J. Phys. Chem. 1995, 99, 2788-2796
2788
Effect of Copolymer Sequence Distribution upon Surface Structure and Interfacial Tension As Predicted by a Self-consistent Field Lattice Model Arvind Hariharan and Jonathan G. Harris* Department of Chemical Engineering, The Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received: September 2, 1994; In Final Form: November 12, 1994@
We study the structure and interfacial tension of the free surface of copolymers with different sequence distributions using a compressible mean-field lattice theory. The results of the analysis show that the degree of surface segregation caused by the differences in the interaction energies between the two components depends strongly on the type of copolymer architecture. For a given chemistry, diblock copolymers exhibit larger degrees of surface segregation than triblock copolymers. In completely random and altemating copolymer melts the surface enrichment is negligible. These trends hold for all copolymer compositions. Although each architecture displays a unique structure at the free surface and is characterized by different degrees of surface segregation, the bond orientation and shape characteristics are similar to that of a homopolymer melt. A study of the surface tension of the various copolymers shows that the architectures with the maximum degree of surface segregation have the lowest surface tension.
1. Introduction The surface and interfacial behavior of copolymer melts is a topic that has been studied rigorously both e~perimentallyl-~ and The motivation for studying these materials arises from a desire to understand the effects of the unique molecular architecture characterizingthem. It is the microscopic connectivity between the two types of components making up the copolymer that leads to the formation of interesting microstructures and surface behavior. While several theories have been devoted to studying the surface behavior of symmetric diblock copolymer melts,"-' less is known about asymmetric diblock copolymers and the effects of chain architecture on the free surface. In one noteworthy work, Theodorou examined the behavior of incompressible diblock and triblock copolymer melts in the vicinity of an interacting and impenetrable Our interests lie in understanding the structure of the free surface.13 In this paper, we will examine the effects of chain architecture on the free surface by comparing the surface structure and thermodynamics of diblock, triblock, multiblock, and random copolymers melts. We will confine ourselves to situations in which the copolymer melt is disordered in the bulk. In a previous p~blication,'~ we described the theory and methods we use in this work. There we developed the theory of the free surface of copolymer melts and applied it to the case of a symmetric diblock copolymer melt. This theory is essentially different from previous t h e o r i e ~ ~ . ~ - ~ % ' have that examined the surface behavior of diblock copolymer melts in that we relax the assumption of incompressibility that past theories have traditionally used. Therefore in our case, 1914315
where @A(r) and @B(r) are the volume fractions of the two components of a copolymer making up the melt at point r and QP(r)(not a constant) is the total volume fraction of polymer. In the theory, QP(r)is determined by the thermodynamic state of the system. Recently, Tang and Freed developed a theory @
Abstract published in Advance ACS Absrructs, February 1, 1995.
0022-3654/95/2099-2788$09.00IQ
of compressible diblock copolymers and examined the influence of the free volume on the static structure factor.16 We formulated the problem by modeling the free volume of the copolymer melt in the framework of the self-consistent field lattice mean-field theory of polymer solutions of Scheutjens and Fleer.12,13~i7-'8 We assumed that each lattice site under consideration is either occupied by a segment of the copolymer chain or is unoccupied. The total number of unoccupied sites determines the amount of free volume in the melt and hence the bulk density of the melt. We then derived the equation of state of the system in the spirit of Sanchez and Lacombe to determine the bulk polymer d e n ~ i t y 'at~ liquid-vapor .~~ coexistence. In the earlier publication, we demonstrated that using the liquid density at which the predicted pressure is 0 results in an error that is imperceptible since the vapor pressure of truly polymeric (long chain) systems is of the order of atm. At conditions in which the polymer exists as a melt, the system phase separates into a vapor phase rich in holes (or unoccupied sites) and a polymer melt phase which is enriched in polymer segments. The interface between the two therefore constitutes a realistic description of the free surface of the copolymer melt. l 3 Unlike phenomenological theories' or incompressible polymer melt ours produces an a priori prediction of the free surfaces' structure and properties, such as interfacial tension, derived purely from fundamental segment- segment interactions. In the next section we summarize the formalism. We then present the theory's predictions of interfacial structure and surface tension for polymers of several different architectures.
