General Diffusion Model for Polymeric Systems Based on Microscopic

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General Diffusion Model for Polymeric Systems Based on Microscopic Molecular Collisions and Random Walk Movement Hidenori Ohashi and Takeo Yamaguchi* Chemical Resources Laboratory, Tokyo Institute of Technology, R1-17, 4259 Nagatsuta-cho, Midori-ku, Yokohama-city, Kanagawa 226-8503, Japan S Supporting Information *

ABSTRACT: Molecular diffusivity in polymeric systems is often the determining factor in performance and efficiency of devices and systems and thus is one of the most important dynamic properties. In polymeric systems, various penetrants, including gas, solvent, and solute molecules can diffuse, and therefore, a general model that can broadly provide their molecular diffusivity could strongly accelerate the designing of polymeric devices. On the other hand, from the microscopic viewpoint, the molecular diffusive motion can be regarded as a random walk movement subject to an enormous number of molecular collisions with neighboring molecules. Therefore, in the present study, both of the microscopic descriptions common for all penetrants in polymeric systems have been unified into one model. The resultant model is the first general diffusion model that can describe diffusivity of all penetrants, including gas, solvent, and solute molecules in rubbery polymeric systems. The applicability of the microscopic model is demonstrated to be acceptable, except in pure polymer near its glass transition temperature.

1. INTRODUCTION Diffusivity in polymers is one of the most important dynamic properties, determining performance and efficiency of most polymeric devices and systems. In polymeric systems, various types of penetrant molecules can diffuse; for instance, gas molecular diffusivity is a crucial property for gas separation membranes and barrier membranes,1−3 solvent molecular diffusivity is important for cast-drying coating processes,4−8 and solute molecular diffusivity affects drug delivery systems and diffusion-controlled polymerization,9−13 and so on. As described, possible penetrants in polymeric systems are wideranging from gas and solvent to solute molecules (here, gas, solvent, and solute represent materials in gaseous, liquid, and solid states around ambient temperature, respectively), and the design of polymeric systems would be accelerated14 if there was a facile and general methodology to predict molecular diffusive properties of such various penetrants in polymeric systems. Among others, a theoretical model has the possibility to offer such a general methodology, and several descriptions of friction,15,16 obstruction,17,18 molecular,19,20 and free volume21,22 concepts have been incorporated into theoretical models.23−25 Nevertheless, such a general model has not been achieved until now, and that is the incentive for this work. In the present study, different physical descriptions of molecular diffusion at a molecular level have been taken. In a microscopic sense, molecular diffusive phenomenon is a molecular random walk movement produced by an enormous number of molecular collisions with adjacent molecules. The description is common in all types of fluid systems of gas and liquid, including complex polymeric systems. To achieve a comprehensive model, the microscopic descriptions of both molecular random walk movement and molecular collisions have been unified into the model in this study. While there might be several ways to assemble the microscopic notions in the model, one promising © 2013 American Chemical Society

approach has been taken in the present study: the notion of random walk movement has been integrated into the previously proposed unique diffusion model based on molecular collisions.26,27 The collision-based diffusion model has been developed by considering the free space around a molecule produced by the enormous number of molecular collisions with adjacent molecules. The model has been applied to solvent diffusivity in polymeric systems, and the applicability has been found to be acceptable. The utilization of the collision-based model is favorable in that the predictive ability of the model can be utilized. The current unification approach enables the model to describe diffusive features of all types of penetrants, including gas, solvent, and solute in polymers, and therefore to be the first general diffusion model for rubbery polymeric systems. The applicability of the microscopic model unifying both the notions of molecular collisions and molecular random walk movement has been demonstrated by applying the model to some experimental molecular diffusivities in polymeric systems.

