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Entropy in self-assembly

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Summary

Colloidal systems show beautiful examples of how entropy can lead to self-assembly of ordered structures, challenging our perception of disorder. In fact, dispersion of hard colloidal particles, systems in which by default entropy is the only thermodynamic driving force, displays both translational and orientational order on increasing density. Entropy is also a fundamental concept for describing effective interactions between colloidal particles. In several cases, entropy maximization generates strong attractive forces, capable of inducing condensation and sometimes crystallization. These entropic forces can even be exploited to drive colloids in specific locations or to orient them in the build-up of supracolloidal aggregates. Depletion interactions and combinatorial contributions are two important manifestations of these forces. Entropy also plays a leading role in systems exploring the bottom of their potential energy surface. In patchy colloids, particles interacting with highly anisotropic and localized potentials, ground-state structures are often degenerate in energy, leaving entropy to decide the thermodynamically stable polymorph. A striking result is the possibility of generating colloidal “liquids” thermodynamically more stable than colloidal “crystals” even at vanishing temperature.

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References

  1. Alder B. and Wainwright T., J. Chem. Phys., 27 (1957) 1208.

    Article  ADS  Google Scholar 

  2. Onsager L., Ann. New York Acad. Sci., 51 (1949) 627.

    Article  ADS  Google Scholar 

  3. Glotzer S. C. and Solomon M. J., Nat. Mater., 6 (2007) 557.

    Article  Google Scholar 

  4. Pawar A. B. and Kretzschmar I., Macromol. Rapid Commun., 31 (2010) 150.

    Article  Google Scholar 

  5. Duguet E., Désert A., Perro A. and Ravaine S., Chem. Soc. Rev., 40 (2011) 941.

    Article  Google Scholar 

  6. Yi G.-R., Pine D. J. and Sacanna S., J. Phys.: Condens. Matter, 25 (2013) 193101.

    ADS  Google Scholar 

  7. Ravaine S. and Duguet E., Curr. Opin. Colloid Interface Sci., 30 (2017) 45.

    Article  Google Scholar 

  8. Bianchi E., Capone B., Coluzza I., Rovigatti L. and van Oostrum P. D., Phys. Chem. Chem. Phys., 19 (2017) 19847.

    Article  Google Scholar 

  9. Bianchi E., Patchy colloids: A theoretical and numerical perspective on functionalized units for self-assembly, in Frontiers of Nanoscience, Vol. 13 (Elsevier) 2019, pp. 37–60.

  10. Pawar A. B. and Kretzschmar I., Langmuir, 24 (2008) 355.

    Article  Google Scholar 

  11. Chen Q., Bae S. C. and Granick S., Nature, 469 (2011) 381.

    Article  ADS  Google Scholar 

  12. Wang Y., Wang Y., Breed D. R., Manoharan V. N., Feng L., Hollingsworth A. D., Weck M. and Pine D. J., Nature, 491 (2012) 51.

    Article  ADS  Google Scholar 

  13. Sacanna S., Irvine W., Chaikin P. M. and Pine D. J., Nature, 464 (2010) 575.

    Article  ADS  Google Scholar 

  14. Gong Z., Hueckel T., Yi G.-R. and Sacanna S., Nature, 550 (2017) 234.

    Article  ADS  Google Scholar 

  15. Jo I.-S., Lee S., Zhu J., Shim T. S. and Yi G.-R., Curr. Opin. Colloid Interface Sci., 30 (2017) 97.

    Article  Google Scholar 

  16. Biffi S., Cerbino R., Bomboi F., Paraboschi E. M., Asselta R., Sciortino F. and Bellini T., Proc. Natl. Acad. Sci. U.S.A., 110 (2013) 15633.

