Article pubs.acs.org/JPCB
Investigation of the Effect of the Tortuous Pore Structure on Water Diffusion through a Polymer Film Using Lattice Boltzmann Simulations Tobias Gebac̈ k,*,†,§ Mariagrazia Marucci,‡,§ Catherine Boissier,‡ Johan Arnehed,‡ and Alexei Heintz†,§ †
Department of Mathematical Sciences, Chalmers University of Technology and Gothenburg University, SE-412 96 Göteborg, Sweden ‡ AstraZeneca R&D Mölndal, SE-431 83 Mölndal, Sweden § SuMo Biomaterials, VINN Excellence Centre, Chalmers University of Technology, SE-41296 Göteborg, Sweden ABSTRACT: Understanding how the pore structure influences the mass transport through a porous material is important in several applications, not the least in the design of polymer film coatings intended to control drug release. In this study, a polymer film made of ethyl cellulose and hydroxypropyl cellulose was investigated. The 3D structure of the films was first experimentally characterized using confocal laser scanning microscopy data and then mathematically reconstructed for the whole film thickness. Lattice Boltzmann simulations were performed to compute the effective diffusion coefficient of water in the film and the results were compared to experimental data. The local porosities and pore sizes were also analyzed to determine how the properties of the internal film structure affect the water effective diffusion coefficient. The results show that the top part of the film has lower porosity, lower pore size, and lower connectivity, which results in a much lower effective diffusion coefficient in this part, largely determining the diffusion rate through the entire film. Furthermore, the local effective diffusion coefficients were not proportional to the local film porosity, indicating that the results cannot be explained by a single tortuosity factor. In summary, the proposed methodology of combining microscopy data, mass transport simulations, and pore space analysis can give valuable insights on how the film structure affects the mass transport through the film. varying film composition,8 process parameters,6 and the polymers’ functional related characteristics.9 The design process of a new porous coating film can be improved if the mass transport properties of the coating film and the relationship between the film microstructure and the film mass transport properties are properly understood. Computing Deff in porous structures using computer simulations can be done in several ways. One way is to use simulations of particles undergoing Brownian motion.10,11 Another way is to solve the diffusion equation, which is the approach chosen here. This assumes that the diffusing molecules are very small compared to the pores. Studies solving the diffusion equation in porous materials have been performed using the finite element method12 and the finite volume method.13,14 These studies consider computer-generated porous structures simulating real material structures, but real material data has also been used directly in diffusion simulations.15 However, the simulated porous structures can largely differ from the real structure of materials of interest for specific application.
1. INTRODUCTION Polymer films are used as coatings in a number of applications. For example, polymer films are often used in the pharmaceutical field to coat tablets and pellets and optimize the plasma drug concentration by modifying the drug release profile. They are also used in agriculture to coat fertilizers and optimize the fertilizers release profile. In a permeable coating system, the solute release occurs mainly by solute diffusion and the release rate is mainly determined by the effective diffusion coefficient (Deff) of the solute in the coating.1 Water/solute Deff gives a representation of the average mass transport properties of the film and its determination has traditionally been done experimentally performing diffusion experiments on free films or release experiments from coated systems.1−3 For a porous polymer film made of a matrix with a very low permeability toward water and the solute and by interconnected pores, the mass transport properties of the film depends on the film porous microstructure, i.e., pore size and shape, porosity, and pore connectivity.4−7 The microstructure of a porous film can be very complicated and largely different from straight channels.6 A further complication is that polymer films may have a structure that differs across the film cross-section. Films with different structures and mass transport properties can be created by © 2015 American Chemical Society
Received: February 27, 2015 Revised: April 1, 2015 Published: April 2, 2015 5220
DOI: 10.1021/acs.jpcb.5b01953 J. Phys. Chem. B 2015, 119, 5220−5227
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The Journal of Physical Chemistry B An improvement in the field requires the use of experimental determined 3D structures. The lattice Boltzmann method, which was used in this work, is more often used for flow simulations in complex geometries16,17 due to its ease of implementation and parallelization, but can also be used for diffusion simulations.18 Because of the recent development of accurate zero flux boundary conditions for general boundaries,19 the lattice Boltzmann method can be used to solve the diffusion equation with nonpermeable boundaries in very complex 3D geometries. In this work, the dependence of Deff on the microstructure of polymer films made of ethyl cellulose (EC) and hydroxypropyl cellulose (HPC) was investigated using lattice Boltzmann simulations in real film structures. The 3D film structures were obtained using confocal laser scanning microscopy (CLSM). Films made of EC and HPC have often been used as coating to control the drug release profile from oral formulations.20 Although EC and HPC can be codissolved in a common solvent, phase separation occurs during the film formation process as the solvent evaporates.21 This results in films presenting a phase separated microstructure with domains rich in EC and domains rich in HPC.6,8 The films become porous when immersed in water due to HPC-leaching if the initial HPC concentration is above the percolation threshold,8 and a complex porous structure is created.6 The lattice Boltzmann simulations were also complemented by investigations of properties of the pore space of the polymer films, such as local porosities and pore sizes, and conclusions were drawn concerning the relationship between Deff and the pore space properties.
