Stratification in Monolayers of a Bidisperse Melt Polymer Brush As

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Stratification in Monolayers of a Bidisperse Melt Polymer Brush As Revealed by Neutron Reflectivity Werner A. Goedel,*,† Clarisse Luap,† Ralf Oeser,‡ Peter Lang,§ Christian Braun,§ and Roland Steitz‡,§ Max-Planck Institut fu¨ r Kolloid- & Grenzfla¨ chenforschung, Rudower Chaussee 5, 12489 Berlin, Germany; Hahn-Meitner-Institut Berlin, Glienicker Strasse 100, 14109 Berlin, Germany; and I.-N.-Stranski-Institut fu¨ r Physikalische & Theoretische Chemie, Technische Universita¨ t Berlin, Strasse des 17 Juni, 10623 Berlin, Germany Received December 8, 1998; Revised Manuscript Received June 16, 1999

ABSTRACT: The volume fraction profile of long chains within a monolayer of a dense bidisperse polymer brush is investigated using neutron reflectivity. Monolayers of polymer brushes are prepared by spreading equimolar mixtures of short nondeuterated and long perdeuterated polyisoprenes with quaternary ammonium headgroups onto the surface of a Langmuir trough filled with H2O or H2O/D2O mixtures. The chain lengths of the shorter and longer chains are approximately 485 and 970 repeat units, respectively. The neutron reflectivity of the monolayer-covered water surface is recorded at two different mean areas per polymer chain (220 and 310 Å2). The analysis of the reflectivity data yields a total brush thickness that is proportional to the surface concentration and is in accordance with the bulk density of the polymer. The inner structure can be described by a stratified model: The bottom region is formed by a mixture of long and short chains. On top of this region is a second part composed only of segments of the longer chains. The content of long chains in the bottom region is higher than that predicted by analytical self-consistent field (ASCF) theories. Thus, the parts of the long chains within the mixed region are less stretched than predicted by ASCF theories. The structure approximately scales with the total brush height, although small deviations occur. The width of the transition zone between the two regions seems to be independent of the brush height.

Introduction In a polymer brush, flexible polymers are bound with one end to an interface with a lateral spacing considerably less than the dimensions of the undisturbed polymer coil. To reduce mutual interactions, the polymer chains stretch away from the interface. In general, the polymer brush can either contain solvent (swollen brush) or be solvent-free (melt brush or dense brush). The most prominent examples of a “dense brush” are phase-separated block copolymers. In these systems, two incompatible polymers are connected by a chemical bond and phase separate into nanometersize domains. The size and shape of these domains is a result of the balance between the interfacial tension between the domains and the loss of entropy associated with the stretching of the closely packed polymer chains.1-6 Although there is considerable interest in these systems,7 it has been very advantageous to study polymer brushes as monolayers at flat surfaces, for example, by spreading hydrophobic polymers with ionic headgroups at the surface of a water-filled Langmuir trough.8-14 In such a monolayer, it is relatively easy to determine and tune the surface concentration of the polymer chains and to give the system a preferred orientation in space. In addition, the plane of tethering in these monolayers usually is less diffuse than in the case of phase-separated block copolymers. * Current address: Werner A. Goedel, Organic Chemistry 3, University of Ulm, D-89069 Ulm. Tel: ++49 731 502 3470. Fax: 502 2883 or 3472. E-mail: [email protected]. † Max-Plank Institut fu ¨ r Kolloid- & Grenzfla¨chenforschung. ‡ Hahn-Meitner-Institut Berlin. § I.-N.-Stranski-Inst. fu ¨ r Physikalische & Theoretische Chemie.

Initial theoretical and experimental investigations of swollen and dense polymer brushes concentrated on polymers of uniform chain length. However, the majority of commercial polymers has a broad molecular weight distribution. In addition, it has been shown that polymers of different chain lengths have a tendency to form mixed brushes15-17 and that domain sizes and shapes of phase-separated block copolymers can be finetuned by mixing block copolymers of different chain lengths.15,18-20 More recent theories21-24 include bidisperse and polydisperse brushes. Besides giving a background for the evaluation of equilibrium morphologies and minimum free energies of mixed brushes, these theories predict a stratified structure of the brush: Close to the tethering plane, the brush is composed of a region composed of segments of short and long chains. Above that region is a second one, which is composed only of segments of the long chains. It is further predicted that if the tethering density is changed, the stratified structure undergoes affine deformation. Yet, only a few experimental papers have concentrated on the distribution of the long and short chains within swollen25,26 or dense27 bidisperse brushes. In this paper, we investigate the inner structure of a bidisperse dense brush composed of an equimolar mixture of long and short chains, which differ in chain length by a factor of 2, and the dependence of the inner structure on the tethering density: (i) A mixture of polyisoprenes with ionic headgroups, in which the long chains are deuterated, is applied to a water surface and forms a monolayer of a dense polymer brush.

10.1021/ma981900r CCC: $18.00 © 1999 American Chemical Society Published on Web 10/07/1999

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chains, NR and Nβ, respectively. Because φR ∼ NRxR and φβ ∼ Nβxβ, one obtains the following:

φR NR xR ) 1 - φR Nβ 1 - xR φR )

Figure 1. Schematic drawing of the different scenarios of a mixed brush.

