MULTISCALE MODELING AND APPLICATIONS IN NANOTRIBOLOGY SU ZHOUCHENG NATIONAL UNIVERSITY OF SINGAPORE 2012 MULTISCALE MODELING AND APPLICATIONS IN NANOTRIBOLOGY SU ZHOUCHENG (M.ENG) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2012 Acknowledgement Acknowledgement The author would like to express his sincere gratitude to all of the kindhearted individuals for their precious advice, guidance, encouragement and support, without which the successful completion of this thesis would not have been possible. Special thanks to the author’s supervisors Prof. Tay Tong-Earn and A/Prof. Vincent Tan Beng Chye, whom the author has the utmost privilege and honor to work with. Their instruction makes the exploration in multiscale modeling technique a wonderful journey. Their profound knowledge on mechanics and strict attitude towards academic research will benefit the author’s whole life. The author would also like to thank Dr Zhang Bing, Dr Sun Xiushan, Dr Liu Guangyang, Dr Yew Yong Kin, Dr Muhammad Ridha, Mr Chen Yu and Mr Mao Jiazhen for their invaluable help. Many thanks to his friends Ms Li Sixuan, Dr Ren Yunxia, Mr Song Shaoning, Mr Jiang Yong, and Mr Chen Boyang for making the research environment a lively place. Last but not least, the author expresses his utmost love and gratitude to his parents, wife, sister and brother for their understanding and support during the course of this project. i Table of Contents Table of Contents Acknowledgement . i Summary . . vi Nomenclature . ix List of Figures xii List of Tables . xix Chapter Introduction and Literature Review 1.1 Introduction . 1.2 Review of Multiscale Modeling Approaches . 1.2.1 Quasi-continuum Method 1.2.2 Handshake Method 1.2.3 Bridging Scale Method 10 1.2.4 Coarse Grained Molecular Dynamics 13 1.2.5 Other Relevant Multiscale Modeling Techniques . 14 1.3 Review of Studies on Nanotribology 16 1.4 Objectives and Significance of the Study . 19 Chapter Multiscale Modeling Using Pseudo Amorphous Cell . 22 2.1 Conceptual Modeling of Pseudo Amorphous Cell . 23 2.2 Formulation of PAC 27 2.2.1 Determination of T Matrix . 28 2.2.2 Calculating Cell Stiffness Matrix of PAC for Pair-wise Potentials . 36 2.3 Coupling between Atomistic Region and PAC 42 2.4 Conclusion 48 ii Table of Contents Chapter Computational Implementation . 49 3.1 Multiscale and Molecular Simulator . 50 3.1.1 PAC Implementation for Unit Cell 51 3.1.2 Solvers for Molecular Mechanics and Multiscale Simulations . 53 3.2 Pre-processors . 61 3.2.1 Polymer Modeling . 62 3.2.2 Input Generator for Multiscale and Molecular Simulator 66 3.2.3 Indenter Generator for Multiscale and Molecular Simulator . 68 3.3 Conclusion 69 Chapter Validation of PAC Multiscale Modeling 71 4.1 Multiscale Simulation of Nanoindentation on a Polymer Substrate . 71 4.2 Distinguishing Features of PAC-based Multiscale Modeling 76 4.2.1 Non-locality 77 4.2.2 Inhomogeneity 79 4.3 Conclusion 84 Chapter Extension to Three Dimensional Modeling 85 5.1 Three Dimensional Multiscale Modeling . 86 5.1.1 Determination of Transformation Matrix 86 5.1.2 Stiffness Calculations . 93 5.2 Three Dimensional Multiscale Simulations of Nanoindentation 97 5.2.1 Nanoindentation on a Polymer Substrate . 97 5.2.2 Nanoindentation on a Crystalline Substrate . 104 5.3 Conclusion 108 Chapter Extension to Complex Force Fields . 109 6.1 Numerical Calculation of Hessian Matrix 110 6.2 Cell Stiffness Construction . 111 iii Table of Contents 6.3 Validations 120 6.3.1 Nanoindentation on a PE Substrate 121 6.3.2 Nanoindentation on a Silicon Substrate . 125 6.4 Chapter Conclusion 129 Multiscale Simulation of Nanoindentation on Thin-film / Substrate System 130 7.1 Description of Models and Simulations 132 7.2 Simulation Results and Discussions . 138 7.2.1 Analyses of Indentation Forces 138 7.2.1.1. Effect of Tip Rounding Radius . 141 7.2.1.2. Effect of Included Angle . 142 7.2.1.3. Effect of Interface Strength 143 7.2.1.4. Substrate Effects . . 144 7.2.2 Pile-up Effect . . 146 7.2.3 Atomic Interface Delamination 150 7.3 Conclusion 154 Chapter Multiscale Simulation of Nanoscale Sliding 155 8.1 Sliding on 2D Crystalline Substrate 156 8.2 Sliding on 3D Polymer Substrate 158 8.2.1 Description of Models and Simulations . 159 8.2.2 Results and Discussions . 160 8.2.2.1 Penetration Depth and Mechanisms . 161 8.2.2.2 Other Frictional Behaviors . 