Healing of polymer interfaces: Interfacial dynamics, entanglements, and strength

Self-healing of polymer films often takes place as the molecules diffuse across a damaged region, above their melting temperature. Using molecular dynamics simulations we probe the healing of polymer films and compare the results with those obtained for thermal welding of homopolymer slabs. These two processes differ from each other in their interfacial structure since damage leads to increased polydispersity and more short chains. A polymer sample was cut into two separate films that were then held together in the melt state. The recovery of the damaged film was followed as time elapsed and polymer molecules diffused across the interface. The mass uptake and formation of entanglements, as obtained from primitive path analysis, are extracted and correlated with the interfacial strength obtained from shear simulations. We find that the diffusion across the interface is significantly faster in the damaged film compared to welding because of the presence of short chains. Though interfacial entanglements increase more rapidly for the damaged films, a large fraction of these entanglements are near chain ends. As a result, the interfacial strength of the healing film increases more slowly than for welding. For both healing and welding, the interfacial strength saturates as the bulk entanglement density is recovered across the interface. However, the saturation strength of the damaged film is below the bulk strength for the polymer sample. At saturation, cut chains remain near the healing interface. They are less entangled and as a result they mechanically weaken the interface. Chain stiffness increases the density of entanglements, which increases the strength of the interface. Our results show that a few entanglements across the interface are sufficient to resist interfacial chain pullout and enhance the mechanical strength.

Figure 1

Snapshots showing the evolution of the interface due to interdiffusion and shear after bonds crossing the plane $z=0$ are cut. Beads are colored based on their positions before the cut: $z>0$ (blue) and $z<0$ (yellow). For clarity only a portion of the sample, $40a$ by $60a$ in the $y$-$z$ plane and $10a$ deep along $x$ is shown. Snapshots (a)–(c) depict the interface at interdiffusion times $t=0.01M\tau$, $0.5M\tau$, and $7M\tau$, respectively. Snapshots (d)–(f) show the corresponding states after a large shear strain $\gamma =12$ along $y$. Chains have initial length $N=500$ and $T=0.2u_0/k_B$.

Figure 2

Mean-squared displacement for the center-of-mass $g_3\left(t\right)$ (solid lines) and the center 10 individual beads of the chain $g_1^{\mathrm{in}}\left(t\right)$ (dashed lines) during self-diffusion in bulk systems with $N=500$ and $k_{\mathrm{bend}}/u_0=0$ (black squares), $0.75$ (red circles), and $1.5$ (green triangles).

Figure 3

The probability $Np\left(N\right)$ that a monomer is in a cut segment of length $N$ for $k_{\mathrm{bend}}/u_0=0$ (black squares), $0.75$ (red circles), and $1.5$ (green triangles). Uncut chains are not included in the statistics.

Figure 4

Snapshots of the primitive paths for the states in Figs. 1–1. Beads are colored based on their positions before cutting: $z>0$ (blue) and $z<0$ (yellow).

Figure 5

Density profiles $\rho \left(z\right)$ of monomers belonging to segments of the indicated length for fully flexible chains with $k_{\mathrm{bend}}=0$ at (a) $t=0.01M\tau$, (b) $1M\tau$, and (c) $16M\tau$. There are roughly equal numbers of monomers in segments of all length (Fig. 3), so the total number of monomers with $3N_e<N<500$ is roughly 3 times larger than the number in the other length ranges which all have width $N_e$.

Figure 6

(a) Mass uptake per unit area $M\left(t\right)$ as a function of interdiffusion time $t$ for healing samples with $k_{\mathrm{bend}}/u_0=0$ (black squares), $0.75$ (red circles), and $1.5$ (green triangles). A dashed line shows the $t^{0.2}$ scaling that curves follow at $t>t_e$. Also shown is $M\left(t\right)$ for a welding sample with $k_{\mathrm{bend}}/u_0=0$ (open black squares). (b) Mass uptake for all monomers (black squares) and monomers in segments with $1\le N<N_e$ (red circles), $N_e\le N<2N_e$ (green triangles), $2N_e\le N<3N_e$ (blue stars), $3N_e\le N<4N_e$ (cyan diamonds), and uncut chains (magenta crosses) during healing of a $k_{\mathrm{bend}}=0$ sample.

Figure 7

(a) Average interdiffusion depth $\langle d\rangle$ of monomers from one side into the opposite side for the same systems as in Fig. 6. (b) Separate averages for monomers in segments of the indicated lengths in healing of $k_{\mathrm{bend}}=0$ chains and for all chains during welding (open squares). For healing, separate results are shown for all monomers (black squares) and monomers belonging to chains with $N<N_e$ (red circles), $N_e\le N<2N_e$ (green triangles), $2N_e\le N<3N_e$ (blue stars), $3N_e\le N<4N_e$ (cyan diamonds), and $N=500$ (magenta crosses).

