Diblock Copolymer Surfactant Transport across the Interface between

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Diblock Copolymer Surfactant Transport across the Interface between Two Homopolymers Benedict J. Reynolds,† Megan L. Ruegg,† Thomas E. Mates,‡ C. J. Radke,*,§ and Nitash P. Balsara*,| Department of Chemical Engineering, UniVersity of California, Berkeley, California 94720, Materials Department, UniVersity of California, Santa Barbara, California 93106, Earth Science DiVision, Lawrence Berkeley National Laboratory, UniVersity of California, Berkeley, California 94720, and Materials Sciences DiVision and EnVironmental Energy Technologies DiVision, Lawrence Berkeley National Laboratory, UniVersity of California, Berkeley, California 94720 ReceiVed March 2, 2006. In Final Form: July 18, 2006 Dynamics of adsorption and desorption of a diblock copolymer to an interface between two homopolymers was measured using dynamic secondary-ion mass spectrometry (SIMS). Thin films were constructed consisting of a layer of saturated polybutadiene with 90% 1,2-addition (sPB90), followed by a layer of saturated polybutadiene with 63% 1,2-addition (sPB63), and finally by another layer of the sPB90 homopolymer. A sPB90-sPB63 diblock copolymer was initially included only in the top sPB90 layer of the film at a volume fraction of 0.05. The thin films were annealed at ambient temperature for times ranging between 0.2 and 108 h, and the concentration profiles of the diblock copolymer through the films were measured using SIMS. The dynamics of adsorption and desorption of the diblock copolymer at the two sPB90-sPB63 interfaces was gauged by comparing the different transient concentration profiles. The sorption process was modeled as diffusion in an external field, generated from self-consistent field theory (SCFT). All parameters for the model were determined independently. Although the model neglects the dynamics of conformational change, experimental results matched theory very well.

Introduction There have been numerous experimental and theoretical investigations into the dynamics of surfactant adsorption and desorption.1-12 These studies are motivated, in part, by the technological importance of surfactant-stabilized materials such as emulsions, drug delivery systems, foams, and paints. Surfactant transport to interfaces is typically described by diffusion to the interface, coupled with the kinetics of adsorption, although more sophisticated models have been proposed.2 At short length scales, these dynamics are believed to be controlled by the kinetics of adsorption and desorption, whereas at larger length scales the dynamics are controlled by diffusion. Accessing the short time and length scales where the kinetics of adsorption may become important is a major experimental challenge. There is no consensus on the magnitude of adsorption and desorption rate constants for most surfactants. * Address correspondence to either author. E-mail: nbalsara@ berkeley.edu (N.P.B.) or [email protected] (C.J.R.). † Department of Chemical Engineering, University of California, Berkeley. ‡ Materials Department, University of California, Santa Barbara. § Earth Science Division, Lawrence Berkeley National Laboratory. | Materials Sciences Division and Environmental Energy Technologies Division, Lawrence Berkeley National Laboratory. (1) Chang, C. H.; Franses, E. I. Colloid Surf. A 1995, 100, 1. (2) Diamant, H.; Ariel, G.; Andelman, D. Colloid Surf. A 2001, 183, 259. (3) Ferri, J. K.; Stebe, K. J. AdV. Colloid Interface Sci. 2000, 85, 61. (4) Joos, P.; Serrien, G. J. Colloid Interface Sci. 1989, 127, 97. (5) Langevin, D. Curr. Opin. Colloid Interface Sci. 1998, 3, 600. (6) Lin, S. Y.; Tsay, R. Y.; Lin, L. W.; Chen, S. I. Langmuir 1996, 12, 6530. (7) Macleod, C. A.; Radke, C. J. J. Colloid Interface Sci. 1994, 166, 73. (8) Miller, R.; Fainerman, V. B.; Leser, M. E.; Michel, M. Curr. Opin. Colloid Interface Sci. 2004, 9, 350. (9) Pan, R. N.; Green, J.; Maldarelli, C. J. Colloid Interface Sci. 1998, 205, 213. (10) Svitova, T. F.; Wetherbee, M. J.; Radke, C. J. J. Colloid Interface Sci. 2003, 261, 170. (11) Shin, J. Y.; Abbott, N. L. Langmuir 2001, 17, 8434. (12) Ward, A. F. H.; Tordai, L. Nature (London) 1944, 154, 146.