165 8.3 Conclusion 168 Chapter Conclusions and Recommendations . 170 9.1 Conclusions . 170 iv Table of Contents 9.2 Recommendations for Future Work 173 References . 175 Appendix A Calculating the Stiffness Matrix of Pair-wise Potentials 191 Appendix B Strain Contour Calculation . 197 v Summary Summary Recent advances in nanotechnology necessitate the development of multiscale modeling techniques, which involve more than one length scale or time scale. In the past two decades, various concurrent multiscale approaches have been developed to simulate multiscale phenomena starting at nanoscale. However, most of existing multiscale approaches are developed to model crystalline materials. None of them are capable of scaling up the modeling of amorphous materials. In this thesis, a concurrent multiscale modeling approach to scale up modeling of amorphous materials using the Pseudo Amorphous Cell (PAC) is proposed. In this method, the domain of interest is firstly constructed as a tessellation of identical Amorphous Cells (ACs), each containing atoms equilibrated with periodic boundary conditions. For regions of small deformation, the number of degrees of freedoms (DOF) is then reduced by computing the displacements of only the vertices of the ACs instead of the atoms within them. The reduction is achieved by determining, a priori, the atomistic displacements within such Pseudo Amorphous Cells (PAC) associated with orthogonal deformation modes of the cell. The vertices of any PAC cell thus behave like nodes in the Finite Element Method (FEM) and are also referred to as nodes in our multiscale models. For regions experiencing large deformation, full atomistic details are retained. Hence, the atomistic domains coexist with continuumlike (PAC) domains. Seamless coupling between atomistic regions and PAC regions is developed mathematically and implemented in a computational code. vi Summary A two dimensional multiscale modeling approach using PAC is developed in detail. This method is computationally implemented in a C++ computer code. It is then validated by multiscale simulation of nanoindentation on a polymer substrate. A good agreement between the multiscale simulation and the pure molecular mechanics simulation is achieved for both indentation force and strain contours, indicating this method is capable of scaling up the modeling of amorphous materials. The accuracy of this method is attributed to the inclusion of non-local effects in the PAC regime and the ability to relate atom and nodal displacement accurately without the assumption of uniform deformation of the amorphous cells. Subsequently, this multiscale approach is extended to modeling materials in three dimensions. It is validated through multiscale simulations of nanoindentation on polymeric as well as crystalline substrates to show its general applicability for both amorphous and crystalline materials. In order to simulate more complex materials, the proposed multiscale approach is also successfully extended to include more sophisticated interatomic force fields with a number of complementary matrices. 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We note that (A.2) corresponds to the component of force in the  direction acting on bead m Also, (A.3) corresponds to the element in the stiffness matrix corresponding to  direction of bead m and  direction of bead n . From (A.1), we obtain, 191 Appendix A: rij xi  Calculating the Stiffness Matrix of Pair-wise Potentials xi  x j rij , rij x j  x j  xi rij , rij yi  yi  y j rij , rij y j  y j  yi rij (A.4) From (A.4), we obtain,  rij xi2  rij yi2    rij xi x j  rij yi y j  rij xi yi  rij x j yi  rij x 2j  rij y 2j       ( y j  yi ) r3 (A.5) ( x j  xi ) r3 ( y j  yi ) r3 (A.6) ( x j  xi ) r3  rij x j y j  rij xi y j   ( xi  x j )( yi  y j ) r3 ( xi  x j )( yi  y j ) (A.7) r3 If we arrange bead displacements and bead forces as vectors, i.e., u xi   f xi   i  i uy  f  and u j f   yj  , then (A.2) and (A.4) give, u x   fx   i  i u y   f y  192 Appendix A: Calculating the Stiffness Matrix of Pair-wise Potentials  xi  x j      yi  y j  f rij rij  x j  xi     y j  yi  (A.8) Eqs. (A.3) - (A.7) give,  xi  x j     2  yi  y j  K 2 xi  x j rij rij  x j  xi     y j  yi   yi  y j      x j  xi   yi  y j rij rij  y j  yi     xi  x j    yi  y j x j  xi y j  yi  (A.9) x j  xi y j  yi xi  x j  Similarly, the stiffness matrix for the three