**3. Fundamentals of transient liquid phase bonding**

TLP bonding requires that the base metal surfaces are brought into intimate contact with a thin interlayer placed between the bonding surfaces. The interlayer can be added in the form of a thin foil, powder or coating [27, 28] which is tailored to melt by eutectic or peritectic reaction with the base metal. The liquid filler metal wets the base metal surface and is then drawn into the joint by capillary action until the volume between components to be joined is completely filled.

The driving force of TLP bonding is diffusion. A process which can be described by Fick's first and second laws. The first law describes diffusion under steady-state conditions and is given by equation 4:

$$J = -D\frac{\partial \mathbb{C}}{\partial \mathfrak{x}}\tag{4}$$

Fick's second law describes a non-steady state diffusion in which the concentration gradient changes with time and can be expressed as shown in equation 5:

$$\frac{\partial \mathbf{C}}{\partial t} = D \frac{\partial^2 \mathbf{C}}{\partial \mathbf{x}^2} \tag{5}$$

Equation 6 shows a general solution for Equation-5 using separation of variables is [29]:

$$\mathcal{C}(\mathbf{x},t) = \frac{1}{2\sqrt{D\pi t}} \int\_{-\infty}^{+\infty} f(\xi) e^{\left(\frac{\xi-\chi}{4Dt}\right)} d\xi \tag{6}$$

Where the error function solution for equation 6 is shown in equation 7:

$$\mathbf{C}(\mathbf{x},t) = \frac{\mathbf{C}\_0}{2} \left[ \text{erf}\left(\frac{\mathbf{x} - \mathbf{x}\_1}{\sqrt{4Dt}}\right) - \text{erf}\left(\frac{\mathbf{x} - \mathbf{x}\_2}{\sqrt{4Dt}}\right) \right] \tag{7}$$

If the following boundary conditions are applied to Equation-7, the concentration as a function of time can be calculated using Equation-8:

$$\begin{aligned} \text{C(x, 0)} &= \begin{cases} \mathcal{C}\_0 & \text{(x > 0)} \\ 0 & \text{(0 < x)} \end{cases} \text{ and } \quad \text{C(x, 0)} = \begin{cases} \mathcal{C}\_0 & \text{x = +\infty} \\ 0 & \text{x = -\infty} \end{cases} \\ \text{C(x, t)} &= \mathcal{C}\_0 \Big( 1 - \text{erf} \left( \frac{\mathbf{x}}{\sqrt{4Dt}} \right) + \text{erf} \left( \frac{k\mathbf{x}}{\sqrt{4Dt}} \right) \Big) \end{aligned} \tag{8}$$

TLP bonding can be conducted by one of two distinct methods. Method-I employs a pure interlayer which forms a liquid through eutectic reaction with the base metal and Method-II employs an interlayer with a liquidus temperature near the bonding temperature [2]. Method-II is most commonly used as it reduces the overall process time by decreasing the volume of solute to be diffused from the interface before the liquid is formed. On the other hand, method-I can be considered to be more effective in TLP bonding as the eutectic reaction is able to displace surface oxides during bonding. TLP bonding process was first divided into five discrete stages by Duvall et al. [30]. These stages were: heating, melting, dissolution of the base-metal, isothermal solidification and homogenization of the excess solute at the bond-line. Zhou later condensed the TLP process to three stages: base-metal dissolution, isothermal solidification and homogenization [7]. In later work by MacDonald and Eager [9] the second and third stage described by Duvall were combined to give four stages; heating, melting and parent metal dissolution, isothermal solidification and homogenization. Zhou [7] reclassified his earlier work to include a heating stage. The second stage was also expanded to be dissolution and widening.
