Preston Equation
By Stephen W. Attaway, Ph.D.

How fast should you run your lap? How much pressure should you use when polishing? One person will recommend that you run the lap at fast speeds, sweeping the stone across the lap. Another will tell you to go very slow so that the stone stops on the lap. One person will advise you that the only way to get a good polish is to use a lot of pressure, while another will state that a light touch is needed.

Here, I will introduce the concept of rate of polish. In 1927, Preston introduced an empirical formula for determining how fast material is removed from a surface. The formula basically states that the rate of removal of material during grinding and polishing is proportional to the pressure times the velocity.

r = k P v

In the above equation, P is the pressure or force per unit area, v is the velocity of the lap relative to the stone, and r is the rate of material removed (cutting depth per unit time). The constant, k, is known as the Preston coefficient. This coefficient reflects the type of lap, the type of material, the type of grit, etc.
This simple relation means that the rate of surface removal (or weight loss) increases as the downward force on the stone increases or if the velocity of the lap increases. Researchers have found that for the most part, the rate of surface removal is independent of polishing particle size.

As you know, the size of a facet can greatly affect the cutting rate. According to the Preston equation, if we always run the lap at the same speed and use about the same pressure, then the area will control the rate of cutting. The area increases with the square of the stone dimension. This simple geometric relation means that a table of a 5mm. stone will cut four times faster than a table on a 10mm. stone. A table on 20mm. stone will take eight times longer to finish than one for a 5mm. stone!

One trick I have learned when cutting a large stone is not try and cut the whole table at one time. In the rough grinding stage, I will use the cheater or the mast height to rock the stone from side to side. That way, the contact area will be reduced and the grinding rate will increase.

When Nancy and I first began faceting, we often received conflicting advice on how fast or slow the lap should spin when polishing. Some like to go slow and bear down with all the weight they can muster. Others like to go fast with just a little weight. If the Preston equation holds, we might see that these have the same effect. (I am sure that there are some limits to the Preston equation. So, donít blame me when your stone chips or falls off because you were going too fast or pressing down too hard!)

There does seem to be one effect that Prestonís equation does not explain. For some stone and lap combinations, Nancy has found that the best polish is obtained by slowly sweeping the lap back and forth so that the stone comes to a stop at the end of each arc. Maybe this effect is caused by the fact that static friction is always higher than the dynamic friction. When the stone comes to a stop at the end of the arc, the stone grabs better than when it is sliding.

Another instance where the Preston equation could break down is when the lap is spinning very fast with ample water. Just as a tire can lose contact with the road through hydroplaning, a large stone can also hydroplane over the lap. Most faceters can feel when this happens. For the rough grinding phase, you can hear the loss of contact.

I had the opportunity to see how computer chip wafers are polished. During the process, the polishing rate is increased by using a foam-like polish pad. The small asperities in the foam pad locally reduce the contact area and increase the polishing rate. This effect is so important to the wafer polishing process that the surface of the foam pad is continuously roughened with a diamond scrubber. This implies that rough surfaces like the one found on mylar sheets could greatly enhance the polishing rate.

While it is hard to believe that such a simple relation as the Preston equation could describe the broad range of complex interactions that occur while polishing and grinding, most of the published data indicate that this relation is fairly accurate.