1 Introduction

Highly porous materials are applied as thermal insulators due to the fact that the intrinsic thermal conductivity of still air is well below 0.03 W m−1 K−1, a value far lower than the thermal conductivity of any nonporous solid. Examples for well-established insulation materials are mineral wool and foams. However, both types of systems are limited by the fact that the effective pore sizes are in the micron range and thus do not profit from the suppression of the gas phase transport at ambient conditions according to the prediction by Knudsen [1, 2]. In contrast, aerogels are highly porous materials with pore sizes that can be tuned from the micron to the nanometer range. As a consequence, silica and organic aerogels show the lowest thermal conductivities at ambient conditions with values between 0.01 and 0.02 W m−1 K−1 [3, 4].

The total thermal conductivity of porous materials can often be described in terms of a superposition of the different thermal conductivity parts [5] describing the heat transfer via the solid backbone, radiation [3] and the gaseous transport as predicted by Knudsen. The latter depends on the structural parameters pore size and porosity of the insulation material. The superposition approach correctly predicts the total thermal conductivity in fully connected materials such as foams [5], where all mass in the system is similarly contributing to solid phase heat transfer. On the other hand, fibrous mats, bed of granules and aerogels show additional effects usually subsumed in a so-called coupling term [5,6,7]. The main reason for these effects is the characteristic heat transfer within a not fully connected backbone fraction of those materials, induced via point contacts (e.g., in bed of granules, fiber and, sol–gel systems) or dangling bonds (e.g., in sol–gel-based materials). The superposition approach is strictly valid only when the thermal transport in the different phases present (here: gas and solid) can be considered as parallel.

Typically, analysis of the gas pressure dependence of the thermal conductivity in open porous materials is performed from vacuum conditions to 0.1 MPa (ambient pressure). In this regime, aerogels usually show only the onset of the gas phase contribution [5, 8, 9]. It was therefore not clear until recently, whether aerogels show coupling and how much it would amount to. In a detailed investigation of this question Swimm et al. [7, 10] elucidated the coupling effects in aerogels experimentally.

From the theoretical point of view, several groups proposed different approaches to describe the total thermal conductivity of aerogels [7, 9, 11,12,13,14,15,16]. However, no model so far was able to consistently describe several sample/gas combinations without a severe adjustment of a large set of parameters. Recently, Swimm et al. [10] proposed an empirical, so called scaling model that proved to be applicable over six orders of magnitude in gas pressure to various aerogel/gas combinations by just fitting one free parameter at given pore size.

In this study, we use the large set of experimental data from refs [7, 10], covering a systematic change in pore size as well as solid phase and gas phase conductivity. The goal is to evaluate the results in view of practical implications, both, in terms of using the gas-pressure dependence of the thermal conductivity to determine the average pore size of aerogels as well as allowing for a targeted optimization of aerogel-based thermal insulations for applications with different given gas pressures and temperatures.

2 Modeling

Aerogels provide exceptional thermal insulation properties due to their high porosity and nanopore size, but also due to other specific structural properties. Figure 1 illustrates schematically the different key parameters for low-density, aerogel-type materials:

  1. 1.

    The fully connected fraction of the solid backbone determines the heat transfer, when the material is evacuated.

  2. 2.

    The nanoporosity introduced by the presence of not fully connected mass reduces the gas contribution to the total heat transfer because of the Knudsen effect [1]).

  3. 3.

    Increasing coupling effects between solid and gas phases contribute to the thermal heat transfer with rising gas pressure, especially for structural situations such as dead ends (see red circles in Fig. 1) and necks [17].

