ARTICLE pubs.acs.org/IECR
Effects of Rotation of Benzene Rings on Diffusion of Solvents in Polymer Melts Makoto Asai,*,† Megumi Awata,‡,§ and Yasuhiro Koike§,^ †
Faculty of Science and Technology, ‡School of Integrated Design Engineering, Graduate School of Keio University, and ^Keio Photonics Research Institute, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, 223-8522, Japan § Exploratory Research for Advanced Technology and Solution Oriented Research for Science and Technology (ERATO-SORST), Japan Science and Technology Agency (JST), K2 town campus E-tower, 7-1, Shinkawasaki, Saiwai-ku, Kawasaki, Kanagawa, 212-0032, Japan ABSTRACT: Diffusion of a solvent through a polymer is one of the important phenomena in the industry. Such diffusion phenomena, however, have complicated multibody problems, and in general, they are difficult to tightly control. In this study, it was our objective to investigate the effects of the structure of the solvent on the diffusion in polymers, in particular, the effects of the structure on concentration dependence of a mutual diffusion coefficient. Knowledge pertaining to chemical structures and diffusion is very useful for controlling the diffusion phenomena because it allows the design of a solvent possessing the desired diffusion. The diffusion of diphenylsulfide and dibenzothiophene in poly(methyl methacrylate) (PMMA) was investigated. Diphenylsulfide and dibenzothiophene have nearly identical molecular structures and differ in a benzene ring with or without rotation within the molecule. The results indicated that the dependence of the mutual diffusion coefficient was significantly affected by the rotation of the benzene ring present in the molecule.
1. INTRODUCTION Diffusion is one of the phenomena that humankind has been familiar with since ancient times. Ever since its industrial importance was recognized as an elementary process of chemical reactions, countless studies have been conducted. Also, since an equation describing the diffusion phenomenon was known to be expressed as a form identical with that of time evolution of the existing probability of Brownian particles, much basic research in which diffusion is regarded as a probability process has been carried out. Among the research, the diffusion phenomenon of solvents through polymers has complicated multibody problems, and it is considered as one of the difficult physical phenomena to analyze. In particular, since polymers can transform into glass, rubber, and molten states in a low, medium, and high temperature range, respectively, the ratio of a relaxation time for a polymer chain to a characteristic process time for diffusion of a solvent (Deborah number) at certain temperatures and solvent concentrations can be significantly changed, complicating the problem even more.1 On the other hand, in recent years, as a technique for forming a highly advanced light-harvesting structure within polymer materials represented by graded-index polymer optical fibers,2 diffusion control of solvents in a highprecision manner is desired.3-5 The aim of this study was to investigate the effects of chemical structures of solvents on diffusion. Knowledge pertaining to their chemical structures and diffusion becomes extremely useful to controlling the diffusion phenomenon because such knowledge allows us to synthesize a solvent which manifests the desired diffusion. It is empirically known that there is a significant difference in concentration distribution created in the polymer between a solvent with a planar configuration and that with some degrees of freedom in rotation within the molecule, even though r 2011 American Chemical Society
the molecules have nearly identical molecular volumes and chemical structures. This means that there is a significant difference in mutual diffusion coefficients between the molecules. Consequently, poly(methyl methacrylate) (PMMA) was doped with diphenylsulfide (DPS) and dibenzothiophene (DBT) to determine the mutual diffusion coefficient in its molten state, and the differences between DPS and DBT were discussed. Each chemical structure is shown in Figure 1. Although DPS and DBT have nearly identical chemical structures and molecular volume, there is a structural difference between them: DPS has a structure allowing two benzene rings singly bonded with a sulfur atom to rotate, whereas DBT has a fixed planar structure, allowing no rotation of the benzene rings. Meares reported that the activation energy for diffusion of a low molecular weight molecule in a polymer in a rubber state is proportional to the cross-sectional area of collision of the solvent.6 Since it is our interest to study in the molten state, Meares’ report will not be applied. By using this report as a reference, actually, we find that with such a small difference (2.0 Å2; see Figure 1) in the cross-sectional area of collision between DPS and DBT it should be difficult to cause a large difference between their diffusion phenomena.
2. PLASTICIZATION Generally, solvent added to a polymer induces plasticization, resulting in decreased glass transition temperature (Tg) of the polymer. Plasticization affects the diffusivity of solvent within a polymer. Therefore, plasticization of DPS and DBT on PMMA Received: October 13, 2010 Accepted: January 21, 2011 Revised: January 13, 2011 Published: February 15, 2011 3280
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Figure 1. Structures of solvents. We estimated the cross-sectional area of collision as van der Waals square Sv.