    Article  ADS  Google Scholar 

  17. Feng L., Dreyfus R., Sha R., Seeman N. C. and Chaikin P. M., Adv. Mater., 25 (2013) 2779.

    Article  Google Scholar 

  18. Cai J., Townsend J. P., Dodson T. C., Heiney P. A. and Sweeney A. M., Science, 357 (2017) 564.

    Article  ADS  Google Scholar 

  19. Cai J. and Sweeney A. M., ACS Central Sci., 4 (2018) 840.

    Article  Google Scholar 

  20. Liu H., Kumar S. K. and Sciortino F., J. Chem. Phys., 127 (2007) 084902.

    Article  ADS  Google Scholar 

  21. Fusco D. and Charbonneau P., Phys. Rev. E, 88 (2013) 012721.

    Article  ADS  Google Scholar 

  22. Likos C., Riv. Nuovo Cimento, 37 (2014) 125.

    Google Scholar 

  23. Asakura S. and Oosawa F., J. Chem. Phys., 22 (1954) 1255.

    Article  ADS  Google Scholar 

  24. Frenkel D., Physica A, 263 (1999) 26.

    Article  ADS  Google Scholar 

  25. Lekkerkerker H. N. and Tuinier R., Colloids and the depletion interaction, Vol. 833 (Springer) 2011.

  26. Dinsmore A., Warren P., Poon W. and Yodh A., EPL, 40 (1997) 337.

    Article  ADS  Google Scholar 

  27. Dinsmore A., Yodh A. and Pine D., Nature, 383 (1996) 239.

    Article  ADS  Google Scholar 

  28. Mason T., Phys. Rev. E, 66 (2002) 060402.

    Article  ADS  Google Scholar 

  29. Kraft D. J., Ni R., Smallenburg F., Hermes M., Yoon K., Weitz D. A., van Blaaderen A., Groenewold J., Dijkstra M. and Kegel W. K., Proc. Natl. Acad. Sci. U.S.A., 109 (2012) 10787.

    Article  ADS  Google Scholar 

  30. Buzzaccaro S., Colombo J., Parola A. and Piazza R., Phys. Rev. Lett., 105 (2010) 198301.

    Article  ADS  Google Scholar 

  31. Gnan N., Zaccarelli E. and Sciortino F., Nat. Commun., 5 (2014) 3267.

    Article  ADS  Google Scholar 

  32. Zilman A., Kieffer J., Molino F., Porte G. and Safran S., Phys. Rev. Lett., 91 (2003) 015901.

    Article  ADS  Google Scholar 

  33. Likos C. N., Soft Matter, 2 (2006) 478.

    Article  ADS  Google Scholar 

  34. Tanaka F., Polymer Physics: Applications to Molecular Association and Thermoreversible Gelation (Cambridge University Press) 2011.

  35. Rubinstein M. and Colby R. H., Polymer physics, Vol. 23 (Oxford University Press, New York) 2003.

    Google Scholar 

  36. Frenkel D., Nat. Mater., 14 (2014) 9.

    Article  ADS  Google Scholar 

  37. Escobedo F. A., Soft Matter, 10 (2014) 8388.

    Article  ADS  Google Scholar 

  38. Piazza R., Soft matter: the stuff that dreams are made of (Springer Science & Business Media) 2011.

  39. Battimelli G. and Ciccotti G., Eur. Phys. J. H, 43 (2018) 303.

    Article  Google Scholar 

  40. Pusey P. N. and Van Megen W., Nature, 320 (1986) 340.

    Article  ADS  Google Scholar 

  41. Carnahan N. F. and Starling K. E., J. Chem. Phys., 51 (1969) 635.

    Article  ADS  Google Scholar 

  42. Boublík T. and Nezbeda I., Collect. Czech. Chem. Commun., 51 (1986) 2301.

    Article  Google Scholar 

  43. Hall K. R., J. Chem. Phys., 57 (1972) 2252.

    Article  ADS  Google Scholar 

  44. Foffi G., McCullagh G. D., Lawlor A., Zaccarelli E., Dawson K. A., Sciortino F., Tartaglia P., Pini D. and Stell G., Phys. Rev. E, 65 (2002) 031407.

