2. Given that the modulus of the complex number $z$ is 1, if $z=z_{1}$ and $z=z_{2}$ when $|z+1+\mathrm{i}|$ respectively attain the maximum and minimum values, then $z_{1}-z_{2}=$ $\qquad$ .
The point $Z$ corresponding to the complex number $z$ lies on the unit circle, and $|z+1+\mathrm{i}|$ represents the distance from point $Z$ to the point $(-1,-1)$. It is easy to calculate that $z_{1}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} \mathrm{i}, z_{2}=-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2} \mathrm{i} \Rightarrow z_{1}-z_{2}=\sqrt{2}+\sqrt{2} \mathrm{i}$.