Attenuated Total Reflectance Mode for Transport through Membranes

Abstract

This chapter is an introductory tutorial to attenuated total reflectance (ATR) mode of Fourier-transform infrared spectroscopy and how it can be used to measure transport through polymer membranes. In addition to covering the experimental set-up and time-resolved data processing, it will present the fundamental equations for analyzing the data in order to obtain diffusion coefficients. The chapter will present several example systems in which FTIR-ATR has been used to determine transport, including water diffusion through polyelectrolytes for fuel cells and block copolymers for water purification as well as ion transport through polymer electrolytes for lithium batteries. Perspectives on future applications in which the technique could provide fundamental understanding will also be covered.

1. Introduction

Fourier transform infrared spectroscopy (FTIR) revolutionized chemical analysis with light-based spectroscopy. The ability to pass all wavelengths of mid-infrared light through a sample simultaneously rather than one wavelength at a time increased sampling rate by orders of magnitude. The IR source beam is split into a reference beam that reflects off a moving a mirror, thereby changing the path length in time, and a beam that passes through the sample. FTIR takes advantage of constructive and destructive interference between the reference and sample beams that is referred to as an interferogram. Due to the interference changing in time, the interferogram can be converted into intensity versus wavenumber via a Fourier transform. Wavenumber is inversely proportional to wavelength and the mid-IR region is from 4000 to 400 cm⁻¹. Intensity is converted to transmission by dividing the sample measurement by a background, which is a measurement without a sample present. This results in an infrared spectrum with a baseline at or near 100% in wavenumber regions where no light is absorbed by the sample, i.e. where the frequency of light does not correspond to the frequency of vibration of any of the molecular functional groups of the sample. In wavenumber regions where light is absorbed a “peak” is observed. The peak is typically a Gaussian decrease in transmitted intensity centered around the frequency of a particular vibration of a functional group containing covalent bond(s). The combination of functional groups in a sample results in a spectrum that can be used to identify the chemical make-up of unknown samples or detect the presence of particular species. Thus, FTIR provides a rapid and reliable means for sample identification and/or detection. Greater detail regarding the fundamentals of FTIR, prediction of functional group absorbance, and tables of wavenumber values can be found elsewhere [1].

One of the limitations of FTIR is that if samples are too thick, an insufficient amount of light passes through the sample for detection. This is due both to absorbance and scattering. This can be overcome by preparing thin samples, on the order of microns, or, if the sample is a powder, it can be mixed with a salt, such as KBr, and pressed into a transparent pellet. Inorganic salts contain only ionic bonds that do not absorb infrared light. So, if the sample is dilute in the pellet, then sufficient transmission is achieved. An alternative to sample processing is to use attenuated total reflectance (ATR) to ensure that sufficient light intensity is transmitted. As shown in Figure 1, the infrared beam is directed into the crystal at an angle less than the critical angle, typically 0° with respect to the normal of the face through which it enters. The beam then totally internally reflects at the top surface of the crystal because the incident angle is greater than the critical angle. The critical angle is a function of ratio of the index of refraction of the crystal and the medium above the reflecting surface. Crystals of high refractive index are commonly used, such that the critical angle is small. For ATR set-ups without adjustable angles of incidence, the incident angle on the top face is usually 45° as shown in Figure 1. After reflecting at the top surface of the crystal, the infrared beam exits the crystal and is directed to the detector. The crystal is housed on an optical set-up that directs the source beam to the crystal and the exiting beam to the detector with a combination of mirrors and sometimes focusing lenses. The mirrors and lenses are adjustable so that the amount of infrared energy transmitted to the detector (without a sample present) can be maximized.

Figure 1.

Schematic of an ATR crystal with an infrared beam entering from the source side, totally internally reflecting, and exiting to the detector. In this schematic the angle of incidence is 45°, a common fixed incidence angle, and there is a single reflection. Not shown are the FTIR spectrometer, the mounting that houses the crystal, the sample that is placed on top of the crystal, nor the anvil that presses the sample to the crystal ensuring intimate contact which is essential for high-quality reproducible data collection.

