Thermodynamic limits on cell size in the production of stable polymeric nanocellular materials

Journal Pre-proof Thermodynamic limits on cell size in the production of stable polymeric nanocellular materials Sergio Estravis, Alan H. Windle, Martin van Es, James A. Elliott PII:

S0032-3861(19)31042-0

DOI:

https://doi.org/10.1016/j.polymer.2019.122036

Reference:

JPOL 122036

To appear in:

Polymer

Received Date: 8 September 2019 Revised Date:

21 November 2019

Accepted Date: 27 November 2019

Please cite this article as: Estravis S, Windle AH, van Es M, Elliott JA, Thermodynamic limits on cell size in the production of stable polymeric nanocellular materials, Polymer (2019), doi: https://doi.org/10.1016/ j.polymer.2019.122036. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Thermodynamic Limits on Cell Size in the Production of Stable Polymeric Nanocellular Materials CRediT author statement:

Sergio Estravis: Writing – Original Draft, Conceptualization, Investigation, Methodology Alan H. Windle: Supervision Martin van Es: Supervision James A. Elliott: Writing – Reviewing and Editing, Conceptualization, Supervision, Funding acquisition,

1

Thermodynamic Limits on Cell Size in the Production of Stable Poly-

2

meric Nanocellular Materials

3

Sergio Estravisa, c, *, Alan H. Windlea, Martin van Esb, James A. Elliotta

4

a.- Department of Materials Science & Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge, CB3 0FS,

5

United Kingdom

6

b.- SABIC T&I, Urmonderbaan 22 6167RD Geleen, Netherlands

7

c.- Department of Advanced Materials, International Centre of Advanced Materials and Raw Materials (ICAMCyL), León,

8

Spain (Present address)

9 10

*author to whom correspondence should be addressed: [email protected]

11

KEYWORDS: Nanofoams, Modelling, Molecular Dynamics

12

ABSTRACT: The reduction of cell size to the nanometre scale is one of the main current focuses in polymeric cellular materials

13

research. The recent achievement of cell sizes in the range of 10-200 nm naturally gives rise to the question of what are the main

14

limiting factors in obtaining further improvements. This paper explores the theoretical limits of this reduction in scale size, and

15

presents an application of atomistic modelling strategies for the study of nanofoams produced by the gas dissolution technique.

16

The main conclusion is that, as the cell size scale becomes closer to the molecular sizes, atomic-scale interactions should be con-

17

sidered, and conventional rules for foams usually applied to continuous media should be reassessed due to the increasingly dis-

18

crete nature of the polymeric material. The gas-polymer equilibrium for a range of different polymers (HDPE, PP, PVDF, PEI, PS,

19

PES, PC and PMMA) were calculated using molecular dynamics (MD), in order to generate structures in the nanometric range and

20

to study their stability. The results show that the minimum stable cell size is governed by the behaviour of the internal pressure in

21

the polymer within small gas-filled voids, which may explain the differences between cell sizes observed experimentally in differ-

22

ent polymers.

23

1

INTRODUCTION

2

The production of polymeric nanoporous materials with both high internal porosity (>80%) and low cell sizes (<100 nm) is current-

3

ly of great interest due to their potentially exciting properties such as high impact resistance1, low thermal conductivity2,3 and high

4

stiffness-to-weight ratio1. Although at present there are a wide variety of materials possessing this type of structure (e.g. zeo-

5

lites4,5, MOFs6,7, PIMs8, nanoporous membranes9,10, nanobubbles11,12), traditional techniques employed for producing them, such

6

as phase inversion13, track etching14–16 or pattern transfer17, are restricted to the production of thin films and/or low porosity

7

materials, typically oriented to ultrafiltration18, and which present several problems related to the use of toxic solvents and/or

8

scalability. Solid state foaming is an alternative technique able to produce samples with high porosity19–21 and is not restricted to

9

ultra-thin films22,23, with the possibility of producing materials at a higher scale24. Although the terms “polymeric nanofoams” or

10

more accurately “polymeric nanocellular materials” refer to low density materials (usually produced by gas expansion within a

11

matrix), and “nanoporous materials” refers to open cell materials employed in membranes, during the present article all these

12

terms will be employed with no distinction.

13

Solid state foaming has been successfully employed previously for the manufacturing of microcellular materials25,26, even in a

14

continuous process27, and in the last few years it has also been applied to the development of nanocellular materials1–3,19–24,28,29. In

15

this technique, a gas, usually CO2, saturates the polymer under high pressure conditions. Once saturation is produced, the sample

16

is heated beyond its glass transition temperature in order to allow the expansion of the infused gas. This final step is carried out in

17

one or two stages, depending on the Tg of the polymer and how it varies with the saturation of CO2.

18

Considering the state-of-the-art in experimental production of nanofoams by solid state foaming, the materials produced show

19

cell sizes as low as 10 nm

20

foamed with this technique. The employment of copolymers and blends pursues the dual objective of generating nanostructures

21

with zones of higher CO2 affinity and increasing the number of heterogeneous nucleation points (i.e. increasing the porosity while

22

reducing the cell size)20,29,31–34. An alternative strategy to achieve higher porosities with nanometric cell sizes has been the intro-

23

duction of nanostructured phases inside the solid polymer before foaming. These nanometric phases act as heterogeneous nucle-

24

ation points, increasing the final number of cells and allowing to achieve lower densities35, 36. However, it appears that nanofoams

25

with porosities higher than 80% and cell sizes lower than 50 nm are not achievable using current materials, it being difficult to

26

produce stable nanoporous materials from most thermoplastic polymers in this density-cell size range.

