MMA mixtures at elevated pressures

Fluid Phase Equilibria 200 (2002) 147–160

Refractive index and swelling of thin PMMA films in CO2 /MMA mixtures at elevated pressures Ulrich Fehrenbacher a , Thomas Jakob b , Thorleif Berger b , Wolfgang Knoll b,∗ , Matthias Ballauff a,1 a

b

Polymer-Institut, Universität Karlsruhe, Kaiserstr. 12, 76128 Karlsruhe, Germany Max-Planck-Institut für Polymerforschung, Ackermannweg 10, D-55128 Mainz, Germany Received 2 July 2001; accepted 17 January 2002

Abstract Waveguide spectroscopy has been applied to optically study the influence of the concentration (0–1.5 M) of methylmethacrylate (MMA) in supercritical CO2 on the refractive index and swelling of PMMA as a function of pressure (up to 52 MPa at 333 K). The refractive index of pure CO2 is compared to literature in order to validate the technique in a supercritical fluid environment. The MMA content was varied, which resulted in an increased refractive index of the fluid in conjunction with a decreased refractive index of the PMMA film. This behavior is a direct result of a larger sorption and therefore swelling of the polymer layer and was measured at different pressures. With increasing pressure an increase of the refractive index of the solution and a decrease of the one of the PMMA layer was observed. The obtained refractive indices were modeled assuming an ideal mixture. The comparison with the experimental results showed that the qualitative behavior agrees well with the calculations. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Waveguide spectroscopy; Supercritical CO2 ; PMMA; MMA; Refractive index; Swelling

1. Introduction The interest in exploiting the use of supercritical fluids as an ecological alternative to conventional solvents has increased in recent years. Due to its unique properties as solvent and transport medium some special applications have become feasible [1–3]. Supercritical fluid extraction [2,4] and polymerization reactions [5] have even been realized technically [6]. Along these lines, the thermodynamic behavior of glassy polymer/gas systems are of general interest. The excellent solving characteristics of compressed gasses can lead to extensive swelling [7–9] and ∗

Corresponding author. Tel.: +49-6131379160; fax: +49-6131379360. E-mail addresses: [email protected] (W. Knoll), [email protected] (M. Ballauff). 1 Co-corresponding author. Tel.: +49-7216083150; fax: +49-7216083153. 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 2 ) 0 0 0 2 3 - 7

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significant lowering of the glass transition temperatures [10–12]. References about the sorption and swelling behavior of polymers in CO2 can readily be found in the literature [7,8,13–17]. Most studies, however, have been carried out at low or moderate pressures [13–17]. The industrial application of polymerization reactions presumes the knowledge of material parameters at elevated temperatures and pressures. Recently dispersion polymerizations have been studied by utilization of light scattering methods [18–21]. The refractive indices of the polymer swollen by the respective monomer have in these cases been extrapolated. Precise measurements of the refractive indices of polymer/monomer systems, however, are the necessary pre-requisite for the evaluation of turbidimetric data that has been used to elucidate the kinetics of polymerization of methylmethacrylate (MMA) in supercritical CO2 [21]. So far, the influence of a monomer on the swelling of the respective polymer in supercritical CO2 has not yet been studied. The techniques used in the past to determine the swelling behavior of polymers are either based on visual inspection [22] or electrical resistance measurements [23]. Here is presented an optical method to simultaneously quantify the swelling of, in this case, a thin polymer film and the refractive indices of the solvent and polymer. In this paper, the characterization of thin PMMA films in contact with (fluid) CO2 or CO2 /MMA mixtures by means of waveguide spectroscopy, a highly sensitive tool for characterizing interfaces and thin films [24,25], is reported. At first, the refractive index of CO2 as a function of pressure is compared to literature data in order to validate the presented optical technique in this context. The refractive indices of the solvent and the swollen polymer are presented thereafter and compared to calculations based on an ideal mixture. A second part concerns the influence of pressure and MMA monomer concentration on the swelling behavior of the polymer. 2. Theoretical considerations In the case of the CO2 /MMA/PMMA system studied, it is extremely difficult to relate the refractive index of the entire system to those of the respective components. Many unknown parameters due to interaction processes amongst the constituents prevent a clear interpretation and modeling of the obtained refractive index data. We therefore made an attempt to model the mixture by treating our system in the simplest way, i.e. as an ideal mixture. Based on the Clausius–Mossotti equation, the Lorenz–Lorentz equation [26] describes the refractive index (n) as a function of material’s parameters and the specific volume (υ sp ), which itself is a function of the thermodynamic variables pressure (p) and temperature (T)
2  n −1 NA α RLL = υsp (p, T ) = , (1) 2 n +2 3ε0 where RLL denotes the Lorenz–Lorentz constant or simply refractivity, NA the Avogadro’s number, α the polarizability and ε0 the dielectric constant. The specific volume changes with pressure and temperature according to 0 (1 + a(T ) + k(p)), υsp (p, T ) = υsp

