Sum of Square Roots

What is the minimum nonzero difference between two sums of square roots of integers? More precisely, find tight upper and lower bounds on $r(n,k)$, the minimum positive value of

$$\left| \sum_{i=1}^k \sqrt{a_i} - \sum_{i=1}^k \sqrt{b_i} \right|$$

where $a_i$ and $b_i$ are integers no larger than $n$. Bounds should be expressed as a function of $n$ and $k$. Examples:

$$r(20,2) \approx .0002 = \sqrt{10}+\sqrt{11}-\sqrt{5}-\sqrt{18}$$ $$r(20,3) \approx .000005 = \sqrt{5}+\sqrt{6}+\sqrt{18}-\sqrt{4}-\sqrt{12}-\sqrt{12}$$