27

Although the existence of these limits in the production of nanofoams is quite evident from the experimental literature, a detailed

28

explanation of the physical processes involved is still lacking. This article presents an account of the limitations in the reduction of

30

with a wide range of porosities. Pure polymers, block copolymers and polymer blends have been

2

1

cell size to the nanometric range. The starting point is a simplified calculation of the change in surface energy that occurs when

2

cell size is reduced at constant relative density, and the expected variations in the foam morphology (nucleation density and wall

3

thickness) that follow from this reduction. Due to the nanoscopic size scales involved, these calculations are complemented by

4

molecular dynamics (MD) simulations of mixtures of different polymers and CO2, which allow a more rigorous evaluation of the

5

energetics without the approximations of continuum mechanics. Although other theoretical approaches have been made to the

6

study of the nanofoams question, they either used a lattice model37–42 or have been more focused on explaining the final proper-

7

ties of nanofoams43. This molecular approach is necessary for nanofoams, as opposed to more conventional cellular materials,

8

due to the much smaller scale of the cell structure. With this approach, it has been possible to study the phase equilibrium be-

9

tween gas and polymer for the different materials under study, through the creation of nanometric cavities inside them. It is

10

important to emphasize that this study does not aim to present a detailed description of the processes involved in the creation of

11

the nanocellular structures (either by nucleation or spinodal decomposition). Rather than this, the purpose is to answer two fun-

12

damental questions: what determines the stability of a nanoscale void at the smallest scale, and how does the chemical structure

13

of polymer affect this stability. The results of this investigation will help to select the most optimal polymers to reach the lower

14

cell sizes.

15

The study starts by describing the energetic limitations to the reduction of the cell size in porous materials from a continuum

16

approach, considering the energy associated to the creation of new surfaces. Different cellular morphologies are studied (includ-

17

ing spherical and truncated octahedron, open and closed cells). These calculations are complemented by the morphological limi-

18

tations associated with the cell size reduction. The obtained results justify the employment of MD simulations to the study of the

19

nanofoams, as the scale reached by these cell structures is in the molecular range. These MD simulations provide a better under-

20

standing of nanofoams and allow the direct calculation of a parameter that quantifies the opposition of the polymer to a variation

21

of its volume. The parameter is the internal pressure, which in turn governs the creation of a void (or equivalently a nano-cell)

22

inside the polymer. The internal pressure of a polymer presents a considerably variation respect to the size of the void, resulting

23

essential in the understanding of the stability of cells. Furthermore, it plays a very important role during the whole foaming pro-

24

cess, as it will be explained in the following discussion.

25

THEORETICAL CONSIDERATIONS

26

I. Surface energy variation with cell size reduction

27

(Further details about the calculations in this section can be found in the supplementary information, Part 1: Surface Energy Vol-

28

ume Ratio (SEVR) calculations)

3

1

Cell size reduction has clear implications for the energy of the system if we consider the amount of surface that has to be gener-

2

ated to produce nanocellular materials. Previously, the concept of line tension has been studied as the scale of cells is reduced to

3

the nanometre range, both in the case of nanobubbles44 and in the case of nanofoams produced with the help of nucleating

4

agents like silica36. Line tension is a thermodynamic concept introduced by Gibbs45 as an analogue to the surface tension: while

5

surface tension is defined as the excess free energy per unit surface of an interface separating two phases, line tension emerges

6

as the equivalent concept when three different phases coexist. In the case of nanofoams, it has been employed to have a better

7

understanding of the heterogeneous nucleation process, making possible the improvement of cell nucleation efficiency by the

8

modification of the nanoparticles surface. However, as it has been mentioned, this magnitude emerges when three different

9

phases coexist, like in the nucleation process of nanofoams. This article is focused in the stability of polymeric nanofoams, where

10

only two phases coexist, so only surface energy will be considered.

11

In order to quantify the energies involved, this article introduces a quantity with dimensions of energy density, or equivalently

12

pressure, called the surface energy to volume ratio (SEVR). SEVR can be defined as the ratio of total surface energy (S × γ, where S

13

is the total internal surface of a foam in a volume V and γ is the surface tension) and volume (V) [Eq. 1] with units of J·cm−3 (so

14

that corresponding pressures can be numerically expressed in MPa). SEVR =

·

(1)

15

To determine the amount of surface, different approaches can be made depending on the accuracy of the estimation required.

16

The simplest one is to consider spherical, isolated cells. This approach is typically more accurate for high-density foams. Consider-

17

ing this, the SEVR can be expressed as [Eq. 2]:

=

6· 1−

∅

(2)

18

where ∅ is the diameter of the cell, ρf is the density of the foam and ρs is the density of the solid. As shown in Figure 1, there is a

19

considerable increase in the SEVR as the cell size is reduced to the nanometer range, with values around 200 MPa for a density of

20

0.05 g·cm‒3 (95% porosity).

