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Aggregates of rod-coil diblock copolymers adsorbed at a surface C. Nowak, T.A. Vilgis Max-Planck-Institut fu¨r Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany The behaviour of rod-coil diblock copolymers close to a surface is discussed by using extended scaling methods. The copolymers are immersed in selective solvent such that the rods are likely to aggregate togain energy. Therodsareassumed toalign only parallel toeachother, suchthatthey gainamaximumenergybyformingliquidcrystallinestructures. Ifanaggregateofthesecopolymers 6 adsorbswiththerodsparalleltothesurfacetherodsshiftwithrespecttoeachothertoallowforthe 0 chainsto gain entropy. It is shown that this shift decayswith increasing distance from thesurface. 0 Theprofileofthisdecayawayfromthesurfaceiscalculatedbyminimisationofthetotalfreeenergy 2 of the system. The stability of such an adsorbed aggregate and other possible configurations are n discussed as well. a J PACSnumbers: 82.35.GhPolymersonsurfaces;82.35.JkCopolymers,phasetransitions,structure;36.20.Ey 3 Conformation(statisticsanddynamics) ] t I. INTRODUCTION tem. That means the width of the system in y-direction f o isequaltotheroddiameterd. Thissystemcanbeviewed s as a narrow slice of a system with infinite extension in . Rod-coil copolymers in selective solvents show a rich t y-direction. Each of the diblock copolymers under con- a phase behaviour. Depending on the chain length and m the solventquality they may form cylindrical micelles or siderationiscomposedofastiffrodoflengthLanddiam- eterdtowhichafullyflexiblechainofN monomerswith - lamellar sheets. Similar phases can be found in melts of d monomer size b is grafted. The solvent is characterised rod-coil copolymers. These systems are therefore widely n by anenergypenaltyγ per unitareaofa rodexposedto studied inthe literature[1, 2, 3, 4, 5, 6, 7, 8,9, 10]. Less o the solvent. The energy gain κ per unit area of a rod extensive are the studies on a single multiblock polymer c − for being in contact with the surface has to be chosen [ composed of stiff rods which are connected by flexible such that an aggregate actually adsorbs to the surface chain spacers, see [11, 12, 13, 14]. It was shown, that 1 without dissociating into single rod-coil copolymers. these polymerscanalsoformmicellarandmulti-micellar v The paper is organised as follows. In a first naive ap- structures. Thestructuralbehaviourofdissolvedrod-coil 7 proach one would assume a constant shift of the rods 3 copolymers in the presence of a surface is far less under- with respect to each other. This assumption leads to an 0 stood. Rod-coil polymers grafted to a repulsive surface 1 are shown to form ’turnip’- or ’jellyfish’-like micelles on artefact in the shifting behaviour, as we show in the ap- 0 top of the surface [15]. However,to our knowledge there pendix. The calculations in the appendix are presented 6 in some detail because some of the intermediate results existsnostudyofrod-coilpolymersinthepresenceofan 0 areusedinsectionIIandIII.Inamoresophisticatedap- attractive surface. / t In this paper we consider rod-coil diblock copolymers proachtheshiftisallowedtovarywithdistancefromthe a surface suchthat it candevelopa profile. This approach m in selective solvent close to a surface which is highly at- to the problem is presented in section II. Some remarks tractive for the rods and neutral to the flexible parts of - on the stability of the adsorbed structure and the corre- d the copolymer. Further, the solvent is assumed to be n poor for the rods, such that they align and tend to form spondingrangeofvaluesforκaremadeinsectionIII.In o aggregates, and good for the chains. In addition it is section IV we finish with a brief discussion of the results c of the foregoing sections. assumed that the rods have a certain chemical modifi- : v cation, such that they prefer to be parallelorientedwith i respecttoeachratherthanantiparallel. Theaggregation X II. PROFILE OF THE SHIFT behaviour of such rod-coil copolymers, showing parallel r a alignment of the rods only, has been investigated exper- imentally and computationally, see [16, 17, 18]. We as- Whenanaggregateofrod-coilcopolymersbecomesad- sumetheenergypenaltyforantiparallelalignmentoftwo sorbedbyastronglyattractivesurface,anadditionalcon- rodstobemuchhigherthantheenergypenaltyforthese finement for the corona of the free chains may introduce rods being fully exposed to the solvent. In aggregates new effects. One possibility is that the entropy penalty of these copolymers the flexible parts therefore stick out oftheconfinementofthecoronachainsduetothewedge in one direction only, see Fig.