(ii) The composition profile normal to the plane of the monolayer is investigated by neutron reflectivity.7 (iii) The area per chain is varied by lateral compression of the monolayer using a Langmuir trough, and the neutron reflectivity experiments are repeated at different areas per chain. In the following, we shortly review different theoretical scenarios and the corresponding composition profiles, present neutron reflection experiments on the floating monolayers, and compare the experimentally derived scattering length density profiles to the theoretical predictions. Theoretical Composition Profiles In all of the following scenarios, the brush is composed of longer R-chains with a chain length NR and shorter β-chains with a chain length of Nβ repeat units, chemically identical to the R-chains. The length ratio of the two chains is given by ) Nβ/NR. All chains are tethered with one end to the lower interface of the brush at z ) 0 and are stretched along the z-coordinate normal to that interface. The total number of chains, n, per surface area, A, is given by σ ) n/A, the number per unit surface of longer (shorter) chains is given by σR (σβ), and the molar fraction of the longer (shorter) chains is given by xR ) σR /σ (xβ ) 1- xR ) σβ /σ). The polymer is regarded as incompressible, and the volume of a repeat unit is set to unity. As a consequence, the total height of the brush is given by the number-average chain length and the number of chains per unit surface:

H ) σ[xRNR + (1 - xR)Nβ]

(1)

At each point in the brush, the volume fraction of the longer and shorter chains φR and φβ, respectively, are correlated by:

φR ) 1 - φβ

(2)

In all scenarios, except for the first one, the brush is composed of a “mixed region” of height h, in which Rand β-repeat units are mixed, and a region composed only of R-segments on top of the mixed region. A sketch of the different scenarios considered is given in Figure 1. Homogeneous Mixture. If one ignores the differences in chain length, one expects the two polymers to mix homogeneously throughout the complete layer (see Figure 1a). Thus, the overall composition of the layer is given by the average volume fraction of the longer and shorter chains, which is related to the mole fraction xR, and the chain length of the longer and the shorter

xR xR + (1 - xR)

(3)

(4)

Uniformly Extended Chains. In the simplest stratified scenario, we assume that in both regions the chains are uniformly stretched. We assume further that the conformations of the segments of the longer chains within the mixed region are identical to the conformations of the shorter chains (see Figure 1b). As a consequence, the volume fraction of the longer chains in the mixed region is identical to the mole fraction, xR:

φR ) xR for 0 e z < h φR ) 1 for h e z e H

(5)

The ratio of the height of the mixed region to the total height of the monolayer, h/H, is given by the ratio of the volume of the mixed region to the total volume of the monolayer. The former is proportional to the chain length of the shorter chains, and the latter is proportional to the number-average of the chain length. Thus, h/H is given by

σNβ h ) ) H σ[xRNR + (1 - xR)Nβ] xR + (1 - xR)

(6)

Analytical Self-Consistent Field (ASCF) Theories. Milner, Witten, and Cates21 and Birshtein, Liatskaya, and Zhulina.22 In these theories, the mutual interaction of the polymer chains is represented by a mean field that gives rise to a nonuniform stretching of the chains. As has been pointed out by Milner, Witten, and Cates21 and the conformations of polymer chains are similar to the flight path of a particle starting at rest at the location of the free end and being accelerated in a field toward the “bottom” of the brush. This field can be described by a kinematic potential U(z), which is a function of the position in the brush, z. The chain lengths of the polymers of the brush are equivalent to the flight time of the hypothetical particle. The stretching of the chains is equivalent to the velocity of the particle. In these models, deviations of the chain conformation from the most probable conformation are not taken into account. Thus, they can only be applied to strongly stretched chains. In the case of a brush of uniform chain length, the flight time of the equivalent hypothetical particle in the kinematic potential is independent of the starting position. Hence, the kinematic potential is harmonic (U ∝ z2). In the case of a bimodal brush, there are two chain lengths present. Hence, the description has to accommodate two different flight times of the hypothetical particle. This can only be achieved if the ends of the shorter and the longer chains segregate into separate regions. The ends of the shorter polymers are at z < h, and the ends of the longer chains are at locations z > h. On the basis of the theory of Milner, Witten, and Cates21 and one can derive an expression for the region height and for the volume fraction of the longer chains (see Appendix):


Stratification in Monolayers 7601

h ) λx1 - xR2 with λ ) H xR + (1 - xR) 2 φR(z) ) ‚arccos π

x

(1 - xR2)λ2 - (z/H)2

(7)

for z < h

λ2 - (z/H)2

φR(z) ) 1 for h e z e H

(8)

This expression is identical to the expression obtained independently by Birshtein, Liatskaya, and Zhulina.22 (see Appendix). As in the case of the uniformly extended brush, changing the height of a mixed brush by lateral compression stretches the whole profile proportional to the film thickness. If the profile is expressed in coordinates relative to the total film thickness, its shape is unaffected by the actual dimensions of the system. At the location of the headgroups (z ) 0), the volume fraction is equal to

2 φR(z ) 0) ) arccos x(1 - xR2) π 2 ) arcsin(xR) < xR (9) π This value is smaller than the molar fraction of the longer chains. This is due to the fact that at z ) 0 the longer chains are stretched more than the shorter chains. Spontak.23 The scenario described by Spontak23 is similar to that described above, with one difference: it is assumed that in the mixed region all R-chains have identical conformations and contribute the same number of repeat units Γ*NR to the mixed region (Γ* e 1). It is found that Γ* depends only on the composition of the bidisperse brush. Values of Γ* have been evaluated numerically. One can calculate the height of the mixed region, h, and the volume fraction profile (see Appendix):

h Γ*NRxR + Nβ(1 - xR) Γ*xR + (1 - xR) ) ) H NRxR + Nβ(1 - xR) xR + (1 - xR)

(

2 z2 Λ2 φR(z) ) xRλ 2 - 2 π H H

)

(10)

-1/2

for z < h


(11)

with

λ)

(

xR + (1 - xR)

)

( )

and Λ ) h csc

π Γ* 2

Comparison of the Different Scenarios. In Figure 2, the theoretical volume fraction profiles derived from the different scenarios are depicted for the case of NR ) 2Nβ and three different molar fractions of the longer chains xR ) 0.2, 0.5, and 0.8. In the case of small molar fractions of the longer chains, the profiles are very similar; the profile derived from Milner, Witten, and Cates21 and Birshtein, Liatskaya, and Zhulina.22 (MWC/ BLZ theory) is in especially close agreement to the profile derived from Spontak. Compared to the uniformly extended model, the mixed region in the analytical self-consistent field scenario has a smaller height and a smaller volume fraction of long chains. This is due to the fact that in the ASCF scenario the longer

