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Langmuir 2005, 21, 1840-1847
Nanostructure of a “Carpet”-like Dense Layer/ Polyelectrolyte Brush Layer in a Block Copolymer Monolayer at the Air-Water Interface Emiko Mouri,† Kozo Matsumoto, and Hideki Matsuoka* Department of Polymer Chemistry, Kyoto University, Kyoto 615-8510, Japan
Naoya Torikai Neutron Science Laboratory, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan Received February 19, 2004. In Final Form: November 9, 2004 The “carpet”/brush double layer structure in the polyelectrolyte layer in the amphiphilic diblock copolymer monolayer at the air-water interface was quantitatively studied by in situ neutron reflectometry in addition to X-ray reflectivity measurements. As a result of the higher contrast between polyelectrolyte [poly(methacrylic acid)] and solvent (D2O) for the neutron, the brush structure could be estimated more accurately as a function of surface pressure, that is, brush density. The thickness of the carpet layer, which is thought to be formed to reduce the interfacial free energy between water and the hydrophobic layer, was almost constant at 10-20 Å at any surface pressure studied. Growth was clearly observed in the whole brush length with increasing surface pressure, and it was estimated to be almost 60% of the full-stretch length of the ionic polymer chain. Furthermore, by the comparison of density profiles by neutron and X-ray reflectometry, an anomalous hydration was suggested.
* To whom correspondence should be addressed. † Present address: Department of Applied Chemistry, Kyushu Institute of Technology, Kitakyushu, Fukuoka 804-8550, Japan.
the advantages of utilizing a monolayer system to study the polymer brush10-23 is that a well-characterized polymer, especially in chain length and its distribution, can be used, and, thus, detailed discussion on polymer structure near the “anchoring” point, that is, the airwater interface in this system, is possible. The unique nanostructure in a polyelectrolyte brush, “carpet”-like dense PMAA layer/diffuse brush PMAA layer formation has been reported.12 The double layer formation is rarely observed in polymer brush systems although it is common for adsorption of free polymers.24,25 However, in addition to our previous studies, some polymer brush systems have been reported to show a double layer formation: poly(styrenesulfonic acid) brush in a poly(ethyl ethylene)-b-poly(styrenesulfonic acid) monolayer by Ahrens et al.18 and poly(N-isopropylacrylamide) (PNIPAM) brush by Yim et al.26 We have also reported recently that the “carpet”-like dense layer was found in the poly(hydrogenated isoprene)-b-poly(styrenesulfonic acid) monolayer system. The thickness was independent of the surface
(1) Milner, S. T. Science 1991, 251, 905. (2) Israe¨ls, R.; Leermakers, F. A. M.; Fleer, G. J. Macromolecules 1994, 27, 3087. (3) Zhulina, E. B.; Birshtein, T. M.; Borisov, O. B. Macromolecules 1995, 28, 1491. (4) Ejaz, M.; Yamamoto, S.; Ohno, K.; Tsujii, Y.; Fukuda, T. Macromolecules 1998, 31, 5934. (5) Tran, Y.; Auroy, P.; Lee, L. T. Macromolecules 1999, 32, 8952. (6) Currie, E. P. K.; Sieval, A. B.; Fleer, G. J.; Cohen Stuart, M. A. Langmuir 2000, 16, 8324. (7) Biesalski, M.; Ru¨he, J. Macromolecules 2002, 35, 499. (8) von Werne, T.; Patten, T. E. J. Am. Chem. Soc. 1999, 121, 7409. (9) Zheng, G.; Sto¨ver, H. D. H. Macromolecules 2002, 35, 6828. (10) Mouri, E.; Wahnes, C.; Matsumoto, K.; Matsuoka, H.; Yamaoka, H. Langmuir 2002, 18, 3865. (11) Mouri, E.; Matsumoto, K.; Matsuoka, H. J. Appl. Crystallogr. 2003, 36, 722. (12) Mouri, E.; Matsumoto, K.; Matsuoka, H. J. Polym. Sci., Part B 2003, 41, 1921. (13) Mouri, E.; Matsuoka, H. Encyclopedia of Nanoscience and Nanotechnology; Marcel Dekker: New York, 2004; pp 2519-2529. (14) Mouri, E.; Furuya, Y.; Matsumoto, K.; Matsuoka, H. Langmuir 2004, 20, 8062. (15) Mouri, E.; Kaewsaiha, P.; Matsumoto, K.; Matsuoka, H.; Torikai, N. Langmuir 2004, 20, 10604.
