Self-Consistent-Field Study of Compressible Semiflexible Melts Adsorbed on a Solid Substrate and...

Self-Consistent-Field Study of Compressible Semiflexible MeltsAdsorbed on a Solid Substrate and Comparison with AtomisticSimulations

Kostas Ch. Daoulas,†,‡,⊥ Doros N. Theodorou,*,†,‡ Vagelis A. Harmandaris,‡Nikos Ch. Karayiannis,‡,§ and Vlasis G. Mavrantzas‡,§

School of Chemical Engineering, Department of Materials Science and Engineering, NationalTechnical University of Athens, 9 Heroon Polytechniou Street, Zografou Campus, GR 15780 Athens,Greece; Institute of Chemical Engineering and High-Temperature Chemical Processes, ICE/HT-FORTH, GR 26500 Patras, Greece; and Department of Chemical Engineering,University of Patras, GR 26504 Patras, Greece

Received February 1, 2005; Revised Manuscript Received May 2, 2005

ABSTRACT: The present work addresses the problem of a self-consistent-field (SCF) description of aspecified polymer melt/solid substrate interfacial system. Key points of the employed method are thecoarse-grained representation and the numerical treatment of the continuous-space SCF theory. Comparedto other works on polymer adsorption, the main difference of the current approach is the description ofthe polymer coil connectivity through the wormlike chain model, which, after incorporating local stiffness,reproduces two characteristic lengths of the studied polymer: the mean-squared end-to-end distance andthe contour length. As a test case, polyethylene melts adsorbed on a graphite substrate are considered;recent atomistic simulations of the same systems are used to evaluate the theoretical approach. Forcomparison and elucidation of some effects of chain stiffness on conformational properties of adsorbedmolecules, an alternative (and more common) representation of chain connectivity through the Gaussianmodel, reproducing the mean-squared end-to-end distance, is also considered. Results refer to local andglobal chain conformational properties, with an emphasis on the latter. In particular, predictions for theshape of chains are obtained, while the conformations of adsorbed molecules are quantified in terms oftails, loops, and trains. For small chain lengths, both the Gaussian and the wormlike chain models deviateconsiderably from the simulation data. At intermediate chain lengths, however (such as C400), the predictivepower of the wormlike model is very good for several conformational properties. On the contrary,predictions from the Gaussian model, especially for the case of loops, deviate considerably from simulationsover a broader range of molecular lengths.

1. Introduction

The theoretical description and understanding ofproperties of polymer melt/solid interfaces are of crucialimportance for a variety of technological applications.For instance, the ability to control the quality ofproducts obtained through extrusion and film-blowingoperations is closely related to understanding the mech-anism behind the cohesive and adhesive failures takingplace at the polymer melt/solid interface at high produc-tion rates. Controlling the strength of polymer/solidinterfaces is also of crucial importance for the technologyof adhesives and polymer coatings and for the design ofhigh-performance polymer-solid composite materials.

When developing a theoretical approach for thedescription of the equilibrium structure and the me-chanical properties of a polymer/solid interface, it shouldbe taken into account that this interfacial systemincorporates a variety of length scales, starting from theangstrom scale when addressing phenomena in closevicinity to the solid substrate, passing to the nanometerscale when considering polymer density variations, and

extending to the submicrometer scale when addressingglobal polymer chain conformational properties andfracture phenomena.

In this scope, the most sophisticated theoretical toolwould comprise a multiscale scheme, combining ab initiocalculations for a detailed description of the phenomenataking place in close proximity to the solid substratewith atomistic and coarse-grained simulation techniquesto address the nanometer and submicrometer lengthscales, respectively. If one is interested mainly in large-length scale properties, however, then a coarse-grainedapproach can by itself provide significant qualitativeand quantitative information. In this case, one of thechallenges is to choose the coarse-grained representationin such a way that the essential factors determining thephysical behavior of the system are still preserved atthis level, while a proper correspondence of the char-acteristic time, energy, and length scales of the coarse-grained model and the prototype system is established.

One of the most common approaches used for thederivation of equilibrium properties of polymer/solidinterfacial systems at a coarse-grained level is the self-consistent-field (SCF) method. This approach is par-ticularly powerful and has been used extensively for theprediction of the equilibrium properties of variousinterfacial macromolecular systems. Apart from this,there is a tendency to employ SCF results as a startingpoint for the study of these systems away from equilib-rium. For instance, SCF results obtained for microphase-separated block copolymer systems in combination with

* To whom correspondence should be addressed at NTUAthens: Tel (+30) 210 772 3157; Fax (+30) 210 772 3112; [email protected].

† National Technical University of Athens.‡ ICE/HT-FORTH.§ University of Patras.⊥ Present address: Department of Physics and Department of

Chemical and Biological Engineering, University of Wisconsin,Madison, WI 53706.

7134 Macromolecules 2005, 38, 7134-7149

10.1021/ma050218b CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 07/12/2005

a density-biased Monte Carlo have been employed tocreate initial configurations for the study of theirloading/unloading behavior with molecular dynamics.1The mechanical properties of grafted polymer/free poly-mer melt interfaces have been successfully addressedthrough polymer entanglement network models,2 whosegeneration relies heavily on SCF results.

Although the SCF approach has been used extensivelyin the past for understanding the equilibrium behaviorof polymer solutions and melts in the presence of solidsubstrates, usually only abstract model systems havebeen considered. Although the effect of various molec-ular parameters and thermodynamic state variables onthe properties of the interfacial polymeric system couldbe addressed qualitatively or semiquantitatively (e.g.,at the level of scaling relations), the question of estab-lishing a direct correspondence between the coarse-grained model, serving as a basis for the SCF method,and a specified polymeric system has not been coveredextensively. To the best of our knowledge, for homopoly-mer melt/solid interfaces only few lattice-based SCFinvestigations have been devoted to this question,3 whilehardly any consideration has been given to it withcontinuous-space SCF.

The essence of the SCF method4 is in the replacementof the ensemble of the interacting polymer chains witha system of noninteracting polymer molecules subjectto some position-dependent chemical potential fields.These fields determine the conformations of the polymermolecules, thereby dictating the spatial distribution ofthe polymer. On the other hand, the chemical potentialfields depend on the polymer distribution. Thus, thewhole approach boils down to determining the fields insuch a way that they are consistent with the spatialdistribution of polymer that they create. One of theessential points of the above scheme is the determina-tion of chain conformations in the mean field, a problemwhich is ultimately linked to the type of coarse-grainedmodel employed for the representation of the chainconnectivity. In the majority of mean-field descriptionsof interfacial polymeric systems, the chain connectivityis introduced through the Gaussian model. This consid-ers the polymer chains as fully flexible and infinitelyextensible, continuous threads or “paths” and offers theadvantage of facilitating the analytical or numericalevaluation of the chain conformations through thesolution of the Edwards equation in the mean field.However, the Gaussian model is incapable of describingconformational properties on the length scale of thepersistence length, while it additionally fails in account-ing for the distal correlation between segments posi-tioned along the chain backbone arising from the chainself-avoidance.5 In this direction, considerable improve-ment can be achieved through the implementation ofsingle-chain mean-field theory approaches6,7 where thecharacterization of chain conformations in the meanfield can be performed using Monte Carlo techniquesafter accounting for the chain connectivity in theframework of more realistic molecular models (forexample RIS) through a “discretized”, bead bond, rep-resentation of the polymeric chain architecture.

Despite the advantages offered by the single-chainmean-field methods, it is frequently more beneficial toemploy continuous thread chain representations due tothe relative simplicity of their numerical treatment andthe possibility of addressing long-chain length limits ata rather low computational cost. In the realm of

continuous thread chain models, a more realistic coun-terpart to the Gaussian representation is the wormlikemodel, which treats the molecules as semiflexiblethreads of prescribed contour length. The mathematicalformalism for the implementation of this model in theframework of SCF, even if more complicated in com-parison to the Gaussian one, is still quite straight-forward.8,9 Nevertheless, the wormlike model has notmet such a wide application.

Several works have shown that the consideration ofchain rigidity can affect the predictions for variousproperties of interfacial polymeric systems. For instance,in the case of block copolymer melts chain rigidityincreases the temperature of the order-disorder transi-tion.10 In tethered polymer/free polymer melt systemsthe correct incorporation of chain rigidity improvessignificantly the predictive power of the mean-fieldtheory regarding the properties of the brush/free meltinterface.11 In a polymer adsorption problem, the prop-erties of the molecules are expected to be determinedthrough a balance between the enthalpic gain obtainedwhen the molecule gets close to the attracting substrateand the conformational loss due to the presence of theboundary. The use of a molecular model with internalrigidity and a constrained total contour length can havea substantial effect on the entropic part of this balance,leading to quantitatively different results when com-pared to the Gaussian case. In addition, the details ofthe molecular model can influence the way in whichadsorbed chains emanate from the substrate, affectingthe large length scale structure of the adsorbed layer(formation of loops and tails). Looking from anotherperspective, it can be said that, through its representa-tion of internal chain rigidity, the wormlike chain, whencompared to the Gaussian one,12 incorporates an ad-ditional length scale on the level of the chain persistencelength; this can enhance its predictive capabilities withrespect to some properties. For very long chains, thelarge length-scale properties of both the Gaussian andwormlike models are expected to converge to the “exact”properties, as extracted, for example, by atomisticsimulation.