Common materials of construction for ATR crystals include diamond, ZnSe, KRS-5, silicon, and germanium. Despite the cost, diamond is an excellent ATR material due to its robustness. It is stable to high temperature, in contact with corrosive substances (including both high and low pH), and in contact with abrasive samples. Other than cost, which limits diamond ATR crystals to single reflection, the only other drawback is its inherent absorbance of IR between approximately 2700 and 1800 cm⁻¹. Although noise is greater in this region of the spectrum, this is actually only a minor drawback because the diamond absorbance is part of the background and thus subtracted from the sample spectrum. Moreover, there are few important sample peaks found in this wavenumber range. If robustness is not important, then the best choice for maximizing throughput of infrared energy is ZnSe. This material is available as single-reflection crystals as well as multiple reflection crystals that increase the sensitivity of detection. Commercial KRS-5 is rarely used today due to toxicity. Silicon and germanium have intermediate properties (including cost and energy transmission) for select applications. For example, due to lower infrared energy transmission, germanium can be used to solve absorption saturation problems that sometimes occur in multi-reflection ZnSe experiments.

ATR mode obviates the need for sample preparation because sampling occurs via an evanescent wave that exists outside the crystal and therefore within the sample. The evanescent wave is a non-propagating (i.e. standing) wave that forms due to interaction between the incident and reflected wave. Although it is non-propagating, the evanescent wave can interact with the sample such that infrared energy can be absorbed by the sample and an infrared spectrum generated. The intensity of the evanescent wave is exponentially decaying such that it is most intense at the crystal surface. The depth at which the intensity decays to $e^{-1}$ is defined as the depth of penetration, $d_p$. It is determined by the physics of the total internal reflection. For reflection at an interface between two dielectric (i.e. non-metallic) materials it is a function of the angle of incidence, $\theta$, refractive index ratio of crystal, $n_{\mathit{ATR}}$, and sample, $n_{\mathit{\text{sample}}}$, and wavelength of light, $\lambda$ [2].

$$d_p=\frac{\lambda }{2\pi n_{\mathit{ATR}}\sqrt{{sin}^2\left(\theta \right)-{\left(\frac{n_{\mathit{\text{sample}}}}{n_{\mathit{ATR}}}\right)}^2}}$$E1

For internal reflection to occur $n_{\mathit{\text{sample}}}<n_{\mathit{ATR}}$. As shown in Table 1, the refractive indices of air, water, and polymers are all less than that of materials commonly used to construct ATR crystals. By way of example, the critical angle for total reflection within diamond with water on top is ${\theta }_c={sin}^{-1}\left(\frac{n_{\mathit{\text{sample}}}}{n_{\mathit{ATR}}}\right)=33°$. This is measured from the normal to the interface, such that $\theta >{\theta }_c$ for total internal reflection. Keeping with this example, at the highest wavenumber (4000 reciprocal centimeters) $d_p=0.4\mathrm{\mu m}$. Due to practical limitations of most mid-infrared optics, the lowest wavenumber at which reliable data can be obtained is 650 cm⁻¹. At this wavenumber, $d_p=2.3\mathrm{\mu m}$. This demonstrates that the only significant drawback to ATR mode is that the sampling depth is a function of wavenumber, which causes a slight skewing of relative intensity of different peaks when compared to transmission mode. Since most libraries of infrared spectra have been constructed in transmission mode, it is best practice to use transmission mode for identification of unknown compounds. However, the aforementioned skewing of relative peak intensities is only slight, such that library searches from ATR spectra are usually successful. Moreover, instrument software often has automated algorithms to correct for this effect, which can also be rigorously accounted for manually if absolute peak intensities are required.

Material	Refractive Index	Reference
Air	1.0	[3]
Water	1.3	[3]
Polymer	∼1.5	[4]
Diamond	2.38	[5, 6]
ZnSe	2.45	[3]
Silicon	3.4	[7]
Germanium	4.0	[7]

Table 1.

Approximate refractive indices of selected samples and ATR materials at room temperature and at wavenumbers between 4000 and 650 cm⁻¹ (wavelengths between 2.5 and 15 μm).

