Lattice cluster theory for dense, thin polymer films Karl F. Freed Citation: The Journal of Chemical Physics 142, 134901 (2015); doi: 10.1063/1.4916383 View online: http://dx.doi.org/10.1063/1.4916383 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Lattice cluster theory of associating polymers. IV. Phase behavior of telechelic polymer solutions J. Chem. Phys. 136, 194903 (2012); 10.1063/1.4714563 Thin film morphologies of π -conjugated rod-coil block copolymers with thermoresponsive property: A combined experimental and molecular simulation study J. Chem. Phys. 132, 214901 (2010); 10.1063/1.3428761 Dynamics in Self‐Assembling Polymer Films AIP Conf. Proc. 708, 245 (2004); 10.1063/1.1764126 Mixtures of lattice polymers with structured monomers J. Chem. Phys. 120, 6288 (2004); 10.1063/1.1652432 Pragmatic analysis for the range of validity of the lattice cluster theory J. Chem. Phys. 110, 1307 (1999); 10.1063/1.478183

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THE JOURNAL OF CHEMICAL PHYSICS 142, 134901 (2015)

Lattice cluster theory for dense, thin polymer films Karl F. Freed James Franck Institute and Department of Chemistry, University of Chicago, Chicago, Illinois 60637, USA

(Received 16 February 2015; accepted 18 March 2015; published online 3 April 2015) While the application of the lattice cluster theory (LCT) to study the miscibility of polymer blends has greatly expanded our understanding of the monomer scale molecular details influencing miscibility, the corresponding theory for inhomogeneous systems has not yet emerged because of considerable technical difficulties and much greater complexity. Here, we present a general formulation enabling the extension of the LCT to describe the thermodynamic properties of dense, thin polymer films using a high dimension, high temperature expansion. Whereas the leading order of the LCT for bulk polymer systems is essentially simple Flory-Huggins theory, the highly non-trivial leading order inhomogeneous LCT (ILCT) for a film with L layers already involves the numerical solution of 3(L − 1) coupled, highly nonlinear equations for the various density profiles in the film. The new theory incorporates the essential “transport” constraints of Helfand and focuses on the strict imposition of excluded volume constraints, appropriate to dense polymer systems, rather than the maintenance of chain connectivity as appropriate for lower densities and as implemented in self-consistent theories of polymer adsorption at interfaces. The ILCT is illustrated by presenting examples of the computed profiles of the density, the parallel and perpendicular bonds, and the chain ends for free standing and supported films as a function of average film density, chain length, temperature, interaction with support, and chain stiffness. The results generally agree with expected general trends. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4916383] I. INTRODUCTION

Thin polymer films find applications to a wide range of technologies, including high resolution photoresists, optical coatings, dielectric insulators, and more, and ultrathin films are of interest for use in nano-fabrication. However, as the thickness of polymer films descends below ∼100 nm, data from experiments and simulations for the equilibrium and dynamical properties of thin polymer films1–7 exhibit marked changes from those of the bulk, in part, because of the confinement of chains to dimensions smaller than or comparable to the radius of gyration in the melts and because of polymer-substrate interactions. Since several different factors, such as the thickness of the film, the surface energy (free standing vs. supported films), and the method of preparation, intertwine in determining the departures from bulk properties, the explanation for these deviations remains unresolved. For instance, the glass transition temperature Tg and glass fragility are examples of technologically relevant molecular properties that exhibit sensitivity to thickness, substrate, etc., as well to molecular details, such as monomer structure, stiffness, and interaction energies,8 and considerations of the shifts in Tg with thickness also discuss the influence of the decrease in density9 and the increase in the number of chain ends10 near the surface. These complex phenomena should be addressable with the inhomogeneous lattice cluster theory (ILCT). The present work extends the LCT11–14 to describe the dependence of the equilibrium properties of thin polymer films on molecular details, including the thickness of the film, the polymer-substrate interaction, molecular weight, the monomer molecular structure, stiffness, and interactions. The 0021-9606/2015/142(13)/134901/9/$30.00

subsequent evaluation of the variation of the configurational entropy density of thin films with molecular details then will enable the extension of the generalized entropy theory15–18 to describe how the glass transition temperature and fragility vary with these molecular details in thin polymer films. Whereas the LCT for bulk polymer properties is already fairly complicated mathematically, the extension of the theory to describe inhomogeneous thin films is, of necessity, considerably more complex and lengthy. The greater complexity may be illustrated as follows: while the leading order LCT is essentially equivalent to Flory-Huggins (FH) theory, the leading order ILCT developed here for a thin polymer film with L layers already involves the numerical solution of 3(L − 1) coupled nonlinear equations. Consequently, the theory is considered in stages; this first paper is devoted to setting out the general formulation of the ILCT for thin polymer films and to describing in detail the predictions of the nontrivial leading order approximation for the distributions of monomer mass, chain ends, and ratio of out-of-plane to inplane bonds. Moreover, just as Flory-Huggins theory remains heavily used, many may prefer to apply the leading order for more qualitative predictions. Thus, we leave to subsequent works the further details of the derivation of the diagrammatic expansion of the ILCT, the methods for evaluating the new diagrams, applications to the dependence of the properties of thin polymer films on monomer molecular details, etc. Previous self-consistent field theories of polymers at interfaces19–23 are formulated in terms of the configurations of an ensemble of single chains24 in the self-consistent environment of their surroundings that are composed of the same set of chains. The present extension of the LCT to

