Vinay Vaibhav The Institute of Mathematical Sciences Student Discussion Group on Glasses August 29, 2020 CLASSICAL NUCLEATION THEORY 

Resources Metastable Liquids: Concepts and Principles Pablo G. Debenedetti Lecture Notes in Physics: Nucleation Theory V. I. Kalikmanov Video Lecture on Classical Nucleation Theory (School on Nucleation aggregation and growth- 2010, ICTS-JNCASR) David Reguera Computational Methods for the study of Nucleation (School on Nucleation aggregation and growth- 2010, ICTS-JNCASR) Charusita Chakravarty 

Phase Transitions, Metastability and Nucleation Phase diagram: At any given T, P and $mu$ which phase is thermodynamically stable — No information about kinetics of phase transformation https://www.chem.libretexts.org van der Waals equation of state Liquid-gas or gas-liquid phase transition P = kBT v b a v2 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 One component system 

Phase Transitions, Metastability and Nucleation P V T C Nucleation: Potential barrier Spinodal decomposition: No potential barrier Unstable Metastable Binodal line Spinodal line 

Phase Transitions, Metastability and Nucleation Nucleation: An activated process; transforming a metastable phase to a thermodynamically stable phase; formation of the first embryos of new phase — Most first-order phase transitions occur via nucleation Homogeneous nucleation: In the absence of impurity, surface etc.; achieved thermally; ideal case Heterogeneous nucleation: Impurity, surface, grain boundary etc. provide the place for nucleation site; more practical Goal of Nucleation theory: Rate at which embryos grow to a critical size 

Example: Condensation of Supersaturated Vapor Surface term Volume term Surface + Volume n G AAAB8XicdVDJSgNBEO2JW4xb1KOXxkTwFGaioMeggh4jmAWTIfR0KkmTnp6hu0YIQ/7CiwdFvPo33vwbO4vg+qDpx3tVVNULYikMuu67k1lYXFpeya7m1tY3Nrfy2zt1EyWaQ41HMtLNgBmQQkENBUpoxhpYGEhoBMPzid+4A21EpG5wFIMfsr4SPcEZWum22L4AiYxeFjv5gls6diegv4lXmv5ugcxR7eTf2t2IJyEo5JIZ0/LcGP2UaRRcwjjXTgzEjA9ZH1qWKhaC8dPpxmN6YJUu7UXaPoV0qn7tSFlozCgMbGXIcGB+ehPxL6+VYO/UT4WKEwTFZ4N6iaQY0cn5tCs0cJQjSxjXwu5K+YBpxtGGlLMhfF5K/yf1csk7KnnX5ULlbB5HluyRfXJIPHJCKuSKVEmNcKLIPXkkT45xHpxn52VWmnHmPbvkG5zXDxgfj+A= Nucleation: Thermal fluctuations allows the generation of small aggregates of liquid phase Small clusters are energetically unfavourable; dissolve back to individual molecules Clusters greater than a critical size are thermodynamically favourable and tend to grow to form a new phase Nucleation kinetics: Rate at which critically sized clusters are formed? G(n) = n µ + A(n) AAACFHicdZDLSgMxFIYz9VbrbdSlm2ArVIplpgq6EeoFdFnBXqBTSiZN29AkMyQZoQx9CDe+ihsXirh14c63MW1H8Hog5Of7zyE5vx8yqrTjvFupmdm5+YX0YmZpeWV1zV7fqKkgkphUccAC2fCRIowKUtVUM9IIJUHcZ6TuD87Gfv2GSEUDca2HIWlx1BO0SzHSBrXtQs47J0wjeJEXu/AY7kEBE+LxCBbgieGeoj2Ocm076xQPnHHB38ItTm4nC5KqtO03rxPgiBOhMUNKNV0n1K0YSU0xI6OMFykSIjxAPdI0UiBOVCueLDWCO4Z0YDeQ5ggNJ/TrRIy4UkPum06OdF/99MbwL68Z6e5RK6YijDQRePpQN2JQB3CcEOxQSbBmQyMQltT8FeI+kghrk2PGhPC5Kfxf1EpFd7/oXpWy5dMkjjTYAtsgD1xwCMrgElRAFWBwC+7BI3iy7qwH69l6mbamrGRmE3wr6/UD0Kia7g== 

Nucleation Kinetics Growth of clusters is modelled as a series of chemical reactions Assumptions: Clusters grow/decay because of attachment/detachment of individual molecule under isothermal condition; a monomer colliding with a cluster stick to it with unit probability; Markov process i.e., no memory + — Cluster of size n Cluster of size n+1 Cluster of size n+1 Cluster of size n f(n+1,t) f(n,t) ↵(n + 1, t) AAAB+HicdVDLSsNAFJ3UV62PRl26GWyFihKS2FrdFd24rGAf0IYymU7s0MkkzEyEGvolblwo4tZPceffOH2ID/TAhcM593LvPX7MqFS2/W5kFhaXlleyq7m19Y3NvLm13ZRRIjBp4IhFou0jSRjlpKGoYqQdC4JCn5GWP7yY+K1bIiSN+LUaxcQL0Q2nAcVIaaln5otdxOIBKvFD50gdFHtmwbbKlfKZ7UJNTtxqufJFHMueogDmqPfMt24/wklIuMIMSdlx7Fh5KRKKYkbGuW4iSYzwEN2QjqYchUR66fTwMdzXSh8GkdDFFZyq3ydSFEo5Cn3dGSI1kL+9ifiX10lUcOqllMeJIhzPFgUJgyqCkxRgnwqCFRtpgrCg+laIB0ggrHRWOR3C56fwf9J0LefYcq7cQu18HkcW7II9UAIOqIIauAR10AAYJOAePIIn4854MJ6Nl1lrxpjP7IAfMF4/AOydkfc= f(n,t) f(n+1,t) (n, t) AAAB83icdVDJSgNBEO2JW4xb1KOXxkSIIGFmTIzegl48RjALZELo6fQkTXp6hu4aIYT8hhcPinj1Z7z5N3YWcUEfFDzeq6Kqnh8LrsG2363U0vLK6lp6PbOxubW9k93da+goUZTVaSQi1fKJZoJLVgcOgrVixUjoC9b0h1dTv3nHlOaRvIVRzDoh6UsecErASF7e8xmQgjyB43w3m7OLpXLpwnaxIWdupVT+Ik7RniGHFqh1s29eL6JJyCRQQbRuO3YMnTFRwKlgk4yXaBYTOiR91jZUkpDpznh28wQfGaWHg0iZkoBn6veJMQm1HoW+6QwJDPRvbyr+5bUTCM47Yy7jBJik80VBIjBEeBoA7nHFKIiRIYQqbm7FdEAUoWBiypgQPj/F/5OGW3ROi86Nm6teLuJIowN0iArIQRVURdeohuqIohjdo0f0ZCXWg/VsvcxbU9ZiZh/9gPX6Ac1DkOI= : Concentration of clusters containing n molecules at time t : Rate of attachment of single molecule to a cluster of size n at time t (condensation) : Rate of detachment of single molecule from a cluster of size n at time t (evaporation) (n, t) AAAB83icdVDJSgNBEO2JW4xb1KOXxkSIIGFmTIzegl48RjALZELo6fQkTXp6hu4aIYT8hhcPinj1Z7z5N3YWcUEfFDzeq6Kqnh8LrsG2363U0vLK6lp6PbOxubW9k93da+goUZTVaSQi1fKJZoJLVgcOgrVixUjoC9b0h1dTv3nHlOaRvIVRzDoh6UsecErASF7e8xmQgjyB43w3m7OLpXLpwnaxIWdupVT+Ik7RniGHFqh1s29eL6JJyCRQQbRuO3YMnTFRwKlgk4yXaBYTOiR91jZUkpDpznh28wQfGaWHg0iZkoBn6veJMQm1HoW+6QwJDPRvbyr+5bUTCM47Yy7jBJik80VBIjBEeBoA7nHFKIiRIYQqbm7FdEAUoWBiypgQPj/F/5OGW3ROi86Nm6teLuJIowN0iArIQRVURdeohuqIohjdo0f0ZCXWg/VsvcxbU9ZiZh/9gPX6Ac1DkOI= ↵(n, t) AAAB9HicdVDJSgNBEO2JW4xb1KOXxkSIIGFmTIzegl48RjALJEOo6XSSJj09Y3dPIIR8hxcPinj1Y7z5N3YWcUEfFDzeq6Kqnh9xprRtv1uJpeWV1bXkempjc2t7J727V1NhLAmtkpCHsuGDopwJWtVMc9qIJIXA57TuD66mfn1IpWKhuNWjiHoB9ATrMgLaSF62BTzqQ06c6ONsO52x84Vi4cJ2sSFnbqlQ/CJO3p4hgxaotNNvrU5I4oAKTTgo1XTsSHtjkJoRTiepVqxoBGQAPdo0VEBAlTeeHT3BR0bp4G4oTQmNZ+r3iTEESo0C33QGoPvqtzcV//Kase6ee2MmolhTQeaLujHHOsTTBHCHSUo0HxkCRDJzKyZ9kEC0ySllQvj8FP9Pam7eOc07N26mfLmII4kO0CHKIQeVUBldowqqIoLu0D16RE/W0Hqwnq2XeWvCWszsox+wXj8Al5yRVg== f(n, t) AAAB73icdVDJSgNBEO2JW4xb1KOXxihEkDAzJkZvQS8eI5gFkiH0dHqSJj09Y3eNEEJ+wosHRbz6O978GzuLuKAPCh7vVVFVz48F12Db71ZqYXFpeSW9mllb39jcym7v1HWUKMpqNBKRavpEM8ElqwEHwZqxYiT0BWv4g8uJ37hjSvNI3sAwZl5IepIHnBIwUvMgyMtjODroZHN2oVgqntsuNuTULRdLX8Qp2FPk0BzVTvat3Y1oEjIJVBCtW44dgzciCjgVbJxpJ5rFhA5Ij7UMlSRk2htN7x3jQ6N0cRApUxLwVP0+MSKh1sPQN50hgb7+7U3Ev7xWAsGZN+IyToBJOlsUJAJDhCfP4y5XjIIYGkKo4uZWTPtEEQomoowJ4fNT/D+puwXnpOBcu7nKxTyONNpD+yiPHFRGFXSFqqiGKBLoHj2iJ+vWerCerZdZa8qaz+yiH7BePwDKKo8o 

Nucleation Kinetics Master equation for the evolution of cluster: @f(n,t) @t AAACDnicdVBLSwMxGMzWV62vqkcvwbZQQcpuW2x7K3jxWMG2Qncp2TTbhmazS5IVyrK/wIt/xYsHRbx69ua/MdtWfKADgWHm+5LMuCGjUpnmu5FZWV1b38hu5ra2d3b38vsHPRlEApMuDlggrl0kCaOcdBVVjFyHgiDfZaTvTs9Tv39DhKQBv1KzkDg+GnPqUYyUlob5UtH2BMKxHSKhKGLQK/NTdZJ8CSqBxWG+YFZqjVa9dQY1aTaqdXNBzEYLWhVzjgJYojPMv9mjAEc+4QozJOXAMkPlxOmVmJEkZ0eShAhP0ZgMNOXIJ9KJ53ESWNLKCHqB0IcrOFe/b8TIl3Lmu3rSR2oif3up+Jc3iJTXdGLKw0gRjhcPeZGOGMC0GziigmDFZpogLKj+K8QTpOtRusGcLuEzKfyf9KoVq1axLquFdn1ZRxYcgWNQBhZogDa4AB3QBRjcgnvwCJ6MO+PBeDZeFqMZY7lzCH7AeP0AnDObyw== = (n 1, t)f(n 1, t) ↵(n, t)f(n, t) (n, t)f(n, t) + ↵(n + 1, t)f(n + 1, t) 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 Cluster of size n-1 Cluster of size n Cluster of size n+1 (n 1, t) AAAB9XicdVDLSgNBEJyNrxhfUY9eBhMhgobdJJjkFvDiMYJ5QBLD7GQ2GTI7u8z0KmHJf3jxoIhX/8Wbf+PkIahoQUNR1U13lxsKrsG2P6zEyura+kZyM7W1vbO7l94/aOogUpQ1aCAC1XaJZoJL1gAOgrVDxYjvCtZyx5czv3XHlOaBvIFJyHo+GUrucUrASLfZrsuA5OS5cwan2X46Y+eL5WqpeoENqZQLJXtB7HIVO3l7jgxaot5Pv3cHAY18JoEKonXHsUPoxUQBp4JNU91Is5DQMRmyjqGS+Ez34vnVU3xilAH2AmVKAp6r3ydi4ms98V3T6RMY6d/eTPzL60TgVXoxl2EETNLFIi8SGAI8iwAPuGIUxMQQQhU3t2I6IopQMEGlTAhfn+L/SbOQd4p557qQqZWWcSTRETpGOeSgMqqhK1RHDUSRQg/oCT1b99aj9WK9LloT1nLmEP2A9fYJv/uRVg== (n, t) AAAB83icdVDLSgNBEJz1GeMr6tHLYCJEkLCbBDe5Bbx4jGAekF3C7GSSDJmdXWZ6hbDkN7x4UMSrP+PNv3HyEFS0oKGo6qa7K4gF12DbH9ba+sbm1nZmJ7u7t39wmDs6busoUZS1aCQi1Q2IZoJL1gIOgnVjxUgYCNYJJtdzv3PPlOaRvINpzPyQjCQfckrASF7BCxiQoryEi0I/l7dLFbderV9hQ2puuWovie3WsVOyF8ijFZr93Ls3iGgSMglUEK17jh2DnxIFnAo2y3qJZjGhEzJiPUMlCZn208XNM3xulAEeRsqUBLxQv0+kJNR6GgamMyQw1r+9ufiX10tgWPNTLuMEmKTLRcNEYIjwPAA84IpREFNDCFXc3IrpmChCwcSUNSF8fYr/J+1yyamUnNtyvlFdxZFBp+gMFZGDXNRAN6iJWoiiGD2gJ/RsJdaj9WK9LlvXrNXMCfoB6+0T4HWQ5A== ↵(n, t) AAAB9HicdVBdSwJBFJ21L7Mvq8dehjQwiGVXJfVN6KVHgzRBF5kdZ3VwdnabuSuI+Dt66aGIXvsxvfVvGj+Cijpw4XDOvdx7jx8LrsFxPqzU2vrG5lZ6O7Ozu7d/kD08aukoUZQ1aSQi1faJZoJL1gQOgrVjxUjoC3bnj67m/t2YKc0jeQuTmHkhGUgecErASF6+S0Q8JAV5Aef5Xjbn2KVKrVy7xIZUK8WysyROpYZd21kgh1Zo9LLv3X5Ek5BJoIJo3XGdGLwpUcCpYLNMN9EsJnREBqxjqCQh0950cfQMnxmlj4NImZKAF+r3iSkJtZ6EvukMCQz1b28u/uV1Egiq3pTLOAEm6XJRkAgMEZ4ngPtcMQpiYgihiptbMR0SRSiYnDImhK9P8f+kVbTdku3eFHP18iqONDpBp6iAXFRBdXSNGqiJKLpHD+gJPVtj69F6sV6XrSlrNXOMfsB6+wSqzpFY ↵(n + 1, t) AAAB+XicdVDLSgNBEJz1GeNr1aOXwUSIKGE3CW5yC3jxGME8IAlhdjJJhszOLjO9gbDkT7x4UMSrf+LNv3HyEFS0oKGo6qa7y48E1+A4H9ba+sbm1nZqJ727t39waB8dN3QYK8rqNBShavlEM8ElqwMHwVqRYiTwBWv645u535wwpXko72EasW5AhpIPOCVgpJ5tZztERCOSk/jSvYKLbM/OOPmiVylVrrEhZa9QcpbE8SrYzTsLZNAKtZ793umHNA6YBCqI1m3XiaCbEAWcCjZLd2LNIkLHZMjahkoSMN1NFpfP8LlR+ngQKlMS8EL9PpGQQOtp4JvOgMBI//bm4l9eO4ZBuZtwGcXAJF0uGsQCQ4jnMeA+V4yCmBpCqOLmVkxHRBEKJqy0CeHrU/w/aRTybjHv3hUy1dIqjhQ6RWcoh1zkoSq6RTVURxRN0AN6Qs9WYj1aL9brsnXNWs2coB+w3j4BWVmSIw== Rate at which n-clusters become (n+1)-clusters (Nucleation rate) per unit volume: J(n, t) = (n, t)f(n, t) ↵(n + 1, t)f(n + 1, t) 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 @f(n,t) @t = J(n 1, t) J(n, t) 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 Steady state nucleation rate: Does not depend on time and size of the cluster Steady state: J(n,t) = J 

Nucleation Kinetics: Kinetic Theory Condensation rate can be evaluated using kinetic theory: (n) = ⌫A(n) AAACAHicdZDLSsNAFIYn9VbrLerChZvBVqibkLTFtAuh4sZlBXuBppTJdNIOnUzCzEQopRtfxY0LRdz6GO58GydtBRX9YeDjP+dw5vx+zKhUtv1hZFZW19Y3spu5re2d3T1z/6Alo0Rg0sQRi0THR5IwyklTUcVIJxYEhT4jbX98ldbbd0RIGvFbNYlJL0RDTgOKkdJW3zwqeD5RqMjP4AX0eAIvU4SFvpm3rbJbq9TOoYaqW6rYC7DdGnQse648WKrRN9+9QYSTkHCFGZKy69ix6k2RUBQzMst5iSQxwmM0JF2NHIVE9qbzA2bwVDsDGERCP67g3P0+MUWhlJPQ150hUiP5u5aaf9W6iQqqvSnlcaIIx4tFQcKgimCaBhxQQbBiEw0IC6r/CvEICYSVziynQ/i6FP4PrZLllC3nppSvV5ZxZMExOAFF4AAX1ME1aIAmwGAGHsATeDbujUfjxXhdtGaM5cwh+CHj7RPpopP/ $\nu$ (impingement rate) is the number of gas molecules hitting the unit area in unit time and A(n) is the surface area of a cluster containing n molecules Cluster of size n 4/3⇡R3 = n4/3⇡r3 AAACB3icdVDLSgMxFM3UV62vUZeCBFvBVZ3pDE67EApuXFaxD2hryaRpG5rJDElGKEN3bvwVNy4UcesvuPNvTF+gogcC555zLzf3+BGjUlnWp5FaWl5ZXUuvZzY2t7Z3zN29mgxjgUkVhywUDR9JwignVUUVI41IEBT4jNT94cXEr98RIWnIb9QoIu0A9TntUYyUljrmYc49dWArovD61oHnkMNFLXSd65hZK+94Jbd0BjUpegXXmhHLK0E7b02RBXNUOuZHqxviOCBcYYakbNpWpNoJEopiRsaZVixJhPAQ9UlTU44CItvJ9I4xPNZKF/ZCoR9XcKp+n0hQIOUo8HVngNRA/vYm4l9eM1a9YjuhPIoV4Xi2qBczqEI4CQV2qSBYsZEmCAuq/wrxAAmElY4uo0NYXAr/J7VC3nby9lUhW3bncaTBATgCJ8AGHiiDS1ABVYDBPXgEz+DFeDCejFfjbdaaMuYz++AHjPcvERyVmQ== R = rn1/3 AAAB+HicdVDLSsNAFJ3UV62PRl26GWwFVzFpi2kXQsGNyyr2AW0sk+mkHTqZhJmJUEO/xI0LRdz6Ke78G6cPQUUPXDiccy/33uPHjEpl2x9GZmV1bX0ju5nb2t7ZzZt7+y0ZJQKTJo5YJDo+koRRTpqKKkY6sSAo9Blp++OLmd++I0LSiN+oSUy8EA05DShGSkt9M1+8hudQQH6bOqflabFvFmyr7NYqtTOoSdUtVewFsd0adCx7jgJYotE333uDCCch4QozJGXXsWPlpUgoihmZ5nqJJDHCYzQkXU05Con00vnhU3islQEMIqGLKzhXv0+kKJRyEvq6M0RqJH97M/Evr5uooOqllMeJIhwvFgUJgyqCsxTggAqCFZtogrCg+laIR0ggrHRWOR3C16fwf9IqWU7Zcq5KhXplGUcWHIIjcAIc4II6uAQN0AQYJOABPIFn4954NF6M10VrxljOHIAfMN4+AV9PkZQ= A(n) = 4⇡R2 = 4⇡r2n2/3 = s1n2/3 AAACHHicdZDLSgMxFIYz9VbrrerSTbAKdVPnUmy7ECpuXFaxF+iNTJq2oZnMkGSEMvRB3Pgqblwo4saF4NuY6QVU9IfAl/+cQ3J+N2BUKtP8NBJLyyura8n11Mbm1vZOenevJv1QYFLFPvNFw0WSMMpJVVHFSCMQBHkuI3V3dBnX63dESOrzWzUOSNtDA077FCOlrW7aObrI8hN4DvOtgMKbjh0jjFlo5p3IPnUm2pNda3E76qYzZs4plPKlM6ihWLDz5gzMQglaOXOqDJir0k2/t3o+Dj3CFWZIyqZlBqodIaEoZmSSaoWSBAiP0IA0NXLkEdmOpstN4LF2erDvC324glP3+0SEPCnHnqs7PaSG8nctNv+qNUPVL7YjyoNQEY5nD/VDBpUP46RgjwqCFRtrQFhQ/VeIh0ggrHSeKR3CYlP4P9TsnOXkrGs7U87P40iCA3AIssACBVAGV6ACqgCDe/AInsGL8WA8Ga/G26w1Ycxn9sEPGR9foJmcxA== u Cylinder with unit cross-sectional area and height u ⌫ = R 1 0 u.1.(N/V ). (u)du AAACF3icdVBNSysxFM348dTqe1ZdurlYhbqZN1OLtQtBcONKFGwVOrVk0owNZjJDciOU4r9w419x40IRt7rz35h+CO/JewdCDufcQ3JPnEthMAg+vKnpmdkfc/MLhcWln7+WiyurTZNZzXiDZTLTFzE1XArFGyhQ8otcc5rGkp/H14dD//yGayMydYb9nLdTeqVEIhhFJ3WK/makLOxDJBR2gkt3JdgH60PoQ/n4d3PbhyjvibLdhq6FzU6xFPg7tXq1vguO7NUq1WBMglrdZYIRSmSCk07xPepmzKZcIZPUmFYY5NgeUI2CSX5biKzhOWXX9Iq3HFU05aY9GO11C1tO6UKSaXcUwkj9MzGgqTH9NHaTKcWe+e4NxX95LYvJXnsgVG6RKzZ+KLESMINhSdAVmjOUfUco08L9FViPasrQVVlwJXxtCv8nzYof7vjhaaV0UJ3UMU/WyQYpk5DUyAE5IiekQRi5Iw/kiTx7996j9+K9jkenvElmjfwF7+0TUIecOA== Maxwell velocity distribution ⌫ = pv p 2⇡m1kBT 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 (n) = pv p 2⇡m1kBT s1n2/3 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 Evaporation rate is a priori not known 

Nucleation Kinetics: Constrained Equilibrium Constrained equilibrium: Assume there exists an equilibrium distribution of droplets in bulk metastable phase — transient period is smaller than the metastable lifetime feq (n) = Constrained equilibrium distribution of cluster of size n Also, J(n) = 0 for all n: J(n, t) = (n, t)f(n, t) (n) feq(n) feq(n+1) f(n + 1, t) 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 ↵(n + 1) = (n) feq(n) feq(n+1) 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 J(n, t) = (n, t)feq(n)[ f(n,t) feq(n) f(n+1,t) feq(n+1) ] 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 Summing over clusters of different sizes: PN n=1 J(n,t) (n,t)feq(n) = f(1,t) feq(1) f(N+1,t) feq(N+1) 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 Boundary condition: f(1,t) feq(1) ! 1 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 f(N+1,t) feq(N+1) ! 0 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 N ! 1 AAAB9XicdVBNS8NAEN3Ur1q/qh69LLaCp5K0xbS3ghdPUsHaQhPLZrtpl242YXeilNL/4cWDIl79L978N24/BBV9MPB4b4aZeUEiuAbb/rAyK6tr6xvZzdzW9s7uXn7/4EbHqaKsRWMRq05ANBNcshZwEKyTKEaiQLB2MDqf+e07pjSP5TWME+ZHZCB5yCkBI90WL7EHMfa4DGFc7OULdqni1qv1M2xIzS1X7QWx3Tp2SvYcBbREs5d/9/oxTSMmgQqiddexE/AnRAGngk1zXqpZQuiIDFjXUEkipv3J/OopPjFKH4exMiUBz9XvExMSaT2OAtMZERjq395M/MvrphDW/AmXSQpM0sWiMBXYPDqLAPe5YhTE2BBCFTe3YjokilAwQeVMCF+f4v/JTbnkVErOVbnQqC7jyKIjdIxOkYNc1EAXqIlaiCKFHtATerburUfrxXpdtGas5cwh+gHr7ROulJH0 as 

Nucleation Kinetics: Constrained Equilibrium J(n, t) = [ P1 n=1 1 (n,t)feq(n) ] 1 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 Now since we know condensation rate, to know the nucleation rate we need to know equilibrium cluster distribution Equilibrium cluster distribution: Probability of occurring an equilibrium cluster distribution p(n) = feq(n) f1 / exp[ W (n) kBT ] 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 W(n) is the minimum work required to form a n-molecule cluster feq(n) = f1 exp[ W (n) kBT ] 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 f1 is the concentration of single molecule clusters i.e., metastable bulk phase because for such clusters W(1) = 0 

Nucleation Kinetics: Summary up to now J(n, t) = [ P1 n=1 1 (n,t)feq(n) ] 1 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 feq(n) = f1 exp[ W (n) kBT ] 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 (n) = pv p 2⇡m1kBT s1n2/3 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 Nucleation rate: condensation rate: Equilibrium cluster distribution: How to find W(n)? 

Nucleation Kinetics: Thermodynamics Use thermodynamics to find W(n) Laws of thermodynamics: Gibbs-Duhem equation: Gibbs free energy: U(S, V, N) = TS + µN pV 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 dU = TdS + µdN pdV AAACAnicdVBNS0JBFJ3Xp9mX1SraXNIgiOQ9ldRFILRpFUY+FVRk3rxRB+d9MDMvkIe06a+0aVFE235Fu/5N40dQUQcuHM65l3vvcULOpDLND2NhcWl5ZTWxllzf2NzaTu3s1mUQCUJtEvBANB0sKWc+tRVTnDZDQbHncNpwhhcTv3FLhWSBX1OjkHY83PdZjxGstNRN7WdcG86h5t7ACbS9CNwrOIXQrWe6qbSZzRfLhfIZaFIq5grmjJjFMlhZc4o0mqPaTb233YBEHvUV4VjKlmWGqhNjoRjhdJxsR5KGmAxxn7Y09bFHZSeevjCGI6240AuELl/BVP0+EWNPypHn6E4Pq4H87U3Ev7xWpHqlTsz8MFLUJ7NFvYiDCmCSB7hMUKL4SBNMBNO3AhlggYnSqSV1CF+fwv+knsta+ax1nUtXCvM4EugAHaJjZKEiqqBLVEU2IugOPaAn9GzcG4/Gi/E6a10w5jN76AeMt0/5SJST 0 = SdT Ndµ + V dp AAACAXicdVDLSsNAFJ3UV62vqBvBzWArCNKStMW0C6HgxpVU7AvaUCaTSTt08mBmIpRQN/6KGxeKuPUv3Pk3Th+Cih64cDjnXu69x4kYFdIwPrTU0vLK6lp6PbOxubW9o+/utUQYc0yaOGQh7zhIEEYD0pRUMtKJOEG+w0jbGV1M/fYt4YKGQUOOI2L7aBBQj2IkldTXD3IGPIf5G7cB8/DK7fkxPIUtN8r19axRKFnVcvUMKlKximVjTgyrCs2CMUMWLFDv6+89N8SxTwKJGRKiaxqRtBPEJcWMTDK9WJAI4REakK6iAfKJsJPZBxN4rBQXeiFXFUg4U79PJMgXYuw7qtNHcih+e1PxL68bS69iJzSIYkkCPF/kxQzKEE7jgC7lBEs2VgRhTtWtEA8RR1iq0DIqhK9P4f+kVSyYpYJ5XczWyos40uAQHIETYAIL1MAlqIMmwOAOPIAn8Kzda4/ai/Y6b01pi5l98APa2ycKjZQN G(p, T, N) = U TS + pV = µN 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 dG = µdN SdT + V dp AAAB/nicdVDLSsNAFJ34rPUVFVduBltBEEPSFtMuhIILXUnFvqANZTKZtEMnD2YmQgkFf8WNC0Xc+h3u/BunD0FFD1w4nHMv997jxowKaZof2sLi0vLKamYtu76xubWt7+w2RZRwTBo4YhFvu0gQRkPSkFQy0o45QYHLSMsdXkz81h3hgkZhXY5i4gSoH1KfYiSV1NP3897lOewGCfSu4emtVz9penG+p+dMo2hXSpUzqEjZLpTMGTHtCrQMc4ocmKPW09+7XoSTgIQSMyRExzJj6aSIS4oZGWe7iSAxwkPUJx1FQxQQ4aTT88fwSCke9COuKpRwqn6fSFEgxChwVWeA5ED89ibiX14nkX7ZSWkYJ5KEeLbITxiUEZxkAT3KCZZspAjCnKpbIR4gjrBUiWVVCF+fwv9Js2BYRcO6KeSqpXkcGXAADsExsIANquAK1EADYJCCB/AEnrV77VF70V5nrQvafGYP/ID29gl63JPd 

Nucleation Kinetics: Assumptions of CNT Assumptions of classical nucleation theory: ➤ Using thermodynamics at the few molecule level ➤ Liquid phase (embryo) is homogeneous and spherical in shape ➤ The density of the liquid phase is same as bulk phase ➤ Capillarity approximation: The interface between the bulk phase and liquid phase is sharp ➤ No impact of curvature on surface tension ➤ Liquid is incompressible 

Nucleation Kinetics: Thermodynamics Consider hypothetical (vapor + liquid drop) system Supersturated liquid phase (l) lnterface (s) N = Nv + Nl + Ns AAAB/nicdVDLSsNAFJ3UV62vqLhyM9gKghCStph2IRTcuCoVrC20IUymk3bo5MHMpFBCwV9x40IRt36HO//GaVpBRQ/M5XDOvdw7x4sZFdI0P7Tcyura+kZ+s7C1vbO7p+8f3Iko4Zi0ccQi3vWQIIyGpC2pZKQbc4ICj5GON76a+50J4YJG4a2cxsQJ0DCkPsVIKsnVj0pNeAmb7gSeq8qyKkquXjSNil2v1i+gIjW7XDUXxLTr0DLMDEWwRMvV3/uDCCcBCSVmSIieZcbSSRGXFDMyK/QTQWKEx2hIeoqGKCDCSbPzZ/BUKQPoR1y9UMJM/T6RokCIaeCpzgDJkfjtzcW/vF4i/ZqT0jBOJAnxYpGfMCgjOM8CDignWLKpIghzqm6FeIQ4wlIlVlAhfP0U/k/uyoZVMaybcrFRXcaRB8fgBJwBC9igAa5BC7QBBil4AE/gWbvXHrUX7XXRmtOWM4fgB7S3T0KykxU= U = Uv + Ul + Us AAAB/nicdVDLSsNAFJ34rPUVFVduBltBEELSFtMuhIIblxVMW2hDmEwn7dDJg5lJoYSCv+LGhSJu/Q53/o3TtIKKHpjL4Zx7uXeOnzAqpGl+aCura+sbm4Wt4vbO7t6+fnDYFnHKMXFwzGLe9ZEgjEbEkVQy0k04QaHPSMcfX8/9zoRwQePoTk4T4oZoGNGAYiSV5OnHZQdeQcebwAtVWV5F2dNLplG1G7XGJVSkbldq5oKYdgNahpmjBJZoefp7fxDjNCSRxAwJ0bPMRLoZ4pJiRmbFfipIgvAYDUlP0QiFRLhZfv4MnillAIOYqxdJmKvfJzIUCjENfdUZIjkSv725+JfXS2VQdzMaJakkEV4sClIGZQznWcAB5QRLNlUEYU7VrRCPEEdYqsSKKoSvn8L/SbtiWFXDuq2UmrVlHAVwAk7BObCADZrgBrSAAzDIwAN4As/avfaovWivi9YVbTlzBH5Ae/sEbmSTMQ== Uv(r) = TvSv(r) + µvNv(r) pvVv(r) 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 Ul(r) = TlSl(r) + µlNl(r) plVl(r) 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 Us(r) = TsSs(r) + µsNs(r) + A(r) 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 W(n) = free energy of (vapor + liquid drop) - free energy of pure vapour G = (pv pl)vl + A + Nl[µl(pl) µv(pv)] + Ns[µs µv(pv)] 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 Equimolar interface: Incompressible liquid phase: G = (pv pl)vl + eAe + Nl[µl(pl) µv(pv)] 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 µl(pl) = µl(pv) + vl(pl pv) 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 G = eAe + Nl[µl(pv) µv(pv)] 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 G = µ n + eAe 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 

Nucleation Kinetics: Thermodynamics G(n) = µ n + eAe(n) 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 Ae(n) = s1n2/3 AAAB/3icdVDLSsNAFJ34rPVVFdy4GWyFuolJWky7ECpuXFawD2hrmEyn7dDJJMxMhBK78FfcuFDErb/hzr9x+hBU9MCFwzn3cu89fsSoVJb1YSwsLi2vrKbW0usbm1vbmZ3dugxjgUkNhywUTR9JwignNUUVI81IEBT4jDT84cXEb9wSIWnIr9UoIp0A9TntUYyUlrzMfu7cI3l+DM+g9GzIbxLnpDDOeZmsZRbccrF8CjUpuU7RmhHLLUPbtKbIgjmqXua93Q1xHBCuMENStmwrUp0ECUUxI+N0O5YkQniI+qSlKUcBkZ1kev8YHmmlC3uh0MUVnKrfJxIUSDkKfN0ZIDWQv72J+JfXilWv1Ekoj2JFOJ4t6sUMqhBOwoBdKghWbKQJwoLqWyEeIIGw0pGldQhfn8L/Sd0x7YJpXznZSnEeRwocgEOQBzZwQQVcgiqoAQzuwAN4As/GvfFovBivs9YFYz6zB37AePsElPaT3g== G(n) = µ n + es1n2/3 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 Critical cluster size: µ + 2 3 s1 e(n⇤) 1/3 = 0 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 n⇤ = 32⇡ 3 v2 l 3 µ3 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 G⇤ = G(n⇤) = 16⇡ 3 v2 l 3 µ2 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 Height of nucleation barrier: Surface term Volume term Surface + Volume n G AAAB8XicdVDJSgNBEO2JW4xb1KOXxkTwFGaioMeggh4jmAWTIfR0KkmTnp6hu0YIQ/7CiwdFvPo33vwbO4vg+qDpx3tVVNULYikMuu67k1lYXFpeya7m1tY3Nrfy2zt1EyWaQ41HMtLNgBmQQkENBUpoxhpYGEhoBMPzid+4A21EpG5wFIMfsr4SPcEZWum22L4AiYxeFjv5gls6diegv4lXmv5ugcxR7eTf2t2IJyEo5JIZ0/LcGP2UaRRcwjjXTgzEjA9ZH1qWKhaC8dPpxmN6YJUu7UXaPoV0qn7tSFlozCgMbGXIcGB+ehPxL6+VYO/UT4WKEwTFZ4N6iaQY0cn5tCs0cJQjSxjXwu5K+YBpxtGGlLMhfF5K/yf1csk7KnnX5ULlbB5HluyRfXJIPHJCKuSKVEmNcKLIPXkkT45xHpxn52VWmnHmPbvkG5zXDxgfj+A= 

Nucleation Kinetics: Nucleation rate J(n, t) = [ P1 n=1 1 (n,t)feq(n) ] 1 AAACLHicdVBNSxxBFOzR+JFN1NUcvTRZAyvoMrMurnsQBC8hJwNZFXbGoaf3jTb29Izdb4SlmR/kxb8SEA9KyDW/I70fASNJQUNRVY/Xr5JCCoO+/+zNzb9ZWFxaflt7935lda2+vnFq8lJz6PNc5vo8YQakUNBHgRLOCw0sSyScJdfHY//sFrQRufqGowKijF0qkQrO0Elx/XjrS1Pt4DY9pIPQlFls1WFQXYRCpTiiYaoZt0FlwwSQTYNpbOGmaqptWkUXdjeotuJ6w2/tdXud3j515KDb7vhT4nd7NGj5EzTIDCdx/SEc5rzMQCGXzJhB4BcYWaZRcAlVLSwNFIxfs0sYOKpYBiayk2Mr+skpQ5rm2j2FdKK+nLAsM2aUJS6ZMbwyr72x+C9vUGJ6EFmhihJB8emitJQUczpujg6FBo5y5AjjWri/Un7FXEHo+q25Ev5cSv9PTtutYK8VfG03jjqzOpbJJvlImiQgXXJEPpMT0iec3JHv5Ik8e/feo/fD+zmNznmzmQ/kL3i/fgMyT6Zg In continuum approximation: feq(n) = f1 exp[ W (n) kBT ] 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 (n) = pv p 2⇡m1kBT s1n2/3 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 J = f1 [ R 1 1 dn (n)exp( G kBT ) ] 1 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 Use steepest descent to solve the integral: G(n) ⇡ G(n⇤) + 1 2 d2 G(n) dn2 |n⇤ (n n⇤)2 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 J = f1 (n⇤)Ze G⇤ kBT 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 Z = q | G00 (n⇤)| 2⇡kBT 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 Zeldovich factor: n G AAAB8XicdVDJSgNBEO2JW4xb1KOXxkTwFGaioMeggh4jmAWTIfR0KkmTnp6hu0YIQ/7CiwdFvPo33vwbO4vg+qDpx3tVVNULYikMuu67k1lYXFpeya7m1tY3Nrfy2zt1EyWaQ41HMtLNgBmQQkENBUpoxhpYGEhoBMPzid+4A21EpG5wFIMfsr4SPcEZWum22L4AiYxeFjv5gls6diegv4lXmv5ugcxR7eTf2t2IJyEo5JIZ0/LcGP2UaRRcwjjXTgzEjA9ZH1qWKhaC8dPpxmN6YJUu7UXaPoV0qn7tSFlozCgMbGXIcGB+ehPxL6+VYO/UT4WKEwTFZ4N6iaQY0cn5tCs0cJQjSxjXwu5K+YBpxtGGlLMhfF5K/yf1csk7KnnX5ULlbB5HluyRfXJIPHJCKuSKVEmNcKLIPXkkT45xHpxn52VWmnHmPbvkG5zXDxgfj+A= n* 

Condensation of Ideal Vapor Gibbs-Duhem equation: 0 = SdT Ndµ + V dp AAACAXicdVDLSsNAFJ3UV62vqBvBzWArCNKStMW0C6HgxpVU7AvaUCaTSTt08mBmIpRQN/6KGxeKuPUv3Pk3Th+Cih64cDjnXu69x4kYFdIwPrTU0vLK6lp6PbOxubW9o+/utUQYc0yaOGQh7zhIEEYD0pRUMtKJOEG+w0jbGV1M/fYt4YKGQUOOI2L7aBBQj2IkldTXD3IGPIf5G7cB8/DK7fkxPIUtN8r19axRKFnVcvUMKlKximVjTgyrCs2CMUMWLFDv6+89N8SxTwKJGRKiaxqRtBPEJcWMTDK9WJAI4REakK6iAfKJsJPZBxN4rBQXeiFXFUg4U79PJMgXYuw7qtNHcih+e1PxL68bS69iJzSIYkkCPF/kxQzKEE7jgC7lBEs2VgRhTtWtEA8RR1iq0DIqhK9P4f+kVSyYpYJ5XczWyos40uAQHIETYAIL1MAlqIMmwOAOPIAn8Kzda4/ai/Y6b01pi5l98APa2ycKjZQN dµ = V N dp = vdp AAACBXicdVDLSsNAFJ34rPUVdamLwVZwVZK2mHZRKLhxJRXsA5pQJpNJO3TyYGZSKCEbN/6KGxeKuPUf3Pk3Th+Cih64cDjnXu69x40ZFdIwPrSV1bX1jc3cVn57Z3dvXz847Igo4Zi0ccQi3nORIIyGpC2pZKQXc4ICl5GuO76c+d0J4YJG4a2cxsQJ0DCkPsVIKmmgnxQ9O0hgA9o+RzjtZOl1Br24ASdeXBzoBaNUserV+gVUpGaVq8aCGFYdmiVjjgJYojXQ320vwklAQokZEqJvGrF0UsQlxYxkeTsRJEZ4jIakr2iIAiKcdP5FBs+U4kE/4qpCCefq94kUBUJMA1d1BkiOxG9vJv7l9RPp15yUhnEiSYgXi/yEQRnBWSTQo5xgyaaKIMypuhXiEVJxSBVcXoXw9Sn8n3TKJbNSMm/KhWZ1GUcOHINTcA5MYIEmuAIt0AYY3IEH8ASetXvtUXvRXhetK9py5gj8gPb2CRO3l6Q= dµv = kBT p dp AAACBXicdVDLSsNAFJ34rPVVdamLwVZwFZK22HYhFN24rNAXtCFMJpN26GQSZiaFErpx46+4caGIW//BnX/j9CGo6IELh3Pu5d57vJhRqSzrw1hZXVvf2MxsZbd3dvf2cweHbRklApMWjlgkuh6ShFFOWooqRrqxICj0GOl4o+uZ3xkTIWnEm2oSEydEA04DipHSkps7Kfj9MHHH8BL2A4FwOnKvmtM0nkI/Lri5vGWWKrVy7QJqUq0Uy9aCWJUatE1rjjxYouHm3vt+hJOQcIUZkrJnW7FyUiQUxYxMs/1EkhjhERqQnqYchUQ66fyLKTzTig+DSOjiCs7V7xMpCqWchJ7uDJEayt/eTPzL6yUqqDop5XGiCMeLRUHCoIrgLBLoU0GwYhNNEBZU3wrxEOkwlA4uq0P4+hT+T9pF0y6Z9m0xXy8v48iAY3AKzoENKqAObkADtAAGd+ABPIFn4954NF6M10XrirGcOQI/YLx9Ap/Gl/4= µl(pv) µl(psat) = vl(pv psat) 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 µv(psat) = µl(psat) AAACCnicdZDLSsNAFIYn9VbrLerSzWgr1E1J2mLahVBw47KCvUAbwmQ6aYdOLsxMCiV07cZXceNCEbc+gTvfxklbRUV/GPj5zjmcOb8bMSqkYbxrmZXVtfWN7GZua3tnd0/fP2iLMOaYtHDIQt51kSCMBqQlqWSkG3GCfJeRjju+TOudCeGChsGNnEbE9tEwoB7FSCrk6MeFvh87k2LkJALJ2Rm8gClgX6Dg6HmjVLHq1fo5VKZmlavGwhhWHZolY648WKrp6G/9QYhjnwQSMyREzzQiaSeIS4oZmeX6sSARwmM0JD1lA+QTYSfzU2bwVJEB9EKuXiDhnH6fSJAvxNR3VaeP5Ej8rqXwr1ovll7NTmgQxZIEeLHIixmUIUxzgQPKCZZsqgzCnKq/QjxCHGGp0supED4vhf+bdrlkVkrmdTnfqC7jyIIjcAKKwAQWaIAr0AQtgMEtuAeP4Em70x60Z+1l0ZrRljOH4Ie01w8ulZnp µv(pv) µv(psat) = kBT ln pv psat 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 µv(pv) µl(pv) = vl(pv psat) kBT ln Sp 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 µ ⇡ kBT ln Sp 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 G⇤ = 16⇡ 3 v2 l 3 (kBT ln Sp)2 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 n⇤ = 32⇡ 3 v2 l 3 (kBT ln Sp)3 AAACL3icdVBdSxtBFJ21ttW0trE++jIYC9aHsB/BmIeCqIiPSo0K2WS5O5mNQ2Zml5lZISz7j/riX/FFRBFf/RdOPgoq9sCFM+fcy9x74owzbVz31pn7MP/x0+eFxcqXr0vfvleXf5zqNFeEtknKU3Ueg6acSdo2zHB6nikKIub0LB7ujf2zS6o0S+WJGWW0K2AgWcIIGCtF1YN12Ss2S/wbh4kCUgR+mLGyCMrZ+zLiPT8cgBDQC8oi3KfcAN4YRrsnIZf4T5T9svp6VK259aDZarS2sCXbTb/hTonbbGGv7k5QQzMcRdXrsJ+SXFBpCAetO56bmW4ByjDCaVkJc00zIEMY0I6lEgTV3WJyb4l/WqWPk1TZkgZP1JcTBQitRyK2nQLMhX7rjcX3vE5uku1uwWSWGyrJ9KMk59ikeBwe7jNFieEjS4AoZnfF5AJsTsZGXLEh/LsU/5+c+nUvqHvHfm2nMYtjAa2iNbSBPNREO+gQHaE2IugvukZ36N65cm6cB+dx2jrnzGZW0Cs4T89k4KgL J = vl( p kBT )2 q 2 ⇡m exp( 16⇡ 3kBT v2 i 3 (kBT ln Sp)2 ) 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 Vapor as ideal gas: Incompressible liquid: At coexistence: Nucleation rate: