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A Chemical Engineer's perspective on a tempest in a tea pot re: COE and Compatibility in Fusing

Updated: Jan 28, 2022

Or How About Some Really Cool Stuff You Can Use From the Coefficient of Linear Expansion Curve (COE)?

One of the things that has mystified me, as I entered the Glass Art world in 2017ish, is the kerfuffle around COE used to indicate compatibility. You know the one. Someone on FB is shouting "COE is not the same as compatibility! something something linear expansion something something other phys props are important, something something". Generally they are shouting at a lot of newbies and non STEM artists and making folks feel bad, insecure and confused. And generally they are self admittedly "not good at math". Well I am good at practical application of math to build stuff (Raise your hand if you've designed a commercial size phosgene scrubber). So that is why I feel the need to weigh in.

My first thought is that this kerfuffle is an odd hill to die on, at least from my perspective which includes decades of chemical plant design, R&D, scale up, construction, statistical data analysis and so on.

If there were some kind of eruption of broken glass to back up the excitement I'd understand but there isn't. I was a member of several large FB fuser groups and the number one issue for breakage was ramping up or down too fast. But hey, let's scare and confuse the purchasing decision makers - the artists. That's bad marketing and unfortunately for me a potential price impact as there is now only one COE 90 dominant supplier. I really don't want to have picked wrong and have to migrate to COE 96.

So to these folks making the "COE is not compatibility" arguement, what's your point?

What exactly, from a practical perspective, are you trying to accomplish from this argument? Elegant useful simplifications are bad? I mean hey, the ASTM standard for generating the Coefficient of Linear Expansion (COE) is in its self a marvel of engineering simplification. It makes use of a simplification beloved to all engineers. The use of a straight line as a proxy for volumetric expansion. Did I mention that Engineers love the use of strategic, elegant simplifications?

Generations of fusers and manufacturers have generated mountains of data showing that COE is a reasonable short hand for compatible. The boundaries created by COE compatible fusing glass are hard to violate for the typical glass artist. If anything the medium is so forgiving no one advises holding different anneals by glass type anymore AND Bullseye now recommends 900F (!) I mean think about that in light of what I am going to show you below. I mean c'mon, we don't even have to worry about heat transfer, beyond the crudest controls!

I don't know the market demographics but I'd bet most artists are doing simple projects, follow the manufacturers' and Art guilds' firing schedules instructions and color well inside the lines. These folks are not getting in trouble and don't need technical information that's neither practical nor useful. As for the boundary pushers, did someone really have to tell those rarified artists pushing the boundaries to test stuff?

To the newbies and non-STEMers: if you want to use COE as a short hand for glass compatibility, I as an engineer I approve. It is close enough for engineering purposes, and I laud your directionally correct abstraction as backed by generations of glass artists' data.

K' now off the soap box and let's get into some useful technical stuff. Would you like to learn something useful about glass COE? Great! Let's start with the COE curve.

I get it, some folks are allergic to math. No worries. I do understand the math and can strategically simplify and get to what's useful. Math is just an tool, an abstraction to usefully model real phenomena. If you are unsure about wading in, jump to the end and look at the firing temperatures to see if it's worth your time to understand the COE graph. Let's not let the tools get in the way of creating art!

Behold, The COE Curve (see image below) where curve just means a line drawn through measured data points. If you google it, you can find many examples of this curve. They all look the same. There is an ASTM (American Society for Testing and Materials) test method folks follow to generate the curve for various glasses. Do you feel different?

So what does this mean and how is it useful?


Awesome, right? I'll add in the Fusing Temperatures for COE 90 after walking you through the curve.

First notice the shape. There are two linear sections (sections that can be described mathematically by straight lines) and two inflection points (bends) and these are very meaningful in our glass fusing process. More on that in a minute and I will tie in the regression graph on the blog for calculating temperature of bullseye glass for a particular viscosity from an earlier blog post.