On the other hand, there are several benefits to ATR mode that outweigh the wavenumber dependence of $d_p$ in most cases. Regardless of the total sample thickness, infrared is absorbed over only about $5d_p\sim 10\mathrm{\mu m}$ at most. Thus, sufficient energy transmission to the detector occurs without any special sample preparation for all but rare instances, such as carbon black that strongly absorbs and scatters infrared light. In addition to ease of sample preparation, temperature is readily controlled in the ATR set-up as is the environment of the sample. Thus, time-resolved FTIR-ATR measurements can be used for reaction kinetics and membrane transport studies, the focus of this chapter.

2. Fundamentals of ATR

As with transmission mode FTIR, a background must be collected when using ATR mode. It is essential that the ATR accessory is in place when collecting the background so that absorption by the ATR crystal is included in the background spectrum. For this reason, the appearance of the background will depend on type of ATR crystal being used. Best practice is for the top surface of the crystal to be empty and dry when collecting the background, although it is possible to have the sample in place and use the initial conditions of the sample as the background in order to generate difference spectra if the main purpose of the experiment is to examine changes in time. This latter approach is not recommended because of the detrimental impact it has on signal-to-noise ratio [3].

After the background has been collected, the sample is placed on top of the crystal. Due to the exponential decay of the evanescent wave intensity, intimate contact between sample and crystal is crucial. This is trivial for liquid samples, but an anvil that generates reproducible force should be used to press solid samples to the crystal surface. An alternative approach that has been used for polymeric samples is to cast a polymer film from solution directly onto the ATR crystal. This approach was used in the pioneering work in which FTIR-ATR spectroscopy was first demonstrated as a powerful technique for quantifying transport in polymer films and is covered in an excellent review [8]. Layer-by-layer deposition has also been used to assemble polymer films on ATR crystals [9]. Care should be taken with polyelectrolyte solutions that can exhibit highly acidic or basic conditions that will damage crystals such as ZnSe. In those cases, diamond can be used, the sample can be physisorbed to the crystal [10], or a pressure-contact method can be utilized [11].

An important consideration in ATR mode is the thickness of the sample. As shown in Figure 2(a), the evanescent wave will extend beyond a thin sample of less than a few $d_p$. For transport experiments in which the conditions at the top surface of the sample are being changed, this is a problem because the functional groups present throughout the sample and outside the sample are all being measured, weighted by the exponential decay of the evanescent wave. This can be expressed quantitatively using the Beer-Lambert Law, which states that for weak to moderate absorption, absorbance ($A$) is proportional to molar concentration ($C$) [12],

Figure 2.

Schematics of evanescent wave and relative thickness of sample for (a) a sample that is less than the ideal thickness for ATR, (b) a sample in which the evanescent wave extends throughout the sample, and (c) a sample much thicker than the depth of penetration in which detection can be approximated as occurring at the crystal surface.

$$A=-log\left(\frac{I}{I_0}\right)=\mathit{\epsilon CL}$$E2

Note that $I$ is the measured intensity, $I_0$ is the intensity of the background, $\epsilon$ is the molar extinction coefficient, and $L$ is the path length of the IR beam through the sample. For ATR mode, the decay of the evanescent wave energy should be incorporated, $I=E^2=I_0{e}^{-\frac{2y}{d_p}}$.

This has been achieved by using the differential form of the Beer-Lambert Law in terms of intensity [12], but it assumes weak absorbance, which introduces 10% error at 80% transmission and greater error with decreasing transmission (increasing absorbance). Moreover, the original reference used natural logarithm to define absorbance, rather than the typical log base 10. This approximation was necessary to yield an analytical solution to transient diffusion [8], but alternate approaches are possible and discussed below. In any case, an exact expression is derived here.

$$A=-log\left(\frac{I}{I_0}\right)$$E3

Relating $\mathit{dA}$ to $\mathit{dI}$ and inserting into Beer-Lambert Law yields.

$$\mathit{dA}=-\frac{\mathit{dI}}{\mathit{Iln}10}=\mathit{\epsilon C}\mathit{dy}$$E4

