Szleifer-2000aa

Macromol.RapidCommun.21,423–448 (2000) 423

Tetheredpolymerlayers:phasetransitionsandreductionof proteinadsorption

I. Szleifer*, M. A. Carignano

Departmentof Chemistry, PurdueUniversity, WestLafayette,IN 47907,U.S.A.

(Received:May 3, 1999;revised:August16,1999)

1 Intr oductionPolymermoleculestetheredat oneof their endsto a sur-face or interface find many applications in a variety offields including colloidal stabilization1,2), biocompatiblematerials3–7) and drug carriers8–12). The main role of thetetheredpolymers is to changethe interactionsof themodified surfaceor interface with the environment.Themost common application is when one desiresto coverthe surface or interface with a protective “steric”layer13,14). This is achieved by choosing polymer mole-culesfor which thesolvent is “good”. Namely, themono-

mersof the polymer prefer, effectively, to be surroundedby solvent molecules ratherthanby polymer segmentsoftheir samekind. In order for the tetheredpolymersto bebetter dissolvedin the solventthey tendto stretch out ofthe surface/interfaceasthe surfacecoverage of polymersincreases abovea certainthreshold. At high enoughsur-facecoverage thepolymersform whatis calleda polymerbrush,wherethechainmoleculesarehighly stretchedoutof the surface. Thesehighly stretchedpolymers are theones that havethe potential of forming a very effectivesteric barrierthatprotectsthesurface/interface.

Feature Article: Thestructuralandthermodynamicprop-erties of tetheredpolymer layers formed by spreadingdiblock copolymersat a solid surfaceor at a fluid-fluidinterfacearestudiedusinga molecularmean-fieldtheory.Therole of theanchoringblock in determiningtheproper-ties of the tetheredpolymerlayer is studiedin detail. It isfound that both the anchoringandthe tetheredblocksarevery important in determiningthe phasebehaviorof thepolymerlayer. Thestructuresof thecoexistingphases,thephaseboundariesand the stability of the layer are foundto dependon the ratio of molecularweight betweenthetwo blocks, the polymer-interface (surface) interactionsand the strength of the interactions between the twoblocks.Thedifferentphasetransitionsfoundarerelatedtoexperimentalobservations.The propertiesof the polymerlayersat coexistencereflect theblock that is thedominantdriving force for phaseseparation.Theability of the teth-eredpolymerlayers,underdifferentconditions,to controlprotein adsorptionto surfacesis also studied.It is foundthat the most importantfactorsdeterminingthe ability ofa polymerlayer to reducethe equilibrium amountof pro-teins adsorbedto a surfaceare the surfacecoverageofpolymer and the surface-polymerinteractions.The poly-mer chain length plays only a secondaryrole. For thekinetic control, however, it is found that the potentialofmean-force,and thus the early stages of adsorption,dependsstronglyon polymermolecularweight.Further, itis foundthat themolecularfactorsdeterminingtheabilityof the tetheredpolymer layer to reducethe equilibriumamountof proteinadsorptionaredifferent thanthosethatcontrol the kinetic behavior. Comparisonswith experi-

mentalobservationsarepresented.The predictionsof thetheory are in very good agreementwith the measuredadsorption isotherms.Guidelinesfor building optimalsur-face protection for protein adsorption,both kinetic andthermodynamic,arediscussed.

Macromol. RapidCommun. 21, No. 8 i WILEY-VCH VerlagGmbH, D-69451 Weinheim2000 1022-1336/2000/0805–0423$17.50+.50/0

Qualitative picture of the different structuresthat a tetheredpolymer layer may adopt. The casesshown include purelyrepulsivesurfaces:a) Mushroomregime,at very low surfacecoverage and b) Brush regime, high surfacecoverage. Forsurfaceswith attractive interactionswith themonomersof theA block: c) Pancakeregime,for low surfacecoverage

424 I. Szleifer, M. A. Carignano

Thebehavior of tetheredpolymer layers hasbeenstud-ied at length in the last twenty years. Thereare a largevariety of different theoretical studiesthat include fullscalecomputersimulations15–18), moleculartheories19) andanalytical approaches20–22). There are also many experi-mental studies,most of the early work concentrated onmeasuring forces betweentethered layers13,23–25). How-ever, therearenow several studieson thestructureof thelayersusing scattering techniques26–28) andmeasurementsof pressure-area isotherms29–31) for polymers spread atfluid-flui d interfaces.

A tetheredpolymer layer in good solventcanhave thefollowing structures,qualitatively pictured in Fig. 1. Atvery low surfacecoverage, and in the casesthat the sur-face is purely repulsive for the monomers of the chain,the polymers are essentially isolated and form the socalled“mushroom” regime.At high surfacecoverage,thecasedescribed abovefor steric repulsion, the chainsarestretchedout of the surfaceforming the “brush” regime.In the casethat the tethered polymer monomers haveattractive interactions with the surface,the low densitylimit is called the “pancake” regime and the chains areadsorbedto the surface.For high surfacecoverage,thereis a small region closeto thesurfacewith high concentra-tion of polymer segmentsfollowed by theremainingunitsof the polymer chains that stretch out of the surface.It isimportant to emphasize, that these regimes are notdividedby sharpboundariesandthat in practicetherangeof surface that they cover, as well as their existence,dependsupon the molecular weight of the polymersamongother variables.For example, it hasbeenshownthat for polyethylene oxide (PEO)in the regime of mole-cular weights usedin biocompatiblematerials and drugcarriers,there is a continuous changeof structurefromthe mushroomto the brush32). It hasbeenalsofound thatPEOsegmentsareattractedto hydrophobic surfaces andto thewater-air interfacebut not to lipid surfaces30,31,33,34).Furthermore, many of the experimental studiesare car-ried out in the intermediateregion of surfacecoveragebetweenthemushroomandthebrush.Therefore, thedivi-sionin differentregimesshould betaken with careandbeusedmostly for descriptive (qualitative) purposes.

In general, it is convenient to describe the stateof thetethered polymer layer under the generic regimesdescribedabove.This is the approach that will be takenthroughout this paper. However, we emphasizethat in ourdescription thewords “mushroom”, “pancake”or “brush”will be merely usedto indicate the averagestructure ofthe layer and not the type of theoreticalapproach thatshouldbe appliedto study the properties of the layer inthis or that regime. A thoroughdiscussion of range ofapplicability of differenttheoretical approachesin thedif-ferentsurfacecoverageregimesasa function of molecu-lar weightcanbefoundin ref.19,32)

Oneof the results that emerge from experimental stud-ies is that, in general, it is very hard to achieve very highsurfacecoverageof polymer. Namely, the brushregimeof highly stretchedpolymersis not easilyreached. One ofthe reasonsis that the steric barrier imposedby the teth-eredpolymer layer to other moleculesalsoexists for thepolymer moleculesthat try to reachthe surface.At thispoint we needto discussthedifferentexperimental meth-odologiesthat havebeenusedto build tetheredpolymerlayers.One approachis to functionalize one of the freeends of the polymer with a moiety that is stronglyattractedto the surface. For example, Klein and cowor-

Fig. 1. Qualitative pictureof thedifferentstructuresthata teth-eredpolymer layer may adopt.The casesshownincludepurelyrepulsive surfaces:a) Mushroomregime, at very low surfacecoverageand b) Brush regime,high surfacecoverage. For sur-faceswith attractive interactionswith the monomersof the Ablock: c) Pancakeregime, for low surface coverage and d)Brushon top of a thin adsorbedlayer. Diblock copolymers: Theanchoringblock (dashedline) is e) completely adsorbed to asolid surface,or f) forming a secondlayer at a fluid-f luid inter-face

Tethered polymerlayers: phasetransitionsandreductionof proteinadsorption 425

kers useda zwitterionic functionalizedpolystyrene(PS)to tether the polymers to mica surfaces23,24). Anotherapproachis to use a functionalizedend of the polymerthat chemically reactswith the surfaceto form a strongchemicalbond. This chemisorptionis much stronger thanthan the physisorption using the zwitterionic end.Usingthe chemical reactionapproachand working with semidilute polymer solutions, Auroy et al.35) achievedveryhigh surfacecoverageof polymers.A third approachis touseblock copolymers, diblock and triblock, to form thetetheredpolymer layer13,25,29,36–38). The ideais to have oneof the blocks as the anchoring group and the other (orothers)will form thetetheredlayer, seeFig. 1. Theuseofa polymer block asan anchoring groupis very appealingsince even a relatively small attractive interaction persegmentof theanchoringblock canresultin many kTs ofanchoringenergy for the whole block. Furthermore,afteradsorption of theanchoring block onecan,in somecases,chemically bind the adsorbed polymer segmentsto thesurfaceachieving very strong andirreversible grafting ofthepolymerlayer39).

One of the purposesof the work presentedhere is todiscusstheeffect of theanchoring block on thepropertiesof the tetheredpolymer layer. In general, the role of theanchoringblock wasnot explicitly consideredin treatingthe behavior of the tetheredpolymer layer. However, thethermodynamic stateand thus, the structureof the teth-eredpolymer layer are strongly coupled to the behaviorof the anchoring block. It wil l be shown that the phasebehaviorandstructureof the tetheredlayer dependuponthe interactions between the two blocks, whether thepolymer layer is at a fluid-solid (surface) or fluid-f luidinterface.The strength of theseeffectswil l turn out to bea function of the interactionsbetweenthe tetheredblockandthebaresurface.

Alexander20) and later Liguore40) predictedthat in thecasesin which the monomers of the tetheredchainhaveattractive interactionswith thesurfacetherewill bea firstorderphasetransitionbetweenthepancakeandthebrushregime. Ou-Yang and Gao41) measuredthe thicknessofPEO layers tethered to polystyrene spheres and theirresultsseemto confirm the theoretical predictions. Bij-sterboschet al.30) andFaureet al.31) measured the proper-ties of PEOtetheredat the water-air interfaceandwhilethe EO monomers are attracted to the interface, theyfoundno evidenceof thepredictedphasetransition. Morerecentlyin ref.42) andthroughoutthis paper, we wil l showthat all the experimental observations can be explainedby introducing the effect of the anchoring block into thepicture. Namely, the anchoringblock turns out to be animportantdeterminant of the phasebehaviorof the poly-merlayer.

Thesecondaimof thispaperis to describe theability oftetheredpolymer layers to preventnon-specific proteinadsorption onto surfaces.The phasebehavior and the

structural properties of the tetheredlayer determinethepropertiesandinteractionsof themodifiedsurface.In par-ticular, we areinterestedin describing theability of poly-mer layers to increasethebiocompatibility of materials7).This may be achieved by the prevention of adsorption ofblood proteins into thematerialssurface43). We wil l showhow the equilibrium adsorption isotherms of proteinsdependuponthepropertiesof the tetheredpolymer layer.Furthermore,we wil l describethe changesin the kineticbehavior of theadsorption inducedby thepresenceof thepolymer layer. Fromthesestudieswe canobtain theopti-mal polymer layers necessary for thermodynamic and/orkineticpreventionof protein adsorption.

The theoretical approach usedthroughoutthis work isthesingle-chainmean-field (SCMF)theoryandits gener-alizationsto studymixturesof polymersandproteins.Thistheory wasoriginally developedto studypacking of sur-factantchainsin micellaraggregates44,45) andlatergeneral-ized to treatpolymersin inhomogeneous environments46).The key idea of this theoreticalapproachis to look at acentral molecule,polymerandprotein, with its intramole-cular interactionsexactly taken into account (within themodelsystemchosento treatthemolecules)andtheinter-molecularinteractionsareconsideredwithin a mean-fieldapproximation. The theory has beenshown to producevery accuratepredictions as compared with full scalemolecular dynamics andMonte Carlo simulations19,47,48),andwith experimental observations31,49). Theability of thetheory to quantitatively predictthepropertiesof thelayersincludes structural and thermodynamic properties.Furthermore,it hasbeenrecently shown that thetheoryisable to predict the adsorption isothermsof proteinsfromsolutionsto surfaceswith graftedpolymers39).