Figure 3. Schematic of measurement system for one-dimensional diffusion experiment.
Figure 2. Relation between solvent concentration and Tg. O, PMMA doped with DPS system; 4, PMMA doped with DBT system.
were measured. Mixtures of MMA and DPS or DBT were polymerized in a hot bath at 90 C for 48 h, and then in a dehydrator at 110 C for 24 h. Di-tert-butyl peroxide and laurylmercaptan were used as a free radical initiator and a chain transfer agent, respectively. Each sample was purified by the reprecipitation technique using dichloromethane and methanol as a solvent and as a poor solvent, respectively. The amount of the remaining monomer of each sample was measured by gas chromatographic analysis (GC 2010, Shimadzu Co., Ltd.). It was determined that the amount of the remaining monomer in every bulk produced in this study was 1.0 wt % and below. The Tg of the fabricated bulk was measured by differential scanning calorimetry (DSC, Shimadzu Co., Ltd.). The rate of temperature increase was determined to be 10 C/min. Plasticization of DPS and DBT is shown in Figure 2. In order to compare plasticization of a single solvent on PMMA, the concentration of a solvent, c, was defined as the number of moles per unit volume of the fabricated bulk. The results indicated that there was no significant difference between the plasticization of DPS and and that of DBT on PMMA.
3. ONE-DIMENSIONAL DIFFUSION EXPERIMENTS Various methods to determine mutual diffusion coefficients have been developed to date. Examples are as follows: gas chromatography in which a carrier gas and a solvent are poured into a column coated with a microscopic layer of a polymer, and then the mutual diffusion coefficient is calculated from the obtained elution curve;7,8 the permeation method which measures the changes in the amount of solvent passing through a
polymer film in time;9,10 the absorption method in which a polymer in vapor is placed in a solvent, and the changes in the amount of the solvent absorbed and released in time is measured.11,12 Each method has strengths and weaknesses and cannot be versatile. It is the status quo that various types of improvements are necessary. In this study, it was decided to apply the concentration distribution measurement technique to realize measurement of the mutual diffusion coefficients ranging up to relatively high concentrations. In order to do so, one-dimensional diffusion experiments were devised.13 The diagram of the experimental concept is shown in Figure 3. The experimental procedures are described below. PMMA bulk doped with a solvent (7.0 10-4 mol/cm3) and PMMA homogeneous bulk without a dopant were polymerized to form a column. The polymerization was conducted in a hot bath at 90 C for 48 h, and then in a dehydrator at 110 C for 24 h. Di-tert-butyl peroxide and laurylmercaptan were used as a free radical initiator and a chain transfer agent, respectively. The amounts of di-tert-butyl peroxide and laurylmercaptan added were adjusted for all the bulk concentrations so that the weight-average molecular weight (Mw) would be 5.0 104. After the bulk columns were polymerized, the upper and lower surfaces were polished to level off. They were placed one above the other and fixed by Teflon rods. The bulk surfaces, then, were covered with a heat-shrinkage tube. These samples were placed and fixed in the already heated dehydrator. Then, they were heated to conduct the diffusion experiment. The heating temperatures T were 463.15, 473.15, and 483.15 K. The thickness and diameter of both bulk columns are 7.0 and 2.2 cm, respectively. When a diameter of a bulk column is small, generally the wall effect would affect the concentration profile of solvent. However, when a diameter is too large, it is difficult to polish its surface to perfectly place two bulk columns. If the surfaces of two bulk columns are placed imperfectly, bubbles appear from the surface during the heating process. Therefore, we have investigated the optimal diameter (=2.2 cm) by testing bulk columns with several diameters. After the heating process, disk plates were sliced off the obtained samples with several millimeters of thickness in the axial direction x of the column (the number of plates sliced from one sample is 10), and the refractive index profiles n(x) were measured using the prism coupling refractive index measurement system (Metricon Co., Ltd.). Finally, the concentration distribution of the solvent c(x) was obtained from the relationship between the 3281
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Figure 4. Concentration profiles. 4, ], PMMA doped with DPS system; , þ, PMMA doped with DBT system. (a) DPS, T = 463.15 K; (b) DPS, T = 473.15 K; (c) DPS, T = 483.15 K; (d) DBT, T = 463.15 K; (e) DBT, T = 473.15 K; (f) DBT, T = 483.15 K.
