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Note: The volume frequency is much more sensitive to larger droplets than is the number frequency.

Note: The volume frequency is much more sensitive to larger droplets than is the number frequency.

tional to d3, and so a volume distribution is skewed more toward the larger droplets, whereas a number distribution is skewed more toward the smaller droplets.

A particle size distribution can also be represented as a smooth curve, such as the distribution function, F(d), or the cumulative function, C(d) (Figure 1.5). The (number) distribution function is constructed so that the area under the curve between two droplet sizes (d, and dj + 5d) is equal to the number of droplets (n) in that size range (i.e., nt = F(d)Sd) (Hunter 1986). This relationship can be used to convert a histogram to a distribution function or vice versa. The cumulative function represents the percentage of droplets that are smaller than d (Figure 1.5). The resulting curve has an S-shape which varies from 0 to 100% as the particle size increases. The particle size at which half the droplets are smaller and the other half are larger is known as the median droplet diameter (dm).

1.3.2.2. Mean and Standard Deviation

It is often convenient to represent the size of the droplets in a polydisperse emulsion by one or two numbers, rather than stipulating the full particle size distribution (Hunter 1986). The most useful numbers are the mean diameter (d), which is a measure of the central tendency of the distribution, and the standard deviation (a), which is a measure of the width of the distribution:

The above mean is also referred to as the mean length diameter (dL) because it represents the sum of the length of the droplets divided by the total number of droplets. If all the droplets in a polydisperse emulsion were laid end to end, they would have the same overall length as those in a monodisperse emulsion containing an equal number of droplets of diameter dL. It

FIGURE 1.5 The particle size distribution of an emulsion can be represented by a histogram, a distribution function F(d), or a cumulative function C(d).

is also possible to express the mean droplet size in a number of other ways (Table 1.2). Each of these mean sizes has dimensions of length (meters), but stresses a different physical aspect of the distribution (e.g., the average length, surface area, or volume). For example, the volume-surface mean diameter is related to the surface area of droplets exposed to the continuous phase per unit volume of emulsion hs

This relationship is particularly useful for calculating the total surface area of droplets in an emulsion from a knowledge of the mean diameter of the droplets and the dispersed-phase volume fraction. An appreciation of the various types of mean droplet diameter is also important because different experimental techniques used to measure droplet sizes are sensitive to different mean values (Orr 1988). For example, analysis of polydisperse emulsions using osmotic pressure measurements gives information about their mean length diameter, whereas light-scattering and sedimentation measurements give information about their mean surface diameter. Consequently, it is always important to be clear about which

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