The SCMF theory is particularly suitedto study shortandintermediatechain length molecules,up to 200–300segments.Therangeof applicability of thetheory aswellas the reasons of why the theory is successful in a widerange of applicationscanbe found in recent review arti-cles19,32,50).

This paper is organized as follows. The next sectionpresentsthe derivationof the theory, how we apply it tothespecific caseof block copolymers, andits generaliza-tion to study protein adsorption. Section III presentsavariety of resultsfor the phasebehaviorand molecularorganization of the block copolymer films. In SectionIVwe show the thermodynamic and kinetic control of pro-tein adsorption that can be achieved for the differentstructuresof the tethered polymer layer. Finally, SectionV containsconcluding remarks with somediscussion onfuturedirections.

2 Theoretical approach

In this section we presentan overview of the theoreticalapproachusedthroughout this work. Sinceseveral publi-


cationsand reviews contain all the necessarydetails tocarry out the calculations19,46), we will concentrateon thespecificapplication of the theory andthe assumptionsofthemodel for theproblemsof interesthere. The first partof this sectiondescribesthe theoreticalapproach for thestructureand thermodynamic behaviorof block copoly-mer layers.The secondpart is the generalization of thetheoryto study theadsorption behavior, andthestructuralchanges, in the tetheredpolymer layers whenthey are incontact with a proteinsolution.

The central quantity in the molecular theory is theprobability distribution function (pdf) of chainconforma-tions. From the knowledgeof this quantity any averageconformational and thermodynamic propertycan be cal-culated. The pdf is determinedby minimization of thesystemfreeenergy. Thus,we needto specifythedifferententropic and energetic contributions in our model sys-tems, then the pdf can be derived and calculations ofequilibrium propertiescanbecarriedout.

2.1 Blockpolymerlayers

We will consider systemscomposedof two types ofblocks that wil l be called A and B. The A block is in agoodsolventenvironment andis thetetheredblock, whilethe B block is the anchoring block. We will contemplatetwo different anchoringmechanismsfor the B block: (i)the caseof a solid surfaceand (ii ) block copolymersspreadat a fluid-fluid interface.

In all caseswe assume thatthesolventof theA block isinfini tely poor for the B block andfor a fluid-f luid inter-facethesolventof theB block is infinitely poor for theAblock. This impliesthe assumption of an infinitely sharpinterfacefor bothblocks.This is not a necessaryassump-tion, however, it greatly simplifies the presentationandcalculationswithout changing in a qualitative way anyofthemainresults.

For thesolid surfaceit will beassumedthattheanchor-ing block is completely grafted to the surface.Namely,we model it as if this block is changing the chemicalstructureof the surface. Therefore,we wil l assume thatits role is to changethe interactionsbetweenthe surfaceand the tetheredblock. However, since we will assumethat it is completely grafted, i. e. all its segments arechemically bound to the surface,the block doesnot haveconformationaldegreesof freedom.

We are interestedin the casesin which the A and Bblockshaveeffectiverepulsionsbetweenthem.Therefore,theneteffect of havingtheB block graftedto thesurface,within our model, is to introducea surfacerepulsion fortheA segmentsthatwill beproportionalto thesurfacecov-erage, i. e.to thenumberof B segmentsonthesurface.

We cannow write theenergeticcontribution to thefreeenergy due to the A and B blocks for the casein whichthe B block is graftedonto a solid surface. We consider

first threecontributions:1. The A-A attractions, which represent effectively the

quality of thesolvent. We define a constant (negative)interactionparametereAA, andthenthe temperature,T,is therelevant controlvariable.Sincewe areinterestedin good solvent conditions for the tetheredblock wewil l havekT A jeAAj. Within a mean-field approxima-tion and accounting for the inhomogeneous distribu-tion of A segments asa function of the distancefromthesurface,thetotal A-A interactionperpolymerchainhastheform

EAA � 12

Z v

0

Z v

0

eAApnA�z�PpbA�z9�Pdzdz9; �1�

where pnA�z�Pdz is theaveragenumberof segmentsoftheA block at a distancez from thesurface. Distancezrefers here andthroughout to the layer betweenz andz+ dz. pbA�z9�P is the averagevolumefraction of seg-ments of type A at distancez9 from the interface.Theintegraloverz is to account for all thesegmentsof thecentral chain andthat overz9 to accountfor themean-field of A monomersinteracting with thecentralchain.In reality theparametereAA arisesfrom the integrationof the van der Waals interactions. The procedure isdescribed in detail in ref.47) In caseof athermal sys-tems (good solvent regime) this contribution is notincludedin thefreeenergy.

2. The A-surface interactions.This is the direct interac-tion betweentheA segmentsandthebare surface.Weassume that the surfaceexerts an interaction with asquarewell potential to theA segmentswith attractivestrength eAS and the rangeof the interaction is d. Thetotal interactionperblock is

EAS�Z d

0

eASpnA�z�Pdz: �2�

3. The interaction betweenA and the surfacebound Bsegments.This is a repulsive interactionanddependson the numberof B segmentspresenton the surface.We assumethat the rangeof this repulsive interactionis thesameasthatof theA baresurfaceattraction, i. e.d, andthestrength of the interaction is eAB. The repul-siveinteraction pertetheredchainmoleculeis

EAB �Z d

0

eABNpolnBpnA�z�Pdz; �3�

where nB is the numberof segmentsof the anchoring(B) block andNpol is thenumberof polymermoleculeson thesurface.For conveniencewe define aneffectiveinteractionparameter by multiplying eAB by the totalareaof the surface and the numberof B segmentsinthe tethering block. Thus, the repulsive interactionterm becomes


EAB � kTZ d

0

vABrpnA�z�Pdz; �4�

with vAB � eABRnB=kT, and r � Npol=R is the surfacecoverageof polymer, with R beingthetotal areaof thesurface.

The free energy of the systemincludesthe energeticcontributionsjust describedandthefollowing threeentro-pic contributions:1. The conformational entropyof the A block. For each

A block wehave

SA � ÿkXfag

P�a�lnP�a� ; �5�

where the sumrunsover all the conformationsof theA chains.

2. The translational entropy of the polymer molecules.This termis of theform (per chainmolecule)

Strans� ÿk ln�rl2�; �6�where l is the length of theA segment.Notethatrl2 isunitless.This term will not be presentin the free energy if theB blocks are chemically graftedto the surface.How-ever, it is neededfor thosecasesin which the chainsaremobile.

3. The translational entropy of solvent molecules.Namely, theentropy persolventmoleculeat z is

Ssolv�z� � ÿk lnbs�z� ; �7�wherebs�z� is thesolventvolume fractionat z.

The total freeenergy of thesystemis obtainedby mul-tiplying thepolymercontribution by thenumber of chainmolecules and integrating the solvent contribution overall z (this is equivalent to summing overall solventmole-cules).

Before we write the total free energy we emphasizeagainthat in this caseof a solid surfacewe areassumingthat theonly role playedby theB blocks is to modify thesurfacesuchthat thereis a repulsive termasexpressed inEq. (4). Namely, the B blocks are grafted to the surfacewithout having conformational degreesof freedom.

It is convenient to write thefreeenergy density, i. e. thefreeenergy perunit areaof thesurface.Further, themostappropriate ensemblefor this systemis a semi grand-canonical, which is canonical for the polymer chains andgrand canonical for the solvent molecules51). The totalfreeenergy perunit areais then

WA

R� r

12

EAA� EAS� EAB

� �ÿ rTSA

ÿZ

bsolv�z�5s

TSsolv� lsolv� �dz; �8�

wherelsolv is the solventchemical potential that mustbeconstantat all z for thermodynamic equilibrium and5s isthe volume of the solvent molecule. We have excludedthe translational entropy of the polymer moleculesbecausethey areassumed to bechemically grafted to thesurfaceby theB blocks.

Inspection of the different contributions to the freeenergy reveals that we are missing a repulsive termbetween the molecules in the system. This term isincludedaspacking constraints.Namely, we assumethatbetweenany two segmentstherearehardcore repulsiveinteractionsand that the available volume is completelyoccupiedby solvent or polymersegments,i. e. incompres-sibility assumption. The incompressibility assumption isnot really needed as hasbeenshown in ref.52) However,using this approximation doesnot modify the resultsin aqualitative way for the systemsof interesthere and it isconvenientfor computationalpurposes.

Thepackingconstraintreads

pbA�z�P� bs�z� � rpnA�z�P 50 � bs�z� � 1; �9�wherethe condition is imposed at all z’s sincethis is theinhomogeneousdirection. 50 (=5s) is the volume of themonomer. Only the A segments of the polymer and thesolvent contribute becausewe assumethat the solvent isinfinitely poor for theB segments.

Now we have a free energy which is a functional ofbs�z� andthe pdf of chainconformationsP�a�. To deter-minethepdf andthesolventdensity profile we minimizethe free energy, Eq. (8), subject to the packing con-straints,Eq. (9). Theminimization is carriedout by intro-ducing a setof Lagrangemultipliers bp�z� conjugated tothe packing constraints. We obtain for the pdf of chainconformations

P�a� � 1q

exp

�ÿbZ

p�z�nA�z; a� dz

ÿ vAA

Z ZnA�z; a�pbA�z9�Pdzdz9

ÿvAS

Z d

0

nA�z; a�dzÿvABr

Z d

0

nA�z; a�dz

�; �10�

whereall the interactionparameters havebeenscaledbythe temperature, i. e. vi � bei and q is a normalizationconstant (single chain partition function). The first termin the exponential represents repulsive interactionsbetweenthe chain in conformationa andthe otherpoly-mer andsolvent molecules.Thesecondrepresentsattrac-tive interactions between the A segmentsor in otherwords is the one representing the quality of the solvent.The third term correspondsto interactionsbetweenbaresurfaceandA segments,andthe last term correspondstotheB-inducedrepulsionsof A segmentswith thesurface.

Thesolvent densityprofile is givenby

bs�z� � expÿbp�z�50 � blsolv� �: �11�


The lastequation enablesusto understand thephysicalmeaning of the Lagrangemultipliers. They arerelatedtothe local (inhomogeneous) osmoticpressurenecessaryinorder to fulfill the thermodynamic condition of constantsolventchemical potential at all distancesfrom the sur-face.Thus,theinhomogeneouspressureprofile is a mani-festation (or a result) of the inhomogeneousdistributionof solventandpolymer segmentsasa function of thedis-tancefrom the surface. The lateral pressures,p�z�, canalsobeobtainedfrom expansion of thesystem’s partitionfunction,this is shownin detail in ref.46)

The only unknownsto determine pdf andsolvent den-sity profiles are the lateral pressureprofiles. Theseareobtainedby introducing the explicit expressions for thepdf, Eq. (10), and the solvent density profile, Eq. (11),into the constraint equations, Eq. (9). Then the inputnecessary to solve for lateral pressuresare: (i) the setofsinglechainconformationsof theA block, (ii) interactionparametersvAA; vAS and vAB , (iii) grafting density r and(iv) the value of the solvent chemicalpotential. It turnsout that the value of lsolv is not necessary to solve theequations due to the incompressibility assumption, seeref.53)

Introducing the input into the constraint equations weendup with a setof non-linearcoupledequationsthatcanbesolvedin a straightforwardway by standardnumericalmethods. It is importantto emphasizethat the set of sin-gle chain conformationsneedsto be generatedonce andthenthesamesetis usedfor all thecalculationsfor differ-entvaluesof surfacecoverageandtheseveral interactionparameters. The output of the theory providesquantita-tive information on how the weight of the differentcon-formations changesdepending upon the thermodynamicstateof the system. Thus,all the resultsshown below foreachpolymer molecular weight havebeenobtainedfromthe samesetof singlechainconfigurations.As it will beseen,depending on the interaction parametersand thesurfacecoverageof polymerthetheory is ableto shift therelative weight of the different conformations. Thisresults in averageproperties that range from a fullyadsorbedlayer to a highly stretchedbrush.

