Alloy Density Calculator Generates Most Interest of All Blog Posts

Folks,

In the nearly 10 years that I have been blogging, I am continually surprised by the interest in a spreadsheet I created that calculates alloy densities.  I get about numerous inquiries a year on this topic.  We just renewed the link to the software, so I thought I would write a summary blog on its use and applicability.

First of all, the algorithm is intended for metals that form an alloy. Examples would be most solders and other metal systems where the metal atoms replace each other in the lattice. So, in addition to solders, copper and nickel would also work. The calculation assumes perfect mixing and that the sum of the initial volumes of the metals equals the total final volume. The correct formula for calculating densities is:

1/Da = x/D1 + y/D2 + z/D3

where

Da = density of final alloy

D1 = density of metal 1

x = mass fraction of metal 1

And the same for metals 2 and 3. 

This formula was derived in a past blog post. People are often surprised that the simple formula Da = xD1 + yD2 + zD3 is not correct.  The reason it is not correct is that density is inversely proportional to volume.  The error with this formula is discussed in another post.  The error, by using this formula, can be quite large. See the graph below for gold and copper. In some cases the error is more than 15%.

One example where the algorithm does not work would be for intermetallic compounds. The reason is that an intermetallic is a compound, not an alloy. Another example where the formula does not work is carbon in iron. The carbon atom is so small that it fits in between the iron atoms.

How accurate is the formula?  Work that I have performed with solder alloys suggests is it about 1-2% accurate. The accuracy can be affected by grain boundaries and the small amount of intermetallics that can form in some solder alloys.  An example is the small amount of  intermetallic “silver plate” (Ag3Sn) that can form in SAC alloys. I hope that many readers continue to find the density calculator useful.

Cheers,

Dr. Ron

Math musings. I read a fun book, The Joy of X. In the book, the author, Steven Strogatz, pointed out that the sum of consecutive odd numbers is always a perfect square. Try it: 1+3 = 4 = 2^2,  1+3+5 = 9 = 3^2 , 1+3+5+7 = 16 = 4^2, and so on.

Solder Alloy Density

Folks,

I have occasionally written on calculating solder alloy density, as there is surprisingly more interest than I thought there would be in this topic. Recently, it occurred to me that it might be beneficial to compare the calculated densities to actual densities of a few alloys to see how accurate the correct formula is (for the derivation of the correct formula see below). The formula assumes “perfect mixing” (i.e., no interactions between the alloy elements). The alloys we investigated were tin-bismuth-silver, tin-silver, tin and tin-bismuth.

To measure the density, I obtained a few alloys from Indium. My student, Evan Zeitchick, determined that a good technique to measure density is to machine the alloy into a rectangular parallelepiped (see photo), weigh it, and calculate its volume from its dimensions.  The results agree with the correct formula to about 1 to 2%. Some people would ask why there is any difference. The reason is that all alloys form different phases, and some form intermetallics. These phases and intermetallics would typically have different densities than that calculated for the alloy. I will have more detail on this work in a future post.

Here is a derivation of the correct density formula:

Many people incorrectly assume that if you have an alloy of x % tin by weight and y % silver, that the density of this alloy would be 0.x*Density tin +0.y*Density silver. This intuitive linear formula is incorrect however, as density has two units (mass and volume).

An easy way to understand the derivation of the correct formula (proposed by Indium  engineer Bob Jarrett) is to consider a 96% tin, 4 % silver example.

Let’s assume I have 1 g of this alloy, 0.96 g is tin and 0.04 g is silver.

The volume of the tin is 0.96 g/7.31g/cc = 0.131327cc

The volume of the silver is 0.04g/10.5g/cc = 0.00381cc

So 1 g of the alloy has a volume of 0.131327 + 0.00381 cc = 0.135137 cc

Hence it’s density is 1g/0.135137cc = 7.39989g/cc

Hence, the general formula is:

1/Da = x/D1 + y/D2 + z/D3

Da = density of final alloy

D1 = density of metal 1, x = mass fraction of metal 1

same for metals 2, 3

The formula continues for more than 3 metals.

I have developed an Excel spreadsheet that calculates density automatically. If anyone wants a copy, send me an email at rlasky@indium.com

Cheers,
Dr. Ron

P.S.: Interesting thought: About 165,000 tonnes of gold have been mined throughout history. If all of this gold was gathered into a cube it would only be about 21 meters on a side. At $1,550/oz, its value would be $8.5 trillion, quite a bit less than the almost $15 trillion debt of the US government. Yikes!

Formula Accuracy

Ken writes:

Dr. Ron, Thanks for your helpful post. I get close (-1.1%) with your formulas for an alloy I am working with. I think the crystal lattice packing factor for some of the individual elements is throwing off the result since it is different than the alloy. I tried to take this into account, but I get an error on the opposite side (+1.6%) of the actual. Any thoughts on if your formula can be made more accurate by taking element and alloy crystal lattice packing factors into account?

The solder alloy calculation assumes that the metals mix with no interaction, much as miscible liquids, of different densities would. There are numerous phenomena that could cause errors. They include:

1. Metals can come from different crystal systems. Lead, silver and copper are face-centered-cubic, whereas tin, the base metal for most solders, is of the tetragonal system.

2. Some metals form intermetallics with tin, such as copper and silver. These intermetallics have different densities than the metals or the resulting alloy.

3. Grain boundaries can leave some (probably small) empty space.

So I think Ken’s 1% accuracy is very good. The biggest mistake one can make however, is the most common: assuming that the density is simply given as the sum of the metal mass fractions times the metal densities. To many, it seems logical, but it is wrong.

My original posting on how to derive the formula for solder alloy density is here.

Cheers,

Dr. Ron