Fig. 1
figure 1

Sketch of aerogel structure between a hot (red) and a cold (blue) plate consisting of a fully connected mass fraction (see blue path via solid backbone) and dead mass. The latter reduces the pore size in the interspace of the fully connected backbone (black circles) to nanopores (blue dashed circles). This reduces the thermal transport within the gas phase, but increases heat transfer due to coupling of gaseous and solid phase thermal transport (indicated by red circles) (color figure online)

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Recently, an empirical scaling model [10] for the gas pressure dependence of the thermal transport in aerogels has been introduced and validated for a wide range of gas pressures from 10 Pa to 10 MPa including a systematic variation of structural and thermal parameters. The gas pressure (p g) dependence of the thermal conductivity (λ g) as described by the scaling model is given by:

$${\lambda _{\rm{g}}}\left( {{p_{\rm{g}}},T} \right) \approx \Phi \cdot {\lambda _{{\rm{Kn}}}}\left( {{p_{\rm{g}}},T} \right) \cdot F$$
(1)

Here T is the temperature, Φ is the sample (meso-/macro) porosity and F a scaling factor. The gaseous thermal conductivity λ Kn(p g, T) is given by the Knudsen formula [2] introduced for gas confined by two parallel walls at distance D:

$${\lambda _{{\rm{Kn}}}}\left( {{p_{\rm{g}}},T} \right)=\frac{{{\lambda _{{\rm{g,0}}}}\left( {{p_{\rm{g}}},T} \right)}}{{1\,{\rm{ + 2}} \cdot \beta \left( T \right) \cdot {l_g}\left( {{p_0},T} \right) \cdot {p_0}{\rm{/}}\left[ {{p_{\rm{g}}}\left( T \right) \cdot D} \right]}},$$
(2)

with the thermal conductivity λ g,0 of still free gas, the gas-type dependent factor β [10], the mean free path l g of molecules in the unconfined gas and a reference gas pressure p 0. In our case, D is the effective mean pore diameter of a porous structure; it is equivalent to the mean free path between opposing solid/void interfaces, e.g., pore walls. The characteristic quantity l g/D, the Knudsen number, determines the gas phase contribution to the heat transfer.

The scaling model empirically describes the gas pressure dependence of the heat transfer as a function of the classical Knudsen term; it has been found to model experimental data in a high quality when using D from independent structural analysis (see Fig. 2). However, in contrast to the superposition approach, which is strictly valid only for pure parallel connection of solid and gas phase heat transport, the scaling factor F for aerogels is not equal to 1 but can rather significantly exceed this value. This model, analyzed in detail in ref. [10], will be applied within the framework of this study to discuss the different practical effects of thermal coupling in aerogels.

Fig. 2
figure 2

Total effective thermal conductivities of different organic aerogels measured in nitrogen and argon at 21 °C. Only the measurement uncertainties of RF 0.6 in Ar are shown as an example. Orange dash-dotted line: Knudsen conductivity for RF 0.6 in Ar according to Eq. 2 multiplied with the sample’s porosity (porosity and pore size taken from Table 1). Solid line: fit curve for RF 7 in N2 according to the scaling model (Eq. 1) with the fit parameter F = 2.0 (porosity and pore size taken from Table 1). Gray box: region, where the thermal conductivity of free gas increases with gas pressure [10] (color figure online)

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3 Experimental

3.1 Thermal conductivity measurements

The thermal conductivity was measured using the transient hot-wire technique [18, 19] in combination with a vacuum/pressure chamber. The set-up allows the determination of the gas-pressure dependence of the thermal conductivity of open porous materials at gas pressures ranging from 10 Pa to 10 MPa at room temperature [7, 10]. Various experimental data are published in refs [7, 10]. In this study, the experimental results determined with the pore gases Argon (λ g,0 = 0.017 W m−1 K−1 at ambient conditions), Nitrogen (λ g,0 = 0.026 W m−1 K−1 at ambient conditions) and Helium (λ g,0 = 0.154 W m−1 K−1 at ambient conditions) are re-evaluated. Further gas parameters can be found in Swimm et al. [10].