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describe thin polymer films, on the other hand, follows the treatment in the LCT of the thermodynamics of polymers in the bulk by being predicated on a wholly different approach. The derivation of the LCT is instead motivated by the observation in what de Gennes25 has called the “Flory theorem,” namely, when two monomer units in a melt find themselves at nearest neighbor sites, they are unable to discern whether they belong to different chains or are distant members of the same chain. Hence, the retention of excluded volume constraints is more important in dense polymer systems than is chain connectivity. Thus, the LCT treats all chain connectivity perturbatively but exactly includes all constraints due to excluded volume.11–14 On the other hand, theories for polymers at lower densities are generally formulated in a manner that precisely maintains connectivity and, thus, that treats excluded volume perturbatively.26 In line with prior descriptions of inhomogeneous polymer systems, the ILCT distinguishes between chain ends and bonds that may lie parallel or perpendicular to the film surfaces. Rather than imposing relative probabilities of parallel and perpendicular bonds as in prior theories for inhomogeneous polymer systems,19,20,27–29 their distributions through the film are computed, along with the mass distribution in the film, from a set of 3(L − 1) coupled non-linear equations for films containing L layers. Illustrative calculations are provided for films of flexible and of semi-flexible linear chains in films with L = 10 layers and comprising chains with polymerization indices equal to 500, so that L ∼ Rg, the radius of gyration of unperturbed chains in the melt, and confinement constraints become significant. The calculations consider a range of temperatures, average film densities, and chain stiffness, as well as the variation of the distributions with the polymersupport interactions. The calculations are performed using values for the molecular parameters that are designed to enhance the departures of film properties from the bulk. Section II sketches in general terms the derivation of the ILCT for the packing of polymers into thin polymer films. The connectivity between the bonds of a chain is represented as a constraint on monomer positions whose value in leading order is taken as the average of the constraint, while the difference from this average provides the corrections that are treated perturbatively in the form of a cluster expansion, i.e., as connectivity constraints. Separate averaged constraints apply for bonds lying within a layer and bonds lying orthogonal to the layers. The derivation concludes with the introduction of combinatorial factors prescribing the number of ways the monomers can be placed on the L layers of the lattice consistent with a specified set of bonds running parallel and perpendicular to the film. The partition function emerges as dependent on the respective distributions {na, da, ua, sa, ea} of the numbers na of monomers in layers a = 1, . . . , L, the numbers of down, up, and same bonds that terminate in layer a, and the number of chain ends ea whose first bond terminates in layer a. Introducing sum rules and the Helfand27,28 constraints leaves 3(L − 1) of these quantities as independent variables. The methods for including interaction energies into the theory are briefly shown in Sec. IV to follow as a special case of the treatment of the chain packing. After combining the energetic contributions from Sec. III with the packing contributions, the sums over

J. Chem. Phys. 142, 134901 (2015)

the independent variables are approximated by the maximum term method, which produces the 3(L − 1) coupled nonlinear equations that are solved numerically in Sec. V for the leading order approximation for films with 10 layers. The leading order contributions due to chain stiffness are appended in Sec. VI, and illustrative calculations describe their influence on the distributions of {na, da, ua, sa, ea} as a function of the layer a.

II. THEORY: PACKING CHAINS INTO A THIN FILM

The system consists of a thin, dense film of polymers, containing c chains distributed throughout L layers, numbered sequentially 1 through L, with A lattice sites in each layer for a total number of lattice sites Nl = L A. Individual chains are taken in this section to have N + 1 fully flexible monomers, each occupying a single lattice site. Section VI provides the generalizations to treat semi-flexible chains, while the extension to describe structured monomers follows similarly, albeit with considerable effort to determine and evaluate the plethora of new diagrams of the theory. The monomers are numbered sequentially from 1 to M = N + 1 and, hence, are endowed with a direction along the chain. However, the sum over all chain configurations contains a given chain configuration twice, with the sequential numbering of monomers in these two copies beginning from opposite ends. The partition function, of course, contains the symmetry number of 1/2 because both directed configurations are identical and should not be counted twice. The perturbative nature of the LCT is most easily illustrated by first constructing the partition function for the packing of chains in the film.30,14,11,12 Consider first a single chain where the first unit (numbered 1) is placed at the position r1,a in layer a, so the next can be placed at any of its z neighbors, where, for convenience, the calculations assume a cubic lattice with z = 6 (with one choice forbidden in the surface layers). Four of these neighbors lie in the same layer, while another two reside in neighboring layers a ± 1, respectively, and these bonds are termed “same,” “up,” and “down” bonds, or for brevity, s-bonds, u-bonds, and d-bonds, respectively, because the vector between r1,a and r2,a′ terminates in the same layer or in one layer up or down. Hence, the condition that monomers are neighbors with any of the six possible orientations is given by the constraint for bond number 1, b1 =