Figure 8

(a) Average contour length $\langle l\rangle$ and (b) average depth $\langle h\rangle$ of loops at the interface for the healing samples $k_{\mathrm{bend}}/u_0=0$ (black squares), $0.75$ (red circles), and $1.5$ (green triangles), and the welding sample with $k_{\mathrm{bend}}=0$ (open black squares). (c) Time dependence of the areal density $\Sigma$ of interfacial loops contributing to $\langle l\rangle$ and $\langle h\rangle$. Only interfacial loops that have more than three monomers are considered.

Figure 9

Density profiles at different interdiffusion times for TCs between chains from (a) either side and (b) opposite sides (interfacial TCs) of the interface in the healing sample of fully flexible chains $(k_{\mathrm{bend}}=0)$. Only interior TCs with the number of monomers to the nearest chain end $n_{\mathrm{end}}$ larger than 30 are included. Inset of (b) shows the development of the areal density of interfacial TCs with interdiffusion time for both healing (filled symbols) and welding (open symbols) samples with $k_{\mathrm{bend}}=0$. Results are shown for TCs with $n_{\mathrm{end}}>30$ (black squares), 50 (blue left triangles), and 70 (magenta right triangles), respectively.

Figure 10

(a) The areal density of interfacial TCs and (b) the ratio of areal and bulk densities versus $\langle d_N\rangle$, the average interpenetration depth of monomers weighted by the segment length $N$. Only interior TCs that are more than 30 monomers from a chain end are included. Results are shown for the healing samples with $k_{\mathrm{bend}}/u_0=0$ (black squares), $0.75$ (red circles), and $1.5$ (green triangles), and the welding sample with $k_{\mathrm{bend}}=0$ (open black squares).

Figure 11

(a) Average shear stress vs. shear strain curves (solid lines and symbols) for bulk samples with $k_{\mathrm{bend}}/u_0=0$ (black squares), $0.75$ (red circles), and $1.5$ (green triangles). Dashed lines and open symbols show the rate of bond breaking $d{\rho }_{\mathrm{broken}}/d\gamma$, where ${\rho }_{\mathrm{broken}}$ is the density of broken bonds. (b) Average bond tension $f$ along the chain as a function of $n_{\mathrm{end}}$ for bulk samples with different $k_{\mathrm{bend}}$ at the shear strain ${\gamma }^c$ where significant bond breaking initiates (solid symbols) and ${\gamma }^{\max }$ where the shear stress reaches its peak value (open symbols). For $k_{\mathrm{bend}}/u_0=0$, $0.75$, and $1.5$, ${\gamma }^c=9$, 7, and 5 and ${\gamma }^{\max }=10$, 8, and 6, respectively. Solid lines are results from fitting the corresponding data points using $f=f_0[1-exp(-n_{\mathrm{end}}/n_{\mathrm{end}}^c)]$. The increase of $n_{\mathrm{end}}^c$ with shear strain $\gamma$ is shown in the inset of (b) for different $k_{\mathrm{bend}}$.

Figure 12

(a) The maximum shear stress ${\sigma }^{\max }$ for monodispersed bulk samples with different chain lengths for $k_{\mathrm{bend}}/u_0=0$ (black squares), 0.75 (red circles), and 1.5 (green triangles). (b) Rescaled plot showing the percentage of the increase with chain length as a function of $N$.

Figure 13

Stress-strain curves from shear tests on different states for healing (solid lines and symbols) and welding (dashed lines and open symbols) of fully flexible chains $(k_{\mathrm{bend}}=0)$. Symbols indicate the peak stress for each curve. Results are shown for interdiffusion time $t=0.01M\tau$ (magenta diamonds), $0.1M\tau$ (blue stars), $0.5M\tau$ (green triangles), $2M\tau$ (red circles), $5M\tau$ (black squares), and $16M\tau$ (dark yellow cross, only for healing). Also shown is the corresponding average stress-strain curve for the bulk (olive dotted line and inverted triangle).

Figure 14

(a) The maximum shear stress ${\sigma }^{\max }$ is shown versus $t$ for both healing (solid squares) and welding (open squares) samples of fully flexible chains. (b) Time dependence of ${\sigma }^{\max }$ for healing samples with $k_{\mathrm{bend}}/u_0=0$ (black squares), $0.75$ (red circles), and $1.5$ (green triangles). Dotted lines indicate the corresponding bulk strengths.

Figure 15

The maximum shear stress ${\sigma }^{\max }$ versus the average contour length $\langle l\rangle$ of interfacial loops for welding of flexible chains (open squares) and healing with $k_{\mathrm{bend}}/u_0=0$ (filled squares), 0.75 (circles), and 1.5 (triangles).

Figure 16

(a) Maximum shear stress ${\sigma }^{\max }$ versus the areal density of interfacial TCs for healing (filled symbols) and welding (open symbols) with $k_{\mathrm{bend}}/u_0=0$ (squares), $k_{\mathrm{bend}}/u_0=0.75$ (circles), and $k_{\mathrm{bend}}$ (triangles). (b) Same plot but excluding entanglements less than 30 monomers from a chain end. (c) Same data showing fraction of increase toward bulk strength against areal density of interfacial entanglements normalized by bulk density.