(5). If such an aggregate defined by the surface and the rod aggregate prevents adsorbs with the rods parallel to the surface, the rods adsorption. Another extreme effect can be the destruc- shiftwithrespecttoeachothertoallowforthe chainsto tion of the aggregate. In between, the balance between gainentropy,see Fig.(1). The nature of this shift will be entropyandenergycanbesuch,thattherodsareshifted examined in the following. withrespectto eachother to allowforthe chainsto gain Forsimplicityweconsideraquasitwo-dimensionalsys- entropy. If this shift does not get to large, the rods stay 2 shift l to brush height is much smaller than depicted in Fig.(2). A chain starts at the rod and ends at the line X(x) x FIG. 1: Aggregate of rod-coil copolymers adsorbed at a surface. The shift of the rods with respect to each other decays 0 with increasing distance from thesurface. together. Such conformations appear possible whenever the defect energy is balanced by the gain of entropy due to reduction of the wedge confinement. The mostnaiveassumptionthatthe shiftmaybe constant,i.e.,itformsalinewithacertainslope,turnsoutto l(x) be unphysical in many respects (see Appendix). There- −D foreitisnecessarytointroduceacurveddeviationofthe shift for each successive rod, which results from a total balance of all entropic and enthalpic contributions. We present in this section the basic model which al- FIG.2: Thissketchshowsthespatialsegmentsorboxesfilled byeach chain. They get larger with increasing distance from lows us to calculate the shift of the rods as a function the surface. The shift at each position x is denoted by l(x). of distance from the surface. From intuition we expect x ranges from −D at the surface to 0 at the last rod. The the rods close to the surface to shift more than the rods splay of thechains is given by u(x)=X(x)−x. further away from the surface, since for the correspond- ing chains close to the surface there is more entropy to X(x) shown in Fig.(2). It is assumed to fill the volume gainthanfortheonesfurtheraway. Thereforeweexpect of the box given by the dashed lines around the chains. an equilibrium conformationsimilar to the one shownin We are aware that this assumption is not valid for the Fig.(1). chains far away from the surface but for these chains The dashed line in Fig.(1) can be interpreted as the thecontributionoftheexcludedvolumetermiscertainly profile of the shift l as a function of the distance x from very small. Hence the assumption does not affect the the surface. The chains are described by a local Flory- total free energy in a significant way. The splay u(x) type model similar to the one used in [19] to describe a is given by X(x) x. To explain how to calculate the finite polymer brush. In this model the free energy of − volume available to each chain, Fig.(3) shows a larger the system is givenby the sum of anelastic term and an sketchofoneofthedashedboxessurroundingeachchain excluded volume term for the chains plus a term which in Fig.(2). quantifies the energy penalty for the additional rod sur- Inasmuch as we are going to consider a quasi two- faceexposedtothesolventduetotheshift. Weconstruct dimensionalsystemonly,thevolumeisgivenbythegrey the free energy such that it is a function of the splay of area in Fig.(3) times the diameter of the rods d. The the chains u(x) - see Fig.(2) - and the shift of the rods (0) area A shaded in light grey is given by h(d+∆u /2), l(x). At all positions x we can safely assume l(x) to be i i small compared to the height h of the brush-like struc- where∆ui =ui−ui−1. ItistheareaA(il) shadedindark ture formed by the chains. If the shift l would be of the grey where the shift of the rods l(x) comes into play. It same orderas h, the chains would hardly see eachother, is given by so there would be no driving force for a shift. Hence 1 A(l) =hq l q . (1) we can assume h to be constant for all x. The height i i− 2 i i is given by the equilibrium height of a polymer brush in good solvent, i.e. h = 4−1/3bN(b/d)2/3. The excluded As can be seen from Fig.(3) the length qi is given by volume parameter v is set to v =b3 for simplicity. ui−1 ui ∆ui q =l =l − (2) i i i Fig.(2) helps to define and explain how the excluded h h volume term of the free energy for the shift geometry is A(l) =l (u ∆u ) 1 li . (3) constructed. Note, that it is only a sketch. The ratio of ⇒ i i i− i · − 2h (cid:18) (cid:19) 3 The excluded volume term in Eq.