(16) Bijsterbosch, H. D.; de Hann, V. O.; de Graaf, A. W.; Mellema, M.; Leermakers, F. A. M.; Cohen Stuart, M. A.; van Well, A. A. Langmuir 1995, 11, 4467. (17) Su, T. J.; Styrkas, D. A.; Thomas, R. K.; Baines, F. L.; Billingham, N. C.; Armes, S. P. Macromolecules 1996, 29, 6892. (18) Ahrens, H.; Fo¨rster, S.; Helm, C. A. Macromolecules 1997, 30, 8447. (19) Ahrens, H.; Fo¨rster, S.; Helm, C. A. Phys. Rev. Lett. 1998, 81, 4172. (20) Peace, S. K.; Richards, R. W.; Taylor, M. R.; Webster, J. R. P.; Williams, N. Macromolecules 1998, 31, 1261. (21) Dewhurst, P. F.; Lovell, M. R.; Jones, J. L.; Richards, R. W.; Webster, J. R. P. Macromolecules 1998, 31, 7851. (22) Faure, M. C.; Bassereau, P.; Lee, L. T.; Menelle, A.; Lheveder, C. Macromolecules 1999, 32, 8538. (23) Wesemann, A.; Ahrens, H.; Steitz, R.; Fo¨rster, S.; Helm, C. A. Langmuir 2003, 19, 709. (24) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymer at Interface; Chapman & Hall: New York, 1993. (25) Lee, L. T.; Guiselin, O.; Farnoux, B.; Lapp, A. Macromolecules 1991, 24, 2518. (26) Yim, H.; Kent, M. S.; Huber, D. L.; Satija, S.; Majewski, J.; Smith, G. S. Macromolecules 2003, 36, 5244.
Introduction The polymer brush, polymer chains with their one end fixed to the surface at a high polymer “graft” density, has unique properties.1-3 It is quite different not only from polymers in bulk but also from those just adsorbed at the surface. The polymer brush grafted on solid surfaces4-7 and on colloidal particles8,9 has been studied intensively. However, the effects of controlling factors (chain length, surface density, pH, ionic strength) on the polymer brush structure have not been fully elucidated. Systematical X-ray reflectometry (XR) studies have been carried out on (1,1-diethylsilacyclobutane)m-block-(methacrylic acid)n [(Et2SB)m-b-(MAA)n] spread monolayer.10-15 Its hydrophilic chains, that is, poly(methacrylic acid) (PMAA) chains, in the spread monolayer form a polyelectrolyte brush from the air-water interface. One of
10.1021/la040028e CCC: $30.25 © 2005 American Chemical Society Published on Web 01/26/2005
Dense Layer/Polyelectrolyte Brush Layer
pressure, chain length, and salt concentration.27 On the other hand, the poly(ethylene oxide) brush in aqueous solution, which is the most frequently studied neutral water-soluble polymer, has not been confirmed to form the double layer structure.16,20-23 The double layer formation has not been confirmed in solvents other than water. Kent et al. studied the poly(styrene) brush in a poly(dimethylsiloxane)-b-poly(styrene) (MPS . MPDMS) monolayer on an organic solvent.28 By changing the temperature and solvent species, they investigated the effect of solvent quality and interaction between PS and PDMS, but they did not observe double layer formation of the polymer brush. The overview here suggests that the ionic character of the polymer chain is thought to be necessary for the double layer formation although more investigation is needed to conclude its origin. Hydration is also supposed to be a key factor because both the polyelectrolyte and PNIPAM26 induce strong hydration. Furthermore, an important and interesting finding of our study was that the “carpet” layer 10-20-Å thick was always formed independent of surface pressure, chain length, pH, and salt concentration.11-15 Ahrens et al. also reported the constancy of the layer thickness with surface pressure in the poly(styrene sulfonic acid) brush.18 An additional significant point clarified by our study was that the “carpet”-like layer and the diffuse brush layer had different contributions to the surface pressure of the monolayer.12,14 Thus, the nanostructure of the polymer brush is not simple but more complicated than those widely imagined and predicted by