Article pubs.acs.org/Macromolecules
Non-Fickian Diffusion of Toluene in Polystyrene in the Vicinity of the Glass-Transition Temperature Florian Mueller, Kai-Martin Krueger,† and Gabriele Sadowski* Laboratory of Thermodynamics, Technische Universität Dortmund, Emil-Figge-Strasse 70, 44227 Dortmund, Germany S Supporting Information *
ABSTRACT: The diffusion process of toluene in polystyrene films is investigated in the vicinity and slightly above the glasstransition of the mixture. A detailed analysis investigates whether the diffusion follows Fick’s law or other effects are superimposing the mass transport process. For that purpose, the diffusion of toluene in polystyrene films of different thicknesses at otherwise constant conditions was observed by gravimetric sorption measurements. Even slightly above the glass-transition of the toluene-loaded polymer a remaining influence of the polymer relaxation on the diffusion process (non-Fickian diffusion) is observed. The diffusion Deborah number concept was applied using an experimentally independent relaxation time for the polystyrene-toluene mixture. The results show quantitatively the same trend but finally deviate by about 1 order of magnitude.
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INTRODUCTION The knowledge of solubilities and of the diffusion processes in polymer/volatile organic compound (VOC) systems allows an optimal control of the polymer production processes in industrial applications. The information, how far and how much VOC dissolves in a polymer film provides the basis for the calculation of production or separation times, e.g. for polymer drying or in membrane separation processes. In a previous work, a gravimetric-sorption apparatus was set up, and sorption isotherms as well as sorption curves (dynamic transport) for toluene in polystyrene were determined.1 It was shown, that the dissolved VOC reduces the glass-transition temperature of the polymer dramatically and shifts the diffusion coefficient to higher values. On the basis of films of one selected thickness, diffusion coefficients above the glass-transition were determined as a function of VOC concentration and temperature. For that purpose, it was assumed that the diffusion process in this temperature region is always Fickian. In this work we now investigate to which extend this assumption can be considered to be true for the polystyrene/toluene system and whether the diffusion coefficients determined from the sorption curves at temperatures being slightly above the glass-transition are reliable. The group of Kishimoto and Matsumoto2 investigated this question already more than 40 years ago for other polymer/ VOC systems. They showed, that non-Fickian sorption behavior can be observed at temperatures even above the glass-transition temperature of the neat polymer. Moreover, Odani3 identified non-Fickian sorption behavior for diffusion of a VOC in polystyrene slightly above the glass-transition temperature of the mixture. On the basis of their investigation and due to the fact that the experimental data for the region © 2012 American Chemical Society
slightly above the glass-transition of the mixture is very limited, a detailed study of the diffusion behavior of the polystyrene/ toluene system is presented in this study. The results should give an idea of the broadness of the non-Fickian region above the glass-transition temperature of the mixture with respect to temperature and concentration. According to Fujita,4 the shape of the sorption curves gives a first clue about the condition of the polymer and the diffusion type. However, the final answer whether the diffusion process is Fickian or not can only be given by comparing reduced sorption curves for different film thicknesses at otherwise identical conditions. The same information can also be obtained by assuming Fickian diffusion and comparing the diffusion coefficients obtained from the sorption curves at different film thicknesses. In case of Fickian diffusion, the reduced sorption curves as well as the derived diffusion coefficients should not depend on the film thickness and thus should be identical. The concept of Deborah numbers proposed by Vrentas and Duda5,6 offers an alternative way to classify the diffusion type in polymers. This concept relates the characteristic relaxation time of the mixture to the characteristic time of the diffusion process. In this work, the analysis of diffusion coefficients and of the sorption curves is being done for the polystyrene/toluene system for different film thicknesses of the polymer at temperatures of 30, 50, and 70 °C. Moreover the characteristic relaxation time of the considered polystyrene/toluene mixture Received: October 19, 2011 Revised: December 14, 2011 Published: January 9, 2012 926
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is estimated and the Deborah numbers for all temperatures and film thicknesses are determined. Finally, the results of the experimental sorption measurements are compared to Deborahnumber calculations.
