|Left: The nodal connectivity map of the various schools of thermodynamics. The Gibbs correspondence table of letter and publication transmissions gives further detail behind these connections.  Right: |
The chemistry class of the Sheffield Scientific School at Yale, founded in 1847, the school, which incorporated both the sciences and the liberal arts, that in some way allowed Gibbs to make his roots and or produced Gibbs in some sense.
The following quote gives an in-context perspective explaining that an example of a pre-Gibbsian school is the Van’t Hoff school (Dutch school): 
“In the early years of this century the pre-Gibbsian (or van’t Hoff’s) school, with its emphasis on dilute and ideal systems, was running out of steam, and the more general examples of phase equilibrium were outside their scope.”
The Gibbs school, launched in 1873, the year of the publication of Gibbs “Graphical Methods in the Thermodynamics of Fluids”, is a precipitate, branch, or product, so to speak, of the Sheffield scientific school, a chemistry-engineer focused division of Yale University initiated in 1847. 
Gibbs entered Yale University at the age of 15 graduating, in 1858, at the age of 18.  He then entered the new Yale graduate school earning the first PhD in engineering in the United States, completed in 1863. Gibbs' PhD thesis was “On the Form of the Teeth of Wheels in Spur Gearing”.  In 1871, two years after returning from a study abroad at various universities in Europe, Gibbs became Yale's first professor of mathematical physics.
Gibbs had very few direct students, one being Mathematician Edwin Wilson, often is often said to have been Gibbs “sole protégé”, and Wilson’s famous student Economist Paul Samuelson, who used thermodynamics in economics. Most of Gibbs students were indirect students, via study of his work.
1. Kipnis, Aleksandr I., Iavelov, Boris, E.,, Rowlinson, John S. (1996). Van der Waals and Molecular Science (Gibbsian school, Van’t Hoff school, pg. 278). Oxford University Press.
2. (a) Pauling, Linus. (1970). “History of Chemical Thermodynamics”, in The Centennial of the Sheffield Scientific School (pgs. 27-32) by Baitsell, George A. and Lawrence, Ernest O. Ayer Publishing.
(b) Sheffield scientific school – Wikipedia.
3. Cropper, William H. (2004). Great Physicists: the Life and Times of Leading Physicists from Galileo to Hawking, (section II: Thermodynamics, pgs. 41-134; ch. 9: “The Greatest Simplicity: Willard Gibbs”, pgs 106-23). Oxford University Press.
4. Gibbs, Willard. (1863). “On the Form of the Teeth of Wheels in Spur Gearing” in Gibbs, W. and Wheeler, Lynde P. (1947). The Early Work of Willard Gibbs in Applied Mechanics. Schuman.
5. Wheeler, Lynde, P. (1951). Josiah Willard Gibbs - the History of a Great Mind (Appendix IV). Woodbridge, Connecticut: Ox Bow Press.