Irreversible adsorption of tethered chains at substrates: Monte Carlo study

THE JOURNAL OF CHEMICAL PHYSICS 124, 094701 �2006�

Irreversible adsorption of tethered chains at substrates:Monte Carlo study

Radu Descasa�

Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germanyand Institut de Chimie des Surfaces et Interfaces, 15, rue Jean Starcky, F-68057 Mulhouse Cedex, France

Jens-Uwe SommerInstitut de Chimie des Surfaces et Interfaces, 15, rue Jean Starcky, F-68057 Mulhouse Cedex, Franceand Leibniz-Institute for Polymer Research Dresden, Hohe Strasse 6, D-01069 Dresden, Germany

Alexander BlumenTheoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany

�Received 5 August 2005; accepted 30 November 2005; published online 1 March 2006�

The irreversible adsorption of single chains grafted with one end to the surface is studied usingscaling arguments and computer simulations. We introduce a two-phase model, in which the chainis described by an adsorbate portion and a corona portion formed by nonadsorbed monomers. Theadsorption process can be viewed as consisting of a main stage, during which monomers join by“zipping” �along their order in the chain� the surface, and a late stage, in which the remaining coronacollapses on the surface. Based on our model we derive a scaling relation for the time of adsorptiont�M� as a function of the number M of adsorbed monomers; t�M� follows a power law, M�, with��1. We find that � is related to the Flory exponent � by �=1+�. Using further scaling argumentswe derive relations between the overall time of adsorption, the characteristic time of adsorption�given by the crossover time between the main and the last stage of adsorption�, and the chainlength. To support our analysis we perform Monte Carlo simulations using the bond fluctuationmodel. In particular, the sequence of adsorption events is very well reproduced by the simulations,and an analysis of the various density profiles supports our theoretical model. Especially the loopformation during adsorption clearly shows that the growth of the adsorbate is dominated by zipping.The simulations are also in almost quantitative agreement with our theoretical scaling analysis,showing that here the assumption of a linear relation between Monte Carlo steps and time is wellobeyed. We conclude by also discussing the geometrical shape of the adsorbate. © 2006 AmericanInstitute of Physics. �DOI: 10.1063/1.2159479�

I. INTRODUCTION

In polymer physics the behavior of polymer chains atinterfaces and surfaces is of great importance and has at-tracted much attention.1–3 Denoting by � the surface stickingenergy per monomer, one distinguishes two situations:reversible adsorption ���kT� and irreversible adsorption���kT�. While the reversible adsorption is rather well un-derstood, there is considerably less knowledge about the ir-reversible adsorption due to the nonequilibrium nature of theprocess. On the other hand, strong adsorption plays an im-portant role in the preparation of thin polymer films and insurface modification phenomena, see Refs. 4–6. Alsobiopolymers such as DNA or proteins interact strongly withthe substrate through hydrogen bonds.

Considering the nature of the bonding of the polymerchain to the surface, the �irreversible� adsorption can be clas-sified as physisorption when the adsorbing energies havetheir origin in hydrogen bonding, in dipolar forces, or in theattraction between charged groups and as chemisorptionwhen covalent bonds are formed between the polymer and

a�

0021-9606/2006/124�9�/094701/8/$23.00 124, 0947

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the surface.7 It has been argued by O’Shaughnessy andVavylonis7,8 that chemisorption can be well understood byassuming that the chain conformations equilibrate betweensuccessive adsorption events. Therefore, the adsorption pro-cess can be considered as being quasistatic. On the contrary,strong physisorption involves true nonequilibrium states.Monte Carlo simulations have been used to study singlechain physisorption by Shaffer.9 It was found that the char-acteristic time of adsorption � is related to the chain length Nby a power law �scaling� relation ��N�. In Ref. 9, the dy-namic exponent � has been found to be around 1.6 for ex-cluded volume chains and around 1.5 for ideal chains. Usinga nonequilibrium force-balance argument, it was first sug-gested by Ponomarev et al.10 that � can be approximated by�=1+�, where � denotes the Flory exponent.

