Surface Segregation in Polydisperse Polymer Melts

Surface Segregation in Polydisperse Polymer Melts

J. van der Gucht,* N. A. M. Besseling, and G. J. Fleer

Laboratory of Physical Chemistry and Colloid Science, Dutch Polymer Institute/WageningenUniversity, Dreijenplein 6, 6703 HB Wageningen, The Netherlands

Received June 4, 2001; Revised Manuscript Received January 23, 2002

ABSTRACT: The effects of polydispersity on the behavior of a polymer melt near a surface areinvestigated. The combined effects of the presence of a surface on the configurational freedom (entropy)of flexible chains and of the constraint that the total density is constant up to the surface yield a strikinglysimple and general relation between the surface excess θN

ex and the bulk volume fraction φNb of a

component of chain length N in a polydisperse polymer melt: θNex/φN

b ) A(1 - N/Nw), where A ≈ 0.195 isa numerical prefactor and Nw the weight-averaged chain length. This relation applies for any molecularweight distribution. Three different methods were used to obtain this law: (i) the numerical self-consistentfield model of Scheutjens and Fleer, which is exact within a mean-field assumption, (ii) an analyticalapproximation of this model, which gives additional physical insight, and (iii) the analogy between thesurface region of a polymer melt and an ideal polymer solution close to the adsorption/depletion transition.

1. Introduction

The behavior of dense polymer systems near surfaceshas been studied in detail before, both theoretically andexperimentally. The interest in such systems derivesfrom their relevance in practical applications. Forexample, many composite materials are based on poly-mers with filler particles dispersed in them. The me-chanical properties of such composites are determinedby the interplay between the particle interfaces and thepolymer chains. A fundamental understanding of thefactors determining the interfacial structure, such asthe chain architecture, the chain length distribution,and the bulk composition, allows in principle for thecontrolled modification of surfaces. In this paper weinvestigate the influence of polydispersity.

The presence of a surface reduces the conformationalentropy of a polymer coil. This entropy loss correspondsto a change in the interfacial free energy, which isimportant for the adhesion properties of a polymer melt.The theoretical description of the surface properties ofdense polymer systems has focused mainly on mono-disperse polymer systems.1,2 Practical polymer systems,however, are always polydisperse. In such polydispersesystems, the entropy loss suffered by the long chains islarger than that by the short chains. Therefore, in amixture of chains of high and low molecular weight withequal chemical structure, one expects the surface regionto be enriched with the shorter species. Apart fromdifferences in chain length, also differences in interac-tion energy with the surface,3-5 differences in chainstiffness,6,7 or different chain architectures8 may drivesurface segregation.

A variety of theoretical methods have been used todescribe polymers near surfaces. Density functionalapproaches, using a square gradient approximation,have proved to give a good description of the surfacesegregation in blends of polymers of different chemicalcomposition.3,4,6 However, the correct incorporation ofthe chain connectivity in such models is not straight-forward. Scheutjens and Fleer9-11 developed a self-consistent-field lattice theory, which has been very

successful in describing polymer adsorption and deple-tion. Their model uses a mean-field approximation inwhich correlations between the occupancy of sites withinthe same layer are neglected. The extension of thismodel to polymer melts and multicomponent systems(including block copolymers) is straightforward.2,12

Hariharan et al.13 used the Scheutjens-Fleer modelto study surface segregation in a bimodal polymer melt.The profiles of both components were calculated, andimplications for the surface tension were discussed. Inthis work we do not restrict ourselves to bimodalmixtures, but we study polymer systems with anarbitrary chain length distribution.

It is known that polymer chains in a melt areeffectively ideal, because excluded-volume interactionsare screened.14,15 Therefore, the analysis for a polymermelt near a surface is completely analogous to that foran ideal polymer solution near a surface. In a recentpaper16 we used a lattice model to investigate a solutionof ideal polymer chains at a surface close to theadsorption/depletion transition. The effect of the chainlength on the adsorption/depletion transition and on thesurface excess at a certain adsorption strength wasinvestigated. The numerical results were compared toa continuum model, and a good agreement was found.Furthermore, the effect of preferentially adsorbing endsegments was investigated.