2. Theory and Definitions In this section, we define some of the important quantities that are used in the subsequent parts of the paper. Our previous paperi3describes the derivation of this theory. To save journal space we will omit the details of the formalism from this work. A short description of the physics follows. We consider a lattice of M layers and L sites per layer. A homopolymer consisting of a single monomer (its length, r;!is 1) represents an empty site or hole. Each site contains one monomer which is either part of a copolymer of length 11 or a 0 1995 American Chemical Society
Structure of the Free Surface of Copolymers
J. Phys. Chem., Vol. 99, No. 9, 1995 2789
homopolymer (i.e., it is a hole). The copolymer consists of rA segments of type A and I-B segments of type B and the fractional composition of each segment type in the copolymer is defined as
so that fA +fB
(3)
=
A homopolymer segment does not interact with any other segments, except by excluding them from its lattice site. This second component plays the role of vacant and noninteracting sites, mimicking the free volume in the copolymer The only nonzero energetic interactions that exist in the melt are characterized by the intracomponent nearest-neighbor attractive energies EAA and EBB between parts of A and B segments, respectively, and EAB the interaction between A and B segments. We define the Flory parameter to be,22,23
x
(4) where z is the lattice coordination number. It is convenient to write E A B in the form
Setting y = 1 generates the familiar geometric mean rule for cross interactions. In this paper, we will consider only cases in which y = 1 and all interaction parameters are negative. In order to define a system completely, the following quantities must be specified: (1) the total chain length of the copolymer (rI); ( 2 ) the temperature T; (3) the number of A segments (rA = rl - Q) which fixes the composition, f ~ of, the copolymer; (4) the identity of each monomer of the copolymer as a type A or type B monomer; (5) the intra- and intercomponent energetic interaction which we will describe in terms of characteristic temperatures
where I and J refer to the segment types Le., A or B. For all calculations in this paper we assumed y = 1; (6) the bulk copolymer density at liquid-vapor coexistence, @P,b, which we compute from the equation of state of the liquid copolymer by setting the pressure to be zero; (7) the number of lattice layers, M. The basic assumption behind this theory is a simple generalization of the Bragg -Williams approximation to an inhomogeneous system. When the interface is planar as in the cases discussed in this work, the segment densities are functions of the direction normal to the surface which we denote as z. A segment in layer i interacts with the smeared out arrangement of segments in layer i and in the two adjacent layer^.^^.^^ The theoryI3 also takes into account chain connectivity in that if the sth segment of a chain is in layer i, the (s 1)th segment must be layer i - 1, i, or i 1. Essentially, the volume fraction @I,; of component I in layer i is
+
directions of the chain, are generated recursively from the free segment probabilities PI,;of the two components. The Kronecker delta function ensures that only segments of the particular type t = A, B contribute to the sum. The formal description of the derivation of this quantity and the definition of the end segments probabilities appear in refs 9 and 11. The determination of the segment density profiles requires the solution of a system of nonlinear equations. These nonlinear equations arise from writing the canonical partition function as a functional of the population of all conformations of the chains. A conformation is defined by the identity of the lattice layers in which each chain segment lies. Reference 13 and the works cited there describe the derivation of the nonlinear equations for the segment density distributions from the mean field approximation to the canonical partition function. In the following sections we make a comparative study of five different architectures of copolymers: (1) diblocks, ( 2 ) triblocks (A-B-A), (3) triblocks (B-A-B), (4)altemating copolymers, and ( 5 ) random copolymers. We will study the effect of changing the compositions of these copolymers and then compare the various architectures. In the case of random copolymers, this theory is evidently not sufficient since it assumes that all the copolymer chains in the melt are identical to each other. If we choose a single Bemoullian sequence to describe a random copolymer, the solutions of the self-consistent field equations are representative of a melt in which every single chain is characterized by that particular sequence. A more exact calculation for the random copolymers requires the modeling of a melt comprised of several chains, each of which is characterized by (in general) a different random sequence. We treat this situation by generalizing the theory to treat a multicomponent system. We define Me¶to be the number of different components in the melt. Each component of the melt is a different random sequence of monomers. All of these copolymers have the same chain length and the same fraction of A-mers. The volume fraction of I units in layer i , @i:q is