Rearranging and inserting the expression for evanescent wave decay of intensity yields.

$$\mathit{dI}=-\mathit{ln}10\mathit{\epsilon CI}\mathit{dy}=-\mathit{ln}10\mathit{\epsilon C}{I}_0{e}^{-\frac{2y}{d_p}}\mathit{dy}$$E5

The left-hand side of the equation can be integrated from the incident intensity, $I_0$, to $I$. Due to the exponential decay, the integral of the right-hand side can be evaluated from 0 to infinity, but is typically evaluated to the top of the sample, $L$, because beyond this the molar extinction coefficient, $\epsilon$, is not well defined. In either case, the integral cannot be explicitly evaluated at this point if $C$ is a function of $y$, which will be the case in transport measurements.

$$\frac{I-I_0}{I_0}=-\mathit{ln}10\int \mathit{\epsilon C}\left(y\right)e^{-\frac{2y}{d_p}}\mathit{dy}$$E6

In terms of absorbance, this is

$$A=-log\left(1-\mathit{ln}10\int \mathit{\epsilon C}\left(y\right)e^{-\frac{2y}{d_p}}\mathit{dy}\right)$$E7

For a known concentration profile, i.e. known $C\left(y\right)$, the integral can be evaluated and an explicit expression for absorbance found. Alternatively, the integral can be evaluated numerically if necessary.

The case of Figure 2(a) where the sample is thinner than a few $d_p$ is not ideal due to the integral in Eq. (7) needing to be evaluated from $0$ to $L$ for sample absorption and from $L$ to infinity for absorption by the medium beyond the sample. There will be much less experimental uncertainty for the case of Figure 2(b) in which the evanescent wave decays by more 99% of $I_0$ within the sample. This means that IR absorption beyond the sample is negligible. However, if there is a gradient in concentration of the absorbing species within the sample, such that $C$ is a function of $y$, then the appropriate expression for $C\left(y\right)$ must be incorporated before evaluating the integral of Eq. (7). In other words, a transport model should be incorporated into Eq. (7) or an approximation thereof in order to accurately predict the rate of change of $A$. The ideal situation, at least for simplicity in transport modeling, is the case of Figure 2(c) in which the sample is at least an order of magnitude thicker than $d_p$. Since 95% of the evanescent wave energy decays within $3d_p$, detection is essentially occurring only at the crystal sample interface for thick samples. This approximation works reasonably well when the sample that is at least $10d_p$, and is basically exact for samples of greater than $100d_p$ . In this case, the absorbance is proportional to concentration at $y=0$, and the integral can be directly evaluated yielding.

$$A=-log\left[1-\frac{\mathit{ln}\left(10\right)\mathit{\epsilon C}\left(0\right)d_p}{2}\right]$$E8

An effective ATR extinction coefficient can be used to simplify the expression, $b=\frac{\mathit{ln}\left(10\right)\epsilon d_p}{2}.$

$$A=-\mathit{log}\left[1-\mathit{bC}\right]$$E9

This demonstrates that the choice of base for the logarithm is arbitrary, but it should be specified. In the limit as concentration, and hence absorbance, go to zero (as assumed by Fieldson and Barbari) [12], Eq. (9) further simplifies to $A=\mathit{bC}$. Thus, in the thick film limit depicted in Figure 2(c), absorbance is proportional to concentration in the limit of weak IR absorption.

3. Membrane transport

3.1 Fickian diffusion

Transport in membranes and polymer films is relevant to a wide range of applications that include barriers, electrolytes, and membrane separations. In the area of barriers, polymer films are used in food packaging to improve quality, extend lifetime, and reduce waste. In these applications tailoring transport of water vapor, oxygen, and other gases like ethylene are important as they control the rate of ripening and spoiling. Polymer films are used as building wraps to exclude humidity and control mold. They are used in a wide range of packing applications to protect products, e.g. to maintain sterilization. In the area of electrolytes, polymers are used as solid electrolytes in hydrogen fuel cells, in batteries, and in dialysis and reverse electrodialysis. In the area of separations, porous membranes are used for filtration. Dense polymer membranes are used as reverse osmosis membranes for desalination. Glassy polymers with high free-volume have proven to perform well in separating gases. In essentially all these applications the rate of transport of different species, mostly small molecules and ions, are important.

