51) has been applied. In Figure 14,

we display the simulation results using Equation (63). Only

states with an adsorption energy per monomer greater

than e¼ 1.4 display scaling according to Equation (63). This

observation is in agreement with the one we have reached

by considering the semidilute surface region, see Figure 11,

where we have found that for e< 1.4 the data do not show

the 2D semidilute asymptotic behavior. Furthermore, for

parameter regions which do not reach semidilute scaling,

one also does not see saturation scaling at higher

concentrations.

Oversaturation Effects

As mentioned above, in the case of tethered chains, the

regime for which ~V � 1 can be physical. We call this

region the oversaturation regime. The regime of over-

saturation can be obtained by increasing the grafting

density beyond the saturation line (see Figure 9).

DOI: 10.1002/mats.200800046


Figure 14. Double logarithmic plot of R2gkG1=2N�1=2 as a function[24]

of ~V. The symbols correspond to different values of e.

Figure 15. Cut through the three-dimensional density distributionof an individual chain at G¼ 4. The left image corresponds to abrush state (e¼0) and the right one to an OSB state (e¼ 2.5).

In the oversaturated state the chains can no longer fully

adsorb on the surface. The single chain order parameter m

is restricted by the available area per grafting point

Macrom

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m � 1

G(64)

as illustrated in Figure 10. Using saturation scaling (as

discussed in the last section) for ~V � 1 the order

parameter obeys

1

G� kð1�fÞ=ffmð~VÞ (65)

To establish Equation (65), we used Equation (60), (61),

and (64). Equation (65) can be satisfied only when fmð~VÞ is

a power law. Then we obtain:

m

m0� ~V

�1and m � kð2n�fÞ=f 1

G� k

G(66)

Figure 16. Double logarithmic plot of the ratio R2g?=R2

gk as afunction of ~

V at different values of e.

where we again used the approximate relation[5,12,23]

n ’ f. Going back to Figure 13, where m/m0 is plotted as a

function of the saturation scaling variable ~V, one can see

that the expected slope [given by Equation (66)] and

indicated in the double logarithmic plot by a straight line is

nicely reached.

We mention here that in the OSB region we expect the

standard brush scaling behavior predicted by Alexan-

der,[46] Milner et al.,[50] and Binder[51] to hold. However, in

OSB the brush forms on the top of the adsorbed layer, as

can be seen in Figure 15. In Figure 15, the density

distribution of monomers in the B state (shown on the left

side) is compared to the OSB state (shown on the right

side). Displayed are cuts through the density distribution

ol. Theory Simul. 2008, 17, 429–453


with two planes, one containing the surface (xy) and one

perpendicular to the surface (xz).

Crossing over from the adsorbed semidilute state to the

oversaturated state, the aspect ratio of the individual

chains changes from oblate to prolate. In Figure 16, we

illustrate this effect by plotting the ratio of the chain’s

extension perpendicular and parallel to the adsorbing

surface as a function of ~V.

Note that the curves in Figure 16 intersect at about

G¼G�� and that for large G/G�� the ratio R2g?=R

2gk becomes

almost independent of e; this can be readily inferred from

Figure 16 from the four data points for largest R2g?=R

2gk

(taken at the same value of G), and enclosed in an ellipse.

Dynamic Properties

In this section, we investigate the fluctuations dynamics of

adsorbed polymer chains. Dynamical problems introduce

time as an additional, relevant variable. For polymers in

the bulk dynamic scaling is well known; it emerges

naturally from exactly solvable models such as the Rouse



444

model for ideal chains[52] and from renormalization group

approaches (see ref.[53]). Usually the longest (characteristic)

relaxation time, t controls the dynamical behavior. We

hence introduce as new dimensionless variable

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V ¼ t=t (67)

Concepts of dynamic scaling have been applied to

various polymer systems such as semidilute solutions or

entangled melts (see ref.[35,52])

Close to the CPA and on time scales much larger than the

segmental relaxation time, the dynamical quantities often

depend on the two relevant variables U and V. Directly at

the CPA, of course, U vanishes and the dependence on V is

particularly evident. An important question now is the

dependence of t on N, which, however cannot be simply

inferred from the dynamical behavior of free chains.