    Article  ADS  Google Scholar 

  45. Israelachvili J. N., Intermolecular and surface forces (Academic Press) 2011.

  46. Bryant G., Williams S. R., Qian L., Snook I., Perez E. and Pincet F., Phys. Rev. E, 66 (2002) 060501.

    Article  ADS  Google Scholar 

  47. Piazza R., Bellini T. and Degiorgio V., Phys. Rev. Lett., 71 (1993) 4267.

    Article  ADS  Google Scholar 

  48. Metropolis N., Rosenbluth A. W., Rosenbluth M. N., Teller A. H. and Teller E., J. Chem. Phys., 21 (1953) 1087.

    Article  ADS  Google Scholar 

  49. Alder B. and Wainwright T., Phys. Rev., 127 (1962) 359.

    Article  ADS  Google Scholar 

  50. Bernard E. P., Krauth W. and Wilson D. B., Phys. Rev. E, 80 (2009) 056704.

    Article  ADS  Google Scholar 

  51. Thorneywork A. L., Abbott J. L., Aarts D. G. and Dullens R. P., Phys. Rev. Lett., 118 (2017) 158001.

    Article  ADS  Google Scholar 

  52. Bernard E. P. and Krauth W., Phys. Rev. Lett., 107 (2011) 155704.

    Article  ADS  Google Scholar 

  53. Kosterlitz J. M. and Thouless D. J., J. Phys. C: Solid State Phys., 6 (1973) 1181.

    Article  ADS  Google Scholar 

  54. Halperin B. and Nelson D. R., Phys. Rev. Lett., 41 (1978) 121.

    Article  ADS  MathSciNet  Google Scholar 

  55. Weber H., Marx D. and Binder K., Phys. Rev. B, 51 (1995) 14636.

    Article  ADS  Google Scholar 

  56. Russo J. and Wilding N. B., Phys. Rev. Lett., 119 (2017) 115702.

    Article  ADS  Google Scholar 

  57. Frenkel D., Lekkerkerker H. and Stroobants A., Nature, 332 (1988) 822.

    Article  ADS  Google Scholar 

  58. Salamonczyk M., Zhang J., Portale G., Zhu C., Kentzinger E., Gleeson J. T., Jakli A., De Michele C., Dhont J. K., Sprunt S. et al., Nat. Commun., 7 (2016) 13358.

    Article  ADS  Google Scholar 

  59. Dussi S. and Dijkstra M., Nat. Commun., 7 (2016) 11175.

    Article  ADS  Google Scholar 

  60. Bushby R. J. and Lozman O. R., Curr. Opin. Colloid Interface Sci., 7 (2002) 343.

    Article  Google Scholar 

  61. Belli S., Patti A., Dijkstra M. and Van Roij R., Phys. Rev. Lett., 107 (2011) 148303.

    Article  ADS  Google Scholar 

  62. Borshch V., Kim Y.-K., Xiang J., Gao M., Jakli A., Panov V. P., Vij J. K., Imrie C. T., Tamba M.-G., Mehl G. H. et al., Nat. Commun., 4 (2013) 2635.

    Article  ADS  Google Scholar 

  63. Greco C. and Ferrarini A., Phys. Rev. Lett., 115 (2015) 147801.

    Article  ADS  Google Scholar 

  64. Allen M. P., Evans G. T., Frenkel D. and Mulder B., Adv. Chem. Phys., 86 (1993) 1.

    Google Scholar 

  65. Zanchetta G., Nakata M., Buscaglia M., Bellini T. and Clark N. A., Proc. Natl. Acad. Sci. U.S.A., 105 (2008) 1111.

    Article  ADS  Google Scholar 

  66. De Michele C., Bellini T. and Sciortino F., Macromolecules, 45 (2011) 1090.

    Article  Google Scholar 

  67. Nguyen K. T., Sciortino F. and De Michele C., Langmuir, 30 (2014) 4814.

    Article  Google Scholar 

  68. Man W., Donev A., Stillinger F. H., Sullivan M. T., Russel W. B., Heeger D., Inati S., Torquato S. and Chaikin P., Phys. Rev. Lett., 94 (2005) 198001.