Transport of water vapor, gases, and ions can be driven by gradients. One of the most common gradients is created by a pressure difference between the opposite sides of the membrane. This pressure difference creates a chemical potential gradient within the membrane, driving transport from the high pressure side to the low pressure side [13]. Although different external gradients can be applied, the universal thermodynamic driving force for transport is that of the electrochemical potential. A gradient of electrochemical potential can be generated by a pressure difference (as already stated), by a concentration gradient, by a voltage gradient, and by a temperature gradient, to name several of the most common driving forces employed in membrane separations. Pressure, temperature, and concentration gradients will universally drive diffusion of all species, but voltage gradients will only drive transport of charged species and their associated solvation shell. This latter mode of transport is commonly referred to as migration for ions and electroosmotic drag for the ions’ solvation shell.

Water is by far the most common diffusant in studies with FTIR-ATR. Since 2000, its diffusion has been examined in asphalt [14], in plasticized polyvinylchloride [15], in cellulose acetate [16], in poly(ethylene terephthalate) [17], in polylactide [18], in polystyrene-poly(isobutylene)-polystyrene block copolymers [19], in fuel cell membranes [10, 20, 21], and in ion-selective membranes for corrosion prevention [22, 23, 24]. Alcohols are the next most common diffusant that has been investigated with FTIR-ATR, presumably due to the strong OH stretching absorbance present in both water and alcohols. In particular, there are several studies of methanol diffusion in various types of polymers [10, 12, 17, 25]. In other studies, FTIR-ATR has been used as a probe to measure changing composition in the receptor compartment of permeation experiments for multicomponent alcohol transport through membranes [26, 27, 28], but this format is beyond the scope of this chapter because the membrane is not in contact with the ATR crystal. Beyond water and alcohols, there is a report of acetonitrile diffusion in cellulose acetate [29]. Finally, the rate of drug release through synthetic skin membranes has been measured with FTIR-ATR [30, 31].

In membrane transport experiments generally and in experiments with FTIR-ATR specifically, it is typically appropriate to assume that transport is 1D because the thickness of the membrane is much less than the lateral dimensions. As shown in Figure 3, we will define the thickness coordinate as $y$. Although not drawn to scale, the control volume shown in Figure 3 extends from the crystal surface at $y=0$ to the top of the sample at $y=L$. Regardless of the material of construction, all ATR crystals are impermeable, which means that a no-flux boundary condition exists at $y=0$. The boundary condition on the top surface will depend on the particular experiment. A common condition is a constant concentration, which can be imposed by placing or flowing a fluid (liquid or vapor) across the top surface of the membrane [8]. If a gradient is imposed in a different way, such as by stacking two membranes of different concentration, then the top boundary condition could also be no flux [32]. Another exciting application of FTIR-ATR is spectroelectrochemistry, in which the driving force for transport is applied potential [33]. If blocking electrodes are used, as in studies of capacitors, then both boundary conditions would be no flux [34]. In battery studies, flux of the active ion could occur at the boundaries if reversible electrodes are used [35]. The flux would match the rate of electrochemical reaction [36]. For transport of neutral combinations of species, the boundary conditions would be no flux in the battery as well.

Figure 3.

Schematic of coordinate system, definition of control volume, and designation of boundary conditions (BCs) for a one dimensional transport experiment conducted with an ATR set-up.