(5) was constructed assuming constant density of monomers for each chain within the box of volume V (Eq.(4)). The monomer density close to the rods is larger than the one further away from the rods and therefore this assumption tends ui to underestimate the excluded volume energy. However, itisthestandardapproximationusedinFlory-typemod- els and has been proven to be sufficient to describe the A(0) behaviour of a finite polymer brush, see [19]. i The totallengthD ofthe rodaggregateperpendicular to the surface is assumed to be D h. This is explicitly (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)h(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) needed in the appendix. ≥ (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) Fig.(2)showsthatthe rodatthe surfacehaszeroshift (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) since there is no other rod underneath with respect to (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) ui(cid:0)(cid:0)(cid:1)(cid:1)−(cid:0)(cid:0)(cid:1)(cid:1)1(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)A(cid:0)(cid:0)(cid:1)(cid:1)(l(cid:0)(cid:0)(cid:1)(cid:1))(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) which it could shift. So for the rod-coil copolymer at (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)i(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) the surface the integrand in Eq.(5) reduces to the one in (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) Eq.(A1)inthe appendix(withσ =1/d2). Thisofcourse (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)q(cid:0)(cid:1)i alsomeansthatthe equationfor the shift(Eq.(8)below) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) li isonlyvalidfromthesecondrodon(ascountedfromthe surface). Tocalculatetheequilibriumshiftl(x)itisnecessaryto computethe Euler-Lagrangeequationsfromafunctional FIG. 3: This figure shows how the volume occupied by each minimisationofEq.(5)withrespecttou(x)andl(x). The chain is calculated as a function of splay u and shift l. Euler-Lagrange equation for the splay u(x) has the first integral Since weassumedl tobe smallcomparedto h,theterm i u(x)2+h2 in the last brackets in Eq.(3) can be approximatedby 1. 2βγl(x)+ In Fig.(3) this corresponds to double counting the area Nb2d of the small triangle with the catheti q and l . N2b3 1 i i + In order to construct a free energy functional that d hd2(2+u′(x))+2dl(x)(u(x) du′(x)) (cid:20) − could be minimised with respect to the shift and splay ′ (h 2l(x))u(x) ls(hxa)p,eus l(x)ua(nxd),u∆(xu) we tdauk′e(xt)h.eTcoanktiinnguuthmelcimonitti:nluiu→m + (hd(2+u′(x))−+2l(x)(u(x) du′(x)))2#=C1. (6) i → i → − limit andadding the twocontributions,the totalvolume Variation of the free energy functional in Eq.(5) with V(x) available to a chain at position x is given by respect to l(x) yields V(x)=d2h(1+u′(x)/2)+dl(x)(u(x) du′(x)). (4) − 2N2b3 u(x) du′(x) 2βγ+ − =0. The elastic term of the free energy is given by a contri- d2 "(hd(2+u′(x))+2l(x)(u(x) du′(x)))2# bution proportional to h2 representing the stretching of − (7) thechainsawayfromtherodsandacontributionpropor- The quadratic Eq.(7) can be solved for the shift l(x). tional to u(x)2 representing the stretching of the chains parallel to the rods. The energy penalty for the addi- 1 b3N2(u(x) du′(x)) 2 hd(2+u′(x)) tional area of a rod at position i exposed to the solvent d2βγ − − l(x)= (8) asa function ofthe shiftli is simply givenby2γlid. The (cid:16) 2(u du(cid:17)′(x)) complete free energy functional can now be constructed. − Inserting the expression for the shift - i.e. Eq.(8) - into 0 0 Eq.(6)resultsinacomplicated,highlynonlineardifferen- 1 βF =2 dxβγl(x)+ dx u(x)2+h2 tial equation which cannot be solved exactly. However, Nb2d −ZD −ZD (cid:2) (cid:3) the overall effect of the shift on the free energy is cer- tainly smaller than the overall effect of the splay. Thus 0 N2b3 1 itisareasonableapproximationtocalculatethesolution + dx 2d hd2(1+u′(x)/2)+dl(x)(u(x) du′(x)) for the splay at zero shift and to use this as an approxi- −ZD (cid:20) − (cid:21) mation for the splay u in Eq.(8). We are awarethat this (5)approximationbreaksdownwhenthesplaybecomesvery smallclosetothesurface. Nevertheless,forzerosplaythe Forsimplicitywedroppedalltermswhichdonotdepend shift should be constant. The shape of the profile of the on either l or u. As already mentioned the excluded shift away from the surface at finite splay can therefore volume parameter v is set to v =b3. be calculated within this approximation. 