In the above framework, the present work is aimedto address the issue of representing a specified polymer/solid interfacial system with a continuous coarse-grained molecular model and subsequently treating itthrough a SCF approach. Polymer molecules will beconsidered as wormlike chains. In parallel, the Gaussianmodel will also be implemented to facilitate the com-parison between these two representations and toelucidate some of the effects of chain stiffness on theconformations of adsorbed molecules. Since, to the bestof our knowledge, the wormlike chain representationhas never been previously employed in a SCF calcula-tion of polymer melt adsorption, one of our basicobjectives is the evaluation of the proposed theoreticalapproach. To this end, the results of recent atomisticsimulations13,14 will be utilized. Without loss of general-ity, all results of the current study will refer to linearpolyethylene melts adsorbed on a graphite substrate.

The article is organized as follows. The next sectioncontains a brief presentation of the atomistic simula-tions of the polyethylene melt/graphite interfacial sys-tems, which serve as a reference point for the SCFcalculations of the current work. Section 3 presents thegeneral SCF theory for the case of adsorbed melts, whilesection 4 is dedicated to the evaluation of the param-

Macromolecules, Vol. 38, No. 16, 2005 Compressible Semiflexible Melts 7135

eters of the mesoscopic representation for the particularcase of polyethylene melts adsorbed on graphite. Somegeneral aspects of the numerical treatment of the SCFformalism are discussed in section 5, while additionaltechnical details are contained in the Appendix. Section6 presents results from the SCF calculations and theircomparison with the atomistic molecular dynamics andMonte Carlo simulation data. Finally, the last sectionsummarizes some basic points of the current work.

2. A Brief Description of the AtomisticSimulations and the Studied Systems

A detailed presentation of the molecular dynamics(MD) and Monte Carlo (MC) atomistic simulations ofthe PE melt/graphite interface, used as a test systemfor the SCF calculations presented in this work, iscontained in refs 13 and 14. For greater clarity, however,we present a brief summary in the following para-graphs.

The atomistic simulations were conducted by describ-ing the PE chains with a united atom model, whichconsiders each methylene (CH2) and methyl (CH3) groupalong the chain backbone as one interaction site. Allintramolecular interactions between sites separated bymore than three bonds and all intermolecular interac-tions were described through a 6-12 Lennard-Jonespotential parametrized using the TraPPE model15 pa-rameters for the CH2/CH2 interaction (i.e., no explicitdistinction was made between the CH2 and the CH3units; they were described by identical pseudoatoms).Regarding the bonded interactions, C-C bond lengthswere kept fixed and equal to 1.54 Å, while the bondangles between successive C-C bonds were allowed tofluctuate around an equilibrium angle, θ0 ) 114°,subject to the van der Ploeg and Berendsen potential.16

Finally, changes of dihedral angles were governed bythe Toxvaerd torsional potential.17 The atomistic rep-resentation of the graphite substrate was achieved bydescribing its interaction with the PE united atomsthrough a method developed by Steele,18 which iscapable of incorporating the exact crystallographicstructure of the graphite. A basal plane of graphiteserved as the adsorbing surface in all calculations.

The simulations were carried out in orthorhombiccells with periodic boundary conditions along the x andy directions, while the system along the z direction wasconsidered as finite. The polymer was allowed to adsorbon the lower face of the simulation cell, where thegraphite substrate was located, thus creating an ad-sorbed polymer film with its upper surface beingexposed to vacuum. The molecular lengths that wereconsidered ranged from C40 up to C400, and it waspossible to create adsorbed films with thicknesses from46 up to 125 Å; that is at least 3 times larger than thecorresponding equilibrium PE chain gyration radius inthe bulk melt. For a thorough equilibration of thesesystems and the subsequent derivation of their struc-tural, conformational, thermodynamic, and dynamicproperties, a hierarchical approach was used. In par-ticular, the equilibration and the derivation of equilib-rium static properties were achieved through a robustMC algorithm based on chain connectivity alteringmoves.19-21 Subsequently, the equilibrated system wassubjected to MD simulations of duration from 20 up to100 ns, depending on the molecular length and size(total number of atoms) of the simulated system. In thisway, apart from deriving the dynamic properties, it was

possible to compare MD predictions regarding the staticproperties of the PE melt/graphite systems against thecorresponding results from MC as a test of the statisticalmechanical consistency of both approaches and of theirergodicity. In this scope, the agreement between MCand MD methods was found to be excellent.

All melts composed of PE molecules with lengths lessthan C400 were strictly monodisperse, while in the caseof the C400 melt a small polydispersity was allowed (inparticular, a flat molecular weight distribution wasrealized with polydispersity index I ) 1.0033) to speedup the equilibration process with the MC method. Allatomistic simulations were conducted at a temperatureof T ) 450 K.

3. SCF Theoretical Description of AdsorbedPolymer Melts

One of the major problems appearing during thedescription of a complex molecular system on a coarse-grained level is associated with the representation ofinteractions between its constituent entities. In our caseof polymer melt/solid interfacial systems the aboveproblem encompasses the following three subprob-lems: (a) the description of bonded interactions througha suitable definition of chain connectivity, (b) theincorporation of nonbonded polymer/polymer interac-tions, and (c) the proper definition of the polymer/solidinteraction potential.

As was mentioned in the Introduction, in the currentwork the basic choice for the representation of chainconnectivity will be the wormlike (semiflexible) chainmodel. However, the commonly used Gaussian modelwill also be considered, in parallel. Comparisons be-tween predictions from the two coarse-grained repre-sentations will help elucidate some aspects of the effectsof local rigidity on global polymer chain conformationalproperties. Using the results of the atomistic simula-tions as a reference point, it will also be possible toestablish the limits of applicability of the Gaussianchain connectivity representation. This is of particularpractical importance since, as will be seen in thefollowing sections, Gaussian chains constitute much“easier” objects to deal with, numerically, than wormlikechains.

Following ref 22, where a compressible melt near anenthalpically neutral (i.e., hard) wall was considered,the nonbonded interactions will be represented at thecoarse-grained level by constraining the polymer densityfluctuations around their mean value, characteristic ofa “bulk” polymer, through a simple harmonic potentialof the form

where φ(r) stands for the polymer volume fraction atpoint r. The function F(r), employed in the definition ofvolume fraction, is the monomer number density at r,while F0 is the monomer number density in the bulk.Finally, the parameter κ denotes the isothermal com-pressibility, defined as κ ) -V-1(∂V/∂p)|n,T. Note thatUcompress has the meaning of a free energy densityrelative to a perfectly homogeneous state at F0, withunits of energy per volume. The potential of eq 1,penalizing density fluctuations, was initially employedby Helfand and co-workers23-25 in their SCF studies ofpolymer/polymer interfaces and block copolymer sys-

Ucompress ) 12κ

[φ(r) - 1]2, φ(r) )F(r)F0

(1)

7136 Daoulas et al. Macromolecules, Vol. 38, No. 16, 2005

tems. Despite its rather primitive form, it has beenshown capable22 of capturing the combined effects ofhard wall and polymer compressibility on chain confor-mational properties, such as the segregation of chainend-segments to the melt/wall interface. On the otherhand, a potential of this form cannot grasp the detailsof the local structure of the melt, such as the densityoscillations close to the solid substrate arising frommonomer packing effects, which are successfully de-scribed through various density functional theory (DFT)approaches.26-33 Taking into account this fact, it shouldbe mentioned that a more precise description of localstructure can be gained following the practice employedin several works31-33 through the incorporation of theDFT excess free energy density functionals into thegeneral SCF formalism.

While presenting the mean-field theory for the coarse-grained polymer melt/solid system, the interaction of a“segment” of a Gaussian or wormlike thread with thesolid substrate will be described in terms of an arbitraryfunctional form US(z) (with units of energy per segment),where z is the distance of the thread “segment” fromthe substrate. For greater clarity, the definition of theexact form of this function will be postponed until ourdiscussion of the mean-field description is completed.

