Fractal modeling of moisture diffusion in wood cell wall

Moisture diffusion in wood remains incompletely understood due to its complex hierarchical structure. This study develops a theoretical fractal moisture diffusion model for the wood cell wall, incorporating its structural geometry, and upscales it to bulk wood using electrical resistance modeling and validation. The model accounts for fractal dimensions, porosity, and pore size distribution. Water vapor sorption data, obtained via dynamic sorption, were used to derive experimental diffusion coefficients, while mercury intrusion porosimetry characterized pore size distributions and calculated pore fractal dimensions. Model validation showed that predicted diffusion coefficients aligned with experimental values at low moisture contents. Fractal diffusivity was negatively correlated with pore size ratio and fractal dimensions of pores and tortuosity but positively correlated with porosity. These findings support the development of predictive models for wood geometric properties and moisture diffusivity based on structural attributes.

The study of molecular moisture diffusion in wood, driven by concentration gradients and quantified by the diffusion coefficient (D), is a fundamental mass transfer phenomenon in various wood processing operations. These include timber, veneer, and particle drying; fiberboard and paperboard manufacturing; thermal modification; biorefinery processing; and combustion for thermal energy production. Moisture transport properties directly impact product quality and processing efficiency. A deeper understanding of these properties is essential for optimizing wood processing technologies, improving product quality, and expanding wood applications.

Despite extensive research, the moisture-dependent phenomena and transport mechanisms in wood remain poorly understood. This complexity arises from its hierarchical structure as a composite material with nanoscale constituents, making comprehensive moisture transport explanations challenging.

Numerous predictive models describe moisture diffusion in wood and wood-based products. Most are designed for specific moisture (M) or temperature (T) ranges, with some refined for applications by adjusting these parameters to predict moisture transport under ambient conditions. However, these models often lack sufficient theoretical justification for the influence of wood pore structural parameters, leading to an incomplete understanding of moisture diffusion in wood.

Wood consists of cell wall substances and interconnected pores of varying diameters, ranging from the microscale to macroscale. To account for this structural complexity in modeling, simplifying assumptions are necessary to minimize parameters while preserving heterogeneity. However, selecting a representative volume element at the smallest structural level remains challenging, and moisture dynamics at the molecular level lack sufficient experimental validation. Given these constraints, this study considers the wood cell level—comprising the cell wall and lumen as the smallest representative heterogeneous unit for moisture transport. The well-defined structural pore boundaries of cell walls and lumens enable localized modeling of physical properties. An upscaling approach is then applied to this level to determine gross wood diffusivity.

Fractal theory offers an effective approach to describing wood’s irregular structure by systematically selecting key parameters while preserving its inherent heterogeneity. Rooted in self-similarity and scale invariance, fractal geometry provides a novel mathematical framework for modeling complex natural structures, enabling the prediction of moisture diffusivity mechanisms in wood. Since moisture diffusion cannot be fully understood without considering wood’s hierarchical structure, developing a universal predictive model that incorporates geometric anatomy is essential for investigating cell wall pore geometry from the diffusion coefficient (D).

Several theoretical moisture diffusion models [1,2,3,4,5] have been proposed, primarily based on porosity and density as key controlling factors. Common approaches include the mixing law correlation model, which applies weighted harmonic and arithmetic means, and serial–parallel electrical analogy principles. These methods estimate D in bulk wood under specific assumptions, such as isotropic behavior in the radial and tangential directions, one-dimensional steady-state moisture transfer, and a spatially periodic microstructure. However, existing models remain inadequate for fully capturing wood’s moisture diffusion behavior.

Fractal methodology is widely applied across various fields, including water sorption modeling in porous materials. In wood research, fractal theory was first used a decade ago to study water sorption on cell wall surfaces and lumen spaces [6,7,8]. However, its application to moisture diffusion modeling remains unexplored. Accurate calculation of the diffusion coefficient (D) requires understanding its relationship with pore structural parameters (_p, _T, and λ). Yet, no documented datasets exist on wood pore geometry to predict D or assess the influence of individual pore parameters on moisture diffusion.

This research examines wood species with varying geometric parameters (_p, _T, V_a, and λ) to predict the diffusion coefficient (D) using a fractal geometry approach. Special emphasis is placed on the influence of individual pore structures and their potential in accurately predicting D. Notably, no existing study has systematically developed a predictive model incorporating multiple wood structural parameters. Given the critical role of geometric structure in moisture transport, this study aims to establish a universal fractal-based model for diffusion coefficient determination D_FM. Experimental evidence of the fractal nature of cell wall surfaces and pore size distributions serves as the primary motivation for developing this model.

Parallel–series electrical resistance model in wood cell wall moisture diffusion

The parallel–series electrical resistance model describes how multiple resistors are connected in a combination of parallel and series within an electrical circuit. In this setup, resistors in parallel share the same voltage, while the total current is the sum of the currents through each resistor. The resistors in series share the same current, and the total voltage is the sum of the individual voltages across each resistor.

