Kinetics and structure of irreversibly adsorbed polymer layersL.C. Jia and PikYin Lai Citation: The Journal of Chemical Physics 105, 11319 (1996); doi: 10.1063/1.472872 View online: http://dx.doi.org/10.1063/1.472872 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/105/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Bulk and interfacial thermodynamics of a symmetric, ternary homopolymer–copolymer mixture: A Monte Carlostudy J. Chem. Phys. 105, 8885 (1996); 10.1063/1.472618 Scaling and scale breaking in polyelectrolytes J. Chem. Phys. 105, 5233 (1996); 10.1063/1.472372 Adsorption of living polymers on a solid surface: A Monte Carlo simulation J. Chem. Phys. 104, 9161 (1996); 10.1063/1.471448 Monte Carlo simulations of the polymer glass transition AIP Conf. Proc. 256, 145 (1992); 10.1063/1.42454 Logarithimically slow coarsening in nonrandomly frustrated models AIP Conf. Proc. 256, 541 (1992); 10.1063/1.42383

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Kinetics and structure of irreversibly adsorbed polymer layersL.-C. Jia and Pik-Yin LaiInstitute of Physics, National Central University, Chung-li, Taiwan 32054, Republic of China

~Received 12 February 1996; accepted 17 September 1996!

The kinetics of formation and the structure of an adsorbed layer formed by homopolymer chainsirreversibly adsorbed onto a flat wall are studied by Monto Carlo simulation using the bondfluctuation model. The rapid initial adsorption is followed by slow kinetics at later times. Thesurface coverage can be described by an empirical law of stretched exponential approach to the finalvalue. The formation time constantt can be interpreted by scaling theories of diffusion inside adense pseudo-brush. The detailed structure of the layer in a good solvent saturated by the adsorbedchains is also measured. The extension of the layer and the distributions of the loops and tails arealso analyzed in light of recent scaling theories. ©1996 American Institute of Physics.@S0021-9606~96!50248-5#

I. INTRODUCTION

Polymer adsorption has been a subject of intensive studybecause of its importance in many areas of science includingphysics, chemistry, biology, material science and tribology.A major reason is that the presence of polymer coating candramatically change the interfacial interactions and hencecan manipulate the surface properties. Most of the previousinvestigations1–31have been mainly about the structure of theadsorbed layer with a finite segment adsorption energy atequilibrium. The adsorption phase transition of a single poly-mer chain has been rather well studied using scaling2,7 andfield theoretical calculations.1,5,10 Both the statics and thekinetics of many chain reversible adsorption in the semi-dilute or dense polymer solutions have also been the subjectof current interests.24–27,30,31On the other hand, the study ofthe structure of the irreversible adsorbed layer is verylimited.32–34Such irreversible adsorption is characterized bythe quenched nature of adsorbed segment in that once a seg-ment is adsorbed, it can no longer move around. Thus theconfigurations of such adsorbed chains are not their mostfavorable ones if they were free to move around and equili-brate. The quenched nature of these layers makes the systemcomplex because the structure would depend on the historyof the layer formation. Furthermore there has been littlestudy on the time dependence of the kinetics of formation ofsuch irreversibly adsorbed layers. The adsorption kinetics isrelevant for the understanding of the formation of such apolymer layer. In this paper, we investigate both the kineticsof formation and the structure of irreversibly adsorbed poly-mer layers as a function of polymer chain lengths usingMonte Carlo simulations of the bond-fluctuation model. Ourresults are then discussed and interpreted using recent scalingarguments.

