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I am writing an academic paper and I do not know whether my expression is formal. The full sentence is

There is a threshold (we call it take-off threshold) of ρc above which the proportion of the successful diffusion increases dramatically (from 0% all the way up to 100%) as ρc increases.

What I want to highlight is that the proportion explodes from 0% to 100% in a very short period. I do not know whether all the way up is a formal expression and whether it can highlight the explosion process.

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    "All the way (up, down, there, here, etc.)" is rather an emphasis that you completely reach some destination without stopping short; it's not really about the speed. For example, "all the way up to 100%" implies "not 98%, not 99%, not 99.5%, but (all the way up to)100%."
    – Brandin
    Aug 2, 2022 at 6:27
  • For a phrase which means to go up very quickly, I would look into "it shoots up". For example, it shoots up from 0% to 100%. However, this phrase does not sound very technical. In technical language, we normally say more explicitly what we mean, e.g. "it increases from 0% to 100% in a very short timespan."
    – Brandin
    Aug 2, 2022 at 6:28
  • Yeah, I understand, thanks Brandin.
    – Gao Roy
    Aug 2, 2022 at 7:44

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I think you're focusing on the wrong words to emphasize your point. It seems as though it's the speed at which it reaches 100% that's more important than how large the range is from 0 to 100.

I would keep it simple and say 0% to 100%.

Then, say what you mean and use the words from your explanation. Consider changing "dramatically" to "rapidly" (or "almost instantly"?), or use the word "explodes" as you did in the explanation.

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