J . Phys. Chem. 1994, 98, 9312-9311
9312
Polymer Molecules at Chemically Random Surfacest K. Sumithra and K. L. Sebastian’ Department of Applied Chemistry, Cochin University of Science and Technology, Cochin 682022, India Received: April 1 1 , 1994; In Final Form: June 1 , 1994”
We consider the adsorption of polymers on the surface of a solid, occupying the region z < 0. The usual approach to the problem is the one due to de Gennes. We show that the propagator of this approach can be represented in terms of path integrals, where the paths are unconstrained and can enter the region z < 0 too. Using this approach, we consider adsorption on a flat, but random surface-where the randomness causes the adsorption energy to be a random function of position. Using the replica trick and variational formalism, we study the size of the adsorbed polymer. To simplify the calculations, we use the ground-state dominance approximation. The calculations revealed a sudden decrease in the size of the polymer, in both the parallel and perpendicular directions, as randomness is increased beyond a certain value. Further, thesizeof the polymer in the perpendicular direction is found to become zero at a larger value of the randomness. To verify whether these are artifacts of the ground-state dominance approximation, we also did exact calculations for test cases. It was found that the sudden change in size was absent in the exact calculations. Increasing randomness leads to a smooth, continuous decrease in the size. Ultimately, however, the polymer was found to collapse in the perpendicular direction.
Introduction
The adsorption of polymer molecules at surfaces has been the subject of many theoretical and experimental investigations. Most of the papers concentrate on the adsorption on planar Long ago, Rubin3 investigated the problem of adsorption of a single polymer chain without excluded volume on a homogeneous surface and showed that there exists a threshold energy for the adsorption of the chain. However, in almost all cases of polymer adsorption, surfaces are neither homogeneous in composition nor smooth. Although the type of heterogeneity (chemical or physical) may strongly influence the adsorption characteristics, these nonideal situations have received relatively little theoretical attention in the past,3-5 and the processes involved are only poorly understood. In a recent investigation, Baumgartner and Muthukumar4 considered the influenceof both physical and chemical roughness on the adsorption of polymers using scaling arguments and Monte Carlo simulations. In this paper, we investigate analytically the problem of polymer chain adsorption at planar but chemically random surfaces. The randomness may be due to impurities on the surface or because the surface is that of an alloy, which has a random distribution of its components on the surface. In an earlier paper,6 we presented a short description of our approach and gave results using the ground-state dominance approximation. In this paper, we give the details, along with results of an exact evaluation of the integrals. Adsorption on a Planar Surface: de Cennes’ Approach We first discuss the de Gennes’ approach1 to the adsorption of the polymers on a flat, uniform surface. Let G(i,i’;N)be the unnormalized probability distribution function for the end vector i , for a chain of length N , whose other end is at 7. G(i,i‘;N) obeys the diffusion equation
( d / d N - (Z/6)Vz)G(i.,?’;N) = 6(N) 6 ( i - i’)
(1)
repulsion of the polymer segments to it. This, according to de Gennes, may be modeled by putting
[d In G ( i , i ’ ; N ) / d ~ ] ,=, ~-co
(3)
The sign of the constant co depends on the temperature. At low temperatures, the attractive interaction between the polymer and the surface dominates, resulting in adsorption. In such cases, co is positive, whereasat higher temperatures the polymer isdesorbed, and correspondingly,co will be negative. It is convenientto expand G(i,i.’;N)in terms of the eigenfunctions t)m(i) of the problem, defined by (4)
where t)m(t)obeys
and are normalized according to
Then G(I,i’;N) is given by
(7) deGennes’prescription can easily beconvertedintoa path integral formulation in which the paths are unconstrained and can enter the region z < 0 too. Path Integral Representation for G(i,i’;N). A path integral expression for G(i,i’;N)is very convenient to use, in our analysis below. To arrive at such a representation, we consider {(i,i’;N), which corresponds to the polymer being not restricted by a hard wall but is attracted to the plane z = 0 by a delta function interaction. It obeys the differential equation
We imagine that there is a surface at z = 0 and adopt the notation (d/dN- (1/6)V2 - 1c06(z)/3){(i,i’;N)= 6(N) 6 ( i - i ’ ) The presence of the wall (surface) can lead to the attraction or
and has the path integral representation
f Submitted in honor of Professor C. N. R. Rao. *Abstract published in Advance ACS Abstracts, August 15, 1994.