Oncewe know the lateral pressure profile we cancal-culateanydesiredaverageconformationalandthermody-namic property. For example, by replacing the pdf,Eq. (10),andthesolventdensityprofile, Eq. (11), into thefreeenergy, Eq. (8), using theexplicit forms for theener-getic and entropic contributions, Eq.(1, 2, 4, 5, 7)respectively, weobtain

bWA

R�ÿ

Z v

0

bp�z�dzÿ r

2

Z v

0

Z v

0

vAApnA�z�P pbA�z9�Pdzdz9ÿ r lnq: �12�The total surfacepressure,which canbe usedto com-

pare with experimental observations and to determine

phaseequilibrium, is obtainedby differentiating the freeenergy with respectto thearea. Namely,

bP � ÿ qWA

qR

� �Npol;Ns;T

�Z v

0

bp�z�dz� r

2

Z v

0

Z v

0

vAApnA�z�PpbA�z9�Pdzdz9

� r2

Z v

0

vABpnA�z�P dzÿ rNA: �13�

Noteagain that thereis no translational termsinceit isassumedthat the polymers arechemically grafted to thesurface.

The discussionup to this point hasbeenconcentratedon the caseof polymermolecules grafted by the B blockto a surface. We now generalize the description to thecasewhere the block copolymers are spread at a fluid-fluid interface.All the contributionsdiscussedabovearepart of this new systemhowever, two more contributionsneed to be considered. First, the translational entropyterm,Eq. (6), mustbe includedbecausethe polymersaremobile on the interface.Second, the B blocks havenowconformational degreesof freedom andinteractionswiththeir own solvent.

Therefore thefreeenergy of theB block wil l be

bWB

R� r

Xfcg

P�c� lnP�c� � r

2

Z v

0

Z v

0

vBBpnB�z�P pbB�z9�Pdzdz9; �14�where c representsthe set of configurations of the Bchains.We wil l assumefor simplicity that there are noattractive interactions betweenB segmentsand the sur-face, and AB (surface) repulsions are alreadyaccountedfor in thecontributionsfrom theA term,seeEq. (4).

At this point it is importantto make anotherdistinctionbetweenthiscaseandthatof thesolid surface. Expression(4) wasderivedassuming that all the B segmentsareonthesurface.In thecaseof a fluid-f luid interfacethenum-berof B segmentsin direct contactwith A, i. e. thosethatarein theclosevicinity of the interface,changesdepend-ing uponthethermodynamicstateof thesystem. For sim-plicity, andasit will bediscussedbelowdueto theparti-cularcases thatwe areinterested in here, we will assumethatvAB is constant.Themainreason is that thechangeinthenumber of B segmentsis not very largeasthesurfacecoveragechangessincewe are only treating the case inwhich the interface is not attractive to the B segments.Therefore,there is a depletion of B segments from theinterfacethat will not increasedramatically if we addi-tionally consider AB repulsion. From the technical pointof view, this approximation implies that the free energiesof A andB areadditive andthereforethesolution is muchsimplerthanin thefully interactingcase.

Thepdf of B chainsis obtainedalongthesamelinesasthat for A, i. e. we considervolume fillin g constraints for


B and its solvent,and thenwe minimize the free energysubjectto thepacking constraint andobtainfor thepdf

P�c� � 1qB

exp

�ÿZ 0

ÿvbp�z�nB�z; c�dzÿ

Z 0

ÿv

Z 0

ÿv

vBBnB�z; c�pbB�z9�Pdzdz9�; �15�

andfor thesolventdensity profile anexpression identicalto Eq. (11). Note that the lateralpressuresherearediffer-ent thanthosefor theA block. Further, we wil l only con-sider the case in which the solvent for B is poor, seebelow.

The B block in the caseof fluid-f luid interface contri-butesto thefreeenergy thefollowing terms

bWB

R�ÿ

Z 0

ÿvbp�z� dzÿ r

2

Z 0

ÿv

Z 0

ÿvvBBpnb�z�P pbb�z9�Pdzdz9ÿ r lnqB � r ln rl2: �16�

And thecontribution to thepressure is

bP �ÿ qWB

qR�Z 0

ÿvbp�z�dz� r

2

Z 0

ÿv

Z 0

ÿvvBB pnB�z; c�P pbB�z9�Pdzdz9ÿ rNB � r: �17�

Thetotal freeenergy andpressurefor theblock copoly-mer at the fluid-fluid interface is obtained by addingEq. (16) to (12) and(17) to (13), respectively.

The Resultssection will showthat the inclusionof thecontribution from both blocks are necessary to explainthe different experimental observations in a variety ofsystems.We will show that there are several types ofphasetransitions andthe phasesat equilibrium havedif-ferent average structures, depending upon the type ofinteraction that dominates in each phase.Furthermore,we wil l demonstrate how the ability of the differentphasesto preventproteinadsorption,both thermodynami-cally andkinetically, dependsupon the structural proper-tiesof thetetheredblock.

The technical detailsof how the conformationsof thechainsaregeneratedandhow thecalculationsarecarriedout can be found in ref.19,53) In the results shownbelowthe set of single chain conformations for the tetheredchains is obtained using the rotational isomeric statemodel.The useof this chainmodel with the theorypro-vides excellentquantitative agreementwith a variety ofexperimental systems19,31,32,39).

2.2 Tetheredlayer in contactwith proteinsolution

We consider protein adsorption onto solid surfaces.Therefore,the contribution to the free energy from thepolymerlayer is asdescribed in Eq. (8). We now addthecontribution from the protein molecules. There is an

importantdifferencebetweenthe contribution to the freeenergy of the protein andthat of the polymerchain. Thisis that the proteins come from a bath (the solution) andthus,we needto considertheprotein in thegrand canoni-cal ensemble. Further, we do not know thedistribution ofthe protein molecules asa function of the distancefromthe surface. The z dependent density of proteins,qpro�z� � Npro�z�=R, is one important quantity that wewant the theoryto beableto predict. In particular qpro�0�providesthenumberof proteinsadsorbed at thesurface.

We considera simplemodelproteinfor which the sol-vent andthe tetheredpolymer chainsexert only repulsiveinteractions.Effectively, this implies that thesolvent-pro-tein, protein-polymer, protein-protein attractions are allequal. The contribution of the protein to the free energyincludes:1. The z dependent conformational entropy of the pro-

tein.This term hastheform

Sconf; pro�z� � ÿkXfkg

Ppro�k; z� ln Ppro�k; z�: �18�

2. Thezdependent translational entropy of theproteins.

Stran; pro�z� � ÿk ln�qpro�z�l3�: �19�3. The bare interaction of the protein with the surface,

Upro-s�z�.4. The repulsion betweenthe protein and the surface

bound segments of the B block (anchoring block).This termhastheform (total contribution)

Epro-B �Z d

0

Zvpro-Brqpro�z�pnpro�z9; z�P dz9 dz; �20�

where, as before, we assume that the range of thisinteraction is d andthe interactionparameter is givenby vpro-B � epro-BRnb=kT, seeEq. (4). pnpro�z9; z�P dz isthe averagenumberof interacting sitesthat a proteinat (reference)position z9 hasat distancez from thesur-face.For thesimplified protein modelthatwe use(seebelow), we have pn�z9; z�P � d�zÿ z9�, and Eq. (20)reducesto

Epro-B �Z d

0

vpro-Brqpro�z�dz: �21�

5. The repulsive interactions betweenthe proteins andtheother moleculesin thesystem. Namely, thesolventandthe tetheredpolymers.Theseareaccountedfor bygeneralizing theconstraint equationsto read

rpn�z�P50 �Zqpro�z9�p5pro�z9; z�Pdz9�bs�z�� 1; �22�

wherethe integralover theproteindensitiesis necess-ary to include all the volume contributions that pro-


teinsat differentz9 make to z. p5pro�z9; z�P dz is theaver-age volume that proteins at z9 contribute to z. q�z9�denotesthe numberof proteinsper unit volume thattheir point of closestproximity to thesurfaceis z9.

Thetotal freeenergy perunit areaof thetetheredpoly-mer-protein-solvent systemis givenby

WR� r�EAS� EABÿ TSA� � Epro-B�

Zqpro�z��Upro-s�z�

ÿ TSconf;pro�z� ÿ TStrans;pro�z� ÿ lpro� dz �23�where the last term appearsbecause we are treating agrandcanonicalensemble for theproteins.

The pdf of tetheredpolymers,the z dependent pdf ofthe proteins, the protein densityprofile, qpro�z�, and thesolvent density profile are obtainedby minimization ofthe free energy density, Eq. (23), subject to the packingconstraints,Eq. (22).

The expressions for the pdf of the tethered chains andthat for the solvent density profile are identical to theonesderivedabove,Eq. (10,11), respectively. The maindifferenceis that now the lateral pressures,p�z�, wil l bedetermined in a systemthat includes the proteinsas itwill be manifested in the packing constraints, Eq. (22),thatarenecessaryto quantifytheLagrangemultipliers.

The minimization gives for the z dependentpdf of theproteins,

P�k; z� � 1qpro�z� exp

�ÿbUpro-s�z� ÿ vpro-BrH�z�

ÿZbp�z9� 5pro�z9; z�dz9

�; �24�

whereH�z� � 1 if z f d andzerootherwise,and

qpro�z� �Xfkg

exp�ÿbUpro-s�z� ÿ vpro-BrH�z�

ÿZbp�z9� 5pro�z9; z�dz9� �25�

is thenormalizationconstant for eachz. The first term inthe exponential of the pdf arisesfrom the surface-proteinattractions, the second are the surface-protein repulsionsinducedby the presenceof the B segmentsandthe thirdcontains theintermolecular repulsions.

Thedensity profile of theproteinsis givenby

qpro�z�l3 � eblproqpro�z�: �26�This last equation shows the interplay that determines

theamount of proteinsthatwill adsorb.Note that thermo-dynamic equilibrium requiresthe chemical potential oftheproteinsto bethesameat all z. Arranging Eq. (26) wegetthat

blpro � lnqpro�z�l3qpro�z� : �27�

Therefore in orderto havea strong adsorption we needthe partition function of the protein at the surfaceto belarge. This can be achieved if the surface-protein attrac-tion is strong, see Eq. (25). However, a large concentra-tion of proteinson the surfacemakes the lateral repul-sions,i. e.p�z�, very largetoo.Therefore,it is thebalancebetweenthetwo thatdeterminestheamount of proteinonthe surfacefor a given bulk concentration. Furthermore,the presenceof tethered polymers further increasesthelateral pressures closeto the surfaceand thus decreasesthe amount of protein adsorbed. This interplay dependsuponmany factorsaswill beshownin theresults section.

Theonly unknownsto determine thetwo pdf’s (proteinand tetheredpolymer) and the two densityprofiles (pro-tein andsolvent) are the lateral pressuresp�z�. They arenumerically obtainedby replacingEq. (10,11,24,26) intothe constraintequations, Eq. (22). The input necessarytosolve theseequations are: the setof singlechainconfor-mationsfor the polymer chains, the set of single chainconformations for the proteins, the surfacecoverageofpolymer, the chemical potentialsof the solvent and theproteinandthe interactionparameters,including thebaresurface-protein interactions.Note again, that in order tosolvefor all the differentconditions the setof conforma-tions for the tetheredpolymersand the proteinsneedtobegenerated only once.

In practice,the resulting equations are solved by dis-cretization of spacereplacing theintegral equationsinto aset of couplednon-linear equations for the lateral pres-sures.The technical details, including detailed equations,canbefoundin ref.54,55)

For simplicity we assume that theproteinin bulk hasasingleconfiguration but it can“denaturate” upon contactwith thesurface.Seesection on resultsof proteins.