Tout largely increased at the end of the spraying process. It had also been shown that the combination of the large increase in temperature at the end of the spraying process and the fact that the last sprayed layer of polymer solution is not further covered by an additional layer of polymer solution resulted in a film surface with HPC-rich domains much smaller than the domains observed inside of the film. In order to obtain a more homogeneous structure across the film thickness, the set value of Tin was changed from 68 to 40 °C as soon as the spraying of the polymer solution started. This was done as larger domains can be obtained at lower Tout, and as the lower the Tin the lower the Tout. Hydrous ethanol 95% alone was sprayed onto the drum before the spraying of the polymer mixture. The value of its flow was equal to the value of the polymeric solution sprayed during film preparation. Tout was monitored and when its value became constant the sucking tube was moved from the beaker containing ethanol to the beaker containing the polymer solution. Tout slightly increased at the end of ethanol spraying as a consequence of the fact that a gap containing air is created in the tube supplying the liquid to be sprayed when the sucking tube is moved from the beaker containing ethanol to the beaker containing the polymer solution and that no energy to evaporated ethanol is required when this air is pumped into the fluid bed. After spraying, the films were kept in the fluid bed for about 20 min to completely dry, and were then peeled off the drum and kept in a desiccator. The film thickness was measured using an electronic micrometer. The sprayed films were approximately 30 μm thick. 2.3. Characterization of the Film Structure. 2.3.1. Characterization of the Phase-Separated Structure of the Films Using Confocal Laser Scanning Microscopy. The phaseseparated 3D-structure of EC/HPC films was studied using confocal laser scanning microscopy (CLSM). The films had not been immersed in water and CLSM could be used as the fluorescent light emitted from EC- and the HPC rich phases largely differed in intensity. The CLSM measurements were performed using a Nikon D-ECLIPSE C1 confocal system equipped with an Eclipse TE2000-E inverted microscope. A 488 nm Argon-ion laser was used for excitation and the fluorescent light was detected using a band-pass filter centered at 605 nm with a bandwidth of 75 nm. The pinhole was set at a diameter of 30 μm. A Nikon CFI Plan Apo VC60XH 60x/NA 1.4/WD 0.13 oil immersion objective was used for the measurements, leading to an optical resolution on the order of 200 nm/500 nm (laterally/axially). A drop of immersion oil was inserted between the coverslip facing the objective and the film, and precautions were taken to avoid the presence of air between the objective, coverslip and sample. The measurements in the body were started from the film’s surface. Optical slices (xy images) in the z-direction (z-stack measurements) were acquired at every 100 nm. The xy pixel size used in these measurements was 76.7 nm × 76.7 nm. CLSM data was first treated in FIJI/ImageJ22 by 3 × 3 × 3 hybrid median filtering23 and then corrected for photo bleaching using the exponential fitting method.24 The microscopy data was further treated to identify the EC- and HPC-rich phases, as described in section 2.7. 2.3.2. Characterization of the Film Structure Using Scanning Electron Microscope. When the EC/HPC films are immersed in water, HPC leaches in water from the films, leaving a porous structure. The morphology of the cross section and of the top and bottom parts of the films after HPC leaching in water was studied using a FEI Quanta 200 scanning electron
2. MATERIALS AND METHODS 2.1. Materials. EC (EC N10CR) was supplied by Dow Chemical Co., USA. HPC grade LF was supplied by Aqualon, USA. Part of the HPC was labeled by CarboMer Inc., USA. Fluorescein was used for labeling at a concentration of 0.005 mol per mol of glucose unit. Tritium-labeled water (Amersham, U.K.) was used in the water diffusion experiments. 