Mathematical Models for Estimating Diffusion Coefficients in Concentrated Polymer Solutions from Experimental Data

Abstract

Diffusion processes in operations involving polymeric materials are of significant interest. Determining experimental values for diffusion coefficients is often challenging. Estimating these coefficients in concentrated polymer solution, polymer films, and membranes relies on experimental tests where the polymer is brought into contact with certain components/solvents. The diffusion coefficient values depend on the diffusion type, which is affected mainly by the nature of the polymer, concentration, and temperature. The literature presents an extensive amount of information regarding the diffusion phenomenon. This paper makes a particular contribution by showing how experimental data obtained from different applications can be processed to determine diffusion coefficients. The manuscript addresses some aspects regarding solvent diffusion in polymers, and illustrates how to determine the diffusion coefficients from experimental data. For specific cases of diffusion, several models for the predictive estimation of diffusion coefficients are also presented. Polymer–solvent systems such as polyvinyl alcohol (PVA)–water, cellulose acetate (CA)–tetrahydrofuran (THF) and cellulose triacetate (CTA)–dichloromethane (DCM) are investigated, with their diffusion mechanisms influenced by changes in structure caused by variations in concentration and temperature. The experimental data obtained through a gravitational technique allow for the highlighting of the diffusion mechanism and the selection of an appropriate mathematical model. A change in the structure of the polymer during the experiment leads to diffusion anomalies. Modeling the experimental data yielded diffusion coefficient values that vary based on the type of system investigated, composition and temperature. Thus, in the case of the CTA-DCM system, the diffusion coefficient at 303 K, at various concentration values, is in the range of 4.5 and 8·10⁻¹¹ m²/s; for the PVA-H₂O system, D = 4.1·10⁻¹² m²/s at 303 K, and D = 6.5·10⁻¹² m²/s at 333 K; while for the CA-THF system, the solvent–polymer diffusion coefficient has a value of 2.5∙10⁻¹² m²/s at 303 K, and D = 1.75∙10⁻¹¹ m²/s at 323 K. Mathematical models can be useful in studies regarding the drying of polymer films with complex structures, providing knowledge for designing or selecting suitable equipment.

1. Introduction

The permeation of atoms, molecules, and nanoparticles through a dense liquid, membrane or polymer network is a complex problem characterized by many factors at the nanometric scale and is important in materials science, nanotechnology, and polymer physics [1,2,3,4]. Among potential applications, the following can be mentioned: the separation of gases and organic molecules in rubbery and glassy polymeric media [4,5,6], barrier materials for coatings [7], solvent transport in biological and polymeric materials [8], nanofiltration [9], and drug release [10]. The permeation coefficient is given by the multiple of the penetrant solubility at equilibrium and the diffusion coefficient. This latter transport property is generally sensitive to many thermodynamic factors.

Determining the diffusion coefficient is of great importance in many operations involving the processing and use of polymeric materials; however, obtaining its experimental value can be challenging due to numerous technical aspects. These difficulties often require the estimation of diffusion coefficients based on mathematical models. The literature presents various models that have been developed, starting from several concepts.

Penetrant diffusion coefficient values are often built on the free volume concept [11,12], which controls molecular transport. This concept, first introduced by Cohen and Turnbull in 1959 [13], is well known in polymer science. The derivation is based on the idea that diffusive displacement is due to voids generated by the redistribution of free volume. The diffusion constant D in a liquid of hard spheres and the “free volume” ν_f is as follows:

$$D=A exp\left[-\frac{\gamma v^{*}}{v_f}\right].$$

(1)

The free volume space is actually the corresponding volume occupied by the molecules and their surrounding space. The random thermal fluctuations of the fluid lead to the continuous redistribution of the free volume. Contrary to the free volume, the occupied volume is independent of temperature. A molecule migrates in the liquid when a free volume hole occurs near the molecule. Therefore, random fluctuation in the local density results in molecular diffusion. According to the Cohen—Turnbull theory, the self-diffusion coefficient in a pure liquid depends on the probability that a hole of critical size is formed due to random fluctuations in free volume [14].

The Fujita model [15] is the first one based on the free volume theory. For its development, a system composed of one solvent, one plasticizer, and one polymer was experimentally investigated. In relation to the polymer concentration, a low value of plasticizer concentration was maintained so that the considered ternary system could be reduced to a pseudo-binary one. Consequently, the mean free volume was caused by the solvent and the polymer. Further, Fujita estimated the free volume based on Cohen and Turnbull’s concept. Thus, assuming a liquid composed of identical molecules, the probability of finding voids of size ν* can be expressed as follows:

$$P\left(v^{*}\right)=A·exp P\left(v^{*}\right)=A\left(-\frac{b{v}^{*}}{f_v}\right).$$

(2)

Fujita assumed that Equation (2) is also valid for binary systems. The probability that a molecule finds a void is closely related to the mobility of the diffuser, m_d:

$$m_d=A\left(-\frac{B}{f_v}\right).$$

(3)

The definition of mobility is given by the Equation (4):

$$D=R ·T ·m_d.$$

(4)

Substituting Equation (3) in (4) results in expression (5):

$$D=A·R·T\left(-\frac{B}{f_v}\right),$$

(5)

Although the Fujita model can be used for the theoretical description of diffusion in polymer–organic solvent systems, in polymer–water systems, it does not provide reliable results due to the numerous interactions between molecules.