3 ResultsAll the results presentedin this section are for genericpolymers,i. e. we usea genericchain model andinterac-tion parametersthatmay represent a largevarietyof pos-sible block copolymers. We have recently shown thatwith theappropriate choice of interactionparametersandchain model, there is very good agreement betweenthetheoreticalpredictionsandexperimental observationsforpressureareaisothermsof PS-PEOspreadat thewater-airinterface31) and for adsorption isotherms of proteins onsurfaces with PEO-PPO-PEO triblock copolymers39).However, the objective of this section is an attempt toshowthevariety of possible structuresandphasetransfor-mationsthat can be found under different experimentalconditionsandhow thosestructuresmay be usedto pre-vent proteinadsorption. Therefore,we presentthe calcu-lations for generic modelsof polymersand proteinsanddiscusstherelationship with experimentalsystems for thedifferent cases.


Thefirst partof this section concentrateson theproper-ties of the layersthemselves, including the structureandphasebehavior. Af ter that we devoteour attentionto theability of thepolymer layersto interactwith proteinsfreein solution.

3.1 Pressure areaisothermsandphasebehavioronfluid-fluid interfaces

We first considerdiblock copolymersspreadat fluid-f luidinterfaces.The caseof solid surfaceswill be consideredbelow. Several experimental systems have beenstudiedsuchas PS-PEO30,31) and other nonionic diblock copoly-mers56,57), aswell asPS-PVP37,38) (PVP is poly(vinylpyri-dinium) which is a polyelectrolyte) at water-air inter-faces.For the PS-PEOdiblock it has been found thatmonomers of PEOhaveaneffectiveattractive interactionwith the interface.This interaction is responsible for hav-ing the pressure-area isothermsshow a “plateau-like”region.Howeverexperimentally, a first orderphasetran-sition was not found30,31). Furthermore, both groups30,31)

havepresentedtheoreticalpredictions that showthat theattraction of PEO monomers to the interface is notenoughto havea first order phasetransition. This is incontrast to earlier predictions of Alexander20) and ofLiguore40) who suggested that tetheredpolymer layersinwhich themonomersareattractedto thesurface/interfaceshouldshowa first orderphasetransitionbetweena pan-cake like structure (2-D adsorbed layer) at low surfacecoverageanda stretchedlike brushconfigurationof high(or moderate) surfacecoverage.As it hasbeenrecentlyshown,and reviewed below, we predict the presenceofphasetransitionsonly whenthe other block is takenintoconsideration and we will show that different type ofphasetransitionsmayoccur.

Severalother diblock molecules at the water-air inter-face were extensivelystudied by Eisenberg and cowor-kers37,38,56,57). They have shown the presenceof aggre-gatesor domainformationat the interface.Theseaggre-gatesmay be two dimensionalmicelles or they alsomayindicate coexistencebetweentwo phases.The pressureareaisotherms in the samesystems exhibit a very welldefined plateau which indicate an apparent first orderphasetransition, and transmission electronmicrographsshowthe formationsof surfacemicelles.Our theoreticalstudiespresented below concentrateonly on the caseofmacroscopic phasetransition andthepossibility of aggre-gate formation on the surfacehas not beenconsidered(seebelow).

We will attempt to considera large variety of systemsto seeunder what conditions oneshould expectonetypeof behavior or another. For all the diblock copolymers ata fluid-f luid interfacewe considerthe B block to be in apoor solvent environment,i. e. the effective solventtem-peratureis below theh temperaturefor that solvent-poly-

mer mixture. We consider that casebecause that wasfound to be a very good description of PSblocks at theair sideof water-air interfaces.For theA block thesolventis good,howeverdifferentsegment-interfaceinteractionswil l bestudied.

The first case that we consider is that in which the Ablock doesnot have any attractive interactions with the

Fig. 2. Pressureareaisotherms,the lateral pressureasa func-tion of the areaper molecule,a� R=Npol . The A block is in agood (athermal)solvent, and the B block is in a poor solvent,T=H = 3/5 (the interactionparametersarethesameasin ref.47)).Both blocks(A andB) havepurelyrepulsiveinteractionwith theinterface.In theupperfigure,we showthecontributionfrom theA block for three different chain lengths nA: 50 (dotted), 75(dashed) and 100 (long dashed)segments.The lower figureshowsthe contributionfrom the B block of 30 segments(solid)and the total pressure of the diblock for the threepossible A-Bcombinations.A long enoughA block stabilizesthe layerat anysurface coverage. In this and the following figures we usethemolecularparametersfor the chainsthat bestfit the experimen-tal pressure-area isothermsof PS-PEOat the air-water inter-face31). However, the resultsare also valid for genericdiblockcopolymersunderthesameconditions


interface,i. e. the interfaceis purely repulsive for both AandB. Fig. 2 showsthe pressure-area isotherms for eachof the blocks and their sumsfor a fixed chain length oftheB block anda variety of lengthsfor theA block. (Forthe individual contributions to the pressure, the B blockincludesthe translational entropy.) The isotherm of the Bblock showsa vanderWaalsloop characteristic of a firstorderphasetransition.The behavior of tetheredpolymerlayersin poor solventhasbeendescribed in several stu-dies28,47,58–61). Thepressure of theA block is alwaysrepul-sive and it is the more repulsive the longer the chainlength. When the two contributions are addedtogether,for shortchain lengthsof theA block thereis a slight shiftof the van der Waals loop but still there is coexistencebetweentwo phases.For largeenoughA blockstherepul-sive contribution of the stretching of the A block isenoughto overcomethe attraction of the B block. As aresult, there is no more van der Waals loop and thediblock layer is stableat anysurfacecoverage.

It will be interesting to determine the necessary mole-cularweightof A block thatwill stabilize theB block, i. e.that will make the polymer layer stableat any surfacecoverage. Clearly, this will depend on the molecularweightof theB block.However, eventheratio of molecu-lar weights wil l be meaningful only if it is found as a

function of the solvent-B interaction,i. e. quality of sol-vent or effective temperature for the B block. Therefore,the bestway to obtain this information is by looking atthe whole phasediagram in the plane of temperature(quality of solvent)vs surfacecoverage.This is shown inFig. 3 for A-B diblock copolymersat fluid-fluid interfaceswith fixed chainlength of the B block anda variety of Amolecular weights.Theeffect of increasing themolecularweight of the A block is to stabilize the polymer mono-layer. The longertheA chain length the lower thecriticaltemperature,andthebinodal is narrower.

Fig. 4 showsthe phasediagramsfor a longer(fixed) Bblock. The shapeand qualitative features of the phasediagramsarethesameasthosein Fig. 3. It is clearthat inorderto have thesamedegreeof stabilization, i. e. scaledcritical temperature,for a longerB block a much longerAblock is necessary.

To summarizetheability of theA block to stabilize themonolayer, Fig. 5 shows the (scaled)critical temperatureas a function of the ratio of chain length betweenthe Aand B blocks. The two different B block systems showvery similar scaled critical temperature as a function ofthe ratio of molecularweights.The relationshipseemstobe linear. We do not havean explanationfor the linearrelationship found in our results. However, this curvemaybevery useful in thedesign of experimental systemsthat are stable at all surfacecoverage. The only inputnecessaryis the critical temperatureof the pureB blockforming a monolayer. While this is not a systemthat canbe realized experimentally (no anchoring group to theinterface) the critical temperature of the monolayerof

Fig. 3. Phasediagrams for the A-B diblock copolymer at afluid-fluid interface. The phasediagram is the binodal in thetemperature-surfacecoverage plane.Namely, the curvesaretheonsetof points that representfor eachtemperaturethe two sur-facecoveragevaluesat coexistence.Insidethe binodal,the sys-tem separatesinto two phaseswith the surfacecoverage of thebinodal at the given temperature.The length of the B block isnB = 30 for all cases.The lengthof theA block is: 0 (solid li ne),50 (dottedline), 100 (dashedline) and 150 (long dashedline).The temperatureis scaledby the critical temperatureof a pureB-block monolayer

Fig. 4. Phasediagramsfor the A-B diblock copolymer at afluid-fluid interface.The lengthof the B block is nB = 50 for allcases.The length of the A block is: 0 (solid line), 50 (dottedline), 100(dashedline) and150(long dashedline)


pureB canberelatedto thebulk critical temperatureandto theh temperatureof theB homopolymer47).

We now considerwhat is the structureof the phasesatequilibrium in the two phaseregion. To understand theorigin of the different averagemolecular structures,it isnecessaryto keepin mind that the driving force for thephaseseparationis the fact that the B block is in a poorsolvent environment. Therefore, one should expect thedilute phaseto havetheB block assmall, compact,singlechains,while the more concentratedphaseis the one inwhich theB block is trying to haveasmuchcontactwithotherB blocksaspossible, i. e. in both phasestheB blockminimizesasmuch aspossiblethe contactwith solvent.This is shownin the two upper graphsof Fig. 6 (pureBmonolayer), the left profile shows the dilute regime(small compactchains) while the right one shows theconcentrated branch (laterally stretched chains). Theother four figures show the structure of the wholediblock, all at the sametemperaturebut for two differentchain lengths of A blocks.The A block structurehasthetypical featuresof a tetheredpolymer layer in goodsol-vent, a small depletion regionvery closeto the interfacefollowed by a stretchedchain.Thedegreeof stretchingoftheA block is much largeron theconcentratedsideof thecoexistencecurve(right graphs)dueto thelargerconcen-trationof chains in thegood solvent regime.

The differencein the B block structurein the dilute(concentrated)branch is that increasing the chain lengthof the A block increases(decreases)the surfacecoverage

at coexistence, thusresulting in a higher (lower) volumefraction of B segments,but always keeping the shapedeterminedby thequality of solvent.

We summarizethecase in which theB block is in poorsolventconditions,theA block is in a good solventenvir-onment, andtheinterfacedoesnotattracteitherof thetwoblocks.We find that if theA block is largeenoughthereisstabilizationof themonolayeratanysurfacecoverage. ForA blocksthataresmalleror of thesamesizeof theB block,our findingsshowthat thereis coexistencebetweentwophasesdrivenby thepoorquality of thesolvent for theBblock. Themain role of theA block is to shift thecriticaltemperature(andthewholephasediagram)towardslowertemperaturesdueto theadditionof theA-A repulsions.Wefind an“empirical” linearrelationship betweenthescaledcritical temperature of the diblock andthe ratio of chainlengths of the two blocks.This relationship may be veryuseful in theexperimental designof stablediblock mono-layers spreadat fluid/fluid interfaces. Thestructureof themonolayers is that of a tetheredchainin thepoor solventregime for theB block attachedto onein thegood solventregimefor theA block.

We now considerthe case in which the A block hasattractive interactions with the interface. All the otherconditions remain the same as the systems describedabove. Namely, the solvent is poor for the B block andgood for the A block but now the A monomers areattractedto theinterface.This hasbeenrecently shown tobe the casefor PS-PEO at the water-air interface wherethepredictionsof the theory showvery good quantitativeagreement with the experimentally measuredpressure-area isotherms.We shortly review the findings of thatwork andthenextendthe calculationspresentedtheretoother systems.ThePS-PEOdiblocksstudiedin ref.31) hada relatively shortPSblock anda longerPEOblock. Thepressure-area isothermshowed a plateau-likeregion andit wasfoundthat theorigin of theplateau wasnot a phasetransitionbut theresultof theattraction of thePEOblockto thewater-air interface.Similar conclusionsweredrawnby theWageningengroup30). The reasonfor theattractionof EO segmentsto the interfaceis theamphiphilic natureof theCH21CH21O group.Thefact thatthesemonomersare solublein water doesnot eliminate the hydrophobicnature of themethylenicgroups.

