Tetrahedral Order of Water. Jonathan Lam Chun Cheong September 16, 2014 Science Center G26

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1 Tetrahedral Order of Water Jonathan Lam Chun Cheong September 16, 2014 Science Center G26

2 Simple Quiz What happens to its volume when a piece of solid melts? A) Expands B) Contracts C) Remains unchange

3 Intuitive Answer Liquid contracts in volume on freezing and expands on melting Molecules are fixed in positions within solid but require more space to move around within liquid Quite true in general: e.g. ethanol, argon, carbon dioxide...etc Equivalent to a positive slope on phase diagram along the fusion curve (cf. ClausiusClapeyron equation)

4 Anomalies of Liquid Water

5 What makes water special? Partial charges, polar molecule Bond angle = degrees (cf degrees in a perfect tetrahedron) The structure of ice is of low packing efficiency, all water molecules are parts of a giant tetrahedrallyoriented hydrogen bonds network This structure weakens when ice melts to water, thus it contracts

6 Hydrogen bonds and tetrahedral network

7 What distinguishes phases? Two phases of a ferromagnet Three phases of a liquid crystal Two phases of water

8 Order parameter An order parameter is a number that indicate in which phase you are It quantifies the order, thus allows a physical model Quantifying the tetrahedrality of water is an interesting and important step toward understanding of water There exists some ways to quantify the tetrahedrality of a system, not perfect

9 A example of order parameter: magnetization <M> = 0 <M> = 1 <M(0)M(r)> B What is the significance of the correlation length? A r

10 Molecular Dynamics Simulations LAMMPS, short for Large-scale Atomic/Molecular Massively Parallel Simulator Open source Let s see some video! (if we have time)

11 Tetrahedral order parameter The existing tetrahedral order parameter is a simple scalar 1 where θαβ is the angle formed by the lines joining the oxygen atom of a given molecule and those of its 4 nearest neighbors α and β. <q> = 0 for completely random configuration; <q> = 1 for perfectly tetrahedral θ12 θ

12 Spatial Correlation Spatial autocorrelation function A(r) is a good starting point to study the order of a system Here, we have two boring autocorrelation functions of ice and water

13 A good tetrahedral order parameters The search of any tetrahedral order parameter was guided by the following idea: It should not change under relabeling the four neighbors It should give information regarding the orientation of the tetrahedron It should respect the tetrahedral symmetry if the tetrahedron is a perfect one

14 Proposed tensorial order parameters For any oxygen atom, let x1, x2, x3, and x4 be the position vectors of the four nearest oxygen neighbors with respect to the chosen oxygen atom, we define where denote tensor product and k is the rank of the tensor. For example, x1 x2 x3 x4 or

15 Benefits of using a tensor Construct invariants from contraction of indices (e.g. the length of a vector) Carry out dot product Definitely carries more information than a scalar

16 Spatial autocorrelation function of the rank 3 tensor

17 Physical significances Rank 1: center of mass Rank 2: moment of inertia Rank 3 and higher: higher moments Analysis show the tensor contractions are the linear combinations of magnitude of the Spherical Harmonics when the mass density function is projected on the Spherical Harmonics where Invariant under rotation (cf. magnitude of the Fourier coefficients) Another Invariant under rotation (cf. dot product)

18 Physical significances It can be shown that where dl some constants. To my surprise, the tensor eventually is related the spherical harmonics

19 Conclusion A tiny step toward building a complete theory of water, solvation...etc In this stage, can model ice in a perturbative manner (e.g. Expresses the Taylor expansion of potential energy in terms or those tensors) Not a perfect order parameter, yet the study of hint gave us some hint on how to look for a better one.

20 Acknowledgement I would like to express my special thanks of gratitude to my supervisors Prof. Phillip Geisseler and Patrick Shaffer in UC Berkeley. They have provided me numerous guidances throughout the project. Also, I would like thank again Prof. Chu and the OPUS@CUPHYSICS program which gave me the golden opportunity to visit UC Berkeley as an exchange student, without the program this project would not possible.

21 What distinguishes phases? Can a vector do the job? We need a two-ends vector, so that it gives the same number up to a 180 degrees flip A tensor will the do the job Three phases of a liquid crystal

22 Tensor order parameter of liquid crystal Want a quantity that encodes the information of the unit vector n Yet does not distinguish n and -n