2.2. Preparation of Free Polymer Films. The films were prepared from mixtures containing 94% w/w hydrous ethanol (95%) and 6% w/w polymers, as it has been shown that at this low polymer concentration EC and HPC do not phase separate [20]. The dry film consisted of 70% EC and 30% HPC. Six percent of the HPC used was fluorescein-labeled. Labeled HPC was used in order to increase the fluorescence contrast between EC and HPC in the confocal laser scanning microscopy (CLSM) analysis of the films phase separated structure. The films were prepared by spraying 20 g of the coating mixture onto a rotating drum inserted into a modified fluid bed chamber. A horizontalmoving spraying nozzle positioned below the drum was used to spray the polymeric mixture. The experimental setup has been described in details elsewhere. The size and shape of the HPCrich domains is largely affected by the process parameters used. The process parameters were chosen in order to have in the bulk of the film HPC-rich domains sufficiently large to be clearly detected with CLSM (i.e., the size of the HPC-rich domains is larger than the limit of resolution of CLSM), but not too large in order to avoid a secondary phase separation inside the formed domains. Fluidizing air flow, atomizer pressure, coating flow, atomizer air flow, rotation rate of the drum and velocity of the spray raster were kept constant during the film spraying process and had values of 40 N m3/h, 2 bar, 12,6 g/min, 1.7 N m3/h, 80 rpm, and 1.3 cm/s, respectively. It had previously been shown that when Tin was kept constant during the spraying process, 5221
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approach the resolution of the CLSM. The following steps were performed in order to obtain a binary 3D image, indicating the solid EC phase and the pores: 1 An FFT high-pass filter was applied to each slice to even out large-scale intensity variations. 2 Suitable sections of the film of about 15.5 × 15.5 μm were chosen manually, including manually determining the top and bottom slice where the film started and ended. 3 The data was up-scaled to double resolution using tricubic interpolation, and then a Gaussian smoothing was applied to each slice, with smaller σ (√2 pixels) used at the top and a larger value (σ = 2 pixels) at the middle and bottom. This was in order to reduce the noise in the image and create a smooth structure, while avoiding to destroy the fine pore structure at the top. 4 All slices in the sections were normalized to have the same average intensity, in order to compensate for loss of intensity with the depth in the film. 5 A global threshold was then computed and applied to achieve a total volume fraction of pores of 0.267. This value was determined quantifying the amount of HPC leached out when exposing the film to water, assuming that all the leached HPC formed the pore space and assuming that the EC- and HPC-rich phases have the same true density and porosity as pure EC and HPC films. The binary data thus achieved was used in the pore space analysis described in section 2.9. In order to also create a smooth surface suitable as input to the lattice Boltzmann simulations, the marching tetrahedra method (implemented in the GTS library, version 0.7.627) was used to create a triangulated isosurface at the level of the computed threshold, as shown in Figure 4. The surface was then scaled to fit the computational grid. 2.8. Lattice Boltzmann Simulations. The lattice Boltzmann method was used to solve the diffusion equation
microscope (SEM). The samples for which the cross section was imaged were embedded in epoxy glue using a piece of 4−6 mm i.d. plastic tubing as a mold. When the glue was dry, these samples were cut using a Leica Ultracut UCT ultramicrotome equipped with a glass knife. The cutting was done in order to obtain a smooth cross section surface. Embedded cut films and nonembedded films were immersed in 600 mL of water overnight, with stirring in order for all the removable HPC to be leached out from the film. The samples were then dried at 40 °C overnight. The surfaces to be observed were coated with a thin layer of gold in a Cressington 108Auto sputter coater before observation with SEM. The SEM analyses of films details were performed with a spot size of 5.0 and an acceleration voltage of 10 kV or 12.5 kV. Overview images were acquired in low vacuum mode at a pressure of 1.2 mbar. The solid state backscattered electron detector (SSD) was used for all images. SEM micrographs were obtained using the software xT Microscope Control. 2.4. Experimental Determination of Deff. Water permeability measurements were performed using a side by-side diffusion cell. The two chambers were separated by the film of interest and the system was thermostated with a water jacket containing water at 37 °C. The area of the film available for diffusion was 0.48 cm2. The chambers were filled at the same time with 15 mL of a 37 °C of water. The water diffusion experiment started when a small amount of tritiated water (10 mL, 400 kBq) was added to the donor compartment. Water in the donor and receiver compartments was well stirred with paddles. Samples of 500 μL were removed from the receiver compartment at regular intervals and replaced by the same amount of phosphate buffer solution. The samples were weighed and analyzed in a liquid scintillator counter (1414 LSC, Win Spectral, Wallac). A sample of 500 μL was taken from the donor compartment 1 min after the diffusion experiment had been started for concentration to be uniform to determine the tritium activity in this compartment. From the tritium activity measurements it was possible to calculate the amount of water that had diffused through the film at different times, and thus the water effective diffusion coefficient of the film. Experiments were performed in duplicate. 