The structure of the PEO block in the dilute regimewas found to be that of an adsorbedmushroom. As thesurfacecoverage of the diblock is increasedsomeof thesegmentsof thePEOdesorbfrom thesurfaceandsolubi-lize into thewaterforming a brushwhoseeffective chainlength is shorterthan that of the PEO block due to theadsorbedsegments.Thestructureof thePSblock is basi-cally the sameas shown above,namely a tetheredlayerin a poorsolventregime.

The studieson ref.31) were done for a relatively shortPS block and the PEO block was longer in all cases.

Fig. 5. The critical temperature of the diblock copolymerlayer, scaledby the critical temperatureof a monolayerof pureB blocks, asa functionof theratio of molecular weightsbetweentheA andB blocks.Circles correspondto nB = 50 andsquares tonB = 30. The straightline is the bestfit to all the points,whichgivesTc;AB=Tc;B = 1 – 0.0454nA=nB


Here,we extend thosestudies to a longerPSblock andalarger range of PEO chain lengths. Throughout the fol-lowing discussionwe will refer to the A block as PEOand to the B block as PS because our predictions are inquantitativeagreement with experimental observations inthat particular system. However, the qualitative featurespresentedaregenericfor the caseof a B block in a poor

solventenvironment and the A block in a good solventenvironmentwith attractionsto theinterface.

Fig. 7 shows the contribution to the pressure-area iso-therms for the PEO block. The main differences com-paredto pressureareaisothermsof theA block that is notattractedto the interface(seeFig. 2) areas follows: Theisothermsfor the blocks with attractive interactionswith

Fig. 6. Polymerdensity profiles,variationof polymersegmentvolumefractionasa functionof thedistancefrom thetetheringsurface, for B (solid line, nB = 30) andA (dottedline) blocks,at thesurfacecoveragecorre-spondingto thecoexistence phasesfor T=Tc;B = 0.86.Plotson theleft correspondto thedilute phaseandthoseon theright arefor theconcentrated phase(notethe differentscale). Thetop setis for pureB block, thesetinthecentercorrespondsto nA = 50,andtheoneon thebottomis for nA = 75


the interface show a “kink” that arisesat the areapermolecule at which the completely adsorbedpolymershave strongenough(two dimensional) lateral repulsionsthat someof the segmentsstart to solubilize into the sol-vent. This changeof slope,which is responsible for theplateau-likeshapeof the isotherms, is found to appear atlarger areasas the PEOchain length increases. The rea-sonis that theadsorbedtwo dimensionalblocks havelar-ger excludedareasas their chain length increases, i. e.longerPEOblocksshowa largervalueof theareafor thekink in the pressure. The changeof position of the kinkresults in a very rich phase behavior as will be shownnext.

Fig. 8a–c show the pressure-area isotherm for thediblock copolymers.The upper graphin eachcaseshowsthe individual contribution to the pressure from eachofthe two blocks, while the lower graph showsthe totalpressure.Comparing the pressure-area isothermsof thethreegraphs,it is clear that thereis a qualitative changein the shapeof the isotherms.The shortestchain lengthsof PEO (Fig. 8a) seemto have a similar effect on thepressure-area isotherm asin the caseof no attractionsofthe A block to the interface,seeFig. 2. The only differ-

Fig. 7. Pressure-area isothermsfor the A block with attractiveinteractionswith the surface (e.g. PEO at the air-water inter-face). Chain lengths are: solid line (nA = 25), dotted line(nA = 40), dashedline (nA = 50), long dashedline (nA = 60), dot-dashedline (nA = 75), solid line-circles(nA = 100), dotted line-squares(nA = 125)anddashedline-diamonds(nA = 150)

(a) (b)


encebeing the larger contribution to the pressuredue tothe stronger repulsions in the PEO caseas compared tothecaseshownin Fig. 2.

For the intermediate PEO chain lengths shown(Fig. 8b), thereis a big shift in theposition of thevanderWaalsloop, comparethe areasof the loop with thoseofFig. 2. The reasonfor the shift is the position of the kinkin the pressure of the PEOblock. The areaper moleculeat which the kink appears increases as the molecularweight of the tethered chain increases. Finally, for thelongestPEO chains considered here (Fig. 8c), thereis ashift in theposition of thevanderWaalsloop,but thereisalsoa changein the shapeof the loop. Again, this is the

direct result of combining the contribution of the PSblock, which is fixed in all the cases,with the pressurecontribution of thePEOblock thatdueto thepresenceofthekink shiftsthepositionandshapeof theloopsdepend-ing uponthemolecular weightof thePEOblock.

It is clearthattheshift in thevanderWaalsloopcausedby thekink in thepressureof the PEOblock wil l dependupon the temperature, i. e. the quality of the solvent forthe PSblock. To summarize the effect, Fig. 9 shows thephasediagramfor a varietyof PEOmolecular weights fora fixed chain length of PS.Thefigure is dividedinto threegraphsin order to have a clearerpictureof the dramaticchangesin the phasediagram causedby the location ofthe kink in the PEOpressureascompared to the originalvanderWaalsloop of thepurePS.

For the shortestPEOblocks,the main role of the PEOis to stabilize the layer. In other words,the critical tem-

(c)

Fig. 8. Pressure-areaisotherms for PS-PEOat the air-waterinterface.PEOis of the type shownin Fig. 7. Individual contri-butionsfrom eachblock are in the upper plots. Total pressuresareshownin the lower plots. In all the cases,the PS(B) blockhas50 segments(upperplots, solid line). (a) short PEO chainlength: dotted line (nPEO= 25), dashedline (nPEO = 40), longdashedline (nPEO= 50). The inset showsthe maximum of thetotal pressure, and its decay to zero for very large areas;(b)intermediatePEO chain length: dotted line (nPEO= 50), dashedline (nPEO= 60), long dashedline (nPEO = 75); (c) long PEOchain length: dottedline (nPEO= 100), dashedline (nPEO= 125),long dashedline (nPEO= 150)

Fig. 9. Phasediagramsfor PS-PEOsystems. In all the cases,chainlengthof thePSblock is 50 segments,andthetemperatureis scaledby the critical temperatureof the purePS layer. PEOchainlengthsareasfollows. Upperfigure: Solid line (nPEO= 0),dottedline (nPEO= 25), dashedline (nPEO= 40). Intermediatefig-ure: Solid line (nPEO= 50), dotted line (nPEO= 60), dashed line(nPEO= 75). Lower figure: Solid line (nPEO= 100), dotted line(nPEO= 125),dashedline (nPEO= 150)


peratureis lower thanin thepurePSlayer. Note however,that the attractive interaction of the PEO segmentswiththeinterfaceresultsin a strongerstabilization thanfor theequivalent molecular weight but without the attractiveinteraction, see Fig. 4. PS-PEO(50-40) alreadyshowsaphasediagramthat is different in shapethan the shorterchain lengths. This effect is more dramatic for 50-50wherethephasediagram(middle graph)shows two criti-cal points.

Thepresenceof thesecondcritical point canbeunder-stoodby looking at the pressure-areaisotherm of the 50-50 diblock. This is shown in Fig. 10. The competitionbetweenthe different slopes of the PS pressureand thePEOpressureresultsin doubleminima for a small rangeof temperatures.The interestingresult is the role of thePEOblock to stabilize both the very dilute and the con-centratedregimes.Namely, the 50-50diblock shows thetransition from a phase diagram dominated by theinstability in the PS block (typical gas-liquid competi-tion), to a phasediagramin which thedilute phaseis sta-bilized due to the attraction of the EO monomers to theinterface resulting in narrow phasediagramsthat do notchangemuch with chain length (in the range nPEO 100–150).

The two branchesin the phasediagramscan be sum-marizedby looking at thecritical temperatureandcriticaldensity as a function of chain length. This is shown inFig. 11. The critical surfacecoverage showstwo distinctbranches.Thelow PEOmolecular weightbranchhaslowcritical surfacecoveragedominatedby the PSinstability.The high surfacecoverage branch is dominatedby the

behavior of thePEOblock, i. e. thelocation of thekink inthe pressure with respectto the van der Waals loop. Inboth branches the critical surface coverage decreaseswith increasing PEOmolecularweight. The critical tem-peratureshows a very strongdecreaseasthe PEOmole-cular weight increasesin thePSdominatedbranch.How-ever, it is not monotonic and variesonly slightly in thehigher PEOmolecular weight branch. We expectthat asthe molecular weight of the PEO keepsincreasing, atfixed PS molecular weight, the diblock layer will bestable at all surfacecoveragesbecause the large PEOpressurewill bedominantat all densities.

It is interesting to look at the structureof the polymerlayer in the different phasesat coexistence.To this end,Fig. 12 shows the density profiles of the PEO and PSblocks for two molecular weightsof PEO but fixed PSchain length.For bothmolecularweightsthedilute phaseshows thePEOin a pancakeconfiguration, i. e.almostallthe segmentsare adsorbedto the interface, while the

Fig. 10. Detailed pressure-areaisothermsof PS-PEO (50-50)diblock copolymers at the air-water interface.The uppersolidline correspondsto T=Tc;B = 0.746, and showsa single mini-mum. Decreasing the temperature, a double minimum isobservedat T=Tc;B = 0.743(dottedline); T=Tc;B = 0.740(dashedline); T=Tc;B = 0.738(long dashedline); andT=Tc;B = 0.735(dot-dashedline). The lowest temperature shown, T=Tc;B = 0.733(lower solid line), hasagainasingleminimum

Fig. 11. Critical grafting density (upper figure) and criticaltemperature(lower figure) of PS-PEOdiblock copolymersat theair-waterinterfaceasa functionof thelengthof thePEOblock


moreconcentratedphaseshows somesegmentsadsorbedfollowed by a brushlike structurewith aneffectivelengthshorterthanthe total PEOblock. Note theabsolutevalueof thevolumefraction in the four graphs.Thecase corre-sponding to the dilute regime of the shorterPEO is 10timessmaller thantheothercases,reflecting the fact thatthis dilute systemis dominatedby the PSblock. For thelonger PEO block, the phase diagram has shifted (andnarrowed) to higher surfacedensitiesdue to the moredominant role of the PEOevenat low surfacecoverage.ThePSshows thesamestructureasdiscussed above.

A questionthat arisesis what type of configurationsare responsible for the brushportion of the structureinthe concentratedphase.Namely, do the segmentson thestretchedportion of thedensity profilesbelongto loop ortail configurations?Fig. 13 shows the density profile ofthe PEO block togetherwith the calculated densitypro-

files arising from tail and loop configurations. It is clearthat almost 100% of the brush-like structure arise fromtail configurations. The reason that loop configurationscontributing to the densityprofile far from the interfacehavevery low probability maybedueto thechainlengthsthat we arestudying. Thus,eventhough the loop config-urationsmay be favoredfrom the entropic point of view,onedoesnot seetheentropic contribution being themostimportantone (as compared to tail configurations) untilthe chain length of the block is very large. The figureshowsPEOwith 100segmentsandour calculationsup ton = 150 showthe sameeffect. We cannotextend conclu-sionsfrom theseresults to very long chainlengths.How-ever, it shouldbekept in mind that themolecular weightsof PEO that we are studying are thoseusedexperimen-tally in mostapplicationsfor biocompatible materials andsurfacemodifiedliposomes.

Fig. 12. Polymerdensityprofile for the PS (solid line) and PEO (dottedline) blocks,at the surfacecoveragecorresponding to coexistencephasesfor T=Tc;B = 0.83. Theplotson the left correspondto thedilute phaseandthoseon theright arefor theconcentrated phase.Thelengthof thePSis nPS= 30 for allthecases.Thetopsetis for nPEO= 50,andthebottomsetcorrespondsto nPEO = 100.Notethatthefigureson top correspondto a PSdominated system,thus,thedilute phaseis very dilute. In contrast,PEOdom-inatesthebehavior for thecopolymer of length30–100resultingin a dilute phaseof muchhigherden-sity (Notethedifferentscalesin thefigure)


The last type of systemthat we presentin this sectioncorresponds to diblock copolymers graftedor tethered ata solid surface.As described in the theory section,wewill only consider the case in which the role of the Bblock (the tetheringblock) is to exerta repulsive interac-tion on thesegmentsof theA block whentheyarein con-tact with thesurface.Further, we wil l only showthecasein which the segmentsof the A block have an attractiveinteraction with the bare surface.Thesesystems wererecentlydescribedin detail in ref.42) andwe review herethemain resultsfor completeness.