dimensional case can be calculated. The square of the distance between two atoms can be expressed as, rij2  x j  xi   y j  yi   z j  zi  (A.10) Let d x  xi  x j , d y  yi  y j and d z  zi  z j , then we have, 193 Appendix A: rij xi rij yi rij   dx rij dy rij d  z z i rij , , Calculating the Stiffness Matrix of Pair-wise Potentials rij x j rij y j rij    dx rij  dy (A.11) rij  dz ,  z j rij From (A.11), we obtain,  rij xi2  rij   rij x 2j  rij  d y2  d z2 r3 ,  rij xi x j  rij  d y2  d z2 r3 d z2  d x2 d z2  d x2   ,  yi y j yi2 y j r3 r3  rij  rij d x2  d y2  rij d x2  d y2   ,  z i z j z i2 z j r3 r3  rij xi yi  rij xi y j  rij xi z i  rij xi z j      rij x j y j  rij x j yi  rij x j z j  rij x j z i   (A.12) dxd y r3 dxd y (A.13) r3  dxdz r3  dxdz r3 (A.14) 194 Appendix A:  rij yi z i  rij yi z j   Calculating the Stiffness Matrix of Pair-wise Potentials  rij y j z j  rij y j z i   d ydz r3 d ydz (A.15) r3 Arranging atom displacements and atom forces as vectors as, u  [uix uiy uiz u zj ]T and f  [ f i x u xj u jy fi y fi z f jx f jy f jz ]T , then Eq. (A.2) and Eq. (A.11) give, d x  d   y   d z  f   rij rij  d x  d y     d z  (A.16) Eqs. (A.3), and (A.11) - (A.15) give, d y2  d z2    dxd y  2   d d K  2  x z2 rij rij  d x  rij  d d  x y  d x d z  d x2   dxd y   d x d z   rij rij   d x2  d d  x y   d x d z  dxd y  dxdz d d z x  d ydz d x2  rij2 d xd y  d ydz dxd y d r d d dxdz x y y ij d ydz dxd y d xdz d d d r d ydz  dxd y d z2  d x2 d ydz d z2  rij2  d xdz  d ydz y ij y z  dxd y dxd z d d ydz  dxd y  d y2 d ydz d z2  dxdz  d ydz  dxd y  dxdz d x2 dxd y d  d ydz d xd y d y2  d z2 dxdz d ydz dxd y d y y  d ydz x  dxd y dxdz   d ydz  d z2  rij2    dxdz   d ydz   d x2  d y2   dxd z    d ydz   d z2   d xdz  d ydz   d z2  (A.17) 195 Appendix A: Calculating the Stiffness Matrix of Pair-wise Potentials Examples of potential functions are,   12      (r )  4        r    r  (A.18) 12  24         (r )   2      r r   r   r   (A.19) 12 2 24         ( r )  26        r r   r   r   (A.20) and   r  (r )  0.5kR ln 1     R0     kr  (r )  r  r R0  (A.21) (A.22)   r R 2 2   ( r )  k   r R 2 r           (A.23) 196 Appendix B: Strain Contour Ca lculation Appendix B Strain Contour Calculation To plot the strains at a point  x0 , y  , we use a moving least squares (MLS) approximation of the displacement field. The approximated displacements u~( x, y) and v~( x, y) around point x0 , y0  are expressed using a polynomial basis, u~  a T p (B.1) For example, if a complete linear basis is used for u~ , then, 1  a0    a   a1  and p   x  y  a2  (B.2) where x  x  x0 and y  y  y0 . Similarly, v~  bT p , where b is a different set of coefficients. The coefficients of the polynomial in (B.1), a, are determined by minimizing the weighted error of u~( x, y) compared to the displacements of atoms within a distance of R from x0 , y0  . The weighted error, e , is given by, 197 Appendix B: Strain Contour Ca lculation e   w(ri )u~xi , yi   ui  (B.3) i Here, u i is the x displacement of atom i , xi , yi  is the position of atom i relative to xi2  yi2 is the distance of atom i from x0 , y0  . x0 , y0  and ri  The weight w(r ) is a function that decreases monotonically from unity at r  to zero at r  R , where r is the distance from x0 , y0  . A possible w(r ) is,  Rr  exp  R     , rR  w(r )   exp    0, otherwise  (B.4) Inserting (B.1) into (B.3) gives,  e   wi aT pi  ui  (B.5) i Minimizing the error with respect to a, gives,   d e2 0 daT    2 wi aT p i  ui pTi  (B.6) i 198 Appendix B: Strain Contour Ca lculation Eq. (B.6) gives, aT  wi pi pTi   wi ui pTi i (B.7) i Solving Eq. (B.7) for aT gives,    a   wi ui p Ti   wi p i p Ti   i  i  1 T (B.8) Therefore, 1     a   wi p i p Ti   wi u i p i   i   i  (B.9) Inserting Eq. (B.9) into (B.1) gives an expression for u~ , from which the strain  x ( x0 , y0 ) can be obtained as, u~   x   aT p  pT x x x ,y ( 0,0) x a x ,y ( 0, ) (B.10) The expression for v~  bT p is obtained in a similar manner. Thereafter,  y x0 , y0  and  xy x0 , y0  are obtained from definition as in (B.10). 199 [...]