3.2 Samples investigated

For a significant variation in pore size and solid phase conductivity we choose two organic aerogels with different average pore sizes of 0.6 and 7.6 µm as well as two carbon (C) counterparts. The organic aerogels were synthesized via a sol–gel process on the basis of resorcinol (R) and formaldehyde (F) [20, 21]. The carbon aerogels were derived from the organic aerogel with smaller pores via pyrolysis at 800 °C and by additionally annealing one carbon aerogel from the same batch at 1800 °C [7, 10]. Since it is well-known that upon pyrolysis of the organic precursors and annealing of the carbon aerogel, respectively the structure on the length scale > 2 nm (including backbone connectivity) is almost unaffected [22, 23], the goal of pyrolysis and annealing was to systematically vary the intrinsic thermal conductivity of the solid backbone for a given backbone structure (see Table 1) [24]. The backbone not only is converted chemically from organic to carbonaceous, but at the same time the backbone phase becomes microporous (pores < 2 nm). The micro pores are expected not to contribute to the gaseous heat transfer. Therefore, the micro porosity is included in the effective thermal properties of the backbone material.

Table 1 Properties of the aerogels investigated: Φ is the porosity not including the micro pores, D is the mean pore size and λ s is the intrinsic thermal conductivity of the respective solid phase as given in literature
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The relevant structural properties of the samples, such as the porosity and average pore size are compiled in Table 1. To calculate these quantities, the bulk density as well as the interfacial area between the skeleton and the meso-/macro pore phase, i.e., the external surface area, were determined. The latter was derived via scanning electron microscope and nitrogen adsorption analyses. For details see refs [7, 10].

In the following, the sample names were chosen to indicate the mean pore sizes of the (corresponding) RF-aerogels (first number) and the respective pyrolysis/annealing temperature (second number). For example, C 0.6/800 denotes a carbon aerogel derived via pyrolysis at 800 °C from the organic aerogel RF 0.6 with an average pore size of 0.6 µm.

4 Experimental results

4.1 Variation of mean pore size and gas type

Figure 2 shows typical experimental results for the total effective thermal conductivity of different aerogels as a function of gas pressure, so-called s-curves in a half-logarithmic representation. The data for organic aerogels RF 0.6 and RF 7 in argon and nitrogen were exemplarily chosen for presentation at this point; related plots for more samples and pore gases investigated show the same characteristics and can be found in refs [7, 10].

In vacuum (l g/D >> 1), the gaseous thermal conductivity is suppressed and the effective thermal conductivity (λ evac) of all samples approaches a minimum value, which includes thermal transport via the solid backbone and radiation. With rising gas pressure, the gaseous thermal conductivity increases (“transition region” at l g ≈ D) and finally becomes fully developed for l g << D. In this range a more or less pronounced plateau is formed. As the thermal conductivity of free gas starts to increase above 1 MPa [10], the plateaus of the experimental curves in Fig. 2 are superimposed by the slope of λ g,0(p g) in this pressure region (see gray box).

The thermal conductivity of sample RF 7 in N2 begins to increase at lower gas pressure than the corresponding curve for RF 0.6; this is due to the ten times larger pore size of RF 7 for which the condition l g < D is fulfilled at about 1/10 in gas pressure compared to RF 0.6. As a consequence, the inflection points of the two curves in terms of gas pressure are shifted about one decade. The two s-curves discussed reach about the same plateau values between 1 and 10 MPa.

Changing the pore gas in the same sample significantly affects the plateau value of the s-curve (compare the curves for RF 0.6 in N2 and Ar), because the thermal conductivities of the free gases greatly differ (see Section 3.1); in contrast, the inflection point and the offset value in vacuum remain unaffected. The height of the step (λ plateau−λ evac) is not equal to the effective thermal conductivity of the respective free gas, but exceeds this value significantly. This is indicated by comparison with the corresponding Knudsen curve according to Eq. 2 for RF 0.6 in Ar (see dash-dotted line in Fig. 2). Moreover, the experimental data clearly exceed the theoretical gaseous thermal conductivity in the whole pressure range. The solid line in Fig. 2 shows exemplarily the high quality of a data fit according to the scaling model (Eq. 1) with the scaling factor F chosen as free parameter at given pore size and porosity; the fit result for F is 2.0.