z 

δ(r1,a − r2,a′ − a β ),

(1)

β

where aβ are lattice vectors in the six possible directions which are assigned as β = 1, . . . , 4 (same), 5 (up), and 6 (down). Of course, certain u-bonds or d-bonds are absent for the surface layers as discussed below. The evaluation of contributions from correlations induced by bonding constraints is facilitated by combining the up and down bonds into “perpendicular,” but the present treatment of the leading order approximation encounters no benefits from this combination. All chain configurations are generated by successively multiplying factors like Eq. (1) for each bond, namely, the product for a single chain yields b1b2b3 · · · bN−1bN, and then the partition function for the individual chains is defined as the analytically

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intractable sum of this product over all monomer positions, subject to excluded volume constraints that prohibit multiple occupancy of individual lattice sites,  Wpack = b1b2b3 · · · b N −1b N . (2) r1, a ,r1, a ,r1, a ,··· 1 2 3

The inequalities in the summation range in Eq. (2) imply that the sum runs over all monomer positions subject to the excluded volume constraints that prohibit multiple occupancy of any site. The description of a polymer melt or a thin polymer film, of course, requires a product of factors like Eq. (2) for each chain along with the excluded volume constraints within and between chains. The perturbative treatment of chain connectivity by the LCT is introduced by use of the average of the bonding constraints for each of the bonds. This separation of contributions into the average ⟨b⟩ and the remainder b − ⟨b⟩ is expressed for each bond symbolically as14,16,30,11,12 b = ⟨b⟩ + (b − ⟨b⟩) = ⟨b⟩ [1 + (b − ⟨b⟩)/⟨b⟩] ≡ ⟨b⟩ [1 + X], (3) where ⟨b⟩ is some suitably defined average of the bonding constraints, which naturally must differ for up, down, and same bonds and where the bond correlation factor X for the bond is defined by the second equality of Eq. (3). The averages ⟨b⟩ may, in principle, be chosen freely, but obviously certain choices produce better approximations. The choice used here for the averages is illustrated by considering first the average probability of placing a u-bond between layers a and a + 1 (other than surface layers) in an otherwise empty lattice. This probability is equal to the ratio of the number of ways of placing the u-bond on the lattice to the number of ways of placing two unattached monomers with one in layer a and the other in layer a + 1. The first monomer of the u-bond can be at any one of the A sites in layer a, while the second monomer of the bond must be placed at the single neighboring site one layer up. Ignoring for now the presence of all other bonds, the total number of ways of placing the pair of bonded monomers is (A)(1), but this number must be normalized by the number of ways of placing two unconnected monomers in adjacent layers without bonding constraints, namely, A ways for the first monomer and A for the second, giving the normalized average probability of placing a bond between adjacent layers as

δ(r1,a − r2,a+1 − aup) = (A)(1)/(A2) = 1/A. (4) The d-bond yields a similar average,

δ(r1,a − r2,a−1 − adown) = (A)(1)/(A2) = 1/A.

(5)

The u- and d-bonds can alternatively be combined into a joint classification of perp

δ(r1,a − r2,a±1 − aperp) = (A)(2)/(A2) = 2/A, (6) where the factor of 2 arises because the second monomer in the perpendicular bond can be in either of the two adjacent layers. An s-bond has four possibilities for the second monomer, which in a z/2 dimensional hypercubic lattice becomes

δ(r1,a − r2,a − asame) = (A)(z − 2)/(A(A − 1)) = (z − 2)/(A − 1).