where n;:q is the number of type I segments in layer i that belong to the chains comprising the particular sequence seq. The bulk melt density of component I , @I,b, is therefore
(9) where the summation, C p ,is over all the sequences that we chosen for the calculation. The derivation of the partition function and the conditions of equilibrium are then easily extended to this multi architecture system. The net result is that the component density, @I,;, in layer i is Qi,,i
+
where t(s) is the type of segment s of the chain. p f ( i , s) and p-(i,r1 -s+l), the end segment probabilities along the two
where
=
c
according to the Scheutjens and Fleer formalism is
One generates the end segment probabilities recursively by
Hariharan and Harris
2790 J. Phys. Chem., Vol. 99, No. 9, I995
utilizing the free segment probabilities, pl,i, and the interface structure using the same methods described above and in ref 13. To define the random sequence, we generated r A distinct random integers on the interval [ 1,rl] from a uniform distribution. The monomers with these sequence numbers became the A-mers and the others became the B-mers. This method ensured that the composition of each component is exactly fA. Additionally we assume that the bulk melt has the same concentration of each of the Me¶components, i.e., @ =. S @l.dMeq. iq
3. Results and Discussion 3.1. Segment Density Distributions. We begin by comparing the segment density profiles of the five architectures that we defined above. One-half of the monomers in the copolymers in the following section will be A-mers ( f ~= O S ) , and the interaction parameters will be F A A= 600 K, F B B= 800 K, r1 = 200 at a temperature of 400 K ( F A A and F B B are within the range of the T* values of typical commercial Any differences between the segment density profiles therefore must arise from the only feature that distinguishes one from the other, Le., the sequence distribution. The five different structures are (A)loo-(B)~oo, A~o-BIwA50, B~o-AIM-B~o,(A-B)lm, and, ABrandomwhere the last architecture denotes a random copolymer WithfA = 0.5. In all the calculations we utilized 100 layers to solve the equations. We found this size to be adequate because bulk properties were attained far from the free surface, and in our previous work system properties were size independent for 100 layer systems.I3 For the sake of convenience in comparing the free surfaces of various architectures, we defined i = 0 to be at the Gibbs dividing ~ u r f a c e * ~of . * ~the total polymer density profile so that all layer numbers on the vapor side of the Gibbs dividing surface are negative and those on the liquid side are positive. For the random copolymer we used 40 sequences. We have found the predicted properties to have only a small dependence upon the number of different sequences for Me4 > 10 and averages over many single-component systems of copolymers with different random sequences to be almost identical to the averages for one multicomponent system, where each component is a different random sequence. Figure la-e displays plots of the segment density profiles of the five types of architectures considered. In each of the figures (a-e) we have plotted, @p,i, the total polymer density and O!,;, (I = A, B), the densities of each of the components A and B, respectively. For each case the total polymer density @p,i decays to a value of @p,b in the slab’s center, where @p,b is the value predicted by the equation of state derived for copolymeric systems. Because the equation of state depends only on the interaction parameters, the temperature, and the fraction of A monomers in the polymer, it predicts the same value of @p,b for dl the systems. Since we consider only situations where the system is in the disordered state in the bulk, this approximation is adequate. In microphase separated systems, such as lamellar phases, it would be necessary to relax this approximation in at least one direction to account for the composition variations normal to the lamellae. From Figure 1 it is clear that, although the bulk volume fractions of these systems are identical, the interfacial structures are clearly different from each other. Each of the component densities exhibits oscillations and different degrees of surface segregation and then decay to a value of ‘/2@P,b = Q1.b (I= A, B) in the bulk since the composition of the copolymer is 0.5 in all cases.
The free surfaces of the symmetric diblock copolymer and the triblock copolymer melts are enriched in component A which has the weaker interactions and therefore are depleted in the other component, B. The random copolymer and the altemating copolymer, on the other hand, exhibit almost no surface segregation. The most surface segregation occurs in the symmetric diblock copolymer melt (Figure la). This is evident from the fact not only that is the free surface almost saturated by component A but also that the oscillations arising from the connectivity of the block components are much stronger and penetrate deeper into the middle of this film (up to ca. 60 layers). In contrast, in the triblocks the oscillations die out so the melt achieves its bulk composition at i = 20. In addition, the period of the oscillation is greater for the diblock copolymer than the triblock as this quantity is related to the length of the blocks making up the copolymer (the lengths of the A sequences in the troblock copolymer molecule are half the size of the A sequence in the diblock copolymer). The random and altemate copolymers clearly do not display any oscillations at all. As discussed in our earlier paper,I3 there are two main factors that drive the surface segregation at the free surface. Firstly, the difference between the characteristic temperatures of the two components produces a driving force that causes the component with the weaker interactions to partition to the surface. This results in the minimization of the increase in the free energy that the system suffers due to the presence of the surface. Secondly, the segmental Flory parameter XAB, which is the net enthalpy of mixing of the two types of components, characterizes the extent of incompatibility between the two components. This incompatibility drives the microphase separation which occurs for sufficiently large positive values of XAB. While the incompatibility is not required for surface segregation, it induces the oscillations in the composition profile which penetrate deep into the melt. In our previous work we demonstrated this effect by comparing density profiles of polymers with identical interaction parameters but different XAB’S.
On the basis of these facts it appears that the diblock melt is closer to phase separation than the triblock melt. The physical reason for this observation is that the diblock has longer sequences of identical monomers and fewer connectivity constraints that prevent phase separation. In the diblock there is only one bond forcing A and B segments to be near each other, while in the triblock there are two such bonds. This delays the onset of phase separation in the triblock (i.e., it will take place at higher chain lengths and lower temperatures than that of a diblock). In this class of two-component copolymers, the diblock copolymer and the altemating copolymer lie on the extremes of the “miscibility spectrum”. Based on our argument, the reason for this is that in a diblock copolymer the two components are connected by the minimum possible number of covalent linkages (one linkage). In a perfectly alternating copolymer, the two components are linked by the maximum possible number of covalent linkages (rJ2 linkages) which force A and B segments to be near each other. Such an alternating copolymer could be considered as a homopolymer with an (AB) repeat unit. A measure of the magnitude of the segregation at the free surface for all these systems is the relative enhancement of a component at the Gibbs dividing surface
where (PI,, (I = A, B) is the volume fraction of component I
J. Phys. Chem., Vol. 99, No. 9, 1995 2791
Structure of the Free Surface of Copolymers 1
cn
c
.-0 Y
> a
0.75
-
0.5
-
0.25
-
.f
i
$9
3 v)
E"
0.25
0 -5
20
45
-5
70
0
Layer ## (i)
5
10
15
20
Layer # (i) 1
1
I I
I
0 -5
20
45
-5
70
0
5
IO
15
20
Layer # (i)
Layer ## (i) 1
I Y
-5
I
20
45
70
Layer ## (i)
Types of Copolymers
Figure 1. (a-e) Segment density distributions for the five different architectures studied: (a) block copolymer, (b) triblock copolymer (ABA), (c) triblock copolymer (BAB), (d) random copolymer, and (e) alternating copolymer. The copolymers have properties r-1 = 200, T = 400 K, F A A = 600 K, and FRR = 800 K. In each of the figures the segment density profiles of component A @A,i (circle), component B @ R . ~ (square), and the whole polymer chain @p.i (up triangle) are compared to each other. Part e compares the relative volume fraction @:.G of component B at the Gibbs surface for each of the types.
and @P.G is the total volume fraction of the polymer at the Gibbs surface respectively. Figure If displays a bar chart of as a function of copolymer type. As can be seen, for alternating M 0.5, denoting that there is practically no copolymers, aisG
surface segregation. For random copolymers, there is a a small but noticeable amount of component A at the Gibbs surface. This slight segregation can occur because of existence of occasional long sequences of a identical monomers. There is
Hariharan and Harris
2792 J. Phys. Chem., Vol. 99, No. 9, 1995 0.008 I
Y
-
0.008
eE %
Y
9 e-
5
e,
0.002
-
2 v)
v)
0.002
d
a 0
-5
I
I
20
45
0 -5
70
25
10
40
Layer # (i)
Layer # (i) 0.008
I
I
eE 'e-
c)
2
v)
-5
20
45
70
20
45
20
45
Layer # (i)
Layer # (i)
-5
-5
70
Layer # (i) Figure 2. End and middle segment density distributions for the five different architectures studied in Figure 1: (a) block copolymer, (b) triblock copolymer (ABA), (c) triblock copolymer (BAB), (d) random copolymer, and (e) alternating copolymer. Each figure illustrates the density profiles of end 1 of component A (circle), end rl of component B (square), and the middle segments (up triangle) are compared to each other.
significant surface segregation in the two triblock copolymer melts, and the most surface segregation appears in the diblock copolymer melts. These examples show a trend in which the surface segregation is stronger in copolymers containing longer sequences of identical segments. This can be explained by the fact that in a copolymer of given composition, long sequences of identical monomers with weaker interactions can form trains
at the surface which have a greater effect on the surface energy than a shorter train of similar molecules. For the perfectly altemating copolymer, the size of the largest sequence is 1 and hence this architecture exhibits the least amount of surface segregation. Finally, we compare the two different triblocks in Figure lb, and c. The total segment density profiles are similar. While
Structure of the Free Surface of Copolymers the number and amplitude of oscillations are evidently identical for both cases, a quantitative comparison of the volume fraction of B at the Gibbs dividing surface, for both the cases in Figure If shows that @E,G(ABA)is only slightly less than @i,G(BAB). This demonstrates that the inversion of the component types in the triblock copolymer does not make a large difference in the magnitude of the segregation at the free surface. The slight difference, however, arises from the fact that entropic effects that drive the ends to the surface add to the energetic effects when the weaker component forms the end blocks. A more detailed study of the structure in the next section will outline the subtle differences between the melt-vapor interface structure of the two triblock architectures. 3.2. End and Middle Segment Density Distributions. In this section we study the more fine-grained features of surface structure by examining the end and middle segment density profiles of the different copolymer species. These quantities provide a more detailed picture of the structure of the chain molecules at the interface than the overall segment density pr0fi1es.I~ Figure 2a-e displays the end and middle segment density profiles of the various architectures we considered. Figure 2a shows the end and middle segment densities of the symmetric diblock copolymer. Here, the end with the weaker interactions probes out into the vapor side of the free surface while the other end lies inside the liquid so that the middle beads are concentrated in the space between them. As one proceeds into the liquid, the chains are aligned in a similar fashion; i.e., the ends of the molecules lie where the density of the corresponding component is highest and the middle beads lie mostly in the regions between the peaks in the A and B monomer densities. The oscillations then decay into the bulk region. The behavior of the end and middle bead densities of the triblock copolymers is quite different. In comparison with the diblock copolymer we note that the both the end segment densities are identical to each other since the triblock (ABA or BAB) copolymer molecule is symmetric about the middle of the chain. The end densities are, however, out of phase with the middle density. A comparison between the two triblock copolymer melts delineates the differences in the arrangement of molecules at the free surface. In the ABA triblock melt we note that the two ends (of type A) partition to the free surface while the middle beads exhibit a peak deeper in the liquid. Thus, on the average a triblock molecule assumes a looplike structure with the ends of the molecule poking out from the liquid into the free surface. On the other hand, in the BAB triblock melt the opposite behavior appears; the middle beads partition to the surface, and the end beads exhibit a peak deeper in the liquid. Therefore, the molecule still assumes a looplike shape except that the loop is oriented so that the ends lie in the liquid and the middle points to the vapor. The altemating copolymer exhibits almost no surface segregation of the ends beyond that seen in a homopolymer melt under these conditions. The ends show a slight tendency to prefer the vapor side of the surface and only the slightest stratification appears. The random copolymer exhibits a slightly greater degree of stratification with a definite separation between the two ends and the middles. The random copolymer is distinct from the alternating one in that the short blocks (contiguous sequences of A or B) appear on the chain and can thus induce such segregation. 3.3. Orientation and Shape Parameters. In order to study the orientation and shape of the chains in the vicinity of the interface we define the bond-order parameter, Si, and the shape
J. Phys. Chem., Vol. 99, No. 9, 1995 2793
Si
I
1
I
I
I
-2
2
6
10
14
Layer # (i) Figure 3. Variation of the segmental order parameter as a function of the distance perpendicular to the film for the five different architectures discussed in Figure 1, Le., diblock (down triangle), triblock (ABA) (circle), triblock (BAB) (square), random (up triangle), and alternating
copolymers (+).
-5
0
5
10
15
20
2.5
Layer # (i) Figure 4. Shape parameter profiles for the five different architectures discussed in Figure 1, Le., block (down triangle), triblock (ABA) (circle),triblock (BAB) (square),random (up triangle), and alternating copolymers (+).
parameter, n,i. The former quantity is defined as11313
where 0 is the angle between a bond and the surface normal. A layer containing only bonds that lie perpendicular to the surface will have an Si = 1, while a layer in which all of the bonds lie perpendicular to the surface will have Si = -0.5, and one with all randomly oriented bonds will have Si = 0. The shape parameter1'-l3is the average number of segments in layer i that belong to a chain passing through that layer. Therefore, large values of n,i imply that the chains are flattened parallel to the surface in layer i and small values imply that the chains are elongated in the normal direction to the surface in that layer. Figures 3 and 4 show that the variations of the chain orientational order parameter and chain shape parameter across the liquid-vapor interface are only weakly affected by changes in the sequence distribution. In fact, Figures 3 and 4 resemble the corresponding figures for homopolymers. The chains are
2794 J. Phys. Chem., Vol. 99, No. 9, 1995
Hariharan and Harris
0 75
0.5
0.25
0
e
0 5
5
15
25
-5
35
I
I
I
I
5
15
25
35
Layer # (i)
Layer # (i)
0.75
0.5
0.5
0.25
0.25
0
0 -5
5
15
25
35
-5
5
Layer # (i)
35
25
15
Layer # (i)
C 5
I
I
1
I
5
15
25
35
-5
0
Layer # (i)
5
IO
15
20
Layer # (i)
,
s
15
25
Layer # (i)
35
0
I
I
I
5
IO
I5
Layer # (i)
Figure 5. Segment density profiles of various copolymer melts with asymmetric compositions. Here, rl = 200, T = 400 K, Pa = 600 K, and 7% = 800 K. Each figure compares the segment density profiles of component A, (PA,, (circle), component B, (PB,, (square), and the whole polymer chain, (Pp,!, (up triangle), are compared to each other. slightly stretched out perpendicular to the interface in the vapor effect is slightly greater in the diblocks and triblocks than in region and parallel to the interface in the liquid region. The the alternating and random copolymer architectures.
J. Phys. Chem., Vol. 99, No. 9, 1995 2795
Structure of the Free Surface of Copolymers
0.8
R
0.6
@B,G 0.4
0.2
0 0
50
IO0
150
200
# of B Segments
0
50
IO0
150
200
## of B Segments
Figure 6. Relative volume fraction, Q,G. of component B at the Gibbs surface as a function of the number of B segments on the chain. The system is characterized by rl = 200, T = 400 K, FAA = 600 K, and FBB = 800 K. Here we compare the diblock copolymer (up triangle), triblock ABA (square), triblock BAB (down triangle), and the random copolymer (circle) consisting of 40 components. The solid line passing through the circles represents the average of the results of 10 different copolymer melts (dotted lines) each Characterized by a single random sequence. Dashed line connects the two points represents the surface composition that would appear if there were no surface segregation.
Figure 7. Variation of the reduced surface tension, Talk& as a function of the number of B segments on the chain for the systems discussed in Figure 6. Here we compare the diblock copolymer (up triangle), triblock ABA (square), triblock BAB (down triangle), and the random copolymer (circle) consisting of 40 components. The solid line passing through the circles represents the average of the results of 10 different copolymer melts (dotted lines) each characterized by a single random sequence. The dashed line indicates the averages of the two homopolymers weighted by the fractions of the two monomers in the system.
3.4. Effect of Composition. To study the effect of composition on the surface structures we consider the copolymers with 200 segments with T*AA= 600 K, PBB = 800 K at a temperature of T = 400 K. For each of four architecturesdiblocks, ABA triblocks, BAB triblocks, and random copolymers-we consider the two situations, one where the number of A segments is 50 c f ~= 0.25) and one where it is 150 (fa = 0.75), respectively. Figure 5a-h displays density profiles corresponding to each of these situations. In the diblock copolymer melt, each of the density profiles exhibits one oscillation before decaying to the bulk densities. The oscillations in this case die off much more rapidly than in the symmetric diblock copolymer of Figure la. This suggests that this asymmetric copolymer is farther from the point of microphase separation than its symmetric counterpart. Furthermore, in all of the diblock and triblock copolymer melts, the component with the weaker interaction parameters appears in higher concentrations at the surface. Finally, the random copolymer melt (parts g-h) displays no perceptible surface segregation and the behavior of the density profiles is similar to that of a homopolymer melt. For a more quantitative analysis of the effects of copolymer composition on the free surface we perform the following exercise. For each of the architectures, we vary r A from a value of 0 to 200 and examine the free surface of the system. We consider melts composed of diblocks, triblocks (ABA and BAB), and random copolymers for this investigation. Figure 6 displays the mole fraction of B mers at the Gibbs dividing surface, @, as a function of the number of B segments on the chain for the four architectures that we considered. The straight line between 0 and 1 shows what would happen if there were no surface segregation. Any deviation from this straight line indicates that the system under consideration exhibits surface segregation. In this case the appearance of points below this line indicates the partitioning of component A to the surface. As expected, in all architectures component A partitions to the surface since it has the weaker interactions. As in the
copolymers with equal amounts of A-mers and B-mers studied in the earlier parts of this paper, the free surface of the block copolymer exhibits the most surface segregation followed by the triblocks and then by the random copolymer. When there are fewer B-mers the difference between the surface compositions of the two triblocks becomes more noticeable. In order to verify the applicability of our treatment to random copolymers, we compared the average density profiles of 10 one-component systems, each having a different random sequence with the same value of r A . Figure 6 includes a plot of the surface composition of the individual random copolymer melts and their average, along with the surface volume fraction we obtained from the random copolymer melt treated as one 40-component system. The 10 dotted lines show the results from the a;,, values of the different single-component melts, each containing one distinct random copolymer. The solid line represents the mean of the surface compositions over the 10 melts, and the symbols represent the surface compositions of the 40-component melts. Clearly the single-sequence melts exhibit irregular variations as expected although their mean exhibits relatively smooth behavior. More importantly, the mean surface composition of the 10 single-component melts is almost identical to the surface composition of the 40-component melt. This enables us to conclude that for purely statistical copolymers of a given composition and having 200 or more monomers, the difference between ensemble averages for a system containing several different random sequences and an average over several different single component melts is imperceptible. 3.5. Surface Tension. In this subsection we examine how the copolymer sequence distribution affects the surface tension, which is the property important to adhesion and wetting studies. For this we consider the reduced surface tension, ra/kBT, where r is the excess surface free energy and a is the surface area per lattice site. Figure 7 displays the reduced surface tensions of the copolymers of Figure 6. The straight line (dashed) in the figure connects the two points representing the two homopolymer melts characterized by F A A = 600 K and PBB = 800 K, respectively.
2196 J. Phys. Chem., Vol. 99, No. 9, 1995
For the four architectures, the surface tension increases from the lower value corresponding to a homopolymer melt with FAA = PBB = 600 K to that for a homopolymer for which PA*= PBB = 800 K. For copolymers containing a fixed number of A-mers and B-mers, we find that the diblocks have the lowest surface tension, followed by the triblocks, and then the random copolymers which have the highest surface tension. This pattem results from the fact that the diblock has the most surface segregation and the random copolymer has the least. The excess amount of A-mers at the surface having weaker interactions decreases the surface tension. Thus while the increase in the surface tension with the addition of component B is relatively slow in the diblock and triblock copolymer, the surface tension of the random copolymer increases in a roughly linear fashion. Figure 7 also shows that the mean of the surface tensions of the 10 random copolymer melts is almost exactly the surface tension of the 40-component melt. Again, the values for the 10 individual systems show some fluctuation about this mean.
4. Conclusions In this paper we have applied our formalism of a compressible lattice model to study the free surfaces of two-component diblock, triblock, random, and altemating copolymer melts. Our model predictions and detailed analyses comparing the various architectures reveal three major features of the surfaces of these melts. First, in all cases we observe that the free surfaces of the melts exhibit an enrichment of the copolymer component that has the weaker interactions. This surface segregation lowers the free energy penalty imposed upon the melt by the presence of the free surface. Second, the surface enrichment of the component with the weaker interactions is greatest in diblock copolymers and smallest in the altemating copolymers. For the interaction strengths we used, having the more weakly interacting monomers in the chain center causes the middle segments of the chain to segregate to the surface. This is true for all compositions of the copolymers that we studied. For a given composition of a copolymer the magnitude of surface segregation depends on the length of the sequences in the chain. Architectures with longer sequences exhibit a greater degree of surface segregation. Thus, the free surfaces of alternating and random copolymers are practically devoid of any surface segregation. Finally, surface tensions of the melts are sensitive to the copolymer architecture. Diblock copolymer melts have a lower surface tension than melts composed of polymers with other sequence distributions. Among the architectures studied, the diblock copolymer has the least surface segregation followed by the triblocks and the random copolymer. This trend holds for all compositions of the copolymer chains.
Hariharan and Harris Inherent in these calculations is the assumption that the mean field approximation is accurate. Likely the usage of the mean field approximation does not affect the trends that we have seen, although the exact magnitudes of the surface tensions of each melt would be affected by corrections to it. One feature of the physics that we have not investigated is the dependence of the effects shown here on chain length and interaction parameters. Particularly interesting would be the examination of cases where the geometric combining rule does not hold, e.g., where XAB is either much larger in magnitude, zero, or even negative.
Acknowledgment. Support for this work has provided by the Division of Chemistry of the Office of Naval Research. We also acknowledge support from the Sloan Basic Research Grant administered by MIT, the donors of The Petroleum Research Fund, administered by the American Chemical Society, and the Herman P. Meissner Chair. We gratefully acknowledge useful discussions with Dr.Doros Theodorou (University of Califomia, Berkeley). References and Notes (1) Russel, T. P.; Anastasiadis, S. H.; Satija, S. K.; Majkrzak, C. F. Phys. Rev. Lett. 1989, 62 (16). 1852. (2) Russell, T. P. Mater. Sci. Resp. 1990, 5, 171. (3) Russell, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dismrsions; Cambridge University Press: Cambridge, U.K., 1989. Helfand, E.; Wasserman, Z. R. Macromolecules 1975, 8, 552. Helfand, E.; Wasserman, Z. R. Macromolecules 1976, 9, 879. Leibler, L. Macromolecules 1980, 13, 1602. Fredrickson, G. H.; Helfand, E. J. Chem. Phys. 1987, 87, 697. Schull, K. R. Macromolecules 1992, 25, 2122. Theodorou, D. N. Macromolecules 1988, 21, 1411. Fredrickson, G. H. Macromolecules 1987, 20, 2535. Theodorou, D. N. Macromolecules 1989, 22, 4578. Theodorou, D. N. Macromolecules 1988, 21, 1422. Hariharan, A.; Harris, J. G. J. Phys. Chem. 1994, 101, 3353. Melenkevitz, J.; Muthukumar, M. Macromolecules 1991.24.4199. Lescanec, R. L.; Muthukumar, M. Macromolecules 1993,26,3908. Tang, H.; Freed, K. F. Macromolecules 1991, 24, 958. Scheutjens, J. M. H. M.; Fleer, G. J. J . Phys. Chem. 1979,83, 1619. Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1985, 18, Sanchez, I. C.; Lacombe, R. H. Macromolecules 1978, 11, 258. Sanchez, I. C.; Lacombe, R. H. J . Polym. Sci. Polym. Lett. 1977, Sanchez, I. C., Panayiotou, C. G., Eds. Thermodynamic Modellinn: Marcel Dekker: New York; 1992. (22) Flory, P. J. Principles of Polymer Chemistry; Comell University Press: Ithaca, New York, 1953. (23) de Gennes, P. G. Scaling Concepts in Polymer Physics; Comell University Press: Ithaca, New York, 1979. (24) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Oxford Science Publications: London, 1989. (25) Hariharan, A.; Kumar, S. K.; Russell, T. P. J. Chem. Phys. 1993, 98, 6517. JP942404H