21

As the density of the foams is reduced, keeping the number of cells constant, the diameter of these cells will grow, and their

22

shape will be closer to a truncated octahedron. Two assumptions can be adopted here: constant thickness among these octahe-

23

drons, and the average cell size approximated as the distance from the centre of the octahedron to the centre of each square in

24

the surface. Then, the expression for the dependence of the SEVR with the density of the foam, the density of the solid and the

25

main cell size will be [Eq.3]

4

≅

6.7 · 1 −

(3)

∅

1

Finally, it is generally found for nanofoams that the open cell content increases as the density is reduced. This increase of open

2

cells can be understood as the way materials reduce the amount of surface and then the total energy of the system. Assuming

3

that the truncated octahedrons have cylindrical edges, a calculation of the surface energy can be also made, showing the next

4

dependence with cell size and densities [Eq. 4]

≅

6 ·

· 1−

1.18 ·

·∅

(4)

5

Given these different approaches to SEVR, it can be useful to compare their behaviour for a given material with a fixed density.

6

Considering amorphous PE with a solid density of 0.9 g/cm3 and a foamed density of 0.05 g/cm3, Fig. 1 shows the dependence

7

with the cell size of the SEVR magnitude. It is possible to observe that for the larger cell sizes, the results for spherical and trun-

8

cated octahedron approaches are quite similar, but as the cell size is reduced, the differences between the two of them emerge.

9

By contrast, the open cell structure gives a considerable reduction of the SEVR for all the cell sizes, but especially for the smaller

10

ones. It suggests that in low density nanofoams, the presence of open cells is a direct way of reducing the surface energy and

11

producing more stable materials. Therefore, it is much more likely to obtain open cell structures as the cell size is reduced, as it

12

has been observed experimentally34, 46.

13 14 15 16

Figure 1. SEVR (in J/cm3) dependence with cell size (nm) for PE and different cellular structures: spherical cells, truncated octahedron with closed cell and truncated octahedron with open cell. The magnitude of these energy densities, in the case of the highest SEVR’s considered here, can be compared (within an order of

17

magnitude) to the chemical energy densities of gunpowder or gasoline. This gives an idea of the amount of energy involved in the

18

creation of nanofoams with the cell size is reduced to the range of nanometre.

5

1

II. Effects of cell size reduction on nanofoam morphology

2

As stated above, the relationship between surface energy and reduction of the cell size to the nanometric range was presented in

3

terms of the magnitude of the SEVR. Although this quantity plays an important role in the creation of nanofoams, other morpho-

4

logical considerations are also relevant as the cell size is reduced. Two of them are the nucleation density and the cell wall thick-

5

ness.

6 7

Figure 2. Surface Energy to Volume Ratio and Nucleation density (cm-3), for PE foam with density ρf = 0.05 g/cm3.

8

Figure 2 shows the variation of the nucleation density (ND), i.e. the number of nucleation points per unit volume, with cell size for

9

a fixed density, together with the SEVR. This property shows a considerable increase with the decrease of the cell size. The ex-

10

pression for nucleation density (ND) is presented in Eq. 547, and it depends on the cell size and the relative density of the foam

11

(porosity). In this plot the range of cell size is restricted to more experimentally accessible values according to experimental state

12

of the art. Although the change in the SEVR magnitude for values close to 10 nm is not extreme, the increase in the number of

13

nucleation points of several orders of magnitude is dramatic. It is clear from this representation that high porosity-low cell size

14

zone implies extremely high surface energies and nucleation densities, and the understanding of these values depending on the

15

material properties and will determine the smaller cell sizes/densities achievable.

ND =

6 · 10 ! 1 # $ ∅" &

%

− 1'

(5)

16

The limitations produced by the cell wall thickness reduction as the cell size decreases are quite considerable too. As Notario48 et

17

al have recently indicated, the equations proposed by Gibson and Ashby49 can be employed for the relationship among densities,

18

cell sizes and wall thickness in truncated octahedrons, both closed and open cell. $ %

= 1.18

( )

(6)

6

$ %

= 1.06

( )

(7)

1

Equation 6 is for closed cell structures, while Eq. 7 is for open cell structures, where ρf is the density of the foams, ρs is the density

2

of the solid materials, t is the wall thickness and l is the edge length. The diameter in a truncated octahedron can be approached,

3

similarly as in the previous paragraph, as 2√2), and then the dependence of the wall thickness with the cell size (Ø) for a fixed

4

density can be plotted as it is shown in Fig. 3 (a).

5 6

(a)

7 8

(b)

9 10 11 12

Figure 3. (a) Cell wall thickness estimation for foamed PE with closed and open cell, truncated octahedron structure, ρf=0.05 g cm−3. (b) Accessible porosity-cell diameter for wall thickness bigger or smaller than 1.5 nm, for an open cellular structure based on truncated octahedron. Estimates from the Figure 3 show the low dimension of the involved structures as the cell size is reduced to the range of nanome-

13

tres: when the value of the cell size is close to 10 nm, the cell wall thickness in the case of open cell structure is reduced to below

14

1 nm in size. This is a clear physical limitation, as the average diameter of a polymeric chain is in the order of 0.5 to 1 nm. Fig.3 (b)

15

shows which cell diameter and porosities can be achieved for a wall thickness of 1.5 nm in a truncated octahedron structure with

16

open walls. Beside these limitations (that suppose a maximum nucleation density between 1018 and 1019 cm−3 for a wall thickness

7

1

between 1.5 and 3 nm) the small size of the walls has implications for the structure of the polymer too: when so small dimensions

2

are present, the use of a classical surface energy for polymer phase should be carefully reconsidered. From this point of view, the

3

employment of atomistic modelling techniques is necessary to accurately characterise the structure and energetics. Considering

4

all these preceding calculations, we now describe the results of molecular dynamics simulations for different polymer and CO2

5

mixtures.