We recall that for a free chain t is related to N according

to a power law[35,54,55]

t ¼ t0Na (68)

with a being the dynamical exponent and t0 being the

characteristic relaxation time of a monomer. For flexible

polymer chains in good solvents and in the absence of

hydrodynamic interactions (situation which is realized in

many simulations), a is given by

a ¼ 1 þ 2n ’ 2:176 (69)

We recall, following ref.,[35,54,55] the main arguments

leading to this result. If we regard the chain as being a

system of N identical, diffusing and interacting particles,

but in the absence of any external interaction (such as

hydrodynamic forces or electrodynamic fields), then the

center of mass motion of the considered system is a force-

free diffusion process. The diffusion constant is then

given by

DCOM ¼ D0=N (70)

where D0 is the diffusion constant of a single particle

(monomer). This result requires the cancellation of all

internal forces according to Newton’s third law. Now,

assuming that the characteristic time scale t is given by

the time needed for the polymer chain to move to a

distance comparable to its own size, R � Nn, leads to the

relation t � R2=D � N1þ2n.

Now, the presence of external forces such as the

adsorption potential invalidate the arguments leading to

Equation (70); thus the dynamical exponent a has to be

determined by other means. Given that a unique

dynamical exponent might appear under critical condi-

tions, we will focus in Section ‘‘Single Chains at the Critical

ol. Theory Simul. 2008, 17, 429–453


Point of Adsorption and the Dynamic Exponent’’ on the

determination of a from the dynamics of chains at the CPA.

Knowing the value of a at the CPA, dynamic scaling

arguments can be applied to provide a basic under-

standing of the dynamics in the adsorbed diluted state;

such arguments and corresponding simulation results are

presented in Section ‘‘Dynamic Scaling in the Adsorbed

State.’’ Finally, in the following sections we consider the

chain dynamics in the semidilute surface state and in the

oversaturated brush-like state.

Single Chains at the Critical Point of Adsorption andthe Dynamic Exponent

In general, an observable O which characterizes some

dynamical property of the system will depend on all

relevant parameters in the scaling limit (length scales

much larger than the segment size and time scales much

larger than the segmental relaxation time). For a single

chain grafted to the surface we thus expect:

O ¼ O0fOðk;VÞ (71)

where O0 denotes some characteristic value of the

observable for given values of k and V. In the mushroom

state dynamical scaling should emulate a bulk chain, apart

from an obvious breaking of symmetry (i.e., the obser-

vables may have different characteristic values with

respect to their direction in space). On the other hand,

for the adsorbed state, k> 0, one can expect the system to

follow the dynamical behavior of a quasi-2D chain.

Indications for a dynamical cross-over from 3D to 2D

behavior during adsorption have already been obtained by

Milchev and Binder.[32] However, in order to explore the

details of the dynamics in the adsorbed state we focus here

on the dynamics of a chain at the CPA. For this one needs in

particular to know the dynamic exponent a.

The End-to-End Vector Correlation Function

An appropriate test of dynamical scaling is provided by the

autocorrelation function of the end-to-end vector in the

direction perpendicular to the surface. This quantity is

given by

C?ðt;N; "Þ ¼ zðtÞ � hzið Þ zð0Þ � hzið Þh ihz2i � hzi2 (72)

In Equation (72), z(t) represents the position of the

monomer at the free end of the chain, in the direction

perpendicular to the surface at time t. Note that for time-

dependent observables averaging has to be carried

out over all the different realizations of the initial

DOI: 10.1002/mats.200800046


conformations of the chain. Thus, dynamic results are very

demanding with respect to computer resources.

At CPA, U vanishes and the behavior is controlled by V

only. Hence, we can write C?ðt;N; "cÞ ¼ f ðt=tÞ. Now, for the

proper choice of the dynamic exponent the explicit N-

dependence of C?ðt;N; "cÞ can be eliminated. Note that the

characteristic value, C0 ¼ C(t¼ 0), is unity and is indepen-

dent of the length of the chain.

In Figure 17, we display results for C? at the CPA using

different values for the dynamic exponent.[23,33] The value

of a is varied on the right hand side of Figure 17 below and

above the value given by Equation (69). In this figure one

can see that small variations of a lead to the loss of scaling.

Best scaling is obtained for a given by Equation (69), results

which are displayed on the left hand side of Figure 17.

Therefore, we assume in the following the relation

Figmoas asidscais a

Macrom

� 2008

aCPA ¼ a ’ 2:176 (73)

to hold.