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EXPERIMENTS
Experimental Setup. The main part of the gravimetric sorption equipment used in this work is a magnetic suspension balance measuring the weight increase of the polymer sample during an isothermal pressure step. The temperature of the cell is controlled by means of an air-bath thermostat. Depending on the viscosity of the mixture, the polymer sample is either hanging as a free film at the measuring hook or is lying horizontally attached to the bottom of a glass bucket. A more detailed description of the experimental procedure is given elsewhere.1 Materials and Film Preparation. Polystyrene was obtained from Gefinex with Mw = 240 000 g/mol, Mw/Mn = 5.65 and a glass-transition temperature of 105 °C. Toluene of 99.9% purity was obtained from Merck and was degassed by three successive freezing-evacuation-melting steps before being filled into the evacuated VOC vaporizer. The polymer films were prepared by casting from toluene/polystyrene solutions on a glass surface with subsequent flattening by means an applicator exhibiting a slit of defined height between glass and applicator surface. Afterward, the films were stored in a vacuum oven to enhance the toluene removal from the films with a temperature and pressure routine, as described earlier.1 Using this procedure, the thickness of the polystyrene films varied at maximum by about ±1 μm. Sorption Measurements. As described earlier,1 the polymer sample is positioned in the measurement cell and equilibrated at the measurement temperature. After evacuation of the cell, a certain VOC pressure is established in the cell and is kept constant. The weight of the polymer sample is observed by means of the magnetic suspension balance until no weight increase is observed for several hours. For the next sorption step, the VOC pressure in the cell is increased also leading to an increasing amount of VOC absorbed. The VOC concentrations at different pressures and constant temperature yield the sorption isotherm (as, e.g., shown in Figure 1) whereas the time
Figure 2. Experimental interval sorption curves for the polystyrene/ toluene system at 70 °C and for a film thickness of 70 μm. The glasstransition was calculated using eq 1.
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RESULTS
Sorption Isotherms. All experimental sorption data obtained within this and from our previous work7 are summarized in Table 1 of the Supporting Information. As an example, Figure 1 shows the sorption isotherms at 30, 50, and 70 °C for film thicknesses between 15 and 70 μm. For each of the three temperatures, the sorption isotherms at different film thicknesses cannot be distinguished within experimental accuracy. Sorption Curves and Diffusion Behavior. Figure 2 shows the sorption curves for all isothermal pressure steps at 70 °C and a film thickness of 70 μm. The VOC-loading WTol (mass of absorbed toluene per mass of the dry polymer) is plotted versus the square root of time (√t). Each sorption step is starting at the final mass uptake of the previous step. To distinguish whether the polymer/VOC mixture is in the rubbery or in the glassy state we use the approach of Mueller et al.8 who fitted the equation of Kelly and Bueche9 to describe the concentration dependence of the glass-transition temperature.
Tg , mixture =
α 2(1 − ϕ1)Tg ,2 + α1ϕ1Tg ,1 α 2(1 − ϕ1) + α1ϕ1
(1)
Tg,1 and Tg,2 are the glass-transition temperatures of the VOC and the neat polymer and α1 and α2 are the thermal expansion coefficients of the pure VOC and the polymer, respectively. These values can be found in Table 1. ϕ1 is the VOC volume fraction, which is related to the VOC weight fraction w1 as follows: w1 = Figure 1. Experimental sorption data for the polystyrene/toluene system at 30, 50, and 70 °C as a function of the polymer-film thickness. Symbols are experimental data; lines are included for better visibility.
v01−1(T , p)ϕ1 v01−1(T , p) + ϕ1(v01−1(T , p) − v02−1(T , p))
(2)
Table 1. Parameters for the Calculation of the GlassTransition Temperature of the Polystyrene/Toluene Mixture Used in Equation 1
dependent weight increase for one pressure step is called sorption curve (as, e.g., shown in Figure 2 for several pressure steps). Samples at 30 and 50 °C were measured as free-hanging films and films at 70 °C were attached to the bottom of a glass bucket, to be able to measure in the region above the glass-transition temperature at lower viscosities of the polymer/VOC mixture. 927
parameter
eq 1
ref
α1 α2 Tg,1 Tg,2
1.067 × 10−3 1/K 5.73 × 10−4 1/K − 156.15 °C 105 °C
12 8 11 8
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v01 and v02 are the specific volumes of pure VOC and polymer, respectively. They are calculated at the pressure p and temperature T of the mixture using an empirical correlation10 for v01 and the Tait equation11 for v02. More details as well as the used expressions and constants can be found elsewhere.1 The VOC concentration which, according to eq 1, corresponds to a glass-transition of 70 °C is marked in Figure 2. The sorption curves below the calculated glass-transition show a different appearance than the curves above which look like Fickian sorption curves and which achieve the features of Fickian sorption given by Fujita4 and Crank.13 A typical representative for a sorption curve above the glass-transition is shown in more detail in Figure 3 for the pressure step from 79 to 87mbar at 70 °C. Figure 4. Diffusion coefficients for toluene in polystyrene at 30, 50, and 70 °C for films of different thicknesses. The data for 30 °C were taken from our previous work1,7 and the data for the 50 °C measurements were taken from Heuwers.14 All data were measured above the glass-transition of the mixture according to eq 1.