6

MOLECULAR DYNAMICS RESULTS

7

I. Simulation Methodology

8

The Materials Studio50 software suite (version 8.0) was employed for the generation of amorphous structures of different poly-

9

mers (PE, PP, PEI, PVDF, PS, PES, PC and PMMA), and for the performing of MD simulations. Structure generation was performed

10

using the Amorphous Cell module, with initial structures built at a relatively low density (0.2 g cm−3) and 298 K. The linear dimen-

11

sions of the cubic cell were around 10 nm before cell relaxation. Geometry optimization was performed during initial structure

12

generation. Periodic boundary conditions were applied in order to avoid any bulk surface effects.

13

MD simulations were performed for both NpT and NVT ensembles, with the Forcite Dynamics Module (Polymer-Consistent Force

14

Field, PCFF), NHL thermostat, Berendsen barostat). The simulation time step was fixed at 1 fs. The amorphous polymer structures

15

previously generated were relaxed in the NpT ensemble at 298K and 1 atm pressure from low density until reaching a stable

16

value close to their nominal density (with this process typically taking around 1 ns to reach equilibrium). The NVT ensemble at

17

different temperatures was employed for the measurements of internal pressure, as will be explained in more detail below.

18

Polymer structures infused with 30wt% CO2, CH4 or Ar were generated similarly to the pure amorphous polymers above, with the

19

gas and polymer being randomly mixed at low densities and relaxed until reaching a stable density value. The starting density was

20

the only difference, depending on the polymer employed, with an initial value that varied from 0.47 to 1 g cm−3.

21

For the study of the stability of the generated nanostructures, CO2 was removed once the equilibrium was reached. Then, struc-

22

tures were relaxed again in the NpT ensemble at 298 K and 1 atm pressure until reaching stable density values, and these results

23

were compared with the pure polymer values.

24

Additionally to Molecular Dynamics simulations, an estimation of some of the properties (shear modulus and surface tension) was

25

performed with group contribution method in Synthia Module. Density functional theory (DFT) calculations presented in the

26

supplementary information were calculated using Gaussian0951 software.

8

1 2

Finally, a summary of the assumptions on which the model is based, together with some clarifications, are presented: -

Totally amorphous polymeric structures are considered: although some of the modelled polymers may also contain crys-

3

talline phases, the present study is focussed only on the amorphous phase, as gas is not able to diffuse inside the crystal-

4

line regions due to their higher degree of molecular compaction.

5

-

As it has been previously mentioned, the objective of these simulations is not to reproduce the final structure of

6

nanofoams nor their production process, but to understand how the polymer reacts against small cavities inside it. Ac-

7

cording to this, the generated structures do not present the typical morphology of nanocells or cell walls.

8

-

9

measured for them. The employment of these unrealistic amounts is justified by the necessity of creating cavities inside

10 11

The amount of gas employed is in most of the materials is sometimes higher than the sorption capacity experimentally

the polymers with a size of several nanometres. -

The instantaneous removal of CO2 does not aim to reproduce an experimental process, in which the CO2 would smoothly

12

diffuse outside the polymer. The objective is just to generate an empty structure and study its stability. Although such

13

fast removal of gas is physically unrealistic, it nevertheless allows a qualitative comparison of the behaviour of the poly-

14

mers under study.

15

Considering all these assumptions and clarifications, it should be noted that the models developed do not have the objective of

16

obtaining a detailed description of the nanofoams themselves nor their production process. The only objective is to study the

17

stability of nanometric voids inside different polymeric materials, in order to make a qualitative analysis of the differences that

18

allow some of them being more suitable for the production of nanofoams.

19

II. Pure polymer and polymer with CO2

20

Figure 4 shows the relaxed structures of one of the systems studied: pure HDPE and HDPE infused with CO2. Contrary to the ex-

21

pected expansion of the system due to the presence of a large amount of gas in a relatively small volume, the polymer and gas

22

mixture reached a density similar to that reached by the pure polymer, but with some separation of the phases. The polymer-gas

23

system showed a non-homogenous CO2 distribution, with zones where it is possible to observe a clearly higher concentration of

24

the CO2 at densities comparable to its liquid state. This resembles the phenomenon of pore condensation, where a substance in a

25

gaseous phase can be in a liquid state under certain conditions inside small pores52.

9

(a)

(b)

1 2

Figure 4. Structure of different systems after NPT relaxation at 298K: (a) amorphous HDPE, (b) amorphous HDPE (black spheres) with 30%wt CO2 (red spheres)

3

A schematic representation of the equilibrium reached by gas and polymer is represented in Figure 5: the tendency of the gas to

4

increase its volume by expansion has to be compensated by the polymer, which will exert a similar amount of pressure but in the

5

opposite direction. An understanding of this pressure is fundamental not only for explaining the equilibrium between the polymer

6

and the gas, but also because it is the pressure that a void generated inside the polymer will have to face within the nanometric

7

range in order to generate a nanofoam. It should be noted that the configurations represented on Fig. 4 and 5 do not reproduce

8

accurately the final structure of a nanocellular foam, as some of the most basic components of these structures as the cell walls

9

are missed. This simplification has some restrictions in the conclusions that can be obtained from this model, as properties like

10

the final cell density are directly limited by the size of the cell walls. On the other hand, the final purpose of the study is to repro-

11

duce how the polymer reacts to a nanometric void located inside of it, and this model is able to provide this information.

(a)

(b)

12 13

Figure 5. (a) Stable mixture of PEI (in black colour) and CO2 (in red colour). (b) Schematic representation of the equilibrium between polymer internal pressure and CO2 pressure.