The governing equation for the control volume depends on the particular experiment being conducted. For the case of Fickian diffusion, driven by a concentration gradient, the transient diffusion equation applies.

$$\frac{\partial C}{\partial t}=D\frac{{\partial }^{2}C}{\partial y^2}$$E10

Due to the solid-like nature of polymer membranes and films, convection is neglected in this equation. Due to the finite control volume, this partial differential equation is most readily solved with a Fourier Series solution using Finite Fourier Transforms (FFT). The basis functions for the FFT are chosen to satisfy the type of boundary conditions. With appropriate nondimensionalization, the boundary conditions can be made homogeneous for all examples discussed in this chapter.

$$\theta =\frac{C\left(y,t\right)-C\left(0,0\right)}{C\left(0,\infty \right)-C\left(0,0\right)},$$E11

$$\eta =\frac{y}{L},$$E12

$$\tau =\frac{L^{2}t}{D}$$E13

Focusing on the situation in which the thick film approximation is appropriate, the initial and equilibrium concentration at $y=0$ are used to make concentration dimensionless. The control volume size, $L$, and diffusion coefficient, $D$, are used to define dimensionless position and time. The experimental set-up will dictate the initial conditions throughout the control volume, $C\left(y,0\right)$, which must be known in order to find a solution.

As shown in Figure 4(a), the most common initial condition is the case where the control volume has a homogeneous initial concentration and a different concentration is applied at the top boundary. This is frequently accomplished by way of a flowing stream of liquid, vapor, or gas [8]. Shown in Figure 4(b), another possible initial condition is to introduce two separate polymer membranes each with a different concentration within the control volume. In this case, the initial concentration profile contains a step change. This has been used with polymer electrolytes to study salt diffusion, which was necessary to exclude liquid solvents [32]. This case is more easily modeled numerically, for example with finite difference methods, but the step change can be Fourier transformed. Finally, it is possible to introduce a concentration gradient via an applied external gradient, such as an electric field or a temperature gradient. If the relevant transport parameter(s) are constant, then the equilibrium condition with an electric field or temperature gradient applied should be a linear concentration gradient. The field or applied gradient can then be turned off or removed and Fickian diffusion alone will result in the control volume returning to a homogeneous concentration over time. This initially linear concentration profile is depicted schematically in Figure 4(c). The definition of the dimensionless parameters and the Fourier Series solutions at $\eta =0$ are also reported in Figure 4 for these three possible diffusion experiments. The solution also as a function of position would include the cosine from the appropriate basis function [37], which goes to one at $\eta =0$.

Figure 4.

Possible initial conditions used in FTIR-ATR diffusion studies. In all cases, $\eta =\frac{y}{L}$, $\tau =\frac{L^{2}t}{D}$, and the boundary condition at $\eta =0$ is no flux. (a) There is a homogeneous initial concentration, $C_1$, throughout the control volume and at $\eta =1$ an infinite source of concentration $C_2$. The basis functions for this case are ${\varphi }_n\left(\eta \right)=\sqrt{2}cos\left(\left(n+\frac{1}{2}\right)\mathit{\pi \eta }\right)$. (b) There are two different regions, each of homogeneous concentration. There is a step change of the initial concentration at $\eta =\alpha$. At $\eta =1$, there is a no-flux boundary condition. The basis functions for this case are ${\varphi }_0=1$ for $n=0$ and ${\varphi }_n=\sqrt{2}cos\left(\mathit{n\pi \eta }\right)$ for $n>0$. (c) The initial concentration profile is linear from $C_1$ at $\eta =0$ to $C_2$ at $\eta =1$. The boundary conditions and therefore basis functions for this case are the same as in (b).

The final solutions in Figure 4 are expressions for concentration as a function of time. Implicit in these expressions are $L$ and $D$. The former must be known and is assumed to be much larger than the depth of penetration of the evanescent wave, i.e. the thick film approximation. The latter can be used as an adjustable parameter to minimize the error between the model and time-resolved FTIR-ATR data. Thus, the diffusion coefficient can be found from time-resolved FTIR-ATR measurements of concentration gradients dissipating via diffusion. In these experiments, $D$ is a mutual diffusion coefficient for flux of the diffusing species in the polymer matrix under the influence of a concentration gradient. It is not a self or tracer diffusion coefficient, but it can be related to these under certain assumptions and considering the concentration dependence of the activity coefficient [38]. The mutual diffusion coefficient can be concentration dependent, but has been found approximately constant in many membrane systems [39, 40].

3.2 Swelling

In Section 3.1, the thickness of the film was assumed to be constant. For many membrane transport situations this is a good assumption. However, if the membrane absorbs large amounts of penetrant, then the thickness can increase with time (or decrease in the case of desorption). This is most likely to occur in experiments of Figure 4(a) in which the average concentration of the control volume is changing with time.

$$⟨C⟩={\int }_0^{1}C\mathit{d\eta }=\frac{{\int }_0^{L}C\mathit{dy}}{{\int }_0^L\mathit{dy}}$$E14

In experiments of Figure 4(b) and (c), $⟨C⟩$ is constant, but the concentration profile is changing with time. In other words, none of the diffusing species enters or exits the control volume, but its distribution within the control volume does change with time. A thickness change would occur in these cases if there were a significant and concentration-dependent volume change upon mixing between polymer and diffusing species. However, in experiments of Figure 4(a) the thickness would change even for the simple case where volume additivity holds, because there is flux into the control volume (or out of it for desorption).

An excellent study of swelling (or dilation) was conducted with FTIR-ATR on several different polymer films with different organic penetrants [41]. Baschetti et al. reported a convenient graphical approach with which to analyze dilation. They plotted the integrated absorbance, which is typically calculated as the area under the peak of interest with a linear baseline subtraction, as shown in the sample time-resolved data of Figure 5(a). This is done at each time point and the value can be plotted versus time, as shown in Figure 5(b). Dilation was assessed by plotting the integrated absorbance of a peak associated with the polymer versus the integrated absorbance of a peak associated with the diffusing species. This plot was found to be linear in rubbery polymers, because dilation proceeds instantaneously with diffusion. On the other hand, a delay in the linear correlation was found in glassy polymers due to polymer relaxation (or plasticization) being slower than diffusion. Several other studies have also considered the effect of polymer relaxation during diffusion, which can result in diffusion appearing non-Fickian [11, 18, 20, 21].

Figure 5.

Example time-resolved data showing the procedure for (a) subtracting a linear baseline and integrating the area under a peak of interest and (b) plotting the integrated area versus time. Only a few representative spectra are shown in (a), but one spectrum was collected for each data point in (b).

The same integrated absorbance shown in Figure 5 can be conveniently normalized from zero to one.

$$\frac{A\left(t\right)-A\left(0\right)}{A\left(\infty \right)-A\left(0\right)}=\frac{C\left(t\right)-C\left(0\right)}{C\left(\infty \right)-C\left(0\right)}$$E15

In the context of the thick film approximation with weak absorbance, the normalized integrated absorbance is equal to the normalized concentration because the constant of proportionality drops out. There are quantitative approaches to determine if absorbance is weak that are essentially calibration measurements. The simplest approach is to plot absolute absorbance versus known concentrations and look for linearity, but it is often convenient to use ratios of integrated absorbance values in order to correct for dilation or other concentration-dependent phenomena that do not fundamentally violate the weak absorbance assumption [10, 42].

3.3 Non-Fickian diffusion

Non-Fickian diffusion is a misnomer because concentration-gradient-driven diffusion always follows Fick’s Law. However, if other phenomena occur during diffusion, then transport can appear to be non-Fickian [43]. A common example is a penetrant diffusing into a glassy polymer and plasticizing it. If the rate of penetrant diffusion within the plasticized polymer is much faster than the rate within the glassy polymer, then Case II diffusion will occur that appears as a sharp front of constant velocity moving through the polymer. The front is the boundary between the glassy polymer and the plasticized polymer. In this case, plasticization of the polymer is rate limiting and it is not diffusion that is being measured at all. General treatment of Case II diffusion has been achieved by treating the plasticization as a swelling [44] or polymer relaxation [45] that is rate limiting.