concentration of the solvent and the refractive index profile, which was determined in advance by using the concentrationadjusted bulk sample. Several measured concentration distributions of the solvent are shown in Figure 4. Boltzmann-Matano Method. The following describes how the concentration dependence of mutual diffusion coefficient was calculated from the obtained concentration distribution c(x). In general, a one-dimensional diffusion equation is described as follows: Dc D Dc ¼ Dm ð1Þ Dt Dx Dx Using the following variable transformation introduced by Boltzmann: η xt 1=2
ð2Þ
dc Dm d 2 c ¼ dη 2η dη2
ð3Þ
eq 1 can be converted to
By integrating both sides of eq 3 with respect to η, the mutual diffusion coefficients Dm can be expressed by eq 4.14 The details of the derivation are described in ref 15. Z c ¼ c0 0 1 dx 0 x0 dc ð4Þ Dm ðc Þ ¼ 2t dc 0 c ¼ 0 c¼c
We calculated Dm using eq 4 based on c(x) derived from the onedimensional diffusion experiments.
4. MUTUAL DIFFUSION COEFFICIENTS Before we analyze the results of the concentration dependence of the mutual diffusion coefficients, we should show the validity of measured diffusion coefficients. Therefore, we calculated concentration distributions of the solvent using eq 1 with measured diffusion coefficients Dm and experimental time t. In Figure 6, the calculated distributions were compared with measured ones. From the results, it was confirmed that both concentration distributions were in good agreement and it indicated that measured diffusion coefficients were reasonable. 3282
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Figure 5. c dependence of Dm. 4, ], PMMA doped with DPS system. , þ, PMMA doped with DBT system. The broken lines indicate calculated (approximated) slopes. (a) T = 463.15 K; (b) T = 473.15 K; (c) T = 483.15 (K).
The concentration dependence of the mutual diffusion coefficients for T = 463.15, 473.15, and 483.15 K are shown in Figure 5. Based on the free volume theory,15-17 diffusion coefficient D is expressed as Bd D ¼ RTAd exp ð5Þ f where R is the universal gas constant. T is the temperature, Ad and Bd are concentration-independent and temperature-independent constants, and f is the free volume fraction per unit volume. Generally, f depends on the concentration of solvent. When the concentration c of solvent is small enough, we assume that the dependence can be expressed as follows. f ðcÞ ¼ f0 ðTÞ þ βc
ð6Þ
Here, f0 is the free volume fraction of homopolymer per unit volume and β is the plasticization factor. Considering that the relation between free volume and glass transition is known to be linear, eq 6 means that the relation between Tg and c is linear. As shown in Figure 2, we confirmed that the c dependence of Tg was linear. Therefore, the results indicate the validity of eq 6. By substituting eq 6 into eq 5, the c dependence of D is expressed as follows. 1 f0 ðTÞ f0 2 ðTÞ 1 ¼ þ ln½DðcÞ=D0 Bd βðTÞBd c
ð7Þ
1 f0 2 ðTÞ R βðTÞBd
ð8Þ
D0 is the diffusion coefficient in the infinite dilution, and R is the concentration-dependent coefficient. We analyzed measured
diffusion coefficients by fitting to eq 5. In Figure 5, it was evident that there was a significant difference between the mutual diffusion coefficients, Dm, of DPS and DBT at any given temperature and the difference became more prominent as the concentration of the solvent was greater. The Dm of DPS was significantly greater than that of DBT. Based on the free volume theory, the greater the free volume of a polymer becomes, the more easily the solvent diffuses through the polymer. However, Tg values of both PMMA doped with DPS and that doped with DBT were shown to be approximately equal (Figure 2), and thus, it was unlikely that there was a significant difference in the average free volumes. Therefore, it was attributed to the fact that DPS has two benzene rings within the molecule that could freely rotate. This could be interpreted as follows: DPS can flexibly alter its shape because two benzene rings present within the solvent molecule can freely rotate. Therefore, the Dm of DPS becomes greater than that of DBT because it is easier for DPS to adjust its shape to that of the free volume. We thought that this physical picture was true in the case where a solvent diffused through the polymer which had approximately equal free volume to that of the solvent volume. On the other hand, when the average free volume of the polymer was not sufficiently large compared to the volume of the solvent, there should not be a very significant difference in diffusion between DPS and DBT. This is because two benzene rings present within the solvent molecule could not freely rotate. In fact, there was hardly a significant difference of Dm between DPS and DBT at the lowest temperature case (T = 463.15 K) or in a range of low concentration of solvent. Figure 7 shows the temperature dependence of R. It was apparent that, as the diffusion temperature rose, there were decreases in R in both DPS and DBT. It is known that the diffusion coefficient indicates the following Arrhenius type 3283
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Figure 6. Comparison of concentration profiles between measured and calculated. 4, ], PMMA doped with DPS system; , þ, PMMA doped with DBT system. (a) DPS, T = 463.15 K; (b) DPS, T = 473.15 K; (c) DPS, T = 483.15 K; (d) DBT, T = 463.15 K; (e) DBT, T = 473.15 K; (f) DBT, T = 483.15 K.