    Article  ADS  Google Scholar 

  69. Torquato S. and Stillinger F. H., Rev. Mod. Phys., 82 (2010) 2633.

    Article  ADS  Google Scholar 

  70. de Graaf J., van Roij R. and Dijkstra M., Phys. Rev. Lett., 107 (2011) 155501.

    Article  ADS  Google Scholar 

  71. Damasceno P. F., Engel M. and Glotzer S. C., Science, 337 (2012) 453.

    Article  ADS  Google Scholar 

  72. Donev A., Cisse I., Sachs D., Variano E. A., Stillinger F. H., Connelly R., Torquato S. and Chaikin P. M., Science, 303 (2004) 990.

    Article  ADS  Google Scholar 

  73. van Anders G., Klotsa D., Ahmed N. K., Engel M. and Glotzer S. C., Proc. Natl. Acad. Sci. U.S.A., 111 (2014) E4812.

    Article  Google Scholar 

  74. Harper E. S., Marson R. L., Anderson J. A., Van Anders G. and Glotzer S. C. Soft Matter, 11 (2015) 7250.

    Article  ADS  Google Scholar 

  75. Escobedo F. A., J. Chem. Phys., 146 (2017) 134508.

    Article  ADS  Google Scholar 

  76. Lee S., Teich E. G., Engel M. and Glotzer S. C., Proc. Natl. Acad. Sci. U.S.A., 116 (2019) 14843.

    Article  ADS  Google Scholar 

  77. Bianchi E., Largo J., Tartaglia P., Zaccarelli E. and Sciortino F., Phys. Rev. Lett., 97 (2006) 168301.

    Article  ADS  Google Scholar 

  78. Mirkin C. A., Letsinger R. L., Mucic R. C. and Storhoff J. J., Nature, 382 (1996) 607.

    Article  ADS  Google Scholar 

  79. Smallenburg F. and Sciortino F., Nat. Phys., 9 (2013) 554.

    Article  Google Scholar 

  80. Sciortino F., Gallo P., Tartaglia P. and Chen S.-H., Phys. Rev. E, 54 (1996) 6331.

    Article  ADS  Google Scholar 

  81. Whitelam S., Tamblyn I., Haxton T. K., Wieland M. B., Champness N. R., Garrahan J. P. and Beton P. H., Phys. Rev. X, 4 (2014) 011044.

    Google Scholar 

  82. Moreno A., Buldyrev S., La Nave E., Saika-Voivod I., Sciortino F., Tartaglia P. and Zaccarelli E., Phys. Rev. Lett., 95 (2005) 157802.

    Article  ADS  Google Scholar 

  83. Moreno A., Saika-Voivod I., Zaccarelli E., La Nave E., Buldyrev S., Tartaglia P. and Sciortino F., J. Chem. Phys., 124 (2006) 204509.