In this paper we use the Scheutjens-Fleer theory todescribe a polydisperse polymer melt near a surface. Wewill show that the description of such a melt is almostidentical to that of an ideal polymer solution at a surfaceclose to the adsorption/depletion transition. It will beshown that there is a very simple expression that relatesthe surface excess of a certain component to its chainlength and bulk concentration for any chain lengthdistribution. The numerical results are compared to asimplified lattice model, for which we can obtain ana-lytical results.

2. Numerical Lattice Theory

A full description of the theory can be found elsewherein the literature.9-11,16 Only the main features areoutlined here. In the Scheutjens-Fleer theory the* Corresponding author: e-mail [email protected].

6732 Macromolecules 2002, 35, 6732-6738

10.1021/ma010964q CCC: $22.00 © 2002 American Chemical SocietyPublished on Web 07/18/2002

polymer chains are described as walks upon a lattice.The space adjoining a solid surface is divided into anumber of parallel lattice layers, numbered z ) 1, 2, ...,M (see Figure 1). The number of sites in each layer isL. The surface area, and hence L, are taken infinitelylarge, so that edge effects in the parallel direction canbe ignored. A reflecting boundary condition is imposedat z ) M in this study, where M is taken large enoughso that near M bulk properties prevail, to avoid bound-ary effects. The surface layer is situated at z ) 0.Segments of the chains are restricted to layers z G 1.This can be considered as a model for a melt at an inertsolid surface but is also a very reasonable model for thepolymer/air interface of a polymer melt or for theinterface between a polymer phase and a coexistingphase of a (very) poor solvent. In principle, it would bepossible to allow for a gradient of air (“holes” ) or thepoor solvent in the model, but we keep the system assimple as possible. The lattice properties are definedthrough a lattice constant λ, which is the fraction ofneighboring sites that a lattice site has in an adjacentlayer. The fraction of neighbors within the same layeris equal to 1 - 2λ. In this paper we use a cubic lattice,with lattice coordination number Z ) 6, and λ ) 1/6. Insuch a cubic lattice the lattice spacing is equal to thesegmental length.

We consider an athermal mixture of polymer mol-ecules of different chain lengths N. The system isincompressible, so that all lattice sites are occupied bypolymer segments. This implies that the followingvolume filling constraint must be satisfied:

for any z. Here φN(z) is the volume fraction of segmentsof chains of length N in layer z. The interfacial regionis in equilibrium with a bulk system of polymers. Foreach component we define the surface excess as

where φNb is the bulk volume fraction of chains of

length N. Obviously, it follows from constraint 1 thatthe sum of θN

ex over all N must be equal to zero.The segmental weighting factor G(z) gives the relative

preference of a detached polymer segment to be in layerz rather than in the bulk:

where u(z) is the local potential of a segment, expressedin units kT. In the present system all segments areidentical, so that we do not need an index for G or u todenote the segment type. The field u(z) contains twocontributions:

The first contribution u′(z), which is independent of thesegment type even when the segments are different,accounts for the volume-filling constraint, eq 1. It isdetermined self-consistently from this constraint. In thederivation it arises as a Lagrange multiplier.9 Thesecond contribution accounts for interactions with thesurface (adsorption energy); the Kronecker delta δz-1is unity for z ) 1 and zero elsewhere. This implies ashort-ranged interaction between segments and surface(“square well” ). In the present case, there is no mixingenergy, since there is only one type of segment and nosolvent.

The displacement of a polymer segment in the surfacelayer by an other segment does not change the energyof the system, so that the adsorption energy gives aconstant contribution LuakT for the surface layer,regardless of the distribution of chain conformations.Hence, the only nontrivial contribution to u(z) is thevolume filling potential u′(z).