(2)

0 where υsp is the specific volume at standard conditions (p0 = 1.013 × 105 Pa, T0 = 298 K), a(T) the isobaric thermal expansion and k(p) the isothermal compression. Both parameters rely on the particular

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equation of state and can, in the simplest case of the ideal gas equation, be written as  aT (T ) =

1 0 υsp



 ∂υsp  (T − T0 ) = αT (T − T0 ), ∂T p

(3a)

and 

1 kp (T ) = − 0 υsp



 ∂υsp  (p − p0 ) = −κp (p − p0 ), ∂p T

(3b)

with α T being the thermal expansion coefficient and κ p the bulk compressibility. The refractive index (nm ) of an ideal mixture consisting of i components is calculated, based on the Lorenz–Lorentz equation, by summation of the refractivities (RLL,i ) of the components times their volume fraction (φ i ): i

n2m − 1  RLL,i φi . = n2m + 2 n=1

(4)

The calculations presented assume an ideal mixture, i.e. the volumes of the individual components add up to the total volume neglecting any volume change upon mixing. Therefore, the Lorenz–Lorentz mixing rule as expressed through Eq. (4) can be applied. The refractive index of CO2 was calculated by extrapolation of the refractivities of CO2 as given by Obriot et al. [27]. The densities needed for this calculation were obtained from the Span–Wagner equation of state [28] and the IUPAC tables [29] at the specific pressures and temperatures. The refractive index of distilled MMA was measured with an Abbé refractometer at λ = 587 nm, T = 293 K and atmospheric pressure and then converted to a wavelength of 632 nm according to Li et al. [30]: n(MMA, 633 nm, 293 K) = 1.414. The density of MMA was taken from the literature to be ρ(MMA) = 0.944 g cm−3 [31]. With these parameters and Eq. (1) the Lorenz–Lorentz constant for MMA could be determined to be R LL (MMA) = 0.265 cm3 g−1 . The thermal expansion coefficient and the isothermal compressibility of MMA at the appropriate conditions were taken from Galland [32], and Sasuga and Takehisa [33]. Also, for PMMA the refractive index, density, isobaric thermal expansion coefficient and isothermal compression were adopted from the literature [26,31,34,35]. Any contribution of the small amount of added inhibitor hydroquinonemonomethylether was neglected. Concerning the ternary mixture CO2 /MMA/PMMA, a pressure independent distribution ratio of one with regard to volume was assumed for MMA in CO2 and PMMA. Within the context of an ideal mixture and without the knowledge of the MMA content in PMMA, this assumption ensured the feasibility of the calculations. In order to obtain the volume fraction of CO2 in the polymer, the specific density of CO2 solved in PMMA was estimated to be 44 cm3 mol−1 [9,36]. Finally, the measured swelling of PMMA due to CO2 , as shown below, was taken into consideration in the calculations. Although the MMA concentration in solution is decreased by its absorption into the PMMA film, this effect was ignored because of the negligible PMMA/solution volume ratio.

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3. Experimental 3.1. Measurement setup A multilayer architecture consisting of a metal substrate, a dielectric thin film, and a superstrate yields different kinds of excitations [24,25]. Plasmon surface polaritons (surface plasmons for short) represent the surface mode with an intensity peak at the metal/dielectric interface. Guided modes can be excited in the case of thick dielectrics with a refractive index higher than that of the surrounding medium. The angle of total reflection always yields the refractive index of the superstrate. Surface plasmons only exist if excited by p-polarized light, whereas, waveguide modes exist for both—s- and p-polarizations. For a simultaneous determination of the film’s thickness and (isotropic) refractive index at least two modes have to be excited. In general, the momentum of incident photons of a given energy (
ω) has to be matched to that of the different excitations, usually by varying the angle of incidence of the laser beam. For all modes, though, the momentum along the propagation direction is larger than that of the light in free space and thus, they cannot be excited in a simple reflection configuration. One way to overcome this problem is to increase the momentum of the incoming light by reflection from a periodically corrugated surface, a grating [24]. The momentum conservation in the plane of incidence at the surface is then given by in out kph ± mG = kph

(5)

out in where kph and kph are the wavevectors of the incoming and reflected light, respectively, and G = 2π /Λ the wavevector of the grating with Λ being the grating constant. Hence, by adding a multiple (m) of the grating wavevector (G) to the wavevector of the incoming light the momentum of the laser beam can be matched to that of the excited mode. The setup in general (Fig. 1a) consists of a linearly s- or p-polarized laser beam at a wavelength of 632.8 nm that is reflected off the sample in the pressure cell, which is mounted on top of θ/2θ-goniometer (Go; Huber, Germany). The detector (D) then measures the reflectivity (R) as a function of the angle

Fig. 1. Experimental setup for surface plasmon and waveguide spectroscopy measurements showing the positions of the chopper (C), the polarizer (P), the mirror (M), the Fresnel rhomb (FR), the lens (L), the detector (D) and the θ/2θ-goniometer (Go) onto which the high-pressure cell is mounted. The pressure is built up using a HPLC pump. The gear pump (GP) provides for a homogeneous solution in the mixing cycle, (T, p) describes the temperature and pressure controllers and (V) a valve.

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Fig. 2. Reflectivity (R) vs. angle of incidence (θ ) of the light for a PMMA layer of 1.1 ␮m thickness in air. The pressure medium was 0.5 M MMA in CO2 at the pressures indicated.

of incidence, i.e. the momentum of the incoming light, taking advantage of the lock-in technique by chopping the incoming beam. Fig. 1b sketches the cross section of the pressure cell together with the high pressure mixing circuit. The latter comprises, besides the cell, a HPLC pump (Jasco Corp. PU 1580, Germany) as pressure generator, a gear pump (GP; Ismatec, Germany) to mix the solvent homogeneously, and several valves to disconnect the pressure cell from the circuit. The tubes to and from the cell were heated in order to avoid temperature sinks while mixing. The cell itself was kept at a constant temperature of 333 K using a thermostat (not shown). Typical reflectivity scans, i.e. reflectivity (R) versus angle of incidence (θ), are shown in Fig. 2. For the selected dielectric layer (∼1.1 ␮m PMMA in air) only waveguide modes [37] appear in the given angle window. Since there exist more than two minima, the isotropic refractive index and the thickness of the polymer layer, as well as the refractive index of the pressure medium, can be extracted from the data. In order to do this in a numerically practicable way, the measured angular scans were transformed to fictitious Kretschmann configuration scans [24] by using G + k0 sin θG = npr k0 sin θp ,

(6)

where k0 denotes the wavevector in vacuo of the incoming light, θ G the angle measured in the grating configuration, npr the refractive index of the fictive prism and θ p the angle in the fictitious Kretschmann configuration. The transformed reflectivity scans were then compared with calculations of a model layer system taking advantage of the transfer matrix algorithm, which is based on Fresnel’s equations [38]. The variables adjustable are the polymer layer thickness and the dielectric constants. These were fit to the measured data by assuming a very thick gold layer (almost no losses due to light coupling out through the gold layer) and approximating the corrugated by a flat surface. Regarding the gold layer, a complex dielectric constant of ε = −12 + i(1.26) were used [39,40]. This choice of parameters results in an total error of the refractive index and the thickness of the PMMA layer of n = ±0.001 and l = ±30 nm, respectively.

