The molecular basis of differential scanning calorimetry: heat capacity and energy fluctuations

Melting and boiling points are convenient and important measures of stability.  But how do you measure a melting point of, for example, nanoparticles that are too small to see?

This screencast shows two Molecular Workbench simulations (you can find them here and here) I made [see credits at the end of the post] to illustrate the connection between phase transitions, changes in heat capacity, and energy fluctuations, and the slides below takes you through the basic ideas behind them.

Slide 1: In the first simulation heat is added to a nano-particle and the resulting temperature increase is measured. When viewing the simulation notice that the temperature increases less during the melting/evaporation.

Slide 2: To analyze the data we first switch the x and y-axes, so that heat added (i.e. the internal energy, U) is plotted as a function of temperature.

Slide 3: The data is a bit noisy (mainly because the simulation heats the particle too fast: from 0 to almost 3000 K in about 200 picoseconds!), so I smooth it by fitting a curve to it.

Slide 4: From the smoothed data I can calculate how fast the energy changes with temperature.  This is the heat capacity (C_v), which peaks at a temperature around 1350 K - the melting temperature of the particle.

Slide 5: This observation forms the basis of differential scanning calorimetry, which measures the temperature as a function of the flow of energy to a system, and determines the melting point by finding the temperature where the heat capacity peaks.

Slide 6: One way to explain why the heat capacity peaks at a phase transition such as melting is through its relation to energy fluctuations: the system changes most during a phase transition ("bonds" between particles are broken and formed), so the energy fluctuates more, meaning that the heat capacity is largest.

Slides 7, 8, and 9: In the second set of simulations the energy is plotted a function of time at 3 temperatures: before the particle melts (500 K), when the particle melts/evaporates (1350 K), and after the particle has evaporated (3000 K).

Slide 10: Results from the 3 simulations are compared.  Clearly the fluctuations are largest when the temperature is 1350 K.  The fluctuations at 3000 K are larger than at 500 K, even though the heat capacities are similar.  This is because the heat capacity is proportional to the average energy fluctuation divided by the temperature squared (slide 6).

Calorimetry

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You may wonder why we don't see two heat capacity peaks: one for melting and one for evaporation.  This is because of the particle is so small (i.e. composed of relative few particles).  For a macroscopic systems (like water) the phase transitions are well defined.  Water is ice at 272 K, melts at 273 K, and is a liquid at 274 K (at 1 atmosphere of pressure); and the heat capacity has a very narrow peak at 273 K. As particles become smaller their phase transitions become less well defined, the heat capacity peak becomes broad, and in some cases (like this one) you get a single heat capacity peak for melting and evaporation.  This means that the phase transition cannot really be classified as melting or evaporation and that is occurs over a relatively large temperature range.  Li and Truhlar have an interesting article on this subject. If you would like to play around with or modify the simulations they can be downloaded here and here, but you need to download Molecular Workbench first. Credits: The simulations are based on models and scripts by Arie Aziman and Carlos Gardena, who based their work on a model by Dan Damelin, i.e. they are made possible, like Molecular Workbench itself, by open source science.