Light Scattering 2—Light Scattering in Crowded Systems

Abstract

Due to scattering volume overlap (dependent light scattering), the efficiency of light scattering by particles embedded in crowded systems, such as paints and paper laminates, is affected by the distances between these particles and their nearest counterparts. This, in turn, is affected by particle concentration. This effect has far-reaching consequences on the opacity of systems crowded with these particles. In this chapter, we discuss different aspects of particle concentration and describe the factors that affect light scattering efficiency in crowded systems such as paints and how these factors can be controlled by the paint formulator. We will also briefly discuss the implications of dependent light scattering on paper laminate opacity.

Appendix

In this appendix, we will show the process for modeling the strength of TiO₂ scattering in paint films that contain large extender particles based on the scattering that we see for the same TiO₂ particles in an unextended paint. This allows us to predict the expected opacities of extended paints based on the measured opacities of unextended paints.

Basis for the model:

The opacity of a paint containing only TiO₂ and resin (which we will refer to as an “unextended paint”) is invariably greater than that of a similar paint for which a portion of resin has been replaced by large extender (which we will refer to as an “extended paint”). The opacity penalty for the extended paint is due to the loss of scattering efficiency per TiO₂ particle because the extender particles crowd the TiO₂ particles together, as discussed in the text.

It is useful to have a means of describing the degree of TiO₂ crowding in a paint. An obvious way would be by the TiO₂ concentration—the higher the TiO₂ concentration, the greater the crowding. We can therefore describe the degree of TiO₂ crowding in an extended paint with a certain PVC value based on the PVC value that would give the same degree of crowding in an unextended paint. As discussed in the text, the effective PVC describes this degree of crowding. For example, a paint with an actual TiO₂ PVC of 20 and large extender PVC of 40 would have an effective TiO₂ PVC of 33.3, as calculated by Eq. 4.9 in the text. This means that the scattering efficiency of the TiO₂ particles in the extended paint (at 20 PVC TiO₂) is the same as the scattering efficiency that the TiO₂ particles would have in an unextended paint at 33.3 PVC TiO₂. Note that the opacity of the extended paint at 20 PVC TiO₂ will not be the same as the opacity of the unextended paint at 33.3 PVC TiO₂ (i.e., at the effective PVC) because, although the TiO₂ particles experience the same amount of crowding, there are more of them present in the unextended paint (since it is at a higher PVC).

In our model, we will develop a way to modify the opacities of the unextended paints by decreasing the scattering strength of each TiO₂ particle to reflect the opacity penalty of extender crowding in the extended paints.

Developing the model:

To demonstrate how this model can be developed, we will use the opacities of an unextended paint system to estimate the opacities of a paint system with 20 PVC large extender.

The model is based on experimentally determined opacity data for the unextended paint system. The first step in this process is to measure the spread rate (or another opacity indicator) of a series of unextended paints that span a range of TiO₂ values. Data for the paint system with the universal grade of TiO₂ that are shown in this chapter are given in the second column of Table 4.2 and plotted in Fig. 4.28.

Table 4.2 Scattering parameters for unextended paints below the CPVC

Fig. 4.28

Measured spread rate values for unextended paint system

Next, a measure of the scattering strengths per individual TiO₂ particle in these paints is calculated by dividing the spread rate (a measure of total scattering) by the TiO₂ PVC (a measure of number of TiO₂ particles). This is shown for the unextended paint in the third column in Table 4.2 and as the blue line and symbols in Fig. 4.29.

Fig. 4.29

Scattering per unit volume for the unextended paint series (blue) and calculated values for the 20 PVC extended paint series (red)

The best fit equation for this particular set of data is

$${\text{Scattering per unit concentration }}\left( {\text{unextended paint}} \right) \, = \, - 0.00{63} \cdot {\text{PVC }} + \, 0.{3634}$$

We next calculate the expected line for the paint series with 20 PVC large particle extender. To do this we will define a factor F that is the ratio of the reciprocal of one minus the extender PVC when expressed as a fraction. In this case, F equals 1.25 (=$\frac{1}{1 - 0.2}$). Multiplying the actual TiO₂ PVC of the extended paints by factor F gives the effective TiO₂ PVC:

$${\text{Effective PVC }} = {\text{F }} \cdot {\text{ PVC}}$$

Because the scattering per unit concentration is linear with PVC, we can calculate the scattering per particle for the extended paint series as simply the scattering per particle at the effective TiO₂:

$$\begin{gathered} {\text{Scattering per unit concentration }}\left( {\text{extended paint}} \right) = - 0.00{63} \cdot ({\text{effectivePVC}}) \, + \, 0.{3634} \\ = - 0.00{63} \cdot \, \left( {{\text{F}} \cdot {\text{PVC}}} \right) \, + \, 0.{3634} \\ = - 0.00{79} \cdot {\text{PVC}} + 0.{3634} \\ \end{gathered}$$

This equation is shown as the red line in Fig. 4.29. We can use this line to predict the opacity values for the extended paints as a function of PVC by multiplying the scattering per unit concentration for the TiO₂ particles in the extended paints, as expressed in the equation above, by the actual TiO₂ PVC (that is, reversing the procedure used to develop Fig. 4.29 from Fig. 4.28). The results of this calculation are shown in Fig. 4.30.

Fig. 4.30

Modeled opacity for extended paint system based on the opacity of the unextended paint system

This model is useful for calculating the expected opacity of paints made with extenders based on the opacity of paints made without extenders. In some situations, we may wish to do the reverse, that is, we may have measured the opacity versus TiO₂ PVC curve for a paint system that has extender in it and wish to calculate the opacity that the paints would have without the extender. One way of doing this is the reverse of the procedure discussed above, that is, to calculate the best fit line for the scattering per unit concentration for the extended paints, then adjust the slope of the line based on the effective TiO₂ PVC.

There is a second way to accomplish this task. To demonstrate this, we begin by plotting one data point from the opacity versus PVC curve for the extended paint that has 20 PVC large extender and 25 PVC TiO₂. This is shown as a red triangle in Fig. 4.31. We can extend a line from the origin, through this data point, and to the TiO₂ PVC of 31.25 (blue circle in Fig. 4.31). This PVC is the effective PVC value of the paint and was arrived at by multiplying the actual PVC (PVC = 25) by factor F (1.25 in this case).

Fig. 4.31

Spread rate data points. Red triangle—extended paint at an actual TiO₂ PVC of 25. Blue circle—opacity of this paint at the effective TiO₂ PVC

As explained in the text, all data points on the line going through the origin have the same TiO₂ scattering strength. The blue circle, therefore, indicates the same TiO₂ scattering strength as the unextended paint would have at a PVC of 31.25, and so this point will be a point on the curve for the unextended paint. For reference, this circle is indicated as a green star in Fig. 4.30.

We can repeat this process for any paint in the extended paint series to recreate the blue line in Fig. 4.30. However, in this case, the X-axis will be the effective PVC of the extended paint, rather than its actual PVC (note that the effective PVC values for the unextended paints equal their actual PVC). This is how Fig. 4.13b was created.