Density Functionals for Polymers at Surfaces - American Chemical

Chapter 18

Density Functionals for Polymers at Surfaces William E. McMullen

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Department of Chemistry, Texas A & M University, College Station, TX 77843-3255

W e derive a n expression for the external field necessary to p r o duce an a r b i t r a r y monomeric density near a planar surface. T h i s result becomes exact i n the l i m i t of weak external fields. We illustrate the u t i l i t y of the f o r m a l i s m by a p p l y i n g it to a p o l y m e r b l e n d interacting w i t h a surface. D u e to monomer-surface cor relations, the monomer densities decay to the b u l k composition more slowly t h a n i n previous phenomenological theories of poly mer adsorption. For our choice of monomer-surface H a m i l t o n i a n , we observe o n l y first-order wetting.

Density functional theories of dense systems often separate i m p o r t a n t thermo d y n a m i c potentials into ideal a n d nonideal contributions. F o r classical, atomic fluids, the division is obvious since the p a r t i t i o n functions of noninteracting, m o n a t o m i c species avail themselves to exact analyses. For example, researchers c u s t o m a r i l y define the ideal free energy functional Fid[p] a n d the interaction free energy Φ so t h a t , i n terms of the t o t a l free energy F[p]

T h e a n a l y t i c a l expression for Fid[p] (1)

a n d the well-known identity

lead to

0097-6156/96/0629-0261$15.00/0 © 1996 American Chemical Society In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

262

CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY

w i t h Λ a n atomic length scale, a n d c(l;[p])


T h e c o m b i n a t i o n c ( l ) — βν(1) o n a n i d e a l gas of atoms.

=

6*[p)/6p(l).

(5)

plays the role of a n effective external field acting

Whereas equation 4 h a r d l y solves the atomic density f u n c t i o n a l p r o b l e m , it does suggest some reasonable schemes (2) for a p p r o x i m a t i n g the interaction t e r m Φ[p]. Some of these y i e l d surprisingly good, mean-field descriptions of dense systems. P a r t of the successes of these theories trace back to equations 2 a n d 4 w h i c h a u t o m a t i c a l l y incorporate the i d e a l , t r a n s l a t i o n a l free energy of atoms into the u n d e r l y i n g f o r m a l i s m . W h a t e v e r mistakes the theories make i n a p p r o x i m a t i n g Φ[p], at least they describe ideal gases correcly. Extensions of the f o r m a l i s m to complicated polyatomic species like p o l y m e r fluids are not readily accomplished. E v e n for a n idealized m o d e l p o l y m e r i n w h i c h monomers do not interact (e.g., freely jointed chains, G a u s s i a n r a n d o m walks, c o n t i n u u m chains, etc.), the monomeric density does not, i n general, reduce to equation 4 w i t h c ( l ) = 0. Imagine, for instance, a linear p o l y m e r subjected to a n external field that acts o n l y o n monomers at one end of the chain. T h e covalent bonds defining the p o l y m e r transfer the response of the monomers at that end to a l l other monomers. T h i s induces a density v a r i a t i o n even i n regions of space where the external field does not act. I n the language of atomic density functionals, we say t h a t f r o m a monomeric point of view, the external field induces a nonlocal density response. O n l y i n the l i m i t that the external field varies i m p e r c e p t i b l y over the volume occupied by a chain can we propose a simple f o r m for the monomer density. In this case, for a chain of Ν monomers that each interact w i t h a n external field ν (S), p ( l ) oc exp[-JV/?i;(l)]

(6)

where we determine the p r o p o r t i o n a l i t y constant f r o m the chemical potential or the average density. G e n e r a l l y speaking, the field-density r e l a t i o n for a n i d e a l p o l y m e r fluid is far more complicated t h a n equation 6, a n d before a t t e m p t i n g to construct a density f u n c t i o n a l for interacting polymers, we must develop methods for describing the single-polymer l i m i t . A t this stage, density f u n c t i o n a l theories exist for describing weakly p e r t u r b e d , b u l k p o l y m e r chains. W e w i s h to extend those theories to p o l y m e r fluids w h i c h , even i n the absence of applied fields, are h i g h l y inhomogeneous. I n the next part of this chapter, we study how i n d i v i d u a l chains respond to a n external field while i n the presence of a surface. T h e surface makes the p r o b l e m difficult since i t breaks the t r a n s l a t i o n a l invariance of the reference, field-free system. In the absence of a surface, our f o r m a l i s m reduces to the p r o b l e m of single polymers i n b u l k a n d reproduces the exist i n g approaches to t r a n s l a t i o n a l l y invariant p o l y m e r fluids (3,4)· O n the other

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.


