Molecular Dynamics Simulations of Thermal Resistance at the Liquid-Solid Interface

Presented the following paper in statmech journal club (https://docs.google.com/document/d/13oAdycroBhekdUbSVp5_1vAqVewPdsws2jXLqZYb2Tc/edit#heading=h.quemo5cimprv) at IMSc:

Molecular Dynamics Simulations of Thermal Resistance at the Liquid-Solid Interface
by Bo Hung Kim, Ali Beskok, and Tahir Cagin
appeared in The Journal of Chemical Physics 129, 174701 ︎(2008︎ ).

Transcript

1. Molecular Dynamics Simulations of Thermal Resistance at the Liquid-Solid Interface Statmech Journal Club Talk Vinay Vaibhav August 20, 2018 Bo Hung Kim, Ali Beskok, and Tahir Cagin; The Journal of Chemical Physics 129, 174701 ︎(2008︎ )
3. Flow at small scale Extensive research on nanoscale fluid flow and velocity slip at the liquid-solid interface Heat transport in nano channels is less explored. Difficulty in handling thermal interactions of molecules with walls
4. Kapitza Resistance
5. Thermal transport at an interface between two dissimilar materials results in a temperature jump Such jump can be explained in terms of a thermal resistance called Kapitza resistance First such temperature jump at liquid-helium-solid interface was measured in 1941 by Kapitza Kapitza resistance provides some insight into the molecular structure of the interface and the microscopic mechanism determining energy transport
6. LIQUID HOT COLD SOLID HOT COLD Temperature For a given thermal gradient, the heat current density will different Nature of heat carriers are different; different thermal conductivity Thermal conductivity is defined as the ratio of heat current density to thermal gradient Heat Transport in a Medium
7. Heat transport from one phase to another LIQUID HOT SOLID COLD Same amount of heat is passed from one phase to another phase Different bulk conductivities produce different slopes in temperature profile Temperature T = JqRK <latexit sha1_base64="FiQDAewys5cFMj5M65XntKA8Qzg=">AAAB/nicbVDLSgNBEOyNrxhfq+LJy2AieAq7uehFCOpB9BIlL0iWZXYymwyZfTgzK4Ql4K948aCIV7/Dm3/jJNmDJhY0FFXddHd5MWdSWda3kVtaXlldy68XNja3tnfM3b2mjBJBaINEPBJtD0vKWUgbiilO27GgOPA4bXnDy4nfeqRCsiisq1FMnQD3Q+YzgpWWXPOg1L2iXGFUR+foxn1A9+4tKrlm0SpbU6BFYmekCBlqrvnV7UUkCWioCMdSdmwrVk6KhWKE03Ghm0gaYzLEfdrRNMQBlU46PX+MjrXSQ34kdIUKTdXfEykOpBwFnu4MsBrIeW8i/ud1EuWfOSkL40TRkMwW+QlHKkKTLFCPCUoUH2mCiWD6VkQGWGCidGIFHYI9//IiaVbKtlW27yrF6kUWRx4O4QhOwIZTqMI11KABBFJ4hld4M56MF+Pd+Ji15oxsZh/+wPj8AaCzk1A=</latexit> <latexit 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sha1_base64="FiQDAewys5cFMj5M65XntKA8Qzg=">AAAB/nicbVDLSgNBEOyNrxhfq+LJy2AieAq7uehFCOpB9BIlL0iWZXYymwyZfTgzK4Ql4K948aCIV7/Dm3/jJNmDJhY0FFXddHd5MWdSWda3kVtaXlldy68XNja3tnfM3b2mjBJBaINEPBJtD0vKWUgbiilO27GgOPA4bXnDy4nfeqRCsiisq1FMnQD3Q+YzgpWWXPOg1L2iXGFUR+foxn1A9+4tKrlm0SpbU6BFYmekCBlqrvnV7UUkCWioCMdSdmwrVk6KhWKE03Ghm0gaYzLEfdrRNMQBlU46PX+MjrXSQ34kdIUKTdXfEykOpBwFnu4MsBrIeW8i/ud1EuWfOSkL40TRkMwW+QlHKkKTLFCPCUoUH2mCiWD6VkQGWGCidGIFHYI9//IiaVbKtlW27yrF6kUWRx4O4QhOwIZTqMI11KABBFJ4hld4M56MF+Pd+Ji15oxsZh/+wPj8AaCzk1A=</latexit> Kapitza