toluene concentrations. No systematic deviation of the data for different film thicknesses is observed. At 50 °C, the diffusion coefficients for different film thicknesses are also very similar. However, the diffusion coefficients at 30 °C obviously vary remarkably with film thickness. In particular, the sorption data for the thinner films lead to smaller diffusion coefficients up to toluene mass fractions of wTol = 0.2 and probably at even higher concentrations. These results indicate that the assumption of Fickian diffusion above the glass-transition of the mixture is valid for 50 and 70 °C but not for 30 °C. Although at this latter temperature only data points above the glass-transition of the polymer/VOC mixture were taken into account and the sorption curves look Fickian,4,13 the diffusion process is obviously still influenced by other effects (i.e., time-dependent swelling) and therefore non-Fickian. Figure 5 shows a comparison of selected sorption curves as well as reduced sorption curves for different film thicknesses at 30 and 70 °C. As expected, the sorption curves for the 15 and 29 μm film at 30 °C (Figure 5a) and for the 30, 50, and 70 μm at 70 °C (Figure 5c) are different for different film thicknesses since for thicker films it takes longer to reach the equilibrium concentration throughout the film than for thinner films. This effect is eliminated for the reduced sorption curves which are obtained from the sorption curves by dividing the mass-uptake by the film thickness. Therefore, in case of Fickian diffusion the reduced sorption curves should coincide at given temperature and pressure step. Comparing the reduced sorption curves in Figure 5, parts b and d, at 30 and 70 °C, respectively, it can be seen, that the curves coincide more or less at 70 °C but do clearly separate from each other at 30 °C. This again supports the conclusion drawn from the comparison of the diffusion coefficients: in contrast to the diffusion at 70 °C, the diffusion process at 30 °C is non-Fickian although the system is above its glass-transition and the sorption curves have a Fickian-like shape. The reduced sorption curves for the 50 °C data (no example shown in Figure 5) behave similar than the 70 °C measurements, as can be expected from Figure 4. A possible explanation for this behavior is the presence of swelling/relaxation effects which can delay the diffusion process in thinner films. The relaxation time of a polymer is independent of the film thickness and only depends on
Figure 3. Experimental interval sorption curve of the polystyrene/ toluene system at 70 °C and a film thickness of 70 μm (pressure step: 79−87 mbar). Symbols are experimental data; line is the calculation using eq 3.
According to the procedure used in our previous work,1 the Crank equation13 (eq 3) was applied to determine the diffusion coefficient for a Fickian-type sorption curve. m(t ) =1− m∞
∞
∑ n=0
⎧ D(2n + 1)2 π 2t ⎫ 8 ⎬ ⎨− exp (2n + 1)2 π 2 d2 ⎭ ⎩
(3)
In this equation d is the film thickness of a free hanging film at the beginning of this pressure step and D is the mean diffusion coefficient. d was calculated for each sorption interval individually based on the equilibrium volume increase of the previous sorption interval with no respect to excess volume and the assumption of one-dimensional swelling only in the direction of diffusion. As it can be seen from Figure 3, the Crank equation is able to describe the experimental data with satisfactory accuracy. All so-determined diffusion coefficients are summarized in Table 1 (Supporting Information) Figure 4 shows a semilogarithmic plot of the diffusion coefficients versus the toluene weight fraction determined at three temperatures and for the different film thicknesses. Here, it was again assumed, that the diffusion above the glasstransition of the mixture can be described as Fickian diffusion. The data for 30 °C were taken from our previous work,1,7 and the data for the 50 °C measurements were taken from Heuwers.14 At 70 °C, the diffusion coefficients determined for different film thicknesses are close to each other, especially for higher 928
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Figure 5. Sorption curves and reduced sorption curves of polystyrene/toluene systems for different film thicknesses at two different temperatures. (a) Sorption curves at 30 °C for films of 15 and 29 μm thickness. (b) Reduced sorption curves at 30 °C for films of 15 and 29 μm thickness. (c) Sorption curves at 70 °C for films of 30, 50, and 70 μm thickness. (d) Reduced sorption curves at 70 °C for films of 30, 50, and 70 μm thickness.