2. THEORY 2.1. Derivation of the Microscopic Model Based on Molecular Collisions. Although the development of the model based on the microscopic notions of molecular collisions has already been detailed,26 the description is summarized here to facilitate further model development. From a microscopic point of view, an enormous number of molecular collisions with adjacent molecules drives molecular diffusive motion. The average picture of a huge number of molecular collisions over time can be approached by statistical arguments. Received: Revised: Accepted: Published: 9940

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sufficient free volume opens up next to a penetrant molecule.28,29 Therefore, when there is sufficient free volume, a penetrant molecule always has enough space for the next jump, and it can move to the next free volume hole without stopping. As a result of the sequence of the motions, the penetrant molecule carries out random walk movement without

Molecular collisions are influenced by the penetrant molecular surface area and the distance between collisions. First, the molecular surface area is the place where molecular collisions with neighboring molecules occur. When the molecular surface area is large, the collision frequency on the penetrant molecule affecting the fluctuation force is large. This results in larger molecular diffusivity. On the other hand, when the distance between the penetrant molecule and the adjacent molecule is large, the mean distance between collisions is large, which also results in larger molecular diffusivity. The free space around a penetrant molecule as the product of the molecular surface area and distance between molecules can be regarded as representative of molecular collisions and was defined as the shell-like free volume (Figure 1).26 In light of the

Figure 2. Schematic illustration of molecular random walk movement to the next free volume hole.

stagnation, as shown in Figure 2. The diffusivity of the random walk movement, Ds,RW, can be described as Figure 1. Schematic illustration of molecular diffusive motion arising from molecular collisions and shell-like free volume as a product of molecular surface area and distance between molecules.

lim Ds ,1 = Ds ,RW =

Vf →∞

⎛ 8RT ⎞1/2 u=⎜ ⎟ ⎝ πM1 ⎠

(1)

πd 3 = v1* 6

(2)

where F1,i/γ and F2,i are the free volume parameters of component i having units of [cm3/(g K)] and [K], respectively. It should be noted that F1,i/γ and F2,i can be interpreted as representing parameters of molecular collisions. For simplicity, penetrant−polymer binary systems are assumed in the present study, and components 1 and 2 are assigned to penetrant molecule and polymer, respectively. For the extension of the model to a multicomponent system, see the previous paper.27 2.2. Integration of Microscopic Molecular Random Walk Movement. When the free volume is large enough (Vf → ∞), eq 1 can be transformed to the following form: D0,1 = lim Ds ,1 Vf →∞

(5)

where R represents the gas constant [J/(mol K)]. On the other hand, d is regarded as the penetrant molecular dimension because the adjacent free volume hole should be in contact with the penetrant molecule, and the distance to the next free volume hole should be equal to the penetrant molecular diameter. d can be calculated with the following formula, assuming that the molecule is spherical:

where ωi, Mi, si, and NA are the weight fraction [g/g], molar mass [g/mol], and molecular surface area [Å2/molecule] of component i, and the Avogadro constant [molecules/mol]. v*1 and D0,1 express molecular volume [Å3/molecule] and preexponential factor [cm2/s] of component 1. Vf,i/γ is the specific free volume [cm3/g] inherent to component i, which is a function of temperature T [K] as Vf , i /γ = (F1, i /γ )(F2, i + T )

(4)

where d, τ, and u represent each jumping distance [m], time required for each jump [s], and velocity of the molecule [m/s], respectively. In the present study, u is regarded as the mean molecular speed and is calculated from the Maxwell distribution of molecular speeds:

concept of the shell-like free volume, the traditional free volume theory28,29 can be reinterpreted as a molecular collision-based diffusion theory, and the following equation describing the penetrant self-diffusivity Ds,1 can be obtained: ⎡ N v * ∑ ω (s / M ) ⎤ i i i i ⎥ Ds ,1 = D0,1 exp⎢ − A 1 ⎢⎣ s1 ∑i ωi(Vf , i /γ ) ⎥⎦

d2 ud = 6τ 6

(6)

Therefore, the preexponential factor of the penetrant molecule can be obtained by combining eqs 3 to 6: D0,1 =

1/3 1/2 1 ⎛ 8RT ⎞ ⎛ 6v1* ⎞ ⎟ ⎟ ⎜ ⎜ 6 ⎝ πM1 ⎠ ⎝ π ⎠

(7)

and can be calculated using only pure component parameters of molecular volume, v1*. This equation is similar to the diffusion coefficient in kinetic theory of gases; however, it is different in that mean free path λ is replaced to molecular diameter d. The preexponential factor calculated by the random walk notion is expressed as D0,calc hereafter. The preexponential factor D0,calc is substituted in eq 1, resulting in the following equation:

(3)

This description means that the preexponential factor of the model can be interpreted as the molecular diffusivity when the amount of free volume is large. The physical image of molecular diffusion in the original free volume theory is that the molecule can move to the next free volume hole only when a hole with

1 ⎛ 8RT ⎞ Ds ,1 = ⎜ ⎟ 6 ⎝ πM1 ⎠

1/2

⎡ N v * ∑ ω (s / M ) ⎤ ⎛ 6v * ⎞1/3 i i i i ⎥ ⎜ 1 ⎟ exp⎢ − A 1 ⎢⎣ s1 ∑i ωi(Vf , i /γ ) ⎥⎦ ⎝ π ⎠ (8)

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solvent molecules relatively close to spherical: benzene, ethylbenzene, chloroform, and carbon tetrachloride. D0,calc was calculated at 25 °C, where all of the solvents are in the liquid state. Considering the fact that D0,exp and D0,calc are temperature independent and dependent, respectively, and the imperfectly spherical shape of these molecules, the agreement between the two values seems to be very acceptable, indicating the validity of the physical description of the preexponential factor with molecular random walk movement between free volume holes. It should also be noted that the above methodology to obtain D0,exp, the comparison between the microscopic model and experimental diffusivity, cannot be utilized to obtain the preexponential factor for gas or solute molecules. This is because the applicable scope of free volume model is inherently limited to the liquid-like condensed phase in which homogeneous free volume distribution can be postulated, and it does not apply to gaseous or solid phases. This means that the microscopic model based only on molecular collisions cannot calculate gas and solute molecular diffusivities in polymeric systems. However, eq 7, which calculates the preexponential factor for molecular random walk movement, is applicable to all penetrants, including gas, solvent, and solute molecules, making the microscopic model a general model. 2.4. Further Simplification of the Microscopic Model in the Infinite Dilution Limit. Considering gas diffusion in a polymer or the situation where a small amount of solute molecule diffuses in a polymeric system, the model equation can be further simplified because it is fundamentally an infinite dilution condition for penetrant, different from most polymer− solvent systems in the finite dilution state. With the substitution of ω1 = 0, eq 8 can be rewritten in the following form:

The above notion of the preexponential factor should also be common for all types of penetrants, including gas, solvent, and solute molecules; the parameters in the microscopic model eq 8 are molecular volume, molecular surface area, and free volume parameters and are all physically known. In other words, the integration of molecular collisions and molecular random walk movement, which is the microscopically correct description of the molecular diffusive phenomenon, enables the model to exclude any physically unknown parameter and predict the diffusivity of all the penetrants, including gas, solvent, and solute in polymeric systems, for the first time. Though the equation of random walk movement in eq 7 seems to be a simple one derived by Cohen−Turnbull original free volume theory, the combination of the random walk movement and the diffusion model for rubbery polymeric systems has never been achieved until now. This is maybe because it is hard to find the association of random walk notion with other concepts like friction or obstruction, which appeared in the previous models. However, in the present study, the consideration in the microscopic molecular collisions leads the resultant microscopic event of random walk movement. That is why the present conception is novel enough. 2.3. Comparison with the Existing Method to Obtain the Preexponential Factor for the Solvent. If the penetrant is a solvent molecule, the preexponential factor can be derived from the pure solvent viscosity simultaneously with the solvent free volume by the existing method30 as below. First, the solvent viscosity ηsolv and solvent self-diffusivity Ds,solv are connected by Dullien’s equation:31 Ds ,solv =

0.124 × 10−16VC2/3 ,solvRT ηsolv Msolv Vsolv

(9)

1 ⎛ 8RT ⎞ Ds ,1 = ⎜ ⎟ 6 ⎝ πM1 ⎠

1/2

where VC,solv is the critical molar volume of the solvent and Vsolv and ηsolv represent the temperature-dependent specific volume [cm3/g] and viscosity [g/(cm s)] of the solvent, respectively. On the other hand, the microscopic free volume theory eq 1 can be applied to the pure solvent, by simply assuming that ωsolv = 1: ⎡ ⎤ * V solv ⎥, Ds ,solv = D0,solv exp⎢ − ⎢⎣ (Vf ,solv /γ ) ⎥⎦

* = V solv

⎡ N v * (s ⎤ ⎛ 6v * ⎞1/3 2,unit / M 2,unit) ⎥ ⎜ 1 ⎟ exp⎢ − A 1 ⎢⎣ s1 (Vf ,2/γ ) ⎥⎦ ⎝ π ⎠ (11)

In other words, in the penetrant infinite dilution condition, a penetrant molecule collides only with the matrix molecules, and therefore, molecular collisions between two or more penetrant molecules can be ignored in practice, as schematically shown in Figure 3.