length: Equivalent thickness of the liquid with the same thermal conductivity that caused equivalent thermal resistance on the interface T = LK @T @n |liq <latexit 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8. Continuum-based models had many limitations and could not fully explain the complexity of thermal resistance at the interface of two distinct phases Many of such limitations have been overcome using MD which considers the role of interaction between liquid and solid
9. Thermal Walls
10. 419 1288 Applied specific boundary conditions to mimic the behaviour
11. Fixed lattice wall model: — Wall molecules are static — Energy not conserved, heat generation, fluid temp rises — Needs thermostat to dissipate the viscous heating
12. Static rough wall model: — Velocity slip found to approach zero — No clear definition of fluid-wall boundary
13. Wall molecules are very heavy ~1010 x massfluid System energy is conserved Ignores the possibility of energy transfer from fluid to wall No-slip condition is valid for high-density liquids as a result of adsorption of the fluid molecules on the wall At lower densities slip occurs and the slip length increases with decreasing number density of the fluid 478 427
14. “Diffuse Reflection” wall model or Thermal wall model Smooth walls which reflect fluid molecules with random thermal velocities based on MB velocity distribution
15. Advanced “Diffuse Reflection” wall model: Smooth walls which reflect fluid molecules with random thermal velocities based on MB velocity distribution Any wall particle reaching the interface reflect with a random thermal velocity corresponding to a specified wall temperature Difficulties in implementation of moving boundaries
16. 104 (a) (b) Thermal wall model1: vx (parallel to confinement) sampled from (a) and vx & vy from (b) Thermal wall model2: All three components of velocity sampled from (b) Model2 leads to spurious physical effects: Fluid temperature different from expected value, deviation from MB velocity distribution, inhomogeneities in density and temperature profiles
17. Vibrating wall model Fluid pressure increases wall separations: volume and density not fixed
18. A constraining mechanism keeps the centre of mass of each of the atomic wall layers constant
19. A thermostat required to maintain the equilibrium temperature of the fluid
20. Wall atoms have been arranged over FCC lattice, (111) plane faces the fluid No interaction between the wall atoms Wall atoms have tethered about their initial lattice positions Each layer of the wall is thermostatted separately (velocity rescaling) Interactive thermal wall model Maintaining thermal equilibrium of fluid was possible; no thermostat needed
21. Confined System Thermostatting via walls (T = 0.71) Confinement along y- direction