2.5. Determination of True Density and Porosity of 100% EC and 100% HPC films. Films containing 100% EC and 100% HPC were produced using the same process conditions as for the 70/30 EC/HPC films. The density and porosity of these films were experimentally determined using He-pyknometry (Micromeritics AccuPyc 1330) and Hg-porosimetry. (Micromeritics III AutoPore 9410), respectively. The density of the EC film was 1.27 ± 0.02 g/cm3, the density of the HPC film was 1.43 ± 0.01 g/cm3. The porosity of the EC film was 3.5 vol %, the porosity of the HPC film was 2 vol %. 2.6. Determination of the Amount of HPC Leachable from the EC/HPC Films. A piece of the film of interest was immersed in 70 mL of purified water. The system was agitated and maintained at a temperature of 37 °C for 24 h. Samples were collected and the HPC concentration in the samples was analyzed using size exclusion chromatography. The leachable fraction of HPC was 0.973 ± 0.7 w/w %. 2.7. Image Processing of CLSM data. In order to transform the data acquired by CLSM into a geometry suitable for computations, some image processing was required. The image processing was performed in FIJI22 and MATLAB (The MathWorks, Inc., Natick, MA). The main concern was to reduce the noise in the structure to create a smooth enough geometry for the computations, while at the same time preserving the structure at the top of the film, where pore sizes
∂c − D0Δc = 0 ∂t
for the concentration c in the pore space, using Neumann (zero normal flux) boundary conditions on the material surface, −D0
∂c =0 ∂n
where n is the outward unit normal, modeling that no diffusion takes place in the solid EC phase. D0 is the free diffusion coefficient in the pore space. On the influx and outflux boundaries (the top and bottom of the film, at z = 0 and z = L), the concentration was set to given values (c1 and c2, respectively) to create a concentration gradient. Reflecting boundary conditions were set on the other box boundaries to create the effect of a periodic structure, while having a continuous material structure over the boundary. After solving the diffusion equation to steady state, the effective diffusion coefficient ∂c
was then computed from the average flux Jz̅ = D0 ∂z in the direction of the concentration gradient as Deff = −(Jz̅ L)/(c2 − c1), where L is the thickness of the film. Six sections of about 15.5 × 15.5 μm, spanning the whole thickness of the film (about 30−40 μm), were taken at different locations in the film, and Deff was computed for each of them on a uniform grid of 300 × 300 × 600−800 voxels. To see how Deff varied with depth, corresponding simulations were also performed in overlapping sections of two of the samples, each of 10 μm thickness and spaced with 5 μm distance. 5222
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The Journal of Physical Chemistry B The lattice Boltzmann method used was a two-relaxation-time method for diffusion18 with boundary conditions implemented using a ghost node scheme as described in detail by Gebäck and Heintz.19 The method is capable of handling the very complicated geometry of the film and has been validated both theoretically and on simpler geometries.19 The computations were performed at the C3SE center for computational science using 64 cores. 2.9. Pore Size and Porosity Computations. The pore size distribution of the film was determined from the thresholded voxel data by fitting spheres into the pore space and recording for each pore voxel the diameter of the largest fitted sphere that contained it. For each slice in the depth-direction, the average sphere diameter over all the pore voxels was then computed. The sphere-fitting was performed using the Euclidian distance function, computed using the algorithm by Felzenszwalb and Huttenlocher.25 Γ probability distributions were fitted to pore size distributions using the MATLAB command gamfit. The porosity of each slice in the depth-direction was also computed from the voxel data, and connected components were found using a flood-fill algorithm, starting from the inlet at the top of the film. All pore-space measurements were implemented in C++.