The entropy is accounted for by assigning an end-point distribution function G(z,s) to walks starting fromsegment 1 in an arbitrary layer in the lattice and endingat a segment s in layer z. The end-point distributionfunctions can be calculated from the segmental weight-ing factors G(z) through the recurrence relation

with the initial condition G(z, 1) ) G(z). The distributionfunction of a segment s in a chain of N segments can befound by considering that segment s joins the two chainparts 1, 2, . . ., s and s, s + 1, . . ., N. If segment s is inlayer z, the former chain part has a statistical weightG(z, s) and the latter G(z, N - s + 1). The statisticalweight of all chains of length N with segment s in layerz is therefore G(z, s)G(z, N - s + 1)/G(z), where thedivision by G(z) corrects for double counting of theweighting factor of segment s. The volume fraction ofsegments s of chains of length N in layer z is propor-tional to this statistical weight. The total volumefraction of chains of length N in layer z is found bysummation over all ranking numbers s:

The factor φNb /N is a normalization factor, ensuring

φN(∞) ) φNb . Equations 5 and 6 with the M boundary

conditions 1 constitute a set of M implicit equationswhich can be solved numerically, giving the concentra-tion profiles φN(z) of all polymer chains.

The entropy S of a polymer mixture near a surface,expressed in the segment concentrations, is given by9

Figure 1. A two-dimensional schematic representation of athree-dimensional cubic lattice. Polymer segments are re-stricted to the layers z G 1.

∑N

φN(z) ) 1 (1)

θNex ) ∑

z)1

M

(φN(z) - φNb ) (2)

G(z) ) e-u(z) (3)

u(z) ) u′(z) + uaδz-1 (4)

G(z, s + 1) ) G(z)λG(z - 1, s) +(1 - 2λ)G(z, s) + λG(z + 1, s) (5)

φN(z) )φN

b

NG(z)∑s)1

N

G(z, s) G(z, N - s + 1) (6)

Macromolecules, Vol. 35, No. 17, 2002 Polydisperse Polymer Melts 6733

Here S* is the entropy of the reference state, for whichwe choose the pure, unmixed components with the samenumber of molecules. Since there is no mixing energy,eq 7 gives the free energy of the system, apart from theconstant contribution of the adsorption energy. Subtrac-tion of the chemical potential terms gives an expressionfor the surface tension σ of a polydisperse polymer melt:9

where a is the area per lattice site.For (incompressible) monomers (N ) 1) φ1(z) ) 1 for

all z > 0, θex ) 0, and u′(z) ) 0 for z > 1, so that eq 8reduces to σma/kT ) -u′(1) ) ua, where σm is the surfacetension of pure monomer. The last equality derives fromeq 4 with u(1) ) 0. Hence, for monomers the surfacetension is fully determined by the adsorption energy.

For a homodisperse melt of chains, again θex ) 0 andσa/kT ) -∑zu′(z), where now u′(z) is nonzero for z > 1because of conformational contributions. When the meltis polydisperse, the Gibbs excess term θN

ex/N contrib-utes as well. With eq 4, eq 8 may now also be writtenas

3. Numerical ResultsThe values of u(z) in all M layers are determined

numerically from the M self-consistent equations. Thesevalues do not depend on the value of the adsorptionenergy ua as they are only determined by the volume-filling constraint. The only effect of varying ua is thatu′(1) is changed according to u′(1) ) u(1) - ua, whereu(1) is independent of ua. The resulting potential fieldfor a mixture of three different chain lengths with equalbulk volume fractions is shown in Figure 2. Thepotential oscillates, because of the impenetrability of thesurface and the volume-filling requirement,9 and is veryshort-ranged. The range is independent of the chainlength. For a melt consisting of infinitely long chains,the sum of u(z) over all layers has a limiting value -B.For a cubic lattice B ≈ 0.1842. For finite chain lengthsthe results are described very accurately by the equation

where Nw ) ∑N NφNb is the weight-averaged chain

length. Note that eq 10 is independent of the precisechain-length distribution; only the weight-average Nwenters.

