Dust grains are aligned with the interstellar magnetic field and drift through the interstellar medium (ISM). Evolution of interstellar dust is driven by grain motion. In this paper, we study the effect of grain alignment with magnetic fields and grain motion on grain growth in molecular clouds. We first discuss characteristic timescales of internal alignment (i.e., alignment of the grain axis with its angular momentum, ${bf J}$) and external alignment (i.e., alignment of ${bf J}$ with the magnetic field) and find the range of grain sizes that have efficient alignment. Then, we study grain growth for such aligned grains drifting though the gas. Due to the motion of aligned grains along the magnetic field, gas accretion would increase the grain elongation rather than decrease, as in the case of random orientation. Grain coagulation also gradually increases grain elongation, leading to the increase of elongation with the grain size. The coagulation of aligned grains can form dust aggregates that contain the elongated binaries comprising a pair of grains with parallel short axes. The presence of superparamagnetic iron clusters within dust grains enhances internal alignment and thus increases the maximum size of aligned grains from $sim 2$ to $sim 10mu m$ for dense clouds of $n_{rm H}sim 10^{5}rm cm^{-3}$. Determining the size of such aligned grains with parallel axes within a dust aggregate would be important to constrain the location of grain growth and the level of iron inclusions. We find that grains within dust aggregates in 67P/Churyumov-Gerasimenko obtained by {it Rosetta} have the grain elongation increasing with the grain radius, which is not expected from coagulation by Brownian motion but consistent with the grain growth from aligned grains.
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