In this section, results obtained using different models are compared when applied on the parameters of the HFM pores structure. Two different approaches were used as mentioned in the previous section, i.e., the approach based on models developed for fibrous filters and a model for membrane filters.

#### 4.1. Fibrous Filters

Figure 5 shows SCE due to the inertial impaction based on different models for a face velocity of 5 cm/s (a) and 20 cm/s (b) in relation to

Stk. Stokes number is a governing parameter of inertial impaction mechanism based on which one can decide if the inertial impaction mechanism dominates at conditions adopted in a filtration process. Moreover, the use of

Stk is more appropriate compared to the relation of efficiency to particle diameter.

Stk relates to the particle diameter itself, particle density, collector diameter, face velocity and other parameters governing the mechanisms taking place during aerosol filtration (Equation (31)). Inertial impaction significantly governs the separation if

Stk > 1, which is mostly true for particles larger than 1 µm at higher face velocities. At such conditions, the particles have higher inertia and easily separate from airflow streamlines and hit the collector (

Figure 2). This is true for higher face velocities even for particles smaller than 1 µm. However, for a lower face velocity of 5 cm/s (

Figure 5a) we can see that impaction efficiency starts to increase significantly for

Stk above 0.1. This is caused by the ultrafine collectors in the membrane structure (

Figure 4a). The average collector diameter is 90 nm, giving a higher

Stk even for smaller particles as

Stk is inversely proportional to collector diameter according to Equation (31), thus giving higher theoretical efficiencies compared to general fibrous filters. The membrane structure is more similar to nanofiber based filters, though it is more dense (

Figure 4, compare e.g., [

19,

37,

77,

78,

79]). Different impaction efficiency models in relation to particle diameter are compared by face velocity in

supplementary material (Figure S1). The courses of impaction efficiency in relation to particle diameter are practically the same as those in relation to

Stk and are shown in

supplementary material (Figure S2). The only model, which significantly deviates from the others is that derived by Ilias and Douglas [

71] which predict high impaction efficiency at low

Stk. This relationship is valid for 0.07 <

Stk < 5. The bottom limit of

Stk is the point where the curve increases in the direction of decreasing

Stk, so it is necessary to omit this part of the curve because it is clear that the impaction regime in this

Stk region does not occur.

Figure 6 shows a single fiber efficiency due to the interception mechanism in relation to interception parameter. Interception may play an important role in nanoparticle filtration if the collector diameters are small [

80] and starts to dominate at an interception parameter of 0.1 [

81]. This is true for most of the models except for that derived by Pich [

63] (Equation (29)) which predicts high interception efficiencies also for very small interception parameters under 0.1 corresponding to particle sizes smaller than 10 nm (

Supplementary Figure S3). The Equation (29) was derived for small fiber Knudsen numbers which is not fulfilled for the given collector diameter. Therefore, the Knudsen number is higher and the model overrates the results to lower particle sizes. Liu and Rubow [

54] derived another relationship (Equation (30)) considering the gas slip effect, which is more appropriate for very small collector diameters. The interception mechanism is independent of face velocity, which is the main difference from inertial impaction and Brownian motion. This is, however, not true for model of Langmuir [

60] where the interception efficiency also dependent on the fiber Reynolds number which is given by the face velocity (

Figure S3a).

Brownian motion (diffusion) is another important mechanism occurring when separating particles from air. Unlike for inertial impaction, this mechanism is enhanced at very small face velocities and for very small particles that are mostly under 100 nm in size. The governing parameter for Brownian motion is the Peclet number, which is a ratio of convection to diffusion transport rate. SCE due to diffusion increases with a decreasing Peclet number, i.e., decreasing particle size (

Figure 7). With increasing airflow velocity, the Peclet number is shifted to higher values which diminishes the capturing effect caused by random motions of particles (compare

Figure 7a,b). Therefore, with increasing velocity, the SCE due to diffusion decreases and is shifted to lower particle sizes. Comparison of efficiency/particle size curves by face velocity calculated using different models are shown in

supplementary material in Figure S4, a comparison of individual models is in

Figure S5. The most appropriate model for SCE due to diffusion is Equation (18). This model developed by Payet et al. [

55] covers even very small particles for which the other models give an efficiency that is higher than 1 (

Figure 7).

Figure 8 shows adhesion efficiency in relation to particle size based on the model developed by Ptak and Jaroszczyk [

72] (Equation (41)). This mechanism is not often considered in theoretical predictions. However, we also use this model to completely describe the mechanical capture of particles in which adhesion plays an important role due to re-entrain and rebound effects. We also calculated single fiber efficiency according to Equation (3), which is the product of collision efficiency (a sum of SCE due to impaction interception and diffusion) and adhesion efficiency presented by the values predicted using Equation (41). Adhesion efficiency is mostly higher for smaller particles and lower face velocities as shown in

Figure 8. This is given by adhesion energy between a particle and a fiber as follows:

where

H is the Hamaker constant and

a_{0} is the adhesion distance. Adhesion energy is directly proportional to the particle size, therefore, higher energy is necessary to keep a larger particle attached to the fiber. It is similar for face velocity, which is mostly assumed the same as the impact velocity of the particle colliding with the fiber surface. The impact velocity should be less than the critical velocity

