Nanolayer Surface Phase Change in Self-Healing Materials

  • Rahul BasuEmail author
Conference paper
Part of the The Minerals, Metals & Materials Series book series (MMMS)


The phase change problem in a semi-infinite medium where temperature and concentration are coupled is solved for variable diffusivity. An integral technique is used to solve for concentration and temperature penetration lengths and velocity of change. Surface conditions are held constant whereas diffusivity is allowed to vary slowly corresponding to phase change in the matrix. An analytic solution for variable properties under these conditions is obtained using the Kirchhoff transformation. Penetration lengths are analytically evaluated. A perturbation analysis allows the boundary layer lengths to be estimated, whence the two parameters can be compared. Dimensions obtained by the simulation show that nanomaterial thicknesses are attainable in the boundary layers. Application to phase change concepts relating to self-healing materials is illustrated for porous surface layers.


Self-healing Self-healing materials Composite materials Surface layers Porous materials 


a, b

Parameters in the Kirchhoff transformation




Specific heat




Diffusion coefficient

L, H

Latent heat

s, l

Solid, liquid


Power exponent in the approximate scheme


Stefan number = latent heat/sensible heat

Ti, To

Initial, final temperature

U, V

Transformed temperature, concentration in the Kirchhoff representation

1, 2

Labels for the interacting phases


Eigenvalue for interface velocity


Diffusivity variation for concentration




Temperature (nondim)


Fusion temperature


Similarity parameter


Recombination or dissolution, small perturbation parameter


Penetration length (thermal slope is zero)


Penetration length (concentration slope is zero)


Thermo-gradient parameter


Position of the moving phase boundary

λ (τ)

Phase change eigenvalue


Thermal diffusivity


Diffusivity of permeating phase


Normalized diffusivity αi/αj


Penetration length for thermal Δ = 2β t0.5


Penetration length for concentration Δ1 = 2β1 t0.5


Separation constant


Temperature-dependent constant for variable diffusivity




  1. 1.
    Gupta N, Shakra M (2016) A two-dimensional mathematical model to analyze thermal variations in skin and subcutaneous tissue region of human limb during surgical wound healing. Appl Math 7:145–158CrossRefGoogle Scholar
  2. 2.
    Liu D, Lee CY, Lu X (1993) Reparability of impact-induced damage in SMC composites. J Compos Mater 27(13):1257–1271CrossRefGoogle Scholar
  3. 3.
    Osswald T, Menges G., (2003) Failure and damage of polymers. In: Osswald T, Menges G (eds) Materials science of polymers for engineers, p 447, Hanser Publishers, Munich, GermanyGoogle Scholar
  4. 4.
    Wang Y, Pham Duc, Ji Chiungian (2015) Self healing composites: preview. Cogent Eng 2:1. Scholar
  5. 5.
    Luikov AS (1975) Systems of differential equations of heat and mass transfer in capillary-porous bodies (review). Int J Heat Mass Transf 18(1):1–14CrossRefGoogle Scholar
  6. 6.
    Ozisik MN (1980) Heat conduction. Wiley, New YorkGoogle Scholar
  7. 7.
    Carslaw HS, Jaeger JC (1959) Conduction of heat in solids. OUP, Oxford, UKGoogle Scholar
  8. 8.
    Broadbridge P, Tritscher P, Avagliano A (1993) Free boundary problems with nonlinear diffusion. Math Comput Model 18(10):15–249CrossRefGoogle Scholar
  9. 9.
    Fujita H (1952) The exact pattern of concentration-dependent diffusion in a semi-infinite medium. Textile Research J 22:757–760CrossRefGoogle Scholar
  10. 10. Accessed Feb 2018
  11. 11.
    Abramowitz M, Stegun I, (eds) (1972) Handbook of mathematical functions. Dover, New YorkGoogle Scholar
  12. 12.
    Crank J (1985) Free and moving boundary problems. Oxford University Press, Oxford, UKGoogle Scholar
  13. 13.
    Soumya H, Philip J (2016) Thermal diffusion in thin plates and coatings: influence of thickness and coating material. Turkish J Phys 40:256–263CrossRefGoogle Scholar
  14. 14.
    Biener J, Wittstock J, Baumann T et al (2009) Surface chemistry in nano scale materials. Materials 2:2404–2428. Scholar
  15. 15. Accessed Jan 2018
  16. 16.
    Wang Y (2014) Dynamic urea bond design of reversible self healing polymers. Scholar
  17. 17.
    Mphahlele K, Roy SS, Kolesnikov A (2017) Self healing polymeric composite material design, failure design, and future outlook: a review. Polymers 9:537. Scholar
  18. 18.
    Nakamura M, Takeo K, Usuda T, Ozaki S (2017) Finite element analysis of Self Healing and Damage Processes in Alumina/SiC Composite Ceramics. Technologies 5:40. Scholar
  19. 19.
    Privman V, Dementsov A, Sokolov I (2018) Modeling of self-healing polymer composites reinforced with Nanoporous glass fibers. Accessed Jan 2018
  20. 20.
    Formia A, Irico S, Bertola F et al (2016) Experimental analysis of Self Healing cement-based materials incorporating extruded Cementitious hollow tubes. J Intell Mater Syst Struct 27(9):2633–2652CrossRefGoogle Scholar
  21. 21.
    Homma D, Mitashi H, Nishikawa T (2009) Self healing capacity of fiber reinforced cementitious composites. J Adanced Concrete Technol 7(2):217–228CrossRefGoogle Scholar
  22. 22.
    Trask RS, Williams GJ, Bond IP (2007) Bioinspired self-healing of advanced composite structures using hollow glass fibers. J Royal Soc Interf 4(13):363–371CrossRefGoogle Scholar
  23. 23.
    Strickland S, Hin M, Sayanagi MR et al (2014) Self healing dynamics of surfactant coating on thin viscous films. arxiv:1309.3537v2
  24. 24.
    Piserchia A, Barone V (2016) Towards a general yet effective computational approach for diffusive problems. J Chem Theory Comput 12(8):3482–3490CrossRefGoogle Scholar
  25. 25.
    Pavlov TR, Staicu D, Vlahovic L et al (2018) A new method for the characterization of temperature-dependent thermophysical properties. Int J Therm Sci 124:98–109CrossRefGoogle Scholar
  26. 26.
    Fernandes AP, Sousa PFB, Borges VL, Guzmaras G (2010) Use of 3D transient analytical solutions based on Greens Functions to reduce computational time in inverse conduction problems. Appl Math Model 34:4040–4045CrossRefGoogle Scholar

Copyright information

© The Minerals, Metals & Materials Society 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringAdarsha Institute of TechnologyBangaloreIndia

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