0022-365419412098-9312$04.50/0
(8)
(9) 0 1994 American Chemical Society
Polymer Molecules at Chemically Random Surfaces
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9313
On performing the average over the random function v(2)
where
S,[i(t)] = (3/21)JON$(t)’ dt
exp{-S]G[i,(N) - i ] (20)
(1 1)
In Appendix A, we show that G(i,i’;N) can be written as G(i,i’;N) = ((i,i’;N)
+ r(Pi,i’;N)
(21) P-
(12)
where Pi = (2,-z). Adsorption on a Random Surface
where SIis given by
S , = ‘/’[1/312
(13)
The function u(2) is a random function of 2. Note that this can be used to model only variation of adsorption energy with position on the surface and would not be a model for the case where the surface is rough. An equation similar to the eq 12 is still valid and may be written as G(i,i’;N) = [(i,i’;N)
+ {(Pi,i’;N)
2
s,”dsJ:dt
B(2&) - 2&)) x
a.B=1
To treat adsorption on a random surface, we now generalize the approach of de Gennes and put
[a In G(i,i’;N)/d~],,~ = -(co + u(2))
1
G(zg(t)) 6(z,(s)) (22) Our interest is only in the adsorbed molecules, for which we would like to calculate the size ( [ F ( N ) - i ( 0 ) l 2 ) .The integral in eq 20 cannot be evaluated analytically. Therefore we adopt the variational formulationof Feynman’s path integral technique.* For this we need a trial action, which should be such that all the integrals can be evaluated analytically. An action for which this can be done is
(14)
with {(i,F‘;N)defined by where where
S,[i(s)] = SO[i(s)]- (lc/3)fG[z(s)]
S,,,[~(s)l = S,[Wl - (1/3)s,Nds v ( W > a z w 1
(16)
It is quite interesting to note that the paths in the path integral in eq 15 can enter the region z < 0 too. To get the normalized probability distribution, we divide G(i,i’;N)by N which is given by
To calculate the mean square end-to-end vector or any other averagequantity, we have to perform the averageover the random function ~(2). To perform this average, it is convenient to take the random function to be Gaussian with vanishing average and to have correlation function ( ~ ( 3~ )( 2 3=) B(2-23,where (...) denotes averaging with respect to the random function. To be specific, we consider randomness with a correlation length Ro and take B(?) = VZ e~p[-?~/R~Zl, where V is a parameter describing the strength of the randomness. Let us denote the normalized probability distribution function by P(i,i’;N). It is given by
P(i,i’;N) = G(i,i’;N)/N
q2/(121N)fdsfdt
ds
+ [2(s)
- 2(t)]*
(24)
Our action in the eq 24 has a delta function like attractive interaction with the surface. This interaction is an effective one and has no ?dependence. On the other hand, there is a “harmonic oscillator”-like term depending on 2. It is more common to use the purely harmonic oscillator term Jf[?(t)]2 dr.9 However, this has the defect of not being translationally invariant. Note that the action in eq 24 is separable as far as z and 3 coordinates are concerned. For the z direction, one has a path integral involving a Diracdelta function in its action. This can be found by changing over to the corresponding diffusion equation, which involves the operator (a/aN- (1/6)82/az2- lcS(z)/3j. On the other hand, the path integral involving ? may appear difficult, as it involves a nonlocal action. This also can be done easily,1°as is demonstrated below: Evaluation of Path Integrals Involving the Nonlocal Action. Let us consider any path integral involving the nonlocal action. We can write it as
(18)
G(i,i’;N)is a functional of v(2) and is still given by eq 14. The quantity of interest is (P(i,i’;N)). To calculate the average of P(i,F’;N), we use the replica trick.’ Thus we are interested in calculating ( {(F,i’;iV)/N) as (P(i,i’;N))can be obtained from this easily. Introducing n replicas, labeled with a = 1, 2, ..., n, one can write