Fig. 14 shows the pressurearea isotherms for thediblocksspreadat thesolid-fluid interfacefor a varietyofstrengths for theA-B repulsions.As vAB increasesthereisa van der Waals loop appearing. Thus, the repulsionsinducea first order phasetransitionbetweena pancake-like configuration and a stretch (brush-like) configura-tion.

This phasetransition is very different from that of theA-B diblock describedabove, in which theB block is in apoor solventenvironmentandthe A block is attractedtothe interface.First, in the case shown in Fig. 14 the driv-ing force for the phase transition is the competitionbetweenattractionsto thesurfaceof theA block andA-Brepulsions. The phase transition in the caseshown inFig. 8 arisespurely from thepoorsolventenvironmentofthe B block whereasthe effect of A block is to suppressthephasetransition.Second,thestructuralchangeson thetwo sidesof the coexistenceare very different. Fig. 15showsthedensityprofilesof theA block at the two sidesof the coexistencecurve for a given value of vAB . The

structuresat coexistenceshow that there is a true “pan-cake” to “brush” transition. In the pancakeregimethe Ablock is completely adsorbedon the surfacewhile thebrush phaseshows a densityprofile with a depletion ofsegmentsin the vicinity of the surfacedue to the A-Brepulsions.

The pancake to brush phasetransition was first pre-dictedby Alexanderandlaterby Liguore.However, bothstudies predictedthat the only necessary conditionsfor

Fig. 13. The density profile (solid line) of a tetheredPEOblock with 100 segments at a surfacecoverager = 0.002A–2 atthe water-air interface, i. e. vAS = –1. Also shownare the contri-butions to the density profile from loop configurations(dottedline) andfrom tail (dashedline) configurations(dashedline)

Fig. 14. Pressureareaisothermsfor diblock copolymers at asolid-fluid interface.Thecopolymer consistsof a tethered chainwith nA = 100segments,andanchoring block completelygraftedon the surface.In the upperfigure vAS = –1, while in the lowervAS = –2. The differentcurvescorrespondto differentAB repul-sion. The solid line is for no repulsion,dotted line vAB = 16,dashedline vAB = 48, long dashedline vAB = 80 anddot-dashedline vAB = 112. Recall that vAB is multiplied by the total areaofthe surface,thus resulting in very large absolutevalues.Therelevantvariableis vABrl2 (seeEq.(4)) which in all casesis oforderunity or lower


the presenceof the phasetransitionwereattractive inter-actionsbetweenthe tethering block and the surface.Wefind that an additionalrepulsive interaction is necessary.Actually, thecase of PS-PEOstudied experimentally (forshortPSblock) did not showa first orderphasetransitionaswould be expected from the predictionsof Alexanderandof Liguore.

An experimental study with thesametype of systemasthat shown in Fig. 14 and 15 is that of Ou-Yang andGao41) in which they adsorbR-PEO-R,(whereR can beeitherhydrophobichydrocarbonchains, or simply termi-nating hydrogens)to large beadsandmeasure the thick-nessof thepolymerlayerasa functionof thesurfacecov-erage. The R block strongly adsorbs to the surfaceof thebeadswhile the PEO is claimed to have an attractiveinteraction with thesurfacebut muchweaker thanthatofR. Theyobservea sharptransition from a pancakeregimeto a brushasthe surfacecoverageincreasesin which thethicknessof the films is four to five timesthebulk radiusof gyration of thePEOblock. We calculatedtheheight ofthe brush as a function of the surfacecoverage for thesamecasesshown in Fig. 14. There is very good agree-ment between these predictions and the experimentalobservations of Ou-YangandGao, further supporting thevalidity of our theory.

A thorough study of the variation of critical surfacecoverageas a function of chain length, strengthof thebare surface-A monomersattraction and strength of theA-B repulsioncanbefoundin reference42).

In this sectionwe presenteda variety of possible sce-nariosof diblock copolymers at fluid-f luid andsolid-fluid

interfaces. In many caseswe found the possibility ofmacrophaseseparation. For liquid-li quid interfaces wefind that if the B block is in a poor solvent environmentthen this is the driving force for phase separation. Thephasediagram of the diblock dependsupon the ratio oflength between A and B blocks and the interactionsbetweenA blocksandtheinterface.Further, in thecaseofsolid surfacesthe repulsions betweenthe A andB mono-mersarenecessary in order to find a macroscopic phasetransitionbetweena pancakeandabrush.

It is important to emphasizethatall of our studieshaveconsidered only the possibility of macroscopic phaseseparation. However, it is possible that the diblock copo-lymers will form micellar aggregateson the surfacesorinterfaces, namelytwo dimensionalmicelles.This indeedhas been observed in a variety of experimental sys-tems37,38,56,57). While our studiesdo not includethis possi-bility, we believe that the phasediagramsthat we predictshould give proper guidelines for the regions of phasespacewheremicrophaseseparation is possible.Forexam-ple, our studiespredictno phasetransition in (short-long)PS-PEOin agreement with experimental observations.However for larger B blocks, the theory predictsa firstorder phasetransition. Some related experimental sys-tems37,38,56,57) haveshown micellar formationat the inter-face. However, one of the blocks was a polyelectrolyteand thus, long range interactions(not considered here)mayalsobeplayinga role in determining thestructureofthesystem.

We believe that our studiescan be used as generalguidelineson the type of diblocks and the regions ofphasespacewhere homogeneousphasesshould form.Clearly, a studyof microphaseseparationwil l be highlydesirableso that one will be able to predict the type ofmorphological and macroscopicphasesthat the systemmayexhibit for a givendiblock copolymer. For example,microphaseseparationhasbeenpredictedin ref.58,60,62–65),aswell asthe formation of surfacemicelles66). A molecu-lar description of surface micelles as the one usedthroughout this paperrequiresextensivecalculations,andwe are planning to pursuethis possibility in the nearfuture.

4 Protein adsorptionAs mentionedin theIntroduction, tetheredpolymerlayersare being used in the design of biocompatible materi-als3,7), including surface modified liposomes for drugdelivery11,12). The idea is to use the tetheredpolymerlayersasa steric barrierthatwill preventthenon-specificadsorption of proteins. The basic idea is that one canform a polymerbrushof highly stretchedchainsthat willnot allow any large molecule from the solution to reachthe surface67). As has been discussedin recent years,experimental systems in generaldo not form a highly

Fig. 15. Density profile of the tethered chains (nA = 100) atboth sidesof the coexistencecurve for vAS = –2 and vAB = 80.The solid line correspondsto the dense(brush)phase,whereasthedottedline is theprofile on thedilute (pancake)phase


stretchedbrush due to the difficulties associated withforming a high surfacecoveragepolymer layer. In reality,the samesteric barrier that wil l not allow proteins toadsorbdoes not allow polymersto reachthe surfaceandthus, it is hard to form a high surfacecoveragebrushusing solution grafting methods. In general, PEO withrelativelow molecular weight is usedfor thesynthesisofbiocompatible materials3). There are several reasons forthechoice of this polymerincludingwatersolubility, highflexibility and availability. It should be noted however,thatotherpolymerscanalsobeused.

We havediscussedin the previous sectionthat PEOisan amphiphilic polymer and therefore, it should beattractedto hydrophobic surfaces.On the other hand,itwill berepelled from polarsurfacessuchasliposome sur-faces.Thus, a complete molecular understandingof theability of polymersin general, and PEOin particular, torejectproteinsshould includethestudy of proteinadsorp-tion on surfaces with grafted polymersas a function of:(i) surfacecoverageof polymer, (ii ) surface-PEOinterac-tion and(iii ) phasetransitionson the surface.Only afterthe understanding of all theseelements one should beableto designthe bestpolymer layer for preventing pro-tein adsorption.

The first resultsthat we showincludethe comparisonsbetween predictions of the theory and experimentalobservationsfor theadsorption isothermsof lysozymeonhydrophobic surfaceswith grafted triblock copolymersPEO-PPO-PEO. The isothermrepresents the amountofproteinadsorbedasa function of the surfacecoverageofpolymer. Thedetails of themodelusedin thecalculationsfor lysozymecan be found in ref.39) where the compari-sons were shown for the first time. It is important toemphasizethat when we refer to protein adsorption iso-thermswe assumethat thesystemhasreachedthermody-namic equilibrium. The kinetic effects, as well as thevalidity of the equilibrium assumption will be discussedlater.

Experimentswere carriedout in thefollowing way: (1)Glasssurfaceswere silanizedto form a hydrophobic glasssurface.(2) The modified glasswasput in contactwith asolutioncontaining the triblock copolymer. The polymeradsorbsto the surfacedueto the strong hydrophobic nat-ure of the PPO block. (3) The PPO was chemicallygraftedto the surface.(4) The surfacecoverageof poly-merwasmeasured.(5) Thesurfacewith thepolymer wasput in contact with a protein solution. (6) The amountofproteinon thesurfacewasmeasured.

The surfacecoverage of polymer wasvaried by usingdifferent concentration of the triblock in solution in step(2). Note that the protein solutionwaskept at fixed con-centration in all the experiments. Further experimentaldetailscanbefoundin ref.39)

Fig. 16 showsthe isotherms for three different chainlengthsof thePEOblocks.Thefigure showsboth experi-

mental observations and theoretical predictions. Thereareseveralfeatures to learn from this figure. First, thereseems to be no dependence of the amount of adsorbedprotein on the chain length of the PEO block. Second,therearetwo very differentregimesin termsof theabilityof the polymer layer to reducethe amount of proteinadsorbed.At very low surfacecoverageof PEOa slightincreaseof surface coverageof polymer reducestheamount of protein adsorbedby a large amount. After acertain surfacecoverage,increasing the amount of poly-merdoesnot seemto bevery effectivein furtherreducingthe amountof proteinadsorbed. Third, thereis excellentagreement betweenthe theoretical predictions and theexperimentalobservations for all the surfacecoverageofpolymer but the highestones.For highestsurfacecover-age,the theory predicts a completeprotectionof the sur-facewhile theexperimentalobservationsstill showsomeproteinadsorption.

The theory enables us to understand at the molecularlevel what are the factors responsible for the behaviorrevealedin Fig. 16. The agreementbetweentheory andexperimentalobservationswasobtainedfor thefollowingconditions. PEO segmentshave an attractive interactionwith the hydrophobicsurfaceof vAS = –2. This is a rea-sonable number since the attraction with the water-airinterface,which is much lesshydrophobic than the sili-nized glass,is half that amounts31). There is a repulsionbetweenthe PEO and PPO segments.Since we do not

Fig. 16. Adsorption isotherm for lysozyme on glass graftedwith triblock copolymers,PEO-PPO-PEO. Amountof lysozymeon thesurfaceasa functionof graftedpolymersurfacecoverage.Thesymbolsareexperimental observations39) while the linesaretheoretical predictions. The circles (solid line) correspondto(nPEO-nPPO-nPEO) 75-30-75,thetriangles(dotted line) to 98-67-98,andthesquares(dashed line) to 128-54-128


haveany experimental information for the value of thisparameter, we havechosenit slightly lower thanthecriti-cal value sothat thepredictedisothermsarein thehomo-geneousregime(seebelow). Furthermore,we havealsoassumedthat the presenceof PPOon the surfaceresultsin a surfacedependent repulsive contribution to the pro-tein. This repulsion doesnot have a strong effect on theadsorption isotherm.The theoreticalresultspresentedinref.39) do not include this repulsion term, and they alsoshowvery good agreement with the experimental obser-vations.