14

In order to know more about the internal pressure of the polymer, other gases (argon and methane) were employed in HDPE at

15

30%wt concentration. These gases were selected due to their pressure dependence with density: argon produces pressure values

16

closer to CO2, while methane produces higher values, always referred to similar value of the density of the gas. Considering this, it

17

can be expected that at least argon will present a stable mixture with HDPE during simulations. Table 1 shows final densities after

18

relaxation of the simulated structures of pure HDPE and HDPE with these different gases. It is possible to conclude that the inter-

19

nal pressure of the HDPE is able to contain the pressure of argon, but not the pressure of methane. It should be noted that the

10

1

difference in the achieved density respect to experimental results of pure HDPE (0.93-0.97 g cm─3) has also been previously ob-

2

served in other MD simulations of this material53. This can be explained by the simplicity of the employed model: only the amor-

3

phous phase has been considered, with no chain size dispersity. It would be possible to achieve more realistic values of density

4

with the employment of larger scale (possibly coarse-grained) models together with realistic values of chain size dispersity54.

5

However, these differences should not affect the qualitative conclusions that can be obtained from the present model.

6

Table 1. Density for HDPE relaxed structure (pure) and condensed with different gasses (30%wt concentration in weight over the

7

final mixture): CO2, Argon and CH4.

Material

Final Density (g/cm3)

Pure PE

0.788

PE+CO2

0.794

PE+Ar

0.751

PE+CH4

0.059

8 9

III. Internal Pressure.

10

Different lattice theories, as for example Sanchez-Lacombe lattice fluid Equation of State or Flory-Huggins lattice model are usual-

11

ly employed to explain the observed behaviour of polymer mixtures and polymer solvates. These are thermodynamic theories

12

apparently suitable for use in the present research. They present the Cohesive Energy Density (CED) as a measure of interaction

13

among molecules of each component, a magnitude that has been also (incorrectly35) understood as the internal pressure of mate-

14

rials. However, there are some disadvantages when these theories are applied to the systems under study:

15

(i) For non-polar liquids, the CED and internal pressure are approximately equal, as repulsion and dispersion interactions

16

will be the main type of interactions present. But for more complex molecules additional interactions are produced, in-

17

ternal pressure requires a more accurate definition55,56 and cannot be approximated by CED.

18 19 20 21

(ii) Lattice theories present a good approach to gas sorption in the case of rubbery polymers, but they do not take into account any additional contribution made by defects or nanovoids present in glassy polymers.

The thermodynamic definition of internal pressure,

, , is

given by Eq. 8. It represents the small change in the internal energy of a

system (-) produced by a small isothermal volume expansion against internal cohesive forces.

11

,

1

= /

01 0 ,

(8)

Applying this expression of internal pressure (eq. 8) to the Sanchez-Lacombe Equation of State (SLEOS) leads to: ,

= 2∗ ( & ∗ )

(9)

2

Where 2∗ is the characteristic pressure of the polymer,

3

to internal pressure gives the variation of internal pressure with the density, a relevant relationship that could be helpful to un-

4

derstand the stability of cells in the range of nanometer. However, as it was previously said, SLEOS is referred to cohesive energy

is the density and

∗

is the packed density of the polymer. This approach

5

densities57, as 2∗ is equal to CED. For this reason, this article will focus in the MD results for internal pressure calculations, more

6

suitable for complex molecules of the polymers.

7

Two methods have been considered for the measure of internal pressure in the generated structures. The first one applies the

8

Maxwell relationship to Eq. 8: ,

= 6/

02 1 −8 06 7

(10)

9

where p is the total, external pressure of the system. Employing this equation and performing NVT simulations at different tem-

10

peratures over the previously relaxed structures it is possible to obtain pairs of pressure and temperature values, which are pre-

11

sented in the supporting information (Part 2: Pressure vs temperature for NVT simulations for different polymers) for the differ-

12

ent polymers studied. However, as it is also explained in the supporting information, this method presents low reproducibility for

13

the calculation of the internal pressure.

14

The other considered method employs the data generated during the relaxation of the polymeric structures and the definition of

15

the internal pressure in Eq. 8. During the relaxation process of polymers in the NpT ensemble, the system evolves at constant

16

temperature. Then, considering the variation of the internal energy of the system respect to its volume makes possible to know

17

the variation of internal pressure of the system during the relaxation process.

18

Plotting the internal energy versus the volume of the system during the MD simulations (shown in figure 6 for PS, HDPE and PEI)

19

reveals several regions defined by differenced slopes. There is clearly a linear zone, with a smaller slope (lower internal pressure,

20

41 MPa), for higher volume values (from 3 to 9×10−19 cm3 approx.). As the volume is reduced, there is a clear increase in the slope

21

that implies an increase of the internal pressure (for volumes lower than 3×10−19 cm3).

12

1 2 3

FIGURE 6. Energy vs volume variation for PS (×), HDPE (•) and PEI (Δ), during relaxation process. Red Lines represent the linear zones of the relaxation process.

4

These absolute values of the volume of the system under study do not have a clear physical meaning, but they can be associated

5

with an equivalent cell size. For doing this, the final volume of the simulated polymer (once it has been relaxed) is taken as refer-

6

ence, and the difference respect to it is calculated for each density value. This difference in volume is employed to calculate the

7

diameter of an equivalent nanovoid with that volume. Then, it is possible to associate the diameter of an equivalent void to each

8

volume during the relaxation process, and the internal pressure variation can be expressed as a function of this void size. Fur-

9

thermore, this procedure allows also comparing the internal pressure of different polymers taking as reference the equivalent cell

10

size.

11

It has to be taken into account that this approach makes a considerable simplifying assumption: the free volume distributed along

12

the polymeric structure has been considered as a unique volume and associated to a spherical shape. However, the underlying

13

idea of employing this approach is having a clearer picture of how the internal pressure evolves respect to an absolute reference,

14

and more in particular how the changes in internal pressure values can affect the process of creation of a nanometric structure.

15

Considering this approach, for the HDPE the change in behaviour of the internal pressure (i.e., the change in the slope during the

16

relaxation process) is observed for an equivalent cell size diameter around 4 nm: bigger sizes imply a small internal pressure, and

17

smaller sizes implies an increase in the pressure. This equivalent cell size-internal pressure concept will be fully introduced in the

18

next paragraph. For PEI and PS data in Fig.6, a linear relationship is observed between energy and volume, again for an equivalent

19

cell size higher than 4-5 nm, and although for small volumes the dispersion of the data is higher, a small increase of the slope is

20

also produced as the volume is reduced. The volume-energy expressions for all the studied polymers are presented in the sup-

21

porting information section part 3: Volume-Energy plots during relaxation (fig. S1-S8).

22

The observed variation of the internal energy with the volume shows that internal pressure cannot be accepted as a fixed value

23

for each polymer: it presents a clear dependence with the variation of the specific volume of polymer.

13

1

Table 2. Internal pressure for the different pure polymers

POLYMER

9: for linear zone (MPa)

PE

41.0 ± 9.7

PP

35.3 ± 14.6

PEI

44.3 ± 11.4

PVDF

13.2 ±6.7

PS

29.9 ± 5.1

PES

35.9 ± 2.5

PC

37.5 ± 4.4

PMMA

41.5 ± 7.1

2 3

Table 2 shows the results of internal pressure calculations for the different polymers studied in the linear zone of the polymer

4

relaxation: an approach to a linear equation has been made for the energy-volume representation, obtaining a constant value for

5

the internal pressure for equivalent cells usually bigger than 4 or 5 nm.

6

The different zones of internal pressure can be partially explained by the cohesive forces, at least for the smaller cell values: at

7

certain distances among molecules, the interactions are small (and so the internal pressure), but as the particles are closer these

8

interactions are stronger (bigger internal pressure). From this point of view, the cut-off distance employed in the simulations

9

could be considered as a possible explanation in this variation of internal pressure, but it has been quickly discarded due to the

10

value employed (12.5 Å).

14

1 2

(a)

(b)

3 4

(c)

5 6

Figure 7. (a) Unstable mixture of PVDF and 30%wt CO2 mixture, relaxed from a low initial density. (b) Stable mixture of PVDF and 30%wt CO2 mixture, relaxed from higher initial density (c) Energy vs volume variation for PVDF, during relaxation process.

7

A clear confirmation of the different zones of internal pressure is presented in Figure 7 for PVDF infused with 30%wt CO2. De-

8

pending on the starting density for the relaxation of polymer-gas mixtures, it is possible to obtain a mixture that is unstable during

9

the relaxation process and produces a polymer-gas phase separation (Fig. 7.a) or a mixture stable during the whole relaxation

10

process (Fig. 7.b). When these polymer-gas mixtures were relaxed from low density values (around 0.470 g/cm3), the generated

11

structures were unstable, and a complete phase separation between polymer and gas was produced (together with a clear de-

12

crease in the final density). On the other hand, when the relaxation of these systems started from a higher density (around 1

13

g/cm3), a stable mixture with a final density closer to the pure polymer was achieved after relaxation. Figure 7.c explains the ob-

14

served difference from the point of view of the internal pressure: the variation of PVDF internal energy respect to volume during

15

relaxation is represented. Beside these volume-energy values, the initial volumes employed in the relaxation of Fig. 7.a and 7.b

16

are also represented, indicating the equivalent density of the system for that volume. In this graph is possible to observe that the

17

slope of the energy-volume representation is clearly higher in the proximity of the 1 g/cm3 values than in the proximity of the 0.47

18

g/cm3 values. Assuming the internal pressure as the slope of the volume-energy representation, it is possible to observe how the

15

1

different values of this internal pressure for the different densities of polymer-gas mixtures explains the capacity of the polymer

2

to contain the gas inside itself during the relaxation process.

3

The values previously presented in Table 2 can give an initial idea about the internal pressure for some polymers, and how it can

4

affect to the creation of nanocellular structures. However, relevant information is missed when the internal pressure of a polymer

5

is associated to a single number, considering that there are different internal pressure zones. Furthermore, differences are small

6

among the studied polymers, and no clear relationship can be stablished with experimental results in the production of

7

nanofoams. An optimal approach to internal pressure in the case of nanofoams implies the knowledge of internal pressure not

8

only for an equivalent cell diameter higher than 4 or 5 nm, but also for smaller cell sizes. This is a relevant issue from the point of

9

view of nanofoams stability: the pressure that a void supports inside a polymer will depend on the size of this void.

10

As it was previously explained, the diameter of an equivalent single cell can be correlated to the volume/density during the poly-

11

mer relaxation. Once more, although considerable simplifying assumptions are made for this approach, it can help to understand

12

how the changes in the internal pressure values can affect to the creation of nanometric structures. Figure 8 shows this idea ap-

13

plied to the relaxation of PES: the variation of the internal pressure with the volume of a single cell has been adjusted to a poly-

14

nomial curve. Then, a more direct idea of how internal pressure evolves it is shown, providing a better understanding of the pol-

15

ymer internal pressure variation for an the equivalent cell size, which is the most outstanding feature of this magnitude. It allows

16

to know when the internal pressure of the polymer will be lower, and therefore the stability of the nanocells higher. In Fig. 8 is

17

possible to observe that, in the case of PES, the equivalent nano-cell has to support higher pressures from the polymer for the

18

sizes under 2 nm, and as the equivalent cell size growths this pressure is reduced. Cells with bigger diameters will have to support

19

clearly smaller internal pressures and then will be more stable. A detailed stability study of the nanostructures generated with

20

CO2 during the simulations is presented in the next section.

21 22

FIGURE 8. Estimation of the variation of the internal pressure with the equivalent cell size, for PES.

23

Finally, the internal pressure effects over the stability of nanocells deduced from the simulations can be linked to the nucleation

24

process of nanofoams. As it has been presented during the introduction, nucleation density is a critical parameter in nanofoams

16

1

production: a higher number of cells is required, with respect to a microcellular material, to produce a nanofoam with an equiva-

2

lent density. Considering the higher energy required during the first stages of a nano-cell growth, the creation of more nanocells

3

will imply more energy that the creation of a smaller number of cells with bigger size. This matches with the observed higher

4

energy demand for the creation of nanofoams compared to conventional microcellular foams.

5

IV. Stability of nanoporous structures

6

The stability of the nanometric structures generated during the polymer-CO2 simulations can be checked once the CO2 is re-

7

moved. Although this computational procedure of obtaining a porous structure differs from the experimental solid-state foaming

8

process, it allows to study the stability of nanometric structures similar to those present in nanocellular materials. After CO2 re-

9

moval, the structures were relaxed again to observe their stability: the polymer presents a tendency to “collapse” the structure to

10

recover the minimum energy configuration, because of its internal pressure. With the help of Eq. 11, the final porosity of each

11

material was calculated.

8;<; =(>(%) = 100 ·

@ABC @DEFGCB

−

HI BCGDJCK BCELM

@ABC @DEFGCB

(11)

12

The equivalent cell diameter at which the transition from high internal pressure to low internal pressure is produced is quite rele-

13

vant here: a smaller internal pressure will make a structure more suitable to be stable at a small diameter. It can provide an esti-

14

mation of the minimum cell size radius that can be achieved in that material.

15 16

FIGURE 9. Porosity after relaxing the structures once the CO2 has been removed.

17

The results in Fig 9 shows that, after removing CO2 and relaxing their structures, most of the polymers reach a density similar to

18

the pure materials. The small porosities observed on most of these materials could be removed with longer relaxation times or

19

annealing processes, and they can be related with the free volume and the aging process58. However, in one of these materials

20

the porosity reached is extraordinarily high for being considered a fluctuation of the free volume: PEI presents a stable nanostruc-

17

1

ture that is able to support the polymer internal pressure. The internal pressure behaviour of this polymer can play a decisive role

2

in the observed porosity. Fig. 10 shows the comparison of behaviour for PEI and PES, two polymers with a similar molecular struc-

3

ture: while the PES presents a high internal pressure for the small values of the equivalent cell size and then a reduction as the

4

equivalent cell size grows, the PEI present clearly a lower internal pressure for equivalent cell sizes.

5 6

FIGURE 10. Dependence of the internal pressure with the equivalent cell diameter, for PEI and PES.

7

This result is completely different from the expected in a continuous classical approach, as it can be observed in the supporting

8

information (Part 4: Cell Stability), where stability calculations obtained from the mechanical properties of the different materials

9

are presented. In the classic approach a higher stability of the nanostructures has been obtained for the different polymers. The

10

theoretical framework in which every result has been obtained can be useful to understand the differences between the previous

11

calculi and the simulation results: while the estimations performed in a continuous framework are based in macroscopic proper-

12

ties, molecular dynamics simulations belong to a discrete framework where the macroscopic properties are not always valid. The

13

variation of the internal pressure is a clear example of this: it presents a high scale-dependence in the nanometric range, with a

14

higher value for the smaller voids values that is reduced as the voids size goes up.

15

As it has been previously indicated, PES has been selected for this comparative due to its molecular structure similarity with PEI

16

(Fig. 11). But, from an experimental point of view, it also presents a different foaming behaviour for the production of nanofoams:

17

although both polymers are able to reach cell sizes down to 30-40 nm, PES nucleation densities59,60 are in 1013-1014 cm-3 range

18

while PEI22 reach values even higher than 1015 cm-3. These differences in the nucleation density can be explained by the different

19

CO2 absorption capacities of both polymers, higher for PEI for a pressure greater than 5 MPa according to Krause et al. via exper-

20

imental parameters for the dual-sorption model46, and now, in a complementary way, by their different internal pressure. But

21

why two polymers with similar chemical structure present so different values of internal pressure needs also to be explained.

22

In order to understand the origin of these differences in the internal pressure, the conformation of the polymeric chains has been

23

analysed. DFT ab initio techniques have been employed for the study of the rotational energy barriers around the torsion of oxy-

24

gen bond in the case of the PES, and involving the nitrogen in the case of the PEI. These results are included in the supporting

18

1

information (Part 5: Ab-initio calculations; distances, angles and torsion distributions), and show that PES presents smaller num-

2

ber of configurations around oxygen bond and requires a higher energy to change from one to another than in the case of the PEI

3

(supporting information, Fig. S18). Although this should limit the movements of PES around its backbone, and then how the pol-

4

ymer interacts with itself to generate the internal pressure, it is clear that the opposite behaviour has been observed. The value of

5

the angles made by single atoms can make a difference here: PES presents a sulphur atom making a small angle respect to the

6

surrounding carbon atoms, which allows the polymeric chain to reach a higher degree of compaction. On the other hand, the PEI

7

molecular structure presents less ability to be compacted, as it can be concluded from the distance and angle distribution showed

8

in the supporting information, where the length stability of the backbone is shown (supporting information, Fig. S19, S20, S21,

9

S22). These results are consistent with the values of the Kuhn length (calculated using Synthia module) for both polymers: PES has

10

a value of approximately 42 nm, while the PEI has a value of almost 81 nm. These differences in the molecular behaviour can lead

11

to the variation of the internal pressure even in the case of very similar chemical structures, affecting to the polymers efficiency of

12

producing nanofoams.

13 14

(a)

15 16

(b)

17

FIGURE 11. Molecular structures: (a) PEI, (b) PES. Colours code: C- grey, H – white, O – red, S – yellow, N – blue.

18

CONCLUSIONS

19

This article presents two different approaches to the theoretical calculation of minimum cell size limits in the production of nano-

20

cellular polymers, one based on a more classical continuous approach and another one base on a discrete approach. The initial

19

1

one, based on the continuous approach, explores the limits on the increase of surface energy, the nucleation density and the

2

narrowing of the cell walls due the reduction of the cell size. The discrete approach, based on Molecular Dynamics simulations,

3

provides a better understanding of the polymer reaction to the creation of a void inside it, and the dependence of this reaction on

4

the size of the void, expressed by the internal pressure magnitude.

5

The continuous approach has shown that the reduction of the cell size under 100 nm implies a considerable increase on the ener-

6

gy required to create new surfaces, and in most of the cases, open cell structures are the only way to achieve stable structures.

7

These results agree with the new morphologies observed in low density and nanometric cell size PMMA61,62 and PC63 nanofoams,

8

different to typical structure of cellular solids.

9

It has been also concluded from the continuous theoretical calculations that the higher nucleation densities imply an additional

10

problem for the production of nanofoams with low densities and nanometric cell sizes, as a higher number of cells with a consid-

11

erable uniform distribution is required. From this point of view, nanostructured polymers can provide an advantage respect to

12

pure polymers, as they present a regular pattern, consequence of their heterogeneous structure, that acts as nucleation points

13

for the nanocells.

14

Further conclusions are obtained when a discrete approach is employed for the study of nanofoams. In a similar way to the case

15

of the Brownian motion, where the effect of forces emerge as the scale is reduced, the reduction of cell size is not just a reduction

16

of the scale length. When the diameter of the cells is in the range of a few nanometers, the internal pressure of the polymers

17

achieves higher values, affecting to the mechanical stability of the cells. The variation of the internal pressure of the polymer with

18

the cell size can help to identify the minimum size that a cell has to achieve to be stable from a mechanical point of view.

19

In summary, it can be concluded that the employment of atomistic techniques has helped to gain a better understanding of the

20

processes involved during the creation of nanofoams and to the limitations in the reduction of the cell size. Thus, polymers with a

21

more stable value of the internal pressure for the different cell sizes are suitable of produce nanofoams with smaller cell sizes.

22

Additionally, this stability of the internal pressure value can be highly conditioned by the Kuhn ratio of the polymeric chain.

23

Finally, the limitations of the present study should also be considered. The diffusion of gas inside the polymer from an external

24

source, a quite relevant process in the solid-state foaming, has not been considered here. The amount of gas and speed of the

25

diffusion can be decisive in the generation of nanofoams. Furthermore, the simulations have been performed only over a short

26

period of time (around 1 ns); longer times of study could help to confirm the results and conclusions obtained from the present

27

article, especially the stability studies.

20

1

Considering these limitations, further research with these techniques will help to understand the changes produced inside the

2

polymer structure by the presence of gas. The extension of these atomistic studies to block copolymers should be also consid-

3

ered, as they are one of the most promising materials for the creation of foams20, 29, 31–34.

4

Acknowledgements

5

The authors are grateful for financial support from SABIC.

6

Supplementary Information

7

Supplementary information attached: Surface Energy Volume Ratio (SEVR) calculations; Pressure vs Temperature for NVT simula-

8

tions for different polymers; Volume-Energy plots during relaxation; cell stability; ab-initio calculations, distances, angles and

9

torsion distributions

10

21

1

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Thermodynamic Limits on Cell Size in the Production of Stable Polymeric Nanocellular Materials Sergio Estravisa, c, Alan H. Windlea, Martin van Esb, James A. Elliotta, * a.- Department of Materials Science & Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge, CB3 0FS, United Kingdom b.- SABIC T&I, Urmonderbaan 22 6167RD Geleen, Netherlands c.- Department of Advanced Materials, International Centre of Advanced Materials and Raw Materials (ICAMCyL), León, Spain (Present address)

Highlights: -

Theoretical limits in the production of nanofoams are explored For cell sizes close to molecular sizes, atomic-scale interactions become more relevant Gas-polymer interactions are explored with molecular dynamics simulations Internal pressure governs the minimum cell size that different polymers are able to achieve

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: