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Ind. Eng. Chem. Res. 2010, 49, 11676–11681
Prediction of Self-Diffusivity in Multicomponent Polymeric Systems Using Shell-Like Free Volume Theory Hidenori Ohashi, Taichi Ito, and Takeo Yamaguchi* Chemical Resources Laboratory, Tokyo Institute of Technology, R1-17, 4259 Nagatsuta-cho, Midori-ku, Yokohama-city, Kanagawa 226-8503, Japan
Shell-like free volume theory for multicomponent polymeric systems was developed. Molecular-diffusive motion in polymeric systems is due to a large number of microscopic molecular collisions with adjacent molecules. To embody this idea, a shell-like free volume is defined as the free space surrounding the penetrant molecule that emerges from molecular collisions. The common description of molecular collisions allows the shell-like free volume theory to include multicomponent systems. An advantage of the theory is it can estimate molecular self-diffusivity in multicomponent polymeric systems using only pure component parameters of the constituents and no arbitrary parameter or empirical equation. The required parameters can be derived from viscoelastic properties and quantum chemical calculations. Moreover, by comparing experimental and theoretical values, we find that, insofar as no strong interaction is included, this model can fairly predict penetrant self-diffusivities in several multicomponent systems with a wide range of compositions and different types of molecules, even though the model contains no adjustable parameters. 1. Introduction Molecular diffusivity in polymeric systems is a very important dynamic physical property that is exploited in wide-ranging applications including separation membranes1-3 cast-drying coating processes,4,5 diffusion controlled polymerization,6,7 drug delivery systems,8,9 and so on, because molecular transport significantly affects the performance and efficiency of the polymeric systems. In addition to diffusion in simple binary systems, which is useful for understanding the fundamental phenomenon, diffusion in complex multicomponent systems is also highly important from the viewpoint of the applications. For example, in the cast-drying polymer coating process, existence of a second solvent greatly enhances the evaporation of the first solvent from a polymer film and increases the productivity.10,11 Similarly, pervaporation flux can be controlled if a cosolvent exists.12,13 On a drug delivery system, diffusive release of the drug from a polymer matrix can be controlled with the existence of a plasticizer or solvent.8 In addition, solute diffusion in polymer solution is definitely a ternary or higher order system and simultaneously contains polymer, solvent, and solute. Therefore, information on molecular diffusivity in multicomponent polymeric systems is very important, and extensive experimental attempts have been made to comprehend the feature,14-32 generating a considerable amount of data on molecular self-diffusivities in systems containing ternary polymers. The methods used to obtain this data include pulsed fieldgradient spin-echo nuclear magnetic resonance,14-16 fluorescence recovery after photobleaching,17,18 forced Rayleigh scattering,19-23 fluorescence correlation spectroscopy,24-26 Taylor dispersion,27,28 and inversed column chromatography.29-32 Various ternary systems are available, such as first and second solvent diffusion in a polymer-solvent-solvent system and solute diffusion in polymer solution. In addition to the experimental work, theoretical research on the expression of molecular diffusivity in polymeric systems has been a priority for material and condition screening because the number of combinations of polymer, solvent, solute (or * To whom correspondence should be addressed. E-mail: yamag@ res.titech.ac.jp. Tel.: 045-924-5254. Fax: 045-924-5253.
