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Niko
Heikkinen
*^{a},
Juha
Lehtonen
^{a} and
Riikka L.
Puurunen
^{b}
^{a}VTT Technical Research Centre of Finland, P.O. Box 1000, FIN-02044 VTT, Espoo, Finland. E-mail: niko.heikkinen@vtt.fi; juha.lehtonen@vtt.fi
^{b}Department of Chemical and Metallurgical Engineering, Aalto University School of Chemical Engineering, Kemistintie 1, Espoo, Finland. E-mail: riikka.puurunen@aalto.fi

Received
20th November 2023
, Accepted 6th February 2024

First published on 14th February 2024

This work presents a diffusion–reaction model for atomic layer deposition (ALD), which has been adapted to describe radial direction reactant transport and adsorption kinetics in a porous particle. Specifically, we present the effect of three particle geometries: spherical, cylindrical and a slab in the diffusion–reaction model. The reactant diffusion propagates as a unidimensional front inside the slab particle, whereas with cylinder and spherical particles, the reactant diffusion approaches the particle centre from two and three dimensions, respectively. Due to additional reactant propagation dimensions, cylindrical and spherical particles require less exposure for full particle penetration. In addition to the particle geometry effect, a sensitivity analysis was used to compare the impact of the particles’ physical properties on the achieved penetration depth. The analysis evaluates properties, such as the combined porosity and tortuosity factor, mean pore diameter, specific surface area, pore volume, and particle radius. Furthermore, we address the impact of the reactant molar mass, growth-per-cycle (GPC), sticking probability, reactant exposure and deposition temperature on the simulated diffusion and surface coverage profiles. The diffusion–reaction model presented in this work is relevant for the design and optimization of ALD processes in porous media with different particle geometries.

Fig. 1 Schematic illustrations of ideal particle geometries: (a) slab with a shape factor s = 0, (b) cylinder (h = nR, n > 100) with a shape factor s = 1 and (c) sphere with a shape factor s = 2. |

The literature presents various analytical and computational models for the ALD process in porous materials and well-defined high aspect ratio features. Gordon et al.^{18} present an analytical model to describe ALD growth inside high aspect ratio holes and trenches, where the reactant exposure required to achieve conformal coating increases quadratically with respect to the aspect ratio. This and other analytical models^{19–21} have been especially suitable for reactant propagation and surface reaction modelling in a diffusion limited growth setting. Computational models to address ALD conformality can be classified^{22} as ballistic transport–reaction,^{23,24} Monte Carlo^{21,25–28} and diffusion–reaction (sometimes “continuum”) models.^{1,4,29–31} Monte Carlo and diffusion–reaction models are applicable from continuum flow (Knunden number, Kn ≪ 1) to free molecular flow (Kn ≫ 1) regimes, whereas ballistic transport–reaction models are used in the Knudsen diffusion regime (free molecular flow).^{24,30,31} Diffusion–reaction models are suitable for simplified particle geometries, while ballistic transport and Monte Carlo models are appropriate for complex structures.^{22,31}

The diffusion–reaction modelling of the ALD process in porous media is based on Fick's law of diffusion.^{32–34} In this work, we assume homogeneous porosity and tortuosity throughout the particle, where reactant travel begins from the particle outer surface and the reactant diffuses through a tortuous pathway towards the centre of the particle. From the surface to the centre of the particle, the reactant travel is affected by the effective diffusion coefficient (D_{e}).^{35–37} Typically, the effective diffusion coefficient related to porous catalyst particles is determined in a Knudsen diffusion (D_{Kn}) regime as D_{e} = D_{Kn}ετ^{−1}.^{35,36} With a Knudsen diffusion assumption, the effective diffusion coefficient depends on the pore dimensions, temperature and reactant molar mass,^{10} porosity (ε) and tortuosity (τ). The diffusion coefficient can also be used in a molecular diffusion regime or a combination of Knudsen diffusion and molecular diffusion.^{35}