The continuum SCF approach is based on a path-integral representation of the partition function. In thepast, this approach has been extensively employed innumerous works; the interested reader is referred toseveral reviews discussing this subject extensively.4,34,35

Here, an effort will be made to present the theory asbriefly as possible, emphasizing only on some essentialpoints specific to the present work. The partition func-tion in the grand canonical statistical ensemble for thecase of monodisperse adsorbed melts takes the form

where kB is the Boltzmann constant and T is thetemperature, while µ, N, and n are the monomerchemical potential, the degree of polymerization of thecoarse-grained molecule, and the number of chains,respectively. N is a normalizing prefactor, and Dra(•)denotes functional integration over all possible con-formationss“paths”sof the ath chain, while φ(r) is thevolume fraction operator, given by

where s is a scaled variable measuring how far alongthe contour length of the chain a considered segmentlies; it ranges from 0 (chain start) to 1 (chain end). Thefunctional P[ra(•)] accounts for the chain connectivity,and in the case of the Gaussian model it takes the form

where Re2 is the mean-squared end-to-end distance of

the polymer chain in the bulk. It can be seen that, inthe case of the Gaussian model, only one characteristicconformational parameter of the atomistic polymerchain enters the coarse grained model: Re

2. For thewormlike chain model the connectivity functional isdefined through8,9,36,37

where η is a dimensionless bending modulus, while u(s)is a dimensionless vector, tangent to the contour of thewormlike chain at “segment” s. u(s) can be defined afterinvoking the chain contour length L as u )(1/L)(dr/ds). Obviously, the product of the delta functionsconstrains the modulus of the tangent vector to unity,so that the total contour length is preserved. Thedetermination of the prefactor η/2N can be betterunderstood if one considers the works of Saito et al.8and Chen.36 In particular, Saito et al. proved8 that themean-squared end-to-end distance, Re

2, of the wormlikechain fulfills the Kratky-Porod relationship:38

By choosing the N/2η parameter so that eq 5 isfulfilled, it is possible through the wormlike chain modelto reproduce both the contour length, L, and the mean-squared end-to-end distance, Re

2, of the atomistic poly-mer molecule. A better understanding of the quantityN/2η can be obtained if the limiting case of eq 5 isconsidered, when N/2η . 1. Then Re

2 ≈ L/D f N/2η )L2/Re

2, and it can be seen that the ratio N/2η is indeeda measure of the flexibility: the smaller the mean end-to-end distance with respect to the contour length, themore flexible the chain. Obviously, for a freely jointedchain, η ) 0.5. Another convenient representation of theN/2η parameter can be gained if one recalls39 that theratio Re

2/L for a polymer molecule, i.e., its Kuhn length,equals twice its persistence length, lp. Since also L )Na, where a is the bond length of the coarse-grainedmolecule, one gets9 η ) lp/a so that

Following a standard field theoretical ap-proach,4,34,35,40,41 it is possible to obtain from eq 2 thegrand canonical free energy (grand potential), Ω, withinthe framework of the mean-field approximation. Afterrecalling the translational invariance of the systemalong the xy plane, the following relationship will hold:

where V is the volume of the system and Lz is its sizein the z direction, while the field functional Q[W]denotes the single-chain, Gaussian or wormlike, parti-tion function in the field W(z). For further convenience

¥ ) ∑n)0

∞ 1

n!exp[µNn

kBT ]Nn∫∏a)1

n

Dra(•)P[ra(•)]

exp(-1

2κkBT∫[φ(r) - 1]2 dr -

F0

kBT∫US(z) φ(r) dr)

(2)

φ(r) )

N∑a)1

n ∫0

1δ(r - rR(s)) ds

F0

(3)

P[rR(•)] ) exp[- 32Re

2∫0

1(drds)2

ds] (4a)

P[ra(•)] ) ∏s

δ(u2 - 1) exp[ -η

2N∫0

1(du

ds )2

ds] (4b)

Re2 ) (exp(-2DL) - 1 + 2DL

2D2L2 )L2, DL ) N2η

(5)

N2η

) L2lp

, when N2η

. 1 (6)

ΩLz

nbulkkBT) -

NLzQ[W] exp[µN/kBT]nbulk

-

∫W(z) φ(z) dz + 12κkBT

Vnbulk

∫(φ(z) - 1)2 dz +

VF0

kBTnbulk∫US(z) φ(z) dz (7)


in eq 7, we have introduced the parameter nbulk denotingthe total number of chains that would be contained inthe volume V in absence of any substrate (i.e., in thecase of a “bulk” polymer melt). The effective field W(z)and the volume fraction, φ(z), are coupled to each otherthrough a set of equations depending on the modeladopted for the chain connectivity. For the case of thewormlike chains, after recalling the translational in-variance along the xy plane, these equations are for-mulated as

where the quantity q(z, s, cos θ), when multiplied bythe surface of the xy plane, equals the single-chainpartition function of the (0,s) portion of the chain underthe restrictions that (a) the s segment is found at adistance z from the substrate and (b) the cosine of theangle formed by the z axis and the tangent vector at sis equal to cos θ. The q(z, s, cos θ) propagator can becalculated through an equation, introduced8 by Saito etal., which is the analogue of the Edwards equation forwormlike chains:

with initial condition9,10,37 q(z, 0, cos θ) ) 1. Theboundary condition at z ) Lz is conveniently determinedafter setting the value of the field W(Lz) ) 0 so thatq(z ) Lz, s, cos θ) ) 1. After defining the W(z) field aszero at Lz and considering that far from the substratethe polymer volume fraction should be unity, i.e., φ(Lz)) 1, the value of the multiplying prefactor in eq 8b isdetermined as 2πNV exp[µN/kBT]nbulk

-1 ) 1/2.The boundary condition at z ) 0 requires some further

discussion. Commonly,42-44 when considering questionsrelated to polymer adsorption, the Edwards equationis supplied with an effective boundary condition. Thispractice originates from the work of de Gennes45 andamounts to splitting the total effective field felt by apolymer segment into a part originating from its inter-action with the substrate and a mean-field part ac-counting for the interactions with the rest of the polymersegments. (The latter is sometimes denoted42 as “themolecular field”.) Then, the surface potential is disre-garded, and its presence is taken implicitly into accountwhen solving the Edwards diffusion equation in theremaining mean-field component, by imposing on thepropagator Q an effective boundary condition:(1/Q)(∂Q/∂z)|z)0 ) -c. The parameter c is the inverse ofthe extrapolation length, specifying the segment/surfaceinteractions.42,44 In our case, however, we will preservethe surface component of the effective field W(z). To thisend the propagator equation, eq 9, has to be suppliedwith a “true” boundary condition and not with aneffective one. This boundary condition is the so-called“absorbing boundary condition”, requiring q(z ) 0, s,cos θ) ) 0. We would like to complete this discussion ofboundary conditions by referring the interested reader

to the work22 of Wu et al. considering a polymer meltnear an enthalpically neutral substrate, where a de-tailed presentation is given of how, beyond some dis-tance from the wall, the microscopic absorbing boundarycondition effectively reduces to the well-known for thiscase, reflecting boundary condition.

Equations 8a and 8b in combination with the propa-gator equation, eq 9, supplemented with the properinitial and boundary conditions, constitute a closedsystem of SCF equations describing the polymer melt/solid substrate interfacial system at the level of thewormlike molecular model.

In the case of the Gaussian model the essence of theapproach remains the same. In particular, eq 8a willremain the same while eq 8b transforms to

Finally, instead of eq 9, the propagator will fulfill theEdwards equation

with the same initial and boundary conditions: q(z, 0)) 1, q(0, s) ) 0, and q(Lz, s) ) 1.

Having cast the SCF framework, it is now possible toproceed with the exact definition of the parametersentering the theoretical formalism, including the de-scription of polymer/substrate interactions and thenumerical approach for solving the SCF set of equations.

4. Representation of the PE Melt/GraphiteInterfacial System with the Wormlike ChainMolecular Model

After the set of SCF equations (eqs 8a, 8b, and 9) hasbeen formulated, there are several parameters to bedetermined so that a correspondence between thecoarse-grained and the atomistic PE/graphite systemscan be established. Not all of these parameters have tobe specified, however, since only some combinations ofthem will have an actual physical meaning in theframework of the mesoscopic representation.24,46 Thesecombinations, which are called invariants after Helfandand Sapse,24 will retain values characteristic of theatomistically simulated polymeric system, establishingthe equivalence of its characteristic length and energyscales to those of the coarse-grained system.

Since all atomistic simulations were performed atT ) 450 K, the same temperature will be assumedduring the SCF calculations of the current work. Thecurrent mesoscopic representation will preserve thevolume, V, of the system as well as the number of thechains, nbulk. To this end, both the mesoscopic and theatomistic systems, in the “bulk”, have the same numberof chains per unit volume: nbulk/V, which can betrivially calculated from the molecular weight and thebulk density, Fbulk, of the atomistic system. In allcalculations of this work Fbulk ) 0.766 g/cm3. Thepressure p is also set as invariant, so the compressibilityof the mesoscopic system, being defined as κ )-V-1(∂V/∂p)|n,T, will equal the isothermal compress-ibility of the atomistically studied polymer. Taking intoaccount the atomistic simulation20 results and experi-mental data,47 we employed κ ) 1.43 × 10-9 Pa-1.