The volume fraction profiles of the components in themixture of Figure 2 are shown in Figure 3. It is clearthat the longest chains are depleted from the surface,whereas there is a net adsorption of the shortest chains.The profile of the chains of intermediate length isnonmonotonic. There is a maximum at an intermediatedistance from the surface, whereas close to the surfacethe concentration is less than its bulk value. This

nonmonotonic behavior is due to the volume-fillingconstraint: the chains of intermediate length fill up theholes left by the short and long chains, which haveintrinsically different decay lengths for the profiles. Itmust be noted that the surface segregation due todifferences in molecular weight is small. The enrich-ment or depletion near the surface is, for these chainlengths, only a few percent of the bulk concentration.The surface segregation becomes stronger if the differ-ences in chain length are larger.

The surface excess can be calculated with eq 2. Figure4 shows the surface excess relative to the bulk volumefraction as a function of the relative chain length N/Nwfor several different chain length distributions, asindicated in the legend. It can be seen from this figurethat the relation between θN

ex/φNb and N is linear, for

any distribution. Furthermore, the intercept with thevertical axis is the same for all distributions. Also, fordistributions with the same weight-averaged chainlength Nw but a different number-averaged chain lengthNn, the relation between θN

ex and N is identical. Fromthe condition that the sum over all θN

ex is zero, it followsimmediately that the line must cross the horizontal axisat N ) Nw. The results in Figure 4 can thus besummarized as

with A ≈ 0.195. Only for very short chains (N j 10) a

S - S*

kL) -∑

z∑N

φN(z)

Nln (φN

b ) - ∑z

u(z) (7)

σa

kT) -∑

N

θNex

N- ∑

z

u′(z) (8)

(σ - σm)a

kT) -∑

N

θNex

N- ∑

z

u(z) (9)

∑z

u(z) ) -B(1 -1

Nw) (10)

Figure 2. Potential profile of the polymer segments in amixture of chains of 200, 600, and 1000 segments, all withequal bulk volume fraction (φN

b ) 1/3).

Figure 3. Volume fraction profiles in a mixture of chains of200, 600, and 1000 segments, all with equal bulk volumefraction (φN

b ) 1/3).

θNex

φNb

) A(1 - NNw

) (11)

6734 van der Gucht et al. Macromolecules, Vol. 35, No. 17, 2002

slight deviation occurs from the linear relation. We seethat θN

ex/φNb depends only on N and Nw but not on any

other aspect of the chain length distribution. This resultis remarkably simple. To our knowledge, it has neverbeen mentioned before in the literature. Remarkably,a linear N dependence of θN

ex/φNb is found also for an

ideal (extremely dilute) solution at an interface forwhich the adsorption energy is close to the adsorption/depletion threshold.16 The reasons for this similarbehavior will be explained in the next section.

Substitution of eqs 10 and 11 in eq 9 results in a verysimple expression for the surface tension of a polydis-perse melt:

where Nn is the number-averaged chain length, definedas 1/Nn ) ∑NφN

b /N. The first term is the same as for amonodisperse melt,2 and the second term accounts forpolydispersity. Since A and B are nearly the same, theright-hand side is approximately B(1 - 1/Nn): for apolydisperse system σ is determined mainly by thenumber-averaged chain length Nn. In principle, eq 12allows an experimental confirmation of the results inthis paper, if it would be possible to determine thesurface tension of a melt accurately enough.

Note that the results obtained in this section applyto a polymer melt at a solid surface but give also areasonable description of the polymer/air interface of thepolymer melt, since the interfacial region of a melt isusually very sharp.

