No Access Submitted: 15 October 2020 Accepted: 20 December 2020 Published Online: 11 January 2021
J. Chem. Phys. 154, 024905 (2021); https://doi.org/10.1063/5.0033413
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• Michael P. Howard
• Zachary M. Sherman
• Delia J. Milliron
• Thomas M. Truskett
We extend Wertheim’s thermodynamic perturbation theory to derive the association free energy of a multicomponent mixture for which double bonds can form between any two pairs of the molecules’ arbitrary number of bonding sites. This generalization reduces in limiting cases to prior theories that restrict double bonding to at most one pair of sites per molecule. We apply the new theory to an associating mixture of colloidal particles (“colloids”) and flexible chain molecules (“linkers”). The linkers have two functional end groups, each of which may bond to one of several sites on the colloids. Due to their flexibility, a significant fraction of linkers can “loop” with both ends bonding to sites on the same colloid instead of bridging sites on different colloids. We use the theory to show that the fraction of linkers in loops depends sensitively on the linker end-to-end distance relative to the colloid bonding-site distance, which suggests strategies for mitigating the loop formation that may otherwise hinder linker-mediated colloidal assembly.
M.P.H. thanks Ryan Jadrich for introducing him to Wertheim’s elegant theory. This research was primarily supported by the National Science Foundation through the Center for Dynamics and Control of Materials: an NSF MRSEC under Cooperative Agreement No. DMR-1720595, with additional support from an Arnold O. Beckman Postdoctoral Fellowship (Z.M.S.) and the Welch Foundation (Grant Nos. F-1696 and F-1848). We acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources.
1. 1. M. S. Wertheim, “Fluids with highly directional attractive forces. I. Statistical thermodynamics,” J. Stat. Phys. 35, 19–34 (1984). https://doi.org/10.1007/bf01017362, Google ScholarCrossref
2. 2. M. S. Wertheim, “Fluids with highly directional attractive forces. II. Thermodynamic perturbation theory and integral equations,” J. Stat. Phys. 35, 35–47 (1984). https://doi.org/10.1007/bf01017363, Google ScholarCrossref
3. 3. M. S. Wertheim, “Fluids with highly directional attractive forces. III. Multiple attraction sites,” J. Stat. Phys. 42, 459–476 (1986). https://doi.org/10.1007/bf01127721, Google ScholarCrossref
4. 4. M. S. Wertheim, “Fluids with highly directional attractive forces. IV. Equilibrium polymerization,” J. Stat. Phys. 42, 477–492 (1986). https://doi.org/10.1007/bf01127722, Google ScholarCrossref
5. 5. L. Rovigatti, F. Bomboi, and F. Sciortino, “Accurate phase diagram of tetravalent DNA nanostars,” J. Chem. Phys. 140, 154903 (2014). https://doi.org/10.1063/1.4870467, Google ScholarScitation, ISI
6. 6. E. Locatelli, P. H. Handle, C. N. Likos, F. Sciortino, and L. Rovigatti, “Condensation and demixing in solutions of DNA anostars and their mixtures,” ACS Nano 11, 2094–2102 (2017). https://doi.org/10.1021/acsnano.6b08287, Google ScholarCrossref
7. 7. E. Bianchi, J. Largo, P. Tartaglia, E. Zaccarelli, and F. Sciortino, “Phase diagram of patchy colloids: Towards empty liquids,” Phys. Rev. Lett. 97, 168301 (2006). https://doi.org/10.1103/physrevlett.97.168301, Google ScholarCrossref