chains in the mixed region are stretched more than the shorter chains, whereas the uniformly extended model assumes an identical degree of stretching for both chains. As the molar fraction of the longer chains increases, differences between the scenarios become more pronounced. It becomes evident that the height of the mixed region and the volume fraction of long chains derived from the Spontak theory are even lower than the predictions based on Milner, Witten, and Cates21 and Birshtein, Liatskaya, and Zhulina.22 This can be attributed to the fact that in the former theory the parts of the longer chains within the mixed region are assumed to have identical conformations, whereas this condition is not implemented in the latter case. As has already been shown for monodisperse brushes,13 this constraint leads to a higher mean degree of stretching of the affected chains. This in turn leads to a lower volume fraction of these chains in the mixed region. Thus, to distinguish between the scenarios, one should investigate mixtures rich in long chains. On the other hand, the compositional difference between the “top region” and the “mixed region” and thus the contrast in neutron reflection experiments decrease with increasing molar fraction of longer chains. Thus, a molar fraction of xR ) 0.5 is most suitable to distinguish between the different scenarios and, therefore, was used in the experiments depicted below. Experimental Section Materials. Nondeuterated (h-PIN+) and perdeuterated polyisoprene (d-PIN+) with tertiary ammonium headgroups were synthesized by living anionic polymerization initiated with 1-(4-dimethylaminophenyl)-1-phenylhexyllithium following published procedures.28-30 Structure and characterization of the polymers are given in Table 1. Molar masses of both polymers were evaluated from size exclusion chromatography using a calibration with nondeuterated polyisoprene standards. The number of repeat units was calculated by dividing the number-average molar mass of the polymer by the molar mass of a repeat unit. In the case of the perdeuterated polyisoprene, it was assumed that the retention time is identical to the one of a nondeuterated polyisoprene of the same number of repeat units. To achieve quaternization of the dimethylamino headgroup, 0.3 g of the raw polymer was dissolved in a mixture of 1 mL iodomethane and 0.3 mL nitromethane. The solution was stirred at room temperature in argon atmosphere for 3 days. The quaternized polymer was precipitated in a mixture of 30 vol % of 2-propanol and 70 vol % of methanol, redissolved in tetrahydrofuran, and reprecipitated three times. H2O was purified with an ion exchange filter system equipped with a photooxidation unit operating at 185 nm wavelength (Milli-Q+185, Millipore). Resistivity exceeded 18 MΩ cm-1. Total organic carbon was less than 5 ppb (checked with the TOC monitor, Anatel10, Millipore). D2O (Aldrich), ethanol (Aldrich) and chloroform (Baker) were used as received. Experimental Setup. Chloroform solutions, which contained 10 wt % of ethanol and 0.05 wt % of the polymer d-PIN+ and binary mixtures of d-PIN+ and h-PIN+ (1:1 mol/mol), respectively, were spread onto the free water surface of a [14 cm × 35 cm] Langmuir film balance (Riegler & Kirstein, Wiesbaden, Germany) equipped with a Wilhelmy plate system for measuring the lateral pressure. The number-average molecular weight was used to calculate the mean area per molecule. All experiments were conducted at a temperature of 25 °C. In the case of mixed monolayers, lateral homogeneity was checked via Brewster angle microscopy. Neutron reflectivity experiments were performed on the polymer monolayers at a lateral pressure of 4 mN/m and 20 mN/m. The Langmuir trough was mounted within an airtight

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Figure 2. Predicted volume fraction profiles for repeat units of the longer chains in a mixed brush following different scenarios for a chain length ratio of ) Nβ/NR) 0.5 and different xR mixing ratios () 0.2, 0.5, and 0.8). Table 1. Characterization of the Polymers number-average molar mass Mn [g/mol]

polydispersity Mw/Mn

no. of repeat units

h-PIN+-485

33 000

1.04

485

d-PIN+-970

74 000

1.1

970

name

chemical formula

box with aluminum windows on the sample stage of the neutron reflectometer V6 at the Berlin Neutron Scattering Center (Hahn-Meitner-Institut). A detailed description of the instrument is given by Mezei, Goloub, Klose, and Toews.31 The incident neutron beam of wavelength λ ) 4.66 Å was reflected from the water surface and was subsequently counted with a 3He detector. The wave vector Q ) [4π/λ]sin θ was calculated from the angle of the neutron beam with the water surface. The resolution of the instrument was set to 1 × 10-3 Å-1 (beam of rectangular cross section, height ) 0.5 mm, width ) 40 mm) for 0.005 Å-1 < Q < 0.06 Å-1 and 2 × 10-3 Å-1 (width ) 1.0 mm) otherwise. Scans were recorded in single θ/θ steps with sampling times per step from 120 to 1200 s for 0.005-0.085 Å-1. For each individual reflection angle, off-specular scattering and superimposed instrument background were measured by two 3He detector tubes offset from the specular position by (0.475°, respectively. To make sure that the properties of the monolayer did not change during the measurement, the neutron reflectivity was measured a second time in the range of 0.005 Å-1 < Q < 0.033 Å-1. (No changes in the reflectivity profile did occur.) The theoretical scattering length density of h-PIN+ () 0.268 × 10-6 Å-2) was calculated from the chemical composition, the volume of a repeat unit of 123.7 Å3,13 and tabulated values of the scattering length of the corresponding elements.32 The scattering length density of d-PIN+ () 6.94 × 10-6 Å2) was estimated from the analysis of the reflectivity experiments on monolayers of pure d-PIN+ (experiment 1; see below). Data Reduction and Analysis. The signal of the offspecular detector tubes was independent of the reflection angle. To eliminate any nonspecular contributions to the experimentally recorded reflectivity, the signal of the offspecular channels was averaged and subtracted from the signal of the specular channel. Data sets were normalized on sampling time and incident intensity. To account for the change of beam area intercepted by the sample, an additional “footprint” correction was applied. Analysis of reflectivity measurements was conducted with the aid of a home-written software program, Parratt32,33 which