Theoretical predictionsexplain thedifferentregimesofthe adsorption isotherm shownin Fig. 16. At low surfacecoverageof polymer, the PEO segmentsare in the pan-cake regime. Namely, all the polymer segments are onthe surfaceandincreasing the surfacecoverage increasestheareaoccupiedby PEOby a largeamount. This resultsin a blockageof adsorption sitesfor the protein and thedecreaseof protein adsorption with surfacecoverageisvery sharp.

Theabrupt changein slopeof theadsorption isotherms(both experimental and theoretical) corresponds to thesurfacecoverage at which for the pure polymer surface,i. e. that in the absenceof the proteins, the PEO-PPOrepulsion and the steric PEO-PEO repulsion on the sur-facearelargeenoughto inducethedesorption of some ofthe PEO segments.A way to quantify the surfaceavail-ability for protein adsorption is to determinethe numberof adsorbedethyleneoxide segmentsasa function of thepolymer surfacecoveragein theabsenceof protein mole-cules.This is shownin Fig. 17. The volume fraction ofadsorbedsegmentsshowsamaximum at theexactsurfacecoveragewhere the protein adsorption isotherm showsthe change in slope. This reaffirms that the mechanismfor the sharp reduction of protein adsorption in the lowsurfacecoverage regime is basedon blocking surfaceadsorptionsites.

Theregion of slow variationof protein adsorptionwithsurfacecoverageis characterizedby a differentmechan-ism of the polymer layer to prevent protein adsorption.As Fig. 17 shows, the volumefraction of adsorbedpoly-mer segments decreases,however there is also a brushformedby thepolymer segments that arenot adsorbed tothe surface.In order to adsorbthe protein it hasto pullsomeof the non adsorbedsegmentsfurther from the sur-face.This processis accompaniedby a lossof conforma-tional entropy of the polymer in the brush.However, theproteinsgain a large amountof attractive interaction bybeingin contactwith thesurface. Theresultof that is thatas long as the distancebetweengrafting points is largeenoughto accommodateproteinson the surface,proteinswill adsorbby stretchingthepolymersegmentsout of thesurface.This mechanismto reduceprotein adsorption ismuch lesseffective than occupying adsorption sitesandtherefore, theslopeis smaller at high surfacecoverage54).

A more clear explanation of this effect wil l be shownbelow by treating the casesof attracting andnon-attract-ing surfaces(to thepolymers)separately.

Theunderstanding of thestructureof thepolymerlayerenablesus to understand the mechanismof reduction ofproteinadsorption by PEO-PPO-PEOgrafted on a hydro-phobic surface.There is one point however, that needsfurther explanation. A careful look at the comparisonbetweentheexperimentalobservationsandthetheoreticalpredictions showsthat even though there is very goodagreement betweenthe two, for relatively high surfacecoverage the theory predicts zero protein adsorptionwhile the experimental observationsshow finite adsorp-tion. We believe that the reason for this discrepancy isthat in the theoretical calculations,we haveassumed thatthe strengthof the PEO-PPOrepulsions is smaller thanthat necessary for a phasetransition. However, experi-mentalobservations in the relatively high surfacecover-age regime are probably in the coexistence regionbetweenpancakeand the stretchedbrush. PreliminaryAFM observations of the structureof the polymer modi-fied surfacein the absenceof proteins show the forma-tions of domainsof stretchedbrushesand domains ofpancakelikestructures68).

Experimental observations are for a systemin whichmany different interactions play an important role indetermining the structure of the polymer layer and thus,the resulting interactions with the protein molecules.While the theorycanpredict this behavior andexplain itin great detail from the microscopicpoint of view, a bet-

Fig. 17. Amount of EO segmentsadsorbed on the surfaceasafunction of the areaper molecule.The systemcorrespondstonPEO-nPPO-nPEO= 75-30-75in the absenceof proteins.Note thatthe surfacecoverageof maximumamountof adsorbedEO seg-mentscoincideswith the changein slopeof the adsorptioniso-thermsshownin Fig. 16


ter understanding of the generic features that determinethe ability of a polymer layer to preventprotein adsorp-tion canbeobtainedby looking at theeffect of theindivi-dual interactionsseparately.

The resultspresentedbelow arefor a modelsysteminwhich the protein can exist in a single configuration inbulk, but uponcontact with the surfaceit can“denature”into a differentconfiguration. The conformation that canexist both in bulk and on the surfacewill be called thesphericalconfiguration.This is modeled asa spherewithdimensionssimilar to thoseof lysozyme andwe considerthe surface-protein interactions to be thoseas calculatedfrom atomistic models of lysozyme with hydrophobicsurfaces69). The interactionpotential is shownin Fig. 18.The denatured conformation is a disk-like configurationwhosevolume is the sameasthe sphericalconfiguration.The height of the disk is 4/5 the radiusof the sphericalconfiguration. Theinteraction of thedisk with thesurfaceis twice that of the sphericalconfiguration.Note that thedisk configurationcanonly exist in contact with the sur-face and thus only the interaction at contact is needed.This model for theproteinis bornof experimental obser-vationson lysozymesuggestingthat uponadsorption onthe surfaceit denatures into a structureresembling thatsuggested here70). It is important to emphasize that thestrengthof the disk-surfaceinteraction is chosenarbitra-rily andthis particularchoice is suchthat in the absenceof grafted polymer, thedisk configuration is preferredonthesurfaceover thesphere.Our maininterestis to seetheinterplay betweenthe two possible adsorbed configura-tions of the protein asa function of the surfacecoverageof polymer andthe interactionsbetweenthepolymerandthesurface.

The first questionthat we addressconcerns the effectof thepolymer molecular weight. Thecomparisonof the-oretical predictions with experimental observationsdemonstrates that in the range of chain lengths studied

thereseemsto belitt le effect.Actually, this independenceof the adsorption isothermon chainlength dependsuponspecific surface-polymer interactions,the strengthof theprotein-surface attraction and the size of the proteinmolecule as comparedto the polymer chain. In ref.54) ithasbeenshown that when the graftedpolymers are notattracted to the surface, there is a molecular weightthreshold above which the equilibrium adsorption iso-therms arecompletely independent of the polymer mole-cular weight. The chain length threshold dependsuponthe size of the protein. Once the height of the polymerlayer is longer than the protein diameter the adsorptionisothermsseemto beindependentof molecular weight.

Theoretical calculations suggest that what determinesthe equilibrium amount of protein adsorbed is not thepolymer chain length but the number of polymer seg-ments thatneedto bestretchedout of thesurfacein orderto accommodatetheadsorbingproteins54). This pointsoutto another interesting finding which shows that in orderto determinetheamount of proteinadsorption on a givengrafted polymer layer, the structure of the layer, whileproviding information on the mechanism of proteinadsorption, cannot be usedto determine the amount ofprotein that wil l adsorb. Namely, the amountof proteinadsorption is determined by the interplay betweenthepolymer, the proteins and their interactions.Moreover,the resulting structureof the polymer with the adsorbedproteins is very different than the polymer layer in theabsenceof protein. This is an important consideration forproteins that havestrong attractive interactions with thesurfaces.In thosecases,theproteinwil l have no problemin deforming thegraftedpolymer layerat anentropiccostthat turnsout to besmall compared to thegainof surface-protein contact54). On the other hand,the structureof thepolymer layer in the absence of the protein containsimportantinformation to determinethekinetic processofadsorption, including quantitative information as will bediscussedbelow.

In case the grafted polymers have attractive interac-tions with the surface,the molecular weight dependenceis different thanfor surfacesthat do not attractthe poly-mer segments. The reasonis that the mechanismof pro-tein adsorption is different. If theprotein needsto removesome of the adsorbed polymer from the surfacethere isan additional penaltyassociated with the loss of contactenergy betweenthe polymer and the surface.This is inaddition to the deformation of the polymer layer. Thesetwo effects combined make theproblemmoredifficult topredict in general terms. For example, in comparisonswith experimentalobservationsshownin Fig. 16 at smallsurfacecoverage,the behavioris dominatedby attractiveinteractionsbetweenPEOandthesurface,andthereis nomolecular weight dependence for the chain lengthsstu-died. However, modeltheoretical predictionshaveshownstrong dependence on molecular weight for a similar

Fig. 18. Interactionpotential betweena lysozymeanda modelhydrophobic(bare)surface69)


modelsystembut with weaker attractive interactionswiththe surface.Again for attractive surfaces,eachcase andrange of parameters has to be considered explicitly todrawconclusionsonmolecularweightdependence.

To understandthe very dramatic effect that the poly-mer-surface attractive interactions hason the adsorptionisothermsin more detail, Fig. 19 showsthe adsorptionisothermsfor a variety of strengths of the surface-poly-mer attraction, vAS. The isothermsareshown for two dif-ferentmolecular weights.Thus,we will discusstheeffectof surface-polymer interactionsand that of molecularweight. Note, that in order to isolate the effect of thepolymer segment-surfaceattractions in all casesthe sur-face-protein interactionsarethesame.It is alsoimportantto emphasizethatwhile thehydrophobicsurfacesusedinthe experiments presentedin Fig. 16 strongly attract theethyleneoxide segments of the PEO chains,there is avery large class of materials in which the PEO is notattractedto thesurface,suchaslipid surfaces33,34).

Therearetwo pronouncedfeaturesthat canbe seeninFig. 19 for eachmolecular weight. First, the stronger theattraction of the polymer segmentsto the surface thelower the amount of protein that can adsorb (at fixedpolymer surfacecoverage).Theeffect is differentfor dif-ferent regimes of surfacecoverage.For low surfacecov-erage, the surface-polymer attractions have a relativelysmaller effect than at intermediatesurface coverage. Inparticular, the surfacecoverageabovewhich no protein

adsorption takesplace dependsstrongly on the strengthof thepolymer-surfaceattraction.

The bestprotective graftedpolymerlayersarethoseinwhich the segments of the chain moleculesareattractedto thesurface.Theability of thepolymersattracted to thesurfaceto better prevent protein adsorption arisesfromthe thermodynamic cost associatedwith desorbing seg-mentsfrom thesurfaceto find adsorptionareafor thepro-tein.This hasto becomparedto theentropiccostof poly-mer deformation that the protein induces in case ofadsorption on surfaceswith graftedpolymersnot attract-edto thesurface.

Second, theshapeof theadsorption isothermsis differ-ent for the weakly polymer-attractive surfacesas com-paredto thestronglyattracting surfaces.Theshapeof theisothermsreflects the interplay betweenspherical anddisk configuration upon adsorption onto surfaces withdifferent graftedpolymerlayers.

An interestingresult shown in Fig. 19 is the depen-denceof the protein adsorption isotherm on molecularweight. It is clear that the molecular weight dependenceis a function of the polymer segment-surfaceattractionandthesurfacecoverageregime.However, in mostcasesthereis no dramaticdependenceon themolecular weight.

Thecompetition betweenthe two adsorbing configura-tions of the protein is shown in Fig. 20 for surfacesthatdo not attractthe polymer (uppergraph)andfor surfacesthat do (lower graph).For surfacesthat do not attractthepolymersegmentsthebehavior is asfollows. Whenthereis no polymer on the surfacethe disk configuration hashigher population. Recall that the bare protein-surfaceinteractionswere chosensothat this will bethecase.Thetotal equilibrium density of proteins on the surface isquitesmaller thanclosepacking dueto the finite surface-protein attraction. As the surfacecoverageof polymerincreasesthe amount of sphericalconfiguration on thesurfacedecreaseswhile the disk remainsmore or lessconstant.Thesurfacedensityof disk configurationsstartsto decreaseonly for surfacecoverages abovewhich thereis essentially no moresphericalconfigurationsadsorbed.