    Article  ADS  Google Scholar 

  84. Sciortino F., Eur. Phys. J. B, 64 (2008) 505.

    Article  ADS  Google Scholar 

  85. Biffi S., Cerbino R., Nava G., Bomboi F., Sciortino F. and Bellini T., Soft Matter, 11 (2015) 3132.

    Article  ADS  Google Scholar 

  86. Kauzmann W., Chem. Rev., 43 (1948) 219.

    Article  Google Scholar 

  87. Noya E. G., Zubieta I., Pine D. J. and Sciortino F., J. Chem. Phys., 151 (2019) 094502.

    Article  ADS  Google Scholar 

  88. Romano F., Sanz E. and Sciortino F., J. Chem. Phys., 132 (2010) 184501.

    Article  ADS  Google Scholar 

  89. Frenkel D. and Ladd A. J., J. Chem. Phys., 81 (1984) 3188.

    Article  ADS  Google Scholar 

  90. Vega C., Sanz E., Abascal J. and Noya E., J. Phys.: Condens. Matter, 20 (2008) 153101.

    ADS  Google Scholar 

  91. Hu H., Ruiz P. S. and Ni R., Phys. Rev. Lett., 120 (2018) 048003.

    Article  ADS  Google Scholar 

  92. van der Meulen S. A. and Leunissen M. E., J. Am. Chem. Soc., 135 (2013) 15129.

    Article  Google Scholar 

  93. Feng L., Pontani L.-L., Dreyfus R., Chaikin P. and Brujic J., Soft Matter, 9 (2013) 9816.

    Article  ADS  Google Scholar 

  94. Angioletti-Uberti S., Varilly P., Mognetti B. M. and Frenkel D., Phys. Rev. Lett., 113 (2014) 128303.

    Article  ADS  Google Scholar 

  95. McMullen A., Holmes-Cerfon M., Sciortino F., Grosberg A. Y. and Brujic J., Phys. Rev. Lett., 121 (2018) 138002.

    Article  ADS  Google Scholar 

  96. Romano F. and Sciortino F., Nat. Mater., 10 (2011) 171.

    Article  ADS  Google Scholar 

  97. Romano F. and Sciortino F., Soft Matter, 7 (2011) 5799.

    Article  ADS  Google Scholar 

  98. Mao X., Chen Q. and Granick S., Nat. Mater., 12 (2013) 217.

    Article  ADS  Google Scholar 

  99. Montarnal D., Capelot M., Tournilhac F. and Leibler L., Science, 334 (2011) 965.

    Article  ADS  Google Scholar 

  100. Smallenburg F., Leibler L. and Sciortino F., Phys. Rev. Lett., 111 (2013) 188002.

    Article  ADS  Google Scholar 

  101. Bomboi F., Caprara D., Fernandez-Castanon J. and Sciortino F., Nanoscale, 11 (2019) 9691.

    Article  Google Scholar 

  102. Böhmer R., Ngai K., Angell C. A. and Plazek D., J. Chem. Phys., 99 (1993) 4201.

    Article  ADS  Google Scholar 

  103. Stockmayer W. H., J. Polym. Sci., 9 (1952) 69.

    Article  ADS  Google Scholar 

  104. Wertheim M., J. Stat. Phys., 35 (1984) 19.

    Article  ADS  Google Scholar 

  105. Tavares J. M., Teixeira P. I. C. and Telo da Gama M., Mol. Phys., 107 (2009) 453.

    Article  ADS  Google Scholar 

  106. Lu P. J., Zaccarelli E., Ciulla F., Schofield A. B., Sciortino F. and Weitz D. A., Nature, 453 (2008) 499.

    Article  ADS  Google Scholar 

  107. Vliegenthart G. and Lekkerkerker H. N., J. Chem. Phys., 112 (2000) 5364.

    Article  ADS  Google Scholar 

  108. Anderson V. J. and Lekkerkerker H. N., Nature, 416 (2002) 811.

    Article  ADS  Google Scholar 

  109. Pham K. N., Puertas A. M., Bergenholtz J., Egelhaaf S. U., Moussaid A., Pusey P. N., Schofield A. B., Cates M. E., Fuchs M. and Poon W. C., Science, 296 (2002) 104.

    Article  ADS  Google Scholar 

  110. Sciortino F., Nat. Mater., 1 (2002) 145.

    Article  ADS  Google Scholar 

  111. Anzini P. and Parola A., Soft Matter, 13 (2017) 5150.

    Article  ADS  Google Scholar 

  112. Lin K.-h., Crocker J. C., Prasad V., Schofield A., Weitz D. A., Lubensky T. and Yodh A., Phys. Rev. Lett., 85 (2000) 1770.