Following the Fujita model, Yasuda et al. [16] assumed that the free volume of a binary system depends mainly on the solvent volume fraction. This hypothesis stands on the assumption that the solvent mobility is significantly higher than that of a polymer. Thus, the effective free volume is mainly caused by the solvent, and the solvent diffusion decreases with polymer concentration. Therefore, the total free volume is due to both the solvent and the polymer, and is expressed by Equation (6):

$$f_V={\varphi }_s{f}_s+\left(1-{\varphi }_s\right)f_p=f_p+{\varphi }_s\left(f_s-f_p\right),$$

(6)

Assuming no interactions between a molecule and polymer, it can be written as follows:

$$\frac{D}{D_0}=exp\left[\frac{B}{f_V^{*}}\left(1-\frac{1}{1-{\varphi }_p}\right)\right],$$

(7)

The model of Yasuda et al. can be used for diffusion studies in dilute or semi-dilute polymer systems, and for diffusing molecules of relatively small size.

This free volume concept has been extended by Vrentas and Duda (1977) [11] for the case of concentrated polymer solutions. Later, Vrentas et al. (1984) [17] applied this model to describe diffusion in ternary polymer–solvent–solvent systems using binary polymer–solvent diffusion data.

According to the model, polymer volume is divided into three elements (Figure 1). (a) The occupied volume corresponds to the polymer volume with no free volume, and is also known as the “van der Waals” volume. It has a constant value, independent of temperature, and can be estimated based on non-bonded contact radii and bond distances. (b) The interstitial free volume originates from the vibrational energy of polymer chains, and it is slightly dependent on temperature. (c) The hole free volume stems from volume relaxation and plasticization due to polymer heating and cooling, and can be affected by solvent adsorption/desorption. The total polymer volume is mainly influenced by the hole free volume. Fast polymer relaxations allow rubbery polymers to achieve the equilibrium volume. Glassy polymers, however, are nonequilibrium liquids where relaxation is slow. Therefore, as more hole free volume is confined, glassy polymers present a larger volume than at equilibrium.

Although polymers may be considered porous in this context, Duda and Zielinski [18] emphasized that the volume confined in glassy polymers is repeatedly distributed. In cases of porous materials, the solvent governs the transport mechanism. Commonly, numerous polymers are nonporous. In this instance, the solvent within the polymer remains undefined, and the transport mechanism cannot be addressed in terms of liquid or vapor.

In the literature, the diffusion coefficient in polymer–solvent systems is called the “self-diffusion coefficient” as well. According to the IUPAC, when the chemical potential gradient is zero, the diffusion coefficient of species i is referred to as the self-diffusion coefficient [19]. Accurately, according to Albright and Milles, this nomenclature is valid in cases of one component diffusion within itself [20].

In terms of the types of diffusion coefficients, the literature mentions self-diffusion coefficients, reciprocal diffusion coefficients, mutual diffusion coefficients, Fick diffusion coefficients, etc. [21,22,23].

In order to describe the diffusion phenomenon in polymers, Vrentas and Duda [11] developed the free volume theory. The mutual diffusion coefficient is the product of a solvent self-diffusion coefficient and a thermodynamic factor:

D = D₁ · Θ,

(8)

$$D_1=D_0·\mathit{exp}\left(\frac{-\left({\omega }_1{\widehat{V}}_1^{*}+{\xi }_{1p}{\omega }_p{\widehat{V}}_p^{*}\right)}{{\widehat{V}}_{FH}/{\gamma }_{1p}}\right),$$

(9)

$$\Theta =(1 - {\Phi }_1^2)(1-2{\chi }_{1p}{\Phi }_1).$$

(10)

For ternary polymer–solvent 1–solvent 2 systems, the self diffusion coefficient is given as follows:

$$D_1=D_{01}·\mathit{exp}\left(-\frac{{\omega }_1{\widehat{V}}_1^{*}+\frac{{\xi }_{1P}}{{\xi }_{2P}}·{\omega }_2{\widehat{V}}_2^{*}+{\xi }_{1P}·{\omega }_P{\widehat{V}}_P^{*}}{\frac{{\widehat{V}}^{FH}}{\gamma }}\right),$$

(11)

$$D_2=D_{02}·\mathit{exp}\left(-\frac{{\omega }_1{\widehat{V}}_1^{*}·\frac{{\xi }_{2P}}{{\xi }_{1P}}+{\omega }_2{\widehat{V}}_2^{*}+{\xi }_{2P}·{\omega }_P{\widehat{V}}_P^{*}}{\frac{{\widehat{V}}^{FH}}{\gamma }}\right),$$

(12)

$${\widehat{V}}^{FH}={\omega }_1{K}_{I,1}\left(K_{II,1}-T_{g,1}+T\right)+{\omega }_2{K}_{I,2}\left(K_{II,2}-T_{g,2}+T\right)+{\omega }_P{K}_{I,P}\left(K_{II,P}-T_{g,P}+T\right),$$

(13)

where D_0i is the pre-exponential factor of component i; ω_i represents the weight fraction; and ${\widehat{V}}_i^{*}$ is the specific critical hole free volume necessary for a jump of component i. The total hole free volume, ${\widehat{V}}_{FH}$, is a function of parameters K_I,i and K_II,i − T_gi. The parameters D₀ and ξ are very sensitive. By strongly varying these two parameters, specific dependences of the diffusion coefficient can be described.