(7)

The −1 in the denominator appears because of the excluded volume constraints upon the placement of two unconnected monomers in the same layer, but for historical reasons,30–33 the denominator of Eq. (7) has been taken as A. When the film constitutes semi-flexible chains, other choices for the averages may be desirable.29 Suppose that layer a contains ua u-bonds, da d-bonds, and sa s-bonds that terminate on layer a. Then, the contribution from that layer to the product of bond averages ⟨b⟩ in Eq. (3) becomes )s ( ) ua ( ) da (  1 z−2 a 1 . (8) bj a = A A A b ∈a j

Since a u-bond cannot terminate on layer a = 1, the number u1 of such bonds must vanish, while the opposite layer a = L cannot receive d-bonds, which implies that dL = 0. However, the final equations become much more compact by retaining factors involving u1 = dL = 0. Collecting the contributions from all layers yields )s L  L ( ) ua ( ) da (  

1 z−2 a 1 . (9) bj a = A A A a=1 b ∈ a a=1 j

Since the system contains many chains, the products in Eqs. (8) and (9) run over all chains, and the numbers ua, da, and sa henceforth refer to the totals for all chains. Because the bonds in each separate category {ua, da, sa} are indistinguishable, Eqs. (8) and (9) must be multiplied by the number of ways the bonds in each layer a can be partitioned into those associated with up, down, and same bonds. Our convention is to associate those bonds to the layer a whose end monomer is in a, and thus, the number of bonds in a equals na − ea, where ea is the number of first monomers in a, i.e., the number of first monomers is equal to the number of end monomers at the beginning of chains. The individual bond orientation factors in Eqs. (8) and (9) must be multiplied by the total number of ways of partitioning the na − ea bonds of layer a into ua u-bonds, da d-bonds, and sa s-bonds, which is given by the multinomial theorem as L  (na − ea )! , u !d !s ! a=1 a a a where dL and u1 are understood to vanish. The summand wpack of the packing partition function Eq. (2) is now interpreted as containing the product of factors bα,j (bond j in chain α) for all the bonds in the system. Because the chains that begin from a given layer a (i.e., have monomers numbered 1 in layer a) are indistinguishable, the denominator also contains an additional factor of the product of ca! (which equals ea!), along, of course, with the symmetry number factor 2e a . These factors of 1/(ea! 2e a ) parallel similar factors for polymers in the bulk, except for the fact that because the layers are distinguishable, wpack requires a product of factors of 1/(ea! 2e a ) over all layers. Note that the distributions {na, ea} are taken as specified for now, but later we introduce sums over all possible values for the distribution of bonds {na, ea, ua, da, sa}. The expression for Wpack in Eq. (2) for the many chain system still contains a sum over all monomer positions, subject to stringent excluded volume prohibitions of multiple occupancy of lattice sites. In addition, the expression for

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J. Chem. Phys. 142, 134901 (2015)

Wpack contains a product over the bond correlation factors,  (1 + Xbond), where a correlation written symbolically as bonds

factor Xbond applies for each bond and separately for up, down, and same bond types. Thus, the product yields a cluster expansion11,12,14,30   (1 + Xbond) = 1 + Xbond bonds

bonds



+



Xbond Xbond′ + · · ·,

(10)

bond,bond′ bonds

where the first term involving unity on the right hand side provides the leading order contribution, while the remainder yields the perturbative corrections. The leading order LCT for homogeneous bulk systems is essentially equivalent to Flory-Huggins theory; however, the present extension to inhomogeneous thin films is sufficiently complicated that here we limit consideration to the already complicated leading order treatment where all factors of Xbond are neglected. A subsequent work will describe these contributions, resting heavily on details from the leading order theory. However, the determination of the perturbative corrections requires a complete revision of the previous methods for evaluating individual contributions generated by the factors of Xbond because the prior methods begin with the requirement of an isotropic bulk system at equilibrium. Hence, the contribution from the product of the Xbond terms in Eq. (10) into up, down, and same contributions will be reserved for a future work. Inserting the leading order approximation from Eq. (9), along with the combinatorial factors into the many chain generalization of the summation of Eq. (2), the resulting summand involves a product of averages ⟨b⟩ which is devoid of the summation variables, the spatial positions {rαj} of the individual monomers. Consequently, the sums may be evaluated by just counting the number of ways the monomers can be inserted sequentially into the lattice, site by site in each layer and layer by layer, strictly respecting excluded volume constraints. Hence, the first monomer in layer a = 1 can enter in any of the A sites of that layer, the next monomer 2 of layer a = 1 in any of the remaining A − 1 sites of layer 1, etc. The total number of possibilities for layer 1 is thus A!/[(A − n1)!], where n1 monomers are in the first layer, etc., and the overall contribution is thus the product of factors for each layer. However, because the partition function contains contributions from two chains with identical configurations (with a normalizing factor of 1/2) but with opposite ends labeled as monomers 1 and

Wpack =



L 

u a, d a, s a,e a a=1

(a) The total number na of monomers in layer a equals the sum of the numbers ua + da + sa of all bonds (from directed chains) whose terminal monomer ends in layer a plus the number ea of bonds beginning in layer a, na = (ua + da + sa) + ea.

a=1

Because Eq. (12) contains a product over layers, the summand can be rearranged (multiplying and dividing by factors of