... concurrent multiscale modeling approach which applies to both crystalline and amorphous materials is developed It can then be used to perform multiscale simulations on nanotribology in depth In the 3 Chapter 1: Introduction and Literature Review following two sections, a literature review of concurrent multiscale modeling and applications regarding to nanotribology will be provided in detail 1.2 Review of Multiscale. .. Multiscale Modeling Approaches This section will review existing multiscale modeling techniques including Quasicontinuum method, handshake method, and coarse grain molecular dynamics, bridging scale method and other methods Their applications will also be presented The limitations of the existing multiscale modeling techniques will then be identified To overcome these limitations, a complete new multiscale. .. components and the substrate embedded within such systems gives rise to nanoindentation and frictional sliding Nanoindentation is also routinely employed to investigate the properties of material Hence, the nanotribology is relevant to a wide range of important applications Although, the multiscale modeling has seen applications in nanotribology, few can reallistically model indentation, sliding and friction... Microelectromechanical Systems (MEMS) and Nanoelectro-mechanical Systems (NEMS) Most existing multiscale modeling techniques are employed to study cracks, dislocations or diffusion The emerging application of multiscale modeling is to model nanoindentation [9], contact and sliding [10] in nanotribology Nanotribology is an area of active research in nanotechnology Nanoscale components are used in MEMS and Nano-Electro-Mechanical... sliding To simulate phenomenon at nanoscale or microscale, researchers have developed 1 Chapter 1: Introduction and Literature Review a variety of multiscale modeling techniques aiming at high accuracy with considerable efficiency Generally, there are two families of multiscale modeling techniques, namely, hierarchical modeling techniques and concurrent modeling techniques In hierarchical modeling, ... as inputs in lower resolution simulations The concurrent modeling differs from the hierarchical modeling in the way that more than one length scale or time scale coexists in the same model and the simulations at different levels are performed simultaneously The latter multiscale modeling technique is more promising because it can couple multiscale phenomena more accurately Various concurrent modeling. .. description in the atomic limit, where nodes coincide with atoms In the handshake method, a single total energy is defined for the entire system One main limitation of this approach is that defining this energy for general geometries and interatomic potentials is not straightforward This can be overcome by coupling atomistic and continuum regions through constraints on displacements rather than constructing... Bridging of the coarse and fine scales is realized by transparently exchanging information between coarse and fine 11 Chapter 1: Introduction and Literature Review scale regions A schematic for the bridging scale method is shown in Figure 1.3 As can be seen, the fine scale has been reduced to being active in a small portion of the domain, while the impedance force acting on the fine scale accounts for... energy to continuum potential energy in order to develop a non-linear continuum constitutive model based on the interatomic potential used for atomistic simulations The total potential energy of the QC model is obtained by summing the energies of all atoms in the atomistic region and at the interface and all elements in the continuum domain The QC method leads to some non-physical effects in the transition... molecular domain are eliminated by introducing forces equivalent to the lattice impedance; this entails the evaluation of inverse Laplace transform in time, and for multidimensional problems, a Fourier transform in space The bridging scale method has been successfully used in modeling buckling of multiwalled carbon nanotubes [39] The dynamic bridging scale method was applied to study wave propagation, and crack . MULTISCALE MODELING AND APPLICATIONS IN NANOTRIBOLOGY SU ZHOUCHENG NATIONAL UNIVERSITY OF SINGAPORE 2012 MULTISCALE MODELING AND APPLICATIONS IN NANOTRIBOLOGY. Multiscale Modeling 71 4.1 Multiscale Simulation of Nanoindentation on a Polymer Substrate 71 4.2 Distinguishing Features of PAC-based Multiscale Modeling 76 4.2.1 Non-locality 77 4.2.2 Inhomogeneity. Mechanics and Multiscale Simulations 53 3.2 Pre-processors 61 3.2.1 Polymer Modeling 62 3.2.2 Input Generator for Multiscale and Molecular Simulator 66 3.2.3 Indenter Generator for Multiscale and