Langmuir 1999, 15, 3935-3943
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Attraction between Surfaces in a Polymer Melt Containing Telechelic Chains: Guidelines for Controlling the Surface Separation in Intercalated Polymer-Clay Composites Ekaterina Zhulina,* Chandralekha Singh, and Anna C. Balazs* Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261 Received October 8, 1998. In Final Form: March 11, 1999 Using numerical and analytical self-consistent field (SCF) calculations, we investigate the interactions between two closely spaced surfaces and the surrounding polymer melt, which contains a volume fraction of end-functionalized polymers. The functionalized polymers contain two reactive “stickers”, one at each end of the chain, that are highly attracted to the surfaces. The surfaces model clay crystallites, or sheets. Through the calculations, we determine the free energy profiles as the surfaces are pried apart and the polymer fills the gap. The free energy vs distance plots reveal a distinctive minimum, even at very small volume fractions of the end-functionalized chains. Thus, the presence of these telechelic polymers promotes the formation of thermodynamically stable intercalated composites, where the polymers enhance the separation between the silicate sheets. However, the telechelic chains prohibit the creation of the more desirable exfoliated structures, where the sheets are uniformly dispersed throughout the polymer matrix. The results provide guidelines for significantly enhancing the separation between the sheets, thus possibly making properties of the intercalated composites more like the exfoliated material.
1.0. Introduction The blending of polymeric melts and inorganic clays can yield composites that exhibit dramatic increases in tensile strength and heat resistance and decreases in gas permeability when compared to the pure polymer matrix.1-11 The unique properties make the composites ideal materials for products that range from high-barrier packaging for food and electronics to strong, heat-resistant automotive components. Fabricating these materials in an efficient and cost-effective manner, however, poses significant synthetic challenges. To understand the challenges, it is helpful to consider the structure of the clay particles. The inorganic clays (montmorillonite being a prime example) consist of stacked silicate sheets; each sheet is approximately 200 nm in length and 1 nm in thickness.2 The spacing between the closely packed sheets is also on the order of 1 nm. Thus, there is a large entropic barrier associated with the molten polymers penetrating this gap and, hence, becoming intermixed with the clay. Recently, we used both numerical and analytical selfconsistent field (SCF) models to isolate a robust scheme for creating stable dispersions from polymers and clays that are immiscible.12 Specifically, we showed that adding a small fraction of end-functionalized polymers to a melt (1) Okada, A.; Kawasumi, M.; Kojima, Y.; Kurauchi, T.; Kamigaito, O. Mater. Res. Soc. Symp. Proc. 1990, 171, 45. (2) Yano, K.; Uzuki, A.; Okada, A.; Kurauchi, T.; Kamigaito, O. J. Polym. Sci., Part A: Polym. Chem. 1993, 31, 2493. (3) Kojima, Y.; Usuki, A.; Kawasumi, M.; Okada, A.; Kurauchi, T.; Kamigaito, O. J. Polym. Sci., Part A: Polym. Chem. 1993, 31, 983. (4) Uzuki, A.; Kawasumi, M.; Kojima, Y.; Okada, A.; Kurauchi, T.; Kamigaito, O. J. Mater. Res. 1993, 8, 1174. (5) Miller, B. Plast. Formulating Compd. 1997, 30. (6) Vaia, R. A.; Jandt, K. D.; Kramer, E. J.; Giannelis, E. P. Macromolecules 1995, 28, 8080. (7) Vaia, R. A.; Sauer, B. B.; Tse, O. K., Giannelis; E. P. J. Polym. Sci., Part B: Polym. Phys. 1997, 35, 59. (8) Messersmith, P. B.; Stupp, S. I. J. Mater. Res. 1992, 7, 2599. (9) Krishnamoorti, R.; Vaia, R. A.; Giannelis, E. P. Chem. Mater. 1996, 8, 1728. (10) Vaia, R. A.; Giannelis, E. P. Macromolecules 1997, 30, 7990. (11) Vaia, R. A.; Giannelis, E. P. Macromolecules 1997, 30, 8000.
of chemically identical, nonfunctionalized chains can lead to the formation of “exfoliated” hybrids. In exfoliated mixtures, the clay sheets are uniformly dispersed within the polymer matrix and the composite exhibits the optimal properties. In our studies, the functionalized polymers contain a surface-active or “sticker” site at just one end of the chain. The synthesis of such functionalized polymers can, however, yield a fraction of polymers that contain stickers along the length of the chain, or stickers at both ends of the chains.13 The later chains are referred to as telechelic polymers. To determine how the presence of the multisticker chains affects the equilibrium behavior of the polymer-clay mixture, we now employ our SCF models to consider a melt that contains a fraction of the telechelic chains. In a subsequent paper, we will investigate the behavior of the system when the melt contains both the one-sticker and two-sticker functionalized polymers. As we show below, the presence of even a small volume fraction of the telechelic chains within the melt can lead to the formation of intercalated composites. In intercalated structures, polymer chains penetrate the host layers and enhance the separation between the silicate sheets. The resulting composite has a well-ordered multilayer morphology, with alternating polymer/inorganic layers. If, however, the distance between the sheets can be significantly enhanced, the intercalated hybrid might yield the superior properties exhibited by the exfoliated composites. Our results indicate how to tailor the system to maximize the separation between the surfaces in these intercalated hybrids. Surprisingly, we also find that despite the strong attraction between the stickers and the clay sheets, there is a range of molecular weights for the nonfunctionalized chains where the polymer/clay mixture is immiscible and (12) Balazs, A. C.; Singh, C.; Zhulina, E. Modeling the Interactions Between Polymers and Clay Surfaces Through Self-consistent Field Theory. Macromolecules 1998, 31, 8370. (13) Kawasumi, M.; Hasegawa, N.; Kato, M.; Usuki, A.; Okada, A. Macromolecules 1997, 30, 6333.