(11)

Combining this relation with the Helfand transport constraint ua = da−1 reduces the number of independent variables in the leading order approximation to the partition function as the set {na, da, ea}. (b) The sum of all ea, a = 1, . . . , L, is the total number c of chains in the film. (c) The sum of all na, a = 1, . . . , L is the total number c(N + 1) of monomers in the system. Given this notation and the constraints, the packing partition function involves 3L − 3 independent summation variables for a film with L layers. Given these considerations, we continue to evaluate the sums over all monomer positions, subject to the bond numbers {na, da, sa, ea} and the constraints that {ua = da−1}. The first layer has na monomers to be placed in layer 1 with A lattice sites, which can be accomplished in A!/(A − n1)! ways. The final expressions are rendered more compact by introducing the vanishing number d0 = 0 and rewriting the results for layer 1 as (A − d0)!/d0!(A − n1)!. The next layer 2 must have u2 = d1 u-bonds that must end on the specific sites on layer 1 adjacent to those from which the down bonds emerge. Thus, the first u-bonds enter in d1! ways, and the remainder may be placed in (A − d1)!/(A − n2)! ways. All the subsequent layers, except the last, yield similar factors of da−1!(A − da−1)!/(A − na)!, but the factor for the Lth layer can be made to have similar form by introducing factors of d L = 0 in an appropriate fashion. Combining everything together yields

( ) ua ( ) da ( )s [Aa − d a−1]![na − ea ]! 1 1 z−2 a , [Aa − na ]!2e a ea !d a ![na − ea − d a − d a−1]! A A A

subject to the constraints of fixed total numbers of bonds and chain ends in the film, L L   c= ea and cN = (na − ea ). (13) a=1

N + 1, we must impose the restriction, noted by Helfand,27,28 that this symmetry requires that the number of u-bonds ua ending on layer a equals the number of d-bonds da−1 leaving layer a to enter layer a − 1. Many of the equations are rendered more compact by defining d0 = dL+1 = 0 because d-bonds cannot end or begin on non-existent layers 0 and L + 1, respectively. The condition ua = da−1 is readily inserted into Eq. (8), Eq. (9), and the combinatorial factor after Eq. (9) to reduce the number of variables, which can be further diminished by the following considerations. The remainder of the calculation of the leading contribution uses several symmetry relations and sum rules.

(12)

da−1!) into a form of the product of exp[Sa/kB], where Sa is the packing entropy per layer with the specified distributions of layer density and bond types. However, the focus here is on deriving overall film properties and distributions and avoiding such arbitrary divisions. After introducing Stirling’s approximation and using the constraints in Eq. (13) to eliminate nL and eL by expressing

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them in terms of {cN, {na, a = 1, . . . , L − 1}} and {c, {ea, a = 1, . . . , L − 1}}, respectively, the sums are approximated using the maximum term method on the summand in Wpack of Eq. (12), with the constraints of Eq. (13) used to eliminate nL and eL. This procedure yields 3(L − 1) coupled nonlinear equations ∂ ln wpack/∂na = ∂ ln wpack/∂ea = ∂ ln wpack/∂d a = 0, a = 1, . . . , L − 1,

(14)

where wpack now designates the whole summand in Eq. (12). The set of Eq. (14) can only be resolved numerically. Inserting the solution to Eq. (14) into the expression for wpack in the summand of Eq. (12) yields the packing entropy density S = (1/AL) ln [wpack]. Before considering the solutions, we now introduce the contribution from the presence of attractive van der Waals interactions between neighboring monomers and perhaps of additional attractive or repulsive interactions between the substrate and the polymers in surface layers 1 and/or L of the film.

III. THEORY: INCLUSION OF INTERACTION ENERGIES

Monomer units on nearest neighbor sites have an attractive interaction ε (using the convention that ε > 0 for attractive interactions), while the surface layers 1 and L may have interactions εs with a supporting surface (with εs > 0 designating repulsive interactions). More generally, if the system comprises the film plus some entrapped solvent, then three independent interaction parameters exist, namely, εpp + εss – 2 εps,εpp – εss, and εss. For simplicity, the model we use corresponds to the conventions ε = εpp + εss − 2 εps and εpp − εss = εss = 0. The present methods can be applied using any other convention for the energetic parameters. The contribution from interaction energies emerges as a multiplicative factor wE multiplying the summand of Wpack in Eq. (12).12,14 Introducing the notation ⟨ij⟩ to indicate that the neighboring lattice sites i and j are both occupied by monomer segments, the contribution wE to the partition function from monomer-monomer interactions appears in the  form of the Boltzmann factor w E = exp[ β ε], where all ⟨i j⟩