theories. On the basis of these findings, in this study we have performed more detailed investigations of the (Et2SB)m-b-(MAA)n monolayer on water by the neutron reflectometry (NR) technique. A partially deuterated diblock copolymer was synthesized for NR measurements. The diblock copolymer with deuterated hydrophobic segment forms a monolayer on D2O with a much higher contrast in scattering length density. This makes it possible to investigate the PMAA brush structure more quantitatively. In fact, much clear fringes derived from the PMAA brush layer were observed. The double layer formation was certainly confirmed by complemental use of XR and NR measurements, and the nanostructure change was discussed as a function of the surface pressure. In addition, the existence of hydration water around the PMAA chain in the brush was suggested. Experimental Section Materials. A deuterated diblock copolymer and D2O were used to obtain higher contrast in NR. The deuterated diblock copolymer, (1,1-diethylsilabutane-d10)23-block-(methacrylic acid)49 [(Et2SB-d10)23-b-(MAA)49], was synthesized basically by the method described in ref 29. A deuterated Et2SB-d10 monomer that has two deuterated ethyl groups was used instead of the Et2SB monomer to construct a system with higher contrast by NR. Deuterated Et2SB-d10 monomer was synthesized by the Grignard reaction of deuterated ethyl bromide (EtBr-d5) and 1,1-dichlorosilacyclobutane. Using this monomer as the first monomer, diphenylethylene (DPE) as the capping agent, and tert-butyl methacrylate (t-BuMA) as the second monomer, living anionic polymerization was carried out. By hydrolysis of t-BuMA, the target polymer was obtained. In this process, the Et2SB homopolymer and DPE were carefully removed by gel permeation (27) Kaewsaiha, P.; Matsumoto, K.; Matsuoka, H. Langmuir 2004, 20, 6754. (28) (a) Kent, M. S.; Lee, L. T.; Farnoux, B.; Rondelez, F. Macromolecules 1992, 25, 6240. (b) Kent, M. S.; Lee, L. T.; Factor, B. J.; Rondelez, F.; Smith, G. J. Chem. Phys. 1995, 103, 2320. (c) Kent, M. S.; Majewski, J.; Smith, G. S.; Lee, L. T.; Satija, S. J. Chem. Phys. 1998, 108, 5635. (d) Kent, M. S.; Majewski, J.; Smith, G.; Lee, L. T.; Satija, S. J. Chem. Phys. 1999, 110, 3553. (29) Matsumoto, K.; Wahnes, C.; Mouri, E.; Matsuoka, H.; Yamaoka, H. J. Polym. Sci., Part A 2001, 39, 86.
Langmuir, Vol. 21, No. 5, 2005 1841 Table 1. Characteristics of (Et2SB-d10)23-b-(MAA)49 m:na
Mnb
Mw/Mnc
23:49
7700
1.05
a
Determined by 1H NMR. b Calculated from the degree of polymerization. c Determined by GPC.
chromatography (GPC) because the obtained solution by polymerization was a mixture of the diblock copolymer and excess DPE. Characteristics of the polymer thus obtained are summarized in Table 1. Preparation of the Monolayer. The polymer monolayer was prepared by spreading a 1 mg/cm3 tetrahydrofuran (THF) solution of diblock copolymer on a water (H2O and D2O) surface in the Langmuir-Blodgett (LB) trough in the reflectometer. D2O (99.9%D) purchased from Cambridge Isotope Laboratories, Inc. (Andover, U.S.A.), was used as received. The conductivity of D2O was 1 µS/cm, and the pH was about 8. After evaporation of the THF solvent, the monolayer was compressed up to the surface pressure desired at 28 °C for XR and 22.5 °C for NR. NR measurements were performed at a lower temperature to prevent the evaporation of the subphase during the measurements because a longer accumulation time was needed for NR. We checked by XR that the temperature difference between 25 and 28 °C did not affect the monolayer structure. XR measurements were carried out for both monolayers on H2O and D2O to confirm the absence of an isotope effect on the monolayer structure. NR measurements were carried out for the monolayer on D2O. NR