Although diffusion and mass transport coefficients can be determined through relatively simple experiments, following the mass variation over time, the particular case of polymer–solvent systems is highly complex. Diffusion in these systems is influenced by numerous factors. Consequently, the literature reports plenty of experimental data and computational models. We consider that many aspects still remain unexplained, and research in this field is ongoing.

The experimental data obtained through a gravitational technique allow the highlighting of the diffusion mechanism and the selection of an appropriate mathematical model. A change in the structure of the polymer during the experiment leads to diffusion anomalies. Three different polymer–solvent systems are considered, CA-THF, PVA-H₂O, and CTA-DCM, in order to calculate diffusion coefficients at various compositions and temperatures. Mathematical models can be useful in studies regarding the drying of polymer films with complex structures, providing knowledge for designing or selecting suitable equipment.

2. Diffusion Coefficients from Experimental Data

2.1. Methodology

The experimental determination of the diffusion coefficients in polymeric membranes/films involves conducting experiments in which the polymeric films/membranes are brought into contact with the component whose diffusion coefficient is to be determined. The values of the diffusion coefficients depend on the diffusion type, which is influenced by the nature of the used polymer (the type of polymer) and the diffusion temperature, which influences the amorphous or crystalline state of the polymer. It is also very well known that the glass transition temperature, T_g, takes different values during diffusion and depends on the polymer purity. A change in concentration can lead to important changes in T_g. As a result, in the diffusion experimental test, the change in the concentration of the polymer membrane/film must take place over the narrowest possible range. The range of concentration variation can be controlled by establishing equilibrium conditions between the fluid phase and the polymer membrane/film within which the diffusion of a certain component takes place.

Diffusion experiments mostly monitor either the variation of the mass of the polymer film over time or the variation of the amount of diffusing component over time.

The first step in the processing of experimental data is the graphical plot of the diffusing component mass as a function of the diffusion time. The shape of this simple dependence can provide the first insights into the diffusion mechanism. Subsequently, the experimental data are processed using literature models that account for various factors such as different operating conditions, diverse behaviors of the system during diffusion, the type of polymer used, structural transformations occurring in the membrane/film during diffusion, and the type of membrane interaction with the diffusing component, among others. Figure 2 presents a scheme of the proposed methodology.

Diffusion processes in one dimension are expressed as follows:

$$\frac{\partial C}{\partial t}=D\frac{{\partial }^{2}C}{\partial x^2},$$

(14)

where D is the mutual diffusion coefficient and C is the concentration. Having the initial and boundary conditions corresponding to a 2l (−l to l) thickness of a film exposed to an infinite reservoir [24]:

$$\begin{gathered} C=C_0 \mathrm{when} 0<x<l, t=0 \\ C=C_1 \mathrm{when} x=l, t\ge 0 \\ \frac{\partial C}{\partial x}=0 \mathrm{when} x=0, t\ge 0 \end{gathered}$$

(15)

Crank [24] reported the solution of Equation (14) based on the boundary conditions (15):

$$\frac{C-C_0}{C_1-C_0}=1-\frac{4}{\pi }{{\displaystyle \sum }}_{n=0}^{\infty }\frac{{\left(-1\right)}^n}{2n+1}exp\left[\frac{-D{\left(2n+1\right)}^2{\pi }^{2}t}{4l^2}\right]cos\left[\frac{\left(2n+1\right)\pi x}{2l}\right],$$

(16)

Equation (16) [24] can be further derived for the specific case of gravimetric sorption:

$$\frac{M_t}{M_{eq}}=1-\frac{8}{{\pi }^2}{{\displaystyle \sum }}_{n=0}^{\infty }\frac{1}{{\left(2n+1\right)}^2}exp\left[\frac{-D{\left(2n+1\right)}^2{\pi }^{2}t}{l^2}\right],$$

(17)

where M_t and M_eq are the amount of penetrant diffused into the polymer at a given time, t, and the penetrate amount at equilibrium, respectively.

In a particular case of short time, Equation (17) becomes the following:

$$\frac{M_t}{M_{eq}}=\frac{2·D^{\frac{1}{2}}}{l}\sqrt{t}\left\{\frac{1}{\sqrt{\pi }}+2{{\displaystyle \sum }}_{n=1}^{\infty }{\left(-1\right)}^{n}ierfc\frac{nl}{\sqrt{Dt}}\right\},$$

(18)

and can be further approximated as follows:

$$\frac{M_t}{M_{eq}}=\frac{2}{\sqrt{\pi }}\frac{D^{\frac{1}{2}}}{l}\sqrt{t},$$

(19)

The diffusion coefficient can be determined by means of Equation (20), determining the initial slope, S, of the dependence M_t/M_eq and $\sqrt{t}$:

$$D=\frac{\pi }{4}{S}^2{l}^2,$$

(20)

When diffusion occurs in films/membranes with different thicknesses, a more accurate dependence is M_t/M_eq to $\sqrt{\frac{t}{l^2}}$. The determined initial slope, S_l, of the dependence is as follows:

$$D=\frac{\pi }{16}{S_l}^2,$$

(21)

where $S_l$ is the following:

$$S_l=\frac{d\left(\frac{M_t}{M_{eq}}\right)}{d{\left(\frac{t}{l^2}\right)}^{\frac{1}{2}}},$$

(22)

where $S_l$ is the slope of the first part of sorption curve during the initial time period. As shown in Equations (18)–(20), the diffusion coefficient depends on both the film thickness and the initial slope of the linear part of the absorption curve. This provides information on the rate of absorption of the solvent vapor mass on the polymer membrane.