φ(z) )NV exp[µN/kBT]

nbulk∫0

1q(z,s) q(z, 1 - s) ds (10)

∂q∂s

)Re

2

6∇2q - W(z)q (11)

W(z) )V[φ(z) - 1]κkBT nbulk

+F0VUS(z)nbulkkBT

(8a)

φ(z) )2πNV exp[µN/kBT]

nbulk∫-1

1 ∫0

1q(z, s, cos θ) q(z,

1 - s, -cos θ) ds d cos θ (8b)

∂q∂s

+ L cos θ ∂q∂z

) N2η

1sin θ[cos θ ∂q

∂θ+ sin θ ∂

2q∂θ2] -

W(z)q (9)


Following the previous section, the length scales ofthe mesoscopic and the atomistic molecular models arerelated through two invariant parameters: the contourlength of the chain, L, and the mean square end-to-enddistance of the chain, Re

2. Both of them can be evaluatedas

where nb, lb, θ0, and C denote the number of chemicalbonds, the bond length, the equilibrium bond anglealong the chain backbone, and the polyethylene char-acteristic ratio, respectively. In the current work, thevalues lb ) 1.54 Å and θ0 ) 114° have been used, whileC was calculated taking into account its dependence onthe number of carbon atoms in the PE molecule, N, asC ) a0 + [a1/(N - 1)] + [a2/(N - 1)2] + [a3/(N - 1)3].The coefficients a0, a1, a2, and a3 are constants, beingequal48 at T ) 450 K to 8.27, -43.59, -246.02, and3182.6, respectively.

After the parameters L and Re2 are known, the ratio

N/2η can be evaluated either by solving numerically theKratky-Porod relationship, eq 5, or by using the limit-ing eq 6, if the condition N/2η . 1 holds. For all systemsstudied here, the values of N/2η calculated from theKratky-Porod equation, eq 5, and from its approxima-tion, eq 6, were found to be practically the same,particularly for the larger molecular weights. Forinstance, for the relatively “short” C78 melt, the exactPorod-Kratky equation gives N/2η ) 6.24, while theapproximation leads to N/2η ) 6.78.

The last step for the accomplishment of the meso-scopic representation of the PE/graphite system is thedefinition of the polymer/substrate interaction potential,Us(z). In this scope, we recall the z dependence of theatomistic PE monomer/graphite interaction potential,Us

at(z), and its effect on the polymer density, bothpresented in a collective graph in Figure 1 for anadsorbed C250 melt. For small z, the Us

at(z) potential isrepulsive, accounting for the excluded-volume interac-

tions between the polymer monomers and the atoms ofthe graphite lattice (in Figure 1 the thickly drawn firstvertical bar shows where the first layer of graphitecarbon atoms ends), while further in it assumes the formof a localized attractive well. Practically, the range ofthe well is around 15 Å, resulting in the emergence ofthree sharply structured layers of adsorbed polymersegments, positioned between the thin vertical bars ofFigure 1. Beyond z ) 15 Å the local density tends toassume a constant value, equal to the density of the bulkmelt. To represent the strong polymer/graphite repul-sion within the first 2 Å in the SCF calculations, weshift the origin of the z-axis [where q(z ) 0, s, cos θ) )0] with respect to the atomistic system by 2 Å; thus,the strongly repulsive part of the Us

at(z) is approxi-mated with an impenetrable wall. Taking into accountthe localized character of the attractive part of theUs

at(z) potential, we represent it on the coarse-grainedlevel through a simple square well, which is completelydefined when its width, w, and depth, U0, are known.The width is determined after observing that, in theatomistic density profile of Figure 1, the most significantadsorption peak is the first one. In addition, an analysisof the atomistic simulation results reveals13 that aconsiderable percentage of the atoms found in the nexttwo peaks (70% for the second and 44% for the third)belong to chains passing from the first adsorption layer.Taking into account all this information, the width wis set equal to the width of the first adsorption peak,that is w ) 4.5 Å.

To determine the depth of the square well potential,U0, we require the coarse-grained PE/graphite systemto have the same adsorption energy, Eads, per unitsurface as the atomistic one (i.e., we choose the adsorp-tion energy per unit surface as an invariant of themesoscopic representation). This requirement is ex-pressed as

where the value Eads ) -0.25 kBT/Å2 was determinedfrom the atomistic simulation. A careful considerationof the SCF equations (eqs 8a, 8b, and 9) and the integralconstraint of eq 13 reveals that the polymerizationdegree, N, of the coarse-grained chain never appears byitself but always in combination with some otherquantity; thus, N is not an invariant of the representa-tion. For instance, in eqs 8b and 13 it appears incombination with the U0 parameter. In this way, oneactually has to determine the product NU0 entering eqs8b and eq 13 and not the U0 parameter by itself.Obviously, the determination of NU0 from eq 13 requiresknowledge of the volume fraction profile, φ(z), as derivedfrom SCF theory. Thus, the NU0 reproducing the desiredadsorption energy has to be determined through aniterative scheme: one starts from some initial guessvalue, solves the SCF set of equations as described inthe following section, determines φ(z), and evaluatesEads. If the result is not the desired one, the NU0 iscorrected by some small quantity and the procedure isrepeated, until the correct adsorption energy is obtained.The value of NU0 has to be defined only for onemolecular weight, C78 for example: For the rest systemsit is derived by multiplying this “base case” NU0 value

Figure 1. z dependence of the atomistic PE monomer/graphiteinteraction potential, Us

at(z) (solid line), plotted for a CH2 unitlying directly above the center of a graphite hexagon and itseffect on local polymer density (shown in arbitrary units, bythe small solid circles). The large, thickly drawn first verticalbar shows where the first layer of graphite carbon atoms ends,while the average positions of the CH2 units in the threeadsorbed polymer layers are situated between the remainingthree vertical bars (shown as thin lines).

Eads ) -0.25kBT/Å2 ) F0∫US(z) φ(z) dz )nbulkNU0

V ∫0

wφ(z) dz (13)

L ) nblb sin(θ0/2), Re2 ) Cnblb

2 (12)


by the ratio of the molecular weight of the melt understudy to the molecular weight of the base case.

5. Solution MethodThe SCF equations formulated in the current work

(i.e., eqs 8a, 8b, and 9 or, alternatively, eqs 8a, 10, and11 if the Gaussian model is invoked) can be solvedfollowing a simple relaxation technique similar to theone employed in ref 49. An initial guess for the fieldW(z) is made (usually a random one) which is substi-tuted into the propagator equation (eqs 9 or 11).Subsequently, the equation is solved numerically so thatthe propagator, q, is found, and the correspondingspatial distribution of polymer, given by the volumefraction φ(z), is calculated from eq 8b or eq 10 (depend-ing on the model). Then a new value for the field isestablished as Wnew(z) ) W(z) + λ[W(z) - W(z)], wherethe W(z) is evaluated after substituting the calculatedvolume fraction φ(z) into eq 8a. The new field valueserves as an input for the next iteration step, and thewhole procedure, described previously, is repeatedagain. The iteration continues until convergence isachieved, being commonly confirmed by monitoring thebehavior of the free energy (eq 7); in the current workthe iterative scheme was considered converged when therelative free energy change between two sequentialiterations was on the order of 10-12. The magnitude ofthe relaxational parameter λ is determined empiricallyso that the stability of the calculations is preserved. Asa rule of thumb, it is mentioned that its magnitude isinversely related with the value of the V/κkBTnbulkprefactor in eq 8a: the larger the molecular weight ofthe studied melt, the smaller the λ that can be em-ployed. For example, for a C78 system a value of λ )0.001 can be used, while in the case of a C1000 melt ithas to be reduced to 0.0005 to maintain stability.

The central component in the above iteration schemeis the numerical solution of the propagator equation (eq9 or eq 11). For the Gaussian molecular model, the one-dimensional Edwards diffusion equation can be solvedthrough a Crank-Nicholson approach,50,51 for instance.The case of the wormlike chain model presents moredifficulties and will be discussed more extensively in thefollowing.

For the wormlike model we proceed with the solutionof the propagator equation, eq 9, expanding the propa-gator function, q(z ) 0, s, cos θ), in terms of Legendrepolynomials after the recommendations of refs 10 and37:

This expansion is substituted into eq 9, which isafterward multiplied by the lth Legendre polynomial37

(l ) 0, 1,..., ∞) and integrated over the variable cos θ.Taking into account the properties of Legendre poly-nomials (i.e., orthogonality and recurrence relation-ships), it is possible to prove that the expansion coef-ficients satisfy the following set of partial differentialequations:

with initial and boundary conditions

where the δR,â is the Kronecker symbol. In eq 15a alllengths are expressed in units of 2ηL/N, which, for largemolecular lengths, will be twice the persistence length(see eq 6).

The propagator differential equation, eq 9, is nowreplaced by the set of eqs 15, which has to be solvednumerically at each iteration.

