Existing methods rely on equations that use cumbersome mathematical adjustments for each temperature change — painstaking work that costs time, computational resources and money. The adjustments also tend to lack physical justification, according to the researchers. 

The new approach eliminates the need for all the adjustments, said principal investigator Michael Webb, assistant professor of chemical and biological engineering. He and graduate student Satyen Dhamankar found a way to neatly account for complex temperature-dependent behavior using a single, self-contained and physically motivated framework. 

Their work was published in ACS Macro Letters on June 14.

Intuitive versus non-intuitive behaviors

While everyday experience leads us to expect rising temperatures in a system to lead to more randomness — like sugar disappearing into water as the water is heated — there are also scenarios where the opposite happens. Solutes can come crashing out of solution upon heating, or proteins can unfold when the temperature is lowered. 

Webb and Dhamankar wondered why this should be the case. Why do some materials behave in one manner and others behave in another?

"That’s what our theoretical framework addresses,” Webb said.

Under complex conditions, many polymers used in common commercial applications — whether in vats of solution for manufacturing or in drug capsules dissolving in the stomach — behave in ways that defy intuition. As they are heated in a solution, instead of breaking down, they first clump together. As the temperature increases further, they finally unravel into individual strands. 

Prior models that seek to predict this behavior have tackled the temperature dependence of state changes by including ad-hoc fitting parameters that also depend on temperature to account for the system’s overall disorder. 

It’s a tedious workaround that engineers have been trying to eliminate for decades. Based on a variety of physical observations and suggestions from the literature, Webb and Dhamankar hypothesized that such behavior depends on molecules interacting and organizing in preferred orientations – like two people facing each other during conversation. Rather than consider the precise molecular details, the researchers borrowed the idea of magnetic spin states to arrive at a much simpler approach.

“Our theoretical framework concretely suggests that the simplest way to obtain the ‘unintuitive’ phenomena is if there are at least two energy scales for monomer-solvent interactions, and we capture these with the spins,” Webb said.

Using spin to model polymer behavior

Rather than focusing solely on the polymers, they looked at the whole polymer-solvent system. They found that certain solvents allow the complex structures of the polymer to stretch out and move around. Without the solvent, the polymer shrinks into itself. 

“Water forms shells around these molecules and, when heated, the shell melts away, allowing the big molecule to collapse,” Dhamankar said. 

By taking a hard mathematical look at the entire system, they developed the first model to capture diverse complex behaviors with no explicitly temperature-dependent parameters. 

The key to this finding is in borrowing a concept from magnetism, a property of particles that scientists call spin. In previous models, polymers were chains of monomers that varied in length but not much else. By borrowing from magnetism, each monomer unit of the polymer chain and each molecule of solvent surrounding the chain can be viewed as having one of two distinct spin states (either up-spin or down-spin). When a solvent molecule has the same spin state as it’s adjacent monomer, it forms a low energy interaction. When the spin states disagree — say the solvent molecule is up-spin and the monomer its next to is down-spin — it forms a high energy interaction. 

Systems that don’t have the spins cannot exhibit certain thermally induced responses.

As a solution heats up, particles switch rapidly between spin states, disrupting the order of the low-entropy configuration or, as Dhamankar put it, melting the solvent shell. By giving each particle a spin, complex polymer behavior can be modeled without the need for individually defined fitting parameters. The result is “a self-contained theoretical framework,” Webb said. They no longer have to account for mysterious external parameters. “We know everything that goes into the model.”

Webb said that while they take the implementation of spins as analogous to these magnetic systems, they believe that the existence of these spin states or their correlated effects on energy also has physical origins and represents characteristics of hydrogen bonding.

Webb said he is eager to see the theory taken even further, using spins to investigate the impacts of other factors, including polymer composition and pH. 

The model is notable in part due to its robustness but also because of how their theory demystifies the role of entropy in polymer-solvent systems, pointing the way to a deeper mathematical understanding of polymer chemistry.