#### 1. What is the FSP?

When a small amount of liquid water is poured onto oven dried timber, a new equilibrium will eventually be established. Observations show that when equilibrium is attained, the added water is not visible in the voids. Instead the water must, by default, be located inside the cell wall matrix. Hence, the mass of the cell walls increases, as does the volume. This is observed macroscopically as swelling of the timber sample. Concurrently, the strength of the timber progressively declines as the moisture content increases. As water continues to be added, and the moisture content increases further, the system eventually reaches an equilibrium state where liquid water begins to accumulate in the voids. When that occurs, the cell walls cease swelling and the strength of the timber no longer changes with the moisture content. The moisture content (defined by wood scientists as the ratio of liquid mass to dry mass) at which this occurs is called the FSP. According to Skaar (1988), p. 35, the FSP was originally described by the American forester, H. D. Tiemann, as ‘the moisture content at which the cell cavities contained no water, but the cell walls were fully saturated with liquid moisture.’ Precise definition of the FSP has proved elusive possibly because different measurement techniques can give slightly different estimates of the FSP (Stamm, 1971). Despite that difficulty, the basic ideas underlying the FSP have been repeatedly confirmed over the last 100 years.

#### 2. The FSP as a chemical phenomenon

What is the nature of the forces that hold the water preferentially in the cell walls? Obvious candidates are chemical bonding and/or capillary (i.e. surface) forces. This question can be addressed by first determining the FSP of intact timber samples. After that, the samples are pulverised to create a larger surface area (but the total mass is kept constant), and the FSP determined again. The results from those experiments show that the FSP remains more or less the same, regardless of whether the timber sample is intact or pulverised (Stamm, 1964). This implies that there is a distinct number of binding sites for water in the timber, and that the number of binding sites is (more or less) independent of the macroscopic surface area of the sample. On those observations, the forces are interpreted as being primarily chemical forces. Further support for that conclusion comes from the fact that heat (known as the heat of wetting) evolves when water is added to dry wood, which also implies the existence of chemical reactions (Stamm, 1964). In essence, the phenomenon can be thought of as one of hydration. (Later, we show that this conclusion is consistent with a whole range of thermodynamic and chemical evidence.)

This does not mean that engineers and material scientists do not recognise capillary forces. Rather, they recognise that for the typical dimensions of conduits in timber (of the order of ≈1–100 µm), the reduction in vapour pressure estimated using the Kelvin capillary equation is very modest (relative humidity ≈99%) (Browning, 1963; Stamm, 1964). Cell walls are assumed to be porous at atomic scales, but at these scales it is assumed that water makes its own space by virtue of the large chemical forces involved (Stamm & Hansen, 1937; Stamm, 1964). Consequently, wood scientists have not applied the Kelvin capillary equation for spaces smaller than about 5 nm (e.g. Browning, 1963). That view is consistent with recent molecular dynamics simulations of water in small pores showing a ‘switching’ behaviour between liquid and vapour phases (Beckstein & Sansom, 2003).

In summary, in the wood and fibre science literature, water is considered to exist in timber in three states: (1) as vapour in the gas-filled voids; (2) as ‘free’ or ‘bulk’ liquid water in the voids; and (3) as ‘bound’ water in the cell wall matrix (Stamm, 1967a). At the FSP, all of the liquid water is bound water, as this represents the maximum amount of water that can be taken up from the vapour phase by a unit mass of timber at a given temperature (Browning, 1963). (More details of the temperature dependence are given in Section III.2.) The integrated mixture of cell wall material and ‘bound’ water is recognised as a distinct phase and has been called a ‘solid solution’ by Stamm (1964). An alternative and perhaps more informative term for the ‘solid solution’ is a ‘gel’ (Skaar, 1988).

#### 3. Estimating the bulk phases in wood

There are important practical advantages of separating the water in timber into the three above-mentioned phases. These arise from the observations that (1) the FSP is an estimate of the maximum amount of water that can be ‘bound’ to the cell wall matrix, and that (2) the density of dry cell wall material is very conservative at ≈1.5 g cm^{−3} (Stamm, 1964). Hence, given an estimate of the fresh volume, fresh mass, dry mass and FSP of a piece of timber, it is straightforward to estimate the volume fractions of the three bulk phases (gas, liquid, solid) in a timber sample, and this is done routinely by engineers (e.g. Rijsdijk & Laming, 1994) and by material scientists. In routine applications, the default value of the FSP is often assumed to be 30% (i.e. the ratio of liquid mass to dry mass is 0.30). At the outset, we emphasise that a universal FSP at 30% is, at best, an approximation, and in later sections we examine this approximation in greater detail. Before that, we show how this framework is routinely used to estimate the volume occupied by the three bulk phases in timber.

The calculations used to estimate the three phases in timber have remained more or less the same over the years, but widely different terminology has been used (e.g. Stamm, 1964; Siau, 1984; Skaar, 1988). The a-u-s-*V* scheme (Roderick *et al*., 1999a; Roderick & Berry, 2001), as summarised below, uses exactly the same concept, and calculations, but presents a simplified terminology for use in estimating the three bulk phases. The details follow.

The total volume of a timber sample (*V*, m^{3}) is the sum of the volumes of the three phases:

- (Eqn 1 )

where *V*_{a} (subscript ‘a’ for air, the ‘gas’ phase), *V*_{u} (subscript ‘u’ for solution, the ‘liquid’ phase) and *V*_{s} (subscript ‘s’ for structure, the ‘solid solution’ phase, or, more simply, the ‘solid’ phase) are the three phases. The mass (*m*, kg) of the sample is the sum of the dry mass (*m*_{d}) and liquid mass (*m*_{q}):

- (Eqn 2 )

where the mass of gas is ignored. The FSP is used to allocate the water between the solid and liquid phases. To do that for timber, it is usually assumed that all of the dry mass is in cell walls, and hence is part of what we have called the solid phase. (This could be modified for use in living trees by accounting for the small amount of dry matter in the vacuoles and cytoplasm of cells, i.e. the liquid phase.) The FSP (α_{f}) is defined as:

- (Eqn 3 )

where *m*_{q,x} is the maximum possible mass of liquid that can be located in the ‘solid’ phase of timber. The mass of liquid in the ‘solid’ phase (*m*_{q,s}) is estimated as:

- (Eqn 4 )

and the mass of the liquid phase (*m*_{u}) is then:

- (Eqn 5 )

The volume of the liquid phase is given by:

- (Eqn 6 )

where ρ (kg m^{−3}) is the density. Similarly, the mass and volume of the ‘solid’ phase (*m*_{s}, *V*_{s}) are:

- (Eqn 7 )

- (Eqn 8a )

In routine applications the density of the ‘solid’ phase (ρ_{s}) is unknown and is usually estimated using the linear mixture approximation:

- (Eqn 8b )

where the density of the dry matter (ρ_{d}) is assumed constant at ≈1.5 g cm^{−3}, and the density of water bound to the cell wall material is assumed to be equal to the density of pure liquid water (ρ_{w}, 1 g cm^{−3}). From a mechanical and thermodynamic perspective, the linear mixture assumption is wrong and leads to a flawed model and hence flawed understanding, because one should use the relevant partial specific (or molar) volumes for the calculations, and because the partial specific volume of water varies with the moisture content of the timber (see Section III.3). However, when used to estimate the volume fractions of the bulk phases, the approximation would lead to errors of perhaps a few per cent at most (see Section III.3). This is adequate for many practical purposes. Finally, the volume of the gas phase is then estimated by difference (Eqn 1). Although the mathematics appears tortuous, in practice the calculations are very simple. See Fig. 1 for an example.

In Fig. 1, the FSP was assumed to be 0.30, and this is a good starting approximation for most timber and other cellulose-based materials (Stamm, 1964). However, wood scientists have found that the FSP does vary. Typically, the FSP decreases in denser wood (Skaar, 1988). The relation between wood density and the FSP is convenient because in practical forestry applications, the most commonly used intensive variable is the mass concentration of dry matter ([*D*], kg m^{−3}):

- (Eqn 9 )

Foresters (and wood scientists) call [*D*] the basic density. The underlying significance of [*D*] for foresters is that many of the mechanical properties and commercial attributes of timber vary directly with [*D*] (Desch, 1973; Zobel & van Buijtenen, 1989). This is easy to understand, because if the density of the dry cell wall material is approximately constant (≈1.5 g cm^{−3}), then it follows that, as [*D*] increases, the volume fraction of the ‘solid’ phase increases and the volume fraction of the remaining phases (i.e. gas and liquid) must therefore decline. It is also easy to show that [*D*] should be significant from a biological perspective, because if a plant has a low value of [*D*], a given increment of dry mass will produce a larger change in volume than for a plant with a high value of [*D*]. Hence, plants with a low value of [*D*] should, all else being equal, increase their volume faster than those with a high value of [*D*] (Roderick, 2000).

Using the above knowledge, it is straightforward to develop a simple overall understanding of how the volume fractions of the phases, and the amount of bound water, varies in timber with different [*D*] (Fig. 2). It is interesting that engineers and biologists might view Fig. 2 with possibly different perspectives. The fundamental starting point for biologists is the liquid phase because that is the ultimate source of the biochemical activity in living plants. However, the fundamental starting point for engineers interested in the strength of wood is the ‘solid’ phase because that is the principal source of the mechanical strength. Ecologists are often simultaneously interested in both aspects because they both impose constraints on plant architecture and function (Niklas, 1992; Roderick & Berry, 2001).