Mean Square Displacement

More information about the dynamical behavior at the

CPA can be obtained considering the displacements of

different monomers of the chain and of the center of mass

of the chain. We will consider such displacements in the

directions perpendicular and parallel to the substrate. To

analyze the dynamical scaling behavior at the CPA we

recall that the characteristic extension of the chain, Rg,

follows the same behavior as that of free chains: Rgk � Nn

and Rg? � Nn (however with different numerical pre-

factors). With Rg being the only relevant length scale, the

ure 17. Scaling plots of the autocorrelation function of the endnomer in the direction perpendicular to the substrate,[33] C?,

function of t/t at the CPA given by ec ¼ 1.01. On the left hande t¼Na is assumed, where a equals the free chain dynamicalling exponent, a¼ 2.176. On the right hand side of the figure assumed to be either a¼ 2.0 or a¼ 2.3.

ol. Theory Simul. 2008, 17, 429–453


displacement function g(t) takes the form

gðtÞ=R2g ¼ f ðt=tÞ (74)

Note that the displacement functions for grafted chains

are limited from above and that their values at the CPA and

in the limiting case t ! 1 have to be proportional to Rg2.

In Figure 18, various displacement functions,[33,55] see

Table 1, are displayed using the scaling form of Equation

(74). In these plots the estimate for the dynamical

exponent given by Equation (73) has been used.

At the CPA the polymer chain is in an anisotropic state.

This can be inferred from Figure 18(a)–(c), by observing

that the mean square displacement in the direction

perpendicular to the surface reaches its lower plateau

value at earlier times than the component parallel to the

surface.

For the displacement functions of particular monomers,

a dynamic scaling behavior can also be investigated in the

framework of Equation (74). By this we can determine the

pertinent exponents in the subdiffusive region[32,56] given

by t � t. In this region the motion of individual monomers

cannot depend on the total chain length. Using Equation

(74) we thus obtain

gðtÞ / ta1 (75)

with

a1 ¼ 2n=ð1 þ 2nÞ � 0:54 (76)

Figure 18(b) and (c) show that the expected behavior for

short times is well obeyed for the diffusion both of the

middle as well as the end monomer. This is indicated by

straight lines in the corresponding plots of Figure 18(b) and

(c). A more detailed discussion can be found in ref.[33]

Dynamic Scaling in the Adsorbed State

In Figure 19, we present the results for the displacement

function, g4?, at the value e¼ 1.4 of the attraction strength,

well above the CPA value (i.e., at ec¼ 1.01). The correspond-

ing plot at CPA was presented in Figure 18(c). From

Figure 19 we infer that single variate scaling according to

Equation (74) is clearly violated.

In order to understand the dynamics in the adsorbed

state, the concept of adsorption blobs can be used. As

shown in Figure 1, a blob of size L contains a subchain of g

monomers, the relation between L and g being given by

Equation (14). Here, we extend the picture of blobs to

dynamical properties of the adsorbed chain. Our key idea is

to assume that the monomers within an adsorption blob



Figure 18. Scaling plots at the CPA for various N and various displacement func-tions:[33] (a) parallel and perpendicular components of g3; (b) same for g1; (c) same forg4; (d) parallel components of g1 and g2. In (a)–(c) we introduced for comparison alsothe free chain results for N¼ 50, 100, and 160.

446

obey critical dynamical behavior (this means that the

blobs represent a local state at the verge of adsorption).

Adopting the previous result obtained for the dynamical

scaling exponent a, we rewrite Equation (68) as

Tab

g1

N/2

g2

N/2

g3

the

g4

Macrom

� 2008

tg � g2nþ1 (77)

where t was replaced by tg and N by g. In this way tg, given

by Equation (77), represents the characteristic relaxation

time of a blob.

le 1. Displacement functions gk.

Mean square displacement of the center monomer

Mean square displacement of the center monomer

relative to the center of mass

Mean square displacement of the center of mass of

chain

Mean square displacement of the end monomer(s)

ol. Theory Simul. 2008, 17, 429–453


To eliminate the unknown variable g,

we use Equation (13) to rewrite Equation

(77):

tg � L2nþ1n � ðR2

?Þ2nþ1

2n (78)

In the last relation, we used as the

characteristic size of a blob the end-to-end

distance R? of a single chain in the

direction perpendicular to the surface.

Note that large loops are very improbable

in the AD region of the phase diagram (see

Figure 9). Therefore, the characteristic

relaxation time in the direction perpendi-

cular to the surface is given directly by tg:

t? / tg (79)

In Figure 20, the data of Figure 19 are

rescaled according to Equation (79).

Obviously, these results are in good

agreement with the scaling argument

given above. However, the validity of the

scaling exponents used here must be taken

with care. It has to be noted that the

relaxation time of the motion perpendi-

cular to the surface is rather short, since

this corresponds to the relaxation of

individual blobs. This short relaxation time

together with the fact that our coarse

graining in time is quite large (5000 MCS,

see the discussion about the simulation

details) already brings us close to the plateau. Therefore,

the scaling region in Figure 20 is rather small and it

extends only over a factor of two to three in the

displacement function.

