Tortuosity in electrochemical devices: a review of calculation approaches

Figures & data

Figure 1. Illustration of a tortuous path of length Δl through a porous microstructure of thickness Δx, where the shortest tortuous path is used to calculate the tortuosity of the sample.

Figure 2. Comparison of Bruggeman exponents and scaling parameters for different battery layers referenced in Table 1.

Table 1. Comparison of Bruggeman exponent and scaling parameter for battery layers fitted to experimental results.

Material	γ	α	Reference
Battery electrode LiMn2O4	1	3.3	Doyle et al. [48]
Battery separator PVdF	1	4.5	Doyle et al. [48]
Battery separator PVdF	1	2.4	Arora et al. [32]
Battery electrode MCMB 2528 and LiMn2O4	1	5.2	Arora et al. [32]
Battery electrode LiFePO4 and LiCoO2	1.8	1.53	Thorat et al. [47]
Battery graphite electrode	0.115	3.2111	Kehrwald et al. [51]
Battery graphite electrode	0.1146	3.159	Kehrwald et al. [51]
Battery electrode LiCoO2	2.5	1.27	Zacharias et al. [53]
Battery electrode LiCoO2	2.6	1.28	Zacharias et al. [53]
Battery separator Celgard 2400	0.667	2.43	Cannarella et al. [52]
Battery separator Celgard 3501	0.58	3.33	Cannarella et al. [52]
Battery separator GMB 500 mAh	1.77	1.77	Cannarella et al. [52]

Figure 3. Mass balance over a Wicke Kallenbach diffusion cell to extract the diffusion flow rate across a porous sample.

Figure 4. Comparison of experimental- and image-based tortuosity values at different temperatures and for varying H₂ concentrations in N₂ reproduced with permission from Elsevier [72].

Figure 5. Illustration demonstrating that high-resolution tomography is necessary to extract microstructural features which affect diffusive mass transport.

Figure 6. Geometric tortuosity distribution of the pore phase of the LiCoO₂ battery cathode of yz (A), xz (B) and xy (C) planes reproduced with permission from Elsevier [114].

Figure 7. Illustration of the pore centroid method calculation approach which measures the distance d_(n) of the centres of mass between two 2D image slices.

Figure 8. Results of the TauFactor solver by Cooper [118] running across the pore phase of a porous sample showing the binary image map, the initial, linear concentration distribution and the concentration distribution at steady state.

Table 2. Tortuosity values for pore, Ni and YSZ phase of an SOFC anode calculated using the random walk method and LBM [103].

		Random walk method	Lattice Boltzmann method
Pore phase	x	1.43	1.42
	y	1.41	1.44
	z	1.33	1.35
Nickel phase	x	4.70	4.66
	y	5.43	5.43
	z	2.63	2.63
YSZ phase	x	5.28	5.26
	y	3.87	3.85
	z	3.14	3.14

Table 3. Tortuosity values for graphite and pore phase using the random walk method and finite volume method [137].

		Random walk method	Finite volume method
Graphite phase	x	1.57	1.56
	y	1.92	1.89
	z	2.59	2.57
Pore phase	x	1.42	1.42
	y	1.19	1.18
	z	2.39	2.37

Figure 9. RVE analysis of the tortuosity factor for the pore and LSCF phase of an SOFC cathode as function of electrode thickness reproduced with permission from Elsevier [147].

Figure 10. Temperature distribution across the porous phase of an YSZ porous support membrane of an oxygen transport membrane.

Table 4. Comparison of tortuosity values along each dimension for pore phases of porous membranes calculated using flux-based algorithms.