According to O’Shaughnessy and Vavylonis,7 one candistinguish two major scenarios for the irreversible adsorp-tion of single chains. Starting from an anchored conforma-tion, Fig. 1�a�, the different ways in which the processevolves in time are depicted in Figs. 1�b� and 1�c�. In thefollowing we denote the first way �shown in Fig. 1�b�� assimple zipping. Here, the spontaneous formation of new ad-

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094701-2 Descas, Sommer, and Blumen J. Chem. Phys. 124, 094701 �2006�

adsorbate by simple zipping �the sequential adsorption of amonomer next to the last adsorbed monomer� controls thekinetics. An alternative scenario is displayed in Fig. 1�c�where the kinetics is controlled by a combination of sequen-tial adsorption and of spontaneous nucleation of new adsorp-tion spots. Following the notation in the literature7 we denotethis as accelerated zipping. In Fig. 1 we display as sketchesthe theoretical model; next to the sketches we present resultsfrom the Monte Carlo simulation runs to be discussed later.In previous works evidence has been obtained9,10 that strongphysisorption is dominated by simple zipping kinetics. Theformation of larger loops which would create new nucleationpoints far away from the last adsorbed monomer is very un-likely, see also Ref. 11.

In this work we want to clarify the mechanism of strongphysisorption. We propose a nonequilibrium model for theirreversible adsorption process. Using scaling arguments wecalculate the dynamical exponent of the zipping process, theoverall time of adsorption, and the characteristic time scaleof adsorption. By means of Monte Carlo simulations, we testour model and the predicted scaling laws. By averaging overa large number of statistically independent chains, we ana-lyze the density profiles, the formation of loops, and thegrowth of the adsorbed part. In particular, we find that zip-ping proceeds sublinearly in time, the rate of adsorption de-creasing steadily.

The paper is structured as follows. In Sec. II we presenta scaling model for the process of irreversible physisorption.In Sec. III we display and discuss the results of Monte Carlosimulations obtained by using the bond fluctuation model;we summarize our conclusions in Sec. IV.

II. TWO-PHASE MODEL FOR IRREVERSIBLEPHYSISORPTION

Generally, the process of irreversible adsorption of asingle polymer chain can be formally subdivided into twostages, namely, into nucleation and into the growth of theadsorbate. Here, nucleation denotes the process by whichnew contact points of the chain with the surface are created.In the present case one nucleation site is always given by theanchor point of the chain. In general, for a given chain,nucleation can happen several times, always when a mono-mer which is not nearest neighbor to an already adsorbate

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acceleration of the adsorption process. An important dynami-cal step in irreversible physisorption is given by the simplezipping of monomers next to the last adsorbed unit. Here, wepropose a model for the physisorption which is sketched inFig. 2.

Before the adsorption process starts, the chain is in amushroomlike state, the monomer with index zero beinggrafted to the surface. Then, the rate of zipping of the firstmonomers is determined by the time needed by these mono-mers �which are located in the region of around a segmentlength around the grafting site� to touch the surface. Thus,the first zipping events are limited by the monomeric timescale of fluctuations. Within this time scale, the other mono-mers in the chain will not have time to adsorb and will re-main in an equilibrium state. As the process goes on, thedragging in of the monomers becomes limited by the tensionwhich builds up between the last adsorbed monomer and thenonadsorbed part of the chain. Thus, the chain can be viewedas consisting of two nonequilibrium phases: the adsorbate onone hand and the nonadsorbed phase on the other hand,which we call the corona. The two phases are connected bya stretched part which we call the stem and which is continu-ously growing in the course of the adsorption process. Ac-cording to this model, the adsorption process is limited bythe transport of monomers from the corona to the surface

FIG. 1. The two physisorption mechanisms ��a� chainbefore adsorption�: �b� simple zipping and �c� acceler-ated zipping �spontaneous nucleation of new adsorptionspots�.

FIG. 2. Sketch of the proposed model for physisorption. The chain consistsof two phases: the adsorbed phase �adsorbate� and the nonadsorbed phase

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094701-3 Irreversible adsorption of tethered chains at substrates J. Chem. Phys. 124, 094701 �2006�

through the stem, a process which does not disturb the shapeof the corona. Thus, a cooperative motion of many mono-mers is required to pull down the next monomer. Eventually,depending on the length of the chain, the corona will get tobe small enough to be moved as a whole by the zippingprocess and will hence finally collapse on the surface.