second solvent), temperature, and composition is extremely large. The free volume theory is an attempt to explain diffusion in polymeric systems. The interesting feature of the free volume approach is additivity of the free volume and the possibility of including each component characteristic within each incremental element of the free volume.33,34 This feature allows the model to treat multicomponent systems using the free volume or other parameters extracted from single or binary system experimental data. Hence, the usage of the free volume approach has the possibility of greatly reducing the experimental or simulative burden to obtain diffusivity in the system. By applying the free volume theory, we have recently developed a novel model referred to as the “shell-like free volume theory”.35 Generally, molecular diffusive motion in polymeric systems is due to an enormous number of microscopic molecular collisions between neighboring molecules.36,37 To embody this notion, a shell-like free volume is defined as the free space surrounding the penetrant molecule that emerges from the molecular collisions. With such a free volume redefinition based on the molecular surface, the unique model, which can eliminate any arbitrary or empirical parameters and can estimate self-diffusivity of solvent molecular diffusivity using only singlecomponent parameters of the constituents, was achieved. All of these parameters can be derived using the viscoelastic properties and quantum chemical calculations of each component. Although the scope of the model has been limited to binary systems, the features of the model appear to be highly suitable for material and condition screening, especially in more complex multicomponent systems, which is in contrast to most other multicomponent models that require adjustable parameters or empirical equations to correlate or estimate experimental diffusivity.14,16,23,26 Because the microscopic concept of molecular collisions that is included in the model is a common concept for any polymeric system, the shell-like free volume theory is extended in the present study to include more complex and application-relevant multicomponent systems. Moreover, to verify the extended theory, we compare its predictions with the experimental data.
10.1021/ie101299q 2010 American Chemical Society Published on Web 10/12/2010
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Figure 1. Schematic illustration of the shell-like free volume. The shelllike free volume is the free space around penetrant molecule originated by molecular collisions with surrounding molecules.
Figure 2. Illustration of the shell-like free volume in a multicomponent polymeric system. The penetrant molecular surface area is spen., and the molecule contacts component i over a surface area fraction σi (i ) 1, 2, 3, respectively). The free volume thickness δi is assigned by the component i.
2. Theory On a microscopic level in polymeric systems, molecules can diffuse to their next positions by colliding with surrounding molecules. Thus, the free space around a diffusing molecule is available for molecular motion, and a shell-like free volume can be defined as the free space surrounding a penetrant molecule,35 as shown in Figure 1. This volume can be expressed as the product of molecular surface area and free volume thickness. An increase in the free volume thickness indicates an elongation of the mean free path, and an increase in the molecular surface area results in a larger collision frequency and weaker fluctuation force on penetrant. Thus, the notion of the shell-like free volume, which is the product of molecular surface area and free volume thickness, is consistent with the microscopic concept of molecular collisions. Detailed discussion and justification of the shell-like free volume was explained in a previous paper.35 The shell-like free volume was adopted as the free volume per molecule in the fundamental equation of the Cohen-Turnbull free volume theory.38,39 It expresses the self-diffusivity Ds,pen of a penetrant molecule as
(
Ds,pen. ) D0,pen. exp -
Vˆ*pen. Vf,SLFV,pen.
)
(1)
where D0,pen. is the preexponential factor (cm2/s), Vˆ*pen. is the molecular core volume (Å3/molecule), and Vf,SLFV,pen. is the shelllike free volume (Å3/molecule) of penetrant molecule. As mentioned above, the shell-like free volume Vf,SLFV,pen. can be expressed as the product of molecular surface area and free volume thickness and is calculated according to the procedure described below. To comprehensively explain the calculation, we assume a ternary system for the derivation. However, the procedure can easily be applied to higher order systems, although we find few examples in the literature. In a ternary system, the suffix i can take the value 1, 2, or 3 to indicate one of the constituents such as solute, solvent, or polymer. Because a penetrant molecule makes contact with other molecules at its surface, the free volume thickness of each component δi (Å) is distributed to the penetrant molecule in proportion to the surface area fraction σi (Å2/Å2) of component i, as shown in Figure 2. Thus, the shelllike free volume for the penetrant can be expressed as Vf,SLFV,pen. ) spen.(σ1δ1 + σ2δ2 + σ3δ3)
(2)
where spen. (Å2/molecule) is the molecular surface area of penetrant molecule. The free volume thickness δi of component i can be calculated by dividing the free volume by the molecular surface area of component i with the adjustment of unit:
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δi )
(Vf,i /γ) NA(si /Mi)
(3)
where si, Mi, and Vf,i/γ express the molecular surface area (Å2/ molecule), molecular weight (g/mol), and free volume (cm3/g) of component i, respectively, being an inherent property of component i. The symbol NA denotes Avogadro’s number (molecules/mol). The surface area fraction σi of component i can be defined as the ratio of molecular surface area summed over component i to molecular surface area summed over all molecules included in the system. Explicitly, it is expressed as σi )
ωi(si /Mi) ω1(s1 /M1) + ω2(s2 /M2) + ω3(s3 /M3)
(4)
where ωi is the weight fraction (g/g) of component i. From eqs 2 to 4, the shell-like free volume Vf,SLFV,pen. is
[
Vf,SLFV,pen. ) spen.