II. THE MODEL AND SIMULATION DETAILS

The bond-fluctuation model~BFM!35 is a lattice modelfor polymer chains as employed in our simulation. In threedimensions, each monomer occupies eight sites of a cubeon a simple cubic lattice. The bonds connecting successivemonomers are taken from the set of 108 bonds gen-

erated from the set:$(2,0,0),(2,1,0),(2,1,1),(2,2,1),(3,0,0),(3,1,0)% by symmetry operations of the cubic lattice. Thebond lengths can fluctuate between 2 andA10. Self-avoidinginteraction between monomers is enforced by the require-ment that no two monomers can share a common site. TheMonte Carlo procedure starts by choosing a monomer at ran-dom and attempts to move it in one of the six directions6x,6y,6z at random. The trial move will be acceptedwhen both self-avoidance is obeyed, and the new bond vec-tor still belongs to the allowed set. In the simulation of irre-versible adsorption process,Np monodispersed homopoly-mer chains, each consisting ofN monomers, are placed in aL3L3H box (L532,H53N11 in our simulations! inwhich periodic boundary conditions are applied in thex andy directions. The irreversibly adsorbing surface(L3L plane!is at thez51 impenetrable plane. Once a monomer touchesthis adsorbing surface, it will be fixed there and will not ableto move above or laterally within the surface. The other sur-face atz5H is simply taken to be an impenetrable hard wall.Before the adsorption process begins all the polymer chainsare allowed to equilibrate for a long period of time inside thebox with the z51 and z5H planes being non-adsorbingimpenetrable surfaces. After the system is well equilibrated,then the irreversible adsorption at thez51 plane is turned onat t50 and the adsorption process begins. The monomerscan then stick on thez51 plane. The adsorption is irrevers-ible and short-range in the sense that once a monomer hitsthe adsorbing wall it will be anchored there forever.

To investigate the kinetics of adsorption, we monitor thetime dependence of the surface coverages(t), which is thenumber of monomers stuck on the adsorbing wall per unitarea.s51 corresponds to the case when the adsorbing sur-face is fully covered with monomers. To ensure the adsorb-ing wall is well saturated by polymer chains, a sufficientnumber of chains are placed in the system withNp.L2/(4N). ~The factor of 4 is due to the fact that in BFMthe minimum separation between two monomers is 2.! Thesystem is left for a sufficiently long period of time so that thesurface coverage saturates without further increase. Since theadsorption is irreversible, we perform a quenched average

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over several different realizations~typically five to ten! ofadsorption processes and obtain the average time dependenceon s(t).

After a long enough period of adsorption, the adsorbingwall is saturated by polymer chains with some free polymerchains not being adsorbed and floating above the adsorbedlayer. The structure of the adsorbed layer is then investi-gated. Experimental and theoretical interests have been fo-cused on the structure of the irreversibly adsorbed layer satu-rated with polymers without the presence of any freepolymer chains. This condition can be achieved in our casesimply by removing these residual free polymer chainswhich corresponds to washing the layer with pure solventexperimentally. We are then ready to probe the structure ofthis irreversibly adsorbed layer. This adsorbed layer is al-lowed to settle down for some time and thermal averages ofthe structure on the layer are then measured over an extendedperiod of time. In our simulation, the chain length rangesfrom N56 toN560. And 10<Np<100, such that the totalnumber of monomers is fixed to atN3Np5600.

III. KINETICS OF LAYER FORMATION

The kinetics of adsorption of homopolymer chains ontoa solid surface has received considerable attentionrecently.24–27The formation of an irreversibly adsorbed layeris a complex process because the equilibrium state is nolonger universal.32 It depends on the history of the sample.The dynamics of this process is largely an unsolved problem.The kinetics of homopolymer adsorption is divided into twostages: First is the fast diffusion and adsorption of the earlyadsorbed chains onto the wall. Then when the wall is moreor less covered with polymers, the subsequent adsorption isfollowed by the much slower chain movement of the latecoming chains through the adsorbed layer. Figure 1 showsthe average surface coverages as a function of time forvarious chain lengths. The total number of monomers in thesystem in each case is fixed to beN3Np5600 and there is

always a sufficient number of chains to saturate the adsorb-ing wall in all cases. The kinetics of formation indeed agreeswith a fast initial process and a much slower one at latetimes. In fact the approach to the surface coverage at satura-tion (t→`), ss , can be described empirically by thestretched exponential function

s~ t !5ssS 12expS 2S t2t0

t̃D aD D for t.t0 , ~1!

wheret̃ anda are fitting parameters.t0 is the time at whichadsorption starts to take place (s50 for t,t0) which is alsothe time scale for the polymers to swim to the adsorbingsurface. Since the height of the boxH } N and the averageconcentration is rather dilute in the bulk, thus the diffusionconstantD } 1/N andt0;H2/D } N3. Values oft0 extractedfrom the data ofs(t) for different chain lengths are shownon a log-log plot in Fig. 2 verifying the simple Rouse typediffusion.