When diffusion of penetrant in polymer reveals a non-Fickian behavior, the approach of diffusion coefficient estimation on the basis of the initial stage and constant diffusion coefficient is widely accepted by the scientific community [25,26,27,28,29,30].

2.2. Fick or Non-Fick Diffusion

Polymers in a solid state can reach two different states. At values below the glass transition temperature, the polymer is in crystalline state, a hard and brittle material. The glass transition temperature is the temperature at which a polymer changes from a crystalline to an amorphous state [31]. The concept of glassy state was presented in detail by Arya et al. [32].

Most of the properties, such as diffusivity, thermal conductivity, specific volume, dielectric constant, refractive index, and hardness, vary considerably for polymers in amorphous and crystalline states, respectively. The difference between the polymer physical properties in amorphous and crystalline states is caused by the mobility of the macromolecular chain.

Theories and models describing diffusion in polymers rely on chain mobility at the molecular scale. When a polymer absorbs a solvent, its molecules rearrange until a new equilibrium conformation is established. In an amorphous state, the diffusion follows a Fick-type mechanism. In this case, the polymer chain segments are active enough to reach structural equilibrium instantaneously.

In crystalline polymers, the equilibrium chain conformation establishes much slower due to the limited mobility of the polymer segments. In this particular case, different types of segmental motion generate different diffusion mechanisms under various conditions.

Diffusion mechanisms in crystalline polymers can be classified as [31] (1) case I (Fickian) diffusion, and (2) case II (non-Fickian) diffusion, anomalous diffusion.

Sorption kinetics are shown by the concentration distribution of the penetrant in the polymer or by the change in polymer weight due to absorption. Differential or interval sorption occurs when the difference between the activity of the external penetrant and that of the equilibrium penetrant in the polymer has a very low value. This results in small variations in the penetrant concentration in the polymer. This kind of kinetics allows one to calculate values of diffusion coefficients throughout a narrow range of concentrations, and the obtained data highlight the variation of diffusion coefficient as a function of concentration. Even though the method (Quartz Crystal Balance, NMR, IGC, etc.) used is different, the literature presents diffusion coefficients obtained from sorption–desorption experiments.

Case II is particular to sorption processes of small molecules in crystalline polymers or during the interdiffusion of polymers when only one polymer is in a crystalline state [33,34].

Characteristic of case II sorption are the following key features: (i) the necessity of an induction time for the formation of the fluid penetration front; (ii) the consistent movement of the front at a constant velocity; (iii) the inverse relationship between the induction time and the front penetration rate; and (iv) the nearly stepwise distribution of penetrant concentration within the swelling polymer.

The following observations are also associated with case II sorption: (i) the fronts serve as a distinct indication of the heterogeneity within the swelling polymer; (ii) the rate controlling process occurs at the front; (iii) case II sorption is nonlinear and irreversible; and (iv) case II sorption may be treated as a quasi-steady transport process.

When a glassy polymer interacts with a highly active organic swelling agent, it undergoes partial absorption almost instantaneously, filling the vacant free volume at its surface. Subsequent absorption of the diffusant induces swelling or the plasticization of the polymer. This transformation in the polymer’s state is facilitated by alterations in its conformation, affecting its chains or segments. The dynamics of this conformational change are dictated by the mobility of the polymer chain, which is influenced by both intramolecular and intermolecular forces. These forces, in turn, can be manipulated by factors such as experimental temperature, diffusant type, and concentration. The resulting conformational changes, crucial for concentration kinetics, do not occur instantaneously, requiring a significant time interval for the diffusant’s surface concentration to reach equilibrium.

The heterogeneous nature of the swelling polymer can be segmented into three distinct regions: a swollen zone, a transitional swelling area, and an unswollen region. Isotropic conformational shifts in polymer chains due to penetrant swelling lead to lateral expansion parallel to the polymer’s surface. However, this expansion is constrained by the unswollen core through neighboring unaffected chains and segments. The most energetically favorable path for this conformational change lies perpendicular to the surface. Consequently, changes in polymer conformation and sample dimension are anisotropic. The swelling region manifests as an interface or front separating the swollen zone from the unswollen region. Nevertheless, this demarcation is not strictly geometric. In the case of poly(methyl methacrylate) (PMMA)/methanol systems, the estimated thickness of the swelling region is approximately 100 nm, as established by Durning et al. [34].