β = 1/kB T is the thermal energy. The nearest neighbor condition may be expressed more explicitly by representing the constraintin terms of Kronecker delta functions as w E = exp[ β εδ(ri − r j − ai j )], where aij is the lattice vecall ⟨i j⟩

tor connecting the nearest neighbor sites ri and rj. The factor of b E = δ(ri − r j − ai j ) behaves exactly like a bond vector factor b and may be of either of the up, down, or same categories. The factor wE is further represented in terms of Mayer f-functions f = exp(βε) − 1 that are approximated by a high temperature expansion, which (following the detailed derivation in Appendix A of Ref. 14) yields the obvious lowest order contribution, (1/A) ln(w E ) =

L  ( βε/2A2)[n2a (z − 4) + na na−1 + na na+1], a=1

(15)

where the three terms in brackets enter from bE components of the same, down, or up bond varieties; for example, nana+1 is the lowest order approximation to the number of nearest neighbor contacts between units in layers a and a + 1, etc. The sum in Eq. (15) contains terms involving n0 and nL+1, which are defined to vanish for free standing films but can be chosen to describe the contribution from interactions between monomer units in layers 1 and L with external surfaces by setting n0/A → −(2ε s /ε),

n L+1/A → −(2ε S′ /ε),

(16)

where both external surfaces may be taken as different.

IV. THE FREE ENERGY

Combining the energetic contribution wE from Eq. (15) with the packing term wpack from Eqs. (12) and (13) and introducing the constraint from Eq. (13) into W give the total leading order partition function as the multiple sum in Eq. (12) of the product w({pa}) = wparkwE, where {pa} = {na, da, ea} designates the 3(L − 1) independent parameters. The maximum term method of Eq. (10) is then applied to the full summand w({pa}), ∂w({pa })/∂pb = 0,

b = 1, . . . , 3 (L = 1),

(17)

to obtain the solutions {pa ∗} = {na ∗, d a ∗, ea ∗} and the free energy as F/L Ak BT = −(1/L A) log[wpack({pa ∗})w E ({pa ∗})]. (18) Basic thermodynamic properties that involve first derivatives of the free energy may be evaluated as usual, for example, ( ) 3(L−1)  ( ∂(w{pa }) ) ∗ ( ∂pb ) ∗ ∂(w{pa ∗}) ∂(F/k BT) =− + , ∂T ∂T ∂pb ∂T b=1 (19) where the superscripted asterisk implies that the quantities are evaluated for the solutions {pa∗} to Eq. (18) which,  in turn, implies that the term following the in Eq. (19) vanishes identically, term by term. The evaluation of second derivatives of the free energy, however, requires the retention of contributions from second derivatives of w with respect to the {pa∗}, derivatives that may be evaluated by numerical or (lengthy) analytical methods according to convenience. The calculated {na∗/A} provide the density profiles in the film as illustrated in Sec. V for symmetric films (either free or supported) which have the symmetry property that na∗ = nL+1−a∗, ea ∗ = eL+1−a∗, and da ∗ = dL−a∗,

(20)

where dL/2∗ is unique for L even, while n(L+1)/2∗ and c(L+1)/2∗ are unique for L odd. The symmetry property is convenient to ensure convergence has been achieved in the solution to the large number, 3(L − 1), of coupled nonlinear equations. Once convergence has been verified, introduction of the above noted symmetry constraints into the summand of the expression for the free energy and subsequent application of the maximum term approximation combine in reducing the problem to the numerical solution of 3L/2 − 2 or 3(L − 1)/2 coupled equations for L even or odd. This reduced number of equations, from 27 to 13 for a 10 layer system, is still considerable for

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J. Chem. Phys. 142, 134901 (2015)

FIG. 1. Deviation φa − φ of the density φa = na/A from the average φ as a function of φ for various T = 200 (red), 300 (blue), 400 (green), 500 (black) for L = 10, M = N + 1 = 500, ε = 50. (a) through (e) for layers 1 through 5, respectively.

films with L = 10 that are used to illustrate the predictions of the theory. V. COMPUTED MASS DISTRIBUTIONS

All the calculations in this section consider flexible linear chains of molar mass equal to M = 500 monomers on a cubic lattice with coordination number z = 6 and with parallel surfaces that may be in contact with a substrate or be free of external supporting surface(s). The interaction parameters are taken to be large to accentuate departures of film properties from the bulk, but the basic behavior and trends remain the same for the different parameter sets. The calculations consider symmetric films, and the computed results display the requisite symmetry of Eq. (20). Figure 1 presents calculations of the deviation φa − φ between the surface density φa = na/A in each layer a = 1,

