Article pubs.acs.org/JPCC
Interface Adhesion Properties of Graphene Membranes: Thickness and Temperature Effect Yan He, Wangbing Yu, and Gang Ouyang* Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education and Department of Physics, Hunan Normal University, Changsha 410081, People’s Republic of China ABSTRACT: The interface adhesion plays a crucial role in many applications of graphene, and is critical to graphene-based nanoelectronic and nanomechanical devices. However, in contrast to that of graphene membrane at ambient temperature, the question as to whether we could address the interface adhesion energy and local separation between membrane and underlying substrate under high temperature still remains unresolved. Herein, we develop a theoretical approach to deal with the temperature and thickness dependence of interface adhesion properties of graphene membranes based on the atomic-bond-relaxation consideration. Theoretical analyses indicate that the adhesion energy and interface separation can be modulated by the membrane thickness and external temperature. The unique adhesion properties can be ascribed to the size- and temperature-dependent Young’s modulus of graphene and lattice strain caused by the thermal expansion coefficient mismatch. Our theoretical predictions are in agreement with the experimental measurements and simulations, which provide the possible method on tunable adhesion properties of graphene for possible applications.
1. INTRODUCTION Graphene, as a two-dimensional material, has received key attention in recent years owing to its charming properties from both fundamental scientific issues and potential applications such as nanoelectronic, biological and nanomechanical devices.1,2 For such applications, knowledge of the interaction between graphene and the substrate is an important issue. Up to now, various methods have been employed to address the adhesion energy between graphene membranes and the underlying substrate, both experimentally3−7 and theoretically.8−15 For example, using a pressurized blister test, Koenig et al.4 measured the adhesion energy of graphene on SiO2 and found 0.45 ± 0.02 J/m2 for monolayer graphene and 0.31 ± 0.03 J/m2 for 2−5 layers. Moreover, the adhesion energy and interface separation between graphene and substrate can be tuned by some factors, including surface roughness, stacking modes, contamination, substrate styles, and so on.12−14,16−18 Physically, the thickness-dependent adhesion properties can be attributed to the strain induced by high surface-to-volume ratio and coordination deficiencies of graphene membranes and interface condition.10,19 However, there is an inevitable issue that the thermal effect will take place when graphene makes as a candidate material for heat control in high-end nanoelectronic and nanomechanical devices. When the temperature of a graphene system is raised, two questions should be clarified: one is the changes of adhesion energy and interface separation, the other is the physical mechanism regarding thermal strain caused by the mismatch of thermal expansion coefficients (TECs) of graphene and its underlying substrate. In fact, the thermal effect for graphene will change a variety of properties, such as Young’s modulus,20,21 Raman shift,22,23 TEC,24−26 and so on. Currently, the thermal issue on graphene has always been © 2015 American Chemical Society
ambiguous as the discrepancies exist in different approaches. For instance, Bao et al.27 and Yoon et al.28 measured the TECs of graphene are, respectively, −7 × 10−6 and −8 × 10−6 K−1 at room temperature (RT). Mounet et al.25 found the TEC of graphene is negative in whole temperature range by using firstprinciple calculations. However, as reported in refs 24, 26, 29, and 30, the TEC of graphene is negative in low temperature, while positive at high temperatures. In particular, knowledge of the adhesion properties as functions of temperature and membrane thickness is important. In order to determine the temperature and thickness dependence of adhesion properties, it would be necessary to address the unique interface adhesion energy and relevant local separation. In this contribution, we propose a quantitative method to investigate the interface adhesion of graphene membranes and show that the adhesion energy and interface separation are determined by the temperature and membrane thickness. Furthermore, the underlying mechanism on anomalous adhesion properties of graphene membranes under elevated temperature is clarified based on the atomic-bond-relaxation consideration from the perspective of atomistic origin.
2. PRINCIPLE 2.1. Thermal Expansion Coefficient of Graphene. Considering the effect of finite temperature of an unsupported multilayer graphene, the Helmholtz free energy is given by F = Vtot − TS
(1)
Received: November 6, 2014 Revised: February 6, 2015 Published: February 23, 2015 5420
DOI: 10.1021/jp511100d J. Phys. Chem. C 2015, 119, 5420−5425
The Journal of Physical Chemistry C
Article
where εsT and εfT denote the thermal strain of substrate and graphene, as and af represent the lattice constant of substrate and membrane. Furthermore, taking into account the combination role from surface, interface, and temperature effect in the graphene on the substrate, the total strain in the graphene membranes can be deduced as
where T is temperature, S is the entropy, and Vtot is the total potential energy at zero temperature and can be expressed by a function of the type
Vtot = zNV (dij , θijk)/2
(2)
where z and N are the coordination numbers (CNs) of an atom and the total atom number, dij is the distance between atoms i and j, θijk represents the angle between the bond i−j and the neighbor bonds, and V(dij,θijk) is the interatomic potential and can be accessed in terms of Tersoff-Brenner potential for carbon.24,31 According to the quasiharmonic approximation, the entropy is32 ⎡ ⎛ hωiκ ⎞⎤ S = −kBN ∑ ln⎢2 sinh⎜ ⎟⎥ ⎢⎣ ⎝ 4πkBT ⎠⎥⎦ κ
εf = ∑s < n (εs + εTf )(cs + εTf )d0 + (ε// + εTf )(t f − ∑s < n ds) tf (7)
where εs = cs − 1 is the lattice strain occurrence in the surface layers, tf is the membrane thickness, ds and d0 denote the bond length in the sth layer and in the bulk. cs = 2/(1 + exp((12 − zs)/8zs))19 and ε∥ = εm(εm + 1)d0/tf10 are the bond contraction coefficient and the lattice strain in interface layers, respectively. Theoretically, the total free energy Utotal is negative at the equilibrium state, which is equal to the interface adhesion energy, that is, Γ = −Utotal8,9 In our case, we focus on the interface adhesion properties between graphene and its underlying substrate. Thus, the total free energy of graphene membranes is
(3)
where kB is the Boltzmann constant, h is the Planck’s constant, ωiκ is the vibration frequencies of atom i and can be determined from the local dynamic matrix 1/mi∂2Vtot/∂xi∂xi of atom i by |ω2iκ I − (1/mi)(∂2Vtot/∂xi∂xi)| = 0, where I is the identity matrix, and mi and xi are the mass and Cartesian coordinates of atom i, respectively.24 Strikingly, with the reduction of the size of a specimen, the surface and interface atoms will become more and more important in system. According to the atomic-bond-relaxation (ABR) consideration,10,19,33,34 the bond of undercoordinated atoms located at the surface or edge part will be shorter and stronger compared to that of the bulk counterpart. Considering the discrepancy between the surface or edge and the core interior, the Helmholtz free energy can be expressed as F* =
1 2
∑ NszsVs(dij*, θijk) + s