Figure 3. Sketch of the two-box model of the scattering length density used for the simulation of the experimental neutron reflectivity profiles. is based on the implementation of the Parratt formalism34,35 for the simulation of reflectivity data of stratified structures. The monolayer of pure deuterated polyisoprene, d-PIN+, was represented by one box with roughness σtop at the air/film interface and roughness σbottom at the film/subphase interface. The mixed monolayer of d-PIN+ + h-PIN+, was modeled by a two-box model as outlined in Figure 3. This model is described by the four scattering length densities (subphase, (b/v)sub; bottom, (b/v)bottom; and top region of the brush, (b/v)top; and air (b/v) ) 0), three interfacial widths (bottom, σbottom; inner, σi; and top interface, σtop) and two lengths (height of bottom region, h, and total brush height, H). The errors of the parameters were estimated the following way: while all other parameters are allowed to readjust in the fitting procedure, the parameter in question is varied until χ2 increases by 30%. In addition, the reflectivity was simulated for the mixed monolayers using the method of simulated annealing36,37 using a program developed by Marczuk.38 In this method, the monolayer is divided into a set of small boxes. In each step, the scattering length density of one of the boxes is varied, and



Figure 4. Lateral pressure-area isotherms of monolayers of the pure polymers h-PIN+ and d-PIN+ and an equimolar mixture of both polymers. Arrows indicate the lateral pressures corresponding to the experiments A and B. the change in the mean square deviation, ∆χ2, is calculated. In the case of a decrease in χ2, the change is accepted; in the case of an increase in χ2, the change is accepted with a probability that decreases with increasing ∆χ2. This process is repeated until changes in χ2 converge to a preset value. For the simulation of reflectivity profiles based on the theoretical volume fraction profiles as predicted by analytical mean field theory (MWC/BLZ theory), the continuous functions were approximated by subdividing them into series of discrete boxes. The individual histograms were chosen such that the width of the subdivisions was small enough to closely approximate the analytical profiles.

Results Monolayers of the pure deuterated polymer d-PIN+ and of equimolar mixtures of d-PIN+ and h-PIN+ were investigated on H2O and on a H2O/D2O mixture of a scattering length density of 4.36 × 10-6 Å2. The lateral pressure-area isotherms of the pure polymers and of the mixture are depicted in Figure 4. To investigate the dependence of the inner structure of the monolayer on the area per molecule, neutron reflectivity experiments were performed at a surface pressure of 4 mN/m (large area per molecule and low monolayer thickness; see arrows in Figure 4) and at 20 mN/m (small area per molecule and larger monolayer thickness). Neutron Reflectivity Experiments. The neutron reflectivities as a function of wave vector Q of monolayers of pure d-PIN+ on H2O are depicted in Figure 5a. The corresponding experiments conducted on equimolar mixtures of d-PIN+ and h-PIN+ on H2O and H2O/ D2O mixtures are given in Figures 6a and 7a, respectively. Monodisperse Polymer Brush (Experiment 1). The neutron reflectivity of monolayers of pure d-PIN+ shows sharp Kiessig fringes superimposed onto the decay of the reflectivity following Fresnel’s Law (see Figure 5a). The fringes are evenly spaced and have a nearly constant amplitude in the whole investigated Q range. The amplitude of the fringes is independent of the area per molecule. However, the spacing of the oscillation decreases with decreasing area per molecule. This signature indicates a uniform layer of constant scattering length density with two comparatively smooth interfaces and a thickness that increases with decreasing area per molecule. As expected, the reflectivity profiles can be simulated by a scattering length density profile based on a onebox model (see Figure 5b). The parameters of the

Figure 5. Experiment 1A,B. Neutron reflectivity (a) and corresponding scattering length density profiles (b) of monolayers of pure d-PIN+ at the surface of pure H2O recorded at 20 mN/m (A/n ) 250 Å2) and at 4 mN/m (A/n ) 350 Å2).

simulated neutron reflectivity profile are given in Table 2. If one averages the scattering length densities in experiment 1A,B, one obtains a value of b/v(PIN+) ) 6.94 × 10-6 Å2. On the basis of bulk density, one can calculate from this scattering length density a degree of deuteration of 99%. This value is well in agreement with the degree of deuteration given by the supplier of the monomer (98%). Bidisperse Polymer Brush (Experiment 2 and 3). As in the case of monolayers of pure d-PIN+, the neutron reflectivity profiles of the mixed monolayers on H2O show pronounced Kiessig fringes (experiment 2; see Figure 6a). The spacing of the fringes is larger by a factor of 1.3 than in the case of experiment 1. The amplitude of the fringes is slightly reduced, and the critical angle of total external reflection ()total reflection edge) is shifted to lower Q values. In contrast to the monodisperse brush, the amplitude of the fringes varies with the wave transfer vector, Q. For example, the first reflection minimum is far less pronounced than the second one.

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Figure 6. Experiment 2A,B. Neutron reflectivity (a) and corresponding scattering length density profiles (b) of monolayers of an equimolar mixture of d-PIN+ and h-PIN+ at the surface of pure H2O recorded at 20 mN/m (A/n ) 220 Å2) and at 4 mN/m (A/n ) 310 Å2).

Figure 7. Experiment 3A,B. Neutron reflectivity (a) and corresponding scattering length density profiles (b) of monolayers of an equimolar mixture of d-PIN+ and h-PIN+ at the surface of an H2O/D2O mixture recorded at 20 mN/m (A/n ) 220 Å2) and at 4 mN/m (A/n ) 310 Å2).

The pronounced fringes similar to those of the monodisperse brush indicate that the neutron reflectivity is dominated by interference of neutrons reflected from the top and bottom interfaces. The shifted total reflection edge and the reduced intensity of the fringes indicate a reduced overall scattering length density, whereas the increased spacing is indicative of a reduced film thickness. Because the intensity of the fringes varies with Q, one can expect additional contributions to the reflectivity due to gradients in the scattering length density within the monolayer. The reflectivity was simulated based on a two-box model, as depicted in Figure 3. This model is described by the four scattering length densities (subphase, bottom and top region of the brush, and air), three interfacial roughnesses (bottom, inner, and top interface) and two lengths (height of bottom region, h, and total brush height, H). Of these parameters, the two lengths, the scattering length density of the bottom region and the width of the inner roughness, were varied

to yield the best possible accordance between simulated reflectivity and experimental data. To reduce the number of fitting parameters, the scattering length density of the top region was fixed to the value of pure PIN+, and the top and bottom interfacial roughness was fixed to a value of 5 and 3.5 Å, respectively. These values are close to the values previously established by X-ray reflectivity.39 The resulting parameters are summarized in Table 2, and the corresponding scattering length density profiles are depicted in Figure 6b. In addition, the reflectivity was simulated using the profiles predicted by ASCF theory using the same the monolayer thickness as in the two-box model (for clarity, only the profile based on the MWC/BLZ scenario is included in Figure 6). This simulation predicts more pronounced deviations from a uniform height of the fringes than was experimentally observed. Compared to simulations based on ASCF theory, the simulation using the two-box model yields a better agreement between simulation and experimental data.



Table 2. Parameters Obtained in the Simulation of the Neutron Reflectivities Using a Two-box Model experiment number substance subphase lateral pressure [10-3 N/m] σtop [Å] σi [Å] σbottom [Å] (b/v)sub [10-6 Å-2] (b/v)top [10-6 Å-2] (b/v)bottom[10-6 Å-2] (b/v)average [10-6 Å-2] theoreticala (b/v)average [10-6 Å-2] height of mixed region, h [Å] total height of monolayer H [Å] h/H, experimental h/H, theoretical uniform ext. brush (eq 6) MWC theory21 (eq 7) (RGβ/h)4/3 a

1A

1B

d-PIN+ H2O 20 ( 1 5

d-PIN+ H2O 4(1 5

3.5 -0.56 6.92 ( 0.25 6.92 ( 0.25 6.92 ( 0.25 6.94

3.5 -0.56 6.96 ( 0.20 6.96 ( 0.20 6.96 ( 0.20 6.94

469 ( 7

342 ( 3

2A

2B

3A

3B

d-PIN+/h-PIN+ H2O 20 ( 1 5 49 ( 11 3.5 -0.56 6.94 3.29 ( 0.40 4.79 ( 0.25 4.7 240 ( 15 408 ( 6 0.59 ( 0.05

d-PIN+/h-PIN+ H 2O 4(1 5 46 ( 10 3.5 -0.56 6.94 3.21 ( 0.50 4.65 ( 0.20 4.7 186 ( 10 290 ( 3 0.64 ( 0.05

d-PIN+/h-PIN+ H2O/D2O 20 ( 1 5 49 ( 20 3.5 4.36 6.94 3.58 ( 0.70 5.08 ( 0.35 4.7 233 ( 50 420 ( 10 0.55 ( 0.13

d-PIN+/h-PIN+ H2O/D2O 4(1 5 55 ( 15 3.5 4.36 6.94 3.51 ( 0.50 4.67 ( 0.203 4.7 198 ( 30 299 ( 6 0.66 ( 0.11

0.67 0.57 0.48 ( 0.04

0.67 0.57 0.68 ( 0.05

0.67 0.57 0.50 ( 0.14

0.67 0.57 0.62 ( 0.13

On the basis of a degree of deuteration of PIN+ of 99%.

To verify the scattering length density profiles obtained in experiment 2, the measurements were conducted again in experiment 3 on a mixed H2O/D2O subphase with a scattering length density close to the bottom region of the monolayer. One observes a dramatic loss of reflected intensity from the mixed films at the air/water interface (experiment 3; see Figure 7). This is to be expected, as the subphase was chosen such as to be close to the scattering length density of the bottom region. In addition, the fringes are “inverted”. Maxima of the fringes occur at Q values of minima in experiment 2 and vice versa. This indicates a 180° phase jump of one of the contributions to the interference pattern. Thus, one can conclude that in experiment 2 the scattering length density in the bottom region is well above the scattering length density of pure H2O whereas in experiment 3 it is close to, but below, the scattering length density of 4.36 × 10-6 Å-2 of the mixed H2O/ D2O subphase. As in the previous case, a simulation based on the two-box model agrees quite well with the experimental data. As in experiment 2, a simulation of the reflectivity data based on the profile derived from ASCF theory exaggerates the variations of the amplitude of the Kiessig fringes. Within the accuracy of the experiment, the profiles obtained in experiments 2 and 3 are identical. The resulting parameters are summarized in Table 2, and the corresponding scattering length density profiles are depicted in Figure 7b. Although the two-box model gives a quite reasonable agreement between simulation and experiment, the actual profile might have a shape that cannot be modeled with a simple two-box model. To test this hypothesis, the reflectivity and corresponding scattering length density profiles were simulated using the modelindependent method of simulated annealing.36,37 The scattering length density profiles thus obtained give a satisfying agreement with the experimental reflectivity data. As an example, in Figure 8 the two-box model of the mixed monolayer on pure H2O at 4 mN/m is compared to a simulation obtained with simulated annealing. Although the scattering length density profile obtained with the latter method has a noisy appearance, the agreement with the experimental data is as good as that of the two-box model. Within the accuracy of both methods, one obtains identical scattering length density profiles with both methods.

Figure 8. Comparison between the simulated reflectivity and scattering length density profile obtained using a two-box model and by the method of simulated annealing, respectively. Within the accuracy of the methods, both scattering length density profiles are identical.

Discussion The reflectivity profiles of the monodisperse polymer brushes can be satisfyingly simulated by a one-box model (see Figure 5). The monolayer thickness increases with decreasing area per molecule. If one assumes that the polymer behaves like an incompressible polymer melt, one can expect the thickness to be equal to the

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Figure 9. Film thickness derived from neutron reflectivity, in accordance with the theoretical film thickness calculated from surface concentration and density of the polymer. O, 3, and 4: data taken from Baltes, Schwendler, Helm, Heger, and Goedel.39 b: expt 1. (: expts 2 and 3. s: theoretical prediction.

surface concentration divided by the density of the polymer.40 In Figure 9, the monolayer thickness obtained from neutron reflectivity is compared with the theoretically expected thickness and with literature values obtained from other polyisoprene monolayers by X-ray reflectivity. One observes a good agreement between the experimental and the expected thickness. This holds true for the pure (solid circles) as well as the mixed monolayers (solid diamonds). Thus, one can regard the assumption of an incompressible polymer melt as fulfilled. In the simulations, the roughness of the top and bottom interfaces was chosen to be close to earlier data obtained via X-ray reflectivity of polyisoprene monolayers.39 Further variation of the roughness did not improve the quality of the simulation. When evaluating the structure of the monolayer at the bottom interface, one may not ignore the fact that the nondeuterated headgroup has a lower scattering length density than the deuterated repeat units. Thus, one expects an additional bottom layer of approximately 10 Å thickness and a volume fraction of 75-80% of deuterated hydrocarbons.41 One might take this layer explicitly into account or include it in an increased roughness of the bottom interface. We note however that the quality of the simulation is not very sensitive to variations along this line of thought, and thus the data do not allow complete resolution of this detail. The reflectivity profiles of the bidisperse brushes on H2O cannot be described by a one-box profile. A satisfying description was obtained using a two-box profile. Simulations based on this two-box profile show better agreement with the experimental data than do the predictions based on analytical self-consistent field theory (see Figures 6 and 7; only the MWC/BLZ scenario is shown; for clarity, the even more deviating Spontak scenario is not included in the diagram). One can expect that the inner structure of the bidisperse brush is independent of the nature of the subphase. Thus, one expects the experiments 2 and 3 to yield identical results. The simulated profiles of experiment 3 are indeed similar to those of experiment 2. The observed differences between experiment 2 and 3 are comparable to the error of the fitted parameters and thus reflect the general inaccuracy in interpreting the reflectivity data. The reflectivity data can be simulated assuming a laterally homogeneous monolayer. This is in accordance with a lack of lateral structure as observed via Brewster angle microscopy and in agreement with the miscibility


of symmetric styrene-isoprene block copolymers of comparable chain length disparities.17 Experiments 2 and 3 confirm a stratified structure of the bidisperse brush. The empirical profiles are between the uniformly extended profile and the profile derived from ASCF theory. The scattering length density of the bottom region is above the value of the ASCFbased model. The bottom region is thicker than that predicted by the ASCF-based model but thinner than that predicted by the uniformly extended model. Although the uniformly extended model does not predict a diffuse inner interface between the bottom and top regions, the two-box profile yields a comparatively wide transition zone between the two regions. From these features, one can conclude that the parts of the longer chains within the mixed region are stretched a bit more than the shorter chains, but not to such an extend as predicted by ASCF theory. It is instructive to compare the experiments conducted at different areas per chain. Both ASCF theory and the uniformly extended chain model predict affine deformation: the h/H ratio and the composition of the bottom region should be independent of the brush thickness. ASCF theory predicts a relatively smooth transition of the scattering length density. The thickness of this transition zone should be proportional to the total brush thickness, too. As can be seen from Figures 6 and 7 and Table 2, within the accuracy of the experiment, the scattering length density of the top and bottom regions are invariant with respect to changes in the brush thickness. On the other hand, in violation of the predicted affine deformation, the thickness of the transition zone between the top and bottom regions, σi, seems to be unaffected by the change in brush thickness. In addition, the h/H ratio seems to decrease slightly with increasing film thickness. Thus, one can conclude that the composition of the bottom region and the width of the transition zone between both regions are influenced by forces not included in the ASCF model. In the case of phase-separated block copolymers, additional blurring of the compositional profile due to the non-negligible width of the interface of tethering has been proposed.26 In our case, however, the plane of tethering is comparatively thin, and thus its width can be neglected. Therefore, it is very likely that these forces are thermal fluctuations. Birshtein, Liatskaya, and Zhulina22 give a criterion for the minimum brush thickness needed to render thermal fluctuations negligible.22 According to this criterion, the Gaussian dimension of the shorter chain RGβ has to be smaller than the height of the bottom region:

( ) RGβ h

4/3

,1

The corresponding values for each experiment are given in Table 2.42 The values are smaller than one; however, the difference is less than 1 order of magnitude. Thus, one can conclude that fluctuations may still be important in the monolayers investigated here. One might be interested to know whether this behavior is unique to the monolayer brush on a water surface or whether analogous behavior has been observed in block copolymers. Mayes, Russel, Deline, Satija, and Majkrzak27 investigated mixtures of phase-separated polystyrene/poly(methyl methacrylate) block copolymers. Two different combinations of symmetric block


copolymers of different total molar mass were investigated: 110-30 and 300-100 kg/mol. In the case of the smaller molecular weight mixtures, the width of the mixed region was comparable to the width of the interfacial roughness between the incompatible domains. Thus, these experiments cannot be compared to the monolayers investigated here. In the case of the longer chains, two mixing ratio with a mole fraction of longer chains xR = 0.05 and xR = 0.1 were investigated. In these cases, the concentrations of segments of shorter chains were indeed higher close to the tethering interface than they were in the inner region of the domain. However, the regions close to the tethering interface had a much higher content in long chains than that which can be expected from ASCF theory. No regions within the brush were composed purely of long chains. If one assumes validity of ASCF theories and estimates a value for h by applying eq 7, one obtains for these systems values of (RGβ/h)4/3 between 0.5 and 0.8. The same holds if the above criterion is applied to the compositions and domain sizes reported in other papers.15,17 Thus, the results reported here are in agreement with previous results on phase-separated block copolymers. If one compares experiments 2A and 3A to experiments 2B and 3B, one gets the impression that the thicker brush deviates less from the ASCF scenario than does the thinner one. This is in accordance with the fact that the values of (RGβ/h)4/3 decrease with increasing film thickness. However, given the limited accuracy of the simulation, one should regard this interpretation with caution. The results obtained on dense polymer brushes are in accordance with similar observations obtained by Levicky, Koneripalli, Tirell, and Satija26 in the case of swollen brushes. In these experiments, a bidisperse brush of polystyrene of chain length comparable to the one investigated here, but with 3-5 times more area per chain, was swollen with good and θ solvents. In both cases, the authors observed a less-pronounced segregation and a smaller degree of stretching of the longer chains than that predicted by ASCF theory. The deviations from ASCF theory were less severe in the case of good solvent conditions than in θ-conditions. This was explained by the fact that the total brush height and thus the overall stretching of the chains is significantly larger in the case of a good solvent than in the case of a θ solvent.


most likely is dominated by thermal motion. Deviations from affine deformation (e.g., changes in the relative height of the mixed region upon changing the brush height) might occur. However, the accuracy of the present experiment is not sufficient enough to definitively settle this question. Acknowledgment. We thank M. Antonietti and H. Mo¨hwald for fruitful discussions and support. We thank Piotr Marczuk for making available his data analysis program. This work was financed in part by the Deutsche Forschungsgemeinschaft. Appendix Calculation of Volume Fraction Profiles from Milner, Witten, and Cates21 Theory. In this theory, the conformations of polymer chains are similar to the flight path of a particle starting at rest at the location of the free end and being accelerated in a field toward the “bottom” of the brush. This field can be described by a kinematic potential U(z), which is a function of the position along the coordinate normal to the plane of the brush, z. The number of chains per unit surface g(U)∂U that have their free ends at potentials between U and U + ∂U is given by (eq 6 in Milner, Witten, and Cates21):

g(U)∂U )

x2 1 π U x max - U

(12)

with the kinematic potential Umax at the upper border of the brush given by (eq 7 in Milner, Witten, and Cates21):

Umax )

π2σ2 8

(13)

and the kinematic potential at h is given by (eq 9 in Milner, Witten, and Cates21):

U(h) ) Umax[1 - (xR)2]

(14)

at locations below h (U < U(h)) the kinematic potential is proportional to z2 (derived from eq 19 in Milner, Witten, and Cates21):

z)

x8 N xU(z) for U e U(h) S z e h π β

(15)

Conclusion Long and short chains in a bidisperse brush are not homogeneously distributed within the brush. Close to the plane of tethering, segments of short and long chains form a mixed region. With increasing distance from the plane of tethering, the local volume fraction of segments of longer chains increases, until a top region composed purely of the longer chains is reached. In the systems investigated here, the mixed region has a volume fraction of longer chains larger than that predicted by analytical self-consistent field theory. Thus, the overall stretching of the parts of the longer chains within that regions seems to be smaller than that theoretically predicted. The description according to Spontak deviates even more from the experimental data than the description according to Milner, Witten, and Cates21. The width of the transition zone between the mixed region and the top region is independent of the height of the brush and

thus, one can obtain h by inserting eqs 13 and 14 into eq 15:

h)

x8 N xU(z ) h) ) σNβx1 - xR2 π β

(16)

From the contribution to the volume fraction at position z, of a chain with free end at z′, φ(z,z′) and the number of chains g(z′)∂z′ that have the free ends in the interval z′ to z′+∂z′, one can obtain the local volume fraction of segments of the shorter chains by integrating over all positions of the free ends between z and h.

φβ(z) )

h g(z′)φ(z, z′)∂z′ ∫z′)z

(17)

φ(z,z′) is given by the inverse stretching of the chain at z (eq 1 in Milner, Witten, and Cates21):

7608

Goedel et al.


φ(z, z′) )

1 1 x2 xU(z′) - U(z)

(18)

2Nβ π

1

x(z′)

2

2 arccos π

φR(z) )

by inserting eq 15 one obtains

φ(z, z′) )

tioned in the Introduction

for z, z′ < h

- (z)

one can calculate the fraction g(z)∂z of the shorter chains with free ends between z and z + ∂z according to

∂U ∂z

g(z) ) g(U) differentiation of eq 15 yields

∂U π2 z ) for z < h ∂z 4 N2

λ2 - (z/H)2

for z < h


(19)

2

x

(1 - xR2)λ2 - (z/H)2

(8)

with λ ) /(xR + (1 - xR)). Calculation of Volume Fraction Profiles from Spontak23 Theory. In this theory, it is assumed that in the mixed region all R-chains have identical conformations and contribute the same number of repeat units Γ*NR to the mixed region. The local stretching of an R-chain, ER, within the mixed region is given by (eq 7b and eq 26 in Spontak23)

ER(z) )

(20)

π xΛ2 - z2 for z e h 2NR

(26)

β

With

and thus one obtains

g(z) )

1 Nβ

z

x(σNβ)2 - z2

for z < h

(21)

(π2 Γ* )

Λ ) h csc

and

Inserting eqs 19 and 21 into eq 17, one obtains

φβ(z) )

∫z′)z x(σNβ)2 - (z′)2 x(z′)2 - (z)2

2 π

z′

h

1

∂z′

for z < h (22)

Integration yields

2 φβ(z) ) arcsin π

xh2 - z2

x(σNβ)2 - z2

2 arcsin π

x

)

(σNβ)2(1 - xR2) - z2

for z < h (23)

(σNβ)2 - z2

In the limit of a uniform brush (xR f 0), eq 23 reduces as expected to φ1 ) 1. The volume fraction profile is independent of the chain length of the longer chains, NR. If one applies eq 2 to eq 23, one obtains

2 φR(z) ) arccos π

x

(σNβ)2(1 - xR2) - z2 (σNβ)2 - z2

this expression is equivalent to eq 25, which can be derived for a chain with repeat units of unity length and volume from the eqs A1.18, 22, and 25 of Birshtein, Liatskaya and Zhulina22:

φR(z) )

1 1 + arcsin 2 π

(

1 - 2(1 - xR2) +

( )

z 1σNβ

( )

2

To calculate the height of the mixed region, h, one has to replace the term Nβ in eq 6 by Γ*NRxR + Nβ(1 - xR) and obtains the expression already mentioned in the Introduction:

h Γ*NRxR + Nβ(1 - xR) Γ*xR + (1 - xR) ) ) H NRxR + Nβ(1 - xR) xR + (1 - xR)

z σNβ

)

φR(z) )

It is convenient to insert eqs 1 and 16 into eq 24 and express the volume fraction in terms of reduced coordinates z/H ) position in the film divided by total film thickness. One obtains the expression already men-

xRσ ER(z)

)

2 xRσNR for z e h πx 2 Λ - z2

(27)

from eq 1 one obtains an expression for σNR:

σNR )

H xR + (1 - xR)

(28)

and thus, one obtains the expression already mentioned in the Introduction:

(

)

2 z2 Λ2 φR ) xRλ 2 - 2 π H H

1/2

for z < h


2

for z < h (25)

(10)

At locations above h, the brush is composed purely of R-segments. At locations below h, the local concentration of R-repeat units φR at an arbitrary position z in the mixed region (0 e z e h) is given by the inverse stretching, the mole fraction xR of the R-chains, and the total number of chains per unit surface σ by

for z < h (24)

Γ* ) (1 - xR)(1 + 0.3056xR - 0.2968xR2)

(11)

with λ ) /(xR + (1 - xR)) given by eq 6. References and Notes (1) (2) (3) (4) (5) (6)

Helfand, E. Macromolecules 1975, 8, 552. Semenov, A. N. Sov. Phys. JETP 1985, 61, 733. Semenov, A. N. Macromolecules 1993, 26, 6617. Ohta, T.; Kawasaki, K. Macromolecules 1986, 19, 2621. Milner, S. T. J. Polym. Sci., Part B 1994, 32, 2743. Bates, F. S.; Fredrickson, G. H. Annu. Rev. Phys. Chem 1990, 41, 525.

Macromolecules, Vol. 32, No. 22, 1999 (7) Russel, T. P. Physica B 1996, 221, 267. (8) Kim, M. W.; Chung, T. C. J. Colloid Interface Sci. 1988, 124, 365. (9) Christie, P.; Petty, M. C.; Roberts, G. G.; Richards, D. H.; Service, D.; Stewart, M. J. Thin Solid Films 1985, 134, 75. (10) Goedel, W. A.; Xu, C.; Frank, C. W. Langmuir 1993, 9, 1184. (11) Lenk, T. J.; Lee, D. H. T.; Koberstein, J. T. Langmuir 1994, 10, 1857. (12) Mirley, C. L.; Koberstein, J. T. Langmuir 1995, 11, 1049. (13) Heger, R.; Goedel, W. A. Macromolecules 1996, 29, 8912. (14) Goedel, W.; Heger, R. Langmuir 1998, 14, 3470-3474. (15) Hadziioannou, G.; Skoulios, A. Macromolecules 1982, 15, 267. (16) Kane, L.; Satkowski, M. M.; Smith, S. D.; Spontak, R. J. Macromolecules 1996, 29, 8862. (17) Hashimoto, T.; Yamasaki, K.; Koizumi, S.; Hasegawa, H. Macromolecules 1993, 26, 2895. (18) Hashimoto, T. Macromolecules 1982, 15, 1548. (19) Spontak, R. J.; Fung, J. C.; Braunfeld, M. B.; Sedat, J. W.; Agard, D. A.; Kane, L.; Smith, S. D.; Satkowski, M. M.; Ashraf, A.; Hajduk, D. A.; Gruner, S. M. Macromolecules 1996, 29, 4496. (20) Sakurai, S.; Irie, H.; Umeda, H.; Nomura, S.; Lee, H. H.; Kim, J. K. Macromolecules 1998, 31, 336. (21) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1989, 22, 853-861. (22) Birshtein, T. M.; Liatskaya, Y. V.; Zhulina, E. B. Polymer 1990, 31, 2185. (23) Spontak, R. J. Macromolecules 1994, 27, 6363. (24) Borovinskii, A. L.; Khoklov, A. R. Macromolecules 1998, 31, 1180. (25) Kent, M. S.; Factor, B. J.; Satija, S. K.; Gallagher, P. D.; Smith, G. S. Macromolecules 1996, 29, 2843. (26) Levicky, R.; Koneripalli, N.; Tirell, M.; Satija, S. K. Macromolecules 1998, 31, 2616. (27) Mayes, A. M.; Russel, T. P.; Deline, V. R.; Satija, S. K.; Majkrzak, C. F. Macromolecules 1994, 27, 7447. (28) Hshieh, H.; Quirk, R. P. Anionic Polymerization; Marcel Dekker: New York, 1996; p 283-285.

Stratification in Monolayers 7609 (29) Quirk, R. P.; Zhu, L.-F. Br. Polym. J. 1990, 23, 47. (30) Quirk, R. P. Makromol. Chem., Macromol. Symp. 1992, 63, 259. (31) Mezei, F.; Goloub, R.; Klose, F.; Toews, H. Physica B 1995, 213/214, 898. (32) Sears, V. F. Neutron News 1992, 3, 26. (33) Program written by Braun, C. HMI Berlin, 1997. (34) Parratt, L. G. Phys. Rev. 1954, 95, 359. (35) Ne´vot, L.; Croce P. Rev. Phys. Appl. 1980, 15, 761. (36) Kunz, K.; Reiter, J.; Goetzelmann, A.; Stamm, M. Macromolecules 1993, 26, 4316. (37) Kirkpatric, S.; Gelatt, C. O.; Vecchi, M. P. Science 1983, 220, 671. (38) Program written by Piotr Marczuk, Technische Universita¨t Berlin, 1998. (39) Baltes, H.; Schwendler, M.; Helm, C. A.; Heger, R.; Goedel, W. A. Macromolecules 1997, 30, 6633. (40) In this calculation, it does not matter whether the surface concentration is expressed as chains, repeat units, or mass per area as long as the density is expressed in the equivalent way, e.g chains, repeat units, or mass per volume, respectively. (41) This estimate is based on the height (10 Å) and cross-sectional area (60 Å2) of the headgroup derived from CPK models and the area per headgroup (250-350 Å2) during the experiment. (42) RGβ was calculated according to RGβ ) (number of repeat units)0.5 × 6.23 Å; see Heger and Goedel.13 Note: in this calculation, the experimental value of h out of the two-box profile was used, not the value predicted by ASCF theory. Theoretically predicted values for h are smaller than the values used here; however, the difference is not large enough to change the estimate significantly.

MA981900R

Stratification in Monolayers of a Bidisperse Melt Polymer Brush As

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