Your question about Vonnegut's Cat's Cradle is a little vague, in that I don't know if you mean actual advancements in science or advancements in science in the setting of the novel.
The work uses the day the atomic bomb was dropped on Hiroshima as a backdrop. The bomb was certainly an actual development. You can find quotes about that in paragraphs 5-7 on the first page of the novel.
If you mean advancements in the novel, anything referring to ice-nine would fit your assignment. You could look at chapter 20, for instance: it's titled "Ice-nine." Chapter 115 also deals with ice-nine, and includes when the disaster caused by ice-nine occurs.
Chapter 29 is one of the great chapters of the novel. It's the episode that reveals the phallic symbol at the graveyard--the phallic symbol, on the mother's grave. That should help you with a quote about life without a strong father figure.