Simplified particle geometries, which are especially relevant in diffusion–reaction modelling, are well known in the literature related to heterogeneous catalyst particles.^{35–37} The catalyst regeneration process is a typical example of gas diffusion and reaction in a spherical particle, where carbon formation (fouling) inside the catalyst particle is removed through oxidation.^{35} Here, oxygen would be analogous to ALD reactant, diffusing into the catalyst pores and reacting with fouled carbon. Due to the spherical particle geometry, the oxygen demand decreases towards the particle centre. The diffusion–reaction regeneration process is commonly described with a shrinking core model (SCM), where the shell volume of a spherical particle decreases towards the particle centre.^{35,37}

In this work, a shape factor is used to determine partial differential equation for the respective particle geometry.^{36,38} Here, symbol r presents the radial coordinate along the characteristic dimension R. With cylindrical and spherical particles, R is the particle radius, whereas with slab particle geometry R is half of the slab thickness. The radial coordinate r is defined to be zero at the outer surface of a particle, and R at the centre for a sphere/cylinder or half thickness for a slab particle. With the slab particle, the ALD reactant is propagating from under- and upside directions towards the centre of the slab. With the cylinder particle, the reactant propagates in a radial direction, whereas the axial propagation direction is neglected. With the spherical particle, the reactant is propagating from all directions towards the centre. In the literature, the shape factor is determined as the external surface area divided by the particle volume, which is then multiplied by the particle radius R.^{35,37} The shape factor is a common simplification to describe the effect of the particle geometry on a gaseous and liquid flow through a packed catalyst bed (for example, Thoenes–Kramers mass transfer correlation).^{35} Furthermore, relevant to our work, the shape factor can be also used to describe particle geometry effect on the intraparticle gas diffusion.^{37} Most relevant previous work on the particle geometry effect on the intraparticle diffusion, specifically in the ALD process, has been presented by Lee et al.^{39} They presented a model to address particle geometry effect on the minimum exposure time for the complete ZnO coverage of planar, cylindrical and spherical porous alumina monoliths. Additionally, Elam et al.^{6} and Detavernier et al.^{40} have also shown that the spherical particle geometry has a significant effect on the decreasing reactant consumption towards the particle centre.

Our work adapts slab, cylinder and spherical shape factors in an ALD diffusion–reaction model. With partial differential equations for different particle geometries, we demonstrate the effect of the shape factor on the diffusion and surface coverage profiles. We discuss the radial reactant propagation for each particle geometry and evaluate the effect of ALD process parameters and porous material properties on the achieved coating penetration depth. We implement parameters for the growth-per-cycle (GPC, from areal number density of the metal adsorbed), sticking probability, particle porosity and tortuosity, pore diameter, specific surface area, pore volume and particle radius into the coupled partial differential equations. The modelling principles in this work might be relevant also to thin film preparation methods beyond ALD, for example, sequential infiltration synthesis^{41} and vapor-phase infiltration.^{42}

(1) |

In the eqn (1), the symbol s is the shape factor with a relationship s = a − 1, where factor a has values of one, two and three for a slab, cylinder and sphere, respectively.^{35–37} The factor a is (A_{Par}/V_{Par})R,^{35,36} where the particle radius R (m) is multiplied by the ratio of the particle outer surface area (A_{Par}, m^{2}) and the catalyst particle volume (V_{Par}, m^{3}). Factor a related to the slab particle geometry is (A_{Par}/V_{Par})L, where L is the half thickness of the slab, corresponding to L = R/2. Table S1, ESI† presents some example calculations to determine the shape factor s for relevant particle geometries. (Note: one should not confuse A_{Par} and V_{Par} with the specific surface area (A_{O}, m^{2} g^{−1}) and pore volume (V_{P}, m^{3} g^{−1}) of the porous media.) The ratio A_{Par}/V_{Par} represents the surface area per volume of a non-porous particle. The adsorption loss term incorporates the self-limiting nature of the ALD process in the diffusion–reaction model.