4. Comparison to an Ideal Polymer Solution at aSurface near the Adsorption/DepletionTransition

It is well-known that chains in a polymer melt areideal, because the excluded-volume interactions arescreened.14,15 In Figure 2 we have seen that the potentialu(z) in a polymer melt has an oscillatory profile, as acombined effect of the impenetrability of the surface andthe volume-filling constraint. These oscillations are veryshort-ranged: only the first one or two layers have apotential that is significantly different from zero. Wemay therefore expect the surface behavior of polymer

chains in a melt to be similar to the behavior of an ideal(extremely dilute) polymer solution near a surface closeto the adsorption/depletion transition. In the former casethe profile u(z) (Figure 2) may be approximated as asquare well potential: u(z) ≈ 0 for z > 1 and a well depthu(1) which is determined by the volume-filling con-straint (and which is independent of the adsorptionenergy ua). For the latter case, the potential u(z) is againzero outside the first layer, and the well depth isdetermined by the interaction with the surface. Thevolume filling is unimportant for dilute polymer solu-tions.

For adsorption of polymers from a binary solution, adimensionless adsorption energy parameter øs can beintroduced:17

where upa is the (short-ranged) interaction energy be-

tween a polymer segment and the surface (the same asua in eq 4) and u0

a that between a solvent molecule andthe surface, both in units kT. The transition betweenpositive adsorption (θN

ex > 0) and negative adsorption(depletion, θN

ex < 0) occurs at a threshold value of øs,denoted as øsc. A segmental adsorption energy øs ) øscjust compensates the conformational loss due to thepresence of the surface. For øs < øsc the polymer isdepleted from the surface, and for øs > øsc it adsorbs. Ina recent paper,16 we have investigated the behavior ofideal polymers at a surface near the adsorption/deple-tion transition. It was shown that øsc is a (weak) functionof the chain length. For infinite chain length øsc equalsthe critical adsorption energy, given by9,18

and with decreasing chain length, the adsorption/depletion transition shifts to slightly lower øs values:

with C ≈ 1/5. The surface excess of a polymer moleculein dilute solution at a øs value close to the adsorptionthreshold19 was found to be a linear function of the chainlength N:16

In the case of a polydisperse polymer melt there isno solvent. Since all segments are identical, there is noenergetic contribution to the potential. The volume-filling potential (which has an entropic origin) now playsthe role of an effective wall potential u. This effectivewall potential u may be estimated, for example, as thesum of u(z) over all lattice layers. It has a small negativevalue (see eq 10). If -u < øsc

N, the potential is too weakto compensate the loss of conformational entropy, anddepletion takes place (θN

ex < 0). If -u > øscN, the chains

adsorb preferentially (θNex > 0). For chains that have a

zero excess, -u ) øscN. In a polydisperse melt, this is the

case for chains of length Nw (see eq 11). We mayconjecture that the magnitude of the effective wallpotential u is such that it just coincides with theadsorption/depletion point of chains of length Nw. For

Figure 4. Surface segregation in a polydisperse polymer meltfor several chain length distributions: crosses, bimodal dis-tribution with Mw ) 600, Mn ) 438; circles, pentamodaldistribution with Mw ) 600, Mn ) 525; squares, octamodaldistribution with Mw ) 290, Mn ) 162. The dashed linecorresponds to eq 11.

(σ - σm)akT

) B(1 - 1Nw

) + A( 1Nw

- 1Nn

) (12)

øs ) -(upa - u0

a) (13)

øsc∞ ) -ln(1 - λ) (14)

øscN ) øsc

∞ - CN

(15)

θNex

φNb

) 56(C + (øs - øsc

∞ )N) (16)


shorter chains (N < Nw), the wall potential is sufficientfor positive adsorption (θN

ex > 0), whereas for N > Nw itis insufficient (θN

ex < 0).The linear relation between θN

ex/φNb and N in a poly-

disperse polymer melt is almost the same as the onewe found before for a dilute solution near the adsorption/depletion transition, eq 16.16 For the latter case theslope of the line is determined by the difference øs -øsc

∞ , whereas in a melt it is determined by the weight-averaged chain length Nw, because the self-consistentpotential is determined by Nw. The intercept (A in eq11) was found to be 1/6 for the ideal solution, whereasfor the melt we find A ≈ 0.195. This difference isprobably caused by the difference in the potentialprofile: in the ideal solution the potential is nonzeroonly in the first layer, while in the melt it oscillates inthe first few layers.

