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Variational model for a rippled graphene sheet.

Jabr Aljedani1,2, Michael J Chen1, Barry J Cox1

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This study models rippled graphene sheets on metal substrates using the calculus of variations. The model, incorporating graphene bending rigidity and van der Waals interactions, accurately predicts ripple profiles, aligning with molecular dynamics simulations.

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Area of Science:

  • Materials Science
  • Condensed Matter Physics
  • Computational Mechanics

Background:

  • Graphene's unique properties make it suitable for advanced applications.
  • Understanding the behavior of rippled graphene on substrates is crucial for device fabrication.
  • Existing models may not fully capture the complex interactions between graphene and substrates.

Purpose of the Study:

  • To develop a theoretical model for analyzing rippled graphene sheets on metal substrates.
  • To investigate the influence of graphene bending rigidity and van der Waals interactions.
  • To explore different constraint scenarios for graphene and substrate lengths.

Main Methods:

  • Utilizing the calculus of variations to formulate the physical model.
  • Developing a model based on two key parameters: bending rigidity (γ) and van der Waals interaction strength (ξ).
  • Analyzing three distinct cases: transitional, substrate-constrained, and graphene-constrained configurations.

Main Results:

  • Numerical results demonstrate a continuous relationship between total energy per unit length and substrate length across all configurations.
  • The model successfully predicts the profiles of graphene ripples.
  • The findings show excellent agreement with previous molecular dynamics (MD) simulation results.

Conclusions:

  • The proposed model provides a robust framework for studying rippled graphene on substrates.
  • The results offer insights into the mechanical behavior and energy landscape of graphene-substrate systems.
  • This work validates the model's predictive power against established simulation techniques.