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4. Sample preparation and measurement execution Firstly, the grating pattern with a grating constant Λ = 568 nm and an amplitude of approximately 35 nm was prepared by ion-etching a holographically prepared photoresist pattern into silica glass. A 150 nm thick gold layer was then thermally evaporated onto the grating (Balzers BAE 250). PMMA of a molar mass, M w = 350.000 g mol−1 (Polyscience, USA) was dissolved in toluene (Fisher, USA, 99%) at a concentration of 12.5 wt.%. The solution was degassed under vacuum and polymer films prepared by spincasting onto the gold layer at 8000 rpm for 60 s. In order to remove any residual solvent in the polymer and to minimize anisotropy effects due to the spincoating process the samples were dried at 393 K in vacuo for 12 h. The extent of completion of the drying process was checked by measuring the refractive index of the PMMA film in air prior to all measurements. The inhibitor in the MMA (obtained from Röhm AG, Germany) was removed by washing with 10 wt.% aqueous NaOH and, subsequently, with water. After careful drying, the monomer was distilled under reduced pressure. 0.2 wt.% hydroquinonemonomethylether was added to the MMA to prevent polymerization during the measurements. CO2 (99.995 wt.%, Linde AG, Germany) was used without further purification. The measurement execution comprised the following steps. After the assembly of the pressure cell, the refractive index and the thickness of the PMMA film, as a reference, was measured against air. Before the measurement protocol with the MMA/CO2 mixture was started, the tightness of the cell was tested against pure CO2 at 50 MPa. In order to avoid damage of the polymer film in the cell (MMA dissolves PMMA), a known amount of MMA was injected into the gear pump, which was de-coupled from the cell (valve V2 closed, cf. Fig. 1b). Then the mixing circuit was filled with CO2 , heated up to 333 K and the MMA/CO2 solution thoroughly stirred. The solution was subsequently pumped several minutes through the whole circuit (V2 opened) to provide for a homogeneous mixture in the cell. After closing valves V2 and V3 the mixture was left to equilibrate for 20 min in the cell before the actual measurement was started. Higher pressures than the starting pressure were obtained by adding CO2 and then resuming the mixing procedure.

5. Results and discussion In the following, measurements of the refractive indices of CO2 , CO2 /MMA and CO2 /MMA/PMMA mixtures and the swelling of a PMMA film in pure CO2 and MMA/CO2 mixtures are presented. All measurements were conducted at a constant temperature of 333 K. The results are listed in Table 1. 5.1. Refractive index of CO2 and CO2 /MMA mixtures At first the performance of the apparatus was examined by measuring the refractive index of plain supercritical CO2 at pressures up to 52 MPa. For this system there exists a good set of experimental data [27,41] to which the presented results can be compared. Fig. 3 shows our data of the refractive indices of CO2 together with the values calculated according to Obriot et al. [27]. Initially, the refractive index increases rapidly with increasing pressure and then, from about 20 MPa, somewhat slower. This behavior can be readily explained by the increase of density with rising pressure. However, sometimes

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Table 1 Compiled experimental data Pressure (MPa)

σ (p) (MPa)

Swelling

n(PMMA)

n(solvent)

0.0 M MMA in CO2 at 332.9 ± 0.2 K 10.3 0.1 13.0 0.2 17.1 0.1 20.1 0.1 23.1 0.1 27.0 0.1 29.9 0.1 33.0 0.1 37.3 0.1 40.1 0.1 51.8 1.0

1.166 1.191 1.221 1.235 1.250 1.267 1.280 1.291 1.306 1.315 1.355

1.453 1.448 1.443 1.440 1.438 1.436 1.435 1.433 1.432 1.431 1.427

1.072 1.111 1.153 1.169 1.178 1.187 1.193 1.197 1.203 1.207 1.218

0.47 M MMA in CO2 at 332.9 ± 0.1 K 17.6 0.1 20.0 0.3 23.7 0.1 27.8 0.1 31.3 0.2 33.9 0.1 37.2 0.1 40.9 0.4 43.5 0.3 48.2 0.2

