Recent fully atomistic molecular dynamics (MD) simulation of polymer nanocomposites based on polylactic acid (PLA) filled with cellulose nanocrystals (CNC) covalently modified by grafting of lactic acid oligomers (OLA) [AD. Glova et al. Soft Matter 13 (2017) 6627] have revealed a non-trivial structure of the OLA brush. It was found that a segregation of grafted OLA chains into two groups, or populations, occurs due to the presence of partial charges so that a fraction of chains folds back to the surface and acquires a hairpin-like conformation. Motivated by these results, we study the structure of a brush made of polymers with dipole moments oriented along the main chain (dipolar chains of A type) by using the numerical self-consistent field method for lattice models. In full agreement with the results of MD simulation study, the segregation of brush chains in two populations is observed as the strength of dipole-dipole interactions increases: A part of the brush chains folds back to form hairpin-like conformation while the remaining chains are extended and directed towards the brush periphery. An increase of the strength of dipole-dipole interactions and/or grafting density leads to the increase in the fraction of backfolded chains. We develop an analytical mean-field theory for the brush in the "dry" state which demonstrates the thermodynamic advantage of backfolding for a fraction of the brush chains. We find that the key factor leading to the observed non-trivial brush structure is the grafting of the chains in the brush via the same end. In contrast, in the case of the brush with the equal amount of chains grafted via each of the two ends, the dipole-dipole interactions do not result in the backfolding. This prediction for the lattice model is confirmed by additional fully atomistic molecular dynamics simulations of the corresponding OLA brush grafted onto CNC surface and immersed in the PLA melt. (C) 2018 Elsevier Ltd. All rights reserved.