A model of dipolar dimer liquid (DDL) on a two-dimensional lattice has been proposed. We found that at high density and low temperature, it has a partially ordered phase which we called glacia phase. The glacia phase transition can be understood by mapping the DDL to an annealed Ising model on random graphs. In the high density limit the critical temperature obtained by the Monte Carlo simulation is $k_BT_c^G = (3.5pm0.1)J$, which agrees with the estimations of the upper and lower bounds of $k_BT_c^G$ with exactly solved Ising models. In the high density and low temperature limit, we further studied configurational entropy of the DDL in the presence of the neutral polymers. The suppression of the configurational entropy scales as a power law of the polymer length $lambda_p^alpha$ with $alpha geq 1$, which implies that the configurational entropy of water plays essential roles in understanding the hydrophobic effect and the protein folding problem.
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