In this work, we present measurements of the adsorption and desorption dynamics of a block copolymer surfactant at a polymer/ polymer interface. The polymers used in this study are model polyolefins. In previous work,13,14 we showed that the equilibrium behavior of these polyolefins is well-described by standard theoretical methods of polymer physics, in particular by selfconsistent field theory (SCFT). We, thus, have a quantitative understanding of the inter- and intramolecular interactions. These polymers, therefore, represent a model system to investigate dynamical interfacial phenomena. Stabilization of polymeric dispersions is an important field in its own right. Blends of two or more polymers are of technological interest since they allow the combination of desirable properties into a single material. Since polymers are typically extremely immiscible in one another, polymer blends consist of a dispersion of one polymer in another, stabilized by a polymeric surfactant. There is a wealth of studies investigating the efficacy of such surfactants and their equilibrium behavior is now wellunderstood.15-24 Adsorption isotherms of various A/B diblock copolymers at A/B polymer/polymer interfaces have been (13) Reynolds, B. J.; Ruegg, M. L.; Mates, T. E.; Radke, C. J.; Balsara, N. P. Macromolecules 2005, 38, 3872. (14) Reynolds, B. J.; Ruegg, M. L.; Balsara, N. P.; Radke, C. J.; Shaffer, T. D.; Lin, M. Y.; Shull, K. R.; Lohse, D. J. Macromolecules 2004, 37, 7401. (15) Bates, F. S.; Maurer, W. W.; Lipic, P. M.; Hillmyer, M. A.; Almdal, K.; Mortensen, K.; Fredrickson, G. H.; Lodge, T. P. Phys. ReV. Lett. 1997, 79, 849. (16) Janert, P. K.; Schick, M. Macromolecules 1997, 30, 3916. (17) Janert, P. K.; Schick, M. Macromolecules 1997, 30, 137. (18) Lee, J. H.; Jeon, H. S.; Balsara, N. P.; Newstein, M. C. J. Chem. Phys. 1998, 108, 5173. (19) Hillmyer, M. A.; Maurer, W. W.; Lodge, T. P.; Bates, F. S.; Almdal, K. J. Phys. Chem. B 1999, 103, 4814. (20) Matsen, M. W. J. Chem. Phys. 1999, 110, 4658. (21) Thompson, R. B.; Matsen, M. W. Phys. ReV. Lett. 2000, 85, 670. (22) Corvazier, L.; Messe, L.; Salou, C. L. O.; Young, R. N.; Fairclough, J. P. A.; Ryan, A. J. J. Mater. Chem. 2001, 11, 2864. (23) Morkved, T. L.; Stepanek, P.; Krishnan, K.; Bates, F. S.; Lodge, T. P. J. Chem. Phys. 2001, 114, 7247. (24) Wang, Z. G.; Safran, S. A. J. Phys. 1990, 51, 185.

10.1021/la060580z CCC: $33.50 © 2006 American Chemical Society Published on Web 09/23/2006

Diblock Copolymer Surfactant Transport

measured, and the results are in good agreement with SCFT.25-29 In contrast, the dynamics of diblock copolymer surfactants in a polymer blend have received little attention. The experimental interface we investigate is that between two highly immiscible high molecular weight polymers labeled sPB90 and sPB63. The surfactant is a symmetric sPB90-sPB63 diblock copolymer, a linear molecule comprising covalently bonded sPB90 and sPB63 chains. The diblock copolymer is labeled with deuterium. The interface is created by depositing sequentially two thin films of sPB90 and sPB63 on a flat substrate. At t ) 0, a third thin film, composed of a mixture of sPB90 and the diblock copolymer, is placed on the top of the sPB63 film. The thickness of our films ranged from 50 to 600 nm. Diffusion of the diblock copolymer surfactant is monitored by dynamic secondary-ion mass spectrometry (SIMS) until both the sPB90sPB63 interfaces reach equilibrium. In doing so, we exploit the slow molecular dynamics that are characteristic of polymeric systems. All of the polymers used in this study are amorphous and rubbery. Slow dynamics is thus due to chain entanglement. Our experiments enable direct measurement of the time dependence of the surfactant concentration profiles. The polymer system may thus be viewed as a model system that accesses the correct spatial and temporal scales associated with surfactant transport to and across an interface. The results of the SIMS experiments are compared with theoretical predictions where the interfaces and the bulk phases in contact with the interfaces are treated together as a free-energy field, U(z), where z is distance normal to the interface. The diffusion coefficients of the surfactant in sPB90 and sPB63 phases were measured in separate SIMS experiments. The potential well was obtained by SCFT. We assume that momentum correlations are damped on length scales smaller than those over which U(z) changes so that the Smoluchowski equation30,31 describes the transport of surfactants in our system. All of the parameters required to predict the adsorption of the block copolymer at the interface were measured in independent experiments. Comparison between our transport data and theory is thus done with no adjustable parameters. In a previous work, we investigated the adsorption isotherm of the same diblock copolymer at the same polymer/polymer interface and found it to be described well by SCFT using no adjustable parameters.13 In this work, we move beyond equilibrium and investigate the dynamics of such systems. Experimental Methods Polymer Synthesis and Characterization. The polybutadiene materials were synthesized via anionic polymerization in hexane using tetrahydrofuran (THF) as a polar additive to control the percent of 1,2- versus 1,4-addition. They were then saturated in cyclohexane using either hydrogen or deuterium gas in a Parr high-pressure reactor at 95 °C with a 5% palladium on barium sulfate catalyst. In this paper, we use the nomenclature sPB90 and sPB63 to describe saturated polybutadienes (with 90% and 63% 1,2-addition, respectively). The nomenclature hPB90/dPB90 and hPB63/dPB63 distinguishes between hydrogenated and deuterated polymers. The molecular weights and architectures of the polymers were determined on a Waters 2690 gel permeation chromatograph (GPC) with a Viscotek triple detector. The characteristics of the polymers used in this study are summarized in Table 1. Labels for the polymers are based on our targets. Samples wherein the percent of 1,2-addition (25) Shull, K. R.; Kramer, E. J.; Hadziioannou, G.; Tang, W. Macromolecules 1990, 23, 4780. (26) Shull, K. R.; Kramer, E. J. Macromolecules 1990, 23, 4769. (27) Dai, K. H.; Kramer, E. J. J. Polym. Sci. Part B 1994, 32, 1943. (28) Dai, K. H.; Norton, L. J.; Kramer, E. J. Macromolecules 1994, 27, 1949. (29) Dai, K. H.; Kramer, E. J. Polymer 1994, 35, 157. (30) Furth, R. Ann. Phys (Berlin) 1917, 53, 177. (31) von Smoluchowski, M. Phys. Z. 1916, 17, 585.

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Figure 1. Diagram of the constructed polymer film. PS denotes polystyrene. Table 1. Polymer Properties polymer hPB90 homopolymer hPB63 homopolymer dPB63-dPB90 diblock copolymer

MW F (kg‚mol-1) PDI %1,2 (g‚cm-3) 220 187 38-41

1.02 88 1.02 63 1.02 63-92

0.86 0.86 0.91

deviated more than 2% from the targets were discarded. References 14 and 32 give detailed descriptions of the polymer synthesis and characterization procedures. Film Preparation. Figure 1 shows a schematic of the films used. They consist of a trilayer of hPB90/hPB63/hPB90 capped on both top and bottom with a polystyrene (PS) layer with an oxidized silicon wafer as the substrate. The dPB90-dPB63 diblock copolymer was initially dissolved in the topmost hPB90 layer at a volume fraction of 0.05. A Sentech SE400 ellipsometer was used to characterize the film thicknesses. The thicknesses of the two hPB90 layers were 150 nm, whereas the hPB63 layer was either 50, 150, or 600 nm thick. Thicknesses of the actual films used were not measured, but rather films spun from the same solution onto silicon wafers were measured in their place. All thicknesses were measured and controlled to within ( 2% of the target. However, some changes in the film thickness occurred during the process of transferring the layers from the mica, on which they were spin-coated, onto the silicon wafers. A more detailed description of the film-construction process is given in refs 13 and 32. The polybutadienes are rubbery at room temperature, so all annealing of the films was done at room temperature. Annealing time is, thus, the time between making the films and quenching them on the cold-stage of the dynamic SIMS apparatus. Films were studied for annealing times ranging from 0.2 to 108 h. We use the symbol Fxx[yy] to represent each film, where xx is the thickness in nm of the middle hPB63 layer and yy is the annealing time in hours. The present study is based on SIMS measurements of 38 independent films. Secondary Ion Mass Spectrometry (SIMS). The dynamic SIMS measurements discussed in this paper were taken with a Physical Electronics 6650 dynamic SIMS using a 3 kV, 60 nA beam of O2+ ions at 60° off normal incidence, which was rastered over a 0.040.09 mm2 region. A static, defocused, 350-500 V electron beam was used for charge neutralization. Negative ions of H, D, C, and Si were monitored as a function of time from an electronically gated area that was 15% of the rastered area. At room temperature, the polymers under investigation were well above their glass transition temperature, so it was necessary to perform the SIMS measurements on a liquid nitrogen-cooled cold stage, which reached a temperature of approximately -100 °C. More details regarding SIMS depth profiling of polymers can be found in a review by Schwarz et al.33 Reduction of SIMS Data. Dynamic SIMS detected the number of counts of H, D, C, and Si ions as a function of time. The speed with which the ion beam etches a crater in the polymer films is constant throughout the sPB trilayer. Thus, time can be directly (32) Reynolds, B. J. Ph.D. Thesis, 2005. (33) Schwarz, S. A.; Wilkens, B. J.; Pudensi, M. A. A.; Rafailovich, M. H.; Sokolov, J.; Zhao, X.; Zhao, W.; Zheng, X.; Russell, T. P.; Jones, R. A. L. Mol. Phys. 1992, 76, 937.

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Reynolds et al. Table 2. SCFT Parametersa χ lA lB NA

0.00642 0.491 nm 0.749 nm 4230

NB NAb NBb

3600 790 730

a l is the statistical segment length, N is the number of reference unit volumes in a polymer, A is sPB90, B is sPB63, and Ab and Bb are the respective blocks of the diblock copolymer. All parameters are based upon a reference volume of 100 Å3.

Figure 2. Adsorption isotherm as measured in ref 13. The solid line gives the isotherm as calculated by SCFT using the parameters in Table 2. The dashed line indicates the linear approximation to the adsorption isotherm. converted into depth. Since the diblock copolymer is deuterated, the number of D and H counts establishes the volume fraction of diblock copolymer at a given depth. Our protocol for reduction of SIMS data is described in refs 13 and 32.

Theoretical Transient Surfactant Volume Fraction Profiles The model we use to describe adsorption and diffusion of surfactant molecules is that of diffusive transport in an external field, U(z), where z denotes distance normal to the interface. Our proposed model is based on three main assumptions. First, the concentration of diblock copolymer is low enough such that there is no interaction between the different diblock-copolymer molecules (i.e., we are in the linear regime of the adsorption isotherm). Figure 2 gives the adsorption isotherm measured in ref 13. In this work, we do not exceed an adsorbed amount of 5 nm, so our assumption that we remain in the linear regime of the adsorption isotherm is a sufficiently accurate approximation. Second, any conformational change of the diblock copolymer occurs much faster than diffusion. The diblock copolymer is treated as a point particle with its nominal position defined to be the position of the junction point between the A and the B blocks. The number density of polymers with their junction point at position z, made dimensionless by the volume of a diblock copolymer molecule, NABν, is written as F(z, t) (i.e., NAB is the number of segments in the diblock copolymer, based on a reference volume, V ) 100 Å3). Third, we are in the viscous regime (i.e., momentum correlations are damped on shorter length scales than those over which U(z) changes). Given these assumptions, the appropriate diffusion equation is that of Smoluchowski:

[

∂F(z, t) ∂ ∂U(z)/kT ∂F(z, t) ) D(z) + D(z) F(z, t) ∂t ∂z ∂z ∂z

]

(1)

where D(z) is the diblock/copolymer diffusion coefficient and may depend on z, and kT is the Boltzmann constant multiplied by the temperature. To find F(z, t) the only inputs necessary are the initial condition, F(z, 0); the field, U(z); and the diffusion coefficient, D(z). SCFT26,34-36 is known to describe the thermodynamics of adsorption of diblock copolymers to polymer/polymer inter(34) Matsen, M. W. J. Phys.: Condens. Matter 2002, 14, R21. (35) Helfand, E. J. Chem. Phys. 1975, 62, 999.

Figure 3. Free energy of a diblock copolymer molecule with the junction point between the sPB90 and sPB63 blocks anchored at position z. Here z ) 0 nm denotes the PS/hPB90 interface.

faces.13,25-29 In a previous work, we examined the equilibrium behavior of the same diblock copolymer at the same polymer/ polymer interface and found it to be well-described by SCFT using no adjustable parameters.13,32 Accordingly, we used SCFT here to calculate the free-energy field, U(z), appearing in eq 1. The parameters necessary for SCFT, listed in Table 2, are the number of segments in the two homopolymers and the two blocks of the diblock copolymer (determined by gel permeation chromatography), the statistical segment lengths of sPB90 and sPB63, and the Flory-Huggins interaction parameter between sPB90 and sPB63 (determined by small-angle neutron scattering from a homogeneous binary homopolymer blend14). U(z) is thus the free energy of a diblock copolymer molecule with the junction point between the sPB90 and sPB63 blocks anchored at position z. Calculation of U(z) using SCFT is discussed in Appendix A. The calculated free-energy landscape for the F150 film series is shown in Figure 3. The field, which is only determined up to an additive constant by SCFT (see Appendix A), is set to zero in the bulk hPB90 layers. The main feature of U(z) is the deep potential wells at the two sPB90-sPB63 interfaces, indicating that at equilibrium we expect positive adsorption. The slightly larger value of U(z) in the middle hPB63 layer than that in the two hPB90 layers indicates a slight preference of the diblock copolymer for the hPB90 layers over the hPB63 layer (the block copolymer is not perfectly symmetric; see Tables 1 and 2). The PS/hPB90 interfaces, located at the edges of our system (see Figure 1), are modeled as hard walls with no other energy of interaction. The field U(z) increases near these interfaces, indicating that the diblock copolymer is repelled from these positions. The repulsion in the vicinity of these interfaces is of entropic origin and arises naturally in SCFT due to the difference in the statistical segment lengths between sPB90 and sPB63. At z ) 0 nm and z ) 450 nm, U(z) approaches infinity. (36) Evers, O. A.; Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1990, 23, 5221.

Diblock Copolymer Surfactant Transport

Figure 4. Diffusion coefficient profile used in Smoluchowski model calculations.

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Figure 6. Equilibrium volume fraction profile of diblock copolymer across the trilayer film after 108 h of annealing.

With U(z) and D(z) quantified, eq 1 and the initial condition in Figure 5 permit calculation of F(z, t). This is done numerically using a finite-difference method. The calculated density profiles cannot be directly compared with the SIMS instrument measured volume fraction profiles, φAB*(z) (as opposed to φAB(z), the actual volume fraction profile), because F(z) is a concentration profile of the junction point between the A and B blocks, whereas φAB*(z) is the measured concentration profile of the entire polymer chain. To convert F(z) to a diblock copolymer volume fraction profile, φAB(z), the segment density distribution of a diblock copolymer molecule with its junction point anchored at x, Φ(z, x), is required. This is calculated using SCFT as described in Appendix A (i.e., Φ(z, x) ) φAB(z, x, NAb)). The volume fraction profile is then given by

φAB(z) ) Figure 5. Initial condition of the dimensionless number-density profile of diblock copolymer junction points.

The bulk diffusion coefficients of the diblock copolymer in the sPB90 homopolymer and in the sPB63 homopolymer were determined to be 1.1 × 10-14 cm2‚s-1 and 2.5 × 10-14 cm2‚s-1, respectively, from independent experiments, as discussed in Appendix B. Accordingly, the diffusion coefficient profile, D(z), is assumed to be a double-step function as shown in Figure 4. To obtain the initial concentration profile of diblock copolymer through the film, we consider the upper layer. The upper layer (initially a blend of hPB90 and dPB90-dPB63) is spin-coated onto mica before deposition onto the film stack. Thus, the initial condition of the diblock/copolymer concentration profile through this upper layer is that of the diblock copolymer in a hPB90 layer exposed to a mica/hPB90 interface and to a hPB90/air interface. We assume that these two interfaces have no energetic preference for sPB90 or sPB63. This is the same assumption we made for the PS/hPB90 interfaces so the diblock copolymer is similarly repelled from these two interfaces. The initial number density profile F(z, 0) is then calculated, as shown in Figure 5, such that the average value of F(z) through the top layer is 0.05, the experimental volume fraction of diblock copolymer in the layer. The number density of diblock copolymer junction points has units of inverse volume. In this paper, F(z) is the number density multiplied by the volume of a diblock copolymer molecule, NABν, so that it is dimensionless. This dimensionless quantity is easier to work with because it has the same average value as φAB(z), the volume fraction profile of the diblock copolymer surfactant.

∫ dxΦ(z, x)F(x)

(2)

Finally, to allow comparison with experimental data, the theoretical volume fraction profile is convoluted with the instrumental resolution function, R(x), measured previously:13

φAB*(z) )

∫ dxR(x)φAB(z + x)

(3)

We refer to this method of predicting the transient volume fraction profiles as the Smoluchowski model.

Results and Discussion Figure 6 plots the measured volume fraction profile of diblock copolymer through a film after 108 h of annealing at ambient temperature, F150[108]. The thickness of all three films in the sPB trilayer is 150 nm. Open circles give the results of the experimental SIMS measurements, and the solid line gives the volume fraction profile predicted by the Smoluchowski model. Vertical dashed lines indicate the various interfaces in the film. It is clear from the symmetry of the data that the film is at equilibrium. There is no diblock copolymer surfactant present in the glassy PS layers, whereas there is diblock copolymer in all three sPB layers. The most noticeable feature of Figure 6 is the strong adsorption of the diblock copolymer to the hPB90/ hPB63 homopolymer/homopolymer interfaces. In Figure 7, the transient volume fraction profiles of diblock copolymer through selected F150 films are shown after various annealing times. Figure 7a gives the initial condition of the Smoluchowski model, determined by applying eqs 2 and 3 to F(z, 0) (see Figure 5). The diblock copolymer is initially only present in the upper sPB90 layer.

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Figure 8. Calculation of adsorbed amount from a volume fraction profile generated by the model. The open circles indicate the volume fraction 50 nm before and after the peak. The dashed line connects these two points and is used as the background (volume fraction of nonadsorbed diblock copolymer). The gray shaded area is the adsorbed amounts where the area above the dashed line is positive, whereas the area below the dashed line is negative.

Figure 7. Volume fraction profile of diblock copolymer across the trilayer film annealed for (a) 0, (b) 0.2, (c) 1.7, and (d) 10.5 h. The open circles show experimental results, and the solid line is the prediction of the Smoluchowski model.

The minimum accessible experimental annealing time is about 0.2 h, corresponding to the time required to insert the constructed film into the SIMS antechamber, to pump down the antechamber, and to transfer the film onto the SIMS cold stage. Figure 7b graphs the diblock copolymer volume fraction profile after 0.2 h of annealing. Open circles give the profile measured using SIMS, whereas the solid line is the theoretical prediction of the Smoluchowski model. After 0.2 h, the diblock copolymer surfactant has diffused from the upper hPB90 layer and adsorbed at the first hPB90/hPB63 interface. Note the local depletion of diblock copolymer near the A/B interface with the concomitant diblock copolymer adsorption peak, a result reminiscent of the classical Ward-Todai analysis of surfactant adsorption dynamics.12 In Figure 7c after 1.7 h, this same trend continues, but now an adsorption peak appears at the second hPB90/hPB63 interface. Clearly, the diblock copolymer diffuses through the hPB63 layer crossing the first interface to arrive at the second interface. Figure 7d graphs the profile after 10.5 h of annealing. The vast majority of the diblock copolymer is now adsorbed at the two interfaces. Finally, a slow process of diblock copolymer diffusion through the intervening hPB63 layer brings the two interfaces into the equilibrium seen in Figure 6 at 108 h of annealing. We, thus, are able to observe the dynamics of diblock copolymer transport through a nonhomogeneous polymer film and to explain this behavior by diffusion in an external field. The quantitative agreement between theory and experiment seen in Figures 5 and 6 for the 150 nm hPB63 layer also holds for the other film thicknesses (50 and 600 nm). Adsorption dynamics are best summarized by considering the amount of diblock copolymer adsorbed at each interface as a function of annealing time. Calculation of the transient adsorbed amount from the volume fraction profile is addressed in Figure 8. The two open circles correspond to the volume fraction 50 nm before and after the peak and are joined by a straight dashed line, which is used to estimate the volume fraction profile of nonadsorbed diblock copolymer. The adsorbed amount is then the area between the volume fraction profile and the dashed line, and has units of nanometers corresponding to cubic nanometers of diblock copolymer adsorbed per squared nanometers of interfacial area. This is an approximate method for estimating

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the d ) 50 nm data (Figure 9a). In these films, the adsorption peaks overlap, and we thus modified the procedure described in Figure 8 to obtain Γ. Rather than drawing the dashed line between two points 50 nm on either side of the interface, this line is drawn from the minimum in the measured volume fraction profile between the two peaks to a point 50 nm on the other side of the interface. This is why in Figure 9a the calculated amounts adsorbed at the two interfaces are not equal at equilibrium. The proposed Smoluchowski model captures the dynamics of the initial adsorption of diblock copolymer at the first interface as well as the much slower diffusion of the diblock copolymer from the first interface to the second. It does this consistently for all three film thicknesses. This excellent agreement endorses the assumptions used in the Smoluchowski model. The most notable assumption in the field theoretic diffusion model is that the dynamics of conformational change of the diblock copolymer are neglected. In generating the free-energy field, U(z), we averaged over all conformational degrees of freedom. Thus, our model excludes any dynamics of molecular conformational change. Agreement with the experiments suggests that the dynamics of conformational change are fast compared to translational diffusion over even very short distances.