10.1021/la981406g CCC: $18.00 © 1999 American Chemical Society Published on Web 04/30/1999
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the system will phase separate. We show that the unexpected phase behavior is due to entropic effects that occur near surfaces when the melt contains both functionalized and nonfunctionalized chains. In the following, we begin with a brief description of the numerical SCF model. In prior studies,14 we used this method to determine the free energies as a function of surface separation for polymer-coated surfaces in solution, and in the melt.12 Here, we describe our findings for the interactions between solid surfaces immersed in a melt that contains polymers with two surface-active end groups. To analyze the behavior of the system in greater detail, we introduce an analytical SCF model for surfaces in a melt containing these end-functionalized chains and present the results from this theory. Comparisons are made between the numerical and analytical SCF results and the implications of these findings are discussed further in the Conclusions section. We note that our findings on the behavior of polymers in confined geometries are relevant not only to the fabrication of polymer/clay composites but also to other industrially important areas, such as tribology and adhesion.15 2.0. The SCF Model Our numerical self-consistent field (SCF) calculations are based on the model developed by Scheutjens and Fleer.16 In the Scheutjens and Fleer theory, the phase behavior of polymer systems is modeled by combining Markov chain statistics with a mean field approximation. Since the method is thoroughly described in ref 16, we simply provide the basic equations and refer the reader to that text for a more detailed discussion. These calculations involve a planar lattice where one lattice spacing represents the length of a statistical segment within a polymer chain. The planar lattice is divided into z ) 1 to M layers. In the one-dimensional model, the properties of the system only depend on z, the direction perpendicular to the interface. The properties of the system are averaged over the x and y directions; that is, the system is assumed to be translationally invariant in the lateral direction. The probability that a monomer of type i is in layer z with respect to the bulk is given by the factor
Gi(z) ) exp(-ui(z)/kT)
(1)
where the potential ui(z) for a segment of type i in layer z is given by
ui(z) ) u′(z) + kT
∑j(*i) χij (〈φj(z)〉 - φjb)
(2)
The parameter u′(z) is a “hard-core potential”, which ensures that every lattice layer is filled. In the second term, χij is the Flory-Huggins interaction energy between units i and j and φjb is the polymer concentration in the bulk. The expression 〈φj(z)〉 is the fraction of contacts an i segment in the z layer makes with j-type segments in the adjacent layers and is given by the following equation:
〈φj(z)〉 ) λ-1 φj(z - 1) + λ0φj(z) + λ1φj(z + 1)
(3)
Here, the λ’s are the fraction of neighbors in the adjacent (14) (a) Singh, C.; Pickett, G.; Zhulina, E. B.; Balazs, A. C. J. Phys. Chem., B 1997, 101, 10614. (b) Singh, C.; Pickett, G.; Balazs, A. C. Macromolecules 1996, 29, 7559. (15) Gong, L.; Friend, A.; Wool, R. Macromolecules 1998, 31, 3706. (16) Fleer, G.; Cohen-Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T. Vincent, B. Polymers at Interfaces; Chapman and Hall: London 1993.
layers: λ-1 is for the previous layer, λ0 is for the same layer, and λ1 is for the next layer. Since polymers contain more than one segment, we must take into account that the segments of the chain are connected. We define Gi(z,s|1) as the (conditional) probability (up to a normalization constant) that a segment s is located in layer z, while being connected to the first segment of chain i. This Green’s function can be calculated from the following recurrence relation:
Gi(z,s|1) ) Gi(z) {λ-1Gi(z - 1,s - 1|1) + λ0Gi(z,s - 1|1) + λ1Gi(z + 1,s - 1|1)} (4) Clearly, Gi(z,1|1) ) Gi(z) and the terms for s > 1 can be calculated from this relationship and eq 4. In the same way, we can obtain a recurrence formula for Gi(z,s|r), the probability that a segment s is in layer z, given that it is connected to the last (rth) segment of the chain. To obtain the volume fraction of i in the z layer due to segment s, in a chain of r segments, the product of two probability functions is needed: the probability of a chain starting at segment 1 and ending with segment s in layer z and that of a chain starting at segment r and also ending with segment s in layer z. This product must be divided by Gi(z) in order to compensate for the double counting of the sth segment. Hence, the volume fraction is given by
φi(z,s) ) CiGi(z,s|1)Gi(z,s|r)/Gi(z)
(5)
Here, Ci is the normalization constant and is equal to Ci ) θi/ri∑z Gi(z,r|1) where θi ) ∑z φi(z) is the total amount of polymer segments of type i in the system and ∑z Gi(z,r|1)/M is the average of the end segment distribution function for a chain of ri segments. We can also express Ci in terms of φib, the volume fraction in the bulk solution, as Ci ) φib/ri. The total volume fraction of φi(z) of molecules i in layer z can be obtained by summing over s:
φi(z) )
∑s φi(z,s)
(6)
Expressions 1 and 5 and the condition that ∑i φi(z) ) 1 for each layer form a set of coupled equations that are solved numerically and self-consistently. Given that the amount of polymer, θi, the length ri, and χij are specified, we can calculate the self-consistent adsorption profile and the equilibrium bulk concentration. (For a given θi, φib is obtained by equating the two expressions for Ci.) We note that the expression for the excess free energy in terms of the segment density distribution is given by
F(z) )
∑j φj(z) ln Gj(z) + (1/2)∑jk χj,k ∫ η(z - z′)φj(z)φk(z′) dz′
(7)
where η(r - r′) is the short-range interaction function, which is replaced by a summation over nearest neighbors. Summing the above equation over all z yields the total free energy (per unit area). The free energy of interaction between two surfaces, ∆F, as a function of surface separation, H, can be obtained by taking the difference between the total free energies when the layers are in intimate contact and when they are separated by a distance H. Using this SCF model, we consider two planar surfaces that lie parallel to each other in the xy plane and investigate the effect of increasing the separation between the surfaces in the z direction. The planar surfaces represent the clay sheets. The sheets are effectively immersed within a polymer melt. As the separation
Attraction between Surfaces
Figure 1. Free energy (per unit area), ∆F/A, as a function of surface separation, H. The length of the functionalized polymers is fixed at N ) 100 and the length of the nonfunctionalized polymers is given by P ) 300. The plots are for χ ) -75. Results are shown for three different values of φ, the volume fraction of the functionalized chains.