Polymer relaxation in response to osmotic swelling stresses or plasticization primarily occurs within this region. The kinetics of relaxation and plasticization are contingent upon the diffusant’s activity within this region and the morphology of the unswollen polymer proximate to it. Within the swollen region, osmotic swelling pressure combines with differential swelling stress, typically compressive in nature. In instances where the swollen region is homogeneous, diffusion can be elucidated using a modified form of Fick’s law, incorporating stress and concentration-dependent diffusion coefficients. Depending on experimental conditions, the swollen polymer may exhibit either a rubbery or glassy state. Notably, DSC results on PMMA-methanol systems have revealed instances where the swollen PMMA remains in a glassy state at temperatures below 313 K, as shown by Lin et al. [35].

The unswollen glassy region’s homogeneity or heterogeneity is contingent upon the level of tensile stress and plasticization engendered by the diffusant. Heterogeneity arises when the stress exceeds the yield strength of the unswollen polymer, potentially resulting in cracking. Diffusion in this region may follow Fickian or non-Fickian behavior, contingent upon the morphology and degree of mechanical damage present within the unswollen core. Mechanical damage or core failure due to differential swelling stresses may accelerate the migration of the swelling region when diffusional resistance is minimal. Consequently, a spectrum of diffusion patterns may manifest within the same polymer across varying temperatures, diffusant types, and activities.

The diffusion process is strongly influenced by several parameters (including temperature, pressure, solute size, and the viscosity of the medium in which the diffusion occurs). Consequently, diffusion processes in liquids and solids are much slower. Diffusion coefficients are difficult to estimate with theoretical models, since in solids, they can differ by one or more orders of magnitude. Guiding values of the diffusion coefficients in the three states of aggregation are given in Figure 3.

3. Materials and Methods

The experimental data used to derive the diffusion coefficients were obtained in two ways: (i) for the drying of polyvinyl alcohol (PVA)–distilled water and cellulose acetate (CA)-tetrahydrofurane (THF, anhydrous, ≥99.9%, Sigma Aldrich, Saint Louis, MO, USA) films; and (ii) in cases of sorption/desorption of dichloromethane (DMT, anhydrous, ≥99.8%, Sigma Aldrich, Saint Louis, MO, USA) in cellulose triacetate (CTA, Sigma Aldrich, Saint Louis, MO, USA) films using the sorption equipment with magnetic suspension coupling, described in our prior work [26].

The experiments were carried out at 303 and 323 K using cellulose triacetate–dichloromethane films. Solvent concentration was modified according to the solvent activity in a sorption cell. Each experiment was carried out in the presence of a pure dichloromethane vapor (in absence of air) at a fixed temperature in a sorption cell connected to an evaporator. Solvent activity in the sorption cell was adjusted by controlling the temperature in the evaporator. The polymer films were obtained by the slow evaporation of the solvent from a cellulose triacetate-dichloromethane solution. Film thickness was determined according to film weight, film surface, and polymer density [36].

PVA is a synthetic, linear, non-toxic semi-crystalline polymer, biocompatible, biodegradable, and water soluble at temperatures above its glass transition (85 °C). The PVA-H₂O film was prepared by adding a 5% solution of PVA (powder, average M_w 89,000–98,000, Sigma Aldrich, Saint Louis, MO, USA) in distilled water.

Cellulose acetate (CA) is an important cellulose ester described by Fischer et al. [37]. CA powder (average M_w~30,000) was purchased from Sigma Aldrich (Saint Louis, MO, USA).

For the CTA-DCM system, the experimental method and procedures are described in our prior work [38].

The solvents selected in these tests, to solubilize the considered polymers, were THF, DCM, and distilled water, respectively.

The equipment used for the PVA-H₂O system is presented in work [26]. The drying equipment was an in-house set-up and included a drying tunnel with a rectangular section. The weighing of the samples was performed with a digital analytical balance (Precisa Gravimetrics AG, Dietikon, Switzerland) with a 0.1 mg sensitivity. Absolute pressure was measured using a Tecsis pressure gauge (WIKA Group, Kent, UK) connected do a data logger, midi Logger GL800 (Graphtec Corporation, Shenzhen, China). Temperature measurements were carried out using thermocouples (Pt 100) connected to a data logger (Sper Scientific, Scottsdale, AZ, USA 800024 Thermocouple Thermometer, 4 Channel Datalogging).

4. Results and Discussion

4.1. Fick Diffusion (Case I)

In polymer networks, Fick diffusion can be observed if the temperature value is significantly higher than that of the glass-transition temperature (T_g). The polymer–solvent system responds immediately when the polymer in a viscous fluid state allows the solvent to penetrate more easily due to the higher mobility of the polymer chains. Thus, in case 1, the solvent diffusion rate, R_dif, is much lower than the polymer relaxation rate R_relax (R_dif « R_relax), which leads to a strong solvent penetration gradient within the system. Figure 4 exemplifies an ideal shape of Fick sorption curve. The diffusion distance is proportional to t^1/2:

M_t = kt^1/2,

(23)

4.1.1. Diffusion Coefficients from Drying Experiments CA-THF

A drying process for a polymer film can be described on the base of a physical model that considers the polymer film placed on a flat surface, having a thickness δ = δ₀ when t = 0. This model assumes uniform temperature (T₀) (Figure 5) [39].

In order to model the drying process of polymeric films, the drying plot can be split into two distinct zones: (i) the first step of drying, stage I; and (ii) the second zone includes stages II and III.