The interplay in case of surfacesnot attractive for thepolymer can be understoodfrom the cartoonof Fig. 21,top.Themaineffect of adsorbing proteinson thepolymerlayeris to stretchsomeof thepolymersegmentsthatareinthe vicinity of the surfacetowards the solvent. The seg-mentsthat are stretched are thosethat are found in theregionof maximal repulsion with theproteins. Thespheri-cal configurationexerts maximalrepulsionsvery closetotheregionof maximal densityof thepolymer layer. There-fore, the spherical configuration is the first one to beexcludedfrom thesurfacesinceit feelsthemaximal repul-sionsfrom thepolymerlayerandit haslessattractiveinter-actionswith thesurfacethanthediskconfiguration.

The caseof surfacesthat attractthe polymer segmentsis exactlytheoppositeone.Namely, first thedisk config-

Fig. 19. Model lysozyme adsorptionisothermson a surfacewith grafted polymers.The amount of adsorbedprotein as afunction of the surfacecoverageof the tetheredpolymers.Thepolymer(PEO)chainlengthsarenA = 50 (solid lines,opensym-bols) and nA = 100 (dotted lines, filled symbols).The differentsymbolscorrespondto different vAS, i. e. different strengthsofsurface-monomer attraction. vAS = 0 (circles), vAS = –0.5(squares), vAS = –1.0 (diamonds), vAS = –1.5 (triangle up),vAS = –2.0(triangledown)


uration population decreasesas the surfacecoverageofpolymer increases(seeFig. 20, bottom). Only after thereare no more disks adsorbed,the surfacedensity of thesphericalconfiguration startsto decrease.The reason forthis behavioris againrelated to the locationof the maxi-mal pressuresexertedby the polymer layer (seecartoonin Fig. 21, bottom). For surfacesthat attractthe polymersegments,the structureof the polymer layer at low sur-facecoverageis that of adsorbed pancakes.Thesestruc-turesexertmaximal repulsionscloseto the surface.Thisis the regionwherethe disk configurationof the proteinhasall of its volumeresulting in very large repulsionsascompared to the sphericalconfiguration. For the latter,themaximal volumeis at a distancefrom thesurfacethattherearealmostno polymersegmentsat low surfacecov-erage.Therepulsionsbetweenpolymersandthediskcon-figuration are strong enoughto overcomethe very largedifference in attractive interactions between disk andsphericalconfigurations.

The interplay betweendifferent configurations as afunction of the interactions betweenthe polymer seg-mentsandthesurfacecanbeusedto prevent or acceleratesurface induced configurational changes on proteins.Furthermore,a properdesign of the polymer layer maylead to selective adsorption of given protein conforma-tions. Clearly, the theoretical resultsalso show that onecan use polymer layers of different kinds for selectiveadsorption from mixtures of proteins. However, that isbeyond the scopeof the discussion here.It wil l be inter-esting to check the predictions of the theory againstexperimentalobservations, howeverwe arenot awareofexperimentalstudiesof this kind.

The discussion up to this point concentratedon equili-brium adsorption. Namely, all the resultsthat have beenshown assumedthat the systemhas reached thermody-namic equilibrium. This is an approximation that may bequestionable due to the very large energy differencesassociated with protein adsorption. Comparisons withexperimental results suggest that the systems havereachedequilibrium or that they arenot very far from it.Actually, kinetic studieshave shown that the measuredadsorption isotherms shown in Fig. 16 do not changeevenaftervery long periodsof time68). However, in manycasesthe processis kinetically controlledandonewouldexpect that this effect will be evenmore pronounced inthepresenceof a graftedpolymerlayer.

Weconsidernowthepotentialof mean-forcethatapro-tein feelsas it approachesthe surfacein the presenceofgrafted polymers.The potential of meanforce includestwo contributions, the attractive interactions of the baresurfacewith theprotein andtherepulsivecontribution thatarisesfrom theinteractionsof theproteinapproaching thesurfaceandthegraftedpolymers.In caseonewantsto cal-

Fig. 20. The total adsorptionisotherm and the isotherms forthe sphereand disk configurations. The chain length of thegraftedpolymersis nA = 100, theuppergraphcorrespondsto noattraction betweenpolymer and the surface(vAS = 0), and thelower figure correspondsto an attractive surface,vAS = –2. Thesolid line is the total adsorption,the dashedline correspondstothe disk configuration, andthe dottedline to the sphericalcon-figuration

Fig. 21. Disk vs sphericalconfiguration of adsorbed proteins.For the caseof a non-attractive surface(a), strongest repulsionsare at a distance (arrows) from the surfacethat approximatelycoincideswith themaximumcross-sectional areaof thespheres.For the attractive surfacecase(b), strongest repulsionoccursnext to thesurface,where thedisk configurationhasa muchlar-ger areathanthespheres


culate the potential of mean-force in the presenceofalreadyadsorbedproteins,interactionsmust also includetheapproaching protein-adsorbedproteinscontribution.

We considerthepotentialof mean-forceof theproteinswith the surfacewith graftedpolymers in the absenceofany adsorbed protein. These interactions should deter-mine the time scale for the beginning of the adsorptionprocess. Further, they provide an excellent measure fortheability of thepolymer layer to kinetically controlpro-tein adsorption.

Fig. 22 showsthepotentialsof mean-forcefor thesamesystemsshownin Fig. 19at asurfacecoveragerl2 = 0.01.Eachgraphrepresentsamolecularweight. For thissurfacecoveragetheability of thegraftedpolymerlayerto reducethe equilibrium amount of adsorbedproteins is the same

for vAS Aÿ kT. The potentialof meanforce for the shortchainlength(n = 50)showsexactly theoppositebehavior.Namely, the potentials of mean-force are the same forvAS aÿ kT. The reason that the potentialsof mean-forcearethesameis thatthestructureof thepolymerlayer is thesamefor all thesegmentsadsorbedonto thesurface.Thus,theproteins(for the largerattractive interactionsbetweenthe polymer andthe surface)does not feel any repulsiveinteractionuntil it is in contact with thesurface.However,at suchreduceddistancefrom the surface,the baresur-face-protein attraction is strong enoughthat the overallpotential is attractive for all distances(for this polymersurfacecoverage). This wil l imply very fastdynamics(atleastat early stages)that wil l be dominatedby the diffu-sionof theproteinstowardsthesurface.

For weakly attractive and non-attractive surfaces,thepicture is rather different. There is a maximum in thepotentialthat arisesfrom thepolymer segmentsstretchedout of the surface. This is the stericbarrier that the poly-mers form and it determines the kinetic behavior. Notethat the largerkinetic barrier, andthus theslower kineticsof adsorption, is predicted in the caseof non-attractingsurfaces.This is the casepredicted to have the maximalamountof proteinsadsorbedat equilibrium.

Thelongerchainlength showsasimilar behavior, how-ever, thepotentialsaredifferent for all vAS, andtherepul-sionsarestrongerandof a longerrange.This reflects thedifferent structuresof thepolymerchains for thedifferentmolecular weightsandattractive interactionswith thesur-face.It is interestingto notethat the equilibrium amountof protein adsorbed for the two chain lengths withvAS� 0, and the surfacecoverageused in Fig. 21 areidentical. However, thepotentialsof mean-forcearecom-pletely differentin strengthandrange.The time scaleforadsorption of the proteins to the surface with longergraftedchainsis going to be ordersof magnitude largerthanfor theshorterchains54).

Fig. 23 shows thepotentials of mean-forcebut now forrl2 = 0.02. In this case, all the polymer layers show arepulsive barrier. However, the magnitude of the barrier(andtherangeof therepulsion) increasesastheattractionbetweenthe surfaceandthe polymer segmentsweakens.Note that while the barrier is lower for the stronglyattracting surfaces, the potential at contact is muchweaker. This is an early manifestation of the smalleramountof proteinthat will adsorbat equilibrium. Again,we seethe samemolecular weight effect by comparingfiguresA andB. Namely, thelongerthemolecularweightthe longerrangeandstrongertherepulsivebarrierfor theapproaching proteins.

5 Concluding remarksThe propertiesof block copolymers spreadat fluid-f luidand solid-fluid interfaceswere studiedwith a molecular

Fig. 22. Potential of meanforce betweenmodel protein andsurfacewith graftedpolymersas a function of the distance oftheproteinfrom thesurface; nA = 50 (upperfigure) andnA = 100(lower figure) at rl2 = 0.01. The different lines correspondtodifferentvAS: no attraction(solid line), vAS = –0.5 (dottedline),vAS = –1.0 (dashed line), vAS = –1.5 (long dashed line),vAS = –2.0 (dot-dashedline). Note that all the potentialscorre-spond to surfaces with grafted polymers but no proteinsadsorbed


approach. The theory enables the calculation of detailedconformational and thermodynamic properties and hasbeenshownto provideaccurateinformation ascomparedto experimental observationsfor bothtypesof properties.

In this work we studied the effect of the anchoringblock on the phasebehaviorof the polymer layers. Wehave found a variety of different phase transitionsdepending upon the type of interactions between thepolymersegmentsandthepolymer-interfaceinteractions.We find thattheformation of thedifferentphasesdependsuponthe type of polymer-polymerandpolymer-interfaceinteractions,as well as in the ratio of molecular weightbetweenthe blocks.The main conclusion is that we can-not in generalassumethat thesoleeffect of theanchoringblock is to havethetetheredblock attachedto thesurface.The interactions betweentetheredand tethering blockmay inducephasetransitionsamongphasesof very dif-ferentmorphologies.

It is importantto emphasize that in this work we haveonly considered thepossibility of macrophaseseparation.In many casesonewould expecttheformationsof micro-

domains, as has been observed experimentally37,38,56,57)

and predicted theoretically58,60,62–65). Thesedomains maybe the resultof kinetically trappedphaseseparating sys-tems or truemorphological changesinducedby aggregateformationthat represent the two dimensional counterpartof three dimensional micellization. The formation ofaggregatesmay preempt the macroscopic phasesepara-tion predictedhere and thus,a complete studyof all thepossibilities is necessary. However, that is a rathercom-plex undertaking due to the large numberof variables,and the many possiblemorphologies.We believe,how-ever, that our predictions canbe usedto find the regimeof interactionsandmolecular weightsof theblocksform-ing the copolymer in which the layer wil l be homoge-neous.

Thestructuralandphasebehaviorof thepolymer layeris a determining factor in theability of thepolymermole-cules to prevent protein adsorption. We have shown avariety of possible scenariosincluding direct comparisonbetweentheoreticalpredictions and experimental obser-vations.Theability of the theory to quantitatively predicttheproteinadsorption isotherms,including thepossibilityof a phaseseparation in the polymer layer confirmedbyexperimentalobservations, is a further evidencethat thetheory is reliable in predicting the interactions betweentetheredpolymersandproteins.

Themain conclusionfrom thework on proteinadsorp-tion is that in order to estimate the ability of a polymerlayer to prevent protein adsorption, the interactionsbetweenthe polymer and the surfaceplay a key role.Further, the possibility of conformational changes(ordenaturation)of the proteinon the surfacedependsuponthe type of polymerlayer with which theprotein is inter-acting. The thermodynamic control of proteinadsorptionseems to be determined mostly by the structure of thepolymer layer in the very close vicinity of the surface(within a few angstroms). Actually, the most importantvariable determining the amount of protein that canadsorb on the modified surfacesis the polymer surfacecoverage.On theother hand,kinetic control is a functionof the distribution of polymer segments in the wholepolymer layer. Therefore,polymer layers that may bevery good for the kinetic control of protein adsorptionmayshowlargeequilibrium adsorption. This implies thatin orderto design a graftedpolymer layer for preventionof protein adsorption both kinetic and thermodynamicconsiderations must be taken into account dependingupon the desiredapplication. The next natural stepis tostudy in detail thekinetic processof theadsorption.

Acknowledgement: This work is supportedby NSF grantCTS-9624268.I. S. is a Camille DreyfusTeacher-Scholar. Wealso acknowledgepartial supportfrom the PetroleumResearchFundadministeredby theAmerican ChemicalSociety.

Fig. 23. SameasFig. 22but atrl2 � 0:02


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