4 Nowwecalculateanapproximatesolutionofthesplay l/d uforzerol. Iftheshiftl(x)inEq.(6)issetidenticalzero, the differential equation equation reduces to 2 u(x)2+h2 2N2b3 1+u′(x) + =C (9) Nb2d hd3 (2+u′(x))2 1 1.5 (cid:20) (cid:21) ′ 1+u u2+8h2 =C (10) ⇒ (2+u′)2 2 1 which is Eq.(A2) from the appendix as should be. It is convenient to introduce dimensionless variables u˜ = u/(√8h), x˜=x/(√8h) and C˜ =C /(8h2). Eq.(10) can 0.5 2 2 be integrated for arbitraryC˜ which leads to an implicit 2 equation for the splay u˜(x˜) similar to Eq.(A3) x -5 -4 -3 -2 -1 2u˜ 2u˜+ln 2u˜+(1 4C˜ +4u˜2)1/2 0 2 − − FIG. 4: The shift l is plotted as a function of negative dis- ln 2u˜ +h(1 4C˜ +4u˜2)1/2 =4x˜i. (11) tance. The rod furthest from the surface is located at x=0, − 0 − 2 0 compare Fig.(2). Parameters: N =700,b/d=0.1,βγd2 =1 h i Notethat D˜ x˜ 0. TheintegrationconstantC˜ can 2 bedetermi−nedb≤yus≤ingtheboundaryconditionu˜( D˜)= 0 - i.e. zero splay at the surface, see Fig.(2). − Tofind anupper limit forthe thresholdvalueγc wenote thatthemaximumvalueoftheshiftisu =h. Anupper 0 1 −2 −2 estimatefor(u(x) du′(x))/(2+u′(x))2 isthereforegiven C˜ = 1 u˜2sinh 2D˜ +u˜ +u˜2cosh 2D˜ +u˜ − 2 4 − 0 0 0 0 by 4h. The upper limit for γc is thus given by (cid:18) h i h (i12)(cid:19) 4 The chain furthest away from the surface at x = 0 can b3N2 N b 3 βγ 0.4 . (16) topple over completely and is therefore allowed a splay c ≈ 4hd4 ≈ d2 d (cid:18) (cid:19) u =h or u˜ =1/√8. Therewith Eqs.(11,12) reduce to 0 0 ForthesetofparameterschoseninFig.(4)thisyieldsthe 1/√2 2u˜+ln 2u˜+(1 4C˜ +4u˜2)1/2 rough estimate of βγ d2 13. 2 c − − ≈ In this section an attempt was made to describe a h i ln 1/√2+(3/2 4C˜2)1/2 =4x˜, (13) variable shift l(x). For values of the splay u(x) which − − are large enough to dominate the effect of the shift, we h1 i −2 C˜ = 2 sinh 4D˜ +1/√2 . (14) foundasetofequations(8,13,14)whichdeterminelasa 2 8 − (cid:18) h i (cid:19) functionofuanduasafunctionofx. Althoughitisnot It is notpossible atthis stage to resolveEq.(13) withre- possible to resolve Eq.(13) with respect to u, the profile spect to the splay u. However, we know x as a function of the shift can be plotted for a certain set of parame- ofu, u′ as a function of u (see Eq.(10)) andl as function ters, see Fig.(4). In this section we always assumed that of u and u′. Hence for a certain set of parameters, we the adsorbed aggregate is internally stable and does not can plot the shift l as a function of x by either numer- disintegrateinthe sensethatsinglecopolymersleavethe ically resolving Eq.(13) or by showing a parameter plot aggregate. Therefore we are going to discuss in the next of l versus x, using u as a parameter. This is done for sectionunderwhichconditionstheadsorbedaggregateis a characteristic set of parameters in Fig.(4). The plot stable and which other configurations are possible. demonstrates that our assumption for the profile of the splay - Fig.(1) - was indeed reasonable. The shift in- III. STABILITY creasesfromthe furthest rodtowardsthe surfaceuntil it reaches its maximum value in a reasonable form. We are now going to estimate the threshold value of γ In section II we assumed that an attached configura- abovewhichtheenergypenaltyforadditionalrod-solvent tionoff rod-coilcopolymersatasurfaceasitispictured exposure becomes to large for a shift to occur. in Fig.(1) is stable. There are three other possible con- The shift is identical zero if the right hand side of figurations. Eq.(8) is less or equal to zero for all x. In this quasi two-dimensional system only one rod is in contact with the surface. Therefore a possible con- 1 b3N2(u(x) du′(x)) 2 hd(2+u′(x)) figuration is the one shown in Fig.(5), where one single d2βγ − − 0 copolymerisadsorbedatthesurfaceandtheothersform (cid:16) 2(u du(cid:17)′(x)) ≤ a detachedsheet. We callsucha situationdetachedcon- − b3N2(u(x) du′(x)) figuration in the following. The rods in the detached βγ − . (15) ⇒ ≥ h2d4(2+u′(x))2 sheet might also prefer a shifted geometry. We refrain 5 from a discussion of this shift, since this paper is mainly concernedwith the behaviourofa complete aggregatein contact with a surface. FIG. 7: The aggregate dissociates and the individual rods adsorbatthesurface. Thisconfigurationispreferredforlarge κ. small the system might prefer the mushroom configuration even for long rods, since it allows for the chains to gain entropy without much increase in exposure of the rods to the solvent. By