Those effects have been discussed by Vrentas15 for several systems. He used the Deborah-number concept to correlate and predict the presence of non-Fickian diffusion. In the next chapter, the experimental results are therefore compared to the diffusion Deborah-number concept of Vrentas et al.5,17 which gives an estimate of whether Fickian or non-Fickian diffusion is to be expected. One advantage of this concept is that it combines all parameters which affect the sorption characteristics like temperature, concentration and film thickness. Diffusion Deborah numbers. The diffusion Deborah number (Deb)D is defined5 by the ratio of the characteristic relaxation time τm of the mixture and the mutual diffusion time θD according to eq 4:
temperature and VOC concentration whereas the diffusion time increases proportionally with the film thickness squared (second Fick’s law). For thicker films the relaxation of the polymer is fast compared to the diffusion time. Therefore, the relaxation does not influence the Fickian diffusion process. For thinner films with smaller diffusion times, the relaxation effects become more and more important for the transport process. Here, the time which the polymer molecules need to attain their new equilibrium conformation (e.g., density) and to absorb more VOC retards the transport process compared to pure Fickian diffusion. For the reduced sorption curves, where the film-thickness effect according to Fickian diffusion is eliminated, this effect should lead to flatter curves for thicker films. This is indeed what is observed in Figure 5b for the reduced sorption curves at 30 °C where the data points for the 15 μm film lie below those of the 29 μm film. At 70 °C no systematic variation of the reduced sorption curve data is observed (Figure 5d). As already mentioned in the introduction, this influence of the polymer relaxation on the transport process is well-known for conditions below the glass-transition1,16 and slightly above the glass-transition temperature of the mixture.2,3 However, from this investigation it becomes clear that for the polystyrene/toluene system this effect can extend far into the rubbery region above the glass-transition of the mixture, especially for thin films and temperatures far below the glasstransition temperature of the neat polymer.
(Deb)D =
τm θD
(4)
whereas the characteristic diffusion time θD is given by
θD =
L2 D*
(5)
L is the diffusion path length and D* is given by the following expression
D* = x2D1 + x1D2
(6)
x1 and x2 are the mole fractions of the VOC and polymer, respectively. D1 and D2 are the corresponding self-diffusion coefficients. According to Billovits and Durning18 it is assumed 929
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that D* is in the order of magnitude of the binary mutual diffusion coefficient D. As shown in the preceding section, the measured diffusion coefficients D at 50 and 70 °C comply the assumption of Fickian diffusion and can be directly used to calculate the diffusion Deborah numbers at those temperatures. In contrast, the diffusion coefficients estimated above for 30 °C are not real mutual diffusion coefficients, since the diffusion there is not only controlled by a purely Fickian diffusion mechanism and varies with film thickness. Therefore, the diffusion coefficients obtained at 30 °C were extrapolated to infinite film thickness as described by Kishimoto and Matsumoto2 to yield the mutual diffusion coefficients at 30 °C. Figure 6 shows the typical
Here c1 and c2 are the polymer-specific WLF-parameters, which have been already determined for the considered polystyrene, T is the measuring temperature in °C and Tg,mixture is the glasstransition temperature of the polymer/VOC mixture in °C. At conditions, where Tg,mixture is equal to the measuring temperature T, the shift factor a(T,T0,w) is equal to 1, which is equivalent to the relaxation time of the reference state: τ m(Tg,mixture,w(Tg,mixture)) = τ m(T0,w = 0). The reference relaxation time τm(T0,w = 0) was estimated using a correlation given by Pipkin:17
τm =
∞ 1 · t(G(t ) − Ge) dt η0 0
∫
(9)
To apply this approach, the experimental creep-compliance data F(t) of neat polystyrene published earlier8 was modeled by the following viscoelastic spring-dashpot model7,21
F (t ) =
1 + E0
⎡ ⎧ tE ⎫⎤ 1⎢ t ⎨ − i ⎬⎥ + 1 exp − ∑ ⎥ Ei ⎢⎣ η η ⎩ i ⎭⎦ 0 1 3
⎪
⎪
⎪
⎪
(10)
Afterward the creep compliance F(t) was transferred into the shear modulus G(t) using the following assumption:21
G (t ) =
1 3F(t )
(11)
The results of the creep−compliance modeling (eq 10) can be seen in Figure 7. Two different sets of parameters were used: Figure 6. Plot of Ia2 vs d−2 for the system poystyrene/toluene at 30 °C using the procedure of Kishimoto and Matsumoto.2