18. M c M U L L E N

Density Functionals for Polymers at Surfaces

263

h a n d , our f o r m a l i s m is readily adaptable to other inhomogeneous polymeric flu ids (e.g., polymers near a corrugated b o u n d a r y or near the surface of a s m a l l , spherical, colloidal particle) whenever we possess sufficient statistical i n f o r m a t i o n about the field-free problem. W e note that many of the most i m p o r t a n t scientific a n d engineering applications of polymers involve surfaces (5), so a p r a c t i c a l motivation transcending theoretical or m a t h e m a t i c a l curiosity exists for focussing o n the planar polymer-surface geometry. T h e t h i r d section of this chapter describes such a n application to b i n a r y polymer blends. We show that correlations between monomers a n d a surface result i n a surface composition profile that decays to the b u l k more slowly t h a n previously predicted. T h i s fea ture of our results brings density functional theory into better agreement w i t h existing experimental studies (6). T h e last part of this chapter summarizes our methodology a n d results a n d outlines some of the l i m i t a t i o n s of the theory. D e r i v a t i o n of Ideal Density

Functionals

Consider a fluid composed of noninteracting polymers. We do not discount the possibility of nonbonded, intramolecular, monomer-monomer interactions although the most straightforward applications of our theory w i l l involve simple models lacking even those interactions. Besides the usual kinetic- a n d potentialenergy contributions, the H a m i l t o n i a n includes a n external field that acts on the monomers. Use μ to denote the chemical potential of a chain. In terms of the single-chain density operator Ν

M R ) = ]>>(R-ri),

(7)

the g r a n d p a r t i t i o n function reduces to ( 7) In Ξ = Ζ

(8)

where Z = ^Jdr exp N

-βΗ

χ

+ j

dRw(R)pi(R)

,

(9)

H\ is the H a m i t o n i a n of a single chain, Λ—with units of length—results from integrations over the monomer momenta, a n d w = — βν. H\ contains all interactions of the chain w i t h the surface a n d any i n tramolecular interactions. We seek a relation expressing external field w(r) i n terms of the average density given by


264


this result following from equations 7-9. A n t i c i p a t i n g that equation 10 w i l l reduce t o equation 6 when the field varies slowly, we express the density as

=

/

* * e - ' * e x p [ | < M r ) p i (')] ^ ( 1 ) J dr e-^p (l)ex y


N

1

drp (r) 1

P

(11) [w(r) - «;(!)]}

.

W h e n w does not vary over the volume of the polymer, the second line indeed simplifies to equation 6. M o r e generally, we ask: H o w does the presence of a slowly varying field alter the form of equation 6? Ultimately, the answer to this question w i l l enable us to approximately invert the field-density relation i n order that we c a n address the problem of determining the external field that leads to a p a r t i c u l a r density profile. We expand the second line of equation 11 as oo

η

χHMR 0 a n d tends to adsorb t h e m i f c < 0. A l l monomers are confined to the half-space ζ > 0. For this m o d e l , the correlation functions #2(21 > z ) a n d gz(*i,z ,zz) follow straightforwardly from the solution to the modified diffusion equation (9,10) 2

dg(z z \r) u

I

d g(z ,z ;r)

2

2

2

6

dr

1

2

2

; * i , z2

dz\

>

0

(27)

subject to the b o u n d a r y condition (10) 6c

dg(z z \r) u

2

dzi

.

x

(28)

zi=0

T h e quantity g(z\,z \r) is the p r o b a b i l i t y that a chain of length τ has one end at z\ a n d the other at z . Reference (8) describes the derivation of the correlation functions. T h e solution of equation 19a reveals that f\ has a deltafuntion singularity at the o r i g i n so that 2

2

h{z)

=

ft{z)-R%8{z)

(29)

where we use / J to denote the nonsingular part of f\. For finite c, b o t h f a n d fz are well behaved near ζ = 0. S u b s t i t u t i o n of equation 29 for each b l e n d component into equation 23 leads to the b o u n d a r y c o n d i t i o n 2

dP (z)

dP {z)

A

B

dz

dz

= 0

F r o m the definition of PA a n d PB—equation 1 po(z)

2=0

(30)

15—and a boundary condition 12c

dpp(z) dz

at ζ = 0.