The effective diffusion coefficient (D_{e}, m^{2} s^{−1}), is derived from the gas phase diffusion coefficient in porous materials, consisting of both molecular diffusion and Knudsen diffusion factors. In eqn (2), according to the Bosanquet relation, the gas phase diffusion coefficient (D_{i}) is a sum of the molecular diffusion coefficient (D_{mi}) and Knudsen diffusion coefficient (D_{Kn}).^{25}

(2) |

(3) |

(4) |

(5) |

After introducing the corresponding shape factor values to the eqn (1), the final partial differential equations for a slab (eqn (6)), cylinder (eqn (7)) and sphere (eqn (8)) can be written in the form:

(6) |

(7) |

(8) |

(9) |

The model initial condition for θ is presented in eqn (10). At time t = 0, θ is zero for all values of the radial distance (r) within the range from 0 to R.

θ(t = 0, r) = 0, ∀r ∈ [0, R μm] | (10) |

n_{P}(t = 0, r) = 0, ∀r ∈ [0, R μm] | (11) |

n_{P}(t, r = 0) = n_{P0}, ∀t ≥ 0 | (12) |

(13) |

Partial differential equations (PDEs) were solved in Matlab (MATLAB Version: 9.14.0 (R2023a)) by discretizing the first and second derivatives of the particle radius applying finite differences. For the first derivative, dss004 code was applied using five-point fourth order approximation and for the second derivative, dss044 code using six-point fourth order approximation was used.^{48} The spatial domain of the radius was discretized typically using 300 points. The formed system of ODEs was solved using Matlab solver ode15s being applicable for stiff ODE systems.

Parameter | Value |
---|---|

Porosity, ε | 0.5 |

Tortuosity, τ | 1 |

Mean pore diameter, d_{pore} |
2 × 10^{−8} m |

Specific surface area, A_{O} |
200 m^{2} g^{−1} |

Pore volume, V_{P} |
6 × 10^{−7} m^{3} g^{−1} |

Particle radius, R | 3 × 10^{−4} m |

Parameter | Value |
---|---|

a Estimated from ALD reactant vapour pressure of 100 Pa at 423 K. b Baseline reactant exposure duration marked with bold text. | |

Reactant molar mass, m_{R} |
100 g mol^{−1} |

Initial sticking probability, β_{0} |
2 × 10^{−2} |

Areal number density of metal atoms deposited on each exposure cycle at saturation, σ_{P} |
4 × 10^{18} m^{−2} |

Reactant volumetric number density outside the porous material, n_{P0} (t, r = 0) |
1.71 × 10^{22} m^{−3}^{a} |

Temperature, T | 423 K |

Reactant exposure time, t | 100, 500 and 1000 s^{b} |

Fig. 2 The effect of the shape factor (s) as a function of the radial coordinate r for (a) the volumetric reactant number density and (b) the surface coverage. A list of the used parameters is presented in Tables 1 and 2. |

Fig. 2(a) presents the particle geometry effect on the gas-phase reactant number density as a function of the coating penetration depth. The slab geometry demonstrates a linear behaviour as the reactant diffuses through a “plate” with an infinite depth and width, with only one diffusion propagation dimension.^{35,36} The cylinder profile is curved because of two diffusion propagation dimensions and one indefinite dimension (see Fig. 1). The spherical geometry has no infinite dimensions and the reactant diffusion front approaches the particle centre from all directions. Then, the deposition area per shell volume decreases towards the centre of the porous particle.^{35}

Although the shape factor is not included in the equation for the surface coverage (eqn (9)), the cylinder and spherical particle penetration depth increases in the surface coverage profile compared to the slab in Fig. 2(b). This is due to ∂n_{p}(t,r)/∂t and ∂θ(t,r)/∂t equation coupling. Despite the increasing penetration depth for the cylinder and spherical shape factors compared to the slab, all three surface coverage profiles have a similar overall shape and slope at the adsorption front.