Conclusions

Figure 9. Amount of diblock copolymer adsorbed at the upper (filled circles and solid line) and lower (open circles and dashed line) hPB90/hPB63 interfaces. Experimental results (circles) are compared with the predictions of the energy-landscape model (lines). The distances between the upper and lower hPB90/hPB63 interfaces are (a) 50, (b) 150, and (c) 600 nm.

the concentration profile of nonadsorbed surfactant in the interface but is sufficient for capturing qualitative trends. We have used more refined methods to calculate the adsorbed amount, and have found that the above method slightly underestimates the amount adsorbed but that the results are qualitatively the same.32 We prefer to use the method shown in Figure 8 because it is simpler and thus more robust. Transient adsorption where the hPB63 layer, d, is 50, 150, and 600 nm thick is summarized in Figure 9. The filled and open circles give the experimentally measured adsorbed amounts at the first and second interfaces, respectively. The solid and dashed lines are the adsorbed amounts at the two interfaces as predicted by applying the method shown in Figure 8 to the volume fraction profiles produced by the Smoluchowski model. The diblock copolymer is initially only present in the upper layer. In the first stage of surfactant transport, it diffuses and adsorbs at the first interface. After some time, a maximum in the adsorbed amount at the first interface is reached. This maximum for the d ) 150 and 600 nm films is well above the equilibrium value of Γ (Figure 9b,c). The final stage of surfactant transport involves desorption from the first interface, diffusion across the hPB63 layer, and adsorption at the second interface. The two stages of surfactant transport are not clearly distinguishable in

Transient concentration profiles of an AB diblock copolymer surfactant diffusing across a polymer/polymer A/B interface were measured. The polymer system under investigation consisted of three polymers: a saturated polybutadiene with 90% 1,2-addition (hPB90) homopolymer (A), a saturated polybutadiene with 63% 1,2-addition (hPB63) homopolymer (B), and a deuterated dPB90-dPB63 surfactant diblock copolymer (AB). Thin trilayer films were constructed consisting of a layer of hPB90, followed by a layer of hPB63, and then by a second layer of hPB90. A 5% volume fraction of dPB90-dPB63 diblock copolymer was dissolved in the upper hPB90 layer, whereas there was initially no diblock copolymer in the lower two layers. A large number of identical films were constructed and annealed at room temperature for various periods of time ranging from 0.2 to 108 h. Transient concentration profiles of the deuterated diblock copolymer through the films were measured by dynamic SIMS. The diblock copolymer surfactant diffuses from the top hPB90 layer and adsorbs at the first hPB90/hPB63 interface. At later times, the diblock copolymer desorbs from the first hPB90/hPB63 interface, diffuses through the hPB63 layer, and adsorbs at the second hPB90/hPB63 interface. Eventually, the concentration profile through the film equilibrates completely and appears symmetric. This process was modeled by translational diffusion in a field, U(z), where the field is the free-energy of a diblock copolymer molecule with its junction point anchored at position z, as calculated by SCFT. All parameters necessary for the SCFT calculations were determined independently, as were the bulk diffusion coefficients of the diblock copolymer in the hPB90 and hPB63 layers. The transient concentration profiles predicted by the Smoluchowski model are in excellent agreement with the concentration profiles measured experimentally. This agreement between experiment and a model that does not take conformational degrees of freedom into account suggests that the dynamics of conformational change are not important in describing the processes of adsorption and desorption of diblock copolymers at polymer/polymer interfaces. In a companion paper,37 we use classical surfactant transport models to interpret the experimental data presented here. In doing (37) Reynolds, B. J.; Ruegg, M. L.; Radke, C. J.; Balsara, N. P. Langmuir 2006, 22, 9201-9207.

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so, we show the correspondence, or lack thereof, between the Smoluchowski and classical models. Acknowledgment. This material is based upon work supported by the National Science Foundation under Grant 0305711. The support and advice of Rachel Segalman and Edward Kramer are greatly appreciated. This work made use of MRL Central Facilities supported by the MRSEC Program of the National Science Foundation under Award DMR00-80034.

does not influence the self-consistent solution found, the diblock copolymer partition functions are used for finding the field U(z). If we constrain a unit, s, on a chain (of species i) to be held at position z in space, the partition function of the section of the chain from 0 to s is described by

∂qi(z, s) ∂s

)

∑ m

σi,m(s)

[

Appendix A: Self-Consistent Field Theory In this work, we model the transport of the diblock copolymer as dilute particles diffusing in a field, U(z). We select the position of the junction point between the sPB90 and sPB63 blocks of the diblock copolymer as its nominal position and write eq 1. In eq 1, F(z) is proportional to the number density of diblock copolymers with their junction points at position z. In this appendix, calculation of the field U(z) is discussed as is the transformation of F(z) to the volume fraction of diblock copolymer, φ(z). We begin with a description of the self-consistent field method we adopt. SCFT26,34-36 is a mean-field model of polymer blends used when the blend can no longer be modeled as homogeneous such as at an interface. Whenever structure becomes important, but the entropy of fluctuations is negligible, SCFT may be successfully applied. We write the volume fraction profiles of the sPB90 (A) and sPB63 (B) homopolymers as φA(z) and φB(z), respectively. The diblock copolymer is assumed to be at low enough concentration such that its volume fraction is negligible. The essence of SCFT is that a system of many different interacting polymers is replaced by a system of non-interacting polymers in external fields. We, thus, calculate the external field necessary for a self-consistent interface between the A and B homopolymers and then see how the diblock copolymer behaves in this external field. The two monomer types, A and B, experience different fields. The fields experienced by a unit of type A and B are

wA(z) ) χABφB(z) + ξ(z)

