1. A Discrete Mathematical Modeling of Constructing Self- assembly Monolayers Hung-Lin Fu ( 傅 恆 霖 ) Department of Applied Mathematics National Chiao Tung University ( 交大應數系 )

2. The Cyclical Nature of Mathematical Modeling Mathematical Model Mathematical Prediction Mathematical Model Real World Data Real World Predictions Interpretation Translation Testing Prediction

3. Discrete Mathematical Modeling  Use Discrete Mathematics as tools.  What is DM? The mathematics which is not continuous!  For examples: Graph Theory, Discrete Algorithms, Combinatorial Designs, Coding Theory, Group Testing, Enumeration and Discrete Probability.

4. Continuous Models  A mathematical model which describes a phenomenon by using a continuous set up.  For examples, the orbit of a satellite, the curvature of a car in making a sharp turn, the velocity of blood in a vessel and the diffusion of oil leaking from a sinking ship.

5. Self-assembled Monolayer  Self-assembled monolayers (SAMs) are surfaces consisting of a single layer of molecules on a substrate.  A common example is an alkane thoid on gold.  They can be used for simulation of biological membranes and as substrates for cell culture.

6. Why we need an SAM?  Is Bio the future?  How can we take the advantage of biological technology?  Mathematical modeling is still the most important tool to help out!  Create for: The future!

7. chip/semi-conductor Industrialization (drug discovery, diagnotics, etc) ‧‧‧‧‧‧ ‧‧‧‧‧‧ ‧‧‧‧‧ ‧‧‧‧‧‧‧‧‧‧ cell 1990~2007 Life Science electronics protein Bio sensing devices Bio-chip General Vision SAMs Funcational materials Chip coating applications SAMs: Self-Assembled Monolayers outstanding (manufacturing; world leading) DRAM (Korea); CPU (US); logitics (Taiwan) bacteria

8. How to obtain a self-assembled monolayer?  The easiest way: Adding a solution of the desired molecule onto the substrate surface and washing off the excess.  What if we need to use “ Chemical Vapor Deposition ” ? We know that it has poor control over the thickness, i.e., it is multi- layers somewhere and somewhere still vacant.  This is why we need “ Research ” !

9. Measurement of Monolayers  First, we have to know the size of chemical molecules we plan to depose on the substrate.  Currently, we may use AFM (Atomic Force Microscopy) or Contact Angles or “ Imagination ” to check what we have on the substrate. (Also, others.)

10. Diffusion distance Static contact angle measurement Surface morphology (AFM measurement)

11. We may use root mean square to check the roughness. Of course, the mean value shows the average height.

12. Contact angle measurement

13. Diffusion distance X Y Contact angle measurement AFM measurement 1 2 3 12 3

14. Diffusion makes it more interesting  The most well-known model for reaction-diffusion is the Fisher ’ s Equation. (This equation was originally derived for the simulation of propagation of a gene in a population.)  In mathematics, Fisher's equation, also known as the Fisher-Kolmogorov equation, named after R. A. Fisher and A. N. Kolmogorov, is the partial differential equation (n  2)R. A. FisherA. N. Kolmogorovpartial differential equation

15. Solve the equation!

16. How to make it discrete?  Substrate becomes a grid. Normally we set up 200x400 grid points. ( 解析度 )

17. The Molecules come!  From left hand side.  Black cell represents an existent molecule.

18. Cellular Automaton  A cellular automaton is a discrete model studied in computability theory, mathematics and theoretical biology.  The state of a cell at time t is a function of the states of a finite number of cells (called its neighborhood) at time t-1.  Every cell has the same value for updating, based on the values in this neighborhood.  Each time the rules are applied to the whole grid a new generation is created.

19. Game of Life AI EL V In the area of nine positions as in the following only certain amount of people can be alive at the same time, say at most 5. Too crowed!

20. Another Example  We use 1 for black cell and 0 for white cell.  Rule: 000, 001, 010, 011, 100, 101, 110, 111 a b c d e f g h d f The vector (a,b,c,d,e,f,g,h) provides a rule. Therefore, there are 256 possible rules.

21. A New Kind of Science Cellular Automata can be utilized to tackle a remarkable array of fundamental problems in science, from the origins of apparent randomness in physical systems, to the development of complexity in biology, the ultimate scope and limitations of mathematics, the possibility of a truly fundamental theory of physics, the interplay between free will and determinism, and the character of intelligence in the universe.

22. Probability works (Example) 0.50.250.125 … So, it is easy to see that a molecule falls in left cells has higher probability. This phenomenon models the vapor deposition of molecules from the left hand side. The ratio between two different consecutive probabilities measures the velocity of diffusion. 0.90.090.009 … Faster Slower

23. Different Strategies g huf a x yze bvd c We may adjust the probabilities of a, b, c, d, e, f, g, h, x, z, u and v after the cell “ y ” is filled. (How?) Therefore, the probability of each cell filled by a molecule is dynamically changed.

24. Experiments are necessary  If there exists a resisting force between two molecules, then after the cell y is filled the nearby cells x, z, u and v are going to have less probability to be filled.  The cells a, b, …, h with larger distance from y will decrease less comparing to them or even with no effect.  On the other hand, if it is easier for molecules to cluster together, then it goes on the other direction, x, z, u and v will have larger probability to fill after y is filled.

25. Final Episode  How do we define the probabilities for all the cells in a grid with 150x500 cells. 1.Fill one cell in each row in one tick (unit of time). 2.Make sure the total probability of the cells to be filled in each row is equal to “ 1 ”. 3.Randomness plays the most important role. 4.The measure of probabilities will be defined by referring to experiments!

26. 晶片表面化學 Surface chemistry on chip (coating technology development) Self-Assembled Monolayers (SAMs) integration technology on ProbabilityProgapationChip 數學工程物理 / 化學 f(P) ≈ f(D) ≈ f(gap/t)

27.  Let ’ s give it a try! Let ’ s give it a try!

28. Conclusion  From experiments, we are going to receive information for helping us to obtain a mathematical model which is very close to the real one.  Then, with the model, hopefully we are able to predict the outcome of the experiments we are running. Even more, we expect to provide the idea of having a successful experiment: obtaining a real “ monolayer ”.

29. Thank You for Your Attention! I am always happy to come home.