Comparing simulated specific heat of liquid polymers and oligomers to experiments

Specific heat is a central property of condensed matter systems including polymers and oligomers in their condensed phases, yet predictions of this quantity from molecular simulations and successful comparisons with experimental data are scarce if existing at all. One reason for this may be that the internal energy and thus the specific heat cannot be coarse-grained so that they defy their rigorous computation with united-atom models. Moreover, many modes in a polymer barely contribute to the specific heat because of their quantum mechanical nature. Here, we demonstrate that an analysis of the mass-weighted velocity autocorrelation function allows specific heat predictions to be corrected for quantum effects so that agreement with experimental data is on par with predictions of other routinely computed quantities. We outline how to construct corrections for both all-atom and united-atom descriptions of chain molecules. Corrections computed for 11 hydrocarbon oligomers and commodity polymers deviate by <$k_{\text{B}}/10$ within a subset of nine molecules. Our results may benefit the prediction of heat conductivity.

Figure 1

Schematics showing different monomeric structures investigated in this paper. (a) Hydrocarbon structures for $n$-octane ($n=8$, including end groups) and $n$-hexadecane ($n=16$), (b) decene-dimer ($n=2$), trimer ($n=3$), and tetramer ($n=4$), and (c) isohexadecane. (d)–(h) Commodity polymer structures for poly(methyl methacrylate) (PMMA), poly(N-acryloyl piperidine) (PAP), poly(acrylic acid) (PAA), poly(acrylamide) (PAM), and poly(N-isopropyl acrylamide) (PNIPAM), respectively. Note that, for (a) and (b) and (d)–(h), chain ends outside the bracket are terminated with hydrogen atoms.

Figure 2

Normalized mass-weighted velocity autocorrelation function $C\left(\mathrm{\Delta }t\right)/C\left(0\right)$ of hexadecane at temperature $T=300$ K (blue) and at $T=560$ K (red).

Figure 3

(a) and (d) Vibrational spectra $g\left(\nu \right)$, (b) and (e) specific heat corrections $\mathrm{\Delta }{c}_p$, as well as (c) and (f) specific heats for (a)–(c) hexadecane in the top row and (d)–(f) for poly(methyl methacrylate) (PMMA) in the bottom row. In each case, $g\left(\nu \right)$ was obtained at a low (blue) and a high (red) temperature and $\mathrm{\Delta }{c}_p$ deduced from it. The corresponding blue and red curves essentially overlap in (b) and (e). Their differences (diff.) are shown in their insets. Experimental data on $c_p$ for $n$-hexadecane [38] and PMMA [39] are shown in black lines. They are compared with three numerical datasets: classical all-atom simulations (blue circles), results obtained using the harmonic reference method [21] (green triangles down), and from the methodology proposed in this paper (red triangles up).

Figure 4

Specific heat corrections for explicit-atom simulations of all chain molecules investigated in this paper. Only lines are shown at temperatures where commodity polymers could not be equilibrated using feasible computing times.

Figure 5

Specific heat predictions from all-atom simulations of various chain molecules after applying quantum corrections grouped into (a) hydrocarbon oligomers and (b) commodity polymers.

Figure 6

(a) A comparison of spectra of $n$-hexadecane from all-atom and united-atom models at 430 K. (b) The difference spectrum (diff) $g_{\text{diff}}\equiv g_{\text{AA}}-g_{\text{UA}}$ is compared with the explicit-H spectrum $g_{\text{H}}$ obtained as described in Sec. 3b. The latter is shown in its original (orig.) and rescaled (resc.) version in green dotted and solid lines, respectively. (c) Integral over the spectra shown in (b). Here, the data for the crude estimation are obtained by the weighted linear combination of the data shown in Fig. S3. (d) Ignored $c_p$ of $n$-hexadecane in united-atom models retrieved via the three approaches described in Sec. III.B.

Figure 7

Specific heat $c_p$ of $n$-hexadecane as a function of temperature: experimental data (black lines), uncorrected $c_p$ of a classical, united-atoms-based simulation before (blue circles) and after (green triangles down) applying quantum corrections, as well as full estimates for $c_p$ (red triangles up) obtained using Eq. (7), which includes corrections for ignored H atoms. The experimental data on $c_p$ is taken from Ref. [38].