In the Black-Scholes-Merton model, with model option price $V$ as a function of underlying price $S_t$, strike price $X$, risk-free rate of return $r$, time-to-maturity (in year fractions) $\tau$ and implied volatility $\sigma$, our $\Delta$ is defined as 

$$
\Delta\equiv \frac{\partial V}{\partial S_t}=\mathrm{N}\left(d_1\right)
$$
with 
$$
d_1\equiv \frac{\ln S- \ln X +(r+\frac{1}{2}\sigma^2)\tau }{\sigma \sqrt{\tau}}
$$

Let $B\equiv Xe^{-r\tau}$ the discounted strike, then 

$$
\begin{align}
\frac{\partial \Delta}{\partial \sigma}&=\mathrm{n}\left(d_1\right)\left(\frac{\partial d_1}{\partial \sigma}\right)\\
&=\mathrm{n}\left(d_1\right)\left(\frac{1}{2}\sqrt{\tau}-\frac{\ln S - \ln B }{\sigma^2\sqrt{\tau}}\right)
\end{align}
$$

Clearly, this expression can be positive / negative depending on whether 

$$
\frac{1}{2}\sigma^2\tau \lessgtr\ln S - \ln B
$$
i.e. whehther the (logarithmic) moneyness is within 1/2 of the term variance. HTH?