between the sheets is increased, polymer from the surrounding “bath” penetrates the gap between these walls. The polymer melt contains both functionalized and nonfunctionalized species. The functionalized chains contain “stickers” that are highly attracted to the surface. One sticker is located at each end of the linear, functionalized polymers; thus, the chains have a telechelic architecture. Aside from the sticker sites, the functionalized and nonfunctionalized chains are chemically identical. The sticker-surface interaction is given by χss. The interaction parameter between the stickers and all other species is set equal to 0. (Thus, the stickers do not react with themselves or other monomers.) For the other monomers in the system (the nonstickers), the interaction with the surface is characterized by χsurf, which is fixed at 0. Below, we determine the interaction profiles as the separation between the surfaces is increased. 2.1. Results and Discussion To provide a basis of comparison, we previously examined the interactions between two bare surfaces and a melt of nonfunctionalized polymers.12 Here, the monomersurface interaction is characterized by χsurf ) 0 and the length of the polymers is given by N ) 100. In the reference state, the surfaces are in intimate contact. The free energy curves indicate that ∆F rapidly becomes positive as the surfaces are pulled apart, and thus the intermixing of these components is unfavorable. Increasing N just raises the free energy to even greater positive values.12 Hence, in the absence of stickers, the polymer and clay sheets would phase separate. We now consider the case where the melt contains a finite volume fraction of telechelic polymers. The free energy (per unit area), ∆F/A, as a function of surface separation, H, is plotted in Figure 1. The length of the functionalized polymers is fixed at N ) 100, and the length of the nonfunctionalized polymers is given by P ) 300. The plots are for χss ) -75.17 Results are shown for three different values of φ, the volume fraction of functionalized chains. The figure indicates that ∆F rapidly becomes negative as the surfaces are pulled apart, and thus the intermixing of these components is favorable. The strong attraction between the stickers and the surfaces drives the functionalized polymers to bind to the clay sheets.
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Figure 2. Free energy (per unit area), ∆F/A, as a function of surface separation, H. The length of the functionalized polymers is fixed at N ) 100 and the length of the nonfunctionalized polymers is given by P ) 300. The plots are for φ ) 0.05. Results are shown for χ ) -50, -75, and -100.
The enthalpic gain upon the binding of the stickers lowers the overall free energy of the system. The free energy per unit area clearly becomes more negative as the fraction of functionalized chains increases from φ ) 0.01 to φ ) 0.1. It is important to emphasize that the addition of just 1% of the telechelic polymers drives the formation of a stable mixture from the otherwise immiscible components. A distinctive feature of all the curves in Figure 1 is the presence of a minimum in the plots of ∆F/A vs H. Such local minima indicate that the mixture forms an intercalated structure. In particular, the lowest free energy state is one where the polymers have penetrated the gallery and enhanced the separation between the plates by a fixed amount. For exfoliated structures, where the sheets are effectively separated and dispersed within the polymer matrix, the comparable plots show a global minimum at large (infinite) separations.12 The local minimum is due to the presence of chains that “bridge” the two walls, with a sticker on each of the sheets. The bridging effectively binds the two surfaces and gives rise to an attractive interaction between these platelets. The plots in Figure 1 indicate the surface separation at the attractive minimum, or separation between the sheets in the intercalated structure. Each lattice site in the SCF model is comparable to the persistence length of a typical polymer chain, roughly 1 nm. While the exact values for the separation between the sheets may not be quantitatively accurate, the plots nonetheless indicate how the various parameters (φ, χss, P, N) affect the layer spacing. To examine the effect of varying the sticker-surface interaction, we calculated the interaction curves for three different values of χss. The influence of this parameter can be seen in Figure 2, where we plot ∆F/A vs H for χss ) -50, -75, and -100, and N ) 100, φ ) 0.05, and P ) 300. Increasing the surface attraction decreases the free energy; there is a greater enthalpic gain as the more attractive stickers bind to the surface. The figure also indicates that the locations of the minima are shifted to larger surface separations for larger values of χss. (17) Note that in the case of surface adsorption, we must divide χss by the coordination number of the cubic lattice (q ) 6) in order to relate this Flory-Huggins parameter to experimentally relevant values. This can be understood by considering the sticker to be a cube in our lattice model. When a sticker attaches to the substrate and gains an adsorption energy /kT, the sticker only contacts the surface through one face of the cube. The other (q - 1) faces are still in contact with the surrounding melt. Thus, the comparable binding energy (/kT) is equivalent to (χss/ 6).
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Figure 3. Free energy (per unit area), ∆F/A, as a function of surface separation, H. The length of the functionalized polymers is fixed at N ) 100 and the length of the nonfunctionalized polymers, P, is varied. Results are shown for P ) 1, 20, and 300. The value of φ is fixed at 0.05.