We may quantitatively compare the conditions forwhich the surface excess vanishes. For the polydispersemelt we have θN

ex ) 0 for N ) Nw. According to eq 10,the field is

For the same chain length adsorbing from dilute solu-tion θN

ex ) 0 when u(1) ) ∑z u(z) equals -øscN. With eq 15

Numerically, the results are extremely close. The minordifferences are again caused by the fact that in the meltthe potential oscillates.

5. Simplified Lattice ModelWe have seen that the surface segregation in a

polydisperse melt can be described by a strikinglysimple law (eq 11). This law was found by carefulexamination of numerical results of many self-consistent-field calculations. The numerical results could be inter-preted using the analogy with a dilute solution of chainsnear the adsorption/depletion transition.

In this section we use a simplification of the full latticemodel to derive the same results analytically. For thispurpose we consider a polydisperse polymer melt in agap of 2M lattice layers wide between two surfaces, asin sections 2 and 3. In all layers apart from the surfacelayers, segments have a coordination number Z, as inthe bulk, but segments in the surface layer have aneffective coordination number Z(1 - λ). This reducedcoordination number accounts for the loss of conforma-tional freedom of chains near the surface. (The numberof possible directions for bonds to a segment in a surfacelayer is Z(1 - λ) rather than Z.) To be able to obtainexplicit analytical expressions, we assume that it isallowed to use for multilayered gaps an averagedcoordination number for the counting of conformations.This averaged coordination number in the gap is definedas

Using an average coordination number implies that thevolume fraction of polymers of length N in the gap is

assumed not to vary in the normal direction; i.e., φN(z)is constant throughout the gap. This assumption mightbe justified by the observation that the volume fractiondoes not change very much near the surface (Figure 3).Obviously, for a gap of one lattice layer (M ) 1/2), ⟨Z⟩ )Z(1 - 2λ) ) 4. For such a single layer the subsequenttreatment is exact.

The total number of lattice sites in the gap is V )2ML, with L the number of lattice sites per layer. Sincethe whole lattice has to be filled with chains, V )∑NnNN, with nN the number of chains of length N inthe gap. The partition function for a mixture of chainsin a lattice of volume V with coordination number ⟨Z⟩has been derived by Flory:15

It is convenient to evaluate Ω with respect to a referencestate (which we will denote with an asterix). We chooseagain the pure, unmixed components with nN, nN′, ...molecules. The degeneracy for the reference state isgiven by9,15

and Ω* ) ∏NΩ*N. Combination of eqs 20 and 21 givesan expression for the entropy of a polymer mixture in agap:

We assume that the gap is in equilibrium with a bulksystem of polymers with a certain chain length distribu-tion. The equilibrium situation is found from a mini-mization of the grand potential:

where µN - µ*N is the chemical potential difference ofchains of length N with respect to the reference state.This can be found from the bulk partition function Ωb

as

where Ωb is given by eq 20, with the bulk coordinationnumber Z instead of ⟨Z⟩. The result of this differentia-tion can be found in ref 15.

