nanoHUB-U Physics of Electronic Polymers/Lecture 1.10: Experiments and the Interaction Parameter
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[Slide 1 L1.10] Hello, and welcome back to The Physics of Electronic Polymers. I'm Bryan Boudouris, and today we're going to wrap up our discussion regarding the thermal dynamics of simple polymers. With the idea of how we compare the physically measured in experiments, regarding the interaction parameter, to we what we've talked about with respect to the regular solution theory, derivation of the interaction parameter, chi last time. So hopefully, what we'll be able to do at the end of this lecture, is to write the empirical formula that's been found for chi based on numerous, numerous experiments with the interaction parameter in real systems. And the second thing we'd like to be able to do, is really summarize what we mean by Chi, how we can use it, when we should be careful using it, and when we can go ahead and try to extract some real information out of it. So we're going to build from what the theory has done, and go ahead and move on to the experiment and practice of using the interaction parameter today. 

[Slide 2] So when we have this experimental measure of the Flory-Huggins interaction parameter, we can look at it terms of regular solution theory, and Flory-Huggins theory. And what we've seen in practice in terms of the polymer community, is that a common correction to it is that previously we derived it from regular solution theory that chi should be equal to this term right here. V hat over RT times the difference of the squares of the solubility parameters. What folks find that when chi is predicted from regular solution theory to be less than 0.3, it's really important to add a 0.34 to this equation. And this is a completely empirical relationship, but it's found to work out very, very well. Where this 0.34 comes from is really, really not well understood at this point. But it turns out there's additional interactions that you need to correct for it, when you have a very low predicted Chi value, to go ahead and make sure that brings something that's more in line with what's actually experimentally measured. Now what you can do is take a completely different approach to how to think about chi, okay? And instead of thinking about chi in terms of coming from some kind of theory, you can say hey, maybe chi is just some way to capture something that's not due to ideal behavior of the system. Remember everything we've talked about to this point, we've always assumed that we have some kind of ideality upon our mixing. So what if we go ahead and instead of thinking about of the free energy of an ideal system, what if we say that the free energy of mixing of a system can be split into two terms. The free energy of mixing for the ideal part, which I've symbolized by id, and the free mixing of the excess part, or anything that's not ideal okay? And I symbolize that by ex. So, if we do that, we can rewrite this expression now in terms of the heat of mixing it or the enthalpy, mixing for ideal, plus the heat of the excess term, minus the entropic terms as well, both spread out into the ideal and the excess term. So if we do this, we remember that for an ideal solution, the entropic chains I'm mixing is due to changes in combinations. That means mixing our red and blue spheres around, right? And that means that delta Sm ideal should just be what we recovered from regular solution theory. In a similar manner, if we really have no change upon mixing, our enthalpy of mixing should be zero, okay? In an ideal situation. So now we have our ideal terms of this equation determined. 

[Slide 3] Let's think about what our excess terms could be. If we reevaluate this expression right here in our regular solution theory, we see that our delta GM, we already have the ideal part there. So that tells us that in entropic terms, delta SM excess is really equal to zero. And really, what this term here is, in terms of our regular solution theory, is just that chi represents some part of the excess terms associated with the enthalpy of mixing. So, now this is just a correction to the ideal mixing kind of situation. Now, the other thing we can do is think about the free energy of mixing as some kind of effective chi interaction parameter. And I say effective, because it's always coming from experimental data. And that's why I've defined it as chi eff or chi effective. And if we go ahead and do that, that tells us that if we switch our mole fractions to volume fractions, so we move from regular solution theory to Flory-Huggins Theory, we can define our chi effective, just from this equation right here. And say that it's just delta Gm access per kT divided by 1 over the product of the two volume fractions. And if that's the case, then you can start to think about chi in a certain term okay? And you can say that there's two main challenges to chi. There's something that has a temperature dependence, and some parameter alpha, which is just a coefficient, some parameter plus another term beta, okay? And that says that chi effective is divided into something that has a temperature dependence, and something that doesn't have a temperature dependence. And really, what people kind of estimate is that temperature dependent term has to do with enthalpic contributions. And a temperature independent term has to do with entropic contribution. So you'll often see this chi effective fitted as a plot of 1 over T. To extract alpha and beta for a particular system. And then for future systems, people can go ahead and use that chi effective to try to figure out the mixing and demixing behavior of the polymer-solution system, or the polymer-polymer system. So there's a variety of ways to get at chi, this interaction parameter. But it really depends on what's seen in experiment, and have to best match that to theory, and combining the two together really lets folks try to get a full picture of what's going on in terms of this very important parameter. In terms of polymer science, and in terms of electronically active materials. 

[Slide 4] So, what we've described during this unit is five key things. We started off by talking about molecular weights in polymeric systems. And then we moved on about how to express, not just molecular weights, but the size of a polymer chain in space. And we started off with the end-to-end distance, and different kind of mash ups of that end-to-end distance. From there, we talked about the radius of gyration, and how to really get at that radius of gyration. And then we moved on to things that told us whether two material systems would mix, or de-mix. Whether they'll be small molecule and polymer, or polymer and polymer. And really we saw that that had a lot to do with that interaction parameter, chi, so we spent a little bit of time talking about how to think about chi from a theoretical, and from an experimental perspective. And really, all of these lectures together set the mathematical foundation for how we think about semiconducting polymers. How we think about how these materials will combine, both as unique solutions, unique thin films. Whether there be single components, or will it be mixtures of polymers for some kind of organic electronic applications, like solar cells. And what this does, is allow us to move on for more mathematically based things to more qualitative things, so we can think about how to look at the exact physics of certain things. And the polymer of physics that we'll look at next is how to use these materials, and think about how they crystallize. So our next module will really focus on polymer crystallization. And when we focus on polymer crystallization, that'll allow us to set up our organized structures of our crystal, that'll really be key to charge transport in a lot of our electronically active polymers. So I thank you for your attention, I thank you for completing this lecture. I thank you for completing this unit. And we'll be able to talk about polymer crystallization in the next unit of