Correction: Modelling of thermal transport through a nanocellular polymer foam: toward the generation of a new superinsulating material

The authors have noticed that the finite discretization for d θ 1 used in calculating the integrals of eqn (34) and (35) in the published paper had significant e ﬀ ects on the calculation accuracy. In the published paper, the range of θ 1 was divided into 1000 equidistant Δ θ 1 pieces. It was found that the finite discretization was not fine enough, and this led to calculation errors in the cases where the cell wall became extremely thin (<4 nm) either at small cell sizes or at high void fractions. It was found that θ 1 should be divided into several thousands of intervals in discretization to ensure a high calculation accuracy, as shown in Fig. C1. The authors have recalculated the data presented in the published paper by using a much finer discretization with 10 000 intervals. With a much finer discretization, R f increased significantly at high void fractions compared to the original case. Subsequently, both λ rad and λ e ﬀ decreased obviously at high void fractions. Thus, the originally published Fig. 8 should be replaced by the updated version of Fig. 8 provided below. It should be noted that λ con was not a ﬀ ected by discretization. Thus, the data in Fig. was not

The authors have recalculated the data presented in the published paper by using a much finer discretization with 10 000 intervals. With a much finer discretization, R f increased significantly at high void fractions compared to the original case. Subsequently, both λ rad and λ eff decreased obviously at high void fractions. Thus, the originally published Fig. 8 should be replaced by the updated version of Fig. 8 provided below. It should be noted that λ con was not affected by discretization. Thus, the data in Fig. 8b was not changed.
Regarding the data presented in the originally published Fig. 9, λ eff , λ rad , and R f changed obviously at small cell sizes when using a much finer discretization in the calculations. Thus, this figure should be replaced by the updated version of Fig. 9 provided below. In the updated version of Fig. 9c, it should be noted that the behaviour of the radiative thermal conductivity as a function of the cell size changes significantly in comparison with the data presented in Fig. 9c of the published paper. In the updated version of Fig. 9c, the radiative thermal conductivity reduced first to a certain minimum level, and then increased gradually to a certain maximum value with decreasing cell size. This in turn led to the increase in the total thermal conductivity upon  Fig. 9a. This phenomenon was due to the sharp decrease in the reflectance of the single cell wall with decreasing cell size (Fig. 9d), and increasing the number of cell walls could not offset the adverse impact on the total reflectance of IR waves.
As shown in the original versions of Fig. 9a and c, both λ eff and λ rad first decreased, then increased, and finally decreased again, upon reducing the cell size. Thus, both λ eff and λ rad showed maximum peak values. In order to explain the final downward trend of the two variables, it was stated in the published paper that the increase in the rate of the cell wall number (n) was much faster than the declining rate of the wavelength-averaged reflectance of the single cell wall R f À Á . However, this was not true because n and R f were not compared over an equivalent numerical range. In fact, if an equivalent numerical range were to be employed, it would be found that the decrease in R f would be so fast that the increase in n could not offset its adverse impact on the total reflectance of IR waves. Subsequently, the maximum peak values that appeared in the original versions of Fig. 9a and c should not have existed, as shown in the updated versions of Fig. 9a and c. It was inferred that, as the cell size reduced indefinitely, R f would gradually approach zero and T unit net would approach one. According to eqn (41) shown in the published paper, the radiative thermal conductivity would finally approach its maximum value of 4 Â σ SB ÂT 3 Â L, which is 257.5 mW m −1 K −1 .
When calculating with a much finer discretization, R f decreased significantly at high void fractions compared with the originally published data. Subsequently, both λ rad and λ eff increased significantly at high void fractions in comparison with the orig- Fig. 8 Dependence of the thermal transport on the foam's void fraction under various cell size levels. (a) Correlation between the void fraction (ε) and the total effective thermal conductivity (λ eff ). (b) Correlation between the void fraction and the thermal conductivity contributed by thermal conduction (λ con ). (c) Correlation between the void fraction and the radiative thermal conductivity (λ rad ). (d) Correlation between the void fraction and the wavelength-averaged reflectance of the single cell wall R f À Á .

Fig. 9
Dependence of the thermal transport on the foam's cell size under various void fraction levels. (a) Correlation between the cell size (δ c ) and the total effective thermal conductivity (λ eff ). (b) Correlation between the cell size and the thermal conductivity contributed by thermal conduction (λ con ). (c) Correlation between the cell size and the radiative thermal conductivity (λ rad ). (d) Effect of the cell size on the wavelength-averaged reflectance of the single cell wall R f À Á and the polymer slab numbers (n).
inal data. Thus, Fig. 10 in the published paper should be replaced by the updated version of Fig. 10 provided below. Notably, it can be seen in the updated version of Fig. 10b that the maximum value of λ rad could be up to 250 mW m −1 K −1 , which is much larger than the maximum value reported in the original version of Fig. 10b. Consequently, the maximum value of λ eff reported in the updated version of Fig. 10a was also much larger than the maximum value reported in the original version of Fig. 10a.
For the same reason, Fig. 11 in the published paper should also be replaced by the updated version of Fig. 11 provided below. Notably, both λ rad and λ eff increased significantly at high void fractions in comparison with the original data. However, the change in discretization here did not change the variation trend of the variables. Moreover, the minimum effective thermal conductivity (λ Min eff ) calculated for the different cases only changed slightly. Fig. 12 in the published paper should also be replaced by the updated version of Fig. 12 provided below. Overall, the contour isotherms of the thermal conductivity did not exhibit any obvious changes in comparison with the original version of Fig. 12, except for the data at small cell sizes and high void fractions.  Also, the formulas (23)-(33) in the paper need to be corrected because the refraction angle, θ 2 , which should be a complex number, was mistakenly processed as a real number when preparing the original article. However, these errors did not affect the calculation results, because θ 2 had been correctly taken as a complex number in all previous modelling and calculation works.
The formula (29) used to calculate the transmittance of polymer film in the article should be changed to the following: 1 þ ρ gs 2 ρ sg 2 e À8πδwυs=λ þ 2ρ gs ρ sg e À4πδwυs=λ cos ω 2 ð29Þ For the incident radiation wave with the electric vector perpendicular to the plane of incidence, formulas (30) and (31) used to determine τ gs and τ sg should be respectively replaced with the following formulas: τ gs ¼ 2n g cos θ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn g cos θ 1 þ u s Þ 2 þ υ s 2 q ð30Þ Meanwhile, for the incident radiation wave with the electric vector parallel to the plane of incidence, formulas (32) and (33) should be respectively changed to the following: τ gs ¼ 2ðn s 2 þ κ s 2 Þ cos θ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ½ðn s 2 À κ s 2 Þ cos θ 1 þ n g u s 2 þ ð2n s κ s cos θ 1 þ n g υ s Þ 2 q ð32Þ τ sg ¼ 2n g ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u s 2 þ υ s 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ½ðn s 2 À κ s 2 Þ cos θ 1 þ n g u s 2 þ ð2n s κ s cos θ 1 þ n g υ s Þ 2 q ð33Þ In addition to the above corrections, eqn (36) in the published paper should be changed to the following: f ðλÞ ¼ 2c 2 πh λ 5 expðhc=λk B TÞ ð36Þ These errors do not affect the main conclusions of the paper. The Royal Society of Chemistry apologises for these errors and any consequent inconvenience to authors and readers.

Nanoscale Correction
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