Based on this idea, we can put forward a dynamic equa-tion for the zipping process. Let us denote the number ofmonomers in the stem by Ns. In order to drag the next mono-mer onto the surface the whole stem must move by one unit�M, where M denotes the number of adsorbed monomers.Hence we write

�t�M� � Ns�M . �1�

Here, we assume that the friction constant is proportional tothe stem length �and hence that is inversely proportional toits diffusion constant�. Now, the stem extends into a regionof around lNs �l being the statistical segment length�, whichwas occupied at the beginning according to the conforma-tional statistics of the mushroom. Given that Ns is ratherlarge we have

M � Ns1/� − Ns � Ns

1/�. �2�

Here, we have assumed the usual excluded volume scalingfor the mushroom state. We note already at this point that fornot too large values of M, the exponent in Eq. �2� is expectedto be larger than �, since the monomers try to avoid thesurface and stretch towards the bulk. With this equation weobtain after carrying out the integration in Eq. �1�

t�M� = M�, �3�

where

� = 1 + � . �4�

The exponent � is smaller than the Rouse exponent, given by1+2�. Note that this is a necessary condition for the consis-tency of our model. We will call the time region where thismechanism is dominating the main stage of the irreversibleadsorption process. Our model implies that the formation ofthe stem progressively separates the corona from the surfaceand that due to it the nucleation of new adsorption spotsbecomes more and more improbable. This fact will be con-firmed by the simulation results presented in Sec. III. Thus,the process by which the two nonequilibrium phases areformed is self-sustaining: If nucleation events take place atthe beginning, they will be suppressed in the course of ad-sorption. We note that our arguments remain valid if morethan one stem connects the adsorbate with the corona: Weonly have to assume that such events are rare enough not tointerfere with the time scale derived from the two-phasemodel, see below.

At later times, the corona fluctuations become compa-rable with the time scale which corresponds to the draggingof a monomer unit �via the stem� onto the surface. Then,pulling the full corona is faster than increasing the length of

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Nc � Ns � M�, �5�

where Nc denotes the number of monomers in the corona andwe have used Flory’s relation. Note that at this stage the rateof adsorption should be constant. Furthermore, the value ofNs is clearly limited by the radius of gyration of the chain,i.e., one has Ns�N�. The exponent in Eq. �5� indicates that�provided that the chain is long enough� the late adsorptionstage is short compared with the main stage. Thus, we canassume that the overall adsorption time � is controlled bythe crossover time � between the two stages which resultsfrom Eq. �3� for M �N. Hence, we can write

� � N1+� �6�

and

� � N1+�. �7�

Since our model predicts an overall adsorption timewhich is much faster than the overall relaxation time �Rousetime� of the chain, the formation of additional adsorptionsites becomes increasingly unlikely. In fact, the transportthrough the stem consumes the monomers next to the surfacein their natural order along the chain; monomers with a highindex can only reach the surface if they were close to italready at the beginning of the adsorption process.

III. SIMULATION RESULTS AND DISCUSSION

In order to test our theoretical predictions and to analyzethe irreversible adsorption process in more detail, we per-form numerical simulations using the three dimensional �3D�bond fluctuation model �BFM�.12 Since physisorption is veryfast and the kinetics of adsorption plays an important role init, the dynamics is realized by local moves only. Now, thereare works which report that dynamic Monte Carlo simula-tions might fail if the transition probabilities are not chosenproperly, see Refs. 13 and 14. However, in many circum-stances BFM was shown to reproduce quite well the univer-sal static and dynamic behavior of polymers, both in dilutesolutions in good solvents �in the absence of hydrodynamicinteractions� and also in concentrated solutions.15 For a re-cent comparison between the BFM and dynamic self-consistent field calculations, see also Ref. 16. Here, our firstaim is to follow the adsorption process numerically in orderto substantiate the model proposed by us. As it will turn outin the following, however, our BFM results are in very goodagreement with the scaling relations Eqs. �3�–�7�. We willinfer that for the problem discussed here, the BFM performsquite well.

For details of the BFM model for single chain adsorp-tion, we refer to our previous work.17,18 Here, we give only abrief account of its implementation for irreversible adsorp-tion. As simulation volume we use a 100100100 lattice;we implement periodic boundary conditions in the x and ydirections and reflecting walls in the z direction. The volumeis big enough to accommodate the chain conformations inthe nonadsorbed as well as in the adsorbed state. One end ofthe chain is anchored at the origin, �x ,y ,z�= �0,0 ,0�. For thelength of the chain N �number of monomers� we consider

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094701-4 Descas, Sommer, and Blumen J. Chem. Phys. 124, 094701 �2006�

The monomer-surface interaction is modeled by an at-tractive, short-ranged potential �, acting only at a distance ofone lattice unit from the surface. An adsorbed monomer �atz=0� is then taken to be in an infinitely deep potential trap��=�: therefore it can move neither away from the surfacenor along the adsorbing plane. This corresponds to strongphysisorption onto an atomically rough substrate.

