From Chemical Potential to Phase Diagram

For phase transitions, the chemical potential measures the free energy for exchanging molecules between different phases. For a configuration of a system, the Widom insertion method yields the excess chemical potential, which is the chemical potential without the ideal part. However, brute-force insertion is computationally expensive. We have developed FMAP-enabled insertion with acceleration from fast Fourier transform. Both insertion methods require a set of configurations, which can be obtained through Monte Carlo (MC) simulation. The values of excess chemical potential for different densities and temperatures through MC and insertions are accessible from MC for LJ particles with brute-force insertion and MC for LJ particles with` FMAP-enabled insertion.

Input the number density and excess chemical potential data in pairs into the following box. The excess chemical potential should be in units of kT. The following example is from simulation with LJ shifted potetail at a cutoff of 3.0 σ and T=0.65 with FMAP-enabled insertion.

Number Density and Excess Chemical Potential Data Entry Area: Select: Ctrl+A | Copy: Ctrl+C | Paste: Ctrl+V

0.058594 -1.49784 0.117188 -2.82125 0.175781 -3.60747 0.234375 -4.08 0.292969 -4.76423 0.351562 -5.07318 0.410156 -5.31277 0.468750 -5.48594 0.527344 -5.97833 0.585938 -6.36961 0.644531 -6.83577 0.703125 -7.46792 0.761719 -7.26286 0.820312 -6.23518 0.878906 -4.40366

Polynomial fitting

We fit the density dependence of the excess chemical potential data to a polynomial without the constant term. While a high order allows the polynomial to match the data better, it may also overfit. We conclude that the fifth order is optimal. The details of polynomial regression are explained by P. Lutus (link provided in the Acknowledgement section).

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Results Area: Select: Ctrl+A | Copy: Ctrl+C | Paste: Ctrl+V

Equation for excess chemical potential

Equation for chemical potential with ideal gas part added

With the above expression for the chemical potential, we use the Newton-Raphson method to find the intersections with a horizontal line and match the enclosed areas on the two sides. The equality in area ensures the equality in pressure between the dilute phase and dense phase, similar to a construction proposed by Maxwell.

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Initial chemical potentail:

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Match areas

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Condition of coexistence

The numerical results can be verified by Wolfram|Alpha

• The x values of the intersections (red points) between the blue curve and black line are the roots of the polynomial at the desired chemical potential, check the roots
• The left and right areas should be very close in absolute value but opposite in sign, check the values of the left area and the right area
• The spinodal densities (blue points) are the roots of the derivative of the polynomial, check the roots of the derivative

The phase diagram collects binodal and spinodal densities at different temperatures.The preset data for brute-force and FMAP insertions are from calcualtion at L = 8.

Data Entry for brute-force	Data Entry for FMAP binodal	Data Entry for FMAP spinodal

0.65 0.0023218 0.8376466 0.70 0.0043734 0.8190991 0.75 0.0075931 0.7934679 0.80 0.0122802 0.7673708 0.85 0.0190183 0.7374460 0.90 0.0284116 0.7068853 0.95 0.0405430 0.6734460 1.00 0.0583024 0.6361260 1.05 0.0769232 0.6003575 1.10 0.1090452 0.5542660 1.15 0.1508763 0.5019465 0.65 0.002175 0.835378 0.70 0.004121 0.819054 0.75 0.007233 0.795280 0.80 0.011764 0.768773 0.85 0.018263 0.738303 0.90 0.026953 0.706901 0.95 0.038744 0.672090 1.00 0.054570 0.636500 1.05 0.075107 0.596934 1.10 0.103884 0.548459 1.15 0.143142 0.497071 0.65 0.039832 0.717891 0.70 0.050656 0.700126 0.75 0.060759 0.676554 0.80 0.072645 0.643124 0.85 0.086550 0.607168 0.90 0.100949 0.577953 0.95 0.118144 0.546073 1.00 0.136241 0.518620 1.05 0.156409 0.489410 1.10 0.180263 0.455078 1.15 0.211136 0.422408

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The codes for this and other methods for calculating the binodals of phase-separated condensates are available on github

The polynomial fitting is modeled after polysolve by P. Lutus

S. Qin and H.-X. Zhou (2016). Fast method for computing chemical potentials and liquid-liquid phase equilibria of macromolecular solutions. J. Phys. Chem. B. 120, 8164-8174.

K. Mazarakos, S. Qin, and H.-X. Zhou (2023). Calculating binodals and interfacial tension of phase-separated condensates from molecular simulations, with finite-size corrections. Methods Mol. Biol. 2563, 1-35.