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Atacama Large Millimeter/Submillimeter Array (ALMA) has revolutionized thefield of dust polarization in protoplanetary disks across multiple wavelengths.Previous observations and empirical modeling suggested multiple mechanisms ofdust polarization toward HL Tau, including grain alignment and dust scattering.However, a detailed modeling of dust polarization based on grain alignmentphysics is not yet available. Here, using our updated POLARIS code, we performnumerical modeling of dust polarization arising from both grain alignment byMagnetically Enhanced Radiative Torque (MRAT) mechanism and self-scattering toreproduce the HL Tau polarization observed at three wavelengths 0.87, 1.3, and3.1$\,$mm. Our modeling results show that the observed multi-wavelengthpolarization could be reproduced only when large grains contain embedded ironinclusions and those with slow internal relaxation must have wrong internalalignment (i.e., the grain's major axis parallel to its angular momentum). Theabundance of iron embedded inside grains in the form of clusters is constrainedto be $\gtrsim 16$%, and the number of iron atoms per cluster is $N_{\rm cl}\sim 2\times10^3$. Maximum grain sizes probed at wavelengths $\lambda$ = 0.87,1.3, and 3.1$\,$mm are constrained at $\sim$ 60, 90, and 130$\,\mu$m,respectively. Assuming a dust differential settling effect with grain sizesfrom the constraint gives the value of maximum grain size at the disk mid-planeto be millimeter-scaled.

Evidence of grain alignment by magnetically enhanced radiative torques from multi-wavelength dust polarization modeling of HL Tau