Method for estimating the cooperativity length in polymers

The problem of estimating the size of the cooperatively rearranging regions (CRRs) in supercooled polymeric melts from an analysis of the $\alpha$-process in ordinary relaxation experiments is addressed. The mechanism whereby a CRR changes its configuration is viewed as consisting of two distinct steps: a reduced number of monomers reaches initially an activated state, allowing for some local rearrangement; then, the subsequent regression of the energy fluctuation may take place through the configurational degrees of freedom, thus allowing for further rearrangements on larger length scales. The latter are indeed those to which the well-known Donth's scheme refers. Local readjustments are described in the framework of a canonical formalism on a stationary ensemble of small-scale regions, distributed over all possible energy thresholds for rearrangement. Large-scale configurational changes, instead, are described as spontaneous processes. Two main regimes are envisaged, depending on whether the role played by the configurational degrees of freedom in the regression of the energy fluctuation is significant or not. It is argued that the latter case is related to the occurrence of an Arrhenian dependence of the central relaxation rate. Consistency with Donth's scheme is demonstrated, and data from the literature confirm the agreement of the two methods of analysis when configurational degrees of freedom are relevant for the fluctuation regression. Poly($n$-butyl methacrylate) is chosen in order to show how CRR size and temperature fluctuations at rearrangement can be estimated from stress relaxation experiments carried out by means of an atomic force microscopy setup. Cases in which the configurational pathway for regression is significantly hindered are considered. Relaxation in poly(dimethyl siloxane) confined in nanopores is taken as an example to suggest how a more complete view of the effects of configurational constraints would be possible if direct measurements of temperature fluctuations were combined with the proposed analysis.

Figure 1

Average energy ${〈E〉}_{\zeta }$ and its product with $w_{\zeta }$ as functions of the energy threshold $\zeta$. All energies are expressed in kcal/mol.

Figure 2

Mechanical loss patterns of a PET sample cold-crystallized at $T_c=100{}^{\circ }\text{C}$ for 7 h. The dashed lines represent the fast and slow components deconvoluted from the $T=95{}^{\circ }\text{C}$ curve. The inset shows a comparison of the $N_{\alpha }$ estimates based on Donth's method [5] (open triangles) and the present analysis (filled circles).

Figure 3

Temperature dependencies of $\xi$ (filled symbols) and $\mathrm{\Delta }T$ (open symbols) for the bulk (circles) and 7.5 nm confined (triangles) PDMS derived by the analysis of the data of Ref. [8]. The lines are guides for the eye. The approximate values of the calorimetric glass transition temperatures, $T_{g,\text{bulk}}$ and $T_{g,7.5}$, are indicated by the vertical arrows.

Figure 4

Relaxation functions (symbols) corresponding to the parameters of Ref. [22] considered to work out the results of Table 4 and fittings with Eq. (5) (solid lines). The inset shows the VFT lines of Refs. [21] (solid) and [22] (solid and dashed); the dashed part is an extrapolation out of the interval represented in [22].

Figure 5

Schematic representation of a force relaxation experiment performed by AFM. (a) Scheme of the $Z$-piezo movement (lower plot) and of the corresponding cantilever deflection (upper plot) as a function of time. The cantilever deflection is converted to the applied force by considering the force constant of the cantilever. A step toward the $Z$-piezo displacement is applied and, after reaching a maximum value, the force applied by the tip decreases with time. (b) Example of a force relaxation pattern obtained with a cantilever having a spring constant of 40 N/m whose integrated tip has been rounded by focused ion beam milling to decrease the applied pressure and prevent plastic deformations of the polymer. The upper curve represents the corresponding indentation as a function of time.

Figure 6

Comparison between typical approach-retraction force curves in the presence or absence of water. Adhesion is pointed out by the negative force values revealed in the retraction curve (*) collected in the absence of water.

Figure 7

Temperature dependencies of CRR size and temperature fluctuations, $\xi$ (squares) and $\mathrm{\Delta }T$ (circles), respectively, for a sample of PnBMA $(T_g\simeq 298$ K). The dashed and dotted lines are linear fits, but they only serve as guides for the eye. The inset shows the VFT behavior; the Vogel temperature $T_{\mathrm{VFT}}\simeq 226$ K was found by fitting with Eq. (7) and setting ${\tau }_{\infty }$ to ${10}^{-14}$ s.