8. 8. E. Bianchi, P. Tartaglia, E. La Nave, and F. Sciortino, “Fully solvable equilibrium self-assembly process: Fine-tuning the clusters size and the connectivity in patchy particle systems,” J. Phys. Chem. B 111, 11765–11769 (2007). https://doi.org/10.1021/jp074281+, Google ScholarCrossref
9. 9. J. Russo, P. Tartaglia, and F. Sciortino, “Reversible gels of patchy particles: Role of the valence,” J. Chem. Phys. 131, 014504 (2009). https://doi.org/10.1063/1.3153843, Google ScholarScitation, ISI
10. 10. J. Russo, J. M. Tavares, P. I. C. Teixeira, M. M. Telo da Gama, and F. Sciortino, “Reentrant phase diagram of network fluids,” Phys. Rev. Lett. 106, 085703 (2011). https://doi.org/10.1103/physrevlett.106.085703, Google ScholarCrossref
11. 11. W. G. Chapman, K. E. Gubbins, C. G. Joslin, and C. G. Gray, “Theory and simulation of associating liquid mixtures,” Fluid Phase Equilib. 29, 337–346 (1986). https://doi.org/10.1016/0378-3812(86)85033-6, Google ScholarCrossref
12. 12. C. G. Joslin, C. G. Gray, W. G. Chapman, and K. E. Gubbins, “Theory and simulation of associating liquid mixtures. II,” Mol. Phys. 62, 843–860 (1987). https://doi.org/10.1080/00268978700102621, Google ScholarCrossref
13. 13. G. Jackson, W. G. Chapman, and K. E. Gubbins, “Phase equilibria of associating fluids: Spherical molecules with multiple bonding sites,” Mol. Phys. 65, 1–31 (1988). https://doi.org/10.1080/00268978800100821, Google ScholarCrossref
14. 14. W. G. Chapman, G. Jackson, and K. E. Gubbins, “Phase equilibria of associating fluids: Chain molecules with multiple bonding sites,” Mol. Phys. 65, 1057–1079 (1988). https://doi.org/10.1080/00268978800101601, Google ScholarCrossref
15. 15. E. A. Müller and K. E. Gubbins, “Molecular-based equations of state for associating fluids: A review of SAFT and related approaches,” Ind. Eng. Chem. Res. 40, 2193–2211 (2001). https://doi.org/10.1021/ie000773w, Google ScholarCrossref
16. 16. M. P. Howard, R. B. Jadrich, B. A. Lindquist, F. Khabaz, R. T. Bonnecaze, D. J. Milliron, and T. M. Truskett, “Structure and phase behavior of polymer-linked colloidal gels,” J. Chem. Phys. 151, 124901 (2019). https://doi.org/10.1063/1.5119359, Google ScholarScitation, ISI
17. 17. B. A. Lindquist, R. B. Jadrich, D. J. Milliron, and T. M. Truskett, “On the formation of equilibrium gels via a macroscopic bond limitation,” J. Chem. Phys. 145, 074906 (2016). https://doi.org/10.1063/1.4960773, Google ScholarScitation, ISI
18. 18. C. A. Saez Cabezas, G. K. Ong, R. B. Jadrich, B. A. Lindquist, A. Agrawal, T. M. Truskett, and D. J. Milliron, “Gelation of plasmonic metal oxide nanocrystals by polymer-induced depletion attractions,” Proc. Natl. Acad. Sci. U. S. A. 115, 8925–8930 (2018). https://doi.org/10.1073/pnas.1806927115, Google ScholarCrossref
19. 19. M. N. Dominguez, M. P. Howard, J. M. Maier, S. A. Valenzuela, Z. M. Sherman, J. F. Reuther, L. C. Reimnitz, J. Kang, S. H. Cho, S. L. Gibbs, A. K. Menta, D. L. Zhuang, A. van der Stok, S. J. Kline, E. V. Anslyn, T. M. Truskett, and D. J. Milliron, “Assembly of linked nanocrystal colloids by reversible covalent bonds,” Chem. Mater. 32, 10235–10245 (2020). https://doi.org/10.1021/acs.chemmater.0c04151, Google ScholarCrossref