We start the simulations by letting the tethered chainequilibrate for �=0, i.e., in the absence of any surface attrac-tion. Depending on N, this requires several hundred thousandMonte Carlo steps �MCSs�. One MCS corresponds to oneattempted move per monomer in average. After letting thechains equilibrate we switch on the irreversible surface at-traction ��=� and start monitoring the adsorption events.

A. Shapes of the tethered chain duringadsorption

To support our model and the geometric sketches ofFigs. 1�b� and 2 we present in Figs. 3 and 4 a cut through the3D density distribution for chains of length N=100 at vari-ous stages of the adsorption process. The cut is defined bythe plane y=0 which contains the anchor point. In Figs. 3and 4 the z axis points towards the left �the surface is on theright and the corona is on the left side�. We display in Fig. 3snapshots from the main stage of adsorption and in Fig. 4from the late adsorption stage. At each s considered, s being

FIG. 3. Three dimensional illustration of the density distribution of allmonomers. Presented is a cut in the plane perpendicular to the adsorbingsurface through the grafted monomers. The images represent the evolutionin MCSs �from s=0 to s=10 000 MCSs in steps of 2000 MCSs from top leftto bottom right� of the distribution of all monomers. Displayed are averagesover 20 000 independent realizations.

FIG. 4. Same as in Fig. 3, only that here s=20 000 MCSs for the first image,s=30 000 MCSs for the second image, and s=50 000 MCSs for the last

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the number of MCSs, the density is averaged over 20 000independent realizations. In Fig. 3 we show the density start-ing from s=0 in steps of 2000 MCSs �from top left to bottomright�. We observe that monomers are adsorbed from the in-terior of the corona without disturbing its shape. In the lateadsorption stage the corona collapses as a whole, as canbe seen in Fig. 4. Here, the density is displayed fors=20 000 MCSs, s=30 000 MCSs, and s=50 000 MCSs�from left to right�. We conclude from Figs. 3 and 4 that thesimulations agree with our model from Figs. 1�b� and 2,namely, that we have two phases: the corona and the adsor-bate.

Further evidence for the nonequilibrium two-phasemodel can be obtained from a one dimensional density plot,in which the mean number of monomers located in the layerat distance z from the surface is plotted against z. The corre-sponding data are displayed in Fig. 5 for N=100 at various svalues. Before adsorption starts, this profile reflects the equi-librium conformations of a grafted polymer with a maximumfar from the surface. We call this maximum the corona peak.When adsorption starts, a second �adsorption� peak rapidlydevelops at the surface. We observe a shift of the corona asadsorption evolves, in the direction opposite to the attractingwall. This corresponds to a process by which the corona doesnot follow the adsorption process but gets emptied �frombelow� through the stem.

B. Loop formation

We turn now to the analysis of the possible appearanceof big loops, which would lead to the creation of new nucle-ation centers as shown in Fig. 1�c�. For this purpose we nowcompute the evolution of loops during the process. Duringthe simulation run we count the number of loops of a certainlength L at different MCSs. By loops of length L we meanthat between two adsorbed monomers there are exactly Lnonadsorbed monomers. An average over 20 000 differentrealizations is taken. This procedure is repeated for differentvalues of N. The evolution of the mean number of loops

FIG. 5. Mean number of monomers � in the layer at z lattice units from theadsorbing surface.

�divided by N� is then plotted in Fig. 6 for different L values.

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094701-5 Irreversible adsorption of tethered chains at substrates J. Chem. Phys. 124, 094701 �2006�

In Fig. 6�a� we display the evolution of large loops contain-ing L=N /2 monomers; we note that their number is smalland that it rapidly decreases during adsorption. This fact sup-ports our assumption of simple zipping used in the scalinganalysis leading to Eqs. �1�–�7�, since big loops are related tothe spontaneous nucleation of new adsorption spots, a situa-tion which gets less and less likely as the process develops.We conclude that the simulations nicely support the basicfeatures of our model, showing that indeed the dynamics ofphysisorption is mainly due to simple zipping, see Fig. 1�b�.