Assuming that the adsorption blobs are the dynamical

units of the chain, we can extend our scaling analysis to

the motion in the direction parallel to the substrate. The

adsorbed chain is again viewed as a 2D effective chain

made of N/g blobs, where each blob relaxes during the

time tg. Hence, in this case tg replaces the segmental

relaxation time t0 in Equation (68). Using this idea we can

express the characteristic relaxation time for motion

parallel to the surface by using Equation (78) and by

generalizing Equation (69) to 2D:

tk ¼ ðN=gÞ2n2þ1tg � N2n2þ1g�2n2�1L2þ1=n (80)

Furthermore, using Equation (13) we obtain

tk � N2n2þ1L�2n2=n�1=nL2þ1=n � N2n2þ1L�2Dn=n (81)

DOI: 10.1002/mats.200800046


Figure 19. Double logarithmic plot of the reduced displacement ofthe end monomer g4?=N2n as a function of t/t with t¼N2.176, inthe adsorbed regime at e¼ 1.4.

Figure 21. Double logarithmic plot of the motion of the centralmonomer in the direction parallel to the substrate, g1k=R2

1k, as afunction of t=tk, for different values of e in the adsorbedregime.[33] The theoretically expected slope is also plotted. Fora better visualization of the simulation points for differentparameters N and e, some of the data points were left out.
Using the relation between L and k given by Equation
(14), one can write

Figponas

Macrom

� 2008

tk � N2n2þ1k2Dn=f (82)

For t � tk the monomer motion is subdiffusive. The

corresponding exponent a12D, see also Equation (76), is now

given by

a2D1 ¼ 2n2=ð1 þ 2n2Þ ¼ 0:6 (83)

Dynamical scaling implies for the displacement func-

tion of the middle monomer

g1kR2

1k¼ f1kðt=tkÞ (84)

ure 20. Double logarithmic plot of the perpendicular com-ent of the reduced end monomer displacement[33] g?=R2

?a function of t/tg at e¼ 1.4.

ol. Theory Simul. 2008, 17, 429–453


In Equation (84), we use to rescale g1k the average square

distance R21k of the central monomer from the anchor point

in the direction parallel to the surface.

Now, R21k contains the contributions of N and of e in an

implicit form,[7] and thus they do not enter explicitly into

f1k. The corresponding scaling result is displayed in

Figure 21. In this figure, g1k=R21k is plotted versus t=tk for

different N and e values. Figure 21 shows that the

simulation data follow a master curve and that in the

subdiffusive region the slope expected from Equation (83)

is indeed well obeyed. This confirms previously obtained

results by Milchev and Binder.[32] Dynamics of single

polymer chains has been considered also in confined

systems like slits[57] and nanotubes.[58] Moreover, it has

been shown in ref.[57] that the situation of adsorbed

polymers on a surface is similar to that of a polymer

confined in a slit, so that the scaling theory for the

dynamics of the two systems is analogous.

Dynamics in the Semidilute Surface State

We now investigate the dynamics in the region ASD of the

phase diagram (see Figure 9). In this regime, G� � G � G��,

the adsorbed polymer chains overlap strongly. We consider

first the diffusive motion in the direction perpendicular to

the surface. Motivated by the results for the case G ! 0, we

again focus on the time scale tg, which corresponds to the

relaxation of a nearly unrestricted chain of extension R?,

as given by Equation (78). In Figure 22, we present the

simulation results for the displacement g4? of the end-

monomer perpendicular to the surface, using the scaling of

length and time variables at the given values of e and G. For

comparison, we have added the simulation results for



Figure 22. Double logarithmic plot of the motion of the endmonomer in the direction perpendicular to the substrate,g4?=R2

?, as a function of t/tg, for different values of e and G inthe semidilute surface regime. The characteristic relaxation timeof the adsorption blob, tg, is given by Equation (78), derived forthe adsorbed state in the dilute surface regime.

Figure 23. Same as in Figure 22, but here for g4k.

448

G ! 0. We observe from Figure 22 that the dynamics of the

blobs largely controls the diffusive motion in the ASD state

and that the displacement perpendicular to the surface is

well characterized by the relaxation dynamics of the blobs

which form the adsorption layer. It is worth noting that in

the ASD region large loops are formed, fact which has been

discussed in Section ‘‘Many Chains and their Phase

Diagram.’’ These chain parts are implicitly taken into

account by using the relevant length scale R? in defining tg

according to Equation (78).

We turn now to the dynamics in the direction parallel to

the surface. As we have discussed in Section ‘‘Many Chains

and their Phase Diagram,’’ in the ASD state a hierarchy of

blobs appears (see Figure 9 and 10). This leads to a complex

dynamics controlled by several parameters. To simplify

the analysis we apply dynamical scaling on the level of

adsorption blobs only.