Minimization of the grand potential leads to thefollowing chain length distribution in the gap:

where R is a Lagrange multiplier accounting for volumefilling. The chain length-induced surface segregation isrelatively weak, so that R is small. If we assume that

∑z

u(z) ) -0.1842 +0.1842

Nw

(17)

∑z

u(z) ) -0.1823 +0.2

Nw

(18)

⟨Z⟩ ) 12M

[2Z(1 - λ) + (2M - 2)Z] ) (M - λM )Z (19)

Ω ) V!∏N

1

nN!(⟨Z⟩

V )nN(N-1)

(20)

Ω*N )(nNN)!

nN! ( ZnNN)nN(N-1)

(21)

S - S*

kV)

1

Vln( Ω

Ω*) ) -∑N

φN

Nln(φN) +

(∑N

φN

N- 1) ln( Z

⟨Z⟩) (22)

Ψ - Ψ* ) -T(S - S*) - ∑N

nN(µN - µ*N) (23)

µN - µ*N

kT) -(∂ ln(Ωb/Ω*)

∂nN)

nN′*N,T(24)

φN

φNb

) Z⟨Z⟩

e-RN ) ( MM - λ)e-RN (25)


R , 1/N, we can expand the exponential in eq 25:

The condition that the sum of all φN’s must be equal tounity yields R ) λ/(NwM). It follows that the assumptionof small RN is justified for large M. The excess persurface is calculated as

If now we take the limit for infinite gap width, wefind an expression for the excess amount at an isolatedsurface. The result is

which is the same as eq 11, apart from a small deviationin the intercept: the numerical prefactor A in eq 11 wasfound to be around 0.195 for a cubic lattice, whereasthe approximate analytical treatment predicts a pre-factor λ ) 1/6.

Using eqs 22 and 25, we can write for the entropy ofthe polymer mixture in the gap per surface:

This expression can be compared to the expression forthe entropy in the Scheutjens-Fleer theory, eq 7. Thedifference between the two approaches is that we usedan average coordination number in this section, so thatthe profiles are homogeneous for each component. TheLagrange parameter R is related to the potential fieldu(z) in the full lattice model of section 2. The differenceis that the field u(z) can vary along the z-axis, whereasR is averaged over the whole gap. If we take the averageover z in eq 7 and compare the result with eq 29, wefind that the average potential in the SF theory uj ≡∑zu(z)/M is related to the Lagrange multiplier R as uj )R + ln(⟨Z⟩/Z). For infinite gap width this averagepotential goes to Muj ) -λ(1 - 1/Nw), which is almostthe same result as eq 10, again with a slightly differentprefactor.

Equation 25 can be interpreted as a Boltzmanndistribution of chains in a gap. The exponent of theBoltzmann factor then corresponds to the free energyneeded to squeeze a chain of length N in the gap. Thisfree energy is proportional to the length of the chain.This is not surprising, because chains in a melt are ideal(see section 4), and for ideal (Gaussian) chains it is well-known that the elastic free energy upon compression isproportional to N.14,20 Note that eq 25 applies even ifthe gap is much wider than the radius of gyration ofthe chains. In that case, the surface affects only thosechains that have their center of mass closer to thesurface than their radius of gyration. These chains havea reduced conformational entropy, because they canexplore a smaller volume than in the bulk due to theimpenetrable surface. This entropy loss is again pro-portional to the chain length. The competition betweenthe volume-filling requirement and the elastic free

energy of the chains results in the simple relation forthe surface segregation, eq 28.

6. Concluding Remarks

In this paper we have shown that there is a verysimple and very general relation between the surfaceexcess, the bulk concentration, and the chain length ofa component in a polydisperse polymer mixture (eq 11).The results of the full lattice theory (section 2) are exactwithin the mean-field approximation. Furthermore, itis known that for concentrated polymer solutions andfor melts the mean-field approximation is effectivelyexact. In section 4 we demonstrated the similaritybetween a polydisperse melt near a surface and a dilutesolution of ideal chains near a surface for which theadsorption energy is close to the adsorption threshold.In section 5 we used a simplified lattice model to deriveanalytical expressions.

Surface segregation in a polymer melt is fully drivenby entropic factors. The entropy loss of a polymer chainupon confinement next to a surface is larger for longchains than for short ones. This entropy loss determinesthe adsorption threshold øsc

N, the minimum adsorptionenergy for chains to adsorb from solution. In a recentpaper,16 we have shown that the adsorption/depletiontransition for ideal chains is a function of the chainlength. This chain-length dependence of øsc

N may beinterpreted as the factor driving surface segregation ina polydisperse polymer melt. The simplified latticeapproach (section 5) suggests that the surface segrega-tion is related to the compression of chains that havetheir center of mass closer to the surface than theirradius of gyration. For ideal chains this entropy loss isproportional to the chain length.

The surface segregation of chains due to differencesin molecular weight is relatively small. Depletion indilute polymer solutions is, for example, much stronger.However, we believe that the results obtained in thispaper are very interesting from a fundamental point ofview, because of their simplicity and generality. Forideal chains in solution we have also investigated theeffect of chemically different end-segments.16 In practi-cal polymer systems the end segments are usuallydifferent, for example because they carry a residualgroup from a polymerization initiator. It was found thatthe influence of the chain length is strongly enhancedif the end segments adsorb preferentially. For shortchains the end segments are relatively more importantthan for long chains. Hence, we expect that the prefer-ential adsorption of short chains is much larger in thiscase. This would result in a higher value of the prefactorA in eq 11. Preliminary calculations (not shown) suggestthat this is indeed the case. In practical systems,therefore, polydispersity may have a significant effecton the surface properties of a polymer melt.

References and Notes

(1) Helfand, E. Macromolecules 1976, 9, 307.(2) Theodorou, D. N. Macromolecules 1988, 21, 1391.(3) Schmidt, I.; Binder, K. J. Phys. II 1985, 46, 1631.(4) Freed, K. F. J. Chem. Phys. 1996, 105, 10572.(5) Hariharan, A.; K.; K. S.; Russell, T. P. Macromolecules 1991,

24, 4909.(6) Frederickson, G. H.; Donley, J. P. J. Chem. Phys. 1992, 97,

8941.(7) Carignano, M. A.; Szleifer, I. Europhys. Lett. 1995, 30, 525.(8) Wu, D. T.; Frederickson, G. H. Macromolecules 1996, 29,

7919.

φN

φNb

) ( MM - λ)(1 - RN) (26)

θNex(M) ) M(φN - φN

b ) ) λMM - λ(1 - N

Nw) (27)

θNex

φNb

) λ(1 - NNw

) (28)

S - S*

kML) -∑

N

φN

Nln(φN

b ) + R + ln(⟨Z⟩

Z ) (29)


(9) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.;Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapmanand Hall: London, 1993.

(10) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979,83, 1619.

(11) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980,84, 178.

(12) Evers, O. A.; Scheutjens, J. M. H. M.; Fleer, G. J. Macromol-ecules 1990, 23, 5221.

(13) Hariharan, A.; Kumar, S. K.; Russell, T. P. Macromolecules1990, 23, 3584.

(14) De Gennes, P. G. Scaling Concepts in Polymer Physics;Cornell University Press: Ithaca, NY, 1979.

(15) Flory, P. J. Principles of Polymer Chemistry; Cornell Univer-

sity Press: Ithaca, NY, 1953.(16) Van der Gucht, J.; Besseling, N. A. M.; Fleer, G. J. Macro-

molecules 2002, 35, 2810.(17) Silberberg, A. J. Chem. Phys. 1968, 48, 2835.(18) Rubin, R. J. J. Chem. Phys. 1965, 43, 2392.(19) For strong adsorption, øs . øsc

∞ (φ(1) almost independent ofN), there is no energetic reason for chain length preference,and the translational entropy then favors adsorption of longchains. This case is not considered here.

(20) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics;Oxford University Press: New York, 1986.

MA010964Q


Documents

Surface Segregation in Polydisperse Polymer Melts