3. RESULTS AND DISCUSSIONS 3.1. Experimental Characterization of the Films Structure and Experimental Determination of the Effective Diffusion Coefficient of Water in the Films. The films had a phase-separated structure made of HPC-rich domains in an EC-rich matrix. Typical CLSM optical cross section images are reported in Figure 1. The domains rich in HPC are bright, while Figure 2. CLSM images obtained at different positions in the film 1. (A) top of samples 1 from film 1; (B) top of sample 2 from film 1. (C) middle of samples 1 from film 1; (D) middle of samples 2 from film 1; (E) bottom of samples 1 from film 1; (F) bottom of sample 2 from film 1.
Large regions characterized by a homogeneous color were present on the top part of the films. For these regions, it was not possible to detect EC-rich and HPC-rich domains. The CLSM resolution in the xy CLSM images is 200 nm, and the homogeneous color could be due to the fact that the domain size in these regions was below the CLSM resolution. SEM has a better resolution than CLSM, and was used to image the top surface of the films. The films used in the SEM study had previously been immersed in water in order to remove HPC, and the dark area in the SEM images indicates the pores created by the leaching of HPC. From the SEM images (see Figure 3A for film 1) it
Figure 1. Optical CLSM cross sections images of samples 1 (A) and 2 (B) obtained from film 1.
those rich in EC are dark. No secondary-phase separation was observable in the created domains. The size of the domains varied across the film cross section in both film 1 and film 2 (see Figure 1 and Figure 2 for film 1; data for film 2 not shown). However, the structure of the film cross section was much more homogeneous than previously observed for EC/HPC films,6 due to changes ported to the manufacturing process. It has been shown that the lower the manufacturing temperature, the bigger the size of the phase-separated domains.6 The HPC-rich domains became smaller toward the top surface of the films, indicating that phase separation occurred to a lower extent in the upper part of the films. This occurred despite the fact that the value of Tout at the end of the film spraying process was lower than the value of Tout at the beginning of the film spraying process. A polymer film is the result of several polymer solution layers that are sprayed on the top of each other. The mobility of the polymers present in the last layer decreases more quickly compared to the mobility of the polymers present in the body of the film. This is caused by the fact that the last sprayed layer cannot be wetted by further polymer solution. The consequence of this is that smaller domains are created on the top of the films.
Figure 3. SEM images of the top surface of the film 1. Different magnifications were used for parts A and B. 5223
DOI: 10.1021/acs.jpcb.5b01953 J. Phys. Chem. B 2015, 119, 5220−5227
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The Journal of Physical Chemistry B was possible to deduce that the top surface of the films presented pores and that the pores were smaller than 200 nm. The SEM images also revealed the presence of areas where no or very small pores were present. Some SEM overview images of the top surface of the films were also acquired and confirmed the existence of areas characterized by different structures (see Figure 3B for film 1). The size of the domains was rather homogeneous in the xy CLSM images obtained within each sample inside the film. However, the domains size could slightly differ among different film samples (see Figure 1 and Figure 2). The film structures of the two produced films were very similar to each other at a visual inspection (data not shown). The effective diffusion of water in the films was experimentally measured. Deff in film 1 was 7.6 × 10−11 m2/s and Deff in film 2 was 8.4 × 10−11 m2/s. The two measurements were rather close to each other as confirmation of the similarity in films structure.