estimating the free energies of these configurations and comparing them with the one of the attached configuration, it is possible to find the range of κ in which the attached configuration is stable. To achieve this at least approximativelywe calculate the free energy of the attached configuration with zero shift. It gives a slight overestimation of the free energy of the configuration considered in section II. However, we still get a rough FIG. 5: For long chains a detachment of the aggregate from therod adsorbed to thesurface might bepreferable. estimate for the parameter range in which the attached configuration is stable. At zero shift the chains form a finite brush. Itsfreeenergyiscalculatedinthe appendix The energy of the rod-surface contact in the detached andgivenbyEq.(A11). One sideofthe brushisfreeand configurationisthe sameasforthe attachedone. Acon- thereforeallowedasplayofu =h,theothersideiscon- figurationwithdifferentcontactenergyisthemushroom- 0 fined by the surface, i.e. u =0. The length D is given like one as depicted in Fig.(6). This configuration is al- D byfd. Thefreeenergyoftheattachedconfigurationthen reads 1 7 1 βF = 3fdh2 h3 . (17) attach Nb2d − 6 − √2 (cid:20) (cid:18) (cid:19) (cid:21) We use this as a reference energy and add energy gains and penalties due to rod-surface contact or rod-solvent exposure to the free energies of the other configurations. Compared to the attached configuration the contact energy of the mushroom differs by κ(Ld fd2). The − chainscan alsobe describedas a finite brush. Here both ends are free and are therefore allowed a splay of u = 0 u =h. The free energy of the mushroom configuration D is hence given by FIG.6: Anotherpossibleconfiguration: Theaggregatedrods βFmushroom = κ dL fd2 − adsorb perpendicular to the surface. This configuration is 1 7 preferable for large aggregates. + (cid:0) 3fdh(cid:1)2 √2 h3 .(18) Nb2d − 3 − (cid:20) (cid:18) (cid:19) (cid:21) ways preferable to a complete detachment of the aggre- Theattachedconfigurationispreferredtothemushroom gate since in the latter case the system would gain no configuration if F < F . This yields the attach mushroom contact energy. following condition for κ The last possible configuration is a complete dissociation of the aggregate into single copolymers due to the 0.12N2 b 3 βκ> . (19) presence of the attractive surface. These single copoly- Ld fd2 d mersthenadsorbindividually atthe surface,seeFig.(7). − (cid:18) (cid:19) This configuration yields the highest gain of contact en- This is the lower bound for κ. To get the upper bound ergy. However,it is alsothe configurationwith the high- thefreeenergyofthedissociatedcopolymers-seeFig.(7) est energypenalty for exposureof rodsurface to the sol- - has to be estimated. vent. For very high contact energy, i.e. κ γ, the Compared to the attached configuration the dissoci- ≫ system always dissociates. On the other hand, if κ is to ated one yields a contact energy gain of (f 1)κLd. − − 6 But on the other hand it also gives rise to an additional detachedconfigurationhasacontributionfromtwoaddi- energy penalty of 2(f 1)γLd. Within this Flory-type tionalrodsurfacesexposedtothe solvent,whichisgiven − theorytheflexiblechainsoftheindividualcopolymersat by 2Ldγ. The free energy of the chains of the detached the surface can be treated as free ones and their free sheet is similar to the one of the mushroom configura- energy can be neglected. Comparison of F and tion, with D = f 1. The free energy of the chain of attach − 2(f 1)γLd (f 1)κLd yields the upper limit of κ the single copolymer can be neglected as in the case of − − − above which the system dissociates: dissociation. F is hence given by detach 1 b 34 b 2 1 7 βκ<2βγ−(f −1)Ld2 "1.2fdN(cid:18)d(cid:19) −0.12N2b(cid:18)d(cid:19) # βFdetach =2Ldβγ+Nb2d(cid:20)3(f −1)dh2−(cid:18)3 −√2(cid:19)h3(cid:21). (24) (20) The rod length Ldetach with separates the detached and Within this range ofvalues for κ the attachedconfigura- attach the attached configuration can now be estimated. tion can actually be stable. For the parameters chosen in Fig.(4), N = 700,b/d= 0.1,βγd2 = 1 and L/d = 80, f =30, this range is given by 1.18<βκd2 <1.52. Ldetach 0.6 N b 34 +0.06N2 b 3 (25) It is also of interest to keep κ and γ fixed and to in- attach ≈ dβγ d dβγ d (cid:18) (cid:19) (cid:18) (cid:19) vestigate at which combination of molecular properties of the copolymers which configuration is preferred. We However,inL-N spacethe detachedconfigurationmight focus here on the length of the rods L and the number sit in between the mushroom and the dissociated config- of chain monomers N. In the following the critical rod uration. This is indeed the case for a wide range of pa- lengths which separate each two of the possible config- rameters. Therefore equating Eq.