(q2/121~JoNdsJoNdt[2(s) - 2(?)]2]F[2(s)] (25) In the above, F[?(s)J is any arbitrary functional of the path 2(s). We now make use of the identity exp{-(q2/ 12lN)fdsfdr
[a($)- 2(t)I2) =
(q2N/6r1)Jdf exp(-(q2/61)JoNdt in eq 25 to obtain
V - 2(t)]’) (26)
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Sumithra and Sebastian
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994
Er
I = (q2N/6xl)Sdj$~L~;fDk(s)exp[-$JNdt
-
1
(42/60fdt [ V t ) -A2 F [ W l (27) In the above equations, j = Cy1,y2)is a two-dimensional vector. The path integral of the eq 27 involves only a local quadratic action. Therefore, if F[k(s)] is such that the path integral involving a local, quadratic action can be evaluated, then the one involving the nonlocal one can also be evaluated! As a typical example, consider the following probability distribution function for the end vector i for a polymer, whose other end is at i’, if the action were St. Then P,(F,i’;N) = Z(?,i!;N)/Jdi
Z(i,i’;N)
(28)
Explicit expression for Gs(z,z’;N) is available (see refs 11-13). Hence the probability P,(i,F’;N) is known. If c > 0, then the operator -(1/6)@/az2 - lcS(z)/3 has one negative eigenvalue, given by Eb = -lc2/6, with the associated eigenfunction +&) = 4 exp(-clzl). This corresponds to the adsorbed (bound) state of the polymer. As the paths are not restricted to the region z > 0, the normalization that we use is J_”dz $b(Z)’ = 1. In the limit where N becomes very large, the two propagators Ga(z,z’;N) and Ghd(2,k’;N) are dominated by the eigenfunctions having the lowest eigenvalues (ground-state dominance approximation) and one gets G,(z,z’;N) = c exp(-c()zl
Gho(k,k’;N)= (q/x1) eXp[-q(k2
+ 121) - cdv)
+ k’)/(2/)
(38)
- qN/3]
(39)
Remembering that the variational parameters c and q have to be determined so as to best suit the description of adsorbed polymers, we make use of the Feynman’s variational procedure7 for their evaluation. Thus, we expect the integral
where
Using eq 26, it is possible to write Pt(?,i’;N) as
Pt(i,i’;N) =
s) exp(-S{ J d jpN)=iDi( r(O)=i‘
-
[i ( s ) ] )/
to behave like exp[-nF~(N)]for n 0 and large N. We estimate F l ( N ) variationally, in the limit n -,0 and choose c and q so as to get the best approximation for it. These parameters are then used to calculate ( [ i ( N )- i(0)]2). So we now consider Z(n), and write it as
where S{[i(s)]= S,[F(s)] - (k/3)f6[z(s)]
ds
+
(q2/61)fds
[W) -312 (31)
The denominator in eq 30 is simply a normalization factor which ensures that P,(i,i’;Nj is normalized. As the k and z are not coupled in the eqs 30 and 31, one gets
Pt(i,i’;N) = I , (i,i’;N)/ S d i I , (F,i’;N)
(32)
zl(?,?’;N) = G&z,z’;N)Jdj Gh?(k,k’;N)
(33)
with
Ga(z,z’;N) is the propagator for Brownian motion in the presence of a delta function sink and obeys the differential equation
(a/aN - (r/6)a2/azz - I C ~ ( Z ) / ~ ) G , ( ~ , ~=’ ; N ) 6(N)
- z ? (34)
eo(!,?’;N) is the propagator for two-dimensional Brownian
motion in the presence of a parabolic sink of the form (q2/61)(2 - J ) 2 , having origin at 3. It satisfies (d/dN- (1/6)Vi - (q2/61)(k -j)2)@,o(k,k’;N) = 6 ( N ) d(k - 39 (35)
Note that
Go(k,k’;N) = Gho(k-j,k’-?;N)
(36)
where G ~isothe propagator corresponding to the usual harmonic oscillator action and is given by
Gho(k,k’;N)= [4/2n1 sinh(qN/3)]
X
exp[-(q/21 sinh(qN/3))[cosh(qN/3)(k2
+ 2 ” ) - 2kk’jI (37)
