Effect of RH on solute diffusivity could not be directly observed. Therefore, the relation between RH and amount of solute diffusing into cell walls, which could be estimated using dimension of sample, was qualitatively predicted by the model of substance migration (Fig. 2) considering the solute diffusivity. The process of the prediction is shown as follows.

### Solvent evaporation

Solvent evaporated from the samples can be separated into those from the cell cavities and from the cell walls. The amount of solvent evaporated from the cell cavities and cell walls, −Δ*n*
_{C0} and −Δ*n*
_{W0}, respectively, can be formulated as [17]:

$$ - \Delta n_{{{\text{C}}0}} = K(P_{\text{S}} - P_{\text{A}} - P_{\text{C}} )\Delta t, $$

(1)

$$ - \Delta n_{{{\text{W}}0}} = K(P_{\text{S}} - P_{\text{A}} - P_{\text{W}} )\Delta t, $$

(2)

where *K* represents surface evaporation coefficient (>0), *P*
_{S} is saturated vapor pressure of the solvent, *P*
_{A} is vapor pressure of the solvent in the atmosphere, *P*
_{C} and *P*
_{W} are evaporation resistance pressure on the boundary between the atmosphere and the wood elements (cell cavity and cell wall, respectively).

The relation *P*
_{S} − *P*
_{A} = *P*
_{W} = *P*
_{C} exists immediately before the conditioning. In the unsteady state during conditioning, the inequality of *P*
_{S} − *P*
_{A} > *P*
_{W} > *P*
_{C} > 0 exists, because vapor pressure decreases to a level corresponding to RH, and water retentivity of the cell wall is higher than that of the cell cavity due to higher energy to remove bound water from the cell wall [18]. From this inequality, and Eqs. (1) and (2), the following expression can be deduced:

$$ {-}\Delta n_{{{\text{C}}0}} > {-}\Delta n_{{{\text{W}}0}}. $$

(3)

This inequality indicates that more solvent evaporates from the cell cavities than from the cell walls.

### Causal relations among solute diffusion, solute concentration difference, and solvent evaporation

The amount of solute diffusing into cell walls during infinitesimal time Δ*t* can be expressed as:

$$ \Delta n_{{{\text{W}}1}} = K_{1} (x_{{{\text{C}}1}} - x_{{{\text{W}}1}} )\Delta t, $$

(4)

where *K*
_{1} is solute diffusivity (>0), and *x*
_{C1} and *x*
_{W1} represent molar fractions of solute in the cell cavity and cell wall, respectively.

The solute concentration difference between the cell walls and cell cavities changes as solvent evaporates from the wood. Immediately before conditioning, the difference is absent producing the following equation:

$$ x_{\text{C1}} - x_{\text{W1}} = \frac{{n_{{{\text{C}}1}} }}{{n_{{{\text{C}}0}} + n_{{{\text{C}}1}} }} - \frac{{n_{{{\text{W}}1}} }}{{n_{{{\text{W}}0}} + n_{{{\text{W}}1}} + n_{\text{W}} }} = 0, $$

(5)

where *n*
_{C0} and *n*
_{W0} represent moles of solvent in cell cavity and cell wall, respectively, *n*
_{C1} and *n*
_{W1} are moles of solute in the cell cavity and cell wall, respectively, *n*
_{W} is number in moles of sorption sites of solution in the cell wall. The concentration difference changes with time (Δ*t*) after the start of conditioning [Eq. (6)]; the inequality relation of Eq. (6) was deduced from Eq. (3).

$$ x_{\text{C1}} - x_{\text{W1}} = \frac{{n_{{{\text{C}}1}} }}{{n_{{{\text{C}}0}} + n_{{{\text{C}}1}} + \Delta n_{{{\text{C}}0}} }} - \frac{{n_{{{\text{W}}1}} }}{{n_{{{\text{W}}0}} + n_{{{\text{W}}1}} + n_{\text{W}} + \Delta n_{{{\text{W}}0}} }} > 0. $$

(6)

The inequality indicates that a solute concentration in the cell cavities greater than that in the cell walls is caused by solvent evaporation. The inequality Δ*n*
_{W1} > 0 is deduced from Eqs. (4) and (6), which indicates that the solute diffuses from the cell cavities to the cell walls due to the concentration difference.

### Relation between RH and total amount of solute diffusing during conditioning

Let us consider the case where the solution-impregnated samples are conditioned at several RH values.

The solute diffusivity, *K*
_{1}, was assumed in this paper to increase with the amount of solvent in the sample. If it was also assumed that the amount of solvent (water) in the sample increases with RH during conditioning (Fig. 3b), the value of *K*
_{1} also increases with RH (gray dotted line in Fig. 3a).

The solute concentration difference, *x*
_{C1} − *x*
_{W1}, increases as RH or *P*
_{A} (Fig. 2) decreases (gray broken line in Fig. 3a), because Δ*n*
_{C0} and Δ*n*
_{W0}, which have negative values, decrease with *P*
_{A} [Eqs. (1) and (2)], and because the value of *x*
_{C1} − *x*
_{W1} increases as Δ*n*
_{C0} and Δ*n*
_{W0} decreases [Eq. (6)]. This tendency was also supported by our previous study [16].

The concave-downward curve showing the relation of the amount of solute diffusing during Δ*t* to RH (gray solid line in Fig. 3a) can be deduced by Eq. (4), and the relations of *K*
_{1} and *x*
_{C1} − *x*
_{W1} to RH (see “Appendix” section). The total amount of solute diffusing during conditioning was the time integration of the amount of solute diffusing during Δ*t*. Thus, the relation of the total amount to RH (black line in Fig. 3a) was approximately the same shape as the relation during Δ*t*.