Fig. 3 The effect of varying particle size R on the achieved r_{θ=1/2} (penetration depth) with three different exposures of reactant (1 × 10^{4}, 5 × 10^{4} and 1 × 10^{5} Pa s). Three curves for each exposure present the model results for different shape factors: sphere (s = 2), cylinder (s = 1) and slab (s = 0). The simulation parameters are presented in Tables 1 and 2. |

Fig. 3 presents the penetration depth results for each shape factor as a function of the particle radius (R). The distinct line starting from the origin is the penetration depth, where the reactant has reached the centre of the particle with a radius R. For example, for a particle with a radius of 50 μm, the penetration depth is 50 μm for all the presented exposures and shape factors. With an increased exposure, larger particles can be fully penetrated by the ALD reactant. In Fig. 3, full penetration is achieved with a 1 × 10^{5} Pa s exposure and a spherical shape factor when R ≤ 275 μm. A reduced particle size can be penetrated with the same exposure with R ≤ 225 μm and ≤ 161 μm for the cylinder and slab particle geometries, respectively. According to Fig. 3, when the particle radius increases, all shape factors result in increasingly similar values for the penetration depth. This behaviour is expected, as the increasing radius of a particle reduces the factor 1/R and 2/R in eqn (7) and (8). Increasing the particle radius R will diminish the effect from the first order partial derivative, resulting in an asymptotically closing equation with respect to the slab geometry (see eqn (6)).

In the example in Fig. 3, the shape factor has a significant effect on the modelled penetration depth when particle radius is <600 μm. However, the model results are not bound to a specific particle size. Instead, the important particle size range depends on the used exposure. Fig. 4 presents the penetration depth difference between slab and spherical geometries. In the figure, the horizontal axis is the normalized penetration depth for sphere, , where the value 1 indicates a full coating (e.g., when particle radius is 200 μm and the reactant has reached the centre of the particle at 200 μm, R = 200 μm and r_{θ=1/2,sphere} = 200 μm). The vertical axis in the figure is the percentage difference between the penetration depths in the slab and spherical geometry. These values can be found in Fig. 3, where for example, a 1 × 10^{5} Pa s exposure resulted in r_{θ=1/2,sphere} of 229 μm and r_{θ=1/2,slab} of 161 μm. In Fig. 4, these values can be found from the vertical axis percentage difference of and horizontal axis value of .

Fig. 4 The percentage difference between the penetration depths of the slab and spherical shape factors as a function of ξ_{θ=1/2}. The vertical axis is the percentage difference in the penetration depth between the slab and spherical shape factors. The horizontal axis presents the normalized value ξ of penetration depth with the spherical shape factor per particle radius. The small inset is an enlarged view of the same plot. The figure shows the results for 1 × 10^{4}, 5 × 10^{4} and 1 × 10^{5} Pa s exposures at 423 K and the particle radius (R) varying from 50 to 3000 μm in 25 μm step intervals. All other parameters are presented in Tables 1 and 2. |

Fig. 4 illustrates the difference in the achieved penetration depth between the slab and spherical shape factors. When ξ_{θ=½,sphere} ≤ 0.15, the percentage difference in the penetration depth estimate between the slab and spherical shape factors is <5%. In order to achieve a <1% difference, ξ_{θ=½,sphere} should be ≤0.05. Increasing the penetration depth with respect to the particle radius, the selection of the shape factor becomes increasingly relevant. At full infiltration, the penetration depth reaches ∼40–45% difference. This finding corresponds to the work by Lee et al.,^{39} Elam et al.^{6} and Detavernier et al.,^{40} in which the spherical particle geometry has a significant effect on the decreasing reactant consumption towards the particle centre.

Fig. 5 Parameter effect on the volumetric reactant number density profile (n_{P}) with a spherical shape factor and slab shape factor. Panel (a) and (b) show the effect of tortuosity (τ) per porosity (ε), (c) and (d) reactant molar mass variation (m_{r}), (e) and (f) show the reactant areal number density (σ, atoms per nm^{2}), (g) and (h) varying = A_{O}/V_{p} (m^{−1}), (i) and (j) sticking probability (β_{0}), (k) and (l) pore diameter (d_{pore}, m). The parameter effect is addressed by changing one parameter at a time and all other values are baseline parameters with a 500 s reactant exposure duration (see Table 1 and 2). The related surface coverage profiles are shown in Fig. 6. |

Fig. 5 panels (a) and (b) show the effect of tortuosity (τ) and porosity (ε) as a single factor τ/ε for the sphere and slab respectively. The factor τ/ε is a reciprocal value of ε/τ which is present in the effective diffusion coefficient (see eqn (5)). Factor values from 1 to 5 are typical for heterogeneous catalyst support materials, where increasing tortuosity increases the diffusion, i.e. the precursor travel distance inside the porous particle. Similarly, reducing the porosity increases the diffusion resistance due to the retained reactant passage through the porous media. Interestingly, with a restricted diffusion and increasing τ/ε, the achieved r_{θ=1/2} is increasingly similar between slab and spherical particles. With a decreasing τ/ε the difference between the particle geometries becomes apparent, especially with the shrinking core effect from the spherical particle geometry.^{35}

Panels (c) and (d) in Fig. 5 present the effect of the reactant molar mass on the penetration depth (r_{θ=1/2}). The penetration depth decreases with an increasing reactant molar mass due to the decreasing reactant gas-phase mean thermal velocity v_{th} (m s^{−1}). This observation corresponds to the findings reported by Yim et al.^{43}

Panels (e) and (f) in Fig. 5 illustrate the effect of the areal number density (σ_{P}, m^{−2}) of deposited metal atoms upon saturation (directly related to the GPC) on the penetration depth. Here, smaller values increase penetration depth due to decreased amount of reactant adsorption sites. The presented trend is similar to Yim et al.,^{43} where an increasing σ (nm^{−2}) strongly reduces the achieved r_{θ=1/2}. With σ = 1 nm^{−2}, the spheres have been fully coated.

Panels (g) and (h) in Fig. 5 present the effect of varying = A_{O}/V_{p} (m^{−1}), a factor found in eqn (6)–(8). Here, increasing the ratio increases the surface area compared to the pore volume. Thus, the increased surface area per given volume (increasing ) results in an increased number of surface adsorption sites per volume and a reduced penetration depth.

Panels (i) and (j) in Fig. 5 present the volumetric reactant number density profiles that appear the same for all varied values of β_{0}. Similar findings have been reported by Keuter et al.^{34} and Arts et al.,^{51} where varying the β_{0} value has no significant effect on the average coating penetration depth. However, the enlarged subplot reveals minute differences in the profiles with varying β_{0}. When the reactant number density approaches zero (reactant depletion), the decreasing sticking probability (β_{0}) reduces the rate of reactant adsorption on the surface, enabling the reactant to travel deeper into the particle. On the other hand, different findings have been reported by Yim et al.,^{43} where they observed a slight increase in the r_{θ=1/2}, as β_{0} increased. Yim et al.^{43} assumed that this would be a result of their analytical model approximation for the reactant partial pressure, resulting in a penetration depth (r_{θ=1/2}) increase, when the β_{0} value increases.

Lastly, panels (k) and (l) in Fig. 5 present profiles with varied pore diameters. In the presented diffusion–reaction model, the pore diameter is only present in the Knudsen diffusion coefficient (see eqn (5)). A reduced pore diameter restricts diffusion and results in a decreased penetration depth. For both the slab and spherical particle, the r_{θ=1/2} increases linearly with the pore diameter. However, due to the spherical particle geometry, the r_{θ=1/2} increases faster. For example, increasing the d_{pore} from 10 to 40 nm will double the r_{θ=1/2} for the slab. The same increase with spherical particle increases the r_{θ=1/2} 2.5 times.

Fig. 6 presents the surface coverage profiles connected to the volumetric reactant number density profiles in Fig. 5. For example, panels (a) and (b) illustrate the effect of the τ/ε on the surface coverage profile. Panels (i) and (j), show the β_{0} surface coverage profiles revealing a widening distribution as the β_{0} decreases. Interestingly, the spherical and slab particle geometries seem to have similar surface coverage profiles with respect to a varied β_{0}, and only the penetration depth is different. Similar surface coverage profiles for the slab and spherical particle geometries with a varying β_{0} are due to the model using the same ALD process, reactant and particle properties for both the spherical and slab particle geometries. Then, the only difference is the reactant propagation inside the particles. With the spherical particle, the reactant propagation depends on the location (radial coordinate) inside the particle, whereas the slab particle reactant propagation is independent of the location.