Here, we also discuss how the mean number of smallloops develops during adsorption. Figure 6�b� displays thenumber of loops with length L�4, a number which alsodecreases rapidly during physisorption. The crossover to astable plateau of the loop fraction takes place when going toL�3, as displayed in Fig. 6�c�. As the process evolves thenumber of short loops increases, an increase which is non-monotonic. At the beginning, many of these loops form anddisappear, but then as physisorption continues short loopsbecome persistent. We note the presence of a finite fractionof nonadsorbed �frustrated� monomers due to the existenceof small loops over adsorbed trains of monomers.9 Finallywe consider in Fig. 6�d� loops with L�2. The number ofthese loops is continuously increasing during adsorption untilreaching a plateau.

C. Mechanism of adsorption

Having demonstrated that physisorption is controlled bysimple zipping, we now investigate it quantitatively. In Fig. 7we display the mean number of MCSs needed for a group ofmonomers k to adsorb. Let this quantity be sads�k�. In orderto account for the formation of small loops of one and twomonomers, we consider groups of three monomers, k beingthe index denoting the position of the group counted fromthe tethered monomer along the contour of the chain. Thefirst adsorption event for the given group is counted.

The plot of sads as a function of k can be approximated

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sads is not linearly proportional to k. A best fit yields sads

�k1.6, which compares very well with the predicted expo-nent of Eq. �3�, given that from Eq. �4� �=1+��1.6.

Complementing to Fig. 7, we display in Fig. 8 the aver-age number of adsorbed monomers as a function of s. Fromit we note again that the main stage process does not dependon N. The plots can be subdivided into three regions: In thesmall s region, 50–100 MCSs, one sees the adsorption of thefirst monomers. For s above 100 MCSs a power law behaviordevelops, which is complementary to the one observed inFig. 7 and which is again in fair agreement with our pre-dicted exponent of 1 / �1+��, obtained by rewriting Eqs. �3�and �4� as M = t1/�1+��. The best fit through the simulationpoints in this region gives an exponent of 0.62, in very goodagreement with 1/ �1+0.6��0.62. In the last stage of adsorp-tion the curves reach a plateau, since only a finite number ofmonomers adsorbs.

FIG. 6. Mean number of loops vs s forvarious sets of loops. �a� Loops con-taining N /2 monomers, �b� loops con-taining more than four monomers, �c�loops containing more than threemonomers, and �d� loops containingmore than two monomers. The sym-bols represent simulation results withN between 40 and 160 given in �a�.

FIG. 7. Plot of sads vs k. To account for small loops formed during adsorp-tion, we consider groups of three monomers each. k represents the index ofthe group’s position along the chain. The first adsorption event in each group

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094701-6 Descas, Sommer, and Blumen J. Chem. Phys. 124, 094701 �2006�

D. Last adsorption event and scaling

We focus now on the last adsorption event and denote by� the mean value of s at which this happens. We estimate� from an average over 20 000 independent realizations andfor chain length between N=20 and N=200. In Fig. 9 wedisplay in a double logarithmic plot � as a function of N.The figure suggests a power law relation between � and N,namely,

� � N�1, �8�

where we have introduced the dynamical exponent �1. FromFig. 9 we obtain �1�1.66. This relation compares well withthe prediction of Eq. �6�, where ��1+0.6=1.6. From thiswe infer that, roughly, ���, which may be taken to implythat time and number of MCSs s are linearly related to eachother.

Here it is appropriate to discuss the relation betweentime and MCS. Evidently, in ideal fashion, the two should belinearly related. The fact that sometimes MC procedureswere found not to behave in this way13,14,19 prompts us toconsider why in our case the BFM appears to work well. Wethink that this is due to the fact that the restrictions to localmoves in the BFM ensure that the diffusional motion of seg-ments is well reproduced in the simulations; the random at-tempts to move a monomer correspond to random forcesacting on an overdamped particle. Generally, the MC dynam-

FIG. 8. Log-log plot of M vs s for different values of N.

FIG. 9. Log-log plot of � and � vs N, see text for details.

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ics might fail in the presence of strong dynamic correlationsbetween monomers. Such is the case for strong externalfields, when collective moves in the direction of the fielddominate the dynamics.20,21 We have also found �by an ad-ditional analysis� that no strong forces act in our simulationsbetween successive monomers along the chain. Finally, wenote that our simulations are carried out under athermal con-ditions, a fact which excludes possible dynamical artifacts,as have been reported in low temperature simulations.13

E. Crossover region

We return now to the discussion of Figs. 7 and 8. Figure7 shows that sads�k� is independent of N over a wide range,for about 0�k�N /2. After reaching a characteristic valuefor the number of steps, ��N�, value which depends on N,scaling breaks down and the simulation points do not followa master curve anymore. As a technical remark, we note thataveraging over s naturally ignores very large s, i.e., frustratedmonomers. Therefore, the last part of such plots indicatessimply that adsorption finishes after a finite number of steps:All monomers which are not hindered to adsorb will eventu-ally get adsorbed; this also explains why the curve has adecreasing slope for large k. The splitting of the curves inFig. 8 allows us, in principle, to read off the crossover value� directly. However, � can now be related to the s rangeanalyzed in previous work.9,10 In Fig. 10 we display the re-laxation function q�s� of the nonadsorbed part, which isgiven by

q�s� =Mp − M�s�Mp − M�0�

, �9�

where M�s� is the number of adsorbed monomers at steps , M�0� is the number of adsorbed monomers at s=0, andMp is the plateau value of the adsorbed monomers at the endof the process. Here, we can see that the subexponential be-havior �power law� at the main stage crosses over into arapid drop at the late stage. The semilogarithmic plot sug-gests an exponential behavior between the two regions, seethe indicated slope in Fig. 10. Using the best exponential fitin the crossover region of q�s�, we obtain an s range which

FIG. 10. Semilogarithmic plot of the relaxation function q�s� given by Eq.�9� vs s. In this plot the symbols represent simulation points for N=100.

agrees with the � values which one can read off from the

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094701-7 Irreversible adsorption of tethered chains at substrates J. Chem. Phys. 124, 094701 �2006�

splitting of the curves in Fig. 8. The result is also plotted inFig. 9 and may be compared to �. This plot also supportsEq. �7� and shows that the terminal time and the crossovertime differ by a factor of about 1.8.

The coincidence between the � and the exponential fitcan be explained as follows: Let us assume that the mainstage �which is limited by �� results in a value of q1=q���.In the s interval of crossover, adsorption takes place at anearly constant rate q1 /�. Thus, we have quite approxi-mately

q�s� � q1 − q1s/� � q1 exp�− s/�� . �10�

We emphasize that in our model we do not necessarilyexpect to find a region where the rate of adsorption is con-trolled by the fraction of nonadsorbed monomers, i.e., wherethe relation q̇=−q /� is valid. Therefore, the exponential be-havior of Eq. �10� is only an approximation and does nothave a physical basis. In fact strong deviations from an ex-ponential behavior have been already noticed by Shaffer.9

F. Geometric properties of the adsorbed phase

To obtain more information about the way in which theadsorbate develops in the adsorbing plane, we consider the

mean square radius of gyration R̂g2�s� of the adsorbed mono-

mers at s MCSs versus the number of adsorbed monomersM�s�. The result is presented in Fig. 11 in a log-log plot for

various values of N, where R̂g2 is defined through

R̂g2�M� = 1

M�s� i=1

M�s�

��xi�s� − X�s��2 + �yi�s� − Y�s��2�� .

�11�

Here, xi�s� and yi�s� are the components of the position of theith adsorbed monomer on the surface at s MCSs and X�s�and Y�s� are the components of the center of mass of theadsorbate at s MCSs given by

X = 1

M�s� i=1

M�s�

xi�s�� �12�

FIG. 11. Averaged squared radius of gyration of the adsorbed monomers R̂g2

vs the average number of adsorbed monomers M for different values of N.

and

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Y = 1

M�s� i=1

M�s�

yi�s�� . �13�

A close inspection of the curves reveals that the function

R̂g2�M� displays three distinct regions. Let us introduce the

effective exponent �ads �which characterizes the geometry ofthe frozen-in adsorbate� through

R̂g2�M� � M2�ads. �14�

For small values of M �in the early stage of adsorption� the

slope of R̂g2�M� is larger than unity and is related to the

random placement of the first monomers during the forma-tion of the stem. During this stage, the first monomers toattach to the surface are those which at the beginning wereadjacent to the surface. However, it does not make muchsense to attribute a definite power law behavior to this re-gion, which, anyhow, involves only a few adsorbed mono-mers.