(A1)

wB(z) ) χABφA(z) + ξ(z)

(A2)

where χAB is the Flory-Huggins interaction parameter between sPB90 and sPB63, and ξ(z) is a field, independent of monomer type that enforces the incompressibility constraint. The function ξ(z) is expressed in terms of the excess energy field, ∆w(z):

ξ(z) ) ∆w(z) -

φA(z) φB(z) - χABφA(z)φB(z) NA NB

(A3)

and

∆w(z) ) σ(φA(z) + φB(z) - 1)

(A4)

where σ is the inverse of the isothermal compressibility of the mixture (σ is constant). In our calculations, σ is chosen to be sufficiently large, so that the blend is essentially incompressible (i.e., that the sum of the volume fractions approaches one). NA and NB are the number of units (we use 100 Å3 units) in the A and B homopolymer, respectively. The parameter s defines units along the chain of a species and runs from 0 to Ni, for species of type i. In these calculations, there are three species: the A homopolymer, the B homopolymer, and the AB diblock copolymer. We keep the notation general in what follows, so that it can apply to the diblock copolymer as well as to the two homopolymers. Although the diblock copolymer

]

lm2 ∂2

qi(z, s) - wmqi(z, s) 6 ∂z2 i ) {A, B, AB}, m ) {A, B} (A5)

qi(z, 0) ) 1, qi(0, s) ) 0, qi(M, s) ) 0

(A6)

Here lm is the statistical segment length of chains of monomer m, and σi,m(s) is 1 if at position s in a species of type i there is an m unit and 0 otherwise. The initial condition and the boundary conditions are given by eq A6 where z ) 0 and z ) M give the upper and lower PS/hPB90 interfaces, respectively. The partition function of the entire chain, Qi(z, s), can then be written as the product of the partition functions of the complementary chain sections:

Qi(z, s) ) qi(z, s)qi*(z, s)

(A7)

where qi* is similar to qi but is the partition function of the section of the chain from s to Ni. In both cases, position s is anchored at z. The governing equation for qi* is similar to eq A5 but with a negative sign in front of the right side and with the initial condition that qi(z, Ni) ) 1. The volume fractions of the two homopolymers (we ignore the diblock copolymer) are then

φi(z) ) Ci

∫0N dsQi(z, s) i

(A8)

where Ci is a normalization constant. Since the total volume fraction of species of type i in the system, φi, is known, Ci can be obtained from

Ci )

φiM

∫0N ∫0M ds dzQi(z, s) i

(A9)

where the integration is performed from z ) 0 to M (the left and right boundaries of the box). We start with an initial guess for φm(z) and calculate the fields using eqs A1 through A4. From these fields, the partition functions Qi(z, s) are calculated with eqs A5 through A7, and volume fractions are found with eqs A8 and A9. Fields are calculated from the new volume fraction profiles again using eqs A1 through A4 and are combined with the previous fields for numerical stability:

EA(z)new ) λ1EA(z)calculated + (1 - λ1)EA(z)old (A10) ξ(z)new ) λ2ξ(z)calculated + (1 - λ2)ξ(z)old

(A11)

where λ1 is typically 0.1 and λ2 is typically 0.001. λ1 and λ2 are chosen to be as large as is possible without the iterations becoming unstable. The new fields are then used to generate a new concentration profile, and the process is iterated until convergence is achieved. We judge convergence to have occurred when the change in the free energy at each iteration step is well below the required level of accuracy. The error due to incomplete convergence is in all cases smaller than the size of the data points in the plots.

Diblock Copolymer Surfactant Transport

Langmuir, Vol. 22, No. 22, 2006 9199

Once a self-consistent solution is found, we calculate the field, U(z), in which our point polymer diffuses. The appropriate free energy is one which averages over all the degrees of freedom of the polymer chain except for the position of the junction point. This free energy, U(z), is thus related to the partition function of a diblock copolymer chain with the junction point anchored at position z, QAB(NAb, z) by

U(z) ) -ln QAB(NAb, z) kT

(A12)

where NAb is the number of segments in the A block of the diblock copolymer. This partition function is calculated using eqs A5 through A7. The final step is to transform the profiles F(z) into volume fraction profiles φ(z). What is needed is the average volume fraction of a diblock copolymer at position z, whose junction point is anchored at position x, Φ(z, x). φ(z) is then related to F(z) by

φ(z) )

∫ dxΦ(z, x)F(x)

Figure 10. Diagram of the constructed polymer film for measuring the diffusion coefficient of the block copolymer in hPB90. PS denotes polystyrene.