]
ω1(Vf,1 /γ) + ω2(Vf,2 /γ) + ω3(Vf,3 /γ) · NA{ω1(s1 /M1) + ω2(s2 /M2) + ω3(s3 /M3)} (5)
By combining eqs 1 and 5, we arrive at the equation of multicomponent shell-like free volume theory
[
Ds,pen. ) D0,pen. exp -
Vˆ*pen. spen. /Mpen.
∑ ω (s /M ) i
i
i
i
∑ ω (V i
i
f,i /γ)
]
(6)
where Vˆ*pen. is the molecular core volume (cm3/g) of the penetrant and Vˆ*pen. ) NAVˆ*pen./Mpen.. Using the multicomponent shell-like free volume theory as expressed in (6), the self-diffusivity of a penetrant molecule in a ternary system can be predicted. 3. Description of Model In (6), the parameters required for predictive calculation of diffusivity are the molecular surface area si, free volume Vf,i/γ, and molecular volume Vˆ*, i of each component included in the system, and the preexponential factor D0,pen. of penetrant molecule. Free volume is defined as the free space in the system, which results in the molecular dynamic motion. Dynamic viscoelastic properties such as viscosity of solvent and storage elastic modulus of polymer can be interpreted within the free volume formalism. Hence, free volume parameters can be derived from the dynamic properties of the individual components and are tabulated in the literature.33,40,41 Using the free volume parameters K1i/γ [cm3/(g · K)] andK2i - Tgi (K), which are calculated
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by Hong et al., the free volume inherent to component i can be expressed as Vf,i /γ ) (K1i /γ)(K2i - Tgi + T)
(7)
or by using the free volume parameters at a given temperature, Vˆ0i (cm3/g) and (fi/γ) (dimensionless), it can be expressed as42 Vf,i /γ ) Vˆ0i (fi /γ)
Table 1. Single-Component Parameters Used in the Calculation of Penetrant Self-Diffusivities in Ternary Systems Using Multicomponent Shell-like Free Volume Theory K1i/γ K2i - Tgi Vˆ*i Mi [cm3/(g · K)] (K) (cm3/g) (g/mol) TIPBa 1.72 × 10-3 toluene 2.21 × 10-3 polystyrene 5.82 × 10-4
204.4 1.28 × 10-4 92.1 1.87 × 10-4 104.2*2
0.964 0.917 0.850
Vˆ0i Vˆ*i fi/γ Mi [cm3/(g · K)] (K) (cm3/g) (g/mol)
(8)
The microscopic molecular surface is also defined as the isoelectric surface with an electron density of 0.0020 au43 (1 au is defined as 6.748 e/Å3), and the definition is generally accepted.44 Thus, the molecular surface area can be calculated from the molecular structure using the PM3 method, which is a semiempirical, quantum chemical calculation.45 The molecular volume can be defined as the molar volume at 0 K and is calculated using the group contribution method compiled by Haward46 from its molecular structure. The last parameter D0,pen. can be simultaneously deduced from the singlecomponent viscoelasticity with its free volume parameters. The quantities D0,pen. and Vˆ*i for various solvents have been tabulated in previous research.33,40,41 By using these parameters in (6), the penetrant molecular diffusivities in multicomponent polymeric systems can be predicted. Although there are many published reports containing data on solute self-diffusivity in polymer solution,14-32 there are still some limitations regarding the estimation of diffusivity based on the shell-like free volume theory. The first limitation regards the assumption on which the model is based. The shell-like free volume model considers molecular collisions as the origin of molecular diffusive phenomenon and does not include other factors affecting diffusion. Thus, systems containing strong forces such as hydrogen bonding and electrostatic interaction are beyond the scope of this model, although retardation effect by hydrogen bonding formation between penetrant and polymer has been successfully introduced into the current model. This discussion will appear in the future. A second limitation regards the availability of parameters. Although many research groups have measured the solute diffusivity near room temperature, the free volume parameters of some often-used polymers such as polystyrene and poly(methyl methacrylate) must be derived from the viscoelastic properties at temperatures much higher than room temperature, because these polymers have an inherently high glass transition temperature (Tg). In that