The simulation results are presented in Figure 23, which

shows that the data can be fit to a master curve. Depending

on G and e, the curves start bending at long times. As

already discussed in Section ‘‘Dynamic Scaling in the

Adsorbed State,’’ dynamic scaling implies a subdiffusive

behavior on time scales smaller than the relaxation times

of the blobs. In the present case this implies for the end

monomer the following power law:

Macrom

� 2008

g4 / t2n=ð1þ2nÞ (85)

In fact we can observe in Figure 23 at small times a

corresponding, subdiffusive behavior. Following the argu-

ments of Section ‘‘Dynamic Scaling in the Adsorbed State,’’

at times larger than tg a slope of 0.6 should be visible; it is

due to 2D excluded volume effects, which control the

statical and dynamical properties on these scales. In the

ol. Theory Simul. 2008, 17, 429–453


ASD state, however, such a behavior is restricted to the 2D

concentration blobs. On larger scales, the chain statistics is

Gaussian,[52] which leads, instead of Equation (85), to a

subdiffusive behavior with an exponent of a1R¼ 1/2. This

behavior can be indeed observed in Figure 23 at

intermediate times, before the cross-over to a plateau

regime takes place.

From our data, we cannot localize the intermediate

dynamical regime which corresponds to the relaxation of

2D excluded volume blobs. Instead, the 3D relaxation

behavior of Equation (85) seems to cross-over directly into

the subdiffusion regime of ideal chains. A possible reason

for the missing intermediate range could be the fact that in

our simulations the chain lengths are rather short. Another

possible reason for the missing intermediate state may be

due to the relaxation of the large loops formed in the ASD

state. As already noted above, we introduced the

characteristic time scale tg based on the thickness of the

adsorbed layers, a quantity which also accounts for

contributions from large loops. Now, the dynamics of

such loops may interfere with the relaxation of the 2D

excluded volume blobs and may obscure the subdiffusive

regime of an ideal 2D excluded volume chain.

As noted above, the subdiffusive regimes are limited by

the onset of saturation due to grafting. This is visible for

almost all parameters in Figure 23.

Dynamics in the Oversaturated State

For G>G�� the tethered chains interact strongly; thus, by

increasing the concentration first long tails and then a

brush-like state forms above a densely adsorbed layer. In

the following we again consider g4, the mean square

displacement of the end monomer, and monitor its

evolution in time. In Figure 24, we plot the results of

our simulations for several values of e and for two values of

G above the saturation concentration. These values are in

DOI: 10.1002/mats.200800046


Figure 24. Double logarithmic plot of g4 as a function of t. Theresults corresponding to the parallel and to the perpendicularcomponent of g4 are displayed for different values of G and e.

Figure 25. Double logarithmic plot of g4?s as a function of t=tg3D .The various symbols represent simulation results of g4? fordifferent values of G and e.

the region OSB (G¼ 4) and also possibly in OSA (some data

for G¼ 1). Displayed are the results for the components

parallel and perpendicular to the surface. We infer from

Figure 24 that the mean square displacement of the end

monomer is only slightly influenced by e, both for motions

parallel and perpendicular to the surface.

Having in this regime a brush-like state formed on top of

the adsorbed layer, we expect that the dynamics is

controlled by 3D semidilute blobs formed in a similar

way as in the case of ‘‘simple’’ brushes (i.e., where no

attractive interactions between the monomers and the

surface exist). We recall here that the chains of a polymer

brush in a good solvent can be viewed, according to the

picture of Alexander and de Gennes,[46,47] as being linear

coarse-grained chains made out of blobs whose diameter is

around j3D ¼ s�1=2. Inside such 3D blobs the conforma-

tions of the subchains resemble those of single chains in

good solvents. Therefore, denoting the number of mono-

mers in the blob by g3D, the size of the blob scales as

j3D � gn3D. Hence, one can write for the relaxation time of a

blob (see also ref.[9,59,60] which deal with brushes)

Macrom

� 2008

tg3D

� j23D=g3D � s�1�1=2n (86)

see ref.[51] for a summary of the literature on the relaxation

times of chains in polymer brushes. These results are:

t?3D � N3s2=3 (87)

for the motion perpendicular and

Figure 26. Double logarithmic plot of g4k=R2k as a function of

tk3D� N2s�1=6 (88)

t=tk3D. The various symbols represent simulation results of g4kfor different values of G and e.
for the motion parallel to the surface.
ol. Theory Simul. 2008, 17, 429–453


In Figure 25, we present the results of our simulations

for g4 and motion perpendicular to the surface using the

scaling ansatz obtained in the 3D semidilute blob picture.