The second region is the result of adsorption during themain stage process. The effective exponent 2�ads�0.6 repre-sents an object with a fractal dimension of aboutdf =1/�ads�3.3. Clearly, such a supercompact behavior can-not be realized by a dense two dimensional �2D� object. Inorder to understand this result, we recall the fact that thecorona is not much affected by the main stage adsorption.Therefore, the center of mass of the corona does not followthe diffusive motion which the adsorbed monomers have tocarry out in order to occupy the surface. As a result, thismotion experiences a back driving force towards the centerof the adsorbate �origin of the diffusion process�, which re-sults in the compactification of the adsorption layer.

Eventually, the monomers find no more free places ofadsorption on the surface and must develop a random �dense�walk on larger scales. This can be seen in the third regionwhere the slope is approaching unity. Note that in this regionlonger loops are necessary in order to bridge between thealready adsorbed monomers.

In Fig. 12 �extracted from Fig. 6�c�� we notice thatthe number of loops which are longer than three units starts

FIG. 12. Same as Fig. 6�c� for N=100.

to increase after about 30 000 MCSs. This corresponds to

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094701-8 Descas, Sommer, and Blumen J. Chem. Phys. 124, 094701 �2006�

M �60 for N=100, see Fig. 8, a fact which is in good agree-ment with the location of the crossover between the secondand the third region in Fig. 11.

IV. CONCLUSIONS

In this work we have presented a nonequilibrium two-phase model for the strong physisorption of single tetheredchains with excluded volume interactions. Here, the adsor-bate is connected to the nonadsorbed monomers �corona� bya stretched part �stem�. The stem is growing during the mainstage of adsorption, a fact which leads to the decay of therate of adsorption �all monomers of the stem have to bemoved cooperatively to render it possible for further mono-mers to adsorb onto the substrate�. We have given a scalingargument for this process. The dynamic exponent which con-trols the scaling relation between the time of adsorption andthe number of adsorbed monomers is �=1+�. The overalladsorption time and the crossover time are smaller than theRouse time of the chain. In the main stage of adsorption thecorona does not relax but gets successively emptied frombelow, by which the distance between the two nonequilib-rium phases increases and further nucleation events are sup-pressed.

The model is supported by Monte Carlo simulations us-ing the BFM. The shape of the two-phase model is supportedby the evolution of the density distribution in 3D as well asin one dimension �1D� �density profile�. The analysis of loopformation process supports the idea that adsorption is domi-nated by simple zipping and that the nucleation of additionaladsorption spots is highly suppressed. Furthermore, the pro-cess of successive adsorption of monomers starting from thegrafting point does not occur at a constant rate but is con-tinuously slowing down until a certain crossover ��N� isattained. We have shown, comparing the relations between� and � and between � and �, that a linear relation be-tween time and MCS holds rather well.

It has to be noted that the exponents obtained for thetwo-phase model can only be valid for sufficiently longchains. The repulsive wall influences the chain’s conforma-tions, leading to a mushroomlike form. Here, chain partsclose to the anchor point are slightly stretched. This leads toa larger effective dynamical exponent �, a tendency corrobo-rated by our numerical results. In the case of adsorption offree chains �starting with the monomer which first touchesthe substrate�, all of our arguments remain valid. In particu-

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lar, the dynamical exponent � should be closer to the theo-retical result of 1+�, a fact which concurs with the previouswork.9 We found that the adsorbate formed in the processdisplays supercompact scaling over a wide range. This canbe explained by the effect of the corona, which is almostimmobilized during the main stage of the adsorption process.In late stages of the adsorption process, the formation oflarger loops �three to four monomers� increases due to theneed to bridge over densely adsorbed parts.

We note that our assumption that the stem is highlystretched while the corona is unperturbed may also be re-laxed by considering a limited, perturbed zone in the coronaas well as by allowing the stem conformations to fluctuate.According to our simulation results, however, such correc-tions do not seem to be essential for the understanding of theprocess.

ACKNOWLEDGMENTS

We thank PD Dr. Ch. von Ferber for discussions. Thesupport of the Deutsche Forschungsgemeinschaft and of theFonds der Chemischen Industrie is gratefully acknowledged.

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