Notice the Y axis of the graph (up and down direction) is a measure of the expansion or contraction of glass. The X axis (side to side) is in Temperature increasing as you move right and decreasing as you move left. Both Y and X scales are linear which may mean something to some of you and if not, don't worry. **

Remember what I said about strategic simplifications? Well this is a beautiful example. Glass is three dimensional. When it heats or cools it is moving volumetrically, but we can reduce and describe usefully the change in the glass by using a straight line (easy math YAY). In fact we can usefully represent many glass properties and how they behave with heating and cooling with this simple line through some data points.

So what is the famous Coefficient of linear expansion (COE) that we all hear about so much about? It is just the slope of the line in the first linear section of the curve. Y = mx +b. where m is slope (rise over run). Put another way for every degree of heat your kiln puts into the glass, it expands, up to about the strain point, in a constant way and COE is a measure of that.

Why do we care about this first linear section of the graph? Because it describes the motion of our glass during heating up to the strain point or cooling from the strain point to room temperature. In our kilns we sometimes need to think about this motion and how to counter it with respect to the support of our glass.

In a fusible line of "same" COE glass, that volume change is small enough and coordinated enough to not generate too much post anneal stress, no mater what colors you melt and stack. (ya ya the COEs are slightly different by composition: batch to batch, color by color, there are other physprops, re-firing breaks bonds and changes phys props, by firing etc. but those differences are close enough for stuff to work (!) and for that we are truly grateful ! Would anyone want to describe and model all of those differences just to reach the same point as can be reached with an elegant simplification? ).

So on just the face of it, look at what is happening as glass heats and cools through a cycle:

Cool, eh? Sure it's not exactly what we do in the kiln but it has some implications for what we do. Personally I think the balance of forces and then the contraction are really interesting. Expansion is compensated by viscous flow in that range. I work with sintered powder a lot and I find that some strategic holds in this range give me a better result.

What does the fuser need to know? That there is movement by your glass. As you approach and move through the glass transition zone there's more. You might need to consider this as you think about support for your work, spacing or what surface you have your work on and where you might like to place warm up holds or where you may want to go slower on your ramp rate.

Now let's tie in Viscosity and the regression from an earlier post to look at what's happening at firing temperatures. All glass will have a curve shaped like this but each glass will have it's own slightly different temperature profile (even "same" COE fusible different color). Different temp profile means different slope.

What remains the same for every glass type is the viscosity at which the curve changes shape. Viscosity is our universal key. Where good stuff happens can be pegged by viscosity and a firing temp can be associated with that viscosity for any type of glass.

There are many excellent sources for glass viscosities easily found on the internet. Most are for industrial uses but have great application for fusing. I've compiled Table 1 below and I have a second table (see Blog post) with a few more viscosity points that I have regressed firing temperatures against. I used that table to write my firing schedules, including one for glass powder with a top temp in the 8.8- 8 poise range. Cooler than anyone, to my knowledge as of 2018 and wonderfully effective. I even tack fuse with a light fire polish in this range. I like that it's below the crystallization temp.

Now a word about units. I've used a simple, elegant simplification, but now I'm going to explain what I did as succinctly as I can in case you need to know it. Units are really important and you can get yourself in trouble leaving off writing them out and making sure you've used a consistent set of units in your calculation. I've used some simplified language in the blog because it's directionally correct, the math works and its easy to understand. Glass viscosity data is generally given in either 10 exponent or the log base 10 using either Pascale seconds (Pa.s) or poise (dyne second)/cm2 - did you see the physics? no worries if you didn't). Either Log or exponent scale is used literally because it's prettier on a graph that way and more easily understandable (see post script below on transforms for more). That's why it's done. In addition, because it's an elegant simplification to report, either just the exponents or the log 10's; which are the same, that's what I've done and that's what you will see below in the typical industrial table. This simplification works since we don't need to actually use a viscosity in a calculation per se, all we need is an index to accurately reference said viscosity vs a firing temperature. I like Log base 10 (Poise) so that's what I use and you will see short hand for that in my posts, as just the log of viscosity number which is the same as the exponent and the word "poise" so you know the unit system. The unit system is critical to which scale temperature has to be in and so you are not off by an order of magnitude. If you do go out looking for graphs and tables of viscosity for glass on the internet don't get hung up on the log 10's and the 10 to the exponent part. Just make sure you are consistently using either Pa.s or Poise. There's more detail on why we do math transforms below ** Here I'm trying to keep this simple, accessible and not too boring.