To the best of our knowledge, the approach for thesolution of this problem, appearing in various applica-tions of semiflexible chains to interfacial polymericsystems, has not been extensively presented in theliterature. Moreover, although in a recent work52 con-sidering melts of semiflexible diblock copolymers aforward time-centered space (FTCS) scheme was suc-cessfully employed for a similar numerical problem,exploratory runs showed that in the case of adsorbedpolymer melts this approach leads to a strong instabilitynear the solid boundary. In view of this, in this work aLax-Wendroff-like approach53 was developed for thenumerical treatment of eqs 15. The implementation ofthis technique requires the truncation of the Legendrepolynomial series which, taking into account ref 37, inthe current SCF calculations was set at m ) 8 (that is,q8 ) 0). This choice was further justified by testcalculations employing a higher expansion order(m ) 13), which led practically to the same results. Asa general comment it can be said that the stability ofthe suggested method was discovered to be dependenton the relevant magnitude of the descretizations em-ployed for the 1D-coordinate space and the chaincontour length: in the case of large chain lengths(corresponding to large absolute mean-field values closeto the wall and large values of the parameter N/2η) asignificant increase in the chain contour discretizationwas required in order to maintain a resolution levelalong the space coordinate similar to the one used forshorter chain lengths. As an example, we mention thatto provide the calculations in the C1000 melt with a spaceresolution on the order of 1 Å, the discretization alongthe chain contour had to be kept around 6 × 10-4. Asimilar level of space resolution in the case of C5000 meltcould be achieved only when the chain contour discreti-zation was on the order of 1 × 10-4.

We avoid further presentation of these rather techni-cal issues in the main text, referring the interestedreader to the Appendix, where a more detailed discus-sion can be found.

6. ResultsLocal Volume Fraction. The local volume fraction,

φ(z), derived from the SCF calculations for the C250 PEmelt/graphite system is presented in Figure 2, wherefor comparison the atomistic simulation result is alsoshown. The two vertical dashed lines delimit the areawhere the polymer is subjected to the effect of theattractive, square well, part of the coarse-grainedpolymer/graphite potential, US(z). Because of the ab-sence of the microscopic single monomer size lengthscale from the wormlike and the Gaussian chain models,in combination with the simplified description of theintermolecular interactions, the SCF calculation shouldnot be expected to reproduce phenomena taking place

q(z, s, cos θ) ) ∑m)0

∞

qm(z,s) Pm(cos θ) (14)

2ηN

∂ql

∂s) -[ l

2l - 1∂ql-1

∂z+ l + 1

2l + 3∂ql+1

∂z ] - (l + 1)lql -

2ηN

W(z)ql, l ) 0, 1, ..., ∞ (15a)

ql(z ) 0, s) ) 0, ql(z ) Lz, s) ) δl,0,ql(z, s ) 0) ) δl,0 (15b)


on atomistic length scales. The SCF calculations of thiswork cannot reproduce the oscillatory behavior of φ(z)close to the substrate, observed during the considerationof the system with atomistic simulations. Qualitatively,however, the SCF approach provides a reasonablesmoothed picture of polymer density variations in thearea subjected to the effect of the potential US(z) in theform of a single adsorption peak, where the polymervolume fraction is ≈1.1. A similar result is derived whenthe atomistic polymer volume fraction profile is aver-aged on a more “coarse-grained scale”: An average overthe first adsorption peak yields φ ) 1.22. For compari-son, on the same figure we reproduce the φ(z) profilederived with the Gaussian connectivity model, whereinthe well depth of the coarse-grained interaction poten-tial US(z) was also tuned to reproduce the correctadsorption energy. In this case the adsorption peak ismore pronounced, while a strong depletion can beobserved close to the boundary, despite the attractiveinteractions with the substrate. This depletion, whichis much more significant in the case of the Gaussianchain melt than in the wormlike chain model, has anentropic origin, i.e., is due to the loss of conformationsof the polymer chain close to the substrate. Thisdifference between the two models can be explainedafter considering the infinite extensibility of segmentsin the Gaussian chain model. This should allow strongervariations of polymer density on a bond-size lengthscale, thus favoring a smooth transition from completeabsence of polymer near the substrate to a well-formedadsorption peak, a few angstroms away.

Local and Overall Chain Conformational Prop-erties. A typical measure of the effects of the solidsubstrate on the conformations of the polymer chains,when considered on the level of individual bonds, is thebond-order parameter P2(cos θ). More specifically, P2corresponds to the second-order Legendre polynomial,while θ denotes the angle between the bond and thez-axis. This parameter is of particular interest since it

can be measured experimentally through deuteriumNMR experiments,54 so a direct connection with thetheoretical predictions can be established. In the caseof the currently studied adsorbed melts the average ofthis parameter was evaluated for the wormlike chainmodel as a function of bond distance from the adsorbingsubstrate as

where the propagator q(z, s, cos θ) is derived from eqs14 and 15a after substituting the converged solution ofthe SCF scheme, W(z). The function w(z, s, cos θ) denotesthe probability density that the cosine of the angleformed by the tangent vector to a “segment” located atposition s along the wormlike chain backbone is equalto cos θ, under the condition that this segment is foundat distance z from the substrate. (The numerator of wdenotes the total number of “paths” satisfying thiscondition, while the denominator represents the totalnumber of paths crossing z, irrespective of the positionalong the chain contour and the tangent vector orienta-tion where this intersection occurs.) The integrals in eq16 can be conveniently evaluated after substituting theexpansion of the propagator over the Legendre polyno-mials, eq 14, and invoking their orthogonality propertyas well as their recurrence relationships.

The ⟨P2(cos θ)⟩z function derived in this way using thewormlike chain model is presented in Figure 3 for twoadsorbed melts C78 and C400, while, for comparison, theatomistic simulation data for the C78 melt are alsoshown. It can be seen that the bond order parameter isa “local” property, being unaffected by the chain molec-

Figure 2. Polymer volume fraction profile, φ(z), as derivedby the SCF theory after the implementation of the wormlike(solid line) and the Gaussian (thick dashed line) chain con-nectivity models. The vertical, thin dashed lines enclose theregion where the polymer is subjected to the effects of theattractive square well potential in the coarse-grained descrip-tion, while the open circles correspond to the atomisticsimulation data.

Figure 3. Local bond order parameter, ⟨P2(cos θ)⟩, calculatedas a function of distance, z, from the adsorbing substrate byapplication of the wormlike chain SCF approach for twomelts: C78 (solid line) and C400 (open circles). The correspond-ing prediction of the atomistic simulations for the orderparameter of the C-C bonds is denoted by solid circles. Thevertical, thin dashed line denotes the boundary of the regionwhere the polymer is subjected to the effects of the attractivesquare well potential in the coarse-grained description.

⟨P2(cos θ)⟩z ) ∫0

1∫-1

1w(z, s, cos θ) P2(cos θ) d cos θ ds

w(z, s, cos θ) )q(z, s, cos θ) q(z, 1 - s, -cos θ)

∫0

1∫-1

1q(z, s, cos θ) q(z, 1 - s, -cos θ) d cos θ ds

(16)


ular weight. Close to the graphite ⟨P2(cos θ)⟩z attainslarge negative values, characteristic of a parallel ori-entation of the segments of the wormlike chain withrespect to the adsorbing substrate. ⟨P2(cos θ)⟩z increasesmonotonically over a length scale equal to the width ofthe attractive well to become zero a few angstroms awayfrom the domain of polymer/substrate attraction, mani-festing a loss of any orientation at the bond level atdistances that are large compared to the segment sizeand to the range of the potential. The discontinuity of⟨P2(cos θ)⟩z at the border of the attractive well is due tothe functional form of the US(z) potential. Similarresults obtained with alternative, smooth US(z) func-tions reproducing the same adsorption energy with asimilar length scale of polymer/substrate attraction didnot exhibit such a discontinuity, while the behavior of⟨P2(cos θ)⟩z in the rest of the domain remained practi-cally the same. In contrast to the SCF results, theatomistic simulation data exhibit an oscillation betweennegative and positive values of P2(cos θ); the lattervalues are somewhat smaller in magnitude. This oscil-lation is commensurate with the variations of thedensity near the substrate (see also Figure 1), originat-ing in the specifics of local monomer packing. Despitethe intrinsic incapability of the current coarse-grainedmolecular model to account for these packing effects, itcan be observed that the SCF can indeed reproduce theaverage trend of the atomistically derived curve, leadingto very similar values of ⟨P2(cos θ)⟩z close to theadsorbing boundary.