The adsorbed layer starts to form fort.t0 and the in-crease in the adsorbed monomers can be described by Eq.~1!. The fitting of the stretched exponential form in Eq.~1! isshown in Fig. 3 forN512. Furthermore, for small values ofN (N<6), a'1 indicating an exponential approach toss

agreeing with the results in random sequential adsorption ofmonomers on a square lattice.36,37 Figure 4 showslog(ss2s(t)) as a function oft for various chain lengths. Forvery short chains (N56), the data lie on a straight line in-dicatinga'1. While for long chainsa is considerably lessthan 1 but is still of the same order.a becomes monotoni-cally smaller for longer chain length. The value ofa reflectsdifferent kinds of dynamical processes governing the adsorp-tion kinetics. The measured stretched exponential form ofs(t) can be interpreted as the existence of a wide spectrumof time scales38 in the formation of the irreversibly adsorbedlayer. This wide spectrum of time scales corresponds to thelarge range of adsorption times ranging from the very fastinitial adsorption and very slow late stage adsorption. The

FIG. 1. Log-linear plot ofs(t) versust for different polymer chain lengthN. The total number of monomers in each case isN3Np5600.

FIG. 2. Log-log plot oft0 vs N. The dashed line denotes a slope of 3.

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slow process is a consequence of the relatively slow trans-port from the bulk to the adsorbed layer due to the fact thatthe polymer chains have to diffuse through the dense ad-sorbed layer. Figures 1 and 4 show that whenN is longer, thecharacteristic adsorption time increases. It is then meaningfulto analyze the variation of the characteristic adsorption timeas a function of the length of polymer chains. The character-istic formation time,t, of the adsorbed layer can be mea-sured by

t5Eto

`ss2s~ t !

ssdt. ~2!

Assuming the stretched exponential form as in Eq.~1!,one gets

t5t̃

aG~1/a!. ~3!

The systematic increase oft with increasing length ofthe chains is shown in Fig. 5 on a log-log plot. The data forN.10 can be fitted with a power law of the formt} N2.160.1. One can attempt to understand the behavior oftwith N as follows. The kinetics of layer formation is domi-nated by the late stage adsorption process in which a poly-mer chain has to go through an adsorbed layer of heighth inorder to be adsorbed. At this stage, the structure of the layeris already in the ‘‘pseudo-brush’’ regime32 with a brushheight h. A ‘‘pseudo-brush’’ is, so to speak, a highly dis-persed brush composed of loops and tails since a finite por-tion of monomers in each chain is grafted on the adsorbingsurface. A non-adsorbed polymer chain has to diffusethrough a distanceh to reach the wall. The scaling result ofa ‘‘pseudo-brush’’ picture of an irreversibly adsorbed layer32

predicts,

h;N5/6F7/24, ~4!

whereF is the initial average monomer volume fraction ofthe system before adsorption. In our simulation, since thetotal number of monomers is fixed and the height of the boxH } N, thereforeF } 1/N and thus we have

h;N13/24. ~5!

The concentration dependent diffusion coefficientD(N,f)has the following scaling form39

D~N,f!5N21f ~f5/4N!, ~6!

where the scaling functionf (x) is roughly constant for smallx and f (x);x21 for x @ 1 which corresponds to the casewhenN @ Ne , the entanglement length,

39 andD;N22 sig-nifying reptation motion takes place. Therefore, the timeneeded for a polymer chain to diffuse through the layer ofvolume fractionfL is

FIG. 3. Surface coverage,s is plotted as a function of time,t. Number ofchainsNp550. Chain lengthN512. Dashed line is a least square fit of thedata to the stretched exponential form in Eq.~1!.

FIG. 4. log(ss2s(t)) as a function oft for different chain lengths. ForN56, the data lie on a straight line indicatinga;1. a decreases for largerN.

