1. Introduction
Wicking is the surface-driven imbibition process in porous microstructures, in which a non-wetting fluid (gas) is replaced by a wetting one (water), when exposed to a capillary suction pressure. This capillary phenomenon is strongly promoted by an open-pored and porous microstructure, which can be found in many materials, such as textiles, woven fibers, and porous polymeric membranes (PPMs). In particular, PPMs are widely used in lateral flow assays (LFAs), where they function as an autarkic microfluidic pump system that transports a liquid sample, containing analyte and detector particles, toward the detection zone (test and control lines, see
Figure 1a).
Due to this quality, cost-effective and easy-to-use LFAs can be realized, covering a wide range of applications in medical diagnostics, drugs of abuse control as well as the environmental monitoring of contamination in water, soil, and air [
1]. Particularly in medical diagnostics, they make it possible to stem and control major pandemics, such as the severe acute respiratory syndrome (SARS) outbreak of 2003, which happened in Asia, or the current worldwide COVID-19 outbreak. Thus, by further improving LFAs, an evident contribution to global health is made [
2,
3].
However, optimizing and designing PPMs for LFAs is challenging for different reasons, such as the fact that (i) multiple length scales are involved, (ii) convection, diffusion, and reaction mechanisms are simultaneously present, and (iii) effective properties of the complex microstructure are difficult to determine. In particular, the prediction of the fluid flow across multiple length scales is crucial, as the flow condition inherently affects the sensitivity of LFAs [
4]. In order to bridge the different length scales, the use of appropriate upscaling methods is necessary. For this purpose, bottom-up approaches are particularly suitable [
5], whereby the information is passed on from the smallest relevant scale to the next larger scale, by using effective parameters. To enable their application, an accurate determination of the effective parameters is essential. In particular, regarding the wicking behavior in PPMs, the use of simplified analytical flow models requires an accurate effective capillary radius.
Figure 1.
Graphical overview: (
a) A typical design of a lateral flow assay (LFA), with highlighted test and control lines; (
b) scanning electron microscopy (SEM) image of a porous polymeric membrane (PPM); (
c) schematic representation of the pore and the structure spaces [
6]; (
d) Lucas–Washburn assumption of a porous medium, as a bundle of parallel capillaries [
6].
Figure 1.
Graphical overview: (
a) A typical design of a lateral flow assay (LFA), with highlighted test and control lines; (
b) scanning electron microscopy (SEM) image of a porous polymeric membrane (PPM); (
c) schematic representation of the pore and the structure spaces [
6]; (
d) Lucas–Washburn assumption of a porous medium, as a bundle of parallel capillaries [
6].
Many approaches exist to model the wicking process in porous microstructures. The three most common ones are based on (i) the Lucas–Washburn equation, (ii) Darcy’s law, and (iii) the Richards equation and are extensively discussed in the literature [
3,
7,
8,
9,
10,
11,
12]. All approaches require different effective properties of the porous media, such as porosity, permeability, a fiber radius, and a pore radius, while the determination procedure is based on the volume averaging of the properties in a representative volume element (RVE) [
13,
14]. Particularly for the permeability, this method is widely used by solving the Stokes equation in the pore space. Thus, numerous correlations between the porosity, the specific surface area, and a resulting permeability have been reported [
15,
16,
17,
18].
However, little attention has been paid to the averaged pore radius in complex porous microstructures, although it is important for the description of the capillary suction pressure. As a driving force, this pressure is usually modeled using the Young–Laplace equation, in which the pressure difference acting across a curved free surface or interface between two immiscible fluids is expressed as
Here,
describes the surface tension between the acting fluids and
is the equilibrium contact angle. The main assumption behind Equation (
1) is that the capillary pressure is developed in a cylindrical capillary with the radius
. Lucas and Washburn transferred this assumption to a porous medium, by assuming it to be a bundle of aligned capillaries (see
Figure 1d)), each having the same capillary radius [
19]. According to this assumption, the curved free surface in a cylindrical capillary with a mean surface curvature
H can be correlated with the capillary radius
, by
where the capillary radius
can also be considered as the geometric pore radius (see
Figure 1c). As long as the perimeter of the cross section in the capillary is closed and axisymmetric, the Young–Laplace equation can be applied and analytical expressions are available [
20,
21]. However, the cross sections of the complex structures are not always closed, and it is unlikely that they are axisymmetric. Furthermore, flow paths are not aligned in parallel tubes but show tortuous curves. Therefore, these assumptions are generally not satisfied for the Young–Laplace equation for PPMs. Thus, a distinction must be made between the geometric and the effective pore radius [
19] of complex porous structures.