. . . , 5 and the average density φ in the film for a wide range in temperatures, film densities, and the interaction parameter ε. A measure of ε is provided by the high value 0.75 of the interaction parameter 3ε/kBT for the system in the bulk at the lowest temperature T = 200, while the highest temperature of T = 500 yields a moderately large value of 0.30. (Recall that the traditional χ parameter of FH theory for the bulk is 3ε/kBT + 0.5, with the latter 0.5 from expansion of the (1 − φ) log(1 − φ) term.) All chains have 500 monomer units, so, as explained below, Rg in the bulk is comparable to the number of layers in the film. Because the film is held together by rather strong attractive interactions, the missing neighbor effect and the restricted possibilities for bonds at the surface lead, as expected, to a diminished surface density. Figure 1 displays the anticipated behavior that the calculated shifts in density from the average are lowest for the higher density films, which are more

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J. Chem. Phys. 142, 134901 (2015)

uniform, and for the highest temperatures T. Because the large ε induces a strong depletion of density from the first layer, subsequent layers must have an enhanced density to compensate until the density approaches the bulk in the interior of a sufficiently thick film. However, the film described in Fig. 1 is thin enough that the interior of the film with thickness L = 10 may reach the “bulk” limit only for the highest densities and lowest temperatures. For example, the surface layer a = 1 of a 10-layer film with φ = 0.95, M = 500, ε/kB = 50, and T = 200 has a deficiency over the mean density of 4.25%, while layer 5 is not yet converged to the bulk limit of φ = 0.95 and has a 1.02% excess with respect to the average density. This behavior, of course, is expected because of the reduced number of choices for the monomer to follow (i.e., bond) with a monomer in layer 1 and hence of the entropic benefit. The computations also provide the distributions (not shown) for the densities of ends {ea/A} and d-bonds {da/A} and, hence, for up and same bonds. The fraction of d-bonds (and u-bonds) of da/A = {0.037, a = 1, 10; 0.039, rest} for the film with φ = 0.84 is well below the bulk average of ∼φ/6 = 0.14. A portion of this difference appears because of biases introduced into the leading order approximation (but likely eliminated by the first order corrections) due to the imposition of Helfand’s constraints and to other differences in treating parallel and orthogonal bonds. Nevertheless, the average da/A for layer a of the film is diminished below the “bulk” limit, defined as the infinite system produced by replacing the film boundary conditions with periodic boundary conditions. This added decrease in perpendicular bonds signifies the wellknown predominance of more two-dimensional type ordering of the chains in thin films. Other trends displayed by the distributions are similar to those for the {na/A}. For example, in order to facilitate the decrease in density at the surfaces of the free film, there is a small enhancement in both the number of e1 (e10) chain ends10 in the surface layer and also of the number of orthogonal d1 (u10) bonds entering the surface layers. More explicitly, e1 = e10 is slightly elevated by 3.75% over the bulk (film average), while ea for the interior layers are depressed by 0.95%. Presumably, enhancements in chain ends and down (up) bonds are correlated and some of the excess orthogonal bonds are end bonds. In order to determine whether the preferred twodimensional-like conformations are associated with the value of the ratio of the radius of gyration Rg,b/L for the bulk to the film thickness, Rg,b is evaluated for a long non-reversing random walk chain with an average bond angle ⟨cos θ⟩ as34 Rg,b(Eb)2 = ((M − 1)/6)(1 − ⟨cos θ⟩)/(1 + ⟨cos θ⟩). (21) The average over θ is ⟨cos θ⟩ = −1/5 because, given a bond in the bulk, a successive bond has five choices, one of which is parallel to the given bond and the other four are orthogonal and, therefore, do not contribute to ⟨cos θ⟩. Hence, the present case gives Rg,b = 11.2 which slightly exceeds L = 10. Decreasing the polymerization index to M = 100 implies that Rg,b = 5 is now less than L = 10. The da increase by only ∼1.5%, and the chain configurations are again pancaked.

FIG. 2. Deviation φa − φ of the layer density φa from the film average φ = 0.84 for chains in a film with L = 10, M = 500, T = 300, and ε = 100. Curves are each for a different εs = 0 (blue), 25 (red), 50 (green), 100 (black), −25 (magenta), −50 (orange), −100 (pink). Points for layer 1 are scaled by −0.1 to fit with other points.

Figure 2 presents a study of the influence of the polymersubstrate interaction on the density profile in the film by considering the deviation of the layer density φa from the film average φ = 0.84 for chains in a film with L = 10, M = 500, T = 300, and ε = 100 for a wide range of repulsive (εs > 0) and attractive (εs < 0) polymer-substrate interactions. Notice that the points for layer 1 are scaled by −0.1 to fit with the other points. As expected, the more repulsive the interaction, the greater the depletion in the first layer and the elevated enhancement in the subsequent layers. A near cancellation of the influence of finite film dimensions emerges when εs = ε and the substrate surface is energetically equivalent to the polymers in the film, leaving a slight entropic difference between the film surface and the bulk because of the reduced number of conformations available to monomers in the surface layer. Alternatively, because εs appears only in the combination εs/T, Fig. 2 can be interpreted as the variation with T (but separately for repulsive and attractive interactions εs) for fixed εs. These observations simply explain the existence of a compensation temperature8 where the film density becomes uniform as in a melt.