1.273 1.279 1.291 1.304 1.316 1.329 1.332 1.353 1.364 1.383

1.437 1.436 1.434 1.431 1.430 1.429 1.428 1.427 1.425 1.424

1.174 1.180 1.191 1.200 1.205 1.211 1.212 1.217 1.220 1.224

0.95 M MMA in CO2 at 333.0 ± 0.1 K 15.8 0.1 20.4 0.1 24.0 0.1 29.3 0.1 30.0 1.0 33.1 0.1 37.8 0.1 41.8 0.1

1.282 1.333 1.342 1.385 1.377 1.398 1.415 1.436

1.437 1.430 1.429 1.425 1.425 1.424 1.422 1.420

1.169 1.195 1.201 1.214 1.213 1.218 1.223 1.228

1.46 M MMA in CO2 at 333.2 ± 0.1 K 19.2 0.2 21.5 1.0 28.4 0.2 30.9 0.2 33.2 0.3 37.5 0.1 40.8 0.1

1.458 1.422 1.449 1.469 1.473 1.502 1.512

1.422 1.424 1.421 1.420 1.419 1.417 1.416

1.217 1.213 1.224 1.228 1.230 1.235 1.238

the Lorenz–Lorentz relation of Eq. (1) does not fit well the experimentally determined refractivities but the data can rather be described with, for example, a virial expansion of Eq. (1) [27,41]
2  n −1 1 (7) = AR + BR ρ + CR ρ 2 . n2 + 2 ρ

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Fig. 3. Refractive index of CO2 vs. pressure. The open symbols represent measured data, whereas, the solid curve is based on parameters given by Obriot et al. [27] (see Eq. (6)).

In the case of Fig. 3, the theoretical curve was calculated using the virial coefficients AR , BR and CR of Obriot et al. [27] (AR = 6.649 cm3 mol−1 , BR = 1.9 cm6 mol−2 , CR = −287 cm9 mol−3 )2 and the density extracted from the Span–Wagner equation of state [28] and IUPAC tables [29]. As can be seen in Fig. 3, experimental and calculated values match very well within the experimental error of n = 0.001. Yet, at high pressures (>35 MPa) systematic discrepancies between both values arise. Smaller values of RLL than the ones calculated from the virial expansion would fit the experimental data better. After the measurement system was verified the MMA content of the solution was varied in order to gain some information about the swelling behavior of the PMMA film. The solutions of MMA (0.5, 1.0 and 1.5 M) in CO2 were tested. As shown in Fig. 4, the higher the MMA concentration the higher is the pressure at the first data point. At 1.5 M the data around 20 MPa are extremely noisy. These observations are explained by the solubility of MMA in CO2 , which depends on the MMA concentration, temperature and pressure [21,42]. At pressures below 16 MPa, all MMA/CO2 mixtures resulted in heterogeneous solutions. The increasing refractive index with pressure is again induced by the increasing density of the fluids. With increasing MMA content, on the other hand, the refractive index of the mixture rises. This phenomenon is explained by the higher fraction of MMA in solution, which has a larger refractive index compared to CO2 . The calculated refractive indices are depicted in Fig. 4 as open symbols, respectively. Clearly, all calculated data are larger than their corresponding experimental data points. However, the qualitative trend of the measurements is reproduced correctly, in that the displacement between both curves remains the same throughout the whole pressure range. The origin of the displacement itself stays unexplained but will presumably be found in the crude assumptions made at the derivation of the calculations. 2

The virial coefficients were measured at 308, 313 and 323 K. The values of AR were, within the error, all the same and the values for BR and CR were adopted from [41] at 323 K and taken to be the same for all temperatures. We thus took the mean of AR and the given values for BR and CR at our measurement temperature of 333 K.

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Fig. 4. The refractive index of 0, 0.5, 1.0 and 1.5 M MMA solved in CO2 vs. pressure indicated by 䉲, 䊉, 䊏 and 䉱, respectively. The calculated refractive indices corresponding to the particular measurement conditions are represented by the corresponding open symbols and connected with dotted lines for clarity.