Hence, we rescale g4? by j3D2 and time by the 3D blob’s

relaxation time given by Equation (86). Scaling turns out to

be reasonable, when one takes into account that the states

corresponding to G¼ 1 (s¼ 0.01) and to 1.2< e< 1.8 are not

in the oversaturated regime (see Figure 16 and its

discussion).

Considering now in Figure 26 the motion parallel to the

surface we rescale g4k by the mean square end-to-end

distance parallel to the surface, R2k , and the time by the

relaxation time tk3D of the chain in the brush regime, given

by Equation (88). The obtained results follow nicely a

master curve for the set of parameters listed in Figure 26.

Finally, we note that experimentally one of the methods

used to investigate the dynamics of end-grafted polymers



450

is deuterium nuclear magnetic resonance (2H NMR).[61,62]

This method is sensitive to the anisotropy of the molecular

motion and therefore it has been used to study the

segmental motion under variations of the grafting density

and of the temperature.[62]

Conclusion

Grafted polymer chains usually interact with the substrate

they are tethered to. We have shown that this leads to a

multitude of phases, depending on the interaction

strength (or, equivalently, on the temperature) and on

the grafting density. One of the major results of our work

here is the state diagram presented in Section ‘‘The Phase

Diagram.’’ There we have shown that the simple mush-

room-brush transition splits into distinct phases in the

presence of an adsorbing substrate. In particular, a

semidilute surface state appears when adsorbed polymer

chains start to overlap. In order to understand the static

and dynamical properties of such systems we made use

both of scaling methods and of computer simulations.

While scaling models reveal the dominant geometrical,

thermodynamical, and dynamical features, computer

simulations provide powerful tests in terms of master-

curves; from such simulations several parameters can be

directly read-off, the cross-over thresholds and the

universal cross-over functions can be determined, and

many observables a priori not accessible by scaling can be

obtained.

An important precondition for the investigation of

surface states of polymer chains is the knowledge of the

set of critical parameters, namely of the critical adsorption

strength (i.e., of the CPA) and of the cross-over exponent.

Namely, in scaling analysis the interaction energies of the

system are related to the CPA. The cross-over exponent is

related to the size of the adsorption blob (i.e., to the

correlation length). An accurate estimation of the CPA is

mandatory, since even small changes in this quantity have

a large impact on the results. In this work we have

reviewed various methods to find the set of critical

parameters. The basic problem here is to extract such sets

of parameters, formally defined in the limit of infinitely

long chains, from simulations which use chains of finite

length. Based on the assumption that corrections to scaling

are small for chain lengths in the range of a few hundred

monomers, we have applied a best scaling approach in

order to obtain an optimal set of critical parameters. The

simulation results have been found to be consistent with

the subsequent scaling analysis of various surface states.

For grafting densities intermediate between the overlap

state of adsorbed chains and the saturation density of the

adsorbed layer, a semidilute surface state forms. The

semidilute surface state corresponds to a semidilute



solution in two dimensions. Here, the scaling analysis is

based on a hierarchy of blobs; this hierarchy consists of

adsorption blobs at small length scales and of 2D

concentration blobs (formed by coarse-grained chains

made of adsorption blobs) at large length scales. Using

such multi-blob scaling arguments, one tacitly assumes

the decoupling of the interaction effects responsible for

blob formation on different scales. In the particular case of

the semidilute surface state one implicitly assumes that

excluded volume effects between overlapping adsorbed

chains do not effect the free energy of the individual

adsorption blobs. To go beyond this assumption we have

taken into account the influence of the excluded volume

effect in a mean-field picture and have put forward the

concept of saturation scaling. Observables such as the

order parameter (the relative number of monomers in

contact with the surface) which are invariant with respect

to semidilute scaling can be analyzed using our saturation

scaling approach. The reduction of the order parameter

with increase in surface concentration, as observed in

simulations, can be explained using saturation scaling.

In contrast to the free chains adsorbed from solution, by

increasing the grafting density the grafted chains can

display oversaturation. In the oversaturated state not all

monomers can be accommodated in densely packed

adsorption blobs and a fraction of monomers has to

remain in the non-adsorbed state. We have shown that

there is no direct transition from the saturated surface

state to the brush state, but that there exists a narrow

region in between both states where oversaturation does

not yet cause the stretching of the individual chains in the

direction perpendicular to the surface.

The dynamical properties of grafted chains at attractive

surfaces can also be described using scaling concepts.