* NAvsolv Msolv (10)

which relates the solvent self-diffusivity, Ds,solv, pure component parameters of free volume, Vf,solv/γ, and preexponential factor, D0,solv. Therefore, the preexponential factor of the solvent molecule can be calculated from pure solvent viscosity by combining eqs 9 and 10. The preexponential factor calculated from the experimental solvent viscosity by this method is expressed as D0,exp hereafter. In the case of solvent, D0,calc based on the molecular random walk movement can be compared with D0,exp of the existing method. Table 1 shows a comparison of the two values for

Figure 3. Schematic illustration of molecular diffusive motion in the penetrant infinite dilution condition corresponding to eq 11. The penetrant molecule collides only with polymer and not with another penetrant.

Table 1. Comparison between Experimental and Calculated Preexponential Factors for Solvent Molecules

benzene ethylbenzene chloroform carbon tetrachloride

D0,exp [cm2/s]

D0,calc [cm2/s]

4.47 1.54 4.07 2.52

2.65 2.57 1.95 1.81

2.5. Fundamental Limitation of the Microscopic Model. Although the model has broad applicability in nature, it has fundamental limitation as follows. A first limitation regards applicable temperature range of the model. As other free volume theory, the applicability of the current model is limited to liquid-like condensed system in which homogeneous 9942

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distribution of free volume can be postulated. Moreover, the free volume parameters of polymer must be derived from the viscoelastic properties at temperature higher than the glass transition temperature (Tg). Therefore, the model and free volume parameters should be used above polymer Tg. The second limitation is that the present model considers molecular collisions and random walk movement as the origin of molecular diffusive phenomena and does not include other factors affecting diffusion. Therefore, systems containing strong forces, such as hydrogen bonding and electrostatic interaction, are beyond the scope of this model. Among them, the effect of hydrogen-bond formation between penetrant and polymer on diffusivity can be successfully incorporated into the model, and the extension will be discussed in the future. 2.6. Demonstration of the Applicability of the Microscopic Model. To demonstrate the applicability of the microscopic model, some experimental diffusivities of model compounds in polymeric systems above its Tg have been calculated using the model. In the penetrant infinite dilution condition, the penetrant molecule is in general too dilute to detect, and it seems difficult to measure penetrant diffusivity in such a condition. However, a membrane permeation experiment can be used to measure gas molecular diffusivity in polymer because gas pressure can be utilized for the detection, and penetrant self-diffusivity is equal to its mutual diffusivity in the penetrant infinite dilution condition. In the present study, diffusivities of gas molecules such as hydrogen, nitrogen, oxygen, carbon dioxide, and ethylene in polyisobutylene, polystyrene, and cis-polybutadiene32,33 were determined from a membrane permeation experiment. On the other hand, the self-diffusivity of 1,3-dimethyladamantane (DMA) in cispolybutadiene in DMA finite dilution condition by PFGNMR34 was extrapolated to acquire the value in the DMA infinite dilution condition. DMA is a nearly spherical molecule and its preexponential factor is still unknown, and thus, it is chosen to show the model applicability to such a penetrant. The parameters required for the calculation using eq 11 are the pure component parameters of molecular surface area s1 and s2,unit of penetrant and polymeric monomer unit, polymer free volume Vf,2/γ, and molecular volume v1* of the penetrant molecule. The method for calculating the parameters is not in the scope of the present study, so it is summarized in the Supporting Information.

Figure 4. Experimental and calculated gas molecular diffusivity in polyisobutylene. The symbols and lines represent experimental and calculated values, respectively. The calculation was carried out using eq 11, which includes both of the microscopic ideas of molecular collisions and molecular random walk motion.