The effect of the adsorbing substrate on global chainconformational properties can be elucidated after ana-lyzing the shape of the chains in the various parts ofthe PE melt/graphite interfacial system. A direct mea-sure of this property at any distance, z0, from thesubstrate is provided by the number of chains per unitsurface passing through a plane placed at z0 parallel tothe attractive surface.55 In particular, the probability,pcross(z0), that a chain starting anywhere in the systemwill intersect the given plane can be calculated as

where the propagators q1(z, s, cos θ) and q2(z, s, cos θ)describe the conformations of the wormlike chain con-tained below or above the dividing z0 plane, respectively,without intersecting it. These propagators can be ob-tained from eq 9 (in practice, from the equivalent form,eq 15a,b) after substituting the converged solution ofthe SCF scheme, W(z), in the two regions 0 < z < z0and z0 < z < Lz. The initial and boundary conditionsfor the fist region are q(z, 0, cos θ) ) 1, where 0 < z <z0 and q(0, s, cos θ) ) 0, q(z0, s, cos θ) ) 0, while in thesecond region q(z, 0, cos θ) ) 1, where z0 < z < Lz andq(z0, s, cos θ) ) 0, q(Lz, s, cos θ) ) 1. After the pcross isevaluated, the number of chains, n(z0), per unit surfacethat intersect the plane at z0 is obtained as

Following the previous discussion, it can be seen thatthis quantity is expressed only through invariant pa-

rameters establishing correspondence between the me-soscopic and the atomistic representations of the studiedsystem. The term in the brackets in eq 18 equals thetotal number of chains, per unit surface, present in aninterfacial PE melt/graphite system of size Lz along thez direction. To facilitate the interpretation of thesubsequent results, it is mentioned that low values ofn(z) denote a “starved” plane which is intersected onlyby a few polymer molecules, being indicative of a chaintendency to become parallel with the adsorbing sub-strate (i.e., assume “flat” conformations). On the otherhand, high values of n(z) indicate that many chainsintersect the considered plane, a situation favored whenthe polymer molecules assume an elongated shape alongthe z-axis of the system.55

The behavior of n(z) as a function of distance fromthe adsorbing substrate as obtained from SCF calcula-tions with the wormlike chain model and from atomisticsimulations for the C78, C250, and C400 melts is shownin Figure 4. Taking into account the above discussion,the low values of the n(z) parameter observed in thegraphs of Figure 4 in the vicinity of the substrate areindicative of the prevalence of flattened chain conforma-tions. With increasing distance from the adsorbingsubstrate, the conformations of the polymer chainsbecome less flat and the bulk behavior is recovered

approximately at distances equal to 1.5xRg2, with Rg

2

being the mean-squared radius of gyration of thepolymer. In particular, n(z) assumes a constant valuecharacteristic of the bulk polymer. This bulk valuedecreases with increasing chain length N as 1/xN.This power law dependence is typical of polymer chainsin a bulk melt in the long chain limit.

Figure 4 shows that, in the case of the C78 system,the SCF theory and the atomistic simulation exhibit aperceptible difference in the global conformationalproperties of the polymer chains, the molecules of theatomistic system assuming flatter conformations. On

Figure 4. Chain shape profiles as derived by the SCF theoryafter the implementation of the wormlike chain connectivitymodel for the C78 (open circles), C250 (open squares), and C400(open triangles) melts. The solid symbols display the corre-sponding atomistic simulation results, while the dashed lineshows the predictions of the Gaussian model for the case ofthe C250 system.

pcross(z0) ) 1 -

∫0

z0∫-1

1q1(z, 1, cos θ) d cos θ dz + ∫z0

Lz∫-1

1q2(z, 1, cos θ) d cos θ dz

∫0

Lz∫-1

1q(z, 1, cos θ) d cos θ dz

(17)

n(z0) ) pcross(z0)[nbulk

V ∫0

Lzφ(z) dz] (18)


the contrary, for the larger molecular weight systemsthe agreement between the two approaches improvessignificantly, so that the curves follow each other closely.This qualitative difference between low and highermolecular weights, which will be reinforced by theresults presented subsequently, shows that the simpli-fications invoked in the formulation of the coarse-grained model are more crucial in the case of shortmolecules: The finite set of their conformations is toosensitive to the details of the force field to be fullydescribed by the wormlike chain model. In comparisonto the semiflexible model, the Gaussian model fails ina broader region of molecular weights. An example ofthe predictions of the Gaussian model for a C250 systemis also presented in Figure 4. It can be observed thatGaussian molecules are less flat, so that the n(z) profileremains always “higher” than both the simulation andthe wormlike chain calculation results. Interestingly,the Gaussian model fails to reproduce even the bulklimit of n(z) for the relatively low molecular lengthsreported here. In particular, it predicts that more chainsper unit surface should be crossing a plane placed inthe bulk than is actually observed in the atomisticsimulations and in the calculations with the wormlikechains. This discrepancy concerning the bulk behaviordiminishes with increasing molecular length: for ex-ample, in the case of a C1000 melt system the differencebetween wormlike and Gaussian SCF limiting valuesof n(z) in the bulk falls to 8%, compared to a 16%difference in the C250 melt.

Structure of Adsorbed Polymer Layer. One of theissues commonly addressed in a theoretical study ofpolymer adsorption is the characterization of conforma-tions of adsorbed molecules in terms of tails, loops, andtrains. More specifically, a molecule is considered asadsorbed when it has at least one adsorbed segment.In the current SCF calculations, a segment is definedas adsorbed when found within the attractive well ofthe polymer/solid interaction, while in the atomisticsimulations any C-C bond with its middle at a distanceless than 6 Å from the adsorbing graphite plane (thatis in the first adsorption peak) is considered as adsorbed.In this way, practically, both coarse-grained and atom-istic representations employ the same characteristiclength for identifying adsorbed segments (see alsoFigures 1 and 2). After clarifying the concept of “ad-sorbed” segments, the definition of trains, tails, andloops follows straightforwardly along the lines presentedin many previous publications (in refs 56 and 57, forinstance).

The evaluation of the total volume fraction profile,φads(z), of the segments belonging to adsorbed moleculesas well as of the separate contributions of tails and loopscan be carried out after decomposing42 the wormlike orthe Gaussian chain propagator, q(z, s, cos θ) or q(z, s),in two parts, qads and qfree, describing the adsorbed andfree states, respectively, of the polymer molecule. Theqfree propagator can be calculated from eq 9 (in practice,from the equivalent set of eqs 15a,b) or from eq 10,depending on the chain connectivity model, with properinitial and boundary conditions: qfree(z, 0, cos θ) ) 1,where w < z < Lz and qfree(w, s, cos θ) ) 0, qfree(Lz, s,cos θ) ) 1. (The reader is reminded that w is the widthof the attractive well.) In this way the qads part is easilyobtained after subtracting the qfree propagator from thetotal one: qads ) q - qfree. The volume fraction profileof the free, φfree(z), and the adsorbed polymer, φads(z),

as well as the contributions of tails, φtails(z), and loops,φloops(z), are obtained42 as

The results of these calculations regarding the volumefractions φfree(z) and φads(z) for the C78, C250, and C400systems are shown in Figure 5 with the correspondingatomistic simulation data. In the same figures, thepredictions derived with the Gaussian connectivitymodel are also reproduced. Obviously, for the smallestmolecular length a significant part of the atomisticφads(z) profile is dominated by the characteristic oscil-lations originating from monomer packing close to thesubstrate. It can be appreciated that the results con-cerning the global conformational properties of theadsorbed molecules will be subject to the small lengthscale details of the molecular model, so the discrepancybetween the predictions of the wormlike chain modeland the atomistic simulation is quite natural. As oneproceeds to systems of longer chains, the agreementbetween atomistic simulation and wormlike chain SCFanalysis improves considerably, and in the case of theC400 melt the two sets of results are very close to eachother. Regarding the Gaussian chain connectivity model,the general conclusion from the graphs of Figure 5 isthat the results obtained for low molecular weightsexhibit a larger divergence from the atomistic simula-tion data than those of the wormlike chain model.Taking into account the comparison for the longest C400melt, however, it can be appreciated that the predictivepower of the Gaussian model also improves with in-creasing chain length.

Figure 6 shows the φloop(z) and φtails(z) contributionsto the total φads(z) profile as obtained from atomisticsimulations and SCF calculations with both chainconnectivity models for the C78, C250, and C400 melts. Itis observed that, according to the atomistic simulations,for small chain lengths the total adsorption profileshould be dominated by segments belonging to the freetails of adsorbed chains, even close to the substrate.Only in the case of the C400 system is φloop(z) nearlyequal to φtails(z) near the first adsorption layer. Forsystems with longer chains, the loop contribution toφads(z) is expected to prevail over the tail component inthe vicinity of the substrate until a characteristicdistance, usually denoted42 as z*, is reached. Consider-ing the SCF calculations, the implementation of thewormlike chain model leads to a much better agreementwith the simulation data in comparison with the Gauss-ian model. In particular for the longest melt, i.e., theC400, the wormlike chains reproduce the atomistic φloop-(z) and φtail(z) profiles very closely. On the contrary, theimplementation of the Gaussian model is characterized

φfree(z) ) 12∫-1

1 ∫0

1qfree(z, s, cos θ) qfree(z, 1 - s,

-cos θ) ds d cos θ (19a)

φads(z) ) 12∫-1

1 ∫0

1qads(z, s, cos θ) qads(z, 1 - s,

-cos θ) ds d cos θ (19b)

φloop(z) ) 12∫-1

1 ∫0

1qads(z, s, cos θ) qads(z, 1 - s,

-cos θ) ds d cos θ, z > w (19c)

φtails(z) ) ∫-1

1 ∫0

1qfree(z, s, cos θ) qads(z, 1 - s,

- cos θ) ds d cos θ, z > w (19d)


Figure 5. Volume fraction profiles of segments belonging toadsorbed, φads(z), and free, φfree(z), chains as derived by the SCFtheory after the implementation of the wormlike (solid line)and the Gaussian (dashed line) chain connectivity models forthe C78, C250, and C400 melts. The solid symbols display thecorresponding atomistic simulation results.