(18) and Eq.(24) gives urations from each other are calculated as functions of the rod length which separates mushroom and detached N and of the other parameters. Since it depends on the configuration. specific combination of parameters, especially on γ and 4 κ, which configurations are neighbouring in L-N space, Nd b 3 γ −1 Lmushroom fd 1.2 1 (26) allpossiblecriticalrodlengthsarecalculated. Note,that detach ≈ − βκd2 d ! − κ not all possible phase boundaries can exist for one given (cid:18) (cid:19) (cid:16) (cid:17) set of parameters. For each one to exist, the set of pa- ThelengthLdetach whichseparatesthedetachedfrom rameters have to be chosen appropriately. dissociate the dissociated configuration is found to be Eq.(19) can be rearranged such that it gives the critical rod length below which the system changes form the 1.2(f 1)dN b 4/3 0.24N2b b 2 attached to the mushroom configuration. Ldetach − d − d . dissociate ≈ 2(f 2)βγd2 (f 1)βκd2) N2b b 2 − (cid:0) (cid:1)− − (cid:0) (cid:1) (27) Lattach fd+0.12 (21) mushroom ≈ βκd2 d In the conclusions we use these critical rod lengths to (cid:18) (cid:19) calculate one phase diagram in L-N space for a typical ComparisonofFmushroom withthe energyofthe dissoci- set of parameters as an example. atedconfigurationyieldstherodlengthLmushroomwhich dissociate Inthissectionitwasshownthattheconfigurationpic- separates the mushroom from the dissociated configura- tured in Fig.(1) can be stable within a certain range of tion. values for κ. There are three other possible configura- 1.2fdN b 4/3 0.24N2b b 2 fdβκd2 tions as shown in Fig.(5), Fig.(6) and Fig.(7). Lmushroom d − d − dissociate ≈ 2(f 1)βγd2 fβκd2 (cid:0) (cid:1) − − (cid:0) (cid:1) (22) IV. CONCLUSION In case of rather large κ (close to the upper limit) there might exist a rod length which directly separates the at- Adiscussionofrod-coilcopolymeraggregatesadsorbed tached configuration from the dissociated configuration. atasurfaceinatwodimensionalapproximationwaspre- It is found to be sented. The aggregates form because of a selective sol- 1.2fdN b 4/3 0.12N2b b 2 vent, poor for the rods and good for the chains. Due to Laditstasochciate ≈ (f 1d)(2βγ−d2 βκd2)d . (23) their chemical structure the rods only align parallel to −(cid:0) (cid:1) − (cid:0) (cid:1) each other. The surface is assumed to be attractive for Tocalculatethe boundariesofthe detachedconfigura- the rods and neutral with respect to the chains. If the tion(seeFig.(5)),itsfreeenergyhastobe estimated. As aggregate adsorbs with the rods parallel to the surface, already mentioned above, this configuration might also the rods shift with respect to each other to allow for the prefer a shifted geometry. But since we already esti- chains to gain entropy and to therefore lower their con- mated F for zero shift, is is sufficient to calculate finementenergy. Iftheshiftisassumedtobeconstantfor attach also F for a rectangular sheet without shift. Com- all rods, the system shows an artificial behaviour. This detach paredtotheattachedconfigurationthefreeenergyofthe can be seen from the considerations in the appendix. 7 In section II we constructed a model which allows for useful example to define the arising problems in a clear the shift to vary. It was possible to partially solve this way. model and to show that away from the surface the shift Therodsareassumedtoshiftaconstantdistancewith decays. Close to the surface the splay of the chains is respect to each other as shown in Fig.(9). The charac- closetozeroandthereforethe shiftis basicallyconstant. teristic quantity related to this shift is the angleα. This The regionfurther awayfrom the surface is the interest- angle can be calculated by calculating the free energy of ingoneshowingthedecayprofileoftheshift,seeFig.(4). the entire system and minimising it with respect to α. In section III we showed that the configuration con- The additional free energy per rod due to the shift is sidered in section II can actually be stable within a cer- givenby the additionalsurfaceofthe rodexposedto the tain range of contact energies between rods and surface. solvent F = 2γd2tanα. To calculate the free energy rod This range was calculated. Three other possible con- of the chains we treat them as if they would form a fi- figurations were also discussed, see Figs.