4.2 Variation of the thermal conductivity of the solid phase

Figure 3 compares the total effective thermal conductivities of the organic (RF 0.6) and the related carbon aerogels (C 0.6/800 and C 0.6/1800) as a function of gas pressure. Experimental curves taken in an argon atmosphere are shown as an example; corresponding nitrogen data are available elsewhere [10]. The general shape of the curves received is equivalent to those in Section 3.3.1. The inflection points of the three curves appear at about the same gas pressure. Very striking, however, are the different curve offsets (λ evac). They increase from the organic aerogel to the related carbon and rise further with annealing. This behavior can be assigned to the different thermal conductivities λ s of the backbone materials (see Table 1). Other possible reasons for the increase can be excluded: Firstly, carbon aerogels have extremely high specific infrared-extinctions [25], thus, the radiative contribution to the heat transfer is negligible in these samples (even in RF due to the high density). Secondly, the structural characteristic of the aerogel backbone on the length scale > 2 nm is not affected by pyrolysis [24], therefore, structural changes of the backbone are not the reason for a variation of the solid phase thermal conductivity.

Fig. 3
figure 3

Total effective thermal conductivities of different aerogels in argon at 21 °C. The measurement uncertainties of C 0.6/1800 and C 0.6/800 are shown as an example. The gray box indicates the region, where the thermal conductivity of dense free gas increases with gas pressure [10]

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5 Discussion

5.1 Description of experimental gas-pressure dependence with Knudsen type relationship

It is quite common to use the gas-pressure dependence of the thermal conductivity of porous samples to derive the mean pore size D according to the Knudsen model (Eq. 2) [5, 8]. As shown in Fig. 2, however, coupling is clearly visible in the whole pressure range for aerogels.

To investigate the impact of the chosen evaluation approach on the resulting mean pore size, two different fit routines based on the Knudsen model (Eq. 2) are compared to the experimental gas-pressure dependent data of the aerogels RF 0.6 and C 0.6/800 in Fig. 4. Fit 1 is only applied to the onset of the s-curve, i.e., the data points up to 0.01 and 0.02 MPa shown in orange and gray, respectively. Here, the pore size is the only free parameter, while the porosity Φ is taken from independent experiments (Table 1). Fit 2, on the other hand, takes into account all data points available, but has to allow for a second free parameter, the porosity, in order to achieve appropriate fit curves. The gas parameters used for both fit types are given elsewhere [10]. All fit curves presented in Fig. 4 agree well with the experimental data within their respective fit regions. The resulting fit parameters are compiled in Table 2 together with the “true” sample properties received from structural analysis methods. The uncertainties given for the fit parameters refer only to the fit routine.

Fig. 4
figure 4

Experimental data and related fit curves according to the Knudsen model (Eq. 2). Dashed lines (Fit 1): only experimental data up to a maximum gas pressure of 0.01 MPa (RF 0.6) and 0.02 MPa (C 0.6/800) are taken into account (see orange and gray data points, respectively); the pore size is a free fit parameter, while the porosity is taken from Table 1. Solid lines (Fit 2): all experimental data points taken into account; pore size and porosity are free parameters. Gray box: region, where the thermal conductivity of free gas increases with gas pressure [10] (color figure online)

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Table 2 Comparison of the “true” parameters derived from structural analysis (Table 1) with fit results according to the Knudsen model (Eq. 2)
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The mean pore sizes D′ derived via Fit 1 are a factor of 3.2 and 5.5 higher than the values derived from independent structural analysis. This means, if the experimentally covered gas pressure regime is restricted to the pressure regions considered in these examples, the significant errors in the pore sizes determined will remain unnoticed due to the good agreement of fit and experimental curves within the fit ranges. In practice, this problem arises for aerogels with smaller pore sizes (e.g., silica aerogel) in combination with measurements up to atmospheric gas pressure, because the widths of the detected s-curve onsets would then be comparable to the ones in the given examples. Respective significant discrepancies between thermally and conventionally derived pore sizes already appeared in literature [8].