and XR Measurements. XR and NR measurements were carried out for monolayers at various surface pressures controlled with (0.1 mN/m accuracy. The area changed during the measurements to maintain a constant surface pressure of about 3%. The very small shift of the barrier position during the measurements indicates the stability of the monolayer. Moreover, it was confirmed by XR measurement that the monolayer structure at the same surface pressure was independent of the spread volume of the sample solution. NR measurements were performed with ARISA, a TOF (timeof-flight) neutron reflectometer for a free surface, at KEK in Japan.30 Reflectivity can be measured down to 10-6 using the neutrons with the wavelength range from 1 to 6 (8) Å. The LB trough (420 × 120 × 3 mm3) was set at the sample stage, and reflectivity was measured under a specular geometry with two fixed incident angles: 0.46 and 1.3° that covered q ranges of q ) 0.01-0.1 Å-1 and q ) 0.04-0.3 Å-1, respectively. The angular resolution, ∆θ/θ, was kept at 5% by adjusting the width of two incident slits. The typical accumulation time was 6 h at the low angle (0.46°) and 13 h at the high angle (1.3°). The background intensity evaluated by off-specular measurement for D2O was used as the common background for the three data sets presented here. The background intensity, which was measured with a detector at 10-mm off from the specular position for the high angle and 4-mm off for the low angle, was subtracted from each raw data set composed of 4000 TOF channels with 10-µs intervals. After the subtraction of the background, the NR data were rebinned into 5% in ∆t/t of the TOF to improve the data statistics and then normalized to the incident neutron spectrum to be converted into the reflectivity profile. The two reflectivity curves for a different q range measured at the two angles for one sample were connected to one continuous reflectivity curve. In this procedure, the reflectivity in the plateau region below the critical q for total reflection was set at 1. For XR apparatus and data analysis, refer to previous papers.10-12,31-33 Experimental errors in XR and NR measurements were estimated. An excellent reproducibility in XR measurements down to 10-7 in reflectivity, in the q range q < 0.4 Å-1 was confirmed. In NR, measurements for the same sample were repeatedly carried out, and it was confirmed that the profile was (30) Torikai, N.; Furusaka, M.; Matsuoka, H.; Matsushita, Y.; Shibayama, M.; Takahara, A.; Takeda, M.; Tasaki, S.; Yamaoka, H. Appl. Phys. A 2002, 74, S264. (31) Yamaoka, H.; Matsuoka, H.; Kago, E.; Eckelt, J. Physica B 1998, 248, 280. (32) Kago, K.; Matsuoka, H.; Endo, H.; Eckelt, J.; Yamaoka, H. Supramol. Sci. 1998, 5, 349. (33) Matsuoka, H.; Mouri, E.; Matsumoto, K. Rigaku J. 2001, 18, 54.
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Table 2. Scattering Length Density Nb and δ Used as Initial Parameters in a Box Model Fittinga PEt2SB-d10 PMAA-d1a PMAA D2O H2O
Nb [10-6 Å-2]
δ [10-6]
3.43 2.20 1.34 6.39 -0.56
2.99 4.19 4.19 3.57 3.57
a PMAA with deuterium by counterion exchange: COO-H+ + D+ f COO-D+ + H+. δ is definded by n ) 1 - δ - iβ, where δ is the refractive index. β is 2 or 3 orders smaller than δ. β is fixed in the fitting at the value calculated from n and d in bulk state.
well reproducible down to 10-5. The experimental errors in XR and NR were estimated by the following formula, in which the Poisson distribution of intensity is assumed. In Figures 3a and 4a, the error estimated by this formula was presented for the 42-mN/m data.
∆ref ) ref
x
1 1 + I0 I
Figure 1. π-A isotherm of the (Et2SB-d10)23-b-(MAA)49 monolayer at 28 °C on H2O. NR measurements were carried out on D2O at three points, which are indicated by closed symbols.