During the second and third drying stages (actually, the second zone), the solvent transport within the polymer solution is governed by diffusion and described by Fick’s law. When the temperature, film thickness, and diffusion coefficient are constant, the variation of the solvent content over time is given by Crank as follows:

$$\overline{M}=\frac{M_{\left(t\right)}-M_{eq}}{M_0-M_{eq}}=\frac{8}{{\pi }^2}{{\displaystyle \sum }}_{n=0}^{\infty }\frac{1}{{\left(2n+1\right)}^2}\mathit{exp}\left[-\frac{{\left(2n+1\right)}^2{\pi }^2{D}_{A}t}{{\delta }^2}\right],$$

(24)

where $\overline{M}$—normalized solvent mass, dimensionless; $M_0$—initial solution mass, g; $M_{eq}$—solution mass at equilibrium, g; $M_{\left(t\right)}$—solution mass at time t, g; $\delta$—film thickness, m; t—time, s; and D_A—solvent–polymer diffusion coefficient, m² s⁻¹.

Equation (24) derived for n = 3 results in the following:

$$\frac{M_{\left(t\right)}-M_{eq}}{M_0-M_{eq}}=\frac{8}{{\pi }^2}\left[ex{p}^{-D_{A}t{\left(\frac{\pi }{\delta }\right)}^2}+\frac{1}{9}ex{p}^{-9D_{A}t{\left(\frac{\pi }{\delta }\right)}^2}+\frac{1}{25}ex{p}^{-25D_{A}t{\left(\frac{\pi }{\delta }\right)}^2}\right],$$

(25)

A comparison of experimental data-values was calculated with the Equation (25) cellulose acetate (CA) tetrahydrofuran (THF) system. The data were obtained when drying some films with a thickness of 200 μm (dry) in different temperature conditions and at low air flow speeds through the drying tunnel. For each experiment at a temperature of 303 K, the solvent–polymer diffusion coefficient had a value of 2.5∙10⁻¹² m² s⁻¹, and at 323 K, D = 1.75∙10⁻¹¹ m² s⁻¹.

The experimental data presented in Figure 6a show that in the case of the CA-THF system, temperature has a marginal influence, the diffusion coefficients taking relatively close values. Although the diffusion is of type I (Fick), as a result of the structure of the CA film, which does not behave like a compact polymer, a dependence of the diffusion coefficient on temperature is not highlighted in the range considered.

The effect of the THF used to prepare the CA films was found by means of calorimetric thermograms analysis [40]. When THF was used to prepare the CA membrane, the melting point was not observed. This suggests that CA membranes prepared with THF have an amorphous structure

Figure 6b shows the variation of the film weight during drying at 323 K. The shown values resulted from the drying simulation with Equation (25), considering the average value of the diffusion coefficient D = 1.75·10⁻¹¹ m² s⁻¹ and variable thickness of the film (500–200 µm).

The same drying process conducted at 303 K proceeded according to Figure 6c. For this simulation, the value of the diffusion coefficient was D = 1.5·10⁻¹¹ m² s⁻¹, and the thickness of the film varied between 500 and 200 µm.

4.1.2. Diffusion Coefficients from Drying Experiments PVA-H₂O

The experiments in the PVA–water system were conducted at two different temperature values, 303 and 333 K. The air flow rate had a low value so that in the final drying stage, convection could be neglected.

From the experimental drying data in the final period, plotting the solvent mass loss as a function of t^1/2, an average value of the diffusion coefficient of the solvent in the film was calculated according to Equation (21).

Figure 7b,c show the experimental data used to calculate the values of the diffusion coefficient.

The mass variation during the drying of the films is shown in Figure 7. The obtained pattern is described by Fick’s diffusion. The difference that appears in Figure 7a between the experimental and calculated values could be caused by the possible influence of the convection at small time intervals.

As the polymer film changes its thickness during drying, in the modeling of the experimental data, a variation range of film thickness between 125 and 250 µm was considered. The diffusion coefficients had the following values: D = 4.1·10⁻¹² m² s⁻¹ at 303 K, and si D = 6.5·10⁻¹² m² s⁻¹ at 333 K. The mass variation during film drying is shown in Figure 7a–c. Policastro et al. estimated diffusion coefficients for water in polyutrethane and epoxy systems [41]. They employed a technique based on capacitance measurements, which revealed diffusion coefficients for water in polyurethane in the range of 10⁻¹⁴–10⁻¹³ m² s⁻¹. Values of solvent–polymer diffusion is of the same order as those reported in this present study. Negoescu determined experimental diffusion data in PVA-H₂O systems using a pressure decay measuring method. He reported a diffusion coefficient value of 1.42·10⁻¹² m² s⁻¹ [42]. The differences between the experimental and calculated values that occur in the early stages of the drying process are mainly due to the possible influence of the convection phenomenon.

The mathematical model used can predict data with reasonable accuracy. Information on the drying behavior of some polymer films can be relatively easily obtained based on the considered model; thus, the variation of the film thickness during drying or the drying time for a certain system with an imposed film thickness can be determined.

4.2. Two-Stage Diffusion

A model for explicit separation of diffusion and relaxation parameters was proposed by Berens and Hopfenberg [43] as shown in Figure 8. In glassy polymers, the sorption process is viewed as a linear superposition of phenomenologically independent contributions from Fick’s diffusion and polymeric relaxation. In the diffusion–relaxation model, Berens and Hopfenberg regarded the absorption process as consisting of two distinct contributions: a diffusion part, M_F(t), obeying Fick’s laws, and a structural part, M_R(t), stemming from polymer relaxation:

$$M\left(t\right)=M_F\left(t\right)+M_R\left(t\right),$$

(26)

M_F(t) is determined by Equation (26) mathematical solutions. It is assumed that multiple independent relaxation processes may occur, hence, M_R(t) is given by Equation (27):

$$M_R\left(t\right)={{\displaystyle \sum }}_i{M}_{\infty }\left[1-{\mathrm{e}}^{-kt}\right],$$

(27)

where M_∞ is the equilibrium absorption owed to the relaxation process, and k is the relaxation process first-order constant. Multiple groups reported studies on two-stage sorption, a significant non-Fickian characteristic of glassy polymer systems [43,44].

This type of diffusion has been met in systems such as polystyrene-ethylbenzene, CTA-DCM [38], and acetone-PVC [45]. Long and Richman [46] were likely the first group to offer a plausible explanation for two-stage sorption behavior by means of a variable surface concentration model. Corresponding to Fick’s second law, Equation (26) describes diffusion within a membrane, assuming the following initial conditions:

t < 0, C = 0, 0 < x < l,

(28)

where l represents membrane half-thickness. The variable surface concentration model proposes that once the membrane comes into contact with the vapor, the concentration at the membrane surface (x = l) rapidly transitions to C₀ and subsequently reaches a final concentration, C_∞, through a first-order relaxation process. At the membrane center (x = 0), the symmetry of the concentration is assumed. The boundary conditions are as follows:

$$\mathrm{t} \ge 0 \{\begin{array}{cc}C=C_0+\left(C_{\infty }-C_0\right)\left(1-{\mathrm{e}}^{-kt}\right)& x=l\\ \frac{\partial C}{\partial x}=0& x=0\end{array},$$

(29)

where k is the relaxation process rate constant. Constant diffusivity D₀ is assumed in solving Equation (26), using the following dimensionless variables:

$$z=\frac{x}{l}, \overline{C}=\frac{C}{C_{\infty }}, \theta =\frac{D_{0}t}{l^2}, \psi =\frac{k{l}^2}{D_0}, \varphi =\frac{C_0}{C_{\infty }},$$

(30)

where θ is the dimensionless time, Ψ is the inverse of the diffusion Deborah number; and Φ is the equilibrium ratio constant, indicative of the ratio between the equilibrium concentration of the first stage and that of the second stage in the process of sorption.

The integration of the dimensionless concentration over z = 0 to z = 1 renders the fractional weight incorporation as a function of dimensionless θ (Equation (31)).

$$\frac{M_t}{M_{\infty }}=\mathsf{\Phi }\left[1-\frac{8}{{\pi }^2}{{\displaystyle \sum }}_{n=0}^{\infty }\frac{exp\left(\frac{-{\left(2n+1\right)}^2{\pi }^2\theta }{4}\right)}{{\left(2n+1\right)}^2}\right]+\left(1-\mathsf{\Phi }\right)\left[1-\frac{\tan \sqrt{\mathsf{\Psi }}exp\left(-\mathsf{\Psi }\theta \right)}{\sqrt{\mathsf{\Psi }}}-\frac{8}{{\pi }^2}{{\displaystyle \sum }}_{n=0}^{\infty }\frac{exp\frac{-{\left(2n+1\right)}^2{\pi }^2\theta }{4}}{{\left(2n+1\right)}^2\left(1-\frac{{\left(2n+1\right)}^2{\pi }^2}{4\mathsf{\Psi }}\right)}\right],$$

(31)

The initial term on the right-hand side of Equation (31) denotes the traditional Fickian diffusion to the quasi-equilibrium (first stage), representing the weight uptake for the penetrant diffusing down the concentration gradient established by the initial surface concentration. The subsequent term accounts for the penetrant that enters due to the temporal variation of the surface concentration.

$$\frac{M_t}{M_{\infty }}=\mathsf{\Phi }\left[1-\frac{8}{{\pi }^2}{{\displaystyle \sum }}_{n=0}^{\infty }\frac{exp\left(\frac{-{\left(2n+1\right)}^2{\pi }^2\theta }{4}\right)}{{\left(2n+1\right)}^2}\right]+\left(1-\mathsf{\Phi }\right)\left[1-exp\left(-\mathsf{\Psi }\theta \right)\right],$$

(32)

Proposed by Berens and Hopfenberg [43], the diffusion–relaxation model interprets the sorption process in crystalline polymers as a linear combination of contributions from both Fickian diffusion and polymeric relaxation. In their experiments with submicron particles, the initial diffusion-controlled sorption proceeds more rapidly than the long-term relaxation, allowing the entire sorption process to be explicitly divided into two seemingly independent mechanisms. For planar geometries, such as membranes, the fractional uptake with dimensionless time is expressed in Equation (33), assuming a single first-order relaxation process. In this context, θ has the same significance as in the variable surface-concentration model, but it can also be interpreted as the ratio of the equilibrium sorption amount in the unrelaxed polymer to that in the fully relaxed polymer. Meanwhile, Ψ and Φ retain the same meanings as those in the variable surface-concentration model.

$$\frac{M_t}{M_{\infty }}=\mathsf{\Phi }\left[1-\frac{8}{{\pi }^2}{{\displaystyle \sum }}_{n=0}^{\infty }\frac{exp\left(\frac{-{\left(2n+1\right)}^2{\pi }^2\theta }{4}\right)}{{\left(2n+1\right)}^2}\right]+\left(1-\mathsf{\Phi }\right),$$

(33)

In these models, Equations (31) and (32) describe the sorption behavior that integrates Fickian diffusion with a first-order relaxation process. If Φ is known, the fractional uptake (M_t/M_∞) as a function of dimensionless time, θ, depends solely on the value of Ψ.

Figure 9 shows the diffusion behavior of the CTA-DCM system based on representations of the dimensionless mass of solvent removed from the solution as a function of the square root of time. Dot line highlights the experimental results, whereas solid lines represent data calculated with Equation (32). The dependence between the ratio M/M_eq (dimensionless mass variation) vs. t^1/2 was determined based on the solvent concentration profiles in the polymer film. The average values of the solvent concentration are also specified in Figure 9. The experimental data show that for this system, the diffusion takes place in two distinct steps. The diffusion coefficient was determined based on Equation (20), using the slope from step 1. Three ranges of solvent concentration variations were considered, as indicated on Figure 9 with dotted lines, corresponding to the right vertical axe. The average values of the three ranges of solvent concentration are also detailed in

Table 1. Model parameters, Equation (32).

Xm, kg kg−1	Φ, -	D·1011, m2 s−1	k·104, -
0.27	0.48	8.0	4.5
0.36	0.51	6.5	6.0
0.50	0.72	4.5	7.5

. Whereas at the highest solvent concentration (x = 0.5) the behavior is Fickian, (with the slope change occurring at M/M_eq = 0.7), at the two lower considered concentrations (x = 0.27, x = 0.36), the two stages are much more clearly highlighted, with the change in slope becoming visible at the M/M_eq ratios of about 0.5. This behavior shows that diffusion, relaxation, and reconfiguration phenomena occur in the polymer film at different rates.

Although in most diffusion studies the diffusion coefficient decreases as the solvent content in the film decreases, in case of the investigated system, this variation does not occur. Actually, over a wide range of solvent concentration variation, the experimental values of the diffusion coefficient do not differ significantly. In the case of Fickian diffusion (case 1), the diffusion coefficient decreases by up to two orders of magnitude. However, in this system, where the diffusion takes place in two stages, the variation can be considered within the error limit (40–50%), which is considered reasonable for experimental diffusion data. Together with the parameters specified in Table 1, these values were used to predict the variation of the dimensionless mass M/M_eq vs. t^1/2. The shape of the dependencies obtained after the simulation shows that the used model can be extended for other systems with similar behavior.

Mathematical models that describe diffusion phenomena can predict different behaviors of films during diffusion. However, the complex structure of polymer–solvent systems and the large number of variables influencing diffusion processes still require further study in order to highlight the influence of different parameters, as discussed herein.

5. Conclusions

The practical aspects regarding diffusion in polymer films are of great importance in managing the drying processes of polymer films. The use for large-scale energy conversion of materials that include polymer layers of small thicknesses (e.g., photovoltaic cells) is well known.

This study makes a unique contribution by demonstrating how diffusion coefficients can be established based on experimental data from different applications. The manuscript explores several cases concerning solvent diffusion in polymers and demonstrates effective methods for extracting diffusion coefficients from experimental data obtained from two different experimental setups: one involving a drying tunnel for PVA in water and CA in THF films, and the other consisting of sorption equipment with magnetic suspension coupling for the sorption/desorption of DCM in CTA films. For the CTA-DCM system, the diffusion coefficient at 303 K varies between 4.5 and 8·10⁻¹¹ m² s⁻¹ at different concentrations. For the PVA-H₂O system, the diffusion coefficient is 4.1·10⁻¹² m² s⁻¹ at 303 K and 6.5·10⁻¹² m² s⁻¹ at 333 K. Meanwhile, for the CA-THF system, the solvent–polymer diffusion coefficient is 2.5·10⁻¹² m² s⁻¹ at 303 K and 1.75·10⁻¹¹ m² s⁻¹at 323 K.

The obtained data from drying experiments can be correlated using mathematical models that represent various types of behavior in polymer systems. These models offer valuable insights for designing specialized equipment or for scaling-up processes.

Mathematical models for penetrant diffusion in glassy or amorphous polymer films were used to determine diffusion coefficients in polymer films on the basis of experimental data.

The simulation results have demonstrated that the state of polymer has an important influence on the concentration profiles vs. time. Mathematical equations that describe diffusion in polymers under different conditions were used to simulate the drying of polymer films. Data on film thickness, average diffusion coefficient and time were used for the simulations.

The obtained results show that, based on some imposed parameters, the drying time periods can be estimated, which then allows the selection or design of specific apparatus.