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The Boyle temperature, TB, for an n-segment polymer in solution is the temperature where the second osmotic virial coefficient, A2, is zero. This characteristic is of interest for its connection to the polymer condensation critical temperature, particularly for n → ∞. TB can be measured experimentally or computed for a given model macromolecule. For the latter, we present and examine two approaches, both based on the Mayer-sampling Monte Carlo (MSMC) method, to calculate Boyle temperatures as a function of model parameters. In one approach, we use MSMC calculations to search for TB, as guided by the evaluation of temperature derivatives of A2. The second approach involves numerical integration of an ordinary differential equation describing how TB varies with a model parameter, starting from a known TB. Unlike general MSMC calculations, these adaptations are appealing because they neither invoke a reference for the calculation nor use special averages needed to avoid bias when computing A2 directly. We demonstrate these methods by computing TB lines for off-lattice linear Lennard-Jones polymers as a function of chain stiffness, considering chains of length n ranging from 2 to 512 monomers. We additionally perform calculations of single-molecule radius of gyration Rg and determine the temperatures Tθ, where linear scaling of Rg2 with n is observed, as if the polymers were long random-walk chains. We find that Tθ and TB seem to differ by 6% in the n → ∞ limit, which is beyond the statistical uncertainties of our computational methodology. However, we cannot rule out systematic error relating to our extrapolation procedure as being the source of this discrepancy. 

The virial equation of state (VEOS) provides a rigorous bridge between molecular interactions and thermodynamic properties. The past decade has seen renewed interest in the VEOS due to advances in theory, algorithms, computing power, and quality of molecular models. Now, with the emergence of increasingly accurate first-principles computational chemistry methods, and machine-learning techniques to generate potential-energy surfaces from them, VEOS is poised to play a larger role in modeling and computing properties. Its scope of application is limited to where the density series converges, but this still admits a useful range of conditions and applications, and there is potential to expand this range further. Recent applications have shown that for simple molecules, VEOS can provide first-principles thermodynamic property data that are competitive in quality with experiment. Moreover, VEOS provides a focused and actionable test of molecular models and first-principles calculations via comparison to experiment. This Perspective presents an overview of recent advances and suggests areas of focus for further progress.