Figure 4. Free energy (per unit area), ∆F/A, as a function of surface separation, H. The length of the functionalized polymers is varied and the length of the nonfunctionalized polymers is fixed at P ) 300. Results are shown for N ) 50, 100, and 300. The value of φ is fixed at 0.05.
Figure 3 shows ∆F/A vs H for three lengths of P and N ) 100, φ ) 0.05, and χss ) -75. For a given surface separation, the free energy per unit area becomes more negative as P increases. (We note, however, that for small values of φ the dependence of the free energy on P is nonmonotonic, as will be seen in Figure 10.) We also see that the value of P affects the optimal separation between the surfaces; larger values of P enhance the separation between the sheets. Finally, we anticipate that the length of the telechelic chains will also play a role in controlling the layer spacing in the intercalated structures. In Figure 4, we plot ∆F/A vs H for various values of N, the length of the functionalized chain. The other parameters are fixed at φ ) 0.05, χss ) -75, and P ) 300. As expected, increasing the length of the functionalized chains enhances the equilibrium separation between the sheets. We also found that the amount of polymer absorbed on the surfaces increases with the length of the functionalized chains. The above SCF calculations clearly illustrate that the location of the minimum in the interaction profiles, and hence the separation between the sheets, depends on the values of φ, χss, P, and N. Using the analytical model described below, we now analyze the explicit dependence of the surface separation, the adsorbed amount, and the depth of the attractive minima on these critical parameters.
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Figure 5. Schematic plot of the free energy of the system vs H. The plot pinpoints the position of the attractive minimum, Hmin, and the depth of the minimum, ∆W/kT.
3.0. Analytical Self-Consistent-Field Model In formulating our analytical model, we consider a melt of mobile polymers that contains a volume fraction, φ, of end-modified or functionalized chains and a volume fraction, (1 - φ), of nonfunctionalized chains. Every functionalized chain contains two terminal groups, one at each end of the linear polymer. In all other respects, the modified and unmodified chains are chemically identical. The end groups on the functionalized chains are highly attracted to the clay sheet; they do not, however, react with themselves or the other monomers in the melt. In other words, the interactions among all the monomers are identical, and the melt constitutes a simple athermal mixture of polymers. Furthermore, the nonreactive monomers are not attracted to the clay sheets. The functionalized chains are monodisperse; each chain contains M ≡ 2N monomers and the diameter of each monomer is equal to a. We assume that the unmodified chains are also monodisperse and each chain contains P monomers. (To simplify the ensuing discussion, we frequently refer to the functionalized polymers as the N chains and the nonfunctionalized species as the P polymers.) The melt is assumed to be in thermodynamic equilibrium with the clay particles. Each clay sheet is modeled as a planar surface of area Σ. Due to the attraction between the end groups and the sheets, the functionalized chains become terminally anchored to these surfaces and effectively push the sheets apart. We assume that, at any distance 2H between the surfaces, there is an equilibrium between the anchored and free functionalized chains. In other words, the degree to which the functionalized polymers bind to the surface is determined by the distance between the particles. As seen from the numerical SCF results, the free energy of interaction between sheets that are covered with functionalized polymers shows a nonmonotonic dependence on the distance H between the particles. The results of the previous section reveal a bridging attraction between the sheets that gives rise to an attractive minimum at relatively small deformations of the polymer coating. Figure 5 displays a schematic plot of the free energy of the system, ∆F/ΣkT, vs H, highlighting the position of the attractive minimum, Hmin, and the depth of the minimum, ∆W/kT. In this section, we use the analytical self-consistent field model to focus on the characteristics of Hmin and ∆W/ kT. In terms of the equilibrium behavior, these parameters govern the thermodynamic stability of the polymer-clay composites. To carry out this analysis, we must determine
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(that is, at z ) 1). (The plots in Figure 6 are for one of the surfaces; the plots for the other surface are identical to those seen in this figure.) At large separations between the surfaces, the dangling ends of the tails are concentrated at the periphery of the layer, giving rise to the maximum indicated in Figure 6. At larger distances from the surface, the concentration of the active end groups approaches the bulk value ()2φ). We assume that our system is in the strong stretching limit and, thus, we can exploit concepts that apply to the behavior of bidisperse polymer brushes.18,19 In particular, we “cut” all the loops at their midpoints and thus, view these segments as tethered chains of length N. The tails constitute chains of length 2N. On a clay sheet of surface area Σ, we label the number of loops by the parameter nl and the number of tails by nt. We consider relatively high values of N and P and ignore the penetration of mobile polymers into the brush. In other words, we model the polymer coating on the clay as a dense bidisperse brush that has 2nl chains of length N and nt chains of length 2N. To characterize the thickness of the brush, we specify the adsorbed amount Θ of functionalized polymer as Θ ) 2N(nt + nl)/Σ. By stipulating that the volume of the brush is conserved, we have
HoΣ/a3 ) 2N(nt + nl)
Figure 6. Density profile of the reactive end groups (as calculated through the numerical SCF model) at large separations between the surfaces (dashed curve) and at small separations that correspond to the attractive minimum in the free energy (solid line). In both these cases, there is a high density of end groups on the surfaces (z ) 1). The plots are shown for one of the surfaces; the plots for the other surface are identical to those seen in the figure. Results are for N ) 100, P ) 300, and φ ) 0.05. In part a, χ ) -75, while in part b, χ ) -100.