case, the free volume parameters determined for conditions that are far from those of the diffusivity measurement do not work well and should not be used for the calculation. Thus, for our demonstrative calculation in this study, we choose two systems that cover a broad composition range and have a temperature above glass transition temperature of their matrix components without strong molecular interaction. The first system is 1,3,5-triisopropylbenzene (TIPB) diffusing in a TIPB-toluene-polystyrene system measured by Zielinski et al.14 above 100 °C using pulsed field gradient nuclear magnetic resonance (PFG-NMR). In addition, Ferguson and von Meerwall15 measured the self-diffusivity of hexafluorobenzene (C6F6) and n-dodecane (C12H26) in the C6F6-C12H26-cispolybutadiene (PBD) system and also that of C6F6 and nhexatriacontane (C36H74) in the C6F6-C36H74-PBD system at various compositions at 80 °C using PFG-NMR. In the present study, we predict the self-diffusivities in these ternary systems using the multicomponent shell-like free volume theory to verify
-161 -103 -327
hexafluorobenzene n-dodecane n-hexatriacontane cis-polybutadiene
0.658 1.41 1.28 1.17
0.199 0.191 0.147 0.135
D0i (cm2/s)
D0i (cm2/s)
0.500 186.1 1.77 × 10-3 1.10 170.3 1.12 × 10-3 1.06 507.0 3.79 × 10-4 0.974 54.1*2
a TIPB: 1,3,5-triisopropylbenzene. unit of polymer.
b
si (Å2/molecule) 287.4 134.0 98.8*2
si (Å2/molecule) 122.8 281.5 747.7 71.8b
The value is for a monomeric
Figure 3. Comparison of experimental and calculated values of the triisopropylbenzene self-diffusion coefficient in a TIPB (1)-toluene (2)-polystyrene (3) system. Symbols represent experimental values, and solid lines represent theoretical values obtained using the shell-like free volume theory extended to include multicomponent systems. Different symbols refer to different system compositions.
its accuracy. All single-component parameters used for the calculations are shown in Table 1. The free volume parameters for C6F6, C12H26, and C36H74 could only be obtained for binary systems, but we find that in the literature the parameters are almost the same as those obtained from the experimental data on single components. Thus, we used these published parameters. If some additional viscoelastic experiment for single components is obtained, the free volume parameter can be used exactly in the same way as done here. 4. Results and Discussion 4.1. Prediction of 1,3,5-Triisopropylbenzene Self-Diffusivity in a TIPB-Toluene-Polystyrene Ternary System. The result of the calculation of self-diffusivity of TIPB in a TIPB-toluene-polystyrene ternary system is shown in Figure 3. The symbols represent the experimental data, and the solid lines represent the theoretical prediction based on the extended shell-like free volume theory. Note that some self-diffusivity in the system without toluene is also predicted to assert the application range of the theory. It is apparent from the figure that the result of the shell-like free volume theory agrees well with the measured self-diffusivity of TIPB in the system. It should be noted that the theoretical result was obtained without using any arbitrary, adjustable parameter. A slight discrepancy between the predicted and experimental values at lower temperature is attributed to the fact that the temperature of 100 °C is near the glass transition temperature of matrix polystyrene and thus near the limit of the validity of the parameter. This ternary system was analyzed in a previous study using a different free volume model.14 To explain the experimental
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Figure 4. Experimental and theoretical values for the hexafluorobenzene (C6F6) self-diffusion coefficient in the C6F6 (1)-n-dodecane (C12H26) (2)-polybutadiene (PBD) (3) system. Symbols represent experimental values, and solid lines represent theoretical values obtained using the shelllike free volume theory extended for multicomponent systems. Different symbols refer to the volume fraction of C12H26 present in the PBD before the addition of C6F6. The measurement temperature was 80 °C for every case shown in this figure.