(5,6,7). These nite brush graftedto the surface shownas a thick line in considerations allow us to plot a phase diagram of the Fig.(9). For a finite brush the trajectories of the single configurations in L-N space, which - for a typical set of polymer chains are not all perpendicular to the grafting parameters - is shown in Fig.(8). The contact energy surfaceasforaninfiniteone. Thepolymerchainsshowa splay u. Fig.(9) illustrates how this quantity is defined; 120 that is, u(x) = X(x) x. The first chain is allowed a − splay of u = htanα due to the surface, where h is the 0 100 brush height given by h = 4−1/3bNσ1/3b2/3. The last chaincantoppleovercompletelyandisthereforeallowed 80 asplayu =h. The graftingdensity σ is afunction ofα L d as well. It is givenby σ(α)=cos(α)/d2. As in section II / L 60 aFlory-typeapproachisusedtodescribethefreeenergy of the finite brush following the lines of [19]. Each chain 40 fills a box of volume hσ−1(1+u′/2), with u′ = du/dx. The free energy for the finite brush is then given by 20 D D d 1 βF = σ dx u(x)2+h2 +4h2 dx , 200 400 600 800 1000 1200 brush Nb2  2+u′  N Z Z (cid:20) (cid:21) 0 (cid:2) (cid:3) 0  (A1)  FIG.8: Configurations oftherod-coilcopolymers at thesur- where D represents the total length of the brush. For face. Parameters: b/d=0.1,f =30,βγd2 =1,βκd2 =1.3 our system of aggregates of rod-coil copolymers it is given by D(α) = fdcos(α). The first integral of the per unit area κ is chosen such that there exists a region Euler-Lagrangeequationforthesplayu(x)obtainedfrom in L-N space in which the entropy loss of the confined Eq.(A1) is given by chains is compensated by the energy gain due to rod- ′ 1+u surface contact. But it is also chosen to be not much u2+8h2 =C. (A2) larger than the rod-rod contact energy γ, since other- (2+u′)2 wise the aggregate would dissociate. However, for very Eq.(A1) does notdistinguish betweenpositive andnega- long chains the aggregate always dissociates into single tivesplay. Hencewehavetoseparateoursystemintotwo copolymers which then individually adsorb. Neverthe- finite brushes which meet at the chain with zero splay, less, Fig.(8) shows that there is indeed a broad regionin see Fig.(9). This chain has to be determined by an equi- L-N space in which the attached configuration is stable librium condition (total length of the brush D). and the rods shift with respect to each as discussed in It is convenient to introduce dimensionless variables section II. u˜ = u/(√8h),x˜ = x/(√8h) and C˜ = C/(8h2). Eq.(A2) For further considerations in the future it would be of can be integrated for arbitraryC˜ which leads to interest to study the behaviour of a finite three dimensional aggregate adsorbed with the rods parallel to the I = 2u˜ 2u˜ ln 2u˜+(1 4C˜+4u˜2)1/2 D − − − surface. We expect the profile of the shift to form a two h i dimensionalsurface with the innermost rods close to the + ln 2u˜ +(1 4C˜+4u˜2 )1/2 =4(D˜ x˜ x˜). D − D − 0− surface showing the maximum shift. h i (A3) The integrationconstantcannow be calculatedby using APPENDIX A: CONSTANT SHIFT the condition 2I(u˜=0) I(u˜=u˜ ) 0 − 1 Despitethefactthattheassumptionofaconstantshift C˜ = 1 (u˜0+u˜D)2sinh[2D˜ +u˜0+u˜D]−2 4 − aaprepedaisrcsuusnsipnhgysiticianlathnidslaepapdesntdoixcoinntrsaodmicetodreytariel.suIlttsiswae + (u˜(cid:16)0 u˜D)2cosh[2D˜ +u˜0+u˜D]−2 . (A4) − (cid:17) 8 XD then be rewritten in the following form D˜−xõ 1 dx˜ u˜2+ 2(2+u˜′) Z (cid:20) (cid:21) x˜c D˜−xõ ′ 1 u˜ 1 = dx˜ u˜u˜′ u˜2u˜′ + x0 2 − − 4 2 − Z (cid:20) (cid:21) D x˜c x˜ u˜ u˜2 u˜3 D˜−xõ = + . (A9) 4 − 8 4 − 6 =0 (cid:20) (cid:21)x˜c X By constructionu˜(x˜ )=u˜ 1, see Eq.(A5). Therefore c c ≪ the integral in Eq.(A6) in the limits [0,D˜ x˜ ] can be 0 − 0 very well approximated by making use of Eqs.(A8,A9): α X0 D˜−xõdx˜ u˜2+ 1 = D˜ −x˜0 u˜D + u˜2D u˜3D. x0 2(2+u˜′) 4 − 8 4 − 6 Z (cid:20) (cid:21) 0 (A10) FIG. 9: Aggregate of rod-coil copolymers adsorbed at a sur- So far we calculated only one part of the brush. The face. Therodsareallshiftedwithrespecttoeachotherbythe one from the “zero splay chain” to the open end. In samedistance. Theshiftischaracterisedbytheangleα. This Fig.(9) this is the right part from 0 to D x . The left sketch shows how we definethex-rangeand thesplay. Note: 0 − u(x)=X(x)−x,i.e. u0 =X0−x0 anduD =XD−(D−x0). part from 0 to x0 or rather from the “zero splay chain” to the surface can be calculated completely analogous replacingD˜ x˜ withx˜ andu˜ withu˜ . Addingupthe 0 0 D 0 − The free energy in Eq.