MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Thermodynamics of Materials

3.00 Fall 2001

W. Craig Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 3: Due Fri. Sept. 28, Before 5PM in 4-047

Exercise 3.1

1. Consider an adiabatically enclosed room of constant volume ( $4\times6\times3$   m$^3$) that contains an ideal diatomic gas ( $\ensuremath{\overline{C_P}} = \frac{7}{2} R$) initially at 1 atm.

   Calculate the minimum amount of work that must be passed into the room to heat it from 10${}^\circ$C to 25${}^\circ$C.

2. Consider a typical enclosed room in a house that is very well thermally insulated with constant volume ( $4\times6\times3$   m$^3$) that contains an ideal diatomic gas ( $\ensuremath{\overline{C_P}} = \frac{7}{2} R$) initially at 1 atm.

   Calculate the minimum amount of work that must be passed into the room to heat it from 10${}^\circ$C to 25${}^\circ$C.

Exercise 3.2

1. Silicon solidifies into a diamond cubic crystal structure. A single crystal of silicon oriented so that $[100]$, $[010]$, and $[001]$ are parallel to the $x$-, $y$-, and $z$-axes has an anisotropic stiffness tensor, $\ensuremath{\underline{C}}$ ( $\sigma_{ij} = C_{ijkl} \epsilon_{kl}$) with components given by:
   $$\begin{split} C_{1111} = C_{2222} = C_{3333} = 166 \mbox{GPa... ... = 80 \mbox{GPa}\\ \mbox{All other } C_{ijkl} = 0 \end{split}$$

   If such a thin film of single crystal silicon is afixed to a flat substrate (the plane of the substrate being $y$-$z$ and the normal to the substrate is parallel to the $x$-axis) the film will have the following strain state:

   $$\begin{split} \epsilon_{22} = \epsilon_{33} = \ensuremath{{\... ...lon}_\circ}\\ \mbox{All other } \epsilon_{ij} = 0 \end{split}$$

   Calculate an expression for the internal energy density as stored by the elasticity.

2. A common way to estimate the thickness at which a thin film will break off (delaminate) from a substrate is to equate the stored elastic energy with the total interfacial energy. If the interfacial energy per unit area $\ensuremath{{\gamma}_\circ}$   J$/$m$^2$, find an expression that predicts the thickness of the film that will delaminate.

3. Suppose that the single crystal film is re-oriented so that it is a $[110]$-oriented thin film ($x$, $y$, and $z$ are parallel to $[110]$, $[1\overline{1}0]$, and $[001]$). Because the stiffness is a tensor, the components must be transformed into a the coordinate system associated with the substrate coordinates. The transformed stiffness for a $[110]$-orientation is:
   $$\begin{split} C_{1111} = C_{2222} = 194 \mbox{GPa}\\ C_{33... ... = 51 \mbox{GPa}\\ \mbox{All other } C_{ijkl} = 0 \end{split}$$

   The state of strain is independent of the orientation.

4. Find an estimate of the ratio of thicknesses $h_{[100]}/h_{[110]}$ at which a thin film would delaminate.