Figure 6. Volume fraction profiles of segments belonging totails, φtail(z), and loops, φloop(z), as derived by the SCF theoryafter the implementation of the wormlike (solid circles andopen circles) and the Gaussian (solid line and dashed line)chain connectivity models for the C78, C250, and C400 melts. Thesolid and open squares show the corresponding atomisticsimulation results for the tails and loops, respectively. Thevertical, thin dashed line marks the boundary of the regionwhere the polymer is subjected to the effects of the attractivesquare well potential in the coarse-grained description.


by a noticeable overestimation of loop contribution closeto the substrate, predicting a deep depletion of tails closeto it. Even in the C400 system, although the Gaussianmodel grasps the functional form and the correct lengthscales of the φtail(z) profile in the outer region of theadsorbed layer, it rather fails in the inner part of it.This can be understood if one considers that for theformation of loops it is critical for the polymer moleculeto “bend” in order to revisit the immediate vicinity ofthe surface. Since there is no bending penalty in theGaussian model and the bonds are infinitely extensible,it is much easier for the Gaussian coarse-grainedmolecules to return to the substrate compared to theatomistic polymer chains. As a consequence, φloop(z) isconsiderably overestimated for the relatively shortmolecular lengths reported in Figure 6. As one proceedsto adsorbed PE melts with molecular lengths higher

than C1000, differences between the predictions of theGaussian and the wormlike chains concerning thestructure of the layer formed by the adsorbed polymerchains become less significant. To illustrate this, theresults of SCF calculations for the φloop(z) and φtail(z)profiles derived by both models are presented in Figure7 for the case of an adsorbed C10000 melt. It can be seenthat, in this limit, the predictions of the two models arenearly identical although, close to the surface, differ-ences are still perceptible.

The φads(z), φloop(z), and φtail(z) profiles of Figures 5and 6 can be utilized to derive the average fraction ofsegments νtails, νloops, and νtrains of an adsorbed chaincontained in tails, loops, and trains, respectively. Theprofiles are integrated over the z coordinate to obtainthe volumes per unit surface of (a) adsorbed polymer,Γ, (b) adsorbed polymer contained in loops, Γloops, and(c) adsorbed polymer contained in tails, Γtails. Then, thefractions νtails, νloops, and νtrains can be derived58 as νtails) Γtails/Γ, νloops ) Γloops/Γ, and νtrains ) (Γ - Γtails - Γloops)/Γ. The results for various chain lengths, N, as deter-mined from both the wormlike and the Gaussian chainconnectivity models, are presented in Figure 8 as log/linear plots. In the same graph, the atomistic simulationresults for the C78, C250, and C400 melts are also shown.It can be appreciated that, for all the three fractions,the SCF results obtained through the wormlike chainmodel are much closer to the atomistic simulationresults than the predictions from the Gaussian chainconnectivity representation. In the case of wormlikechains the νtails fraction, for large chain lengths, tendsasymptotically to a fixed value estimated to be around0.67.

Nearly the same limiting value for νtails was obtainedby Scheutjens and Fleer,56 who incorporated the discus-sion of adsorbed melts in their lattice SCF study ofpolymer adsorption from solution, as a limiting case ofzero solvent concentration. In this limit they establisheda direct correspondence between the findings of thelattice SCF theory concerning the conformations ofadsorbed polymer chains with the predictions of thetheory of Roe59 referring to isolated adsorbed molecules,in the special case when the energy of monomer-solid

Figure 7. Volume fraction profiles of segments belonging totails, φtails(z), and loops, φloops(z), as derived by the SCF theoryafter the implementation of the wormlike (solid circles andopen circles) and the Gaussian (solid line and dashed line)chain connectivity models for the case of a C10000 melt.

Figure 8. Average fraction of segments belonging to adsorbed chains found in tails, νtail, loops, νloop (left), and trains, νtrain(right), reported as a function of chain length. On the left graph, open circles and squares display the SCF predictions derivedfrom the wormlike chain model for tails and loops, respectively, while filled circles and squares display the SCF predictions fromthe Gaussian connectivity model. Also on the right graph, open and filled circles display the predictions of the wormlike andGaussian chain models, respectively, for the fraction of trains. Solid triangles, on both graphs, show the atomistic simulationresults.


substrate interaction equals the critical adsorptionenergy. According to the conclusions of Roe,59 ≈70% ofsegments of the adsorbed chain should be distributedbetween two long tails. The rest, located in the middleof the chain, should form loops and trains. The averagelength of loops should scale as N1/2, while the trainsshould be of some small finite length being independentof N.

In the framework of the conformational properties ofthe adsorbed molecules presented above, it is expectedthat for large N, apart from νtails, νloops will alsoasymptotically reach some fixed value. This can beappreciated after considering that in an adsorbed meltthe polymer chains are expected, effectively, to followreflected random walk statistics60 (despite the fact thatthe microscopic boundary condition for the chain propa-gator is the absorbing one), so the number of a chain’scontact points with the substrate will scale as N1/2. Sincethe loops are located between the contact points, theirtotal number per chain should also scale as N1/2.Considering the dependence of loop average length onN given by Roe, one concludes that the total number ofsegments located in loops should depend linearly on N.After dividing by total chain length, the asymptoticscaling dependence of νloops on N is obtained as νloops∼N0, i.e., constant.

In the case of the wormlike chains it can be observedfrom Figure 8 that, while νtails reaches “saturation” quitefast (for melts longer than the C1000 it is practicallyconstant), νloops still changes considerably over thestudied range of molecular lengths. The atomisticsimulation results seem to confirm this trend, althoughit was, of course, impossible to reach the molecularlengths considered in the SCF calculations. The Gauss-ian model, contrary to the semiflexible chains, consider-ably overestimates the fraction of νloops for the lowmolecular weight melts, while at the same time itunderestimates the tail contribution. In addition, νloopsin the Gaussian SCF model reaches the “saturationplateau” too fast: it is practically constant for meltslonger than C1000. As a general conclusion, it can be saidthat the discrepancies between the predictions of theGaussian model concerning the conformational proper-ties of the adsorbed chains when compared to thewormlike chain model results and the atomistic simula-tion data are more significant for loops than for tails.This is also supported by the graphs of Figure 6 showingthe φloop(z) and φtail(z) profiles. Also, the agreementbetween the two coarse-grained models is better con-cerning the νtrain fraction; it can be observed that, beyondthe C250 melt, their predictions for this fraction are closeto each other and to the atomistic simulation results.

The data on the tail fraction, νtails, can be furtherutilized for the calculation of the average tail length,after the number of tails per chain is calculated. Toevaluate the number of tails per chain, it is convenientto calculate an auxiliary propagator, q(z, s, cos θ), whichis defined from the solution of eq 9 (in practice, of itsequivalent form eq 15a,b) after substituting the solutionof the SCF scheme, W(z), with initial conditions q(z, 0,cos θ) ) 0, 0 < z e w, and q(z, 0, cos θ) ) 1, w < z < Lz,while the boundary conditions remain the same as forthe q(z, s, cos θ) propagator. This auxiliary propagatorwill be proportional to the number of chain conforma-tions that have one nonadsorbed (i.e., “free”) end, whilethere is no restriction on the location of the second one.It is then possible to calculate the probabilities p0, p1,

and p2 that the adsorbed chain will have no tails, onlyone tail, and two tails, respectively. These are found as

After this, the average number of tails per chain, ntail,is trivially evaluated as ntail ) p1 + 2p2 so that the chainfraction per tail is given as νtails/ntail. This is aninvariant quantity, and the average tail length of anatomistic chain is calculated as Nνtails/ntail.

The predictions concerning the average tail lengthobtained from the above calculation for the case of thewormlike and the Gaussian chain connectivity modelsare shown in Figure 9. On the same figure, the atomisticsimulation results for the C78, C156, C250, and C400 meltsare also reproduced. It can again be observed that thepredictions for the average tail length derived throughthe wormlike chain model are closer to the atomisticsimulation results. Nevertheless, the initial divergenceobserved between the predictions of the two coarse-grained models diminishes for melts consisting of longerchains (not shown on the graph). For example, in thecase of the C10000 melt the predictions of the Gaussianand the wormlike chain models for the average taillengths are 3385 and 3444, respectively.

7. Summary and ConclusionsIn this work, the problem of determining the equi-

librium properties of a polymer melt of specific chemicalconstitution, adsorbed on a certain solid substratethrough a continuum SCF approach, was considered.