3.2. Extraction of 3D Structures from Experimentally Characterized 3D Structures. One example of a 3D structure obtained after processing of the CLSM data is shown in Figure 4. The figure shows half the simulation domain, but the full thickness of the film is shown from the bottom left to the top right. The small pores at the top of the film are seen on the left. Since the size of the smallest pores are close to the resolution of the CLSM, the top part of the data was treated with care in order not to destroy the pore structure. A comparison with the SEM images from the top of the film in Figure 3 show that the pores in the final structure are in the correct size range of about 200−400 nm and that the density of pores is approximately correct. However, there are also regions in the SEM image where the pores are considerably smaller, and where the resolution of the CLSM is too low to capture the pores. 3.3. Simulations of Diffusion and Dependence on Porosity throughout the Film. The effective diffusion coefficients for the films computed using the lattice Boltzmann method are shown in Figure 5. The diffusion coefficients were computed for square subsamples of the film, measuring 15.5 × 15.5 μm, and spanning the whole thickness of the film as seen in Figure 4. The computed values of Deff are within the same range as the experimental values, though there is no perfect agreement. There are also large variations between samples. The major reason for the less than perfect agreement is believed to be that the measured diffusion constant is measured over an entire film (0.48 cm2), while the simulations are made on a much smaller section of the film. Additionally, the sections selected for simulation are not representative for the whole sample, but had to be chosen so that the pores at the top of the film where resolved with CLSM, and thus having pore sizes around 200−400 nm or larger. As can be seen from the SEM data in Figure 3, there are regions where the pores near the top are much smaller−most likely resulting in a much reduced diffusion constant. Note that for a system with parallel channels with different Deff, the total Deff is the average of the local values. It is therefore likely that the total effective diffusion constant is lower than the value computed in the simulations, and the computation of the true value is impossible using the techniques used here, mainly due to limits in resolution of the CLSM. However, even if a different microscopy technique was used to resolve the finest
Figure 4. Section of the film structure after transformation to a surface suitable for diffusion simulations, together with flux lines following the diffusive flux field through the film (some of them in the part of the structure not shown). The blue parts are the fluid parts and the structure has been made partly transparent to show the interior. The whole thickness of the film is shown from left to right.
Figure 5. Simulated and measured values for Deff/D0. On the left, simulation results for the whole thickness of the film are shown and compared to experimental results. The error bars show standard deviations over the samples (N = 6 for the simulations and N = 3 for the experiments). The graph on the right shows simulated Deff/D0 (red, crosses) for 7.7 μm thick subsamples located at different depths of two different film regions together with the porosities (blue, circles) of the same subsamples. The thin red lines show the value of Deff/D0 for the whole thickness of the film. 5224
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Figure 6. Porosity of all pores and connected pores as a function of the depth into the structure. The thin lines are for the individual samples, and the thick lines are averages over all samples. Note that the films differed in thickness.