20. 20. A. Stukowski, “Visualization and analysis of atomistic simulation data with OVITO–the open visualization tool,” Modell. Simul. Mater. Sci. Eng. 18, 015012 (2010). https://doi.org/10.1088/0965-0393/18/1/015012, Google ScholarCrossref
21. 21. R. P. Sear and G. Jackson, “Thermodynamic perturbation theory for association into douby bonded dimers,” Mol. Phys. 82, 1033–1048 (1994). https://doi.org/10.1080/00268979400100734, Google ScholarCrossref
22. 22. R. P. Sear and G. Jackson, “Thermodynamic perturbation theory for association into chains and rings,” Phys. Rev. E 50, 386–394 (1994). https://doi.org/10.1103/physreve.50.386, Google ScholarCrossref
23. 23. A. Galindo, S. J. Burton, G. Jackson, D. P. Visco, Jr., and D. A. Kofke, “Improved models for the phase behaviour of hydrogen fluoride: Chain and ring aggregates in the SAFT approach and the AEOS model,” Mol. Phys. 100, 2241–2259 (2002). https://doi.org/10.1080/00268970210130939, Google ScholarCrossref
24. 24. A. S. Avlund, G. M. Kontogeorgis, and W. G. Chapman, “Intramolecular association within the SAFT framework,” Mol. Phys. 109, 1759–1769 (2011). https://doi.org/10.1080/00268976.2011.589990, Google ScholarCrossref
25. 25. J. M. Tavares, L. Rovigatti, and F. Sciortino, “Quantitative description of the self-assembly of patchy particles into chains and rings,” J. Chem. Phys. 137, 044901 (2012). https://doi.org/10.1063/1.4737930, Google ScholarScitation, ISI
26. 26. L. Rovigatti, J. M. Tavares, and F. Sciortino, “Self-assembly in chains, rings, and branches: A single component system with two critical points,” Phys. Rev. Lett. 111, 168302 (2013). https://doi.org/10.1103/physrevlett.111.168302, Google ScholarCrossref
27. 27. B. D. Marshall and W. G. Chapman, “Thermodynamic perturbation theory for associating fluids with small bond angles: Effects of steric hindrance, ring formation, and double bonding,” Phys. Rev. E 87, 052307 (2013). https://doi.org/10.1103/physreve.87.052307, Google ScholarCrossref
28. 28. B. D. Marshall and W. G. Chapman, “Molecular theory for the phase equilibria and cluster distribution of associating fluids with small bond angles,” J. Chem. Phys. 139, 054902 (2013). https://doi.org/10.1063/1.4816665, Google ScholarScitation, ISI
29. 29. B. D. Marshall, “A general mixture equation of state for double bonding carboxylic acids with ≥2 association sites,” J. Chem. Phys. 148, 174103 (2018). https://doi.org/10.1063/1.5024684, Google ScholarScitation, ISI
30. 30. K. Hoppe, E. Geidel, H. Weller, and A. Eychmüller, “Covalently bound CdTe nanocrystals,” Phys. Chem. Chem. Phys. 4, 1704–1706 (2002). https://doi.org/10.1039/b201219g, Google ScholarCrossref
31. 31. W. Maneeprakorn, M. A. Malik, and P. O’Brien, “Developing chemical strategies for the assembly of nanoparticles into mesoscopic objects,” J. Am. Chem. Soc. 132, 1780–1781 (2010). https://doi.org/10.1021/ja910022q, Google ScholarCrossref
32. 32. R. J. Macfarlane, B. Lee, M. R. Jones, N. Harris, G. C. Schatz, and C. A. Mirkin, “Nanoparticle superlattice engineering with DNA,” Science 334, 204–208 (2011). https://doi.org/10.1126/science.1210493, Google ScholarCrossref
33. 33. S. Borsley and E. R. Kay, “Dynamic covalent assembly and disassembly of nanoparticle aggregates,” Chem. Commun. 52, 9117–9120 (2016). https://doi.org/10.1039/c6cc00135a, Google ScholarCrossref