In contrast, fit 2 yields reasonable values for the mean pore size, showing only shifts of about 15 and 20% towards smaller values compared to the “true” pore sizes, respectively (Table 2). However, the corresponding porosities Φ* found are clearly larger than one, which is a non-physical result. This discrepancy can be ascribed to coupling of gaseous and solid thermal conduction. Thus, fit 2 is actually equivalent to the scaling model that takes into account these coupling effects (see chapter 2), i.e., the pretended porosity Φ* is equal to the “true” porosity scaled with factor F.

Swimm et al. [10] validated a consistent description of experimental gas-pressure dependent data for many different aerogel samples and gases by means of the scaling model with the scaling factor F chosen as free parameter at given pore size and porosity and no further free parameters. Accordingly, the curve shape of the gas-pressure dependent thermal conductivity received for aerogels is equal to the shape of the original gaseous thermal conductivity curve according to Knudsen (Eq. 2), but the s-curve for pure gaseous thermal transport is stretched in vertical direction (see Fig. 4) due to coupling effects. That means, coupling seems to occur uniformly over the whole gas pressure range yielding an increase of the gas-pressure dependent coupling contribution with rising gas pressure. This can be observed in Fig. 2 by comparing the triangular data points to the theoretical dash-dotted curve for pure gaseous thermal transport. This finding is surprising since a dominant gap size between unconnected neighboring backbone elements or a defined neck geometry is to be expected that may yield predominant coupling effects at certain gas pressures. However, the experimental findings do not reveal such an effect; likely the fact that all gas pressure dependent effects contributing within 2–3 decades in gas pressure are smearing out individual effects. In addition, the spread in “gap”-distribution could be wider than expected.

5.2 Extent of thermal coupling

The scaling factor F identified from fitting Eq. 1 to the experimental gas-pressure dependent thermal conductivity indicates, how much the experimental curves exceed the pure gaseous thermal conductivity according to Knudsen theory (Eq. 2), thus representing the extent of thermal coupling between the gas and solid phase of the porous material. For aerogels F-values ranging from 1.3 to 3.3 were found [10], dependent on different parameters that are discussed in the following.

Heat conduction via not fully connected mass (see Fig. 1) is suppressed in the evacuated state and will be “activated” by coupling of solid and gas phase thermal transport with increasing gas pressure in the pores of aerogels. That means, the higher the fraction of fully connected mass, the lower the coupling effect, i.e., the connectivity of the solid backbone influences the amount of thermal coupling [26].

The plateau values of the N2-curves for samples RF 0.6 and RF 7 in Fig. 2 are almost equal, although the mean pore sizes differ by a factor of 10. Consequently, also the plateau height of the s-curve, i.e., the coupling effect, is not influenced by the mean pore size at otherwise similar backbone properties. The pore size only affects the inflection point of the s-curve.

Finally, the experimental data indicate a correlation between the intrinsic thermal conductivities of the two phases in a porous material and the extent of thermal coupling. This behavior is shown for the pore gas Argon in Fig. 5, where the respective offset-values λ evac are subtracted from the original curves in Fig. 3. Obviously, for a given pore gas (constant gaseous thermal conductivity) and backbone structure the plateau height of the experimental gas-pressure dependent curves increases with increasing effective thermal conductivity in vacuum (compare Fig. 3), i.e., intrinsic backbone thermal conductivity λ s (see chapter 3.3.2) of the three aerogels. Corresponding curves taken in a nitrogen atmosphere reveal the same trend but with different plateau values [10] because of the factor 1.5 higher thermal conductivity of free N2 vs. Ar (see chapter 3.1). For a given sample (constant solid phase thermal transport), the scaling factor F decreases with increasing gaseous thermal conductivity [10]. Thus, we conclude, the higher the ratio of the intrinsic thermal conductivity of the solid phase to the gaseous thermal conductivity (λ s/λ Kn) the stronger the coupling effect: F ~ λ s/λ Kn. The larger the difference between the two thermal conductivities, the higher is the impact of a thermal shortcut at small gaps in the backbone, thus the coupling effect increases.

Fig. 5
figure 5

Gas-pressure dependent thermal conductivities λ g for an organic aerogel and two different related carbon aerogels in argon. The curves were derived from the experimental data in Fig. 3 by removing the corresponding offsets λ evac. Dashed line: effective thermal conductivity of free argon. Gray box: region, where the thermal conductivity of dense free gas increases with gas pressure [10]

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5.3 Comparison of experimental data to theoretical limit for two parallel phases

In this section, the total effective thermal conductivity reached in the plateau regime is further analyzed for different aerogels by comparing the experimental data to theoretical values calculated by assuming a pure parallel thermal transport within the gas and the solid phase (see Fig. 6, right). The effective thermal conductivity for the parallel connection is given by

$${\lambda _{{\rm{parallel}}}}={\lambda _{{\rm{g,0}}}} \cdot \Phi +{\lambda _{\rm{s}}} \cdot \left( {1 - \Phi } \right),$$
(3)

with the aerogel porosity Φ, the thermal conductivity λ g,0 of free gas and the intrinsic thermal conductivity λ s of the solid phase. λ parallel represents the maximum value for the effective thermal conductivity that can be achieved in two-phase media [27]. Hence, the effective thermal conductivity of any arbitrary distribution of the two phases at given porosity, such as in aerogels, will not exceed this theoretical limit.

Fig. 6
figure 6

Sketch of two-phase media with identical porosity between a hot (red) and a cold (blue) boundary; left: arbitrary distribution of the two phases, e.g., aerogel structure, right: strictly parallel arrangement of the two phases (color figure online)

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In the following, the theoretical limit values λ parallel (Eq. 3) are compared to the total effective thermal conductivities λ tot measured in different aerogels at about 1 MPa, where the plateau regime of the gas-pressure dependent curves is reached and the gaseous thermal conductivity λ Kn is fully developed. The original experimental data presented in Swimm et al. [7, 10] refer to carbon and RF aerogels in combination with the gases Ar, N2, and He. In Fig. 7, the respective values λ tot/λ parallel are plotted vs. the ratio of the intrinsic thermal conductivities of the two phases (λ g,0/λ s). Most λ tot/λ parallel-values lie significantly below the maximum λ tot/λ parallel = 1. However, with increasing λ g,0/λ s, the experimental total effective thermal conductivity gradually approaches the theoretical limit according to the parallel connection of the two respective phases for identical porosity Φ. Figure 7 reveals a deviation between carbon and RF aerogel data around λ g,0  /λ s = 0.1 that, however, is still within the uncertainty margin. The limit λ tot  /λ parallel = 1 is almost reached for λ g,0  /λ s values of about 0.67, corresponding to RF-aerogels in a Helium atmosphere.

Fig. 7
figure 7

Total effective thermal conductivities λ tot measured for different aerogel samples at a gas pressure of about 1 MPa in different gases (Ar, N2, and He) normalized to the corresponding theoretical limits λ parallel (Eq. 3). The data points are plotted vs. the ratio of the thermal conductivity λ g,0 of free gas to the intrinsic solid phase thermal conductivity λ s. The error bars include the inaccuracy by reading the plateau values from the original measurement curves (see Swimm et al. [7, 10]) and uncertainties in λ s

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We can take advantage of this fact to derive the mean pore size of an RF-aerogel from thermal conductivity data as a function of the gas pressure in Helium (λ g,0/λ s ≈ 0.7), even if the experimentally covered gas pressure range is limited (e.g., only data below atmospheric pressure available) and thus only the onset of the s-curve is visible. In this case, the upper limit value, λ parallel, calculated via Eq. 3 can be used to define the plateau value of the s-curve in thus improving the quality of the fit result. This procedure is exemplarily shown in Fig. 8 for the aerogels RF 7 and RF 0.6 in Helium. For fitting Eq. 1 to the experimental curves, data points up to 0.01 and 0.1 MPa, respectively, were taken into account together with the estimated plateau values λ parallel (colored data points at 10 MPa). The gas-pressure dependence of the thermal conductivity of dense free gas is neglected in this evaluation process. The scaling factor F and the mean pore size D were used as free parameters. The fit result for the scaling factor F is 1.3 for both curves in Fig. 8; this agrees with the respective F-values found for the complete experimental curves (see ref. [10]). The pore sizes derived from the fit are D = 8.8 µm for RF 7 and D = 0.64 µm for RF 0.6. These values are within the respective uncertainty margins of the mean pore sizes determined from structural information (Table 1). Therefore, the suggested procedure represents a good possibility for the evaluation of gas-pressure dependent Helium data.

Fig. 8
figure 8

Evaluation of gas-pressure dependent thermal conductivity curves taken in a Helium atmosphere using only a limited pressure range. The hot-wire data taken into account end at 0.01 MPa for the aerogel RF 7 and at 0.1 MPa for RF 0.6 (respective colored points). The plateau values of the s-curves were estimated via a parallel connection according to Eq. 3 (see colored data points at 10 MPa). Solid lines: fit curves with Eq. 1 taking into account all colored data points; pore size D and scaling factor F were free parameters. Gray box: region, where the thermal conductivity of free gas increases with gas pressure [10] (color figure online)

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6 Conclusion and outlook

Within the framework of this study, the large set of experimental thermal conductivity data compiled in refs [7, 10] was re-evaluated with respect to

  • the impact of thermal coupling effects on the determination of the average pore size of aerogels from the gas-pressure dependent thermal conductivity and,

  • the optimization of structural properties of aerogel for thermal insulations.

All gas-pressure dependent thermal conductivity curves recorded for the different types of aerogels reveal a significant heat transfer contribution due to coupling of the solid and the gas phase within the whole gas pressure range investigated. The gas-pressure dependence is well described via a scaling model introduced in [10], using the Knudsen-type gas pressure dependence for the given pore size and porosity and only one free parameter, the scaling factor F, representing the extent of the coupling effect.

The mean pore size of an aerogel does not affect the scaling factor nor the other way round: Independent of the extent of the coupling effect, the inflection point of a gas-pressure dependent thermal conductivity curve is located at a certain gas pressure and the corresponding mean pore size is in good agreement with data from independent pore size analysis. However, the ratio of the intrinsic thermal conductivity of the solid phase to the gaseous thermal conductivity (λ s/λ Kn) as well as the fraction of not fully connected mass significantly increase the extent of the coupling effect (scaling factor).

Comparing the experimental thermal conductivities at fully developed gas phase transport for different aerogel/gas combinations to the respective theoretical limit values λ parallel for a pure parallel connection of gas and solid phase reveals that for most aerogel/gas combinations the experimental total effective thermal conductivity (λ tot) is significantly below the respective parallel connection limit. On the other hand, an increasing fraction of not fully connected mass at given porosity reduces the thermal conductivity of the solid backbone in vacuum. These results can be used for the optimization of aerogels as thermal insulation for specifically given boundary conditions such as temperature, gas atmosphere and pressure. However, it has to be kept in mind, that the deviation between λ tot and λ parallel is highly sensitive to the ratio of the intrinsic thermal conductivities of the two phases (λ g,0/λ s). For RF-aerogels in a Helium atmosphere (λ g,0/λ s ≈ 0.7), for example, the theoretical limit is almost reached. Therefore, the pore gas used is very critical and has to be taken into account within the thermal optimization process of aerogels. On the other hand, the theoretical limit λ parallel for high λ g,0/λ s is a useful complement of the experimental data for further evaluation.

Future investigations are aiming at explicitly analyzing the impact of the backbone connectivity on the coupling effect by systematically changing the fraction of fully connected mass. The experiments will be complemented by respective thermal simulations of model structures.