And now for an overlay of Firing Temperatures. First let's take a look at another graph adapted from Lehigh University's Richard Brow to give some perspective on what's going on as our glass is heated in the kiln. Brow used temperatures for a different glass but because viscosity points are universal standards it was easy to adapt for our regressed COE 90 data. Forgive the strategic laziness. Rather than re-plot the curve I just photoshopped and marked up a screen shot of the original.

Is it wrong that I find this as exciting as what I pull out of the kiln?

A few things leap out. First the log 10 scale. Big changes for small increments. Glass is not boolean (on / off or with sharp bordered ranges) it's full of long, rangy, somewhat hard to pinpoint, transition-y changes and for some things like the anneal temperature, a point value within a range is chosen based on practicality and in some cases like the strain point, measurability by a standard test protocol. At some point measuring higher viscosity runs into practical detection limits of standard testing, for example. Measuring devices have ranges and limitations. This is why an anneal for Bullseye at 900 F is really not violating any hard stop rules and IS just following the data.

Some practical temperatures and holds are chosen for ease of optimizing manufacturing for industrial production. The anneal range is a great example. You can anneal in seconds (if you could get the heat transfer right) at the top of the range, minutes at 13 poise (specifically 15 defined in ASTM) and hours at the strain point. We are in effect flying blind through an entire stress relief zone. Somewhere there is a crude optimum for us based on our kiln shape / materials and the total of the individual firing glass dynamics. I picture it as horses running around a curve in a race track (the outside of the glass cooling faster) and then hopefully straightening out on the straight stretch to pass together through an anneal and then cool together enough to minimize re-introduction of stresses as things freeze up. This is why I pause my firings at 1000 F on the way down and take the temp down slowly to where I hold at the "anneal temp" and then slowly again through 700F - 800 ish or the general vicinity of 18 poise on the graph (which matches some older conventional wisdom of no thermal shock above about 800F). I find I get the best results this way (I've taken a polarized lens look). I get best results at a hold at 950F but I do believe this is where individual kiln heat transfer is important, so I can not say w/o testing where your best hold would be. I use a top down forced convection. Most people don't. What I can say is people have been doing anneal holds all over the range for decades with predominantly consistent, replicable good results.

It is a balance between results, and time / cost to manufacture, yes?

That said, the reason everyone once choose to anneal around 950 ish - in the minutes zone - but to hold for hour(s) is because of the crudeness of our heat transfer control vs the excellent insulating properties of glass. Note that we also don't anneal based on opaque vs transparent any more. Consider that we have consolidated to one representative Anneal temp for the COE "family" of glass and it still works! Or even COE 96 vs 90 (gasp) when folks moved from color / opacity anneal temps to 950/ 960 for the family, as the recommended respective hold points. What's + or - 20 degrees F or so within this range? Not really worth the effort of it's own program. Crude control, strategic knowledge of glass properties backed up by years of practical results. What does merit separate programs is thickness and variable thickness in the glass.

So where is the end of the anneal range? The strain point corresponds to the lowest temperature in the annealing range at which viscous flow of glass will not occur. Another way to look at it is the max temp at which glass can be used (say for a wood stove) without creep. It's a range and the end marks the end of being able to stress relieve and ushered in where glass can be thermal shocked. Sources list 15.5 - 16 in poise as the viscosity corresponding to the highest temperature a glass can be heated to without destroying the anneal. This suggests the transition to thermal shock zone and a physical end to the anneal zone vs around 800 - 880 ish.

Finally it looks like At (deformation point) is at the boundary between glass viscoelastic transition and Newtonian fluid which makes sense. As an aside, deformation is also reported as a range from 11.7 to 11.3 "poise" scale and the Anneal range is reported to start at 11. Someone could probably update this with some fatter zone lines to the left and right of the transition zone for even more transitions and perhaps that would be useful.

