Since $\overrightarrow {a}$ is perpendicular to ($\overrightarrow {a}$ + $\overrightarrow {b}$), we have $\overrightarrow {a} \cdot (\overrightarrow {a} + \overrightarrow {b}) = \overrightarrow {a}^2 + \overrightarrow {a} \cdot \overrightarrow {b} = 0$.

Let the size of the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$ be $\theta$. Then, from the problem, we get $1 + \sqrt{2}\cos\theta = 0$.

Solving for $\cos\theta$, we get $\cos\theta = -\frac{\sqrt{2}}{2}$.

Given that $0 \leq \theta < \pi$, we find $\theta = \frac{3\pi}{4}$.

Therefore, the answer is $\boxed{\frac{3\pi}{4}}$.

This problem primarily tests the understanding of the definition of the dot product of two vectors, the property of perpendicular vectors, and finding an angle based on the value of a trigonometric function. It is a basic problem.