The parallel–series resistance model can be applied to moisture diffusion in wood to represent the movement of water through various pathways with different resistances. Wood is a complex, heterogeneous material, containing voids like lumens and cell walls made up of cellulose, hemicellulose, and lignin, each contributing differently to moisture transport.

Moisture moves through wood along multiple pathways, shaped by its complex anatomical and chemical structure. These pathways operate simultaneously and can be modeled using a parallel–series resistance model, where different wood regions offer varying levels of resistance to moisture movement.

Main moisture pathways in wood

Cell lumen pathway (bulk flow) Water moves through the large open spaces (lumens) of tracheids and vessels. Dominates when wood is wet (above fiber saturation point, FSP) and fast transport with low resistance.

Pit membrane pathway Moisture moves between cells through pit membranes, controlling flow, moderate resistance, influenced by pit aspiration and clogging.

Cell wall pathway (microfibril network) Water diffuses through the hydrophilic cell wall matrix (hemicellulose and cellulose). Dominates below FSP as bound water transport and higher resistance due to molecular interactions.

Intercellular space pathway Water moves through small voids between cells and minor role but contributes at high moisture levels.

Ray parenchyma pathway Radial moisture moves through ray cells and facilitates water redistribution within the wood.

To better understand the intricate and multifaceted mechanisms governing moisture movement within the wood structure, the following diagram has been included, which visually represents the various routes through which moisture can travel radially, considering the complex interaction between the cell walls, lumens, rays, and pits in the wood’s internal structure, as seen in the 2D cross sections (Fig. 1).

Siau [14] described a classical moisture diffusion model that accounts for the complex transport of water in wood. His model highlights how moisture moves through different regions of the wood structure, combining both series and parallel diffusion pathways. Moisture diffusion in wood, as described in Siau’s [14] model, can be analogized using the electrical resistance model by considering different moisture transport pathways as resistances (Fig. 2).

Since these diverse pathways for moisture transport within the wood operate simultaneously and in an interconnected manner, the overall moisture movement can be conceptualized as a complex system where different routes interact, each contributing to the total resistance in a way that can be mathematically represented as a parallel–series system of resistance, effectively combining both parallel and series elements to account for the varying degrees of moisture flow through the cell walls, lumens, rays, and pits. In a system where resistances are in both series and parallel, the total resistance is calculated by first combining the parallel resistances, and then adding the series resistances as

Since resistance is inversely proportional to diffusivity, the equivalent moisture diffusivity (D_tot) is

This model explains why moisture movement in wood is highly anisotropic, meaning it differs along the longitudinal, radial, and tangential directions.

Fractal geometric model

From a fractal geometry perspective, the pore space and surface of the wood cell wall exhibit statistical self-similarity, meaning that their structural characteristics appear similar across different scales, but only in a statistical sense rather than exact replication [9,10,11,12]. The hierarchical structure of wood spans from macro- to nanoscale, comprising vessels, tracheids, pits, and nanoscale pores in the cell wall. These voids exhibit a power-law distribution, characteristic of fractal structures, where larger pores branch into progressively smaller ones in a statistically self-similar manner. The wood cell wall surface is inherently rough due to its fibrillar and microfibrillar arrangement, maintaining statistical self-similarity across different magnifications, which can be quantified by its fractal dimension. Models such as the percolation theory describe the connectivity and diffusion pathways within these pores. The fractal dimension (D_f) influences diffusion rates, affecting moisture transport, adhesive penetration, and gas exchange, with higher values indicating more tortuous diffusion paths that impact permeability and substance retention [10, 12]. Wood can thus be modeled as a fractal structure. To apply fractal geometry and simplify model development, the following assumptions are made:

1. 1)

   The pore structure of wood cell walls was modeled as tortuous cylindrical pores, with moisture migration occurring only in the radial direction.

2. 2)

   Pores within the wood cell wall are assumed to be isolated, and both the lumen and cell wall have square cross sections.

3. 3)

   For model development, cell wall pores and mixtures are considered equivalent, and the gross wood diffusion coefficient is determined using a series–parallel electrical resistance approach.

4. 4)

   The model accounts only for bound water within cell wall pores and water vapor in lumens.

Based on a fractal geometry interpretation of porous media, the relationship between pore size (r) and the number of pores (N) larger than r is given by [13]

where r_max is the largest pore size in the cell wall, f(r) is the pore size function, K₀ is a constant, and D_p​ is the pore fractal dimension of the wall pores.

Equation (2.1) can be expressed as

where K₁ is a constant, which equals (-\({\mathfrak{D}}_{p}\) K₀).

The cumulative pore volume, V(< r), for pores smaller than r is expressed as

From Eqs. (2.2) and (2.3), the cumulative pore volume V(< r) for pores with a radius smaller than r is given by

where K₂ is a constant, which equals \(C{K}_{1}/(3-{\mathfrak{D}}_{p})\).

Similarly, the total cumulative pore volume V_tot​ is given by

where V_cum.fra​ is the cumulative pore volume fraction for pores smaller than r, and V_ccu.fra​ is given by

Due to the significant heterogeneity in pore sizes within wood cell walls, where r_min​ ≪ r_max​, Eq. (2.6) can be simplified as

Equation (2.8) is derived by taking the logarithm of Eq. (2.7):

where K is a constant that equals \((3-{\mathfrak{D}}_{p})ln{r}_{max}\).

Gross wood moisture diffusion coefficient model

The proposed fractal moisture diffusion coefficient model is developed based on the Siau unit cell model, which simplifies the complex geometry of wood cells into a cuboid with a square cross section, ignoring cell wall heterogeneity. It assumes the cell wall contains tortuous pores of varying sizes. The fractal diffusion model incorporates the irregularities of pore structural parameters using the capillary bundle model and fractal theory for bound water in the cell wall. An upscaling approach based on serial–parallel electrical analogy is then applied to model moisture transport at the macroscopic scale of bulk wood.

Figure 3 a illustrates the geometry of a single wood cell, featuring a bundle of tortuous capillaries within the cell wall, where capillary distribution and tortuosity follow fractal scaling laws. The capillary tubes pass through a two-dimensional (2D) cross section (A_cs) along the cell’s cross and sidewalls. A bundle of fractal tortuous capillaries with varying pore sizes facilitates bound water diffusion in the radial direction, as shown in Fig. 2. Water vapor transport through the lumen is modeled similarly to previous studies [14]. In Fig. 3 b, L_t represents the actual path length for bound water travel in the radial direction, L₀ is the straight-line distance, and r is the pore radius.

According to Fick's law, the mass flux of bound/adsorbed water q(r) through a single tortuous capillary pore in the cell wall (Fig. 1b) is expressed as

where A(r) = πr² is the cross-sectional area of a single pore, D_BR​ is the activated diffusion coefficient for bound water in the radial direction along the pore, and \(\Delta C\) is the water concentration gradient along the radial direction. Pores of various sizes are randomly distributed within the walls, creating moisture diffusion pathways modeled as a bundle of tortuous capillaries with radii r and tortuous length L_t​. Based on the fractal scaling law for tortuous capillaries proposed by [35], the relationship with pore radius is expressed as follows:

where D_T​ is the fractal dimension of tortuous capillaries, ranging from 1 to 2 in a 2-dimensional plane. D_T​ = 1 represents a straight capillary path, while higher values of D_T​ indicate increased tortuosity, with D_T = 2 corresponding to a highly tortuous line filling the plane. Due to the complexity of the tortuous structure, no standard experimental method exists to determine D_T​. To quantitatively describe the complexity of tortuous structures in porous media, we developed an equation for the tortuous fractal dimension. Traditional methods of characterizing porous structures often rely on Euclidean measures, which fail to capture the intricate, multi-scale nature of pore connectivity and transport pathways. Fractal geometry, on the other hand, provides a powerful framework for analyzing such systems, as it accounts for irregularity and self-similarity across different length scales.

The tortuous fractal dimension is influenced by both the porosity (V_a) of the medium and the fractal dimension of the pore distribution (\({\mathfrak{D}}_{p}\)), as these parameters determine the degree of complexity and connectivity of the porous network. Porosity governs the overall void fraction available for fluid flow, while the pore distribution fractal dimension characterizes the spatial heterogeneity and scaling properties of the pore structure.

By integrating these factors, we derived an equation that expresses the tortuous fractal dimension as a function of porosity and the pore distribution fractal dimension. This formulation allows for a more accurate representation of flow pathways in porous media, enabling better predictions of transport properties such as permeability and diffusion. Therefore, [34] developed the following equation for the tortuous fractal dimension:

The derived equation is a function of porosity and pore distribution fractal dimension. This equation enhances our ability to model and analyze complex porous systems, with applications in fields such as geophysics, material science, and biological transport processes.

The total mass flow rate through a 2D cross-sectional area (Fig. 2a) is obtained by integrating the individual mass flux q(r) from the minimum to maximum radius. Using Eqs. (2.1) and (2.10), along with the pore area, the following equation is derived:

The integral was done and simplified step by step: \({\int }_{{r}_{min}}^{{r}_{max}}\frac{\pi .{r}^{2}.{D}_{RB}.\Delta C (-dN)}{{{L}_{0}}^{{\mathfrak{D}}_{T}}{.\left(2r\right)}^{1-{\mathfrak{D}}_{T}}},\)

we simplify the denominator \({\left(2r\right)}^{1-{\mathfrak{D}}_{T}}= {2}^{1-{\mathfrak{D}}_{T}}{r}^{1-{\mathfrak{D}}_{T}}\), so the integral becomes \({\int }_{{r}_{min}}^{{r}_{max}}\frac{\pi .{r}^{2}.{D}_{RB}.\Delta C (-dN)}{{{L}_{0}}^{{\mathfrak{D}}_{T}} . {2}^{1-{\mathfrak{D}}_{T}}{r}^{1-{\mathfrak{D}}_{T}}}\); now we simplify the powers of r:\({\int }_{{r}_{min}}^{{r}_{max}}\frac{\pi .{r}^{1+{\mathfrak{D}}_{T}}.{D}_{RB}.\Delta C (-dN)}{{{L}_{0}}^{{\mathfrak{D}}_{T}} . {2}^{1-{\mathfrak{D}}_{T}}}.\)

Now evaluating the integral,

dN is related to a fractal distribution of pore sizes, and it is given by Eq. (2.1); then Eq. (2.12a) becomes.

The general integral formula is as follows: \(\int {r}^{n}dr= \frac{{r}^{n+1}}{n+1}\), for \(n \ne -1\)

Applying the above general integral formula,

We define \(\lambda = \frac{{r}_{min}}{{r}_{max}}\), which is the minimum pore radius to maximum pore radius ratio. So, with re-arranging of Eqs. (2.12b) it becomes

Handling the \({r}_{max}\) to \({L}_{0}\) transition the term \({\left(\frac{{r}_{max}}{{L}_{0}}\right)}^{{\mathfrak{D}}_{T}}.{r}_{max}\) suggests a geometric scaling relationship. Typically, in fractal porous media, characteristic lengths relate with porosity (V_a) and pore distribution fractal dimension (\({\mathfrak{D}}_{p}\)) as.

\({r}_{\text{max} }\approx {L}_{0}{\left(\frac{(2-{\mathfrak{D}}_{p}){V}_{a}}{\pi {\mathfrak{D}}_{p}(1-{V}_{a})}\right)}^{\frac{1}{{\mathfrak{D}}_{p}+1}}\). Substituting this into the equation and raising it to the power of \({\mathfrak{D}}_{T}\)​ lead to

Substitution of Eq. (2.12d) into Eq. (2.12c) can obtain Eq. (2.13) as

While D_RB ​ itself is not explicitly expressed as a function of porosity in the given equation, in porous media like wood, the effective diffusion coefficient (D_eff) often depends on porosity via empirical or theoretical relationships like \({D}_{eff}= {D}_{BR}f\left({V}_{a}\right)\), where V_a is the porosity of wood voids, \(f\left({V}_{a}\right)\) is a porosity-dependent function that accounts for tortuosity and connectivity of the pore networks.

According to Fick's law, the total mass flow rate through the cross-sectional area can also be expressed as

When we consider the area of a single pore in Eq. (2.14), Eq. (2.15) and Eq. (2.14) are equivalent to each other.

Note: These above two Eqs. (2.14 and 2.15) are not directly the same in terms of their form, but they can be equivalent under consideration of the relationships between A_cs and L₀. In simplified/idealized pore model, which is if we take a cylindrical geometry for simplicity, the cross-sectional area A_cs​ is given by

In simplified models, particularly for porous media such as wood cell walls with regular geometries, the cross-sectional area A_cs​ can be expressed in terms of the characteristic length L₀​.

By comparing Eqs. (2.13) and (2.15), the fractal diffusion of bound water through the wall in the radial direction is expressed as

The radial diffusion coefficient of bound water (D_BR​) is based on the transverse bound water diffusion coefficient (D_BT​), as developed in [15], representing activated/surface diffusion through cell wall pores, \({\mathfrak{D}}_{p}\) is pore fractal dimension, \({\mathfrak{D}}_{T}\) is the tortuous fractal dimension, \(\lambda \) is the pore ratio (minimum pores size divide by maximum pore size), and \({V}_{a}\) is the porosity of wood voids. The relationship between radial and transverse bound water diffusion is described by [16], where the radial diffusion coefficient is 1.5 times the transverse diffusion coefficient (D_BR ​ = 1.5D_BT​).

Equation (2.18) represents a bound water diffusion coefficient along the radial direction (D_BR​) which is a function of temperature (T), universal gas constant (R), and a moisture content (M).

Consequently, the effective bound water diffusion through the wood cell walls is given by

Using the same upscaling method (series–parallel electrical analogy) employed by Siau to determine the total radial diffusion in gross wood, the fractal diffusion for gross wood is derived. Applying Eq. (2.20), the gross wood fractal diffusion coefficient D_FM​ in the radial direction is given by

where \({D}_{FM.CW}\) is given by Eq. (2.17).

The fractal model equation developed in this study (Eq. 2.20) represents an effective fractal diffusion coefficient (D_FM) through wood along the radial direction. The key components are as follows: D_V​: diffusion coefficient in the void which represents diffusion occurring through the larger open pores, such as vessel elements in hardwoods or tracheids in softwoods. These voids allow for relatively unhindered mass transport; D_FM: diffusion coefficient within the cell wall or denser region of the material which presents greater resistance to mass transfer; V_a: apparent porosity (fraction of void space in the material) and higher porosity typically means easier diffusion, while lower porosity indicates more resistance; and \(\sqrt{{V}_{a}}\): a term reflecting a characteristic scaling factor related to the fractal nature of the pore structure.

The model considers diffusion through both the open voids (lumens) and the denser cell wall regions. The numerator represents a weighted sum of the diffusion contributions from both regions, based on their respective fractions in the material. The denominator accounts for the complex connectivity and interaction between the two diffusion paths, incorporating a correction factor related to the porosity and its fractal influence.

This fractal-based model provides a way to estimate the effective diffusion coefficient in wood which is a porous media, considering both pore complexity and the resistance from solid structures (cell walls). The inclusion of​​ \(\sqrt{{V}_{a}}\) suggests that diffusion follows a non-linear scaling behavior, influenced by the microstructural organization of the wood cell wall materials.

This model can help predict how moisture or solutes move through wood, which is important in drying, impregnation (e.g., preservatives), and other wood treatment processes.

Stem sections from 72-year-old Douglas-fir (Pseudotsuga menziesii), 69-year-old western red cedar (Thuja plicata), 82-year-old aspen (Populus tremuloides), and 89-year-old birch (Betula papyrifera) logs were harvested from local forests. The sections were then sealed in plastic bags and stored at 5 °C in a cold room.

In this study, Mercury Intrusion Porosimetry (MIP) was used to analyze the pore structure of different wood species. To prepare the samples, radial boards were cut from both sapwood and heartwood sections of each species. The samples were then labeled as follows: Douglas-fir sapwood (DF-S), Douglas-fir heartwood (DF-H), western red cedar sapwood (WRC-S), western red cedar heartwood (WRC-H), aspen sapwood (A-S), aspen heartwood (A-H), birch sapwood (B-S), and birch heartwood (B-H).

The boards were ripped into 100 mm wide strips, then planed to precise dimensions (100 ×ばつ 100 ×ばつ 5 mm³) in the longitudinal, tangential, and radial directions. The strips were cut into wafer-type samples using a band saw and randomly divided into two categories for sorption testing. Ten replications were performed for each wood sample.

Sorption kinetics analysis by conditioning chamber

Adsorption data were collected from all wood samples (sapwood and heartwood of DF-S, DF-H, WRC-S, WRC-H, A-S, A-H, B-S, and B-H) at 30 °C and 50 °C, and relative humidity levels of 30%, 50%, 70%, 80%, and 90%. Each sample was conditioned in a cabinet for 3 to 7 days, depending on the temperature and humidity, until equilibrium was reached.

Wafer-type samples from sapwood and heartwood had dimensions of 100 ×ばつ 100 ×ばつ 5 mm³ in the longitudinal, tangential, and radial directions, respectively. Due to the thin radial thickness, no edge sealing was applied. Forty samples were randomly divided into two groups for sorption tests at 30 °C and 50 °C, with 20 samples at each temperature, and five relative humidity levels (30%, 50%, 70%, 80%, and 90%). The samples were vacuum-dried at 50 °C for three weeks to achieve a moisture content of 0%, ensuring material structure recovery without damage. After drying, the samples were cooled in a desiccator, and their dimensions were measured at six positions for thickness, three for width, and two for length. The samples were weighed to the fourth decimal place using an electronic balance and placed into conditioning cabinets (Parameter Generation and Control, Inc., USA) set to 30 °C and 50 °C with air temperature tolerance of ± 0.1 °C and stable relative humidity tolerance of ± 0.1%.

Weight gains were recorded gravimetrically, and the dimensionless moisture content (E(t)) was calculated using Eq. (3.1), while the equilibrium moisture content (M_emc) was determined using Eq. (3.2).

where M₀ is the initial moisture content (oven-dry mass), M(t) is the moisture content at the i^th hour, w₀ is the oven-dry weight (grams), and w_∞ is the final sample weight.

A total of 160 sorption measurements (80 per temperature) were conducted. The experimental data (M_emc), calculated using Eq. (3.2), were then fitted to a single hydrate sorption model [17].

How to measure diffusion coefficients using the dynamic method

The moisture content (M) was determined based on oven-dry weight and converted to dimensionless moisture content (E(t)). The fractional moisture gain (E̅(t)) was plotted against the square root of time, and the slope of the linear portion was determined. This slope, along with sample thickness, was used to calculate the gross wood diffusion coefficient (D_Exp) using the following equation:

where a_1/2​ is the half-thickness of the sample.

Sorption kinetics, required for calculating D_Exp​, were obtained from measurements at 3, 6, 12, 24, 48, 72, 96, and 120 h after the samples were placed in the conditioning cabinet, following removal from Ziploc bags. Kinetic measurements were performed on all samples used in the sorption isotherm experiments.

The moisture content (M) corresponding to D_Exp​ was determined using Eq. (3.4) [18] for wood:

where M_bar is the average moisture content, M₁​ is the initial moisture content, and M₂​ is the final moisture content from the dynamic sorption method.

Pore analysis by mercury intrusion porosimetry (MIP)

MIP was conducted using an AutoPore IV 9500 (Micromeritics Instrument Corporation, USA) to determine porosity, pore volume, threshold pressure, pore size, and distribution. Cubic samples were placed in a sample tube and loaded into the chamber. Measurements were initiated by increasing pressure from 0.10 to 61,000 psia, with samples immersed in non-wetting mercury. Prior to each experiment, samples were evacuated, and equilibrium was reached within 10 s for each pressure increment. Pressure increments were automatically adjusted, with slower rates at lower pressures. As pressure increased, mercury gradually intruded from larger to smaller voids. Each run took approximately 30 min. Pore volume and size distribution were calculated from the volume of mercury intruded at corresponding pressures.

Determination of fractal dimension

By utilizing the data obtained from Mercury Intrusion Porosimetry (MIP), the relationship between the pore volume and pore size within the cell wall structure of the wood was thoroughly analyzed. This analysis enabled the calculation of the cumulative pore volume fraction (V_cum.fra), as outlined in Eq. (2.6). To examine the nature of the pore structure, a plot was generated that displayed the natural logarithm of the cumulative pore volume fraction (ln(V_cum.fra)) against the natural logarithm of the pore radius (ln(r)).

If the resulting plot exhibited a linear trend, the slope of this line was interpreted to correspond to the value of the three-dimensional pore fractal dimension (D_p). In such cases, the fractal dimension, D_p, is constrained to a value between 2 and 3, which indicates that the pore structure of the wood cell wall exhibits scale invariance. This scale-invariant characteristic suggests that the pore structure can be effectively modeled as a fractal, reflecting self-similarity across different scales. The fractal dimension (D_p) is then computed using the formula: D_p = 3—slope.

Determination of tortuosity fractal dimension \({\mathfrak{D}}_{T}\)

The tortuosity fractal dimension (D_T) is a key parameter used to describe the complexity of a porous medium’s structure, specifically referring to the convoluted nature of the pathways through which fluids or gases must travel. The determination of D_T is typically carried out through various computational or analytical approaches, as no direct experimental method exists to measure this dimension. Among the most common techniques are the box-counting method, which involves dividing the space into smaller boxes to estimate the number of boxes required to cover the structure at different scales, and the Monte Carlo method, which uses statistical sampling to approximate the tortuosity. Additionally, in some cases, tortuosity can be estimated using analytical expressions, although these are less frequently applied.

In the present study, the tortuosity fractal dimension (D_T) was evaluated using an analytical expression that was specifically developed as part of our ongoing research. This method, described in Eq. (2.11), provides a more direct and efficient means of estimating D_T, allowing for a deeper understanding of the tortuosity characteristics of the material under investigation.

Pore structure analysis of wood

Figures 4 and 5 show the pore size distributions of the studied wood samples, with pore sizes ranging from ~ 1 nm to 106 nm. The differential intrusion curves reveal pore distributions below 104 nm (red dashed line). Except for WRC-S and B-S, the other samples display bimodal distributions. The log differential intrusion curves highlight distributions for pore sizes greater than 102 nm (red dashed line), with all wood samples exhibiting bimodal distributions. From these curves, three pore size categories are identified: macropores (350 μm–15 μm), mesopores (15 μm–0.1 μm), and micropores (0.1 μm–3 nm). Based on these distributions, parameters such as porosity, pore fractal dimension, tortuosity fractal dimension, and pore ratio were calculated and are summarized in Table 1 as input for the fractal diffusion model. As noted by [19], each pore size category (micro, meso, and macro) influences the material's physical properties. Therefore, structural analysis of each pore size distribution will be conducted separately, although the values presented in Table 1 are cumulative for each region.

The fractal dimension D_p​ was determined from the slope of the cumulative pore volume fraction versus pore size, while Va​ and λ were derived from MIP data. The D_p values obtained ranged from 2.87 to 2.96, with the highest value for DF-S and the lowest for B-S, corresponding to porosities between 58 and 76%. These results suggest that the pore structures of both sapwood and heartwood, from softwoods and hardwoods, exhibit complexity and irregularity. The largest D_p value was observed for DF-S, indicating a more irregular pore structure, while the lowest was for B-S. This is consistent with previous studies [20], which reported D_p​ values for cell wall surfaces ranging from 2.5 to 2.99. The fractal dimension model developed here offers a simplified and more accessible method for determining D_p​ from experimental pore data, providing a more accurate description of the geometric complexity of wood cell walls.