Ρ

(31)

on the u n p e r t u r b e d densities, we find άφ

12


(32)

18. M c M U L L E N

269


for the p e r t u r b e d b l e n d . A second b o u n d a r y c o n d i t i o n (33)

φ(ζ —• oo) — 'oo φ {

ensures that φ(ζ) relaxes t o i t s b u l k value far f r o m the surface. These two b o u n d a r y conditions a n d f , a n d / 3 as functions of c enable us t o solve equation 23. 2

N u m e r i c a l l y we observe that the surface correlation functions fi, f , a n d fz decay t o their asymptotic values over a range comparable to RQ—the radius of g y r a t i o n of a single chain. F i g u r e 1 plots these functions for a representative value of c — i V / c / J . A l l s p a t i a l variations i n the surface correlation functions occur w i t h i n 2RQ of the surface, a n d f a n d / 3 vary less d r a m a t i c a l l y t h a n fi. T h e simplest density f u n c t i o n a l theories (11,12) of p o l y m e r adsorption employ a s y m p t o t i c values for the surface correlation functions a n d a b o u n d a r y c o n d i t i o n at ζ = 0 derived f r o m a n assumed, local surface c o n t r i b u t i o n to the free energy. T o assess the effects of the surface correlations, we compare, i n F i g u r e 2, the composition profiles obtained f r o m 23 a n d f r o m the older theory. G e n e r a l l y speaking, the improved theory predicts profiles that decay to their b u l k values more slowly t h a n those predicted b y the older theory. A l t h o u g h the effect is not pronounced, i t accounts, i n p a r t , for a systematic d e v i a t i o n of experimentally measured profiles (6) from the simpler density f u n c t i o n a l theory. O u r calculations agree, i n this sense, w i t h self-consistent-field studies of p o l y m e r a d s o r p t i o n (13) where surface correlations also dilate the surface profile. 2


1

2

2

Superficially, equation 23 exhibits some rather profound differences from analogous density functional theories (11,12) that ignore surface correlations. T h i s motivates a brief discussion of the predictions of our theory w i t h regards to surface phase transitions. Imagine a b i n a r y fluid m i x t u r e o n its b u l k coexistence curve so that the b u l k fluid consists of only one of the p a i r of phases. Suppose that the surface adsorbs most strongly the phase not present i n b u l k . W e refer to this as the A phase. A t low temperature, the surface fluid consists of droplets of A adsorbed t o the surface surrounded b y the other phase—the Β phase. S i m p l e scaling arguments (14) predict that as one follows the coexistence curve to higher temperatures, the droplets disappear i n favor of a macroscopically thick layer of phase A. T h e t r a n s i t i o n f r o m droplets t o a thick layer occurs at the wetting temperature T where the relation (14) w

7SB

= ISA

+ ΊΑΒ

(34)

holds between the surface tension JAB of the AB interface, the free energy *ysA of the surface-A-phase interface, a n d the free energy 7 5 5 of the surface-5-phase interface. For our m o d e l , the wetting t r a n s i t i o n is always first order. F i g u r e 3 plots the adsorption layer thickness (12) (35)


270


0.6 0.4 0.2 Fix) Downloaded by UNIV OF GUELPH LIBRARY on October 9, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch018

0 -0.2 -0.4 0

1

2 x

3

F i g u r e 1. T h e surface correlation functions / i , / 2 , a n d fz (denoted by F(x)) for c = -0.01. RQ is a n ideal-chain radius of g y r a t i o n a n d χ = Z/(2RG). S o l i d curve / i , long-dashed curve / 2 , a n d short-dashed curve fz.