FIG. 5. Log-log plot oft versusN. Dashed line denotes a best fit slope of2.1.

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t;h2

D~N,fL!;

N25/12

f ~fL5/4N!

. ~7!

The monomer concentration inside the layer dependsweakly onN and one can takefL'constant. And finally wehave

t;HN25/125N2.08 for N,Ne

N37/125N3.08 for N@Ne. ~8!

From Ref. 39 the entanglement length occurs atNe.50 to80 for the bond-fluctuation model in the range of concentra-tion of the present study.Ne corresponds to the chain lengthscale at which entanglement effects are important and repta-tion dynamics sets in forN @ Ne . Thus in our present simu-lations, our chain lengths are still far from the regime inwhich entanglement is important and hence one still hast;N25/12 in good agreement with our simulation result.

IV. STRUCTURE OF THE LAYER

A sufficient number of chains are put in the system andare allowed to saturate the adsorption surface until no furthermonomer adsorption occurs, then the final structure of thelayer is monitored. Even though a finite portion of the chainis irreversibly grafted on the wall, the rest of the monomersare still free to fluctuate. Thus the usual time average over anextended period of Monte Carlo steps is performed for themeasurement of structural quantities. In fact, the averageconformation of the adsorbed chains does not depend verymuch on the adsorption process. This has been verified fromthe volume fraction distribution from different realizations ofthe adsorption processes. Although the adsorption process iscomplex, the distribution of the adsorbed chains is simple ifwe wash out the residual free polymer chains.32 The averagefraction of monomer adsorbed for each adsorbed chain isabout 0.53 and is more or less independent of the chainlength forN>10. On the other hand, the saturated surfacecoveragess shows a weakN dependence. ForN51, oursimulation givesss.0.748 which is in good agreement withsimulation36 and virial expansion37 results of random sequen-tial adsorption on a square lattice. For longer chain lengths,ss is found to decrease systematically with increasingN. Forexample, we obtainss'0.732 forN56 andss'0.70 forN560. Figure 6~a! displays the configurations of the ad-sorbed layers formed by long and short polymer chains. Forshort chains, the adsorbing wall is more densely filled whilefor long chains the layer is more diffuse. From the configu-ration of the longer chains@Fig. 6~b!#, it agrees quite wellwith the qualitative picture of a pseudo-brush composed ofloops and tails. This slight decrease ofss with N can beinterpreted from the following picture: for short chains, thelate coming polymers can still move and penetrate around inthe layer quite well and can fill up the empty absorbent siteswithout overcoming a very strong screening from othermonomers belonging to the adsorbed chains. But for systemswith larger N, the late coming chains have to strugglethrough a greater layer height and screening effects. Thus the

long polymers cannot freely move around to fill the vacantadsorbent sites easily and efficiently.

Figure 7~b! shows that the monomer volume fractionprofile f(z) before and after excess free chains is washedaway. After washing with the solvent, we can regard thedistribution of the adsorbed chains as highly polydisperseloops and tails. Our simulation result forf(z) can be fittedwith a power law decay for distances far away from the wall.Figure 8 shows the fitting of the data forz>5 and we obtain

f~z!;z21.3460.1. ~9!

FIG. 6. Polymer chain configurations in the adsorbed layer after washing.~a! N58, ~b! N524.

11322 L.-C. Jia and P.-Y. Lai: Kinetics and structures of adsorbed polymer

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This should be compared with the pseudo-brush picture ofirreversible adsorption which showed32 that

f~z!;z2b, ~10!

with b54/3.1.33 in the proximal region andb52/5 in thecentral region. However, in the present simulation, the chainlengths are not long enough to probe the central region withreasonable accuracy. We also measure the extensionh of theprofile which is defined as the distance from the wall whenthe volume fraction profilef is less than 0.01. We performextensive simulations to measureh for different chainlengths and the results are shown in Fig. 9. ForN>10,h canbe described by a power law of

h;N0.5560.05. ~11!

This result agrees rather well when compared with Guiselin’sscaling prediction32 which used a pseudo-brush picture asgiven by Eqs. ~4! to ~5! for the present study:h;N13/245N0.54.