(A1) cannot be integrated di- resultsforbothpartsinbothintervals,accountingforthe rectlyusingtheimplicitsolutionforthesplayu˜,Eq.(A3). prefactors in Eq.(A1) and converting back to variables Thereforewehavetofindanappropriateapproximation. carrying dimensions we get as a final result for the free It can be shown that 1 4C˜ 0 for all u if D h. energy of the chains forming the finite brush 0 − ≈ ≥ Hence Eq.(A3) can be very well approximated as d βF = σ 3Dh2 h2(u +u ) u˜ brush Nb2 − 0 D 2u˜ 2u˜ ln u˜=4(D˜ x˜ x˜), − D− (cid:20)u˜D(cid:21) − 0− + h (u2(cid:2)+u2 ) 1(u3+u3 ) . (A11) 1 √2 0 D − 6 0 D for u˜ u˜ = (1 4C˜)1/2. (A5) (cid:21) c ≥ 2 − Pluggingintheα-dependentexpressionsforσ,D,u and 0 For 0 u˜ u˜ we can choose a linear approximation. u and adding F we get the α-dependent part of the c D rod ≤ ≤ To calculate the free energy we have to perform the total free energy of the system. following integration 4/3 3 b 1 βF(α)=2fβγd2tan(α)+ fN cos(α)2/3 dx˜ u˜2+ . (A6) 42/3 d 2(2+u˜′) (cid:18) (cid:19) Z (cid:20) (cid:21) N2 b 3 + cos2(α) We first consider the regime 0 u˜ u˜c: 4 d × ≤ ≤ (cid:18) (cid:19) 1 7 tan2(α) tan3(α) 1 1 1 u˜′(x˜ x˜ ) 2 , u˜2 (A7) tan(α)+ ≤ c ≪ ⇒ 2(2+u˜′) ≈ 4 c ≪ 4 (cid:20)√2 − 6 − √2 − 6 (cid:21) (A12) The integral in Eq.(A6) in the interval [0,x˜ ] can there- c fore safely be approximated by This equation is only valid for 0 α π/4, since ≤ ≤ u (π/4) = h = u . In the interval π/4 < α < π/2 0 D x˜c the splay at the surface remains constant at its maxi- 1 1 dx˜ u˜2+ 2(2+u˜′) = 4x˜c. (A8) mum value u0 = h, and the term in square brackets in Z (cid:20) (cid:21) Eq.(A12) reduces to [√2 7/3]. 0 − There are two regimes in each of which the above In the interval [x˜ ,D˜ x˜ ] the approximation for the free energy, Eq.(A12), shows a different behaviour. The c 0 − splay, Eq.(A5), is valid. The integral in Eq.(A6) can regime where the chains are rather short shows a first 9 order like transition. With decreasing γ there is a jump For N N the third term in the free energy, t ≥ from a stable phase with no shift (α = 0) to a stable Eq.(A12), gets equal to or bigger than the second term. phase with a large shift (α 0). The other regime in This means that the increase in splay of the chains close ≫ which the chains are rather long shows a second order to the surface becomes important. Since u scales with 0 transition from a stable phase with α = 0 to a phase tanαlikethecontributionoftherods(F )does,acon- rod with finite α. tinuous transition is now possible. The dependence of As a criterion to distinguished these two regimes the the free energy on α for different values of γ is shown in chainlengthcanbeused. ThechainlengthN whichsep- Fig.(11). t arates these two regimes is given by N 1.4f d 5/3. For N < N the transition is first ordter≈like andb con- The assumption of the shift to be constant is an over- t tinuous for N > Nt. This condition can be inter(cid:0)pr(cid:1)eted simplification. The crossover from one regime (N ≥Nt) such that for N <N the second term in the free energy in which a shift develops continuously to a regime (N < t (Eq.(A12)) dominates the third term. The second term Nt)inwhichthesystemshowsafirstordertransitionlike solelyrepresentsthe effectofdecreasinggraftingdensity, jump from zero shift to large finite shift is an unphysi- whereas the third term also represents the effect of in- cal artefact of this approximation. Only in the limit of creasing splay of the chains close to surface. Therefore, a very large number of copolymers f forming one lamel- if the grafting density effect is dominant, the transition larlikeaggregatetheconstantshiftassumptionmightbe is similar to the tilting transition observed in lamellar reasonable. However, in this limit the effect of the sur- structuresofrod-coilcopolymers,seee.g. [4]. Sincethen face becomes negligible and the system always shows a the minimum in the free energy is essentially given by tilting transition - compare [4] - even if not in contact the balance of tanα (first term) and cosα (secondterm) with the surface. there cannotbe acontinuoustransitionwitha minimum at small values of α. To illustrate this behaviour, the free energy as a function of α for different values of γ is F plotted in Fig.(10). 260 F 255 90 250 80 245 70 240 235 60 230 50 α 40 0.2 0.4 0.6 0.8 α FIG. 11: This plot shows the dependence of the free energy 0.2 0.4 0.6 0.8 1 1.2 1.4 in units of kT on α for different values of γ in the regime N > Nt. 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