Another example of apparent non-Fickian diffusion has been observed when the polymer contains groups that interact with the diffusing penetrant. The interaction can be in the form of a reaction, for example a polyelectrolyte reacting with water forming a dissociated hydronium ion and an anionic functional group. This was observed in dry Nafion^® that was exposed to water vapor in experiments of Figure 4(a) type, where the sulfonate species were dissociated by water [20]. The consumption of mobile water molecules by the reaction caused a delay in the appearance of water molecules at the ATR crystal interface as well as an appearance of a protonated water bending peak in the FTIR spectra as it was formed by the reaction, as shown in Figure 6. This study had relevance to hydrogen fuel cells and desalination membranes. Another example of polymer-penetrant interaction causing apparent non-Fickian transport has been observed in semicrystalline poly(ethylene oxide), PEO [46]. In this case, water molecules are immobilized while they dissolve PEO crystals. The immobilization during water diffusion through PEO results in apparent non-Fickian behavior. It was observed in block copolymers containing PEO that were studied for water vapor transport that could potentially be used for gas drying or carbon dioxide capture. The morphology of the block copolymer was also found to affect transport in terms of the effective diffusion coefficient, but it did not cause non-Fickian behavior [46, 47].

Figure 6.

Experiment of the Figure 4(a) type. Dry Nafion^® at 30°C is exposed to 100%RH at the top surface, and the FTIR-ATR spectra collected through time. Arrows show the change with time, and the dark red spectrum is time zero (dry Nafion). The equilibrium, hydrated Nafion spectrum is light blue. Reprinted with permission from [20]. Copyright 2010 American Chemical Society.

4. Conclusions and outlook

In summary, FTIR-ATR is a powerful technique for time-resolved measurements. The specific application to transport in polymer membranes and films has been covered here. It can be used to simultaneously quantify diffusion coefficients, even of multiple species by tracking their distinct FTIR-ATR absorbances [10]. It can also distinguish polymer swelling and other non-diffusive phenomena that occur in tandem with diffusion and cause apparent non-Fickian diffusion. The ability to directly track these non-diffusive processes, such as swelling, reaction, and crystallite dissolution, makes it possible to determine the physical cause of apparent non-Fickian behavior. The ATR set-up itself is quite amenable to transport measurements due to the ability to experimentally define a one-dimensional transport system with well-defined boundary conditions.

A recent development in FTIR-ATR is spectroscopic imaging that has made it possible to glean additional information. It has been used to investigate water and plasticizer distributions in polyvinylchloride membranes [48]. Another interesting approach is the coupling of photoacoustic spectroscopy with FTIR (FTIR-PAS), in order to control the sampling depth. Both FTIR-PAS and FTIR-ATR have been used to study drug transport in a membrane skin model [30].

Other exciting future directions include the growth of FTIR-ATR spectroelectrochemistry, by introducing electrodes into the set-up, which enables electric fields to be applied to polymer electrolytes and desalination membranes for example. The most common use of FTIR-ATR spectroelectrochemistry has been to study the electrochemical doping of electronically conductive polymers that can be electrochemically synthesized directly on the ATR crystal or an intervening electrode [49, 50, 51, 52, 53, 54, 55, 56]. There is no reason that this technique could not be extended to other types of membranes, including those that cannot be electrochemically synthesized, and to the study of electric-field-driven transport with or without redox reactions.

Surface-enhanced spectroelectrochemistry [57] is also exciting in that surface specific measurements are possible (much shorter sampling length than the depth of penetration from total internal reflection in a purely dielectric ATR set-up). This enables surface-specific phenomena to be examined and much thinner films to be studied while maintaining the thick film approximation. Another advancement in the study of thin films is polarization modulation infrared reflection absorption spectroscopy (PM-IRRAS) that was applied to water transport in Nafion [58]. This is not an ATR technique, but it is well-suited for studying transport in thin films. Traditional FTIR-ATR has also been applied to thin films, but not for studying transport. It was found that the FTIR-ATR intensity scaled linearly with thickness for sub-micron thick membranes [59].

FTIR-ATR spectroscopy is a versatile, powerful technique for studying multicomponent transport in thin and thick membranes. However, it has been applied to a rather limited set of material systems. Thus, its greatest promise from the membrane transport perspective is in the application to new material systems.

Acknowledgments

This chapter was written with funding support from the National Science Foundation, award numbers 1751450 and 1804871. The author wishes to acknowledge Ashley David for supplying example FTIR-ATR spectra and data.

Conflict of interest

The author declares no conflict of interest.