This video will teach you how to create a tailored thermal expansion by pairing positive CTE materials  with Negative CTE ALLVAR Alloys. Using this method you can specify and hit a target CTE for your athermal application.

Transcript of video below

Hello, and welcome to another series on thermal expansion, In this series we’re going to take a look at how we combined negative thermal expansion ALLVAR alloys with positive thermal expansion materials  to create a tailored  thermal expansion.

In this case what I think about in terms of tailored thermal expansion (CTE) in series – and so that’s a little bit different than the infrared optic topic where the parts and components are in parallel – this is a series system. What I mean by series is you have one piece of material butted up against another piece of material. If this material is ALLVAR, for example, it will have a thermal expansion coefficient, and if this material is another material with a positive thermal expansion coefficient you can start to cancel out this positive CTE with a negative CTE, so effectively the other material is going to be positive, which is positive thermal expansion, and alpha ALLVAR is negative and that’s negative thermal expansion.

In a system like this you can get an effective or total thermal expansion coefficient that is between the ALLVAR, or the lower bound, and the upper bound of the CTE. So the thermal expansion of ALLVAR will be the lower bound of the total thermal expansion and the upper bound will be the other material. So that corresponds to this whole bar being 100% ALLVAR or this whole bar being 100% of that other material and then by playing with the relative lengths of these parts you can get a tailored thermal expansion (CTE).

So what does that look like? How do we create a characteristic equation to understand what the thermal strain of a configuration like this will produce? Well, first we have to do some definitions um we have both of these parts at a particular temperature, t1, and they have a particular length – this is the length of ALLVAR at temperature one, T1, and this is the length of the other component or other material at temperature one, T1, and as we heat this system, the ALLVAR – and we’re just going to use for sign – for simplicity right now we’re going to use an expansion. So both of them are going to expand or change shape, that change in shape we’re going to call delta L of ALLVAR, and that’s at t2 and this is going to be the delta l or change in length of the other material at temperature two, T2, and the total length at temperature one is governed by the sum. So the total length at temperature one is equal to the length of ALLVAR and the length of the other material at temperature one, t1. We can see that the total or the change in length of the total bar which would be L1 + delta L_{Allvar and} delta L_other other sorry not l1. So this essentially says that the increase in length or the change in length associated with this total length is equal to the change in length of the ALLVAR component plus the change in length of the other component.

Now we can bring in the notion of thermal expansion that we created or that we covered in another video where the instantaneous thermal expansion, which is defined as the change in strain with temperature the first derivative of strain of temperature. It approximately equals the change in strain versus the change in temperature. This would be equal to the average thermal expansion coefficient, so we’re going to do the development with the average thermal expansion coefficient. If we rearrange this and put in the definition of strain, where strain is equal to a change in length over the original length at the first temperature, then we will get the average thermal expansion coefficient x l1 x delta T is equal to delta L. You can look at the other video to get a rundown of why this is and the difference between instantaneous and average cte. So the change in length – we can plug in here so the total- um oops let’s erase tha.So if we look at l1 total times alpha total average times delta T, so that’s an effective thermal expansion coefficient. It’s not a true material property; it’s an amalgamation of these two materials. That’s going to be equal to L1 ALLVAR * alpha average ALVAR delta T plus L1 othe alpha other average times delta T. So you can see that the delta Ts cancels so it doesn’t actually matter what the change in temperature. That’s not going to be true specifically for the instantaneous CTE, but we’re making the average cte assumption so the delta ts drop out and we’re left with L1 total alpha total is equal to L1 ALLVAR alpha ALLVAR average plus L1 other average other. Now we can take a look and realize that the total length is equal to L1 ALLVAR plus L1 other. You can use this equation directly; I would like to understand, okay, how much ALLVAR am I going to need as a fraction of the total length. So we can rearrange this and L1_other is equal to L1_ALLVAR minus L1_total and we can plug this into here and if we plug it in for L1_other and then multiply it out what we get is L1_total alpha average total is equal to L1_ALLVAR times alpha average all bar plus 1_total alpha other total, minus because of that minus sign, L1_ALLVAR alpha average other. Then if we combine terms to isolate the lengths and bring L1_total term over we get L1_total times alpha average total minus alpha average_other is equal to L1_ALLVAR times alpha average ALLVAR _minus alpha average _other . Then we can divide both sides by L1 _total and divide both sides by this combined term here, and we can get L1 _ALLVAR divided by L1 _total is equal to alpha _total minus alpha_other divided by alpha_ALLVAR minus alpha_other

So this is an equation that you can use to say, “okay how much of a fraction of the total length do I need of ALLVAR if I want a specific effective thermal expansion coefficient? I know the thermal expansion coefficients of ALLVAR and of the other material that I’m going to be using. So this can help you understand or give a ballpark or rough estimate of how much ALLVAR two other material you’re going to need, so this is the fraction of ALLVAR length. Now in the next video we’re going to actually apply this and see how we can use this equation to understand how much ALLVAR, we need but also dial in the coefficient of thermal expansion that we want.