where ∆ref is the error in reflectivity, ref is the reflectivity, I0 is the direct intensity, and I is the reflected intensity. The data were fitted by the calculated reflectivity based on a box model. The model fitting procedure of the NR profile was the same as that of XR.12 First the profiles were fitted with a two-box model, which consisted of a hydrophobic layer and a hydrophilic layer. Additionally, the possibility of the two-box model consisting of a hydrophobic + hydrophilic layer and hydrophilic layer was also examined. Then a three-box model fitting was applied. Fitting parameters were the scattering length density Nb, layer thickness d, and interface roughness σ for each layer. Here, N is number density and b is the scattering length of each component. However, Nb for D2O and Et2SB-d10 were fixed at the bulk value presented in Table 2 in the three-box model fitting to reduce the number of fitting parameters. The model fitting was carried out in the range of smaller q with good statistics, and the fitting quality in this range was examined by the R(%) value:11
R(% ) )
∑(log x - log x ) ∑(log x )
x
i 0
i 2 c
i 2 c
× 100
where xi0 is for the ith data and xic is for the ith calculated data. In this study, NbD2O was fixed at 6.39 × 10-6 Å-2. A smaller value, that is, 6.35-6.37 × 10-6 Å-2, would be more realistic for NbD2O taking into account the exchange of D to H, but we do not have any reason to a choose specific value. The very small difference seen in the fitting curve was also checked in the q range ( H2O > PEt2SB-d10; Nb, D2O > PEt2SB-d10 > PMAA). These profiles mean in principle the same nanostructure of the monolayer. In the Nb profiles, double layer formation of PMAA is more clearly observed than in the δ profiles as a minimum. This means that PMAA is more condensed near the PEt2SB/PMAA interface because Nb of PMAA is much smaller than those for Et2SB-d10 and D2O. The structural parameters thus estimated are listed in Tables 3 and 4. However, as mentioned previously, the parameters of MAA1 in the table do not always agree with those shown in the density profiles in Figure 5. This is because the thickness of the MAA1 layer is comparable with the interface roughness, that is, σMAA1 and σMAA2. For other layers, the values in the table agree with those in the density profiles in Figure 5 because the layer thickness is much larger than its interface roughness. The carpet layer thickness is about 10-20 Å, although the carpet layer thickness at 35 and 42mN/m is estimated to be thinner than the NR result. However, as a whole, including the data on H2O shown in Figure 2b, the carpet layer thickness can be regarded as almost constant independent of the surface pressure. The 10-20 Å “carpet” layer thickness is also supported by the comprehensive data presented in refs 11-15. On the other hand, the brush layer thickness increases with increasing surface pressure. The ratio of the PMAA layer thickness, that is, lPMAA ) dPMAA1 + dPMAA2, to the fully extended PMAA chain [lfull (Å) ) 2.6 Å/unit × 49 units], lPMAA/lfull, increases from 0.45 to 0.61 according to the XR results and from 0.50 to 0.60 according to the NR results. The brush density is calculated from the PEt2SB layer thickness by XR data and the PEt2SB bulk density. Thus, these points correspond to the brush density of 0.64 chain/nm2 (25 mN/m), 0.81 chain/nm2 (35 mN/m), and 0.91 chain/nm2 (42 mN/m). The interface roughness is about 5 Å for the air/Et2SB, Et2SB/MAA1(carpet layer) interfaces and 10-20 Å for the MAA2(brush layer)/D2O interface. This indicates that the PMAA brush is diffused into D2O. For the MAA1/MAA2 interface, the interface roughness is estimated to be larger in NR. The difference might be explained by the existence of hydration water, which will be discussed later. We discussed the origin of this “carpet”-like dense layer the first time we reported this phenomenon.12 We supposed that the “carpet” layer was formed to prevent the contact between the hydrophobic layer and water, and a 10-20 Å separation is enough not to feel each component. However, other aspects might also be needed: The association behavior of polyelectrolytes at the air-water interface is strongly affected by the image charge.34,35 Recently, we found that the micelle was formed without Gibbs monolayer formation in the series of the ionic (34) Wittmer, J.; Joanny, J. F. Macromolecules 1993, 26, 2691. (35) Yim, H.; Kent, M. S.; Matheson, A.; Stevens, M. J.; Ivkov, R.; Satija, S.; Majewski, J.; Smith, G. S. Macromolecules 2002, 35, 9737.
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Table 4. Best-Fit Parameters of NR Profiles for the (Et2SB-d10)23-b-(MAA)49 Monolayera π [mN/m]
NbEt2SB
dEt2SB
σEt2SB
NbMAA1
dMAA1
σMAA1
NbMAA2
dMAA2
σMAA2
NbD2O
σD2O
R
25 35 42
3.43 3.43 3.43
30 50 62
0 5 5
2.28 3.27 2.60
12 14 9
5 5 5
4.93 4.64 4.61
54 60 68
13 5 8
6.39 6.39 6.39
15 19 12
5.4 5.5 7.7
a
Nbx [10-6 Å-2]: Scattering length density of the layer x. Parameters in italics are those fixed in the fitting procedure.
