Completions of Kleene's second model

We investigate completions of partial combinatory algebras (pcas), in particular of Kleene's second model $\mathcal{K}_2$ and generalizations thereof. We consider weak and strong notions of embeddability and completion that have been studied before. By a result of Klop it is known that not every pca has a strong completion. The study of completions of $\mathcal{K}_2$ has as corollaries that weak and strong embeddings are different, and that every countable pca has a weak completion. We then consider generalizations of $\mathcal{K}_2$ for larger cardinals, and use these to show that it is consistent that every pca has a weak completion.