Figure 5. Experimental and calculated gas molecular diffusivity in polystyrene. The symbols and lines have the same meaning as those in Figure 4.

3. RESULTS AND DISCUSSION Equation 11 of the microscopic model was applied to the experimental data above, and the results are shown in Figures 4−6. For all figures, the same range of abscissa and ordinate was used purposefully to clarify the difference in polymeric matrices. Consideration of these figures shows that the model can grasp the whole picture of temperature dependence, dependence on gas species, and dependence on polymeric matrices of gas diffusivity in polymers. Considering that the diffusivity data and the parameters for the calculation were taken for different grades of polymers, it can be said that the model successfully captures the general features of gas diffusivity in rubbery polymers, which is an important dynamic property for designing gas separation membranes and barrier membranes. Further, to show the model applicability to penetrant molecules whose preexponential factor is unknown, the selfdiffusivity of 1,3-dimethyladamantane in polybutadiene is shown in Figure 6 as well as that of gas molecules. DMA is

Figure 6. Experimental and calculated gas and 1,3-dimethyladamantane diffusivity in cis-polybutadiene. The symbols and lines have the same meaning as those in Figure 4.

80 times heavier and 15 times volumetrically bulkier than hydrogen, but the model can express well the diffusivity from DMA to hydrogen. This suggests that the model also has the ability to describe solute diffusivity in polymeric systems. Although the applicability of the microscopic model to solvent molecular diffusion in polymeric systems was not investigated in the present study, it has already been proven in an indirect manner. For solvent molecular diffusion in polymeric systems, the applicability of the molecular collision9943

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Author Contributions

based model using the existing D0,exp has been confirmed as acceptable in previous studies.26,27 Furthermore, in Table 1, it has also been ascertained that the existing D0,exp for solvent does not differ greatly from D0,calc based on the random walk movement between free volume holes, which proves the applicability of the model to solvent penetrant. It should be noted that the calculated self-diffusivity in polystyrene seems to be deviated when the temperature approaches Tg in Figure 5. At such temperature, free volume amount of polymer becomes small and deviation should be amplified by exponential term of the eq 11. This seems to be a possible practical limitation of the model and care must be taken when using the microscopic model. Nevertheless, in polymer−solvent systems, the model can fortunately express well the solvent self-diffusivity even near glass transition temperature26,27 because the free volume amount of solvent is dominant in that situation. The above results demonstrate the applicability of the microscopic model to diffusivity of gas, solvent, and solute molecules in rubbery polymeric systems, except in pure polymer near Tg. It should be noted that the predictive ability of the model is certainly important; however, a more important point of the present study is that the integration of the two microscopic ideas of molecular collisions and random walk movement is of significance to the development of the general diffusion model.

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. All authors contributed equally. Notes

The authors declare no competing financial interest.

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4. CONCLUSIONS On the basis of the principle of molecular diffusive motion, which is the random walk movement resulting from many molecular collisions with neighboring molecules, a model integrating both of the microscopic ideas has been developed. The preexponential factor D0,1 of the collision-based diffusion model is described as a random walk movement between free volume holes, by considering a limiting state of the model. The new methodology to obtain D0,1 is applicable to gas, solvent, and solute molecules, while the existing method can be used only for solvent penetrant. The integration of the microscopic aspects of molecular diffusion, molecular collisions, and molecular random walk movement makes the model the first general model that includes only physically known parameters and therefore can calculate self-diffusivity of all penetrants, including gas, solvent, and solute in rubbery polymeric systems except in pure polymer near Tg. Further investigation of the model applicability is still required, it is demonstrated that the model can express well the dependence of penetrant diffusivity in polymeric systems on penetrant, polymer, and temperature. The above versatile feature strongly suggests that the model will pave the way for the design in a wide range of polymeric systems.

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ASSOCIATED CONTENT

S Supporting Information *

Calculation of parameters required for predictive calculations and comparison between molecular volumes calculated by DFT calculations and semiempirical quantum chemical calculations. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

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*(T.Y.) E-mail: [email protected]. Phone: (+81)45-9245254. Fax: (+81)45-924-5253. 9944

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