Figure 9. Average tail length reported as a function of chainlength. The open and the solid circles display the SCF theorypredictions derived from the wormlike and the Gaussian chainconnectivity models, respectively. The solid triangles show thecorresponding atomistic simulation results.

p1 ) 2∫0

w∫-1

1q(z, 1, cos θ) d cos θ dz

∫0

Lz∫-1

1qads(z, 1, cos θ)d cos θ dz

p2 )

∫-1

1∫w

Lz[q(z, 1, cos θ) - qfree(z, 1, cos θ)] dz d cos θ

∫0

Lz∫-1

1qads(z, 1, cos θ) d cos θ dz

p0 ) 1 - p1 - p2 (20)


Monomer-monomer interactions were introduced byconsidering the polymer melt as compressible withdensity variations being driven by a harmonic penal-izing potential suggested by Helfand and co-workers.23-25

This approach is quite simplified and does not accountfor local monomer packing effects, which can be par-ticularly important to some aspects of structure in theclose vicinity of the adsorbing substrate, such as thespatial variations of the local density. In this directiona possible future improvement could come from “bor-rowing” the interaction potential from a density func-tional theory and subsequently incorporating it into theSCF formalism.4,31-33 The interaction potential betweenthe polymer and the substrate could have been de-scribed by preserving the original functional form of theatomistic polymer/substrate interaction, after tuning theparameters so as to reproduce the total energy ofadsorption per unit surface of the atomistic polymer/solid interfacial system. Instead, in the current workthe simple functional form of square well was employed.The parameters of the latter (width and depth) weretuned to reproduce the total energy of adsorption andthe characteristic length scale of the density variationsof the atomistic system. This proved adequate for ourcase; however, care should be taken when addressingsystems with long-range polymer/substrate interactions,where the square-well approximation may not be ap-plicable.

The chain connectivity was represented through thewormlike chain model, which has the advantage ofreproducing on the coarse grained level both the mean-squared end-to-end distance and the contour length ofthe atomistic polymer molecule. Although this modelhas been used in several studies of interfacial polymericsystems,9,10,37 to the best of our knowledge, this is thefirst time it is implemented in the framework of a SCFtreatment of a polymer adsorption problem. The Gauss-ian chain connectivity model reproducing the mean-squared end-to-end distance in the bulk was alsoconsidered in the SCF analysis, and results from it werecompared to the predictions of the wormlike chainmodel.

After the specification of the coarse-grained repre-sentation, the SCF formalism was cast in terms ofseveral parameters, the invariants of the mesoscopicrepresentation, establishing correspondence betweenthe atomistic polymeric system and its coarse-grainedcounterpart. The formulation is general. It has beenimplemented for the specific case of linear polyethylenemelts adsorbed on graphite, since the availability ofrecent atomistic simulation data13,14 offered a means ofevaluating the SCF theoretical approach. The solutionof the SCF formalism was performed through an itera-tive scheme, requiring at each step the determinationof the conformations of the wormlike chain in the meanfield. These conformations were described through adifferential equation formulated8 by Saito et al., whichconstitutes the analogue of the Edwards equation forwormlike chains. The numerical solution of this formu-lation was achieved through a combined spectral andreal-space approach.37 Instabilities close to the adsorb-ing boundary required the development of a new Lax-Wendroff-type numerical solution method.

During the SCF study various monodisperse meltswere considered, ranging from C78 to C10000, withparticular emphasis on molecular weights allowing adirect comparison with the atomistic simulation data.

After solving the SCF equation set, it was possible tocharacterize the chain conformations through a varietyof descriptors. Both the wormlike and the Gaussianchain SCF approaches as well as the atomistic simula-tions showed that the polymeric chains become signifi-cantly “flattened” near the boundary. Generally, per-turbations in their conformational properties due to thesurface persist over a characteristic length which isroughly 1.5 times the mean gyration radius of theunperturbed polymer molecule in the bulk. It wasobserved that, even for the relatively short chain meltsaddressed in the atomistic simulations (up to C400), SCFpredictions using the wormlike chain model concerningthe degree of flattening of the polymeric molecules aremuch closer to simulation results that those from theGaussian model. In particular, starting from the C250melt, the wormlike SCF model predictions capture thesimulation results quite well, contrary to the Gaussianchain model, which underestimates the degree of chainflattening even for the longest C400 system. It is clarified,however, that in the case of the shortest chain melts(i.e., C78) the results are too much dominated by theatomistic details of the molecular model to be ap-proximated reliably by any of the two coarse-grainedrepresentations that were implemented.

The conformations of adsorbed molecules were furthercharacterized by considering the properties of tails,loops, and trains, as derived by SCF analysis from thetwo coarse-grained models and from the atomisticsimulations. It was observed that the discrepanciesbetween the predictions of the Gaussian chain modeland the atomistic simulation data remain significant inthe entire range of simulated molecular lengths (i.e., upto C400). The failure of the Gaussian model in this regionis particularly evident in the case of loops, whosecontribution to the adsorbed polymer layer it consider-ably overestimates. On the other hand, as regards theproperties of the outer region of the adsorbed polymerlayer formed by the dangling tails of the adsorbedmolecules, SCF predictions from the Gaussian modelimprove fast with increasing molecular weight. At thesame time, SCF predictions from the wormlike chainmodel are much closer to the atomistic simulationresults; in the case of the longest C400 system, agreementbetween wormlike SCF and atomistic simulation pre-dictions is quantitatively very good concerning both tailsand loops. The superior performance of the semiflexiblechain model when compared to the Gaussian one, in allcases considered, can be explained after taking intoaccount that the former model is characterized by anadditional length scale on the level of chain persistencelength. The incorporation of this intermediate lengthscale brings the wormlike chain closer to the atomisticmolecule, improving its predictive capabilities concern-ing the structure of loops. With increasing molecularlength, the effect of the local chain structure (i.e., ofsmall length scale properties) on global chain confor-mational behavior decreases. Thus, for melts consistingof long polymer chains (C10000, for instance), the differ-ence between the chain level structures predicted by thetwo coarse-grained models becomes insignificant.

Acknowledgment. We are grateful to the GeneralSecretariat of Research and Technology of Greece forfinancial support through a PENED program, No.01Ε∆529.


Appendix. Numerical Solution of the WormlikeChain Propagator Equation

As was mentioned in the main text, the principalnumerical difficulty encountered in the current workwas the solution of eqs 15a,b obtained after expressingthe propagator q(z, s, cos θ) in terms of Legendrepolynomials (eq 14).

After the truncation of the Legendre polynomialseries, the set of differential equations (eqs 15a,b)becomes finite and can be solved through an explicitLax-Wendroff-type technique.53 The suggested ap-proach following the standard Lax-Wendroff methodfor hyperbolic equations employs a discretization ofspace and time domains through some regular grid.Next, it relies on the observation53 that, when thepartial derivatives with respect to s and z in eq 15a areapproximated through first-order forward and second-order central finite difference schemes, respectively, theresulting equation will not correspond to the originaleq 15a but to the equation

where ltrunc denotes the moment after which theLegendre polynomials series is truncated (i.e., in ourcase ltrunc ) 7). Thus, in the finite difference analogueof eqs 15a,b, the term (2η/N)(∆s/2)(∂2ql/∂s2) should beadded to the right-hand side of the equation to cancelout the -(2η/N)(∆s/2)(∂2ql/∂s2) component. For its rep-resentation, the second-order derivative (∂2ql/∂s2) isexpressed from eq 15a as

where, for convenience, the operators Dab have beenintroduced to denote the second-order partial derivativesof the function ql with respect to the variables a and b.The functions Dszql-1 and Dszql+1 in eq A.2 can besubstituted with their equivalent form obtained afterdifferentiating the right-hand side of eq 15a with respectto z as

where it is clarified that ql ) 0 for l ∉ [0,ltrunc]. Afterthe above manipulations, the finite-difference schemeis formulated by adding to eq 15a the term (2η/N)(∆s/2)(∂2ql/∂s2) being expressed only through the Ds, Dz, andthe Dzz differential operators. Next, in this “trans-formed” eq 15a an approximation to the Ds, Dz, and Dzzderivatives is performed through first-order forward,second-order central, and second-order central second

differences, respectively, and an algebraic expression forql(s + ∆s, z) is obtained. The repeated application of thisexpression delivers the desired time-marching solutionql(s,z), l ) 0, ..., ltrunc, of the propagator equation (eqs15a,b).

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2ηN

∂ql

∂s) -[ l

2l - 1∂ql-1

∂z+ l + 1

2l + 3∂ql+1

∂z ] - (l + 1)lql -

2ηN

W(z)ql - 2ηN

∆s2

∂2ql

∂s2, l ) 0, ..., ltrunc (A1)

2ηN

Dssql ) -[ l2l - 1

Dszql-1 + l + 12l + 3

Dszql+1] -

(l + 1)lDsql - 2ηN

W(z)Dsql, l ) 0, ..., ltrunc (A2)

2ηN

Dszql-1 ) -[ l - 12l - 3

Dzzql-2 + l2l + 1

Dzzql] -

(l - 1)lDzql-1 - 2ηN

Dz[W(z)ql-1]

2ηN

Dszql+1 ) -[ l + 12l + 1

Dzzql + l + 22l + 5

Dzzql+2] -

(l + 1)(l + 2)Dzql+1 - 2ηN

Dz[W(z)ql+1] (A3)


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