Instead of accurately determining the effective pore radius, it is common to fit the mathematical wicking models to the experimental results and derive the effective pore radius or the capillary suction pressure [
19,
22,
23,
24,
25,
26]. The resulting effective pore radius can be used to accurately describe the fluid flow for one specific PPM; however, prediction attempts for new wicking structures always require further experiments.
Geometry-based approaches to determine an effective pore radius require a detailed representation of the pore space, which can be obtained by scanning electron microscopy (SEM), transmission electron microscopy (TEM), confocal laser scanning microscopy, or X-ray computer tomography (CT) measurements [
27], among others. Alternatively, digital representations can be approached by algorithmically generating both simplified and complex capillary pore spaces [
28,
29]. For the extraction of the effective pore radius, image processing methods are used to capture morphological properties by voxel-based algorithms. Fitting spheres in the pore space is one approach where the geometric mean radius [
30,
31] of the spheres is considered as the effective pore radius (see
Figure 1c).
Another approach focuses more on structural space and assumes that the porous wicks consist of spherical particles. Since the particle sizes vary along the ligaments, an effective pore radius is derived on the basis of the particle size distributions [
9]. As a third approach, it is common to determine the pore space volume and the surface of the structural space, to thus calculate the hydraulic pore radius as an effective radius [
32,
33,
34]. In [
12,
32], a comprehensive summary of different approaches to estimate an effective pore radius for the wicking process is given.
Approaches based on physical two-phase simulations are reported in the literature as well. Both closed capillaries with different cross-sectional shapes [
29] and open-pored porous structures [
35,
36,
37] were generated, while the resulting mean surface curvature of the fluid and gas interface was studied. As a main result of the investigations, it was shown that the resulting mean surface curvature is influenced by the cross section and thus affects the effective pore radius [
29].
In this work, two highly porous nitrocellulose membranes (porosities ≈ 85%) are investigated as polymeric wicks, where each shows a complex microstructure and arbitrary cross sections with open perimeters of the pores. The objective of this work is to use physical two-phase simulations to establish a correlation between geometric properties and an effective pore radius, so as to predict the capillary driven wicking process. For the derivation, high-resolution X-ray computer tomography experiments were conducted, and 3D representations of the two membranes were obtained. In our previous work [
27], a characterization tool was presented, which allows extracting both the pore and the ligament radius distributions of porous microstructures as the basic geometric properties.
Based on a free energy minimization approach [
38,
39,
40], mean surface curvatures in simplified porous 3D geometries with defined geometric properties are calculated and correlated with an effective pore radius. A final validation of the correlation is done by applying an analytical model with the derived effective pore radii and by comparing the prediction to wicking experiments. All applied simulation methods and characterization tools are implemented in the simulation framework
Pace3D [
41]. The presented determination of an effective capillary radius is based on the PhD Thesis [
6], which is written in German. In order to extend the reach to a broader audience, some results of the work are presented in this article.
3. Results
3.1. Simulation of Surface Curvature Formation in Two-Phase Equilibrium Conditions
The validation of the model for two-phase equilibrium conditions is achieved by simulating the rise of a liquid column in capillaries with different cross-sectional shapes. The analytical equilibrium heights are obtained by following Jurin’s law [
20]:
where the capillary driving force is balanced with the gravitational force. Here,
is the density of the liquid and
g represents the gravitational acceleration.
For the numerical experiments, water with a density of
is assumed to make the capillary rise, and a free surface is shared with the air, where the interface has a surface tension of
. According to the conditions on the ground, the gravitational acceleration corresponds to
. Furthermore, different capillaries with a circular, a rectangular, and an open-pored cross section are shown in
Figure 5a–c.
