# American Institute of Mathematical Sciences

April  2015, 8(2): 341-379. doi: 10.3934/dcdss.2015.8.341

## Structure formation in sheared polymer-rod nanocomposites

 1 Laboratory of Mathematics and Complex Systems, Ministry of Education and School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China 2 Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States 3 School of Mathematics and LPMC, Nankai University, Tianjin 300071

Received  July 2013 Revised  November 2013 Published  July 2014

We develop a hydrodynamic theory for flowing inhomogeneous polymer-nanorod composites (PNCs) coupling the Smoluchowski transport equation for the distribution function of the nanorod dispersed in a polymer matrix and the transport equation for the distribution of the polymer in the host matrix. The polymer molecule phase is modeled by bead-spring Rouse chains while the nanorod phase is modeled as semiflexible rods. The polymer-nanorod surface contact interaction and the conformational entropy of semiflexible nanorods are incorporated, resulting in a coupled system of nonlinear, nonlocal Smoluchowski equations for the polymer and nanorod. We then implement the theory to infer rheological properties and predict mesoscale morphologies in fully coupled plane shear flows. Our numerical study focuses on the mesoscale morphology development with respect to the surface contact interaction due to the pretreated surface properties of the nanorods, extending our studies on monodomain polymer-nanorod composites [16]. We find that surface contact interaction dominates the mesoscopic morphology and thereby corresponding rheological properties. When the nanorod favors parallel alignment with the polymer in the host matrix, the only globally stable state is the flow-aligning steady state. When the nanorod prefers to align orthogonally to the polymer in the matrix, however, spatially inhomogeneous structures, time-dependent homogeneous structures, and various spatial-temporal structures emerge in different regimes of the model parameter space and versus strength of the bulk imposed shear. Effective rheological features of the inhomogeneous morphologies are also predicted by the theory.
Citation: Guanghua Ji, M. Gregory Forest, Qi Wang. Structure formation in sheared polymer-rod nanocomposites. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 341-379. doi: 10.3934/dcdss.2015.8.341
##### References:

show all references

##### References:
 [1] Jun Li, Qi Wang. Flow driven dynamics of sheared flowing polymer-particulate nanocomposites. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1359-1382. doi: 10.3934/dcds.2010.26.1359 [2] Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155 [3] Eric S. Wright. Macrotransport in nonlinear, reactive, shear flows. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2125-2146. doi: 10.3934/cpaa.2012.11.2125 [4] Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644 [5] Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561 [6] Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093 [7] Claire david@lmm.jussieu.fr David, Pierre Sagaut. Theoretical optimization of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 286-293. doi: 10.3934/proc.2007.2007.286 [8] Zhenlu Cui, Qi Wang. Permeation flows in cholesteric liquid crystal polymers under oscillatory shear. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 45-60. doi: 10.3934/dcdsb.2011.15.45 [9] Alex Mahalov, Mohamed Moustaoui, Basil Nicolaenko. Three-dimensional instabilities in non-parallel shear stratified flows. Kinetic & Related Models, 2009, 2 (1) : 215-229. doi: 10.3934/krm.2009.2.215 [10] Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395 [11] Scott Gordon. Nonuniformity of deformation preceding shear band formation in a two-dimensional model for Granular flow. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1361-1374. doi: 10.3934/cpaa.2008.7.1361 [12] Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133 [13] Emma Hoarau, Claire david@lmm.jussieu.fr David, Pierre Sagaut, Thiên-Hiêp Lê. Lie group study of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 495-505. doi: 10.3934/proc.2007.2007.495 [14] Tian Ma, Shouhong Wang. Global structure of 2-D incompressible flows. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 431-445. doi: 10.3934/dcds.2001.7.431 [15] M. Bulíček, P. Kaplický. Incompressible fluids with shear rate and pressure dependent viscosity: Regularity of steady planar flows. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 41-50. doi: 10.3934/dcdss.2008.1.41 [16] James Nolen, Jack Xin. Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1217-1234. doi: 10.3934/dcds.2005.13.1217 [17] Houda Hani, Moez Khenissi. Asymptotic behaviors of solutions for finite difference analogue of the Chipot-Weissler equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1421-1445. doi: 10.3934/dcdss.2016057 [18] Mou-Hsiung Chang, Tao Pang, Moustapha Pemy. Finite difference approximation for stochastic optimal stopping problems with delays. Journal of Industrial & Management Optimization, 2008, 4 (2) : 227-246. doi: 10.3934/jimo.2008.4.227 [19] Giovanna Citti, Maria Manfredini, Alessandro Sarti. Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 905-927. doi: 10.3934/cpaa.2010.9.905 [20] Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558

2018 Impact Factor: 0.545