temperature dependence in many cases. E DðTÞ ¼ A exp RT
ð9Þ
where A is the frequency factor; E is the activation energy. Our results also indicated the temperature dependence of the Arrhenius type (Figure 8), demonstrating the validity of the mutual diffusion coefficient evaluated for each concentration of the solvent. Thus, the activation energy E was obtained through the slopes, and the results are shown in Figure 9. The activation energies of DPS and DBT at infinite dilution concentration were 160 and 125 kJ/mol, respectively, and the activation energy of DPS was greater than that of DBT. This result could be interpreted as follows: In spite of the fact that the two benzene rings present in DPS are able to intrinsically rotate freely, they were not able to rotate smoothly because the average free volume was not sufficiently large enough in a polymer in an infinite dilution concentration range. As a result, the activation energy required for the
Figure 7. T dependence of R. O, PMMA doped with DPS system; 4, PMMA doped with DBT system.
diffusion of DPS was increased by the energy necessary for the benzene rings to rotate, and thus, the energy for DPS indicated a greater value than that for DBT. Then, as the concentrations of the solvent increased, both the activation 3284
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Figure 8. Arrhenius plots of each additive amount of solvents. O, 0.0 10-4 mol/cm3; 0, 1.0 10-4 mol/cm3; 4, 2.0 10-4 mol/cm3; ], 3.0 10-4 mol/cm3; /, 4.0 10-4 mol/cm3; , 5.0 10-4 mol/cm3; þ, 6.0 10-4 mol/cm3. The values in the figure represent mean square errors. (a) PMMA doped with DPS; (b) PMMA doped with DBT.
energy of DPS and that of DBT decreased. This was attributed to the fact that the average free volume in the polymer increased due to plasticization as shown in Figure 2. The important point here is that DPS is more inclined to decrease in activation energy than DBT is. As shown in Figure 2, PMMA doped with DPS and that doped with DBT had virtually the same degree of increasing tendency in free volume with the concentration of the solvent. Taking that into consideration, it was thought that the difference in the decreasing tendency of activation energy was also mainly attributable to the difference in molecular structures between DPS and DBT. As mentioned above, in an infinite dilution concentration range, the activation energy of DPS was larger than that of DBT because of the necessary energy for the two benzene rings of DPS to rotate. However, as DPS concentration increased, the average size of free volume in polymer increased due to plasticization, and then the required energy for the two benzene rings of DPS to rotate was decreased. Therefore, DPS was more inclined to decrease in activation energy than DBT was.
5. SUMMARY We investigated the effects of the chemical structure of the solvent on the concentration dependence of the mutual diffusion coefficients of solvent in the polymer melt. The results clearly demonstrated that the concentration dependence of the mutual diffusion coefficients was significantly affected by the rotation of the benzene rings present within the solvent. The presence/absence of the rotatable benzene rings within the solvent and the concentration dependence coefficient R should have a major significance, for example, in designing materials for realizing the optimized index profile in light-harvesting fibers. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We thank E. Nihei for fruitful discussions. We also thank Dr. T. Sassa for his technical advice in writing this paper. This work was partially supported by a Grant-in-Aid for JSPS Fellows.
Figure 9. c dependence of E. O, PMMA doped with DPS system; 4, PMMA doped with DBT system.
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’ NOTE ADDED AFTER ASAP PUBLICATION After this paper was published online February 15, 2011, a correction was made to section 2, “Plasticization”, regarding the amount of monomer remaining in every bulk produced in this study. The revised version was published March 9, 2011.
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