where denotes averaging with respect to the trial action ST. 2,is defined in Appendix B. 2,and the other integrals can be evaluated analytically in the ground-state dominance approximation and hence Fl(N) may be found. The integrals can also be evaluated exactly, though for some of them, one has to perform numerical integration. The details of these calculations are given in the Appendixes B and C. Results and Discussion Ground-State DominanceApproximation. The expressionthat one obtains for FI,in the ground-state dominance approximation, written in terms of the dimensionless variables N = N/I, E = cl, p = ql, Ro = Roll, Eo = col, P = VI, is
P,/R = [ E 2 - 2EE0 + q - (r22v2RR;]/ {3(2 + &3(4
+ &71)1/6
+
(42)
+
Putting tr = gRO2,p= 2 N P / 3 andfltr) =ptr/[(2 @)(6 a)], we can rearrange the above equation-to write an expression for Fl(ii,C), defined to be equal to 6F1/N as P,(ii,E)=
[E2
- 2CE0
+ @ / R ;- 2 f l i i ) ]
(43)
In eq 43, E2 represents the effect of entropy (in quantum mechanical parlance kinetic energy), trying to spread out the adsorbed chain, increasing its thickness, -2i~0,the lowering of energy resulting from adsorption, ii/Ro2, the effect of localizing the chain to dimensions of q-1 in directions parallel to the surface and - E 2 f ( i i ) , the effect of the randomness, which results in a net attraction between the chains, as is indicated by the negative sign. Note that this term has 4,because of the lowering of energy caused by adsorption. We now find the best values of @ and E, which make F l ( t r , ~a) minimum. Finding the value of E
The Journal of Physical Chemistry, Vol. 98, No. 37. 1994 9315
Polymer Molecules at Chemically Random Surfaces
- 01
such that Pl(ir,t) is a minimum gives
=3.0 p1=5.0 1.
e = co/(l -An))
p=IO
(44)
Note that if is negative, there is no bound (adsorbed) state, which violates our basic assumption in deriving the eq 43. Therefore only t > 0 is acceptable to us. This happens only if f(P) < 1. Finding the value of ir for whichflir) is a maximum, one gets P. = dg.So ifflamax)< I, one would never have e < 0. f(P,.) < 1 implies p < PO.where po = 6 4 4 % Hence for p > PO.the solution given by the eq 44 is not acceptable. The reason for this is simple. Asp -PO. 2- m; that is, the thickness ofthe polymer in the perpendicular direction becomes verysmall. I f p > PO,then the value o f t that minimizes Fl(P,c) is m. That is the polymer is collapsed (has a thickness = 0) in the perpendicular direction! Now consider p 0.
collapse in the perpendicular direction
(45)
where g(0)
= P -p,fr/[(2
+ P)(4 + a) -pa]
(46)
with pI = ?02&2p.We give in Figure 1 a plot of g(U) against P, for different values of pI.for p = 10. If pI > 8, then the slope of the curve at ir = 0 is negative and there is minimum at a ir # 0. On the other hand, if pI < 8, then the slope of g(P) at G = 0 is positive and hence tr = 0 is a minimum. However therecould be another, lower minimum, at a nonzero 0. This would happen iftheg(ir)-ir plot cuts the horizontal axis for P > 0. This is true for values of p > 16 2(8 - pl)l12], with 0 S pI S 8. So a s p crosses the value [6 2(8 - pl)I12], with 0 5 pI 5 8, the value of P at which the minimum occurs, viz., Pmln would change discontinuously from thevalueofzero toa nonzerovalue. These results are summarized in the Figure 2, in which we have plotted the values of ir which makes g(P) a minimum, viz., 0.1" against pandpl. Forp>po,thepolymeriscollapsed. Forp pa, the polymer is collapsed in the perpendicular direction. (2)p> I , the polymer has a finite size, in the parallel direction which can be found from eq 41 to be given by ([.?(N)-.?(O)]Z)/P
-
= ./zii;
-
(48)
-
Note that this is independent of N. On the other hand, imagine that Ro m . Then again, pI m, and Pmin dg.Then if (8)112N/6&2