Fig. 6 The parameter effect on the surface coverage profile with a spherical shape factor and slab shape factor. Panel (a) and (b) present the effect of tortuosity per porosity, (c) and (d) show the reactant molar mass variation, (e) and (f) show the reactant areal number density (atoms per nm^{2}), (g) and (h) show a varying = A_{O}/V_{p} (m^{−1}), (i) and (j) show the sticking probability β_{0}, while (k) and (l) show the pore diameter (m). The parameter effect is addressed by changing one parameter at a time and all other values are baseline parameters with a 500 s duration reactant exposure (see Tables 1 and 2). The related volumetric reactant number density profiles are presented in Fig. 5. |

Fig. 7 Achieved penetration depth (r_{θ=1/2}) with three varying parameters (a) the surface area (m^{2} g^{−1}) and (b) pore diameter (nm). The results are shown for particle radiuses of 100, 200 and 300 μm and with reactant exposures of 1 × 10^{3}, 1 × 10^{4} and 1 × 10^{5} Pa s at 423 K. Other parameters are baseline parameters and the used shape factor spherical particle geometry. Values used for A_{O} = 100–1000 m^{2} g^{−1} and d_{pore} = 3–50 nm. All varied parameters and values for the Knudsen number are presented in the ESI† Table S2. |

In Fig. 7(a), the coating penetration depth decreases when the specific surface area (A_{O}) increases. The increasing specific surface area increases the number of adsorption sites per unit volume, resulting in a reduced penetration depth. The pore diameter (d_{pore}) in Fig. 7(b) is included in the Knudsen diffusion coefficient (D_{Kn}) (see eqn (5)). Therefore, the pore diameter has significant effect on the reactant travel inside the particle through the Knudsen diffusion coefficient. Increasing the pore diameter increases the Knudsen diffusion coefficient (D_{Kn}), which will increase the achieved penetration depth via faster reactant diffusion inside the porous media.

Fig. 8 The effect of selecting the volumetric reactant number density (m^{−3}) or partial pressure (Pa) in a diffusion–reaction model as a function of temperature. Panel (a) shows the fixed volumetric reactant number density to 1.94 × 10^{22} m^{−3} and (b) shows the fixed partial pressure to 100 Pa as a function of the achieved penetration depth (r_{θ=1/2}). A full list of the simulation parameters and plotted values are presented in the ESI† in Table S4. |

Our diffusion–reaction model uses an absolute number of reactant molecules (volumetric reactant number density, n_{P0}) to represent the initial reactant quantity at t = 0. Fig. 8(a) illustrates the simulation results with a fixed initial volumetric reactant number density (1.94 × 10^{22} m^{−3}). Here, the absolute quantity of reactant molecules is fixed and the reactant partial pressure increases with the increasing temperature. As the absolute reactant quantity remains unchanged, the penetration depth increases due to temperature assisted diffusion and the increasing rate of surface reactions. As the sticking probability (β_{0}) remains unchanged, the increased rate of surface reactions is related to the increased frequency of reactant collisions with the surface.

If a partial pressure (p_{A0}) would be selected for the initial reactant quantity, this value has a temperature dependency. Fig. 8(b) illustrates profiles when volumetric reactant number density is determined from a fixed 100 Pa partial pressure (p_{A0}) at varying temperatures. For example, at a temperature of 373 K, a 100 Pa partial pressure corresponds to n_{P0} of 1.94 × 10^{22} m^{−3} and at a temperature of 573 K, a 100 Pa partial pressure corresponds to 1.26 × 10^{22} m^{−3} volumetric reactant number density. Here, the achieved penetration depth decreases as a function of temperature. The decrease is due to the decreasing initial volumetric reactant number density.

By comparing the model results for the slab and spherical particle geometries, we found that when the reactant penetration depth (r_{θ=1/2}) is less than 5% of the particle radius (R), all shape factors result is similar estimates (difference within 1%) for the penetration depth. However, when increasing the penetration depth with respect to the particle radius, the difference between shape factors increases. With the slab shape factor, the penetration depth increases linearly as a function of the square root of the reactant exposure. However, this relationship is different for the spherical particle, where the shell volume available for reactant deposition decreases towards the spherical particle centre as r^{3}, where r is the radial distance from the particle surface.