The capillary radius is defined as half of a measurable minimum distance between two boundaries within the cross section. Finally, the size of the capillary radius is varied in the range of 0.25 to 1.5 , while the equilibrium contact angle is kept constant at .
Since we are interested in the equilibrium height, the numerical experiments are conducted by solving only the phase-field equations numerically (Equations (
5) and (
6)), without any coupling to fluid flow models, such as the Navier–Stokes equations. For this purpose, an explicit Euler scheme is solved for the temporal derivatives, and the finite difference method on an equidistant mesh is used for the spatial derivatives. As the capillaries are assumed to be symmetrical, the simulation domain can be reduced to a quarter of the cross section (see
Figure 5a).
A Neumann boundary condition with the normal derivatives of the order parameter
is defined on the symmetry planes. For the differently shaped capillary cross sections,
Figure 5 shows the resulting equilibrium heights
h over the capillary radius
in log–log scale. The symbols indicate the simulations, whereas the solid lines correspond to the analytical prediction. According to Equation (
15),
h decreases with increasing
as also shown by the simulations.
By drifting away from the ideal circular cross section, the equilibrium height changes, even though the geometric capillary radius is equivalent. For the open cross section, the equilibrium height is the lowest, which is due to the deviation of the curvature of the free surface.
Figure 5.
Equilibrium height
h, over capillary radius
r, for three different cross-sectional shapes in log–log scale: (
a) circle, (
b) rectangle, and (
c) open cross section (adapted from [
6]). The simulation results are presented by symbols, while the solid line follows Equation (
15) for the circular cross section. The capillary radius
is defined as half of a minimum measurable distance between two boundaries within the cross section. The red line defines the zero level and the Neumann boundary condition.
Figure 5.
Equilibrium height
h, over capillary radius
r, for three different cross-sectional shapes in log–log scale: (
a) circle, (
b) rectangle, and (
c) open cross section (adapted from [
6]). The simulation results are presented by symbols, while the solid line follows Equation (
15) for the circular cross section. The capillary radius
is defined as half of a minimum measurable distance between two boundaries within the cross section. The red line defines the zero level and the Neumann boundary condition.
First, it should be mentioned that these numerical experiments confirm that the applied phase-field approach is able to predict equilibrium conditions for capillary wetting. Second, the numerical experiments emphasize the importance of taking surface curvatures into account, to thus determine the effective pore radii.
3.2. Methodological Determination of the Effective Pore Radius
In order to determine the effective pore radius for the highly porous membranes, a correction factor F is introduced. This factor accounts for the deviation of the surface curvature within open-pored porous structures, compared to the surface curvature in a cylindrical capillary. It is derived by performing a simulation-based parameter study in simplified geometries, for which the presented phase-field approach is applied.
As shown in
Figure 2, the investigated polymer membranes reveal a highly porous pore system. Since the capillary pressure is the driving force for the wetting dynamics and is governed by the surface curvature, the correction function
F is derived by correlating the geometric properties with the resulting surface curvature. Therefore, a simplified open-pored pore system was generated by means of substitute parameters, which is formed from parallel ligament structures oriented in the direction of wetting. The ligament radius
and the distance between the ligaments
are introduced as substitute parameters for the mean ligament radius and the mean pore radius.
Figure 6a shows a small section (
of the complex membrane structure as an example and links this to the simplified model with the defined equivalent geometry parameters.
Based on the substitute parameters, a simulation study was conducted, in which a wetting scenario was simulated for each geometry. For this purpose, a total of four simplified geometries with a domain size of
were generated, whereby the distance of
= 80 cells was kept constant and the diameter of the ligaments
varied between 10 cells and 40 cells. Thus, the ratio of the geometry parameter
varies between 0.125 and 0.5, which covers the real structural conditions of the CT data, as presented in
Table 2.
For each combination of the structure parameter, a two-phase simulation based on Equation (
5) was performed. By defining a wetting (water) and a dewetting (air) phase with an initial saturation of
, a meniscus develops between the phases, when equilibrium conditions are reached. Since the capillary length
for
Porefil® is
mm and therefore consequently larger than the mean pore radii for diagnostic membranes (1
m to 10
m), it is assumed that gravity has no influence on the shape of the equilibrium meniscus.
Therefore, the gravitational term in Equation (
5) is neglected for the parameter study and a term that ensures the volume preservation of each phase is used instead, as described in [
39]. Furthermore, a periodicity of the investigated geometries is assumed. Since the effective pore radius is a geometric measure, it is assumed to be independent of the contact angle. Therefore, and for validation reasons (see
Section 2.1.1), the equilibrium contact angle
is defined. After reaching the state of equilibrium, the mean curvature of the surface
is determined numerically.
Using the predefined
and the estimated mean surface curvature
for each combination of the structure parameter, an equivalent capillary radius
is estimated as follows:
This equivalent radius
can be considered as the radius of a corresponding capillary with a circular cross section and a closed perimeter, as seen in
Figure 6b. Hence,
fulfills the Young–Laplace assumptions. In order to obtain a general correction factor
F, the equivalent radius is divided by the measured minimum distance between the ligaments
As a result, this correction factor can be directly linked to the ratio of the two structure parameters, as shown in the left diagram in
Figure 7. For an analytical correlation, the results from the simulations of the simplified geometries are fitted to the following expression:
by using a nonlinear least-squares Marquardt–Levenberg algorithm, which is implemented in the scientific graphing utility Gnuplot 5.2 [
51]. The respective best-fit coefficients for the simplified geometries are
and
. The left diagram in
Figure 7 shows an inversely proportional behavior of the correction factor
F, with an increasing ratio
of the structure parameters. By decreasing the ratio, the correction factor strongly increases, which is caused by a progressing deviation between the equivalent capillary radius
and the measurable distance between the ligaments
.
The correction factor for the two investigated membranes is estimated by applying the structure parameter from
Table 1 and Equation (
18). The resulting factors for sample 1 and sample 2 are
and
, respectively. By multiplying the factors with the measured mean pore radius
of the porous structure (as described in
Section 2.2.2)
the effective pore radii of
m and
m are obtained. An experimentally determined correction factor of ∼7, which lies between the geometric and effective pore radius for the porous structure under investigation, is also reported in the literature [
24].
3.3. Validation in an Ordered and Open Cross Section
As a first proof of concept, the equilibrium height
for the open cross section in
Figure 5c is calculated by including the correction factor
F in Equation (
15) as follows:
In this case, the ligaments are arranged in an ordered manner and the ratio of the structure parameter is constant. This results in a correction factor of , which means that the equilibrium height for the open cross section is lowered by a factor of , compared to the circular cross section.
For different minimum ligament distances
,
Figure 7 (right) shows the comparison between the simulation results and Equation (
20). The simulations and the prediction with Equation (
20) show a good agreement, with a maximum deviation of
.
3.4. Prediction of Wicking in the PPMs
For the two investigated porous membrane samples, the measured wicking times
are depicted in
Figure 8. The experiments for both samples are performed with
Porefil®. For each sample, three wicking curves were determined. Since the corresponding measurement curves lie on top of each other and there is a small relative standard deviation for both samples, it is clear that the measurements are reproducible. Therefore, the results are used to validate the presented approach to predict an effective pore radius, since the wetting conditions for
Porefil® are well defined.
To verify the developed correlation between the geometric structure properties and the resulting effective pore radius, the predicted wicking behavior is examined by comparing the experimentally measured curves with the results of the analytical wicking model. Furthermore, to emphasize the accuracy of the correlation, common approaches to determine the effective pore radius are estimated as follows and also used to represent the wicking behavior: The first approach [
30] describes the effective capillary pressure without additional corrections, using the geometric pore radius
from
Table 1.
In contrast, the second approach [
12,
19] describes the capillary pressure using the hydraulic radius
, where the corresponding pore radius
is determined via
. Here, the hydraulic radius is defined with
, where
represents the porosity and
is the specific surface area. Both properties are determined as described in [
27] in the voxel-based representation of the 3D microstructures.