ν derived from the adhesion energy given as follows [

46]:

The Hamaker constant can be calculated as follows [

46,

82]:

where

ε is the static dielectric constant,

n is the refractive index,

h is the Planck constant and

ϑ_{e} is the main electronic absorption frequency typically around 3 × 10

^{15} s

^{−1}. The subscript notation 1, 2 and 3 of

ε and

n indicate the particle, membrane surface and fluid, respectively. The typical value of Hamaker constant ranges between 10

^{−19} and 10

^{−20} [

83]. However, significant influence will also have particle surface charges, which can cause the membrane to act as an electret filter, so the particles may be captured due to electrostatic forces. In this work however, we focus on the mechanical means of filtration only, so this effect is not considered.

#### Overall SCE and overall Filtration Efficiency

Overall SCE is shown in

Figure 9a. This is a typical shape of efficiency/particle size curve with a minimum corresponding to most penetrating particle size (MPPS). The left-hand side of the minimum is governed by the diffusion mechanism while interception and inertial impaction are responsible for the right-hand side. However, the curves in

Figure 9a correspond only to one single filter fiber, i.e., one collector of the HFM structure (

Figure 4a). To get an overall membrane efficiency, it is necessary to recalculate the SCE to a whole membrane structure using Equation (2). The results are shown in

Figure 9b,c. After recalculating, we get 100% removal efficiency for all particle sizes (

Figure 9b).

Figure 9c shows the same expressed as penetration, i.e., the amount of particles which can penetrate through the membrane, which is in order of 10

^{−66} which is practically equaled to zero. The results shown in

Figure 9a are single collector efficiencies calculated using models for diffusion (Equation (18)), interception (Equation (21)), impaction (Equation (35)) and adhesion (Equation (41)). So it is an example of one selected combination of models for individual mechanism. The other was not calculated as it was assumed that the result would be the same or would vary somewhere in the order of 10

^{−70}, which is negligible.

The main reason for these results is the high solidity of the HFM structure, which is 0.48, while most of the fibrous filters have solidity between 0.01–0.3 [

44] and most of the models are developed for this solidity range. Moreover, the membrane collector diameter is very small, giving a very dense structure. If we look at

Figure 4a, we can see collector diameters of about 100 nm in size. The thickness of the membrane wall is 36 µm. This means that there are about 360 such layers in the membrane wall creating a dense network that is very hard for particles to penetrate. Therefore, the results seems to be reasonable. In practice, this membrane could serve as an absolute filter which are used for aerosols which must have 100% removal efficiency. Such aerosols include some radioactive particles, toxic aerosols and viruses.

#### 4.2. CPM

The approach based on membrane pore size instead of membrane fiber diameter is presented in this section. Inertial impaction is stronger for larger particles at higher velocities, which is in accordance with theory. However, the model of Pich [

74] is less accurate as it does not consider the possible sieving effect in membrane filters i.e., complete capture of particles on the membrane surface for particles larger than membrane pore size. This is obvious from

Figure 10. The membrane pore size considered in the calculations is 205 nm (

Table 1). If circular pores are assumed, which is a simplification in the model, we should obtain 100% efficiency for particles above 205 nm regardless of the face velocity. This is not seen to be true from

Figure 10.

More plausible results are obvious for interception efficiency (

Figure 11). The interception efficiency increases up to a particle size of 202 nm with an efficiency of 99.97%. For a particle size of 209 nm (slightly larger than pore size), efficiency is 100% which is reasonable. Therefore, the model proposed by Spurny et al. [

75] (Equation (51)) seems to be accurate for the structure of polypropylene HFMs.

Diffusion is an important part of the overall efficiency. We can distinguish between diffusion capture in pores and diffusion capture on membrane surface (

Figure 3). Prediction models were developed for both (Equations (48) and (53)).

Figure 12a shows pore diffusion efficiency. To talk about diffusion capture within membrane pore structure is possible only for particles smaller than the largest pore size (i.e., smaller than 205 nm). Larger particles will only be a subject to surface diffusion capture (

Figure 12b) which is possible for whole particle size range. From

Figure 12a, a similar problem for the model for impaction efficiency is obvious. While efficiency for impaction should be 100% for particles above 205 nm, pore diffusion should be equaled to zero because no particle larger than 205 nm cannot penetrate the pore structure, so there is no diffusion capture of these particles.

Overall efficiency is predicted based on the models for individual mechanisms and calculated using Equation (56).

Figure 13 shows 99.997% MPPS (290 nm) efficiency at a velocity of 5 cm/s. With increasing velocity, the efficiency for MPPS decreases. However, it is still in the range of 99.7% at a velocity of 20 cm/s. MPPS is shifted to smaller particle size with velocity. It is 250, 225 and 202 nm for 10, 15 and 20 cm/s, respectively. This model gives more realistic results compared to the model for fibrous filters, where unconditional 100% efficiency was obtained for all face velocities.