the values of ∆Fo, the free energy of the system at large surface separations and ∆Fmin, the free energy at the minimum (see Figure 5). We describe these calculations below. (i) Large Separations Between the Surfaces. We start by investigating our system at large separations between the sheets, or H ) infinity. Under these conditions, each individual particle is in equilibrium with the surrounding melt of mobile polymers. We let (>0) be the gain in energy that occurs when a terminal group on the polymer attaches to the clay sheet. (As explained in ref 17, /kT ) -χss/6.) At relatively high values of , the attached chains form a polymer brush. The conformation of this brush consists of “loops” and “tails”. Loops are formed when both terminal groups are attached to the surface of the particles, while tails are due to chains that have only one adsorbed end-group. (The second terminal group is dangling inside the brush.) In effect, we have a bidisperse system. The loops form a sublayer near the surface, while the dangling ends are localized predominantly in the periphery of the brush. Figure 6 shows the typical distribution of reactive end groups (as calculated through the numerical SCF model) at large separations between the particles (dashed curve) and at small separations that correspond to the attractive minimum in the free energy (solid line). In both these cases, there is a high density of end groups on the surfaces
(8)
where Ho is the thickness of the coating and F ) (nt + 2nl)/Σ is the corresponding grafting density of the bidisperse brush. (In terms of our model, we note that Ho ) Θa3.) We let q′ ) nt/(nt + 2nl) be the fraction of the longer (2N) chains. The above parameters form crucial ingredients in our calculation of the free energy of the system. The free energy itself contains a contribution from the elastic free energy of the stretched chains, the entropic contribution due to mixing of the loops and tails within the brush, and, finally, the energetic gain due to the adsorption of the reactive end groups. The first contribution, the elastic free energy of the stretched chains, is given by19
∆Fel/kT ) π2Ho12 (nt + 2nl) (1 + q′3)/8N
(9)
where Ho1 ) (nt + 2nl)Na3/Σ is the thickness of a monodisperse, dense brush of the shorter N chains with a grafting density F. After substituting Ho1 into eq 9 and employing the definition of q′, we arrive at the following expression for the elastic contribution:
∆Fel/kT ) π2N/8Σ2[(nt + 2nl)3 + nt3]
(10)
The entropic contribution due to mixing of the loops and tails within the brush is given by
∆S/kT ) nt ln[nt/(nt + nl)] + nl ln[nl/(nt + nl)]
(11)
Finally, the energetic gain due to the adsorption of the terminal groups of the functionalized polymers yields
∆E/kT ) -(nt + 2nl)/kT
(12)
(18) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1989, 22, 853. (19) Birshtein, T. M.; Liatskaya, Yu. V.; Zhulina, E. B. Polymer 1990, 31, 2185.
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Thus, the total free energy of the system is given by
∆F/kT ) (π2N/8Σ2)[(nt + 2nl)3 + nt3] + nt ln[nt/(nt + nl)] + nl ln[nl/(nt + nl)] (nt + 2nl)/kT - (nt + nl)µ/kT (13) where
µ/kT ) ln φ + (1 - φ)(1 - N/P)
(14)
is the chemical potential of the functionalized chains in the surrounding melt.20 We note that the reference state for the free energy in eq 13 corresponds to the state where the clay particles are totally demixed from the polymer melt. Under the conditions of thermodynamic equilibrium
δ(∆F/kT)/δnl ) 0
(15)
δ(∆F/kT)/δnt ) 0
(16)
By substituting expression 13 into eqs 15 and 16, we arrive at the equations that determine the equilibrium values of nt and nl, or, equivalently, the equilibrium fraction
qt ) nt/(nt + nl)
(17)
of the dangling ends (tails) in the system and the adsorbed amount, Θo ) 2N(nt + nl)/Σ. Those equations read
3π2Θo2(1 - qt)/8N - /kT + ln[(1- qt)/qt] ) 0
(18)
3π2Θo2(1 - qt/2)2/4N + ln(1 - qt) ) µ/kT + 2/kT (19) By rewriting eq 18 as
Θo2/N ) 8[/kT + ln(qt/(1 - qt))]/[3π2(1 - qt)]
(20)
and substituting this expression into eq 19, we arrive at the final equation, which determines the equilibrium fraction of tails, qt, as
2[/kT + ln(qt/(1 - qt)](1 - qt/2)2/(1 - qt) + ln(1 - qt) ) µ/kT + 2/kT (21) The corresponding value of the free energy per unit area is then given by
(both terminal groups of a chain are located at the same surface) or bridges (the terminal groups are located at different surfaces) so that qt ) 0. The numerical SCF results in Figure 6 confirm that such an approximation is indeed reasonable for the chosen range of parameters. By comparing the two distributions for the end groups in Figure 6a (/kT ) 12.5), we find that the fraction qt of dangling ends in the attractive minimum (that is, in the intercalated state) decreases by an order of magnitude relative to the value obtained at large separations between the particles. Increases in the adsorption energy /kT lead to further suppression of the dangling ends. As seen from Figure 6b, at /kT ) 16.7 the amount of the dangling ends is not even seen on the scale of the plot in Figure 6b. We let 2nl and nb be the respective total number of loops and bridges that form in the gap between two sheets that are separated by a distance 2Hmin. As before, we ignore the penetration of the mobile polymers into the adsorbed layer. Consequently, the condition for the conservation of volume gives
Hmin ) Na3(2nl + nb)/Σ
(23)
We now let qb ) nb/(2nl + nb) be the fraction of bridges in the system. By cutting each loop in the middle, we transform a loop into two tails so that the total number of tails on a single surface equals 2nl. We now view the system as two dense brushes of thickness Hmin. Each such brush contains 2nl tails of length N and nb “half-bridges” of length N. (We assume that the midpoint of each bridge is located at H ) Hmin.) Now, the only difference between the tails and bridges is that the midpoints of the bridges are localized in the center of the gap; the free ends of the tails are distributed throughout the brush. As a result, the bridges experience greater stretching than the tails and an increase in the fraction qb of bridges should lead to an increase in the elastic free energy of the system. The formation of bridges, however, is favored by the entropy of mixing of the tails and the bridges. As a result of the competition between these two factors, one finds a noticeable number of bridges in the system.21,22 To estimate qb more precisely, we employ the results of ref 21. Namely, we use the following expression for the elastic free energy of a dense brush containing a fraction qb of bridges
∆Fel/kT ) π2N(2nl + nb)3[(1 - τqb)3 + 12 qb2/π2]/8Σ2 (24)
∆Fo/kTΣ ) Θo/N{π2Θo2[(1 - qt/2)3 + qt3/8]/8N (1 - qt/2)/kT + qt ln(qt)/2 + (1 - qt) ln(1 - qt)/2 µ/2kT} (22)
where the parameter τ is the root of the equation
where qt is determined by eq 21. By solving eq 21 numerically, we find the equilibrium value of qt and the corresponding values of Θo (through eq (20)) and ∆Fo/kTΣ (through eq 22). (ii) Small Separations between the Surfaces. We now analyze the attractive minimum of the interaction curve ∆F(H). Here, we adopt the following model for the system. We assume that when ∆F reaches its minimal value, the fraction of dangling ends (tails) is negligibly small. That is, all the adsorbed chains form either loops
The entropy of mixing of the tails and the bridges yields
(20) Flory, P. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.
(1 - τqb) tan(πτ/2) ) 2qb/π
(25)
∆S/kT ) 2nl ln[2nl/(2nl + nb)] + nb ln[nb/(nb + 2nl)] (26) while the energy contribution is given by
∆E/kT ) - 2(2nl + nb)/kT
(27)
(21) Zhulina, E. B.; Halperin, A. Macromolecules 1992, 25, 5730. (22) Matsen, M. W.; Schick, M. Macromolecules 1994, 27, 187.
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The total free energy of the system is then given by
∆F/kT ) 2π2N(2nl + nb)3[(1 - τqb)3 + 12qb2/π2]/8Σ2 + 2nl ln[2nl/(2nl + nb) + nb ln[nb/(nb + 2nl)] - 2(2nl + nb)/kT µ(2nl + nb)/kT (28) where the factor of 2 in the first term in eq 28 is due to the existence of the two brushes in the gap between the particles and, as before, µ is the chemical potential of the functionalized chains in the melt as determined by eq 14. By minimizing the free energy with respect to nl and nb, we obtain the following equations, which determine the equilibrium fraction qb of bridges and the adsorbed amount Θmin ) N(2nl + nb)/Σ 2
2
2
2
ln[qb/(1 - qb)][(1 - τqb) + 8q /π ]/[τ(1 - τqb) 4qb/π2] ) 2/kT + µ/kT - ln(1 - qb) (29)
Figure 7. Behavior of qt and qb as a function of φ for different values of the attraction energy, . The fraction qt of dangling ends (or tails) at large separations between the surfaces is calculated according to eq 21 and the fraction of bridges qb in the attractive minimum is calculated according to eqs 29 and 25.
and
Θmin2/N ) (4/π2) ln[qb/(1 - qb)]/[3τ(1 - τqb)2 12qb/π2] (30) where τ is related to qb through eq 25. The corresponding value of the free energy per unit area yields
∆Fmin/2kTΣ ) (Θmin/N){π2Θmin2[(1 - τqb)3 + 12qb2/π2]/8N - /kT + qb ln(qb)/2 + (1- qb)ln(1 - qb)/2 - µ/2kT} (31) where qb is determined by eq 29. Results and Discussion Using the above analytical formulas, we can generate plots that reveal the behavior of the system as a function of the key parameters. The plots also allow us to compare the results of the analytical and numerical SCF studies. In our discussion, we focus mainly on the behavior of the depth of the attractive minimum, ∆W/kT ) ∆Fo/kTΣ ∆Fmin/2kTΣ, and its position, H ) Hmin (see Figure 5). These parameters are crucial in dictating the equilibrium structure of the polymer-clay composite. Note that ∆W is expressed in units of energy per unit area; thus, to obtain the total energy, we must multiply this term by Σ, the area of the clay sheet. If ∆WΣ is much larger than kT, the bridging attraction between the sheets dominates the behavior of the system. Because ∆W depends on qt, qb, and Θ, we start with an analysis of the dependence of qt, qb, and Θ on the volume fraction of the functionalized polymers in the melt, φ, and the adsorption energy of the terminal groups, /kT. We choose three representative values of /kT, 8.3, 12.5, and 16.7, (which correspond to the respective χss values of -50,-75, and -100) and focus on the limit of long nonfunctionalized polymers, P . 1. Figure 7 represents the behavior of qt and qb vs φ for the different values of . The fraction qt of dangling ends (or tails) at large separations between the particles is calculated according to eq 21 (H > Ho) and the fraction of bridges qb in the attractive minimum is calculated according to eqs 29 and 25 (H ) Hmin). As seen from Figure 7, in both cases, our model gives noticeable values of qt and qb. Increases in φ, the volume fraction of the functionalized polymer, lead to increases in the total
Figure 8. Dependence of the adsorbed amount Θ on φ. The analytical data is plotted with solid lines, while the SCF results are indicated by the symbols. Results are shown for three different values of the attractive energy, . The plot in part a shows the adsorbed amount at large surface separations. The plot in part b shows the adsorbed amount at the attractive minimum.
amount of adsorbed polymer and to a rapid increase in the fraction of dangling ends, qt; the fraction of bridges qb is affected to a much less extent (qb diminishes only slightly when φ increases and is close to 0.4). The dependence of the adsorbed amounts on φ is illustrated in Figure 8; Figure 8(a) shows the adsorbed amount at large surface separations, Θo, and Figure 8b
3942 Langmuir, Vol. 15, No. 11, 1999
Figure 9. Depth of the attractive minimum, ∆W/kT, as a function of φ, the fraction of functionalized polymers in the melt. The analytical data is plotted with solid lines, while the SCF results are indicated by the symbols.
shows the adsorbed amount at the minimum, or Θmin. Recall that in our model, Hmin ) Θmina3, thus, our comments about Θmin also apply to Hmin, the location of the minimum in the free energy. For both Θo and Θmin, increases in φ enhance the adsorption of functionalized polymers and lead to a corresponding increases in Θ. At relatively high concentrations of functionalized chains in the melt, the adsorbed amount that is attained at large separations between the particles (Θo) exceeds that found for H ) Hmin (or Θmin). However, decreases in φ lead to a reverse of this effect. Namely, Θo becomes smaller than Θmin. The difference between Θmin and Θo is, however, small. This picture is found to be in reasonable quantitative and qualitative agreement with the results of the numerical SCF calculations, which are marked by the symbols in Figure 8. Increasing the length of the functionalized polymers leads to increases in Θo and Θmin. As a result, the position of the attractive minimum, Hmin ) Θmina3, shifts to higher separations between the sheets. Equations 21 and 29 indicate that the fractions of tails and bridges, qt and qb, are only slightly affected by variations in N (through the weak N-dependence of the chemical potential µ, (see eq 14). According to eqs 20 and 30, the adsorbed amounts scale as Θo ∼ Θmin ∼ (2N)1/2. In other words, employing longer functionalized polymers provides larger separations between the sheets in the intercalated composites (as can also be seen in Figure 4). Finally, Figure 9 reveals how the depth of the attractive minimum, ∆W/kT, depends on φ, the fraction of functionalized polymers in the melt. Again, we plot the analytical data with solid lines, while the numerical SCF results are indicated by the symbols. The plots are for the three different values of . As seen from both the analytical and numerical SCF models, the depth of the attractive minimum exhibits a nonmonotonic behavior as a function of φ. The maximal depth is always found at relatively low values of φ, though the analytical and the numerical models give slightly different positions (values of φ) for the maximum. The appearance of a maximum in ∆W(φ) indicates that there is a composition of the melt at which the formation of composites with highly separated or dispersed sheets is the most unfavorable. (We note, however, for applications requiring strong adhesion between surfaces, the characteristics that yield the maximum in ∆W/kT will provide optimal binding between the interfaces.) Increases in the adsorption energy /kT of the terminal groups lead to an increase in the maximal value of ∆W/kT. However, this effect is not strong.
Zhulina et al.
We note that contrary to the numerical SCF results, our analytical model indicates the disappearance of the attractive minimum at high concentrations of functionalized polymer (at φ ) 1, the value of ∆W/kT becomes negative). This, however, is an artifact of the model and is related to our assumption about the absence of dangling ends in the attractive minimum. By neglecting even a small faction of the dangling ends, we overestimate the free energy at H ) Hmin. The proper entropic contribution due to the mixing of tails, loops, and bridges would decrease ∆Fmin/2kTΣ and lead to the retention of the attractive minimum. We emphasize that our analytical model employs other approximations; namely, we assume that our system is in the strong stretching limit, and we neglect the penetration of the mobile polymers into the adsorbed layer. The strong stretching limit implies that the thickness of the adsorbed layer, H, should be at least larger than the Gaussian size of the functionalized chain, a(2N)1/2. In terms of our model, this is equivalent to the condition
Θ > (2N)1/2
(32)
Thus, for 2N ) 100, one anticipates good agreement between the analytical and numerical SCF results at Θ > 10. That is, reasonable agreement will be found at high values of /kT ()16.7) and relatively large values of φ < 1 (see Figure 8). However, reasonable correspondence between the analytical and numerical SCF results is also found at lower values of Θ (Θ > 5), where the strong stretching limit is expected to have broken down. It is thus surprising that our analytical model captures the major features of the system for a wider range of parameters than anticipated, including the subtle effect of the nonmonotonic behavior of the attractive minimum. To obtain proper insight into even lower ranges of Θ, we turn to the numerical SCF model. We now focus on the range of parameters where Θ is close to 1. As indicated by the numerical data in Figure 8, this range of Θ is attained when /kT e 8.3 and φ < 10-4. Figure 9 demonstrates that, for /kT ) 8.3 and φ ) 10-6, the depth of the attractive minimum, ∆W/kT, is still big enough to prevent large-scale separation between the sheets. In other words, the value of ∆WΣ is much larger than the thermal energy kT for realistic values of Σ, and the clay particles will not be able to overcome the bridging attraction due to the functionalized polymer. We thus expect the appearance of the intercalated structures in this range of Θ and P (recall that P was chosen as P ) 300 in the calculations presented in Figures 8). However, as we show below, even the intercalated structures can be prohibited in the polymer-clay mixture if the concentration of the functionalized polymer in the melt is low and the nonfunctionalized polymer has the inappropriate molecular weight. In Figure 10, we plot the values of the two free energies, ∆Fo ) ∆F(H ) Ho) and ∆Fmin ) ∆F(H ) Hmin), as functions of the length of the nonfunctionalized polymer, P. (Here, the volume fraction φ of the functionalized polymer is kept fixed at φ ) 10-4 and /kT equals 8.3 or 12.5.) The former value, ∆Fo, corresponds to the free energy at large surface separations, while the latter value, ∆Fmin, refers to the free energy for intercalated structures. Due to the bridging attraction, the curve ∆Fo(P) always lies above the curve ∆Fmin(P). (Recall that ∆W/kT ) ∆Fo/kTΣ - ∆Fmin/2kTΣ; that is, the difference between the two curves determines the depth of the attractive minimum at any value of P.) Figure 10 indicates that at /kT ) 8.3 (χss ) -50) both curves pass through a maximum that is located in the
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Langmuir, Vol. 15, No. 11, 1999 3943
increase in the adsorbed amount Θ of the functionalized polymer and the corresponding decrease in the amount of the mobile polymer near the surface. The latter tendency wins in the range of large P, and the system starts to form the intercalated structures at P > P*. Increases in /kT will lead to rapid decreases in P*. As can be seen in Figure 10, at /kT ) 12.5 (χss ) -75), both curves lie below the zero level for all values of P. (We note that a similar shape of the curve ∆F(P) is expected for the functionalized polymer with a single end-active group. In this case, however, the intercalated structures will be substituted by the exfoliated ones due to the absence of the bridging attraction.)
Figure 10. Free energy at large surface separations, ∆Fo, and the free energy at the separation corresponding to the attractive minimum, ∆Fmin, plotted as a function of the length of the nonfunctionalized polymer, P. The volume fraction of functionalized polymer is φ ) 10-4 and the length of functionalized polymer is N ) 100. Results are shown for /kT ) 8.3 (χss ) -50) and /kT ) 12.5 (χss ) -75).
range of relatively low values of P. An important feature is that in the vicinity of the maximum, the values of ∆Fo and ∆Fmin are positive. In other words, in the range of P where the ∆Fmin > 0 (that is P < P* ) 65 for the parameters in Figure 10), even the formation of intercalated structures is thermodynamically unfavorable. We recall that the reference state (zero value) for the free energy of the system corresponds to the totally demixed state of the clay particles and the polymer melt. Increases in P > P* lead to a decrease in ∆Fmin < 0. Here, the formation of the intercalated structures becomes thermodynamically favorable. At P > P** ) 150, ∆Fo becomes negative as well. Here, the mixture becomes thermodynamically favorable at surface separations greater than Hmin. However, since ∆Fo > ∆Fmin in the whole range of P, large-scale separation of the sheets is hindered and only the intercalated structures are attained in the polymer-clay composite at P > P*. The value of P* thus determines the lower limit of the molecular weight of the nonfunctionalized polymer where the polymer-clay mixture is expected to mix by forming intercalated structures. Our SCF results indicate that P* rapidly increases with decreases in /kT (the adsorption energy of the terminal groups) and φ (the volume fraction of the functionalized polymer). The origin of this effect can be explained as follows. Decreases in /kT and φ lead to a corresponding decrease in the adsorbed amount of the functionalized polymer, Θ. As a result, a larger amount of the mobile nonfunctionalized polymer penetrates the “presurface” region near the clay sheet. The mobile polymer chains experience conformational restrictions near the impermeable surface of the particle. (Those conformational losses increase with increases in P and prevent the mixing between the clay particles and the nonfunctionalized polymer in the absence of the end-active polymer.) Due to these entropic losses, the free energy of the system could become positive if the adsorbed amount Θ of the functionalized polymer is sufficiently small. (We see in Figure 10 that at χss ) -50, ∆Fmin > 0 when P* > P > 1.) Increases in P lead, on one hand, to the corresponding increases in the entropic losses of the mobile chains near the surface but, on the other hand, to an
Conclusions In summary, our investigations have probed the behavior of telechelic polymers confined between two surfaces within a melt. Previous studies 23,24 were primarily focused on the properties of confined telechelics in good solvents. Some scaling predictions were, however, proposed for the case of bridging chains in the melt.23 In this paper, we considered the bridging attraction between surfaces in the melt in greater detail. Namely, we derived an analytical model that yields the depth of the attractive minimum as a function of the molecular features of the telechelic and nonfunctionalized chains and the concentration of telechelics in the melt. We also performed numerical SCF calculations on a comparable system and found good agreement between the analytical and numerical models. Our results indicate that the surface attraction, which is induced by the adsorption of the bridging polymers, plays a dominant role in the behavior of polymer-clay composites even at very small volume fractions of these telechelic chains within the melt. In modifying the desired polymers, the experimental reactions can produce functionalized groups not only on one end but also on both ends of the chain. Thus, the presence of the bridging polymers (bearing the adsorbing groups on both ends) could prohibit the formation of exfoliated structures and the full-scale mixing of the polymer and the clay particles. In a forthcoming paper, we will investigate the role of bridging polymers in a mixture of mono- and bifunctionalized macromolecules and we will delineate the optimal conditions for forming exfoliated structures in polymerclay composites. We note, however, that our results do provide guidelines for maximizing the separation between the sheets within the intercalated structures. Namely, we showed that increasing φ, χss, P, and N will increase the layer spacing. Though not as optimal as the exfoliated material, intercalated composites with relatively large layer spacings still offer improved properties with respect to the pure polymer melt. Thus, the findings are useful in optimizing the behavior of polymer-clay hybrids. Acknowledgment. The authors thank Drs. David Moll, Rich Fibiger, Juan Garces, and Prof. Frans Leermaker for helpful discussions. A.C.B. gratefully acknowledges the financial support of the Army Office of Research and the NSF through Grant No. DMR-9709101. LA981406G (23) Milner, S.; Witten, T. Macromolecules 1992, 25, 5495 (24) Semenov, A. N.; Joanny, J.-F.; Khoklov, A. R. Macromolecules 1995, 28, 1066.