Figure 5. Experimental and theoretical values for the C12H26 self-diffusion coefficient in the C6F6 (1)-C12H26 (2)-PBD (3) systems. The meaning of symbols and solid lines and the measurement temperature are the same as those for Figure 4.
data, the authors suggested that a TIPB molecule cannot diffuse as a single unit, but instead diffuses as segments. This explanation appears to not coincide with the rigid skeleton of TIPB, which is composed of an aromatic ring and with the directly attached isopropyl groups. Regarding this point, describing the molecular collisions, which do not require discussing the size of the diffusing unit, may be preferable. Because the TIPB-toluene-polystyrene ternary system is the representative of polymeric systems with large oblate solutes and small oblate solvents, we consider the multicomponent shelllike free volume theory to be a valid model for predicting solute diffusion in polymer solutions. 4.2. Self-Diffusivities of C6F6 and C12H26 in the C6F6-C12H26-PBD Ternary System and of C6F6 and C36H74 in the C6F6-C36H74-PBD Ternary System. The self-diffusivity predicted for C6F6 and C12H26 in the C6F6-C12H26-PBD system is shown in Figures 4 and 5, respectively, and that predicted for C6F6 and C36H74 in the C6F6-C36H74-PBD system is shown in Figures 6 and 7, respectively. Again, the symbols represent the experimental data, and the solid lines represent the theoretical values obtained using the multicomponent shelllike free volume theory. Note that the abscissa of the figures is converted to volume fraction using the density data of each component. For the C6F6-C12H26-PBD system, the shell-like free volume theory properly replicates the dependence of the measured self-diffusivity on the C12H26/PBD ratio when this ratio
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Figure 6. Experimental and theoretical values of the C6F6 self-diffusion coefficient in the C6F6 (1)-n-hexatriacontane (C36H74) (2)-PBD (3) system. The meaning of symbols and solid lines are the same as those of Figure 4. The different symbols identify the volume fraction of C36H74 in PBD before C6F6 was added. For every case shown in this figure, the measurement temperature was 80 °C.
Figure 7. Experimental and theoretical values for the C36H74 self-diffusion coefficient in the C6F6 (1)-C36H74 (2)-PBD (3) system. The meaning of symbols and solid lines and the measurement temperature are the same as those of Figure 6.
is below 0.55. However, the model fails to accurately predict the measured result when the C12H26/PBD ratio is 1:0. For the C6F6-C36H74-PBD system, the model accurately replicates the dependence of the measured self-diffusivity on the C36H74/PBD ratio when this ratio is below 0.42, although the predicted values deviate from the measured results when the C36H74/PBD ratio is above this value. Underestimation by the model of self-diffusivity for higher solvent concentrations is a commonly observed characteristic of the free volume theory and has been attributed to the fact that the polymeric laboratory cage is not constructed around the penetrant molecule in the low polymer concentration range. This explanation may also be applicable in the present case because our model is developed on the basis of the original free volume theory. At the same time, the dependence of the self-diffusivity on C6F6 concentration is well-replicated by the model. We can also observe that the predicted self-diffusivity of n-alkane deviates from the measured value, which we attribute to an error in the offset value of the preexponential factor (1). We do not know the reason for this deviation, so further investigation is required to understand this issue. For all of the systems, the discrepancy between predicted and experimental values never exceeds the factor of 2, and the shell-like free volume theory faithfully replicates the difference in magnitude and the tendency of self-diffusivity between C12H26 and C36H74. In particular, it obtains nearly quantitative accordance between experimental and theoretical values for the very long chainlike molecule C36H74 without using any adjustable parameters.
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5. Conclusion
Figure 8. Experimental and theoretical values of the penetrant selfdiffusivities in multicomponent systems using the multicomponent shelllike free volume theory.
The theoretical model used in our previous study35 accurately predicted self-diffusivities of long chain-like molecules in binary systems. These results indicate that the concept of molecular collision is a general one and applies to chain-like molecules as well as oblate molecules. Therefore, the shell-like free volume theory, which takes molecular collisions into account, can successfully predict self-diffusivity in systems that include chainlike molecules of C12H26 and C36H74. The C6F6-C12H26-PBD and C6F6-C36H74-PBD ternary systems are representatives of systems containing oblate molecules, chain-like molecules, and polymer systems and also of solvent-solvent-polymer systems. Thus, we consider that the multicomponent shell-like free volume theory is also valid for predicting the behavior of such systems. In a previous paper,42 the experimental data were interpreted as motion of segments of long chain-like molecules, which seems reasonable as opposed to the TIPB-toluene-polystyrene system, because n-alkane is a flexible penetrant. If we use the ratio of the jumping unit sizes of the constituent molecules (acquired by adjusting experimental self-diffusivities in the C6F6-PBD and paraffin-PBD binary systems), the model can express the ternary self-diffusivities with reasonable accuracy.42 The jumping unit size of a polymer molecule must be obtained by fitting experimental self-diffusivity data in a binary system. In contrast with this approach, the newly derived multicomponent shell-like free volume theory only requires singlecomponent parameters, and this feature is especially attractive for such a complex system. Correlation between experimental and theoretical self-diffusivities in the above ternary systems are summarized in Figure 8. For points close to the diagonal, the estimation is quite accurate. On the basis of Figure 8, we consider that the multicomponent shell-like free volume theory provides relatively accurate predictions for the ternary systems examined in this study, even if these systems are composed of different types of molecules (i.e., oblate, short, and long chain-like molecules and also large as well as small molecules). This result clearly shows that the model may be applied to complicated multicomponent systems comprising molecules of differing shapes. Furthermore, this is the unique model that can predict penetrant self-diffusivity in the polymeric system using only single-component parameters. The combination of these features makes the multicomponent shell-like free volume theory one of the more promising tools for the screening and design of polymeric systems and devices.
The shell-like free volume theory, which includes the concept of microscopic molecular collisions, is successfully extended to multicomponent systems. This extended model can estimate the penetrant self-diffusivities in polymer-containing multicomponent systems using only single-component free volume, molecular surface area, and molecular volume, which can be calculated from the viscoelastic properties, by quantum calculation and by using the group-contribution method, respectively. We used the model to predict the penetrant diffusivities in typical ternary systems and found that the results are acceptably accurate for multicomponent systems with a wide range of composition and comprising a variety of molecules (e.g., large, small, oblate, or chain-like) as far as the system contains no strong molecular interaction. On the basis of these results, we conclude that the multicomponent shell-like free volume theory is an appropriate predictive tool for applications such as various polymeric membrane systems, polymer coating-drying processes, and diffusion in polymer solution. Nomenclature Ds,pen. ) self-diffusivity of penetrant molecule (cm2/s) D0,pen. ) preexponential factor of penetrant molecule (cm2/s) δi ) free volume thickness of component i (Å) fi/γ ) free volume parameter (free volume fraction) of component i K1i/γ ) free volume parameter of component i [cm3/(g · K)] K2i - Tgi ) free volume parameter of component i (K) Mi ) molecular weight of component i (g/mol) NA ) Avogadro’s number (molecules/mol) si ) molecular surface area of component i (Å2/molecule) σi ) surface area fraction of component i (Å2/Å2) T ) temperature (K) Vf,i/γ ) free volume of component i (cm3/g) Vˆ*i ) molecular volume at 0 K of component i (cm3/mol) Vˆi0 ) free volume parameter (molecular volume) of component i (cm3/g) Vf,SLFV,pen. ) shell-like free volume of penetrant molecule (Å3/ molecule) Vˆ*i ) molecular volume of component i (Å3/molecule) ωi ) weight fraction of component i (Å3/molecule) Subscripts i ) any component of penetrant, solvent, or polymer pen. ) penetrant molecule
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ReceiVed for reView June 17, 2010 ReVised manuscript receiVed September 14, 2010 Accepted September 22, 2010 IE101299Q