34. 34. Y. Wang, P. J. Santos, J. M. Kubiak, X. Guo, M. S. Lee, and R. J. Macfarlane, “Multistimuli responsive nanocomposite tectons for pathway dependent self-assembly and acceleration of covalent bond formation,” J. Am. Chem. Soc. 141, 13234–13243 (2019). https://doi.org/10.1021/jacs.9b06695, Google ScholarCrossref
35. 35. N. Marro, F. della Sala, and E. R. Kay, “Programmable dynamic covalent nanoparticle building blocks with complementary reactivity,” Chem. Sci. 11, 372–383 (2020). https://doi.org/10.1039/c9sc04195h, Google ScholarCrossref
36. 36. C. A. Mirkin, R. L. Letsinger, R. C. Mucic, and J. J. Storhoff, “A DNA-based method for rationally assembling nanoparticles into macroscopic materials,” Nature 382, 607–609 (1996). https://doi.org/10.1038/382607a0, Google ScholarCrossref
37. 37. A. P. Alivisatos, K. P. Johnsson, X. Peng, T. E. Wilson, C. J. Loweth, M. P. Bruchez, and P. G. Schultz, ““Organization of ’nanocrystal molecules’ using DNA,” Nature 382, 609–611 (1996). https://doi.org/10.1038/382609a0, Google ScholarCrossref
38. 38. H. Xiong, D. van der Lelie, and O. Gang, “Phase behavior of nanoparticles assembled by DNA linkers,” Phys. Rev. Lett. 102, 015504 (2009). https://doi.org/10.1103/physrevlett.102.015504, Google ScholarCrossref
39. 39. D. Zanchet, C. M. Micheel, W. J. Parak, D. Gerion, and A. P. Alivisatos, “Electrophoretic isolation of discrete Au nanocrystal/DNA conjugates,” Nano Lett. 1, 32–35 (2001). https://doi.org/10.1021/nl005508e, Google ScholarCrossref
40. 40. B. D. Marshall and W. G. Chapman, “Thermodynamic perturbation theory for associating molecules,” in Advances in Chemical Physics, edited by S. A. Rice and A. R. Dinner (John Wiley & Sons, Inc., 2016), Vol. 160, pp. 1–47. Google ScholarCrossref
41. 41. W. Zmpitas and J. Gross, “Detailed pedagogical review and analysis of Wertheim’s thermodynamic perturbation theory,” Fluid Phase Equilib. 428, 121–152 (2016). https://doi.org/10.1016/j.fluid.2016.07.033, Google ScholarCrossref
42. 42. H. C. Andersen, “Cluster methods in equilibrium statistical mechanics,” in Statistical Mechanics, Modern Theoretical Chemistry Vol. 5, edited by B. J. Berne (Springer, 1977), pp. 1–45. Google ScholarCrossref
43. 43.An articulation point of a connected graph is a point that, if removed, will disconnect the graph into two or more graphs.4242. H. C. Andersen, “Cluster methods in equilibrium statistical mechanics,” in Statistical Mechanics, Modern Theoretical Chemistry Vol. 5, edited by B. J. Berne (Springer, 1977), pp. 1–45. An irreducible graph is free of articulation points. For example, a pair of points or any closed cycle of points is irreducible, but three points bonded collinearly are not irreducible because the middle point is an articulation point.
44. 44.AB denotes that A is a subset of B, including the improper subset A = B.
45. 45.A partition of set A is a grouping of the elements of A into one or more non-empty sets using every element exactly once. For example, if A = {a, b, c}, then {{a}, {b}, {c}}, {{a}, {b, c}}, {{a, b}, {c}}, {{a, c}, {b}}, and {{a, b, c}} are all partitions of A. P(A) denotes the set of all possible partitions of A. The last partition, into only a single subset {A}, is called an improper partition.
46. 46.AB = {aA|aB} denotes the set difference, i.e., all elements that are in A but not in B.