    Article  ADS  Google Scholar 

  113. Hertlein C., Helden L., Gambassi A., Dietrich S. and Bechinger C., Nature, 451 (2008) 172.

    Article  ADS  Google Scholar 

  114. Franosch T., Lang S. and Schilling R., Phys. Rev. Lett., 109 (2012) 240601.

    Article  ADS  Google Scholar 

  115. Qi W., Gantapara A. P. and Dijkstra M., Soft Matter, 10 (2014) 5449.

    Article  ADS  Google Scholar 

  116. Filali M., Ouazzani M. J., Michel E., Aznar R., Porte G. and Appell J., J. Phys. Chem. B, 105 (2001) 10528.

    Article  Google Scholar 

  117. Nykypanchuk D., Maye M. M., Van Der Lelie D. and Gang O., Nature, 451 (2008) 549.

    Article  ADS  Google Scholar 

  118. Park S. Y., Lytton-Jean A. K., Lee B., Weigand S., Schatz G. C. and Mirkin C. A., Nature, 451 (2008) 553.

    Article  ADS  Google Scholar 

  119. Wang Y., Wang Y., Zheng X., Ducrot E., Yodh J. S., Weck M. and Pine D. J., Nat. Commun., 6 (2015) 7253.

    Article  ADS  Google Scholar 

  120. Angioletti-Uberti S., Mognetti B. M. and Frenkel D., Nat. Mater., 11 (2012) 518.

    Article  ADS  Google Scholar 

  121. Bachmann S. J., Kotar J., Parolini L., Šarić A., Cicuta P., Di Michele L. and Mognetti B. M., Soft Matter, 12 (2016) 7804.

    Article  ADS  Google Scholar 

  122. Sciortino F., Gang O. and Kumar S., submitted (2019).

  123. Zhang D. Y. and Winfree E., J. Am. Chem. Soc., 131 (2009) 17303.

    Article  Google Scholar 

  124. Martinez-Veracoechea F. J. and Frenkel D., Proc. Natl. Acad. Sci. U.S.A., 108 (2011) 10963.

    Article  ADS  Google Scholar 

  125. Kitov P. I. and Bundle D. R., J. Am. Chem. Soc., 125 (2003) 16271.

    Article  Google Scholar 

  126. Flory P. J., Principles of polymer chemistry (Cornell University Press) 1953.

  127. Stockmayer W. H., J. Chem. Phys., 11 (1943) 45.

    Article  ADS  Google Scholar 

  128. Bianchi E., Tartaglia P., La Nave E. and Sciortino F., J. Phys. Chem. B, 111 (2007) 11765.

    Article  Google Scholar 

  129. Bianchi E., Tartaglia P., Zaccarelli E. and Sciortino F., J. Chem. Phys., 128 (2008) 144504.

    Article  ADS  Google Scholar 

  130. Hill T. L., An introduction to statistical thermodynamics (Courier Corporation) 1986.

  131. Sciortino F., Basic concepts in self assembly, in Soft Matter Self Assembly, Vol. 193 (IOS Press) 2016, p. 1.

  132. Russo J., Tartaglia P. and Sciortino F., J. Chem. Phys., 131 (2009) 014504.

    Article  ADS  Google Scholar 

  133. Walker J. S. and Vause C. A., Sci. Am., 256 (1987) 98.

    Article  ADS  Google Scholar 

  134. Camp P. J., Shelley J. C. and Patey G. N., Phys. Rev. Lett., 84 (2000) 115.

    Article  ADS  Google Scholar 

  135. Rovigatti L., Russo J. and Sciortino F., Phys. Rev. Lett., 107 (2011) 237801.

    Article  ADS  Google Scholar 

  136. Tlusty T. and Safran S., Science, 290 (2000) 1328.

    Article  ADS  Google Scholar 

  137. Dudowicz J., Freed K. F. and Douglas J. F., J. Chem. Phys., 111 (1999) 7116.

    Article  ADS  Google Scholar 

  138. Stell G., Wu K. and Larsen B., Phys. Rev. Lett., 37 (1976) 1369.

    Article  ADS  Google Scholar 

  139. Panagiotopoulos A. Z., J. Chem. Phys., 116 (2002) 3007.

    Article  ADS  Google Scholar 

  140. de Gennes P. G. and Pincus P. A., Phys. Kondens. Materie, 11 (1970) 189.