22. Upper and lower walls are static pair_style lj/cut 2.5 pair_coeff 2 2 1.0 1.0 2.5 pair_coeff 1 2 0.8 0.75 2.5 pair_coeff 1 1 0 0
23. vx vy Fluid Fluid vz Walls static: Temp. profile Walls moving: Temp. profile Walls moving: Temp. profile Couette flow: Upper and lower walls move along x and -x direction with speed 1.0 Overall temp of fluid Walls moving: Den. profile Walls static: Den. profile pair_style lj/cut 2.5 pair_coeff 2 2 1.0 1.0 2.5 pair_coeff 1 2 0.4 0.75 2.5 pair_coeff 1 1 0 0
24. vx vy vz Fluid Fluid Wall pair_style lj/cut 2.5 pair_coeff 2 2 1.0 1.0 2.5 pair_coeff 1 2 2.0 0.75 2.5 pair_coeff 1 1 0 0 Walls moving: Temp. profile Walls static: Temp. profile Overall temp of fluid Walls moving: Den. profile Walls static: Den. profile ? Couette flow: Upper and lower walls move along x and -x direction with speed 1.0
25. vx vy vz Fluid Fluid pair_style lj/cut 2.5 pair_coeff 2 2 1.0 1.0 2.5 pair_coeff 1 2 0.4 0.75 2.5 pair_coeff 1 1 0 0 Poiseuille flow: Fx = 0.5 Flow starts: Temp. profile Before flow starts: Temp. profile Before flow starts: Den. profile Flow starts: Den. profile
26. 550 fluid atoms at density ~ 0.8 in a periodic box Particles interacting via LJ interaction NVT ensemble using Nose-Hoover thermostat at 100 K Thermal conductivity: Green-Kubo calculation Green-Kubo method derived from linear response theory Calculates thermal conductivity using energy current correlation function
27. 578 fluid atoms at density ~ 0.8 Confinement along y-direction PBC along x, z-direction Wall and fluid atoms have same mass Walls are at 90 K and 110 K Fluid is binned along y-direction (10 bins) Interaction strength for fluid-wall and fluid-fluid are same Crystal bonding stiffness Kw = 64 (acoustic mismatch) Local thermal equilibrium: Skewness and kurtosis calculated for velocity distribution Thermal conductivity: NEMD calculation Temperature profile
28. Establishes the reliability of Fourier’s law in nano-confined dimension using interactive thermal wall model
29. Effect of thermal oscillation frequency on the Kapitza length Crystal bonding stiffness K of lattice crystal walls is changed K varied between Kw and 9Kw where Kw =64 Walls are set at 160 K and 90 K Temperature jump and hence gradient is decreasing with increasing crystal bonding stiffness Heat flux also decreases with increasing crystal bonding stiffness Thermal conductivity does not vary Thermal conductivity is material property while temperature jump and hence thermal resistance are the properties of interface
30. The wall at high temperature exhibits higher temperature jumps as crystal bonding stiffness is varied With increased crystal bonding stiffness, thermal resistance increases T = LK @T @n |liq <latexit sha1_base64="FCTPWA99gI0h0Iaiqjbr1+3TVOs=">AAACJXicbZDLSsQwFIZT7463UZdugqPgamjd6EJhUBeCLkaYUWE6lDRzqsE0rcmpONS+jBtfxY0LRQRXvoqZC3j9IfDxn3OSnD9MpTDouu/OyOjY+MTk1HRpZnZufqG8uHRqkkxzaPJEJvo8ZAakUNBEgRLOUw0sDiWchVf7vfrZDWgjEtXAbgrtmF0oEQnO0FpBeWfNPwCJjDboLj0OjqgfacZzP2UaBZO0UXyxKu6C3Ee4xVyK66JYC8oVt+r2Rf+CN4QKGaoelF/8TsKzGBRyyYxpeW6K7bx3P5dQlPzMQMr4FbuAlkXFYjDtvL9lQdet06FRou1RSPvu94mcxcZ049B2xgwvze9az/yv1sow2m7nQqUZguKDh6JMUkxoLzLaERo4yq4FxrWwf6X8ktmY0AZbsiF4v1f+C6ebVc+teiebldreMI4pskJWyQbxyBapkUNSJ03CyT15JM/kxXlwnpxX523QOuIMZ5bJDzkfn13qpUQ=</latexit> <latexit 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sha1_base64="FCTPWA99gI0h0Iaiqjbr1+3TVOs=">AAACJXicbZDLSsQwFIZT7463UZdugqPgamjd6EJhUBeCLkaYUWE6lDRzqsE0rcmpONS+jBtfxY0LRQRXvoqZC3j9IfDxn3OSnD9MpTDouu/OyOjY+MTk1HRpZnZufqG8uHRqkkxzaPJEJvo8ZAakUNBEgRLOUw0sDiWchVf7vfrZDWgjEtXAbgrtmF0oEQnO0FpBeWfNPwCJjDboLj0OjqgfacZzP2UaBZO0UXyxKu6C3Ee4xVyK66JYC8oVt+r2Rf+CN4QKGaoelF/8TsKzGBRyyYxpeW6K7bx3P5dQlPzMQMr4FbuAlkXFYjDtvL9lQdet06FRou1RSPvu94mcxcZ049B2xgwvze9az/yv1sow2m7nQqUZguKDh6JMUkxoLzLaERo4yq4FxrWwf6X8ktmY0AZbsiF4v1f+C6ebVc+teiebldreMI4pskJWyQbxyBapkUNSJ03CyT15JM/kxXlwnpxX523QOuIMZ5bJDzkfn13qpUQ=</latexit> LK = RK <latexit sha1_base64="OTrUHPdo8EVFKPou2wkfEGB6tBk=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDLaCq5J0oxuh6Eaoiyr2AW0Ik8mkHTp5MDMRS+ivuHGhiFt/xJ1/47TNQlsPDBzOuYd753gJZ1JZ1rdRWFvf2Nwqbpd2dvf2D8zDckfGqSC0TWIei56HJeUsom3FFKe9RFAcepx2vfH1zO8+UiFZHD2oSUKdEA8jFjCClZZcs1y9dZuX6N5togHXMR9XXbNi1aw50Cqxc1KBHC3X/Br4MUlDGinCsZR920qUk2GhGOF0WhqkkiaYjPGQ9jWNcEilk81vn6JTrfgoiIV+kUJz9Xciw6GUk9DTkyFWI7nszcT/vH6qggsnY1GSKhqRxaIg5UjFaFYE8pmgRPGJJpgIpm9FZIQFJkrXVdIl2MtfXiWdes22avZdvdK4yusowjGcwBnYcA4NuIEWtIHAEzzDK7wZU+PFeDc+FqMFI88cwR8Ynz+Lr5LR</latexit> <latexit 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sha1_base64="OTrUHPdo8EVFKPou2wkfEGB6tBk=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDLaCq5J0oxuh6Eaoiyr2AW0Ik8mkHTp5MDMRS+ivuHGhiFt/xJ1/47TNQlsPDBzOuYd753gJZ1JZ1rdRWFvf2Nwqbpd2dvf2D8zDckfGqSC0TWIei56HJeUsom3FFKe9RFAcepx2vfH1zO8+UiFZHD2oSUKdEA8jFjCClZZcs1y9dZuX6N5togHXMR9XXbNi1aw50Cqxc1KBHC3X/Br4MUlDGinCsZR920qUk2GhGOF0WhqkkiaYjPGQ9jWNcEilk81vn6JTrfgoiIV+kUJz9Xciw6GUk9DTkyFWI7nszcT/vH6qggsnY1GSKhqRxaIg5UjFaFYE8pmgRPGJJpgIpm9FZIQFJkrXVdIl2MtfXiWdes22avZdvdK4yusowjGcwBnYcA4NuIEWtIHAEzzDK7wZU+PFeDc+FqMFI88cwR8Ynz+Lr5LR</latexit> <latexit sha1_base64="OTrUHPdo8EVFKPou2wkfEGB6tBk=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDLaCq5J0oxuh6Eaoiyr2AW0Ik8mkHTp5MDMRS+ivuHGhiFt/xJ1/47TNQlsPDBzOuYd753gJZ1JZ1rdRWFvf2Nwqbpd2dvf2D8zDckfGqSC0TWIei56HJeUsom3FFKe9RFAcepx2vfH1zO8+UiFZHD2oSUKdEA8jFjCClZZcs1y9dZuX6N5togHXMR9XXbNi1aw50Cqxc1KBHC3X/Br4MUlDGinCsZR920qUk2GhGOF0WhqkkiaYjPGQ9jWNcEilk81vn6JTrfgoiIV+kUJz9Xciw6GUk9DTkyFWI7nszcT/vH6qggsnY1GSKhqRxaIg5UjFaFYE8pmgRPGJJpgIpm9FZIQFJkrXVdIl2MtfXiWdes22avZdvdK4yusowjGcwBnYcA4NuIEWtIHAEzzDK7wZU+PFeDc+FqMFI88cwR8Ynz+Lr5LR</latexit> LK / (Kwall/Kliquid)2 = (!wall/!fluid)4 <latexit sha1_base64="tBLC+R5iIY9L8Zcp4hDT2mwilMk=">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</latexit> <latexit sha1_base64="tBLC+R5iIY9L8Zcp4hDT2mwilMk=">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</latexit> <latexit sha1_base64="tBLC+R5iIY9L8Zcp4hDT2mwilMk=">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</latexit> <latexit sha1_base64="tBLC+R5iIY9L8Zcp4hDT2mwilMk=">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</latexit>
31. Effect of surface wettability on the Kapitza length Wettability means the liquid-solid interaction strength Kw is fixed to 3Kw where Kw =64 Wettability is varied from 0.4 to 4.0 Kapitza length varies exponentially with the interaction strength Strong attractive wall enhances heat transfer at the interface Strong liquid/solid interaction results in layering of fluid molecules which may change thermal conductance LK is proportional to exp(-1.85 Ewf/E)
32. Navier temperature-jump boundary condition For given crystal bonding stiffness, wettability and wall at fixed temperatures, vary the channel width Assume T = f(!w/!f , ✏wf /✏, Tw, @T/@n) <latexit sha1_base64="NMdY7BG+tKB8SH3TVZ/wBhWHzwE=">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</latexit> <latexit sha1_base64="NMdY7BG+tKB8SH3TVZ/wBhWHzwE=">AAACO3icdVBNTxsxEPVCgTR8hXLkYjUggRQlu0h89IAUQQ89QpUAUjZazTqziYXXu7K9RNEq/4sLf4Ibl156aFX12jtOWCpA8CRLz+/NjD0vTAXXxnXvnZnZD3PzC6WP5cWl5ZXVytqnc51kimGbJSJRlyFoFFxi23Aj8DJVCHEo8CK8Opn4F9eoNE9ky4xS7MbQlzziDIyVgsr3Tf8rCgO0RY9otO0nMfYhGDYKEtV8TDUXtjQfRuPG063WCoY1PwVlOAjaavyncmczqFTdujsFfUb2XO/Lvke9QqmSAqdB5c7vJSyLURomQOuO56amm08mMoHjsp9pTIFdQR87lkqIUXfz6e5jumWVHo0SZY80dKo+78gh1noUh7YyBjPQr72J+JbXyUx02M25TDODkj0+FGWCmoROgqQ9rpAZMbIEmOL2r5QNQAEzNu6yDeFpU/o+Od+te27dO9utNo+LOEpkg3wm28QjB6RJvpFT0iaM3JAf5Bf57dw6P50/zt/H0hmn6FknL+D8ewCO7609</latexit> <latexit sha1_base64="NMdY7BG+tKB8SH3TVZ/wBhWHzwE=">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</latexit> <latexit sha1_base64="NMdY7BG+tKB8SH3TVZ/wBhWHzwE=">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</latexit> T = f(!w/!f , ✏wf /✏, Tw)@T/@n <latexit sha1_base64="pumtpjw0gzdQM5/S1Z3Ap1YQX7g=">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</latexit> <latexit sha1_base64="pumtpjw0gzdQM5/S1Z3Ap1YQX7g=">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</latexit> <latexit sha1_base64="pumtpjw0gzdQM5/S1Z3Ap1YQX7g=">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</latexit> <latexit sha1_base64="pumtpjw0gzdQM5/S1Z3Ap1YQX7g=">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</latexit> Since, T = LK @T @n |liq <latexit sha1_base64="9RNFNhmG6KMgqSJsOt5gErqqX/w=">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</latexit> <latexit sha1_base64="9RNFNhmG6KMgqSJsOt5gErqqX/w=">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</latexit> <latexit sha1_base64="9RNFNhmG6KMgqSJsOt5gErqqX/w=">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</latexit> <latexit sha1_base64="9RNFNhmG6KMgqSJsOt5gErqqX/w=">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</latexit> So, LK = f(!w/!f , ✏wf /✏, Tw) <latexit sha1_base64="6mC6tv+l2fAvTVj9WBW1zJv2k1o=">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</latexit> <latexit sha1_base64="6mC6tv+l2fAvTVj9WBW1zJv2k1o=">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</latexit> <latexit sha1_base64="6mC6tv+l2fAvTVj9WBW1zJv2k1o=">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</latexit> <latexit sha1_base64="6mC6tv+l2fAvTVj9WBW1zJv2k1o=">AAACIHicdVDLSgMxFM3Ud32NunQTbIUK0s4IvhZC0Y2giwq2FtoyZNI7bTDzIMlYytBPceOvuHGhiO70a0zbKajogcDJOeeS3ONGnEllWR9GZmp6ZnZufiG7uLS8smqurddkGAsKVRryUNRdIoGzAKqKKQ71SADxXQ437u3Z0L+5AyFZGFyrfgQtn3QC5jFKlJYc8zB/6VzgE+wVmqEPHeL0SinxdpsQScZ1Kul5g9Lktnvt9HbyjpmzitYI+BvZt+zjAxvbqZJDKSqO+d5shzT2IVCUEykbthWpVkKEYpTDINuMJUSE3pIONDQNiA+ylYwWHOBtrbSxFwp9AoVH6veJhPhS9n1XJ32iuvK3NxT/8hqx8o5aCQuiWEFAxw95MccqxMO2cJsJoIr3NSFUMP1XTLtEEKp0p1ldwmRT/D+p7RVtq2hf7eXKp2kd82gTbaECstEhKqNzVEFVRNE9ekTP6MV4MJ6MV+NtHM0Y6cwG+gHj8wtPYqJ8</latexit> We already established, LK / (Kwall/Kliquid)2 = (!wall/!fluid)4 <latexit sha1_base64="tBLC+R5iIY9L8Zcp4hDT2mwilMk=">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</latexit> <latexit sha1_base64="tBLC+R5iIY9L8Zcp4hDT2mwilMk=">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</latexit> <latexit sha1_base64="tBLC+R5iIY9L8Zcp4hDT2mwilMk=">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</latexit> <latexit sha1_base64="tBLC+R5iIY9L8Zcp4hDT2mwilMk=">AAACTXicdVFNT9swGHY6Nlhho2zHXSwKUnfpkgoGOyBV2wWpHECigNR0keO8KRZOnNlvGFWUP7jLpN34F1w4gBDCLdkEaLySpcfPh2w/DjMpDLruuVN7MfPy1ezc6/r8wpu3i42ldwdG5ZpDnyup9FHIDEiRQh8FSjjKNLAklHAYnnyb6IenoI1Q6T6OMxgmbJSKWHCGlgoa0cpO0KN+plWGirZ6QeEjnGHxk0lZlp/+7aX4kYuoLD9+79At2vJVAiP2xPyYjGUVWFsJGk237U6HPgDrrvfls0e9immSanaDxh8/UjxPIEUumTEDz81wWDCNgkso635uIGP8hI1gYGHKEjDDYtpGSVctE9FYabtSpFP2YaJgiTHjJLTOhOGxeapNyP9pgxzjzWEh0ixHSPn9QXEuqS1uUi2NhAaOcmwB41rYu1J+zDTjaD+gbkv4+1L6PDjotD237e11mt2vVR1z5ANZJi3ikQ3SJdtkl/QJJ7/IBbki185v59K5cW7vrTWnyrwnj6Y2ewd0/7Xh</latexit> LK / exp( 1.85✏wf /✏) <latexit sha1_base64="n7HFRdNM41FxrDEL/GelzA1dq/w=">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</latexit> <latexit sha1_base64="n7HFRdNM41FxrDEL/GelzA1dq/w=">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</latexit> <latexit sha1_base64="n7HFRdNM41FxrDEL/GelzA1dq/w=">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</latexit> <latexit sha1_base64="n7HFRdNM41FxrDEL/GelzA1dq/w=">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</latexit> LK = ↵(Tw)(!w/!f )4 ⇥ exp( 1.85✏wf /✏) <latexit 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33. Keeping crystal bonding stiffness, wettability and channel width constant, wall temperature is varied Now we have full functional dependence of temperature jump
34. Using continuum heat conduction equations with the given form of temperature jump Using continuum heat conduction equations with the given form of temperature jump
35. ThankYou