47. 47.Forgiving some abuse of notation, the label of a single site A should be replaced by a set {A} when it represents a set of bonded sites.
48. 48.|A| denotes the number of elements in set A.
49. 49.AB denotes that $A$ is a proper subset of B, i.e., there is at least one element of B that is not in A so AB.
50. 50. T. Boublík, “Hard-sphere equation of state,” J. Chem. Phys. 53, 471–472 (1970). https://doi.org/10.1063/1.1673824, Google ScholarScitation, ISI
51. 51. R. P. Sear and G. Jackson, “The ring integral in a thermodynamic perturbation theory for association,” Mol. Phys. 87, 517–521 (1996). https://doi.org/10.1080/00268979600100341, Google ScholarCrossref
52. 52. M. S. Wertheim, “Thermodynamic perturbation theory of polymerization,” J. Chem. Phys. 87, 7323–7331 (1987). https://doi.org/10.1063/1.453326, Google ScholarScitation, ISI
53. 53. J. D. Weeks, D. Chandler, and H. C. Andersen, “Role of repulsive forces in determining equilibrium structure of simple liquids,” J. Chem. Phys. 54, 5237–5247 (1971). https://doi.org/10.1063/1.1674820, Google ScholarScitation, ISI
54. 54. M. Fuchs and K. S. Schweizer, “Structure of colloid–polymer suspensions,” J. Phys.: Condens. Matter 14, R239–R269 (2002). https://doi.org/10.1088/0953-8984/14/12/201, Google ScholarCrossref
55. 55. S. Plimpton, “Fast parallel algorithms for short-range molecular dynamics,” J. Comput. Phys. 117, 1–19 (1995). https://doi.org/10.1006/jcph.1995.1039, Google ScholarCrossref, ISI
56. 56. G. S. Grest and K. Kremer, “Molecular dynamics simulation for polymers in the presence of a heat bath,” Phys. Rev. A 33, 3628–3631 (1986). https://doi.org/10.1103/physreva.33.3628, Google ScholarCrossref
57. 57. A. Haghmoradi, B. D. Marshall, and W. G. Chapman, “Beyond Wertheim’s multi-density theory: Steric hindrance and associated rings in a two-density formalism for binary mixtures of molecules with two associating sites,” J. Chem. Eng. Data, 65, 5743–5752 (2020). https://doi.org/10.1021/acs.jced.0c00695, Google ScholarCrossref
58. 58. M. P. Howard, Z. M. Sherman, A. N. Sreenivasan, S. A. Valenzuela, E. V. Anslyn, D. J. Milliron, and T. M. Truskett, “Effects of linker flexibility on phase behavior and structure of linked colloidal gels,” arXiv:2011.12512 (2020). Google Scholar
59. 59. E. Zaccarelli, “Colloidal gels: Equilibrium and nonequilibrium routes,” J. Phys.: Condens. Matter 19, 323101 (2007). https://doi.org/10.1088/0953-8984/19/32/323101, Google ScholarCrossref, ISI
60. 60. R. E. Miles, “On random rotations in R3,” Biometrika 52, 636–639 (1965). https://doi.org/10.1093/biomet/52.3-4.636, Google ScholarCrossref
61. 61. C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, and T. E. Oliphant, “Array programming with NumPy,” Nature 585, 357–362 (2020). https://doi.org/10.1038/s41586-020-2649-2, Google ScholarCrossref
62. 62. P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt and SciPy 1.0 Contributors, “SciPy 1.0: Fundamental algorithms for scientific computing in Python,” Nat. Methods 17, 261–272 (2020). https://doi.org/10.1038/s41592-019-0686-2, Google ScholarCrossref
63. 63. S. K. Lam, A. Pitrou, and S. Seibert, “Numba: A LLVM-based Python JIT compiler,” in Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC, LLVM ’15, 2015. Google Scholar