Strength evolution laws in curing of solvent-welded polymers

This paper reports the scaling laws to describe the time-evolution behavior of solvent-mediated strength at the interface between two identical thermoplastic polymers below the glass-transition temperature. Our results suggest that the evolution scales as $\sqrt{t}$, where $t$ is the curing time. It depends on the time evolution of interfacial stiffness and toughness, each of which scales as $\sqrt{t}$. Employing a combination of experiments and continuum scale simulations, we show that the evolution of strength, stiffness, and toughness is controlled by pure diffusion. It can therefore be treated as a Gaussian process. While the “saturation of strength,” which describes the transition of strength evolution into a steady state, does not strictly follow any power-law type behavior, a simple exponential law accurately characterizes both evolution and saturation of strength. This suggests that the longer timescale nonlinear processes (that are overdetermined by the power-law type scaling laws) diminish rapidly in approaching a steady state. Furthermore, the kinetics of the evolution processes is well captured by the dissolution of polymer particles. While dissolution involves a different timescale, it strongly correlates with the solvent-welding process upon normalization. The correlation highlights the equivalence of the dissolution and solvent-joining processes and offers an easier route to determining strength at arbitrary curing times. Additionally, the dissolution rate of polymer particles is shape dependent and governed by the surface-to-volume ratio.

Figure 1

Strength variation as a function of the curing time $t$. The schematic shows the sample configuration and loading direction applied for the uniaxial tensile tests. The data points (shown by the filled circles) are fitted by the exponential equation ${\sigma }_{\text{saturated}}\left[1-exp(-\alpha t)\right]$, represented by the solid line. Here ${\sigma }_{\text{saturated}}=10.06$ MPa is the maximum possible interfacial strength and $\alpha =0.08474$. The goodness of fit is measured as $R^2=0.9817$. The time required to achieve the saturated strength is around $t_c=70$ h.

Figure 2

Plots of the interfacial strength vs (a) $t^{1/1.5}$, (b) $t^{1/2}$, (c) $t^{1/3}$, and (d) $t^{1/4}$, where the unit of $t$ is hour. The left side of the vertical blue line represents the time period over which the evolution of strength takes place and the right-hand side represents a steady state of interfacial strength. Fitting $\sigma ={\alpha }_1+{\alpha }_2{t}^{1/n}$ to the data points, the constants are obtained as ${\alpha }_1=0.52,{\alpha }_2=0.9875$ for $n=1.5$; ${\alpha }_1=-0.55,{\alpha }_2=1.886$ for $n=2.0$; ${\alpha }_1=-2.731,{\alpha }_2=3.981$ for $n=3.0$; and ${\alpha }_1=-4.921,{\alpha }_2=6.061$ for $n=4.0$. Both ${\alpha }_1$ and ${\alpha }_2$ depend linearly on $n$: ${\alpha }_1=3.841-2.2n$ and ${\alpha }_2=-2.136+2.04n$. The vertical dashed blue line represents the transition line at $t=t_t=30$ h.

Figure 3

Left: Strength variation as a function of the curing time $t$. The data points (shown by filled circles) are fitted by the nonlinear equation: ${\sigma }_{\text{max}}\left[1-exp(-\alpha t)\right]$ where the fitting parameters are ${\sigma }_{\text{max}}=10.06$ MPa and $\alpha =0.08474$ with a goodness of fit measured as $R^2=0.9817$. Best-fit curves of the form ${\sigma }_{\text{max}}={\alpha }_1+{\alpha }_2{t}^{1/n}$ for different choices of $n$ are shown by the solid lines. The values of ${\alpha }_1$ and ${\alpha }_2$ are available in Fig. 2. Right: Comparison of the series expansion of the exponential function with a truncation of order $O\left(t^n\right)$. The vertical dashed blue line represents the transition line at $t=t_t=30$ h.

Figure 4

Schematic of the joined polymers with the yellow color representing a state during diffusion of the solvent material. Two spatially varying profiles of $E\left(x\right)$ and $G_c\left(x\right)$ with their minimum showing at the interface. Over time the minimum increases and ultimately reaches the bulk value giving the maximum strength and toughness for the joint.

Figure 5

Interfacial strength for different profiles of $E(x,t)$ and $G_c(x,t)$ for a given choice of $t$, parametrized by $\beta$. Each data point represents the critical stress (or strength) at which the interface failed for a selected profile of $E(x,t)$ and $G_c(x,t)$. The variation in strength for each choice of the exponent $n$ is fitted by a nonlinear equation of the form $c_1{t}^n$. The inset shows the finite element domain with 41 strips at the interface regime. The domain is loaded along the $x$ direction and its stress-strain responses for different profiles are examined, as described in the text. The properties $G_c(x,t)$ and $E(x,t)$ of the strips follow the mathematical profiles described in the text.

Figure 6

Experimental data on the fraction dissolved of the PMMA sample as a function of the normalized curing time, for particles of two different shapes. Fitting $1-exp(-\beta t^{*})$, we get $\beta =5.281890$ for triangle and $\beta =6.2572$ for rectangle, where $t_c=2$ h for the triangle and $t_c=8$ h for the rectangle. The surface-to-volume ratio is $(2\sqrt{3}{a}^2/4+3ah)/(h\sqrt{3}{a}^2/4)=1.51$ for the triangle; and $2(ab+bh+ah)/\left(abh\right)=0.63$ for the rectangle, indicating the surface-to-volume ratio for the triangular sample to be 2.39 times higher than the rectangular sample. The pin support shown in the schematic drawn in the inset is to make sure the PMMA sample continues to be immersed throughout the entire dissolution process.

Figure 7

Experimental data on the fraction dissolved of the PMMA sample as a function of the curing time $t$, for particles of three different shapes and four different sizes. Fitting $1-exp(-\alpha t)$, we get $\alpha =2.641,1.479$, 0.747, and 0.747 for the triangle, square, and the rectangular shapes, respectively.