Figure 6. PMAA volume fraction calculated from Nb and δ profiles with an assumption that the PMAA density is 1.22 g/cm3.
amphiphilic diblock copolymer system.36-38 The concept of the image charge also should be taken into account in this system. Furthermore, Taniguchi et al. reported their simulation result on the polyelectrolyte brush structure.39 Their simulation showed results similar to those we obtained experimentally. They attributed the existence of the dense layer to the distribution of small ions. Small ions should be taken into account especially in a system with salt. To discuss the nanostructure of the monolayer more quantitatively, we converted the density profiles (Figure 5) into PMAA volume fraction profiles. In formulation of this process, we used the following assumptions. (1) F ) 1.22 g/cm3 for bulk PMAA. (2) The carboxylic acid group is in a COOD form, that is, δMAA ) 4.19 × 10-6 and NbMAA ) 2.2 × 10-6 Å-2. (3) The average δ and Nb values for carpet and brush layers can be calculated by
δ ) δMAAφ + δwater(1 - φ ) for XR Nb ) NbMAAφ + NbD2O(1 - φ) for NR where φ is the volume fraction of PMAA in each layer. Figure 6 shows the PMAA volume fraction profiles as a function of distance from the hydrophobic layer/ hydrophilic layer interface thus evaluated by XR and NR. The existence of definitely two discrete layers of PMAA was confirmed in PMAA fraction curves from the NR profile. The high-density layer of about 10-20 Å is formed just below the water surface, and the brush layer exists under the dense layer with a gradual decrease of its density with increasing Z. The carpet/brush double layer formation12 is now more conclusively demonstrated by NR. It is also confirmed by NR that the carpet layer thickness, (36) Matsuoka, H.; Matsutani, M.; Mouri, M.; Matsumoto, K. Macromolecules 2003, 36, 5321. (37) Matsuoka, H.; Maeda, S.; Kaewsaiha, P.; Matsumoto, K. Langmuir 2004, 20, 7412. (38) Matsumoto, K.; Ishizuka, T.; Matsuoka, H. Langmuir 2004, 20, 7270. (39) Taniguchi, T.; Qiang, W.; Fredrickson, G. H.; Sugimoto, M.; Koyama, K. Polym. Prepr. Jpn. 2003, 52, 2438.
which is about 10-20 Å, is independent of the surface pressure, although a smaller thickness was obtained by XR than by NR. The PMAA layer structural change with the surface pressure obtained by NR is also consistent with that obtained by XR: By increasing the surface pressure, the PMAA volume fraction in the brush layer increases while that in a “carpet” layer keeps constant. The increase of the brush thickness with the increase in the surface pressure is also clearly demonstrated in Figure 6. The absolute value of the PMAA volume fraction is different for two measurements in both the “carpet” layer and the brush regions. Especially in the brush region, the PMAA volume fraction φMAA obtained by NR was about 50% smaller than that obtained by XR. We rechecked the assumptions used in the calculation because the values of the polymer bulk density were calculated by a simulation program.10 However, we could not make the two values meet. The difference in the PMAA fraction by XR and that by NR is too big to be attributed to the assumptions in the values of density and to the experimental errors. Because XR and NR are responsible for the contrast in electron density and scattering length density, respectively, if the real bulk density of PMAA is higher and that of PEt2SB is lower than the assumed values, the volume fraction values by XR and NR will be in agreement. However, an extremely large value, such as FΜΑΑ ∼ 1.7, was needed for the two values to meet, which is not physically realistic. Hence, another factor should be considered to clarify these points. One possible factor is hydration. Hydration water (D2O) should contribute to the density contrast although it is difficult to know the density and amount of hydration water. First, the amount of PMAA per unit area was calculated at each surface pressure from the PEt2SB-d10 thickness for XR and NR with an assumption that the density of PEt2SB-d10 is 0.828 g/cm3. These values were compared with those used in the XR and NR fittings, that is, from φMAA. The amount of PMAA obtained by XR is 10-30% more than the calculated amount. The amount of PMAA obtained by NR is 50% less than the calculated amount. The XR result is more realistic. Thus, one assumption is introduced: some fraction (
