![]() Based on molecular dynamics simulations, many models have been proposed to describe heterogeneous hydrogen bond networks to explain physical anomalous properties of liquid water, such as flickering models 23 and percolation models 19, 24. In addition, X-ray small-angle scattering 14, X-ray absorption 17 and emission spectroscopy 18 can provide evidence for inhomogeneous structures of liquid water. The presence of an isosbestic point at approximately 3450 cm −1 of the O–H stretching mode proves the coexistence of two types of local structures in liquid water by Raman spectroscopy 13. Density heterogeneity occurs on a length scale of 10–15 Å at ambient temperature 14, 15. The model of two-state water, which comes from the competition between a high-density liquid (HDL) of distorted hydrogen bond patches and a low-density liquid (LDL) of ordered locally tetrahedral patches, can reproduce the anomalous behavior of thermodynamic properties of water 6, 7, 22. The presence of density fluctuations in hydrogen bond networks in supercooled or liquid water has been widely investigated according to experiments 13, 14, 15, 16, 17, 18 and molecular dynamics 12, 19, 20, 21, 22. Based on hydrogen-bonded networks, liquid water is a mixture of two different types of molecular arrangements: tetrahedral-like and hydrogen-bond distorted structures related with low- and high-density patches 8, 9, 10, 11, 12, respectively. Water anomalies are strongly associated with locally favored structures corresponding to particular long-lived molecular arrangements with local minima of free energy 6, 7. However, at microscopic scale, frequent spatial–temporal fluctuations emerge in the hydrogen bond network with local tetrahedral symmetry, elucidating the main reasons for anomalous properties of water 5. At macroscopic temporal or spatial scale, liquid water can be regarded as a homogeneous substance. Different from simple liquids, liquid water shows numerous anomalous thermodynamic and kinetic behaviors, which are of significant importance to our planet and living systems 3, 4. Water is the most ubiquitous liquid involved in multifarious physical, chemical, and biological processes on earth 1, 2. In addition, the water clusters can elucidate structural and density fluctuations on different length scales in liquid water. Our results highlight the properties and changes of water clusters as the fundamental building blocks of hydrogen bond networks. The water molecules in the small clusters, inside which are the void regarded as hydrophobic objects, have a preference for being more tetrahedral. We show further that the interior of clusters favors high-density patches. Temperature can destroy large clusters into small ones. The local favored clusters and the preferred connections between adjacent clusters correspond to lower energy and conformational entropy depending on cluster topologies. The clusters imply fractal behaviors forming percolating networks and the morphologies of small and large clusters show different scaling rules. The water clusters can cover over 95% of hydrogen bond network, among which some clusters maximally encompass thousands of molecules extending beyond 3.0 nm. Here, we use molecular dynamics simulations of liquid water to study the properties of three-dimensional cage-like water clusters, which we investigate using extended graph-based hierarchical clustering methods. (Horizontal forces cancel.The microscopic structures of liquid water at ambient temperatures remain a hot debate, which relates with structural and density fluctuations in the hydrogen bond network. Their difference is the buoyant force \(F_B\). This pressure and associated upward force on the bottom of the cylinder are greater than the downward force on the top of the cylinder. \): Pressure due to the weight of a fluid increases with depth since \(P = h\rho g\).
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