%0 Book Section %1 statphys23_0259 %A Otsuki, M. %A Sasa, S. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K glassy materials property rheological statphys23 topic-9 %T Order Parameter Equation for Non-linear Rheology of Glassy Materials %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=259 %X The rheological property of glassy materials strongly depends on the density and the temperature. At a high temperature and low density state, the shear stress $\sigma_xy$ behaves as a linear function of the shear rate $\gamma$. On the contrary, at a low temperature and high density state, $\sigma_xy$ behaves as a non-linear function of $\gamma$, and shear thinning, shear thickening and the appearance of yield stress are observed. For this rheological property, the approach using mode coupling theory achieves certain success. In this approach, the non-linear viscosity is expressed in terms of the time correlation function of density fluctuations by combining the generalized Green-Kubo theory and the projection operator method, and then the time correlation function is evaluated within the framework of the mode coupling theory. Although this theory can predict the non-linear rheological property well, its description is rather complicated. In this presentation, we take an alternative approach. We first notice that the shear stress $\sigma_xy$ is determined from the pair distribution function $g(r)$. Second, we consider the system whose size is relatively small. Then, by applying a bifurcation analysis to an evolution equation for $g(r)$ under some approximations, we obtain an order parameter equation that describes the rheological property (M. Otsuki and S. Sasa, J. Stat. Mech., L10004 (2006)). The rheological property described by the order parameter equation is expressed as eqnarray a (T - T_s) \sigma_xy + b \sigma_xy^3 + c = 0, equation eqnarray where $T$ is the temperature of the system, and $a, b, c$ and $T_s$ are constants which can be calculated from the interaction potential. This equation describes the non-linear rheological property as well as the linear rheological property as shown in Figure 1. In particular, when $T$ is close to $T_s$, we find $\sigma_xy$ is proportional to $\gamma^1/3$. Note that this behavior is observed in numerical experiments for systems whose size is relatively small (M. Otsuki, cond-mat/0612136). We conjecture that our theory corresponds to the mean field theory of critical phenomena. Thus, we expect that a more complicated behavior appearing in large systems can be studied by extending our theory so as to describe the large scale modulation of the order parameter.

@incollection{statphys23_0259, abstract = {The rheological property of glassy materials strongly depends on the density and the temperature. At a high temperature and low density state, the shear stress $\sigma_{xy}$ behaves as a linear function of the shear rate $\gamma$. On the contrary, at a low temperature and high density state, $\sigma_{xy}$ behaves as a non-linear function of $\gamma$, and shear thinning, shear thickening and the appearance of yield stress are observed. For this rheological property, the approach using mode coupling theory achieves certain success. In this approach, the non-linear viscosity is expressed in terms of the time correlation function of density fluctuations by combining the generalized Green-Kubo theory and the projection operator method, and then the time correlation function is evaluated within the framework of the mode coupling theory. Although this theory can predict the non-linear rheological property well, its description is rather complicated. In this presentation, we take an alternative approach. We first notice that the shear stress $\sigma_{xy}$ is determined from the pair distribution function $g({\bf r})$. Second, we consider the system whose size is relatively small. Then, by applying a bifurcation analysis to an evolution equation for $g({\bf r})$ under some approximations, we obtain an order parameter equation that describes the rheological property (M. Otsuki and S. Sasa, J. Stat. Mech., L10004 (2006)). The rheological property described by the order parameter equation is expressed as \begin{eqnarray} a (T - T_s) \sigma_{xy} + b \sigma_{xy}^3 + c \gamma = 0, \label{equation} \end{eqnarray} where $T$ is the temperature of the system, and $a, b, c$ and $T_s$ are constants which can be calculated from the interaction potential. This equation describes the non-linear rheological property as well as the linear rheological property as shown in Figure 1. In particular, when $T$ is close to $T_s$, we find $\sigma_{xy}$ is proportional to $\gamma^{1/3}$. Note that this behavior is observed in numerical experiments for systems whose size is relatively small (M. Otsuki, cond-mat/0612136). We conjecture that our theory corresponds to the mean field theory of critical phenomena. Thus, we expect that a more complicated behavior appearing in large systems can be studied by extending our theory so as to describe the large scale modulation of the order parameter.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Otsuki, M. and Sasa, S.}, biburl = {https://www.bibsonomy.org/bibtex/222986125ae6456f3bc91c936af9b20ee/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {02c6d84ec4d7ed6f376274e53bea6619}, intrahash = {22986125ae6456f3bc91c936af9b20ee}, keywords = {glassy materials property rheological statphys23 topic-9}, month = {9-13 July}, timestamp = {2007-06-20T10:16:15.000+0200}, title = {Order Parameter Equation for Non-linear Rheology of Glassy Materials}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=259}, year = 2007 }