5.2. Refractive index of the PMMA film The following section focus on the characterization of the PMMA film. Firstly, the pressure and MMA molarity dependent change in refractive index is examined. Secondly, attention is drawn to the swelling characteristics of the film as a function of the above mentioned parameters. In Fig. 5, the refractive index of the MMA/CO2 -swollen PMMA film is plotted against pressure. Again, two features are noteworthy. First, with increasing pressure the refractive index of the layer decreases due

Fig. 5. The refractive index of the swollen PMMA film plotted against pressure for different MMA concentrations ((䉱) 1.5 M, (䊏) 1.0 M, (䊉) 0.5 M, (䉲) 0 M MMA content in CO2 ). The calculated refractive indices corresponding to the particular measurement conditions are represented by the respective open symbols and connected with dotted lines for clarity.

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to the pressure-induced increase of MMA/CO2 content in the film, both of which have a lower refractive index than the polymer. Second, with increasing MMA concentration the refractive index of the film decreases. This behavior is induced by the swelling ability of MMA with respect to PMMA (Section 5.3). The higher the MMA content, the more swollen the PMMA film, the higher the MMA/CO2 fraction in the film and thus, the lower the apparent refractive index of the polymer. Focusing on the calculated values, marked by the respective open symbols, one can see the qualitative conformity with the measured data, in that the refractive indices decrease with pressure and MMA concentration. However, in this case the slopes of the curves deviate from the measured ones. A smaller dependence on the pressure is predicted. Also, with increasing MMA molarity the calculated values diverge to a much larger extent. It is clear that especially for high MMA concentrations the assumption of an ideal mixture is no longer valid. The smaller slope of the calculated curves could be explained by an extended MMA content in the polymer film with increasing pressure (as compared to the constant volume fraction that was assumed in the calculations). 5.3. Swelling of the PMMA film As explained above, the waveguide spectroscopy yields, besides the refractive indices of the superstrate and the dielectric film, the thickness (l) of this layer, provided that at least two resonances are excited. Fig. 6 shows the measured thickness dependence of the polymer film on the pressure. Clearly, the thickness increases, at first, rapidly and then, from about 17 MPa on, slower. The comparison to data measured against water as pressure medium reveals the hydrophobic property of PMMA [26]. Kleideiter et al. found a decreasing thickness (and increasing refractive index in contrast to the data presented above) with increasing pressure. Since, in a first approximation, no water is solved in the polymer it is simply compressed by the water. Using CO2 as pressure medium the presented measurements demonstrate the sorption of CO2 and therefore, the swelling of the PMMA film. In this work, the PMMA film could not expand in the plane of the surface but only in thickness. The assumption on which the following discussion is based is that the volume change of the film is proportional

Fig. 6. The absolute thickness of a PMMA layer vs. pressure.

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Fig. 7. Swelling of the PMMA layer vs. pressure for varying MMA concentrations ((䉱) 1.5 M, (䊏) 1.0 M, (䊉) 0.5 M, (䉲) 0 M MMA content in CO2 ). The swelling was determined from the thickness variation by assuming that a thickness change corresponds to a volume one (see explanations in Section 5.3).

to the thickness change since the PMMA used is neither cross-linked nor anisotropic. The strain built up inside the sample is assumed to relax. Thus, the swelling describes the volume change (V) of the polymer related to the initial volume (V0 ) and is based on the thickness change (l−l0 ) with l0 being determined prior to every measurement cycle. As can be seen in Fig. 7, the swelling of the PMMA film in the different MMA/CO2 solvents increases with pressure. It also increases with increasing MMA content, resulting, overall, in a reverse behavior compared to the one of the refractive index (Fig. 5). The way the refractive index behavior is explained

Fig. 8. Comparison of the experimental data (CO2 -swollen PMMA at T = 332.9 K) with literature values [8] which were determined at 331.2 K. A correspondence with the sorption data can be observed.

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in Section 5.2 is based on Fig. 7, which illustrates an increased swelling of the polymer due to a higher concentration of MMA. In the literature, there already exist some studies on the sorption and swelling behavior of PMMA in CO2 . Liau and McHugh, for example, found by visual determination that PMMA increasingly absorbs CO2 as a function of pressure which, in turn, results in an increased volume of the polymer [8]. The comparison of the swelling results of Liau and McHugh with our data (Fig. 8) shows large deviations. However, Wissinger and Paulaitis have already noted that the visual determination of the swelling may be accompanied by large errors [7]. In addition, different sample formats have been used. On the other hand, the sorption data of Liau and McHugh correspond very well with the swelling behavior of the measurements presented. While surprising at first, this correlation becomes more reasonable when taking into account that the partial specific volume of CO2 in PMMA stays virtually constant at pressures above 7 MPa [36]. Thus, in the pressure range under investigation the volume change is proportional to the mass and therefore, to the sorption change.

6. Conclusions It has been shown for the first time that thin polymer films with supercritical CO2 as solvent can be characterized by using waveguide spectroscopy. Furthermore, the results suggest the possibility that even thinner films with no waveguide modes present might readily be characterized and that sorption experiments in a supercritical fluid environment can be realized by means of surface plasmon spectroscopy. The study of a spincast PMMA film at the interface with CO2 and CO2 /MMA mixtures has given the following results: Firstly, the refractive index of pure CO2 corresponds well to prior measurements [27] and can be modeled with a virial expansion of the Lorenz–Lorentz equation. Secondly, quantitative deviations up to 1.5% arise when the experimental refractive indices of the three-component system CO2 /MMA/PMMA are compared to those calculated by means of the Lorenz– Lorentz mixing rule for ideal mixtures. Yet, the qualitative behavior of the experimentally determined refractive indices is reproduced correctly. Thirdly, the thickness of the polymer film rises with increasing pressure and MMA concentration in the fluid. A higher percentage of solvent is absorbed and the refractive index of the film decreases hereupon. Lastly, the swelling behavior of PMMA due to sorption of CO2 clearly deviates from other studies [8]. The waveguide spectroscopy, though, as an optical method, seems to be a more accurate method than visual inspection alone. In conclusion, the refractive indices of the presented solvent mixtures cannot be simply described by an ideal mixture at the given conditions. Rather, it is necessary to complement experimental measurements of refractive indices, for example, by means of waveguide spectroscopy, with suitable theoretical models. List of symbols a(T) isobaric thermal expansion AR , BR , CR virial coefficients G wavevector of the grating
Planck constant k0 wavevector in vacuo of the incoming light

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k(p) in kph out kph l m n nm npr NA p p0 R RLL RLL,i T T0 V/V0

isothermal compression wavevector of the incoming light wavevector of the reflected light thickness of the PMMA layer multiple number refractive index refractive index of an ideal mixture refractive index of the fictive prism Avogadro’s number pressure pressure at standard conditions (1.013 × 105 Pa) reflectivity Lorenz–Lorentz constant Lorenz–Lorentz constant of component i temperature temperature at standard conditions (298 K) volume change of the polymer related to the initial volume

Greek letters α αT ε ε0 φi κp λ Λ θ θG θp ρ υ sp 0 υsp ω

polarizability thermal expansion coefficient complex dielectric constant dielectric constant in vacuum volume fraction of the component i bulk compressibility wavelength grating constant angle of incidence angle measured in the grating configuration angle in the fictitious Kretschmann configuration density specific volume the specific volume at standard conditions frequency of the light

159

Acknowledgements We acknowledge financial support from the Deutsche Forschungsgemeinschaft. We also thank Micheal Türk for helpful discussion of the equations of state for carbon dioxide. References [1] J.M.L. Penninger, M. Radosz, M.A. McHugh, V.J. Krukonis, Supercritical Fluid Technology, Elsevier, Amsterdam, 1985. [2] M.A. McHugh, V.J. Krukonis, Supercritical Fluid Extraction: Principles and Practice, Butterworths, Boston, 1994.

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