Calculation method	Sample	Tomography technique	Pixel size	Sample volume	Porosity	Dimension	τ^2	τ	Reference
			[μm]	[μm^3]	[-]		[-]	[-]
Laplace equation	Ni-YSZ based SOFC anode	FIB-SEM	0.0417	105.2	0.195	x	2.10	1.45	Wilson et al. [102]
						y	2.20	1.48
						z	1.90	1.38
Laplace equation	Ni-YSZ based SOFC anode	X-ray	0.0427	250.0	0.300	x	2.94	1.71	Izzo et al. [81]
						y	3.28	1.81
						z	3.15	1.77
Laplace equation	Ni-YSZ based SOFC anode	X-ray	0.008	13.8	0.180	x	1.77	1.33	Grew et al. [120]
						y	1.51	1.23
						z	Not presented
Finite volume method	Mesocarbon microbead based battery electrode	X-ray	0.016	1,100.0	0.451	x	2.01	1.42	Tariq et al. [137]
						y	1.39	1.18
						z	5.61	2.37
Lattice Boltzmann method	Ni-YSZ based SOFC anode	FIB-SEM	0.062	972.4	0.496	x	2.03	1.42	Iwai et al. [103]
						y	2.06	1.44
						z	1.83	1.35
Random walk method	Ni-YSZ based SOFC anode	FIB-SEM	0.062	972.4	0.496	x	2.05	1.43	Iwai et al. [103]
						y	1.99	1.41
						z	1.78	1.33
Random walk method	Mesocarbon microbead based battery electrode	X-ray	0.016	1,100.0	0.451	x	2.03	1.42	Tariq et al. [137]
						y	1.41	1.19
						z	5.72	2.39
Heat flux simulation	LiFePO4 based battery electrode	X-ray	0.020	3,000	0.410	x	2.70	1.64	Cooper et al. [63]
						y	2.19	1.48
						z	3.32	1.82
Heat flux simulation	YSZ based porous support layer	FIB-SEM	0.030	421.9	0.350	x	2.82	1.68	Tjaden et al. [85]
						y	2.72	1.65
						z	3.10	1.76
Heat flux simulation	YSZ based porous support layer	X-ray	0.0325	314.4	0.410	x	3.13	1.77	Tjaden et al. [85]
						y	2.25	1.50
						z	2.96	1.72
Heat flux simulation	LiMn2O4 based battery electrode	X-ray	0.597	10,434,731	0.363	x	8.29	2.88	Shearing et al. [106]
						y	2.31	1.52
						z	4.97	2.23
Heat flux simulation	LiMn2O4 based battery electrode	X-ray	0.065	75,164	0.380	x	6.50	2.55	Shearing et al. [106]
						y	2.22	1.49
						z	3.96	1.99
Laplace equation	LSCF based SOFC cathode	FIB-SEM	0.035	144.7	0.483	x	1.82	1.35	Joos et al. [147]
						y	1.83	1.35
						z	1.88	1.37
Laplace equation	YSZ based porous support layer	X-ray	0.060	46,656	0.470	x	2.30	1.52	Laurencin et al. [150]
						y	2.80	1.67
						z	2.60	1.61

Table 5. Comparison of tortuosity values along each dimension for pore phases of porous membranes calculated using geometric-based algorithms.

Calculation method	Sample	Tomography technique	Pixel size	Sample volume	Porosity	Dimension	τ^2	τ	Reference
			[μm]	[μm3]	[-]		[-]	[-]
FMM	LiCoO2 based battery electrode	X-ray	0.65	17,487,363	0.431	x	1.16	1.08	Taiwo et al. [2]
						y	1.12	1.06
						z	1.02	1.01
FMM	Graphite based battery electrode	X-ray	0.65	28,080,838	0.484	x	1.10	1.05	Taiwo et al. [2]
						Y	1.11	1.05
						z	1.03	1.01
FMM	LiMn2O4 based battery electrode	X-ray	0.65	27,529,606	0.453	x	1.14	1.07	Taiwo et al. [2]
						y	1.11	1.05
						z	1.02	1.01
FMM	YSZ based porous support layer	FIB-SEM	0.030	421.9	0.350	x	1.42	1.19	Tjaden et al. [85]
						y	1.25	1.12
						z	1.23	1.11
FMM	YSZ based porous support layer	X-ray	0.0325	314.4	0.410	x	1.44	1.20	Tjaden et al. [85]
						y	1.25	1.12
						z	1.19	1.09
Pore centroid method	LiFePO4 based battery electrode	X-ray	0.020	3,000	0.410	x	1.46	1.21	Cooper et al. [63]
						y	1.41	1.19
						z	1.58	1.26

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