In addition to the analysis of the shape factors, the presented diffusion–reaction model was used to perform a parameter sensitivity analysis to address the parameter effects on the model results. The varied parameters included porosity (ε), tortuosity (τ), reactant molar mass, areal number density of metal atoms deposited on each exposure cycle upon saturation (σ), the specific surface area (A_{O}), pore volume (V_{P}), sticking probability (β_{0}), and pore diameter (d_{pore}). Furthermore, the single parameter variation was extended to address in detail the combined effect of the specific surface area (A_{O}), mean pore diameter (d_{pore}) and pore volume (V_{P}) on the achieved penetration depth. The general trend was that with a fixed exposure, increasing the specific surface area decreases the coating penetration depth, while increasing the pore diameter decreases the coating penetration depth, and increasing the pore volume increases the coating penetration depth. Lastly, we conceptually compared the effect of the ALD process temperature on the diffusion–reaction model volumetric reactant number density (m^{−3}) and partial pressure (Pa). Our diffusion–reaction model uses the volumetric reactant number density (n_{P0}) to represent the initial reactant quantity. Here, the absolute initial reactant quantity has no temperature dependency, whereas the initial reactant quantity given as the partial pressure (p_{A0}) depends on the temperature. Then, the absolute quantity of initial reactant molecules and the achieved penetration depth decrease as a function of the increasing temperature.

This work presents a diffusion–reaction modelling approach to ALD on porous media with different particle shapes. We used a porous and tortuous heterogeneous catalyst particle as an example substrate. Typically, catalyst particles are simplified as a spherical particle (s = 2). Furthermore, the model can be implemented for the ALD process on many other shapes of porous media, for example, porous membranes (s = 0), high aspect ratio fibres (s = 1) and monoliths (s = 1). The selection of the particle shape factor and corresponding partial differential equation becomes increasingly relevant as the coating penetration depth is approaching the particle centre.

ALD | Atomic layer deposition |

a | Shape factor; a = s + 1(−) |

A
_{O}
| Specific surface area (m^{2} g^{−1}) |

A
_{Par}
| Catalyst particle outer surface area, ideal non-porous sphere (m^{2}) |

β
_{0}
| Initial sticking probability (−) |

D
_{e}
| Effective diffusion coefficient (m^{2} s^{−1}) |

D
_{i}
| Gas phase diffusion coefficient (m^{2} s^{−1}) |

D
_{Kn}
| Knudsen diffusion coefficient (m^{2} s^{−1}) |

D
_{mi}
| Molecular diffusion coefficient (m^{2} s^{−1}) |

d
_{pore}
| Mean pore diameter (m) |

v
_{th}
| Mean thermal velocity (m s^{−1}) |

ε | Porosity (0 ≤ ε ≤ 1) |

GPC | Growth-per-cycle |

h | Height (m) |

k
_{B}
| Boltzmann constant (m^{2} kg s^{−2} K^{−1}) |

m
_{r}
| Mass for a single reactant molecule (g) |

n
_{P0}
| Volumetric reactant number density (m^{−3}) |

p
_{A0}
| Partial pressure of component A (Pa) |

r | Radial coordinate (m) |

r
_{
θ=1/2}
| ALD reactant penetration depth (μm) |

R | Particle radius (m) |

s | Shape factor for a particle; s = a − 1 (−) |

Ratio of surface area (A_{O}) and pore volume (V_{P}) | |

s
_{0}
| Average surface area of adsorption sites (m^{2}) |

SCM | Shrinking core model |

σ | Areal number density of metal atoms deposited on each exposure cycle upon saturation (m^{−2}) |

t | Time (s) |

τ | Tortuosity (–) |

T | Temperature (K) |

θ | Surface coverage (0 ≤ θ ≤ 1) |

V
_{P}
| Pore volume (m^{3} g^{−1}) |

V
_{Par}
| Catalyst particle volume (m^{3}) |

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3cp05639b |

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