Figure 7. Average pore diameters (from the pore size distribution determined by fitting spheres in the pore space) as a function of depth. The thin lines are for the individual samples, and the thick lines are averages over all samples. The profiles agree very well with the CLSM image of the cross sections; see Figure 1
diffusion is even more reduced in the top layer of the film, having a dramatic effect on the total Deff. In addition, pore size distributions were computed as described in section 2.9. Figure 7 shows the average pore diameter at different depths of the sample. It is clear that the pores are smaller at the top of the film, then increasing toward the center of the film and finally decreasing slightly toward the bottom. This is in agreement with the knowledge of the film formation process, as described in section 2.2, and with the CLSM images of cross sections of the film, see Figure 1. In Figure 8, pore size distributions are shown at different depths of one of the samples, adding information about the width of the distributions. The Gamma distribution gives a reasonably good fit, although the long tail predicted by the distribution is truncated in the measured distribution, which may be due to the small sample size. Although the pore size itself does not influence the diffusion directly (as long as the diffusing species is much smaller than the pores), the porosity distributions shown in Figure 6 also indicate a loss of connectivity toward the top of the film, which has a large effect on the effective diffusion constant. This is because in the phase separation process, small regions of HPC (which become pores after leaching) are less likely to be connected to other regions than large regions. 3.5. Insights on the Relationship between 3D Structure and Effective Diffusion. The above results show that we do not obtain perfect agreement between simulated and experimentally measured effective diffusion constants for the whole
pores, direct diffusion simulations in the full pore structure would quickly become prohibitively time-consuming, and alternative approaches would have to be considered. On the right of Figure 5, Deff is also shown for subsamples of thickness 7.7 μm taken at different depths from the same sample. These results show that Deff is higher further into the sample and lower at the top, following a similar trend as the porosity, but not being exactly proportional to it. One can also see that the local Deff values here are in general larger than the Deff for the full sample, meaning that the value for the full sample may not be computed from the local values, suggesting that the top layer with low Deff controls the diffusion and that connectivity effects may be important for the resulting diffusion through the full sample. 3.4. Pore Space Analysis of 3D Structures. In order to analyze the pore structure of the film more closely, the local porosity and pore size distributions were computed at different depths using the same CLSM data used for the lattice Boltzmann simulations. The results for the porosity are shown in Figure 6, where the porosities at different depth show that the porosity decreases toward the top of the film. This may partly be the result of pores becoming smaller than the resolution of the CLSM, and thus not appearing as pores in the data, but is also most likely a real effect, due to the fact that the phase separation process has not progressed long enough to produce HPC domains. In addition, Figure 6 also shows the porosity when counting only connected pores spanning the whole thickness of the film. It is then apparent that the pore volume actually contributing to the 5225
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function of porosity and a single tortuosity. Thus, the presented methodology has provided new insights into how the microstructure determines the mass transport through the film, which can be very useful when designing new films differing in structure and mass transport properties.
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AUTHOR INFORMATION
Corresponding Author
*(T.G.) E-mail:
[email protected]. Telephone: +46 31 7723547. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was funded by SuMo Biomaterials, a VINN Excellence Center funded by Vinnova. This support is gratefully acknowledged.
Figure 8. Pore size distributions at different depths for one 30 μm thick film sample. Gamma probability distributions have been fitted to pore size histograms for the whole sample (black, with histogram shown in gray), and for 4 μm thick slices at the top (red), middle (blue) and bottom (green).
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film. Still, the simulations are of great value because they enable us to investigate what aspects of the film structure are most important for determining the effective diffusion constant. Thus, from the simulation results in Figure 5 it is clear that the structure of the top layer of the film is to a very large extent determining the total Deff for the film. From the pore space analysis in Figure 7, we see that the pores are much smaller in the top layer. Smaller pores do not immediately imply a lower Deff, since diffusion is size-independent (as long as the diffusing species is much smaller than the pores), but in this case the small pores are connected to a lower porosity, which directly reduces Deff in the top layer. Figure 6 also shows that the pores in the top layer are less connected, which reduces Deff even further. Thus, smaller pores, lower porosity and lower connectivity go hand in hand in the top layer of the film, determined by the fact that in the film formation process the phase separation does not proceed as far in the top layer as in the bulk, because of the drying of the film. Another factor that is often mentioned as important for the effective diffusion constant is tortuosity, as in the equation ϕ D0 (1) τ2 26 with the porosity ϕ and tortuosity τ. Obviously, the pores in the film are very tortuous, which in part explains why Deff/D0 in the bulk of the film is so much lower than the porosity (as seen in Figure 5). However, this does not at all explain the very low value for Deff obtained for the whole film, and just using eq 1 to explain our results would be misleading and would not contribute to the understanding of the system. It is also clear from Figure 5 that Deff/D0 is not proportional to the porosity throughout the depth of the film, indicating that a single tortuosity in eq 1 does not explain the results and further reducing the usefulness of eq 1. Deff =
4. CONCLUSIONS In this work, the detailed 3D structure from CLSM data of EC/HPC films was used for diffusion simulations and pore space analysis. The results show that the diffusion through the film is largely determined by the top layer of the film with smaller pores, lower porosity, and lower connectivity. Furthermore, the effective diffusion coefficient was shown not to be a simple 5226
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