0.300

0.275

φ(χ)

°·

2

5

0

0.225

0.200 0

1

2

3

4

Χ F i g u r e 2. C o m p o s i t i o n profiles near a p l a n a r surface of a blend i n the onephase region. χΝ = 2.31 a n d χ = z/(2Rc)S o l i d curve: T h e o r y of the present article using CA = -CJB = -0.025. Dashed curve: Phenomenological theory that ignores surface correlations. a n d fz proceeds readily. T h i s suggests that more sophisticated, even atomistic, models of chains could be simulated to o b t a i n g a n d #3. In that case, our f o r m a l i s m w o u l d enable a description of surface-polymer correlations that dispenses entirely w i t h the somewhat overly simplified E d w a r d s ' H a m i l t o n i a n . O f course, we cannot expect the theory to provide n o n t r i v i a l short-wavelength i n f o r m a t i o n about such systems beyond that contained i n the u n p e r t u r b e d density a n d correlation functions. 2


2

2

R e t u r n i n g to our specific application to a b l e n d , we recall that most phe nomenological theories (11,12) predict second- as well as first-order wetting transitions. We suspect that the absence of second-order wetting i n our calcula tions arises, i n p a r t , from our single-chain-surface H a m i l t o n i a n . Phenomenolog ical theories of surface adsorption generally assume a more h i g h l y parametrized surface H a m i l t o n i a n that accounts for b i n a r y - as well as single-monomer i n teractions w i t h the surface. T h i s H a m i l t o n i a n can exhibit a m i n i m u m i n the surface composition φ(0). W h e n this m i n i m u m coincides w i t h the concentra t i o n of one of the b u l k phases, the wetting t r a n s i t i o n can become second order. T h e H a m i l t o n i a n employed i n our theory contains only a purely repulsive or attractive c o n t r i b u t i o n a n d exhibits no such energy m i n i m u m . However, the differential equation 23 for the concentration profile contains surface correla tions a n d thus terms that do not appear i n the phenomenological analysis. Inspection of equation 23 does not immediately suggest how these can affect the wetting t r a n s i t i o n . O u r calculations show that surface correlations p r i m a r i l y dilate the interfacial profile, but this d i l a t i o n cannot, by itself, induce second-order wetting. F u t u r e applications of the theory should explore the p a rameter space of surface H a m i l t o n i a n s for second-order wetting more carefully. We w o u l d also hope to advance our treatment of intermonomer interactions i n those calculations since collective effects may significantly impact surface b o u n d a r y conditions.

Literature Cited

(1) Evans, R. Adv. Phys. (1989), 28, 143. (2) Evans, R. In Microscopic Theories of Simple Liquids and their Interface Charvolin, J.; Joanny, J.F.; Zinn-Justin, J., Ed.; Elsevier: Les Houches, 1989. (3) McMullen, W.E. In Physics of Polymer Surfaces and Interfaces; Sanchez, I.C., Ed.; Butterworth: London, 1992.



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(4) Tang, H.T.; Freed, K.F. J. Chem. Phys. (1991), 94, 3183. (5) Wu, S. Polymer Polymer Interface and Adhesion; Marcel Dekker: New York, 1982. (6) Jones, R.A.L.; Norton, L.J.; Kramer, E.J.; Composto, R.J.; Stein, R.S.; Russell, T.P.; Mansour, Α.; Karim, Α.; Felcher, G.P.; Rafailovich, M.H.; Sokolov, J.; Zhao, X.; Schwarz, S.A. Europhys. Lett. (1990), 12, 41. (7) McMullen, W.E.; Trache, M. J. Chem. Phys. (1995), 102, 1449. (8) Wong, K.Y.; Trache, M.; McMullen, W.E. J. Chem. Phys. in press (1995). (9) de Gennes, P.G. Scaling Concepts in Polymer Physics; Cornell University: Ithaca, NY, 1979. (10) Nemirovsky, Α.; Freed, K.F.; J. Chem. Phys. (1985), 83, 4166. (11) Nakanishi, H.; Pincus, P. J. Chem. Phys. (1983), 79, 997. (12) Schmidt, I.; Binder, K. J. Phys. (Paris) (1985), 46, 1631. (18) Carmesin, I.; Noolandi, J. Macromolecules (1989), 22, 1689. (14) Cahn, J.W. J. Chem. Phys. (1977), 66, 3367.


Density Functionals for Polymers at Surfaces - American Chemical

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