The monomer distribution of the adsorbed layer is com-posed of loops and tails. The distribution of the loops andtails of the adsorbed chains is also studied. As shown in Fig.10, the concentration of monomers belonging to the tails isvery small near the surface as expected. But whenz isgreater than some distanceh1, the tail contribution domi-nates over the loop contribution. Figure 10 shows that undersaturation condition, the tails of the polymer chains give animportant contribution to the volume fraction profile in theouter layer away from the wall. Therefore, at some distancez5h1 from the wall,23 the tail contribution in the concentra-

FIG. 7. The monomer volume fraction profilef(z) versusz. Chain lengthN520. Open circles: Before washing, andNp530. Filled square: Afterwashing,Np519 chains remain.

FIG. 8. f(z) after washing. Dashed line is a power law fitting which givesf(z);z21.34. The chain lengthN530, and final number of chainsNp511.

FIG. 9. The extension,h versusN in a log-log plot. Dashed line denotes abest fitting slope of 0.55.

FIG. 10. The circle~s! indicates the total volume fraction profilef total .The square~h! denotes the tail contributionf tail . The diamond~L! de-notes the loop contributionf loop . Chain lengthN is 20. The final number ofchainsNp is 19. The inset is just a magnification near the cross-over regimeh1.

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tion profile becomes comparable to the loop contribution, i.e.f tail(h1).f loop(h1). h1 is then measured for various valuesof N and the results are shown in Fig. 11 on a log-log plot.Our data can be fitted with the power law

h1}N0.5160.05. ~12!

This result is in agreement with the theoretical result ofSemenov and Joanny23

h1;N1/2, ~13!

which considered partition functions of the tail and loopmonomers.

V. CONCLUSION

In this paper, we performed Monte Carlo simulations toprobe the kinetics for the formation of an irreversibly ad-sorbed layer as well as the structure of such a layer. Theformation kinetics is initially fast but slow at late stages. Thiswide range of time scales in the kinetics is manifested in thestretched exponential approach to the final coverage. Fur-thermore, our scaling analysis employing a pseudo-brushpicture gives a rather accurate scaling description for theformation time with the chain length. This indicates that theformation time is dominated by the late stage slow adsorp-tion process in which a polymer chain has to diffuse throughthe almost fully formed layer. As for the structure of thelayer, our data on the loop and tail distributions indicate thatthe inner layer is formed mainly by loops with a self-similarprofile whereas in the outer layer it is dominated by tails.The cross-over distance is found to be consistent with thescaling result ofh1;AN. Our data are consistent with thescaling argument using Guiselin’s pseudo-brush picture,however considerably longer chains are needed in order toprobe for the reptation motion for the kinetics as predicted inthe second branch in Eq.~8! and also the much slower decayof the profilef;z22/5 in the central regime as predicted inRef. 32. In principle, our simulation on the structure should

be able to distinguish the different predictions for the struc-tures of reversible and irreversible adsorption models. Forthe case of reversible adsorption, de Gennes’ theory predictsa self-similar concentration profile off;z24/3 and the brushheight scales withh;Nn;N0.59. Unfortunately, the differ-ence in the scaling exponents for the brush heights for thesetwo models is too small (.0.05) to be resolved by thepresent simulation. Although our results for the presentsimulations are closer to Guiselin’s prediction, more exten-sive simulations and larger systems of longer chains are re-quired in order to discriminate clearly from the reversiblemodel.

Finally, we would like to remark that although the bond-fluctuation model has been shown31 to contain an artifact inthe adsorption study of the dynamics due to the non-ergodicbehavior when the chain cross-over from three-dimensions toquasi-two-dimensions when adsorbed on the wall, in thepresent irreversible adsorption, there is no quasi-two-dimensional motion on the adsorbing wall and it is still le-gitimate to use this model without any problem.

ACKNOWLEDGMENT

We thank National Council of Science of Taiwan underGrant No. NSC 85-2112-M-008-016 for support.

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FIG. 11. h1 versusN in a log-log plot. The dashed line denotes a best fitslope of 0.51.

11324 L.-C. Jia and P.-Y. Lai: Kinetics and structures of adsorbed polymer

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