The other relevant effective properties of the membrane structure, such as the mean pore radius, the mean ligament radius, and the permeability, are extracted on the basis of the CT data as described in
Section 2.2.2 and presented in
Table 1. Furthermore, the correction factors
F for samples 1 and 2 are shown in
Table 2 with the help of Equation (
19). The resulting radii for both samples are summarized in
Table 2.
Table 2.
Effective pore radii, depending on the respective approaches [
30] describes the effective capillary pressure without an additional correction, using the geometric radius
. [
19], and [
12] describes the capillary pressure with the effective radius
, using
. The value
is based on the correction factor
F, presented in Equation (
19).
Table 2.
Effective pore radii, depending on the respective approaches [
30] describes the effective capillary pressure without an additional correction, using the geometric radius
. [
19], and [
12] describes the capillary pressure with the effective radius
, using
. The value
is based on the correction factor
F, presented in Equation (
19).
| (
m) | (
m) | (-) | F (-) | (
m) |
---|
Sample 1 | 2.16 | 13.85 | 0.296 | 9.6 | 20.7 |
Sample 2 | 1.73 | 9.26 | 0.416 | 7.75 | 13.4 |
For the considered membrane samples, the comparison between the analytical prediction and the experimental measurements is shown in
Figure 8, where the wicking model is taken from Equation (
11). The symbols represent different experimental results, while the curves represent the predictions with different pore radii.
For both samples, significant differences are evident, regarding the three different approaches. When using the geometric pore radius
, the capillary pressure is strongly overestimated in all cases, resulting in a wicking time that is significantly lower than the experimental value. The prediction using the equivalent pore radius
, based on the hydraulic pore radius
, also overestimates the wicking behavior for both samples. In contrast, by using the corrected effective pore radius
from pore-scale simulations and the correction function in Equation (
19), an excellent agreement between the experiments and the modeled prediction can be observed.
With a deviation of 7 for sample 1 and for sample 2, the wicking time at a wicking length of 4 m is matched, and the wicking course is well approximated. Furthermore, the coefficient of determination was calculated for the statistical validation of all experiments and the respective analytical function. For sample 1 and sample 2, was fulfilled in all cases. This indicates that the wicking course is well approximated, without performing any fit adjustments.
4. Conclusions
In this work, a correlation for the determination of effective capillary pore radii in open-pored porous microstructures was derived. For this purpose, the wetting behavior was first simulated in simplified representative structures, using a two-phase phase-field approach. Thus, depending on the structural properties, the different resulting mean surface curvatures of the menisci were evaluated. Based on this simulation study, on the pore-scale, a function for a correction factor F was derived, which adjusts the deviation in the effective pore radii between wetting in cylindrical capillaries and porous microstructures. To validate the correlation, the macroscopic wicking behavior in two real PPM samples was calculated and compared to experiments, while the correction factor F was calculated based on the mean ligament and the mean pore radius in the respective CT scan.
We demonstrated that the effective radii for the observed samples were about ∼8 times (sample 2) and ∼10 times (sample 1) larger than the geometric mean radius, which was calculated based on the pore size distributions (e.g., see
Figure 3e). From a physical point of view, the deviation between the geometrical mean pore radius and the presented effective pore radius originates from the mean surface curvature of the free surface. It is smaller in open-pored porous microstructures than in ideal cylindrical capillaries where the geometrical pore radius and the capillary radius are assumed to be the same. As a result, in porous microstructures, the capillary pressure is smaller, and hence the wicking is slower. In other words, it is evident that this adjustment of the geometrical pore radius is necessary to accurately predict the wicking behavior. With the derived correlation for effective pore radii, no experimental fitting procedures are required to establish further structure–property linkages.
For future work, not only digital twins of a given porous membrane structure but time-dependent physical simulations of the underlying phase separation [
52] could be used as well. This would provide a time series of 3D microstructures as database, which brings in combination with the presented computer-aided membrane characterization a tool for direct linking between process parameters for the material production and the resulting wicking behavior. Moreover, by entering the data-driven research in membrane science using research data infrastructures, such as Kadi4Mat [
53], far-reaching possibilities for further structure–property linkages and ultimately for the digital design of porous membranes are opened up.