procedure: The initial slope Ia of sorption curves at 30 °C and different concentrations is plotted against the film thickness d. The increasing value of Ia with increasing film thickness implies that the absorption of toluene in polystyrene at 30 °C is not controlled by a purely Fickian diffusion mechanism. By assuming that the hypothetical sorption curve with the initial slope extrapolated to infinitive film thickness is controlled by a purely Fickian diffusion mechanism, the mutual diffusion coefficient D was calculated. The results are listed in Table 1 (Supporting Information) with brackets. As expected, the extrapolated diffusion coefficients were higher than the measured diffusion coefficients. The values are listed in Table 1 (Supporting Information) with brackets. The concentration-dependent relaxation time τm(T,w) was calculated using the following equation proposed by Ferry:19
τm(T , w) = a(T , T0 , w)τm(T0 , w = 0)
Figure 7. Creep−compliance curve of neat polystyrene at 105 °C. Experimental data were taken from Mueller et al.8 Lines are modeling results using eq 10. Model 1 describes the experimentally obtained data, and model 2 describes the theoretical behavior of the rubbery state without viscoelastic flow above the glass-transition.
(7)
Here a(T,T0,w) is a concentration-dependent shifting factor and τm(T0,w = 0) is the reference relaxation time of the neat polymer at reference temperature T0 = Tg. This shift-factor concept is based on the theory of William, Landel, and Ferry.20 The shift factors for the temperature shift as well as for the concentration shift were taken from our previous work.8 The concentration dependence of the shift factor was introduced using the concentration dependence of the glass-transition temperature according to eq 1 from Kelly and Bueche9 which led to the following equation:8
log a(T , T0 , w) = −
parameter set one (model 1) was used to describe the true trend of the creep−compliance curve whereas parameter set two (model 2) was used to describe the theoretical trend of the creep−compliance curve without viscoelastic flow in the rubbery state. The assumption of no-viscoelastic flow in the rubbery state was necessary to allow for convergence of the integral in eq 9. All model parameters of eq 9 are listed in Table 2. On the basis of that, we estimated a relaxation time of τm = 2.72 × 104 s at T0 = 105 °C for the considered polystyrene. For comparison with literature, this value was converted to 129 °C using the WLF equation8,20 which led to a value of τm = 1.78 × 102 s.
c1(T − Tg , mixture) c 2 + (T − Tg , mixture)
(8) 930
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the fact that increasing toluene load of the polymer/VOC mixture has the same effect as increasing temperature of a neat polymer, namely a decreasing relaxation time. Moreover, it can be seen, that the calculated Deborah numbers decrease with increasing film thickness for each temperature. This is due to the fact, that the diffusion coefficients as well as the relaxation times are independent of film thickness, whereas the Deborah number is a function of film thickness as it can be seen from eq 4 and eq 5. Comparing the sorption measurements of this work to the diffusion regimes according to the Deborah number (see Table 1 (Supporting Information)), we find that all calculated Deborah numbers are in the order of one magnitude to high compared with the experimental findings. For the 30 °C measurements all experimental sorption curves were found to be non-Fickian whereas at least for the lower concentrations at 30 °C the calculated Deborah numbers are above 10 indicating Fickian diffusion behavior. The 50 °C and the 70 °C measurements all showed Fickian diffusion. In contrast to this the calculated diffusion regime for 50 °C is all non-Fickian with Deborah numbers between 0.1 and 10. For 70 °C only the calculated low concentration points still remain in the nonFickian region whereas at higher solvent concentrations the calculated diffusion regime is again Fickian with Deborah numbers below 0.1. It can be concluded that the findings in this paper are explainable with the diffusion Deborah number concept. The concept predicts the Fickian or non-Fickian diffusion for the polystyrenene/toluene system qualitatively well, although there are some quantitative deviations since the calculated Deborah numbers are about 1 order of magnitude higher when compared with the experimental observations. The main reason for this discrepancy will probably be the determination of the relaxation time, which is burdened with some assumptions (e.g., concentration, temperature dependence, and shift to the experimental conditions, eq 7 and eq 8) as well as experimental uncertainties in the determination of the relaxation time. These can certainly account for an error of this magnitude. A classification of the diffusion-type slightly above the glasstransition of the mixture only with the Deborah number concept based on literature data might not be accurate enough. A combination with the traditional analysis of reduced sorption curves for different film thicknesses at otherwise identical conditions promises to give the most reliable results.