    ADS  Google Scholar 

  141. van Roij R., Phys. Rev. Lett., 76 (1996) 3348.

    Article  ADS  Google Scholar 

  142. Sear R. P., Phys. Rev. Lett., 76 (1996) 2310.

    Article  ADS  Google Scholar 

  143. Teixeira P. I. C., Tavares J. M. and Telo da Gama M. M., J. Phys.: Condens. Matter, 12 (2000) 411.

    ADS  Google Scholar 

  144. Weis J. J. and Levesque D., Phys. Rev. Lett., 71 (1993) 2729.

    Article  ADS  Google Scholar 

  145. van Leeuwen M. E. and Smit B., Phys. Rev. Lett., 71 (1993) 3991.

    Article  ADS  Google Scholar 

  146. Caillol J.-M., J. Chem. Phys., 98 (1993) 9835.

    Article  ADS  Google Scholar 

  147. Jia R., Braun H. and Hentschke R., Phys. Rev. E, 82 (2010) 062501.

    Article  ADS  Google Scholar 

  148. Rovigatti L., Russo J. and Sciortino F., Soft Matter, 8 (2012) 6310.

    Article  ADS  Google Scholar 

  149. Ronti M., Rovigatti L., Tavares J. M., Ivanov A. O., Kantorovich S. S. and Sciortino F., Soft Matter, 13 (2017) 7870.

    Article  ADS  Google Scholar 

  150. Tavares J. M., Teixeira P. I. C. and Telo da Gama M. M., Phys. Rev. E, 80 (2009) 021506.

    Article  ADS  Google Scholar 

  151. Tavares J. M., Teixeira P. I. C. and Telo da Gama M. M., Mol. Phys., 107 (2009) 453.

    Article  ADS  Google Scholar 

  152. Wertheim M., J. Stat. Phys., 35 (1984) 19; 35.

    Article  ADS  Google Scholar 

  153. Wertheim M., J. Stat. Phys., 42 (1986) 459; 477.

    Article  ADS  Google Scholar 

  154. Tavares J. M., Teixeira P. I. C., da Gama M. M. T. and Sciortino F., J. Chem. Phys., 132 (2010) 234502.

    Article  ADS  Google Scholar 

  155. Russo J., Tavares J. M., Teixeira P. I. C., Telo da Gama M. M. and Sciortino F., Phys. Rev. Lett., 106 (2011) 085703.

    Article  ADS  Google Scholar 

  156. Walther A. and Müller H., Soft Matter, 4 (2008) 663.

    Article  ADS  Google Scholar 

  157. Sciortino F., Giacometti A. and Pastore G., Phys. Rev. Lett., 103 (2009) 237801.

    Article  ADS  Google Scholar 

  158. Bomboi F., Romano F., Leo M., Fernandez-Castanon J., Cerbino R., Bellini T., Bordi F., Filetici P. and Sciortino F., Nat. Commun., 7 (2016) 13191.

    Article  ADS  Google Scholar 

  159. Fernandez-Castanon J., Bianchi S., Saglimbeni F., Di Leonardo R. and Sciortino F., Soft Matter, 14 (2018) 6431.

    Article  ADS  Google Scholar 

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    Francesco Sciortino

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Sciortino, F. Entropy in self-assembly. Riv. Nuovo Cim. 42, 511–548 (2019). https://doi.org/10.1393/ncr/i2019-10165-1

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