The tortuosity fractal dimension (D_T​) values in this study ranged from 1.0357 to 1.0734, with the highest value for DF-H (lower V_a ​ and D_p ​) and the lowest for WRC-S (higher V_a ​ and D_p ​) (Table 1). These results are consistent with those from Monte Carlo simulations on heterogeneous aquifer pores, as reported by [21], which also evaluated tortuos fractal dimension.

Diffusion coefficients predicted from fractal model

Assuming water vapor diffuses through the cell lumens and bound water diffuses through the cell walls, the radial gross wood fractal diffusion model (D_FM) was developed, incorporating structural parameters (i.e., D_T​, D_p​, λ, and V_a​), as shown in Eq. (2.11). The model is based on the following: 1) the bound water diffusion coefficient in the cell wall, a function of fractal dimensions (D_T, D_p​), λ (pore size ratio), moisture content (M), and temperature (T); 2) D_v​, dependent on the specific gravity of the cell wall (G_c.w​), sorption isotherm slope, water vapor diffusion in bulk air (D_air​), and saturated vapor pressure; and 3) porosity (V_a​). Structural parameters (D_p​, λ, D_T​, and V_a​) are provided in Table 1. The model was validated against experimental diffusion coefficient data.

Figures 6 and 7 present the D_Exp and D_FM for all studied wood samples, along with the corresponding moisture content (M) and temperature (T). These comparisons confirm the similarity and reliability between the experimental and predicted moisture diffusivity models. The DFM values, derived from the fractal model, ranged from 1.2 ×ばつ 10−¹¹ m²/s to 3.3 ×ばつ 10−¹0 m²/s at 30 °C, and from 0.5 ×ばつ 10−¹¹ m²/s to 6.3 ×ばつ 10−¹0 m²/s at 50 °C. These values align with previous studies on wood diffusivity [22,23,24,25].

The radial D_FM determined by the fractal model increased with moisture content (M). At high M, the D_FM values were higher than D_Exp, while at low M, they closely matched experimental results due to pore geometry effects, which were neglected in the experimental methods. Experimental results also increased with M, but not as drastically as the D_FM values. At low M (up to 9% M), D_Exp for B-S and B-H was 2.5 to 3 times greater than the fractal model predictions, while at high M (above 9% M), D_Exp was 1 to 2 times higher. For DF-S and DF-H, D_Exp at low M (up to 8% M) was 1 to 4 times greater than the D_FM, while at high M (above 8% M), D_Exp was 0.5 to 1.5 times the fractal model value. Similar trends were observed across all wood samples. Thus, at low M (up to 9%), both D_Exp and D_FM showed comparable trends, demonstrating that the D_FM can successfully predict diffusion coefficients across the moisture range. Similar trends were observed for various wood species [26,27,28,29,30]. Additionally, diffusion coefficients increased by 2–3 times when temperature increased from 30 °C to 50 °C. Overall, the close agreement between the experimental and fractal model values supports the validity of the proposed model.

Statistical analysis of the diffusion coefficients (D) presented in Table 2 was conducted to assess differences between D_FM and D_Exp, as well as D_FM and D_Siau, using the fractal model, experimental method, and Siau's model. The results indicated significant differences between D_FM and D_Exp for some wood types (p < 0.05, Table 2), while no significant differences were observed for others (p > 0.05) at 30 °C and 50 °C. Similarly, significant differences were found between D_FM and D_Siau for certain wood types (p < 0.05), while others showed no significant differences (p > 0.05) at both temperatures.

Effects of structural parameters on DFM

The influence of pore structural parameters, including the fractal dimension of pores (D_p), tortuosity (D_T), pore size ratio (λ), and accessible pore volume (V_a), on the diffusion coefficient in fractal media (D_FM) is analyzed based on data obtained from the mercury intrusion porosimetry (MIP) analysis of all wood samples. D_p is intrinsically linked to pore volume distribution, λ, and V_a, affecting moisture transport properties.

For sapwood samples (WRC-S, DF-S, A-S, and B-S) at 30% relative humidity (H) and 30 °C with a D_T of 1.036, an inverse relationship is observed between D_FM and D_p. This decline in diffusivity with increasing D_p is attributed to enhanced pore fractality, which increases the complexity of the pore structure and impedes moisture movement. When D_p approaches 2, indicating a highly intricate pore network, D_FM trends toward zero, reflecting a significant reduction in effective diffusivity.

A similar trend is observed in the relationship between D_FM and tortuosity (D_T) at 30% H and 30 °C with a D_p of 1.965. As D_T increases, D_FM decreases due to the greater resistance to moisture movement within more tortuous capillary pathways. This behavior is consistent with theoretical expectations, where higher tortuosity values correspond to increased diffusion resistance and reduced moisture transport efficiency.

The impact of the pore size ratio (λ) on D_FM at 30% H and 30 °C, with D_p of 1.965 and D_T of 1.036, further supports this pattern. An increase in λ is associated with a reduction in D_FM, likely due to the presence of a greater disparity between minimum and maximum pore sizes. This results in a more heterogeneous and fractal-like pore network, intensifying resistance to moisture diffusion.

The relationship between V_a and D_FM at 30% H and 30 °C, with D_p of 1.965 and D_T of 1.036, reveals a direct correlation. As V_a increases, the fractal model predicts higher diffusivity, as a larger accessible pore volume reduces moisture transport resistance and facilitates diffusion. This trend is consistent with theoretical expectations, where an increase in V_a provides more effective pathways for moisture movement. At these conditions, the predictions from the fractal model align closely with those of Siau’s model, though deviations occur at low V_a for WRC-S and at high V_a for other sapwood samples. At elevated humidity and temperature, the diffusivity values predicted by Siau’s model diverge significantly from those obtained using the fractal approach.

Statistical analysis revealed strong correlations between structural parameters and D_FM for most wood samples (R² > 0.4), except for WRC-S and DF-S, where λ and D_FM showed weak correlation. A strong positive correlation (R² > 0.90) was observed between D_FM and V_a. Negative correlations were found between D_FM and D_p (R² > 0.85) and D_FM and tortuosity D_T (R² > 0.73).

The data indicate a strong positive correlation between V_a and D_FM, while D_p exhibits a strong negative correlation with D_FM. These two structural parameters primarily influence moisture diffusion in wood. In porous materials, higher V_a and D_p typically signify greater structural complexity. Similar trends have been reported in previous studies on wood and wood composites [31,32,33].

The correlation between D_FM and both tortuosity (D_T) and λ is influenced by pore size variation and arrangement. Greater differences in pore size and irregular pore structures contribute to increased wood complexity. While porosity and pore fractal dimension show a strong correlation with moisture diffusivity, the structural complexity related to pore size distribution and tortuosity also plays a significant role. Thus, the proposed fractal model provides deeper insights into the physical mechanisms of moisture diffusion in wood compared to existing models.

Pore fractal dimensions were determined from MIP data using the proposed model, confirming the fractal nature of wood cell wall structures. Values ranged from 2.87 to 2.96, with the lowest for birch sapwood and the highest for Douglas-fir sapwood, corresponding to porosities of 58–76%. Tortuosity fractal dimensions ranged from 1.036 to 1.073, with the lowest in western red cedar sapwood and the highest in Douglas-fir heartwood.

A fractal diffusion model was developed to predict radial moisture diffusion in wood below the fiber saturation point. Based on Siau’s unit cell approach, the model incorporates geometric heterogeneity using fractal theory, considering pore fractal dimension, tortuosity, porosity, pore size ratio, and other structural parameters. Unlike empirical models, it has no fitting constants and applies to all wood species, providing deeper insights into moisture diffusion.

The model’s predictions aligned well with experimental results, particularly at low moisture content, while Siau’s model overestimated diffusion at high moisture content. Sensitivity analysis revealed the following:

• Positive correlation between diffusivity and porosity (R² = 0.89–0.97).

• Negative correlation between diffusivity and pore fractal dimension (R² = 0.85–0.95), tortuosity (R² = 0.78–0.96), and pore size ratio (R² = 0.30–0.86).

Overall, the fractal model effectively captures moisture transport by considering structural irregularities, offering a promising approach for optimizing wood drying processes.

The datasets used and analyzed in this study are available from the corresponding author upon reasonable request.

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the University of British Columbia (UBC)-4YF Fellowship. The authors thank Professor Jingbo Shi (Nanjing Forestry University) for providing the MIP equipment and the UBC Forestry Research Forest for supplying wood samples.

This study was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the UBC Four Year Fellowship (4YF) for doctoral students.

Authors and Affiliations

1. Department of Wood Science and Engineering, College of Forestry, Oregon State University, 119 Richardson Hall, Corvallis, OR, 97331, USA

   Dessie T. Tibebu

2. Department of Wood Science, Faculty of Forestry, The University of British Columbia, 2424 Main Mall, Vancouver, BC, V6T 1Z4, Canada

   Stavros Avramidis

Contributions

Tibebu Dessie conducted the experiments, prepared wood specimens, calculated diffusion coefficients, developed the model, and authored the manuscript. Avramidis Stavros supported the experiments, provided feedback on the manuscript, and approved the final draft.

Corresponding author

Correspondence to Dessie T. Tibebu.

Competing interests

The authors declare no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions