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I see how bulky bases influence the regioselectivity of E2 and E1 reactions. I do not see very much content on the involvement of bulky bases in substitution reactions.

I know that bulky bases like t-BuOK are generally not involved in substitution reactions because the t-BuO ion would prefer to take a β-hydrogen instead. Are any substitution reactions possible, and if so, what mechanism would be seen in substitution reactions involving bulky bases?

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    $\begingroup$ The clue is in your title - "bulky bases" . Steric hindrance is a very real effect $\endgroup$
    – Waylander
    Nov 3 '18 at 17:03
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It is very tough, but cases of t-butoxide acting as a nucleophile are known, e.g. J. Org. Chem. 2011, 76 (24), 10129–10139:

n-Butyl-tert-butylether. To a stirred solution of 10 g of potassium tert-butoxide (89 mmol) in 100 mL of anhydrous dimethylsulfoxide was added 12.1 g of n-butylbromide (89 mmol) dropwise at 10 °C. The reaction mixture was allowed to warm and stand under stirring for 8 h at room temperature. The mixture was distilled under vacuum at −20 °C. The crude was redistilled at normal pressure to collect 6 g (52%) of the fraction distilled at 75−78 °C as a colorless liquid.

Mechanistically, I don't see any other real possibility by which this occurs apart from an SN2 reaction. Of course, there will be significant amounts of competing elimination, and that is probably why the yield is not amazing.

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