Table 2. Model Parameters for the Spring−Dashpot Model (Eq 10) with Model Parameters for True Trend (Model 1) and for a Material Behavior without Viscous Flow (Model 2)14 parameter
model 1
E0 E1 E2 E3 η0 η1 η2 η3
× × × × × × × ×
4.2 1.0 1.4 2.0 8.0 6.0 1.5 1.0
8
10 104 105 105 109 1011 108 109
model 2 4.2 1.0 1.4 2.0 8.0 6.0 1.5 1.0
× × × × × × × ×
108 104 105 105 1015 1012 108 109
Akovali22 published a value of τm = 8.40 × 103 s at 129 °C for a monodisperse polystyrene with the same molar mass of Mw = 240 000 g/mol. The smaller value for the relaxation time obtained in this work could be caused by the fact, that the PDI of the investigated polystyrene is 5.65. For a given weightaverage of a polymer, a broader molecular weight distribution (higher PDI value) causes smaller relaxation times due to the presence of the short polymer chains. Thus, monodisperse polystyrene with a molar mass of Mw = 117 000 g/mol exhibits a relaxation time of τm = 5.2 × 102 s at 129 °C.22 This indicates that the order of magnitude of the estimated relaxation time is in good agreement with the values published in literature. The finally obtained Deborah numbers for the investigated systems at 30, 50, and 70 °C are graphically shown in Figure 8
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CONCLUSIONS The diffusion of toluene in polystyrene was investigated at different concentrations and temperatures in the vicinity of the glass-transition of the mixture. For this system, it was found that at lower temperatures, non-Fickian diffusion extends far more into the region above the calculated glass-transition of the mixture than at high temperatures. Here a determination of the diffusion coefficient from a single sorption curve alone can lead to wrong values although the sorption curve looks Fickian. A more thorough test at different polymer−film thicknesses and subsequent analyses of the reduced sorption curves and/or of the derived diffusion coefficients reveal the significance of the polymer relaxation for the diffusion process in the polystyrene/ toluene system. The Deborah number concept was applied using a relaxation time of the polystyrene/toluene mixture which was estimated using relaxation data measured in exactly the same system using a completely novel experimental setup which has recently been
Figure 8. Calculated diffusion Deborah numbers for polystyrene/ toluene mixtures at 30, 50, and 70 °C at different toluene concentrations. The hatching indicates a Deborah-number range between 0.1 and 10, which should indicate the region of non-Fickian diffusion.5
and summarized in Table 1 (Supporting Information). According to Vrentas et al.5 the value of the calculated Deborah number qualitatively indicates whether a diffusion is purely Fickian or not. They proposed that a Deborah number between approximately 0.1 and 10 indicates non-Fickian diffusion which means that the characteristic relaxation time τm of the mixture and the mutual diffusion time θD are in the same magnitude. The predicted diffusion regimes (Fickian or non-Fickian) are compared with the experimental observations for each individual sorption step in Table 1 (Supporting Information). It can be seen in Figure 8 that all calculated Deborah numbers decrease with increasing toluene load. This is due to 931
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published.8 The results show qualitatively the same trend but deviate by about 1 order of magnitude. The reason for this might be caused by the uncertainties of determination of the relaxation time.
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ASSOCIATED CONTENT
S Supporting Information *
Table of experimental sorption data and calculated Fickian diffusion coefficients. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION Corresponding Author *E-mail:
[email protected]. Present Address † Evonik RohMax Additives GmbH, Innovation Management Oil Additives, Kirschenallee, 64293 Darmstadt, Germany, E-mail:
[email protected] ■ ■
ACKNOWLEDGMENTS The authors are grateful to the Deutsche Forschungsgemeinschaft for supporting this work with Grant SAD 700/13. REFERENCES
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