(A13)

To calculate the volume fraction of the unconstrained diblock copolymer, φ(z), the partition function of a molecule with segment s anchored at position z, QAB(z, s), is found, (eq A7) and then φ(z) is found by integrating over s (eq A8). To find the volume fraction profile of a diblock copolymer with segment s′ (s′ ) NAb in our model) anchored at position x, Φ(z, x), we calculate the partition function of a molecule with two constraints. Segment s′ is anchored at x, and segment s is anchored at z. We label this partition function Qi′(z, x, s, s′). In the same way that Qi(z, s) ) qi(z, s)qi*(z, s), we can write then

Qi(z, x, s, s′) ) qi(x, s′)qi′(z, x, s, s′)qi*(z, s) (A14) Here qi(x, s′) and qi*(z, s) are the sections on either side of the two anchored points and are calculated as described earlier (eq A5). qi′(z, x, s, s′) is the partition function of the segment between the two anchored points and is calculated in the same manner except with the initial condition, qi′(z, x, s′, s′) ) δ(z - x) where δ(x) is the Dirac delta function. If s > s′, this initial condition is propagated forward as in the calculation of qi(z, s); whereas if s < s′, the initial condition is propagated backward as in the calculation of qi*(z, s):

∂q′i(z, x, s, s′)

)

∂s

∑ m

σi,m(s)

[

]

l2m ∂2

qi′(z, x, s, s′) - wmqi′(z, x, s, s′) 6 ∂z2 s > s′ (A15)

Setting s′ to NAb, we can then find Φ(z, x) by

Φ(z, x) ) C

∫0N

Ab

+ NBb

dsQAB(z, x, s, NAb)

(A16)

where C is a normalization constant that ensures ∫ dzΦ(z, x) ) 1.

Appendix B: Determination of Diffusion Coefficients To determine the diffusion coefficient of dPB90-dPB63 in hPB90, a set of duplex films shown in Figure 10 were constructed where the lower layer was 300 nm thick and initially contained no diblock copolymer, whereas the upper film was 150 nm thick

Figure 11. Volume fraction profile of diblock copolymer through film (as in Figure 10) annealed for 1.0 h. The open circles indicate the SIMS measurements whereas the solid line is a numerical fit using the diffusion equation and is characterized by DAB/At ) 3600 nm2.

and had a volume fraction of diblock copolymer of 0.05. The time between formation of the film and the placement of the film on the SIMS cold stage is referred to as the annealing time, t. Figure 11 shows the volume fraction profile of diblock copolymer through the film after 1.0 h of annealing. Open circles show the SIMS measurements as the diblock copolymer diffuses from the upper layer to the lower layer. The solid line is a numerical solution to the diffusion equation with the known initial condition and a constant diffusion coefficient. After 1.0 h, a value of Dt ) 3600 nm2 is necessary to fit the experimental data. Six such films were made with different annealing times, and their volume fraction profiles were measured with SIMS. A value of Dt was found for each film so that the numerical profiles matched the experimental data at that time. Figure 12 plots the numerical fitting parameter Dt against experimental annealing times t. The six experimental points lie on a straight line, indicating that our analysis is sound. The slope of this line gives the diffusion coefficient of the diblock copolymer in the hPB90 homopolymer, which is measured to be 1.08 × 10-14 cm2‚s-1. The fact that the fitted line almost passes through the origin is an indication that the films quickly cool below their glass-transition temperature when they are placed upon the SIMS cold stage. The calculation of the diffusion coefficient in the hPB63 homopolymer was made in a similar manner. The geometry of the films used to make the measurements differs slightly and is shown in Figure 13. The lower layer was 600 nm thick and contained no diblock copolymer. The middle layer was 300 nm thick and contained a volume fraction of diblock copolymer of

9200 Langmuir, Vol. 22, No. 22, 2006

Figure 12. Comparison of actual annealing times for 6 polymer bilayer films with the DAB/At values necessary to fit the diffusion equation to the experimental results (as shown in Figure 11). The slope of the line gives the diffusion coefficient.

Reynolds et al.

Figure 14. Volume fraction profile of diblock copolymer through film (as in Figure 13) annealed for 0.60 h. The open circles indicate the SIMS measurements whereas the solid line is a numerical fit using the diffusion equation and is characterized by DAB/Bt ) 10700 nm2.

Figure 13. Diagram of the constructed polymer film for measuring the diffusion coefficient of the block copolymer in hPB63. PMMA denotes polymethyl methacrylate.

0.05. The upper layer was 600 nm thick and again contained no diblock copolymer. This arrangement reduced adsorption of the diblock copolymer at the hPB63/PMMA interfaces for short annealing times. Figure 14 shows the volume fraction profile of diblock copolymer through the film in Figure 13 after 0.60 h of annealing. The diblock copolymer, initially constrained to the central layer, diffuses outward. Six films with different annealing times were formed. SIMS measurements were made, and numerical solutions were fitted to the measured diblock copolymer volume fraction profiles. Figure 15 plots the numerical fitting parameter Dt against experimental annealing times t. The six data points again obey a linear relationship. However, unlike the linear fit in Figure 12, here the straight line does not intercept

Figure 15. Comparison of actual annealing times for six polymer bilayer films with the DAB/Bt values necessary to fit the diffusion equation to the experimental results (as shown in Figure 14). The slope of the line gives the diffusion coefficient.

the origin but intercepts the x-axis at -0.6 h. The simplest interpretation of this result is that the hPB63 layer does not vitrify as quickly as the hPB90 layers, and the diblock copolymer continues to diffuse for a period of time after being placed on the cold stage. The glass transition temperature of hPB63 is -45 °C as compared to -30 °C for hPB90.38 LA060580Z (38) Krishnamoorti, R. Ph.D. Thesis, 1994. p 65.