Oxygen index methods, which describes the tendency of a material to sustain a flame, are widely used as a tool to investigate the flammability of polymers. They provide a convenient, reproducible, means of determining a numerical measure of flammability. A further attraction is that the test method uses inexpensive equipment and only requires a small sample size. These methods have been used to systematical investigate the relative flammabilities of fire-retarded materials, frequently comparing the effectiveness of fire-retardants and fire-retardancy mechanisms.
The quintessential feature of oxygen-index methods is that the sample is burnt within a controlled atmosphere. The standard procedure is to ignite the top of the sample, using a gas flame which is withdrawn once ignition has occurred, and to find the lowest oxygen concentration in an upward flowing mixture of nitrogen and oxygen which just supports sustained burning. The criticality criterion typically takes the form of a minimum burning length: either specifying that the sample must burn for a certain length of time or that a specified length of material be consumed. The effectiveness of fire retardants is measured by the change in the critical oxygen concentration that they induce as a function of their concentration.
The limiting oxygen index (LOI), also called the critical oxygen index (COI) or oxygen index (OI), is defined as:
LOI = [O2,cr] Equation (1), ------------- [O2,cr] + [N2]where [O2,cr] and [N2] are the minimum oxygen concentration in the inflow gases required to pass the ``minimum burning length'' criterion and the nitrogen concentration in the inflow gases respectively. If the inflow gases are maintained at constant pressure then the denominator of equation~(1) is constant since any reduction in the partial pressure (concentration) of oxygen is balanced by a corresponding increase in the partial pressure (concentration) of nitrogen. Limiting oxygen index is more commonly reported as a percentage rather than fraction.
Since air comprises about 20.95% oxygen by volume, any material with a limiting oxygen index less than this will burn easily in air. Conversely, the burning behaviour and tendency to propagate flame for a polymer with a limiting oxygen index greater than 20.95 will be reduced or even zero after removal of the igniting source. Self-sustaining combustion is not possible if LOI>100, such values are not physically meaningful.
In (Nelson 2001) we investigated how the introduction of a fire-retardant changes the oxygen index of a material. For this purpose it is useful to assign materials into experimentally meaningful groupings depending upon their oxygen index. The minimum level of retardancy required to increase the classification of a material can then be calculated. From the preceding paragraph two obvious groupings are LOI<20.95$ and LOI>100. We refer to materials satisfying these requirements as being ``flammable'' and ``intrinsically non-flammable'' respectively. Several researchers have suggested that materials with a a limiting oxygen index greater than 28 are generally self-extinguishing (Horrocks et al 1989). We describe materials satisfying 28.00 < LOI < 100 as being ``self-extinguishing''. The threshold LOI=20.95 is of great practical interest and we define materials with a limiting oxygen index of 20.95 as being ``marginally stable''. We follow Fenimore (1975) and refer to materials that are between the marginally stable and self-extinguishing thresholds, i.e. 20.95< LOI< 28, as being ``slow-burning''.
Marginally-stable materials form a natural set for a quantification of the efficiency of fire-retardant mechanisms. We achieve this by finding the value of the relevant continuation parameter to increase the LOI of these materials to 28.0, the transition between slow-burning and self-extinguishing polymers, and to 100, the threshold for intrinsically non-flammable materials.
It should be realised that our classification of materials (flammable, slow-burning, self-extinguishing, intrinsically non-flammable) is specific to the limiting oxygen index test, i.e. a material that is self-extinguishing here is not necessarily self-extinguishing in another test method. The tenet in the limiting oxygen index is that the higher the value of the LOI the `safer' the material. However, we stress that results from one test method do not necessarily agree with another (Emmons 1974). The reasons for this were alluded to in the opening paragraph. Thus throughout this paper an assignment of a material as being self-extinguishing is short-hand for ``self-extinguishing in the limited oxygen index test''.
Additional details of oxygen-index methods and their applications, particularly to assessing the burning behaviour of textiles, are provided in the comprehensive review by Horrocks et al (1989).
A complete description of the mechanisms leading to the establishment of a flame over a burning surface requires consideration of mass and heat transport in both the gas and solid phases. Although the overall phenomena are complicated, two salient processes, one in each phase, must occur if a material is to ignite. The solid must first decompose to release volatiles into the boundary layer. These gases must then mix with surrounding air to produce a flammable mixture, which then either autoignites or is ignited by an external source, such as a pilot flame. Traditionally fire scientists have used highly simplified models which, typically, examine these key processes in isolation. Recently non-linear dynamical systems models have been developed describing these processes (Rychlý and Rychlá 1986; Búcsi and Rychlý 1992; Rychlý and Costa 1995; Rychlý and Rychlá 1996; Nelson 1998)
Rychlý and co-workers have developed a two-phase dynamical systems model describing the transient burning behaviour of polymers in the limiting oxygen index test and the cone calorimeter (Rychlý and Rychlá 1986; Búcsi and Rychlý 1992; Rychlý and Costa 1995; Rychlý and Rychlá 1996). This model has been used to investigate the action of certain types of fire retardants (Rychlý and Rychlá 1986; Rychlý and Rychlá 1996) and it has been established that there is a good coincidence between calculated and experimental values (Rychlý and Costa 1995). It has been validated as a suitability tool to investigate polymer ignitability and burning, capturing the essence of the two test methods.
Nelson et al introduced a revision of the Rychlý limiting oxygen index model. The essential features of the model were retained, some inconsistencies in the modelling of certain physical and chemical processes being eliminated. It was shown that a limiting oxygen index can be defined in a steady-state formulation as an extinction limit point. In (Nelson 2001) the revised Rychlý model was extended to consider two solid-phase fire-retardant mechanisms: non-competitive char formation and dilution by addition of an inert filler. We investigated how effective these mechanisms are at increasing the oxygen-index, paying particular attention to the retardation of marginally-stable materials.