Not in current versions. It used to be the case that min server memory was allocated on startup, but that hasn't been the case for a long time.
SQL will take what it needs, if it needs more and hasn't reached max server memory yet, it will allocate more. It will aggressively ramp up the buffer pool initially, but it won't grab more than it needs.
That said, there's more that needs memory than just the data cache.

The only time it pre-allocates is when large pages are used, as memory allocated that way cannot be resized after allocation.