Now back to our COE curve. Below is a hand drawn not to scale COE curve "hook". Let's tie in both viscosity and bullseye firing temperatures. All viscosities are in "poise" and all temperatures given are in degrees F. The Regression on the blog is based on Bullseye data for clear 1100. In a family of COE compatible fusible glasses, you can pick a representative curve and write firing schedules based on what is representative, and it works!

It's interesting to me that 11 poise is close to the lowest recommended mold top slump temperature (1160F).

For readability's sake the regressed representative COE 90 firing temps (Poise/F):

Td (diametric softening) at 9 (1150F), 10 (1100F) or 11 (1040F)

At deformation point at 11.5 (1018F ) with range @ 11.3 (1025F) to 11.7 (1010F)

Tg or Glass plastic Transition zone from 12 (997F - feel free to round), to 13.5 (940 F)

Anneal Zone 11 (1040F) to 14.5 ASTM (910F) / 14.6 industrial (907F) 14.8 (Bullseye) 900F

Anneal Point Tanneal: 13 (960F)

Bonus 15.5 - 16 top temp w/o destruction of anneal which backwards spells no annealing. 881F - 868F.

Extrapolation to curve intersection 18 (820F). Hangs together with old conventional wisdom of 800 F thermal shock boundary* and fits with legacy data on strain points for COE 90 strain and anneal points.

*Note that absolutely everyone has put in practical overplus for operating (firing) parameters for obvious reasons. Hope this helps de-mystify what is going on with our glass within the warm up and cool down cycles.

Finally it's really cool that gold pinks work and are within this family!

So there you have it. Simple, useful.

The End


Post script:

** A linear scale shows a smaller, proportional change. A log scale (natural or base 10) shows a magnitude change scale. Doing a transform of scales can be really useful. Sometimes log transforms are used to compress an axis to give a more usefully viewable graph. Imagine a melt viscosity at 100 and an anneal at 10000000000000. You'd need a tall monitor to see that graph or you could just write the same thing as exponents: 10^2 and 10^13 respectively or take the log 10 and use 2 and 13. Much better!

Often a transform is done to get a linear graph because everyone loves the simplicity of Y=Mx+B. Generally you need the M (slope) to plug into a design calculation for some useful thing. No one really wants to do a regression fit with some clunky, pain in the ass math for plug and chug (hand calcs), even with a great fit. See a trend here? In fact I rejected the first fit regression in favor of a simpler (constant - exponent) equation because I like the math better. Naturally you can automate what ever you want but I enjoy hand calc, like a lovely tea ceremony but with numbers. yummy delicious numbers.

Here is a short background on the test method for generating these curves. It may or may not help you. The data for these curves is generated using a Dilatomer and there is a weight put on the glass. The weight enhances the signal of what's going on in the glass for detection and measurement purposes because glass is an amorphous liquid that can support weight with out viscous flow; up until the transition temperature when the amorphous liquid starts to flow under weight. In practical design you may want a glass for a hot service like viewing a wood stove fire and you need to know where temperature wise the glass starts to give. As you can see from the graph, the transition happens over a range and not at one temperature point. That's all you really need to know. It's a standard test, there's a weight so it's different from our kiln process but still very useful and even though the test is "standard", different companies use different weights and slightly different terms for the points on the curve. No worries since we all speak Viscosity. You don't need to because I have translated and using my regressed curve to fill in some points for Bullseye glass. Feel free to snooze through the ASTM paper though.

If you are still here reading, this is a warning that I've seen even high profile people do with public math in Twitter. Watch your units! It's easy to make an order of magnitude mistake if you aren't doing dimensional analysis. You can not mix poise with degrees F. You must convert to one consistent system. Calculations must be either all in SI or English or Martian for Elon Musk. I am reporting to you temperatures in deg F because I am conditioned by years of engineering to think in F but I am converting my regression which is in Poise and Deg C. It's an easy way to make a mistake so it is important to check your units.

The strain point corresponds to the lowest temperature in the annealing range at which viscous flow of glass will not occur. Another way to look at it is the max temp at which glass can be used (say for a wood stove) without creep. It's a range and the end marks the end of being able to stress relieve and ushered in where glass can be thermal shocked.

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