VI. SEMIFLEXIBLE CHAINS

Recent simulations of thin films by Savit and Riggleman8 and by Hanakata et al.8 demonstrate the strong influence of chain and substrate stiffness, respectively, on the properties of thin polymer films. A common lattice model of semiflexible chains35 assigns a pair of consecutive parallel bonds as “trans” with zero energy and a consecutive pair of orthogonal, “gauche” bonds with higher energy Eb, called the bending energy.12,36,37 The lowest order description for a melt of semiflexible chains contains a statistical factor for each bond (except an end bond) of the statistical weight for the placement of a monomer to form a subsequent bond.12 Strictly speaking, apart from bonds terminating at the surface layers a = 1 or L, the cubic lattice implies that the statistical factor is Zb = (1 + 4 exp(−β Eb))/5 since each subsequent bond has 5 choices, one parallel to the previous and four orthogonal. Perpendicular surface bonds d1 and uL may be followed only by one of

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four in-plane gauche bonds, while parallel s1 and sL bonds yield one trans and three gauche possibilities, for statistical factors of Zo = 4 exp(−βEb)/4 and Zi = (1 + 3 exp(−βEb))/4, respectively. (Prior work38–40 comparing LCT predictions11,12 with experiments for polymers with carbon backbones uses the more realistic model of two gauche bonds and thus a factor involving 1 + 2 exp(−βEb).) Here, we employ the factors appropriate to a strict cubic lattice. The lowest order model of thin films with semiflexible chains11,12 introduces an overall bending factor wbend into Eq. (12), so the summand in this equation becomes w = wpackwEwbend, FIG. 3. Deviation φa − φ of the layer density φa from the film average φ = 0.84 as a function of bending energy Eb for chains in a film with L = 10, M = 500, T = 300, and ε = 100. Points for layer 1 are scaled by a factor of −0.5 to fit on figure. Points for layer 4 are overlapped by those for layer 5.

wbend = (Z p )d1+u L −(1/3)(e1+e L )(Zi )s1+s L −(2/3)(e1+e L ) ×

L−1  (Z0)d a +u a +s a −e a .

(22)

a=2

Equation (22) emerges because the each bond in the interior layers 2 ≤ a ≤ L − 1 has na − 2 ea pairs of successive bonds that are each assigned factors of Z0, while the surface layers have, e.g., d1 = u10 bonds with a statistical factor of Zp and s1 = s10 bonds with a statistical factor of s1 − e1. However, an extra number of e1 bonds must still be parsed between factors of Zp and Zi. Given the relatively small value of e1 compared to d1 and s1, the result is insensitive to whether decomposition places the extra e1 statistical factors in the s-bond category or uses the bulk averages to place 2/z = 1/3 in the d-bond category and the other 4/z in the s-bond category. The overall w is yet again approximated with the maximum term method. Figure 3 illustrates the behavior of the profiles for the deviation of the layer density φa from the film average φ = 0.84 as a function of the trans-gauche energy Eb for films with L = 10, M = 500, T = 300, ε = 100, εs = 0, and variable Eb. The curves for layers 3 and 5 are almost indistinguishable from each with the resolution of Fig. 3, while the curves for layers 4 and 5 differ from each other by less than ∼1%. The unperturbed radius of gyration for the semiflexible chains in the bulk emerges from Eq. (21) using the average cosine (⟨cos θ⟩) for successive bonds as ⟨cos θ⟩ = −1/[1 + 4 exp(−Eb /kT)],

(23)

because each subsequent bond has five choices, one trans with cos θ = 1 and four gauche with cos θ = 0. Hence, for example, the current system with M − 1 = 499 bonds and T = 300 yields Rg,b(Eb) = 11.1, 15.5, 26.1 for Eb equal to 0, 400, 800, respectively. Computations with shorter chains [M = 100, Rg,b(0) = 4.97] show similar overall trends, whereas those for M = 10 lack an enhancement of d1 = u10 bonds but otherwise yield similar trends. The deviation φ1 − φ increases with growing Eb from Eb = 0 to 800 (while the other φa − φ decrease) because increasing chain stiffness favors an enhancement in two-dimensional type conformations of chains in the film. Assistance with this ordering comes from the 75% decline in d1, an increase of d2 of only 2.9%, and changes in the rest of the da by

Lattice cluster theory for dense, thin polymer films.

While the application of the lattice cluster theory (LCT) to study the miscibility of polymer blends has greatly expanded our understanding of the mon...
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