Adsorption leads to different time scales: While diffusion

processes in the direction perpendicular to the surface are

limited by the blob dynamics, the lateral diffusion can be

understood based on the 2D conformational properties of

the chains. In particular, the diffusion dynamics in the

direction parallel to the surface can be much slower than

the diffusion dynamics of free chains. If the surface

concentration crosses the saturation threshold, the brush-

like state starts to dominate the dynamics of the chains.

For very high grafting densities one may even envisage the

opposite effect. Then the relaxation times in the direction

perpendicular to the surface may increase, while the

relaxation times in the direction parallel to the surface

may decrease gradually.

The study of substrate interactions on grafted polymer

chains reveals an intimate interplay between adsorption

and concentration effects. By using different solvents this

fact may even be used to switch the polymer layer from

one state to the other. The solvent molecules can namely

change the effective polymer/substrate interactions and

DOI: 10.1002/mats.200800046


thus can be used to shift the system from adsorption to a

critical state or even to a situation in which the surface is

repulsive. Such effects are already found and used in liquid

chromatography.[63] By an appropriate choice of para-

meters (such as the chain length and the grafting density)

it is then possible to switch between the brush state and

the semidilute adsorbed state, thus considerably changing

the thickness and the surface properties of the polymer

layer.

Appendix A: Simulation Method

The Bond fluctuation method (BFM) is an algorithm with

local moves which was introduced by Carmesin and

Kremer.[64] Besides studying the static properties of

polymers in the presence of an adsorbing surface, BFM

is also suitable for investigating dynamical properties. The

algorithm was already used in the research on polymers in

the vicinity of surfaces.[65–67] It is well known that the BFM

reproduces the universal static and dynamic behavior of

polymers, both of dilute solutions in good solvents (in the

absence of hydrodynamic interactions) and also in

concentrated solutions.[68] In BFM, the time is measured

in Monte Carlo steps (MCS). One MCS corresponds to one

attempted move in average per monomer (N attempted

moves per chain). Since the MCS is the unit ‘‘time’’ in our

simulations, its correspondence to the physical unit time

has to be discussed. In a coarse-grained model an effective

monomer corresponds to about five chemical monomers. A

typical estimate under usual conditions is that one MCS

corresponds to about 10�11 s.[51] In our work, in all cases

where we plot the data of a simulated dynamical quantity

as a function of time, the time is expressed in MCS units. In

the following we present the main properties of BFM

(focusing on the 3D-case).

Figure 27. Presented is a polymer chain in a three-dimensionalsimulation box. The straight lines between the cubes show theconnectivity of the monomers and the numbers give the index ofthe monomers along the chain contour. A monomer occupieseight lattice sites and it can move according to the BFM rules insix directions (x, �x, y, �y, z, �z). The chain is tethered on thesurface, i.e., the monomer with index 0 cannot move, neitheraway from the surface nor on the surface. In the z-direction theboundaries of the simulation box are defined by the attractingsurface (on which the monomer with index 0 is tethered), and byan opposite reflecting boundary. In the x- and y-directions per-iodic boundary conditions are implemented.

The Bond Fluctuation Method

One of the most important features of the BFM is that it

reproduces (in the absence of hydrodynamic interactions)

the Rouse dynamics. This means that the motion of a

polymer is described by the Brownian stochastic motion of

the monomers which form the chain. Another important

feature of the BFM is its ergodicity. The BFM has been used

for extensive simulations to study the dynamics of

polymeric systems.[68–71]

In the 3D BFM each monomer of the polymer is

represented by a unit cube which occupies eight lattice

sites. An important property of this model is that it allows

for a variable bond length. The bonds between two

successive monomers are chosen in such a way as to

impede the bonds to cut themselves during local monomer



moves. To fulfill this condition a set of 108 vectors are

allowed; they correspond to the basis set given by the

vectors {(2, 0, 0), (2, 1, 0), (2, 1, 1), (2, 2, 1), (3, 0, 0), (3, 1, 0)}

and by all the other vectors which follow from them

through symmetry operations on the lattice.

The dynamics is realized in the following way; a

randomly chosen monomer (from the N monomers of the

system) attempts to make a move in a randomly chosen

direction (from the six possible ones). The move is accepted

subject to the following conditions: The new position

should be unoccupied, i.e., the excluded volume conditions

should be fulfilled and the new bond vectors should belong

to the set of allowed bond vectors (see above). A Metropolis

algorithm[72] is applied if further interactions are imple-

mented. An illustration of the BFM is sketched in Figure 27.

The figure displays a tethered polymer chain with six

monomers on a three-dimensional lattice. Depending on

the studied problem, additional rules may hold.

In the following, we discuss the implementation of the

3D BFM for each particular problem under study.

Simulation Details: Single Chains

To simulate the reversible adsorption of single chains on

an attracting planar surface we consider a 100 100 100

lattice. Periodic boundary conditions are implemented in

the x- and y-directions, and reflecting walls in the



Table 2. Parameters of the simulation box.

s 10 20 25 30 35 40 60

Lx¼ Ly 100 200 250 300 350 400 600

G 4.0 1.0 0.64 0.44 0.32 0.25 0.11

452

z-direction (see Figure 27). Since we study the behavior of

polymer chains in the presence of an adsorbing surface we

implement a Metropolis algorithm. The monomer/surface

interaction is represented by an attractive, short-ranged

potential e0, acting only at a distance of one lattice unit

from the surface. The Metropolis scheme implies that a

monomer located at the surface (z¼ 0) can leave it only

with a probability

Fig(misur

Macrom

� 2008

p" ¼ expf�"g (A1)

Figure 29. Snapshot of the relaxed tethered system of chains forG¼0.44 (i.e., for s¼ 30) and e¼ 2.2. The lightly colored monomersare in contact with the surface.

where e¼ e0/kT. By setting kT¼ 1 we will focus in the

following only on e. A reflecting wall in the z-direction

implies that the monomers cannot cross the wall and all

moves violating this condition have to be rejected. In our

simulation, we fix (tether) one end of the chain to the surface,

as it has been considered also in previous works.[5,38,73]

We consider fixed values of the chain length N

(20 � N � 200) and fixed values of the surface attraction

e (0:0 � N � 2:0). Before setting e for a given value of N, the

chain is allowed to relax in the absence of any surface

attraction. This, depending on N, requires several hundred

thousands MCS. A snapshot with an equilibrated chain

conformation is presented on the left hand side of

Figure 28. Using the equilibrated state another run is

started, where the interaction e is set to a fixed value. The

chain is allowed to relax again during a period of

(about) 108 MCS. To be sure that the chain conformations

are indeed well equilibrated, for large values of e (e> 1.1)

the equilibration periods were extended up to 109 MCS. In

the middle and on the right hand side of Figure 28

snapshots with equilibrated conformations are presented

for e¼ 0.8 and for e¼ 1.4, respectively. The obtained relaxed

conformations are collected in files in steps of 5000 MCS.

After collecting the simulation data we perform their

analysis and evaluate the quantities of interest.

Simulation Details: Many-Chains Systems

For simulations involving many chains in contact with the

surface we proceed as follows: all chains are anchored at

the adsorbing surface and all of them are made of N¼ 100

monomers. The chains are grown up in a regular square

ure 28. Snapshots of chain conformations with N¼ 100 for e¼0ddle) and for e¼ 1.4 (right). The lightly colored monomers are inface.

ol. Theory Simul. 2008, 17, 429–453


array of 10 10 chains with a distance of s lattice units

between the grafting points in the xy-plane. To grow up

the chains we use the (0, 0, 2) vector only. In the x- and y-

directions we implement periodic boundary conditions. In

the z-direction we place the adsorbing surface at z¼ 0 and

a reflecting wall at z¼ 210. Note that even totally stretched

chains (reaching the z¼ 200) do not touch this wall. By

varying s the simulations are performed for several

monomer surface concentrations G [see Equation (28)].

The parameters of the simulation box are presented in

Table 2. Here, Lx, and Ly are the (equal) dimensions of the

box in the x- and y-directions.

We equilibrate in the absence of any interaction with

the surface for each fixed value of G every system during

2.5 107 Monte-Carlo steps. After this procedure the

interaction between the monomers and the surface is

turned on. This interaction is introduced in the same way

as for single chains. We set the interaction e to a prescribed

value between 1.1 and 2.5 for each fixed value of G and

allow the chains to relax again during a period of 2.5 107

MCS.

In Figure 29, we present a snapshot of an equilibrated

system for G¼ 0.44 and e¼ 2.2.

(left), for e¼0.8contact with the

Acknowledgements: The support of theDeutsche Forschungsgemeinschaft and of theFonds der Chemischen Industrie is gratefullyacknowledged. R. D. thanks the IPF Dresden forits hospitality during a research stay.

Received: May 20, 2008; Revised: September 4,2008; Accepted: September 10, 2008; DOI:10.1002/mats.200800046

DOI: 10.1002/mats.200800046


Keywords: Adsorption; Monte Carlo simulations; Polymers; Scal-ing; Surfaces

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Grafted Polymer Chains Interacting with Substrates: Computer Simulations and Scaling

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