The Unity Molecular Formula

One of the most useful and powerful tools for examining a glaze is the Unity Molecular Formula, developed in Germany by Herman A. Seger (1839-94). Seger also developed numbered cones for measuring heat work in kilns, among other things. The unity formula shows the relationship between the number of molecules in the three categories: Fluxes, Stabilizers, and Glass Formers, aka RO/R2O, R2O3, RO2.  From this formula, one may deduce the surface characteristics, melting range, and color response of a glaze. It is easy to adjust a glaze on the molecular level, fine-tuning a troublesome mix in rather short order, and new glazes may be engineered from scratch and converted to a batch of ingredients. Many glazes that seem to have quite different batch recipes often are found to be very similar once the unity formulae are calculated, and a recipe may be reconfigured to substitute for unavailable or discontinued ingredients. It is not difficult to convert a recipe of ingredients to a molecular formula and back again using a long division (or a calculator) and a list of glaze material formulae and weights. There are even a number of software packages for personal computers that will do the math and conversion as quickly as one can type, so using the unity formula is easier than ever.
The unity formula is typically set out in three columns from left to right: the RO/R2O group, the R2O3 group, and the RO2 group. Each constituent oxide in the glaze will have a number indicating the number or fraction of molecules of that oxide present. For example:
 

 This hypothetical glaze has 0.7 molecules of calcium, 0.2 molecules of potassium, and 0.1 molecules of sodium.  In the second and third columns, we see that there are 0.35 molecules of alumina and 3.5 molecules of silica. The unity formula is simply a convenient way of expressing the relative amounts of oxides in a glaze. Therefore, we are forced to accept the notion of fractions of molecules. An actual count of molecules would be a number with about 27 zeros on the end of it!  Note that the amounts in the first column add up to one (unity). It is standard to express glazes with the flux column set to unity. Clays and slips (and sometimes single feldspars) are shown with unity set in the R2O3 group, as there is considerably more alumina than in a glaze.  In this hypothetical glaze, for each single molecule of flux material there is 0.35 molecule of stabilizer and 3.5 molecules of glass former.

 Let's put this idea into practice and see if we can make some more sense of it.

 Calculating the Unity Formula of a Glaze.

Given a glaze recipe and a list of chemical formulae (see table 3-1, coming soon), it is a simple matter to convert a glaze to its unity formula. For example, let’s convert this popular glaze:

40 Custer Feldspar

30 Flint
20 Whiting
10 Kaolin
100 Total

Step 1

Get the molecular formula and molecular weight of each material. See Table 3-1.

Custer Spar  0.68 K2O•  1.05 Al2O3• 7.11 SiO2  molecular weight: 619
                    0.29 Na2O
                    0.03 CaO 
 
Flint           SiO2                         weight  60
Whiting      CaCO3                      weight 100
Kaolin        Al2O3•2SiO2•2H2O  weight 258
 
Step 2
Divide each batch amount by its material’s molecular weight, giving the number (or fraction) of molecules contributed by each ingredient. This amount is known as the molecular equivalent for that material. Carry work to three places.

Custer Feldspar: 40/619 = 0.065

Flint:      30/60  = 0.5
Whiting: 20/100 = 0.2
Kaolin:   10/258 = 0.039

To understand this, think of walnuts and peanuts. A pound of walnuts may only be sixteen nuts (molecules) while a pound of lightweight peanuts represents quite a large quantity of nuts. Similarly, a quantity of a molecularly heavy material will contribute fewer molecules than the same quantity of a molecularly light material. This is an important step in seeing beyond the recipe. Note that the flint and whiting, while representing smaller percentages in the batch,  each contribute more molecules (05. and 0.2, respectively) than the heavy feldspar. 

Step 3

Multiply the number of molecules contributed by each ingredient from Step 2 times the amount of each oxide in its molecular formula from Step 1 to get the amount of each oxide contributed by each ingredient. Total the amounts for each oxide.

                    K2O     Na2O     CaO     Al2O3        SiO2

Ingredient 
Custer        0.044    0.019      0.002      0.068      0.462
Flint                                                                     0.5
Whiting                                  0.20
Kaolin                                                  0.039      0.078

Totals        0.044   0.019       0.202      0.107      1.04

Step 4

Total the amounts in the RO/RO2 group and divide this amount into the amount of each oxide. This will put the fluxes into unity.

So:
0.044 K2O + 0.019 Na2O + 0.202 CaO = 0.265 total fluxes.

RO/R2O                                R2O3                                              RO2

0.166 K2O   (=.044/.265)     0.403 Al2O3 (=.107/.265)    3.92 SiO2 (=1.04/.265)
0.072 Na2O (=.019/.265)
0.762 CaO   (=.202/.265)                             

What may we deduce about this glaze by examining its unity formula? Starting with the fluxes, we see that the glaze has three flux oxides: potassium, sodium, and calcium. Calcium is the major flux, making up just over 75% of the fluxes (not of the entire glaze!). The potassium and sodium are clearly the secondary melters, with about sixteen and seven percent, respectively. For each molecule of flux in the glaze, there is just under a half molecule (0.403) of alumina and nearly four (3.92) molecules of silica. To learn more, we will need a few more tools. 

Silica to Alumina Ratios
To learn about the fired surface of our glaze, we must look at the ratio of silica to alumina. To do this, divide the amount of silica by the amount of alumina.

So:
3.92 Silica / 0.403 Alumina = 9.73
The ratio of silica to alumina is then expressed as 9.73:1.

Generally, in glazes that do not contain boron, a ratio higher than 8:1 indicates a glossy glaze; lower than 5:1 indicates a dry matte glaze. The satin glazes tend to lie in the range of around 6 or 7:1.
This glaze, with a ratio of over 9:1 should have a shiny, glassy surface.

Glaze Limit Formulas
Glaze limit formulas are guidelines for amounts of different oxides that will tend to produce a “good glass”  at various temperature ranges. (The term "good glass" is from early literature written for ceramic engineers.) While not hard and fast boundaries, they are good indicators of over- and under-supply of different oxides in a glaze, and serve as useful guides in developing new glazes and in examining existing ones. Glazes that fall within limits are, by and large, well melted, stable, and predictable. As different oxides go outside the limit formula, the glaze may start to behave in unexpected (often interesting) ways. Keep in mind that there is not a single, authoritative limit formula; do not be alarmed to find different limits than those compiled here.
What may we expect from our sample glaze by comparing it to the limit formulas given? By quickly comparing the molecular amounts of our glaze with the different limits, we can quickly infer that this is a cone 8-10 glaze. It seems to fall right in the middle of nearly all the limits for glazes in this temperature range.
Let’s look more closely, starting with the fluxes. From our calculations, we see that the combined amount of potassium and sodium (often lumped together because of their similar characteristics) is 0.238 molecules. This seems to fall well within the limits suggested by the table. From our previous discussion, we know that sodium and potassium contribute brightness and smoothness to a glaze surface. In addition, they are both good auxiliary melters for calcium. Over supply of KNaO can cause crazing due to high thermal expansion, and can contribute to a soft, easily abraded glaze surface. As this glaze falls toward the middle of the limits, we can expect a reasonably bright surface and color response and some tendency towards crazing.
The calcium content of this glaze exceeds the recommended limits given in table 3-2. From our discussion of calcium as a flux, we can expect this glaze to have a hard, durable surface. It may be a bit of an abrupt melter, though, with little tolerance for under firing.  We might also expect that the over-supply of calcium will give this glaze a tendency to become matte if cooling times are long, or if there is insufficient silica present.
The alumina content of this glaze is comfortably in the middle of the limits given. We can expect a reasonably hard, stable, durable surface. A glaze with this much alumina will have little tendency to run in normal firings, unless applied very thickly or over-fired. Crawling will likely not be a problem with this glaze, either, unless applied too thickly, or to a dirty surface. This glaze is probably not a good candidate for development of vivid copper reds, which favor very low alumina amounts.
The silica, at 3.92 molecules, also falls right in the middle of its limits.  This amount of silica should give a glaze with average hardness and tensile strength, and it should form a decent glassy matrix in the firing. It should also be relatively resistant to attack by acidic foods, given the sufficient alumina and silica.
Remember that using glaze limits and unity formulas are tools that can help us as ceramic artists to examine, modify, troubleshoot, and develop glazes. They are in no way substitutes for knowledge of the oxides and their effects on glazes, and running lots of tests through the kiln.