#
Partition Coefficients and Diffusion Lengths of ^{222}Rn in Some Polymers at Different Temperatures

^{*}

## Abstract

**:**

## 1. Introduction

^{222}Rn, half-life 3.8232 d) is recognized as a severe health-risk factor, being the leading cause for lung-cancer after smoking [1]. Due to its short half-life, thoron (

^{220}Rn, half-life 55.8 s) appears indoors in significant concentrations only under specific circumstances. However, in such cases, thoron could be a health-hazard too [2,3,4]. Therefore, a wide range of methods for radon and thoron measurements are developed dealing with different aspects of the problem—metrological assurance, risk assessment, dose estimation, average activity concentration measurement, mitigation, etc.

^{®}family are polycarbonate-based products by Bayer AG, Leverkusen, Germany) have a remarkably high absorption ability to Rn (and other noble gases) [17,18]. Based on that property of the polycarbonates, several methods for radon and other radioactive noble gases (RNGs) measurement were developed. These methods use the polycarbonate as a passive sampler that absorbs and concentrates the RNG from the ambient media. Some of these methods measure the cumulative activity of the absorbed radon relying on the track-etched properties of the Makrofol DE or another (external) track detector [19,20,21], while others register the alpha-, beta- or gamma-particles of the absorbed radon and its SLP (or other RNG) by active detectors—Liquid Scintillation (LS) counters, gross alpha/beta counters, HPGe gamma-spectrometers and others (see [22,23,24,25] and the references there). To apply these methods, the temperature dependence of the diffusion properties of the polycarbonates should be known. This dependence is studied for Makrofol DE [26,27], but, for Makrofol N, the diffusion properties are known only for a single temperature value [23,28].

## 2. Materials and Methods

#### 2.1. Transport of RNGs in Polymers

- The atoms of the RNG are caught in the polymer matrix at the border ambient media/polymer and, in any moment, the ratio of the RNG concentrations at the surface of the polymer ${c}_{in}$ and, in the ambient media, ${c}_{out}$ is given by the partition coefficient $K=\frac{{c}_{in}}{{c}_{out}}$. It must be noted that the partition coefficient of some polymers could be greater than one (For example, $K\approx 100$ for
^{222}Rn at the border Makrofol N/air at room temperature, which makes it very appropriate for a radon sampler). One possible explanation of this phenomenon could be the presence of free-volume traps in the polymer matrix (see [28] and the references there). In the free-volume trap models, it is considered that there are small voids in the polymer matrix with sizes close to the dimensions of the RNG atoms. The RNG atoms are trapped in these voids, and the concentration of the RNG in the polymer appears to be higher than in the ambient media; - Once the RNG atoms are caught in the polymer matrix, their transport in the polymer is described by the diffusion equation (Fick’s second law) with an additional term that accounts for the radioactive decay:$$\frac{\partial c}{\partial t}=D\left(\frac{{\partial}^{2}c}{\partial {x}^{2}}+\frac{{\partial}^{2}c}{\partial {y}^{2}}+\frac{{\partial}^{2}c}{\partial {z}^{2}}\right)-\lambda c,$$
^{−3}] is the RNG concentration in the polymer sample as a function of the space $x,y,z$ [m] and time t [s] coordinates (Hereafter, the units of the quantities according to the Intentional System of Units (SI) are given in square brackets “[ ]”, when the quantity is introduced for the first time in the text), D [m^{2}/s] is the diffusion coefficient of the atoms of the noble gas in the polymer, and $\lambda $ [s^{−1}] is the decay constant of the RNG. In [29], Equation (1) is solved for some given shapes of the polymer samples, immersed in RNG-containing media. Once the polymer sample is exposed, it absorbs the RNG, and the dynamics of the absorption depends on the exposure conditions, polymer geometry, and on the parameters K [dimensionless] and D. In the present work, plate-shaped specimens are considered exposed to radon in air for time ${t}_{s}$ [s] and left to desorb in infinite radon-free media for time ${t}_{d}$ [s]. In the considered exposure, radon is promptly introduced in the exposure volume and then the activity concentration of radon decreases exponentially (due to radioactive decay) with the decay constant of radon. For plate-shape specimens (specimens for which one of the dimensions is orders of magnitude smaller than the others), the process is considered one-dimensional, and the solution for the RNG activity $A({t}_{s},{t}_{d})$ [Bq] absorbed in the specimen is [29]:$$A({t}_{s},{t}_{d})=\frac{8\lambda {L}_{D}^{2}VK{C}_{A}}{{L}^{2}}\sum _{k=0}^{\infty}\frac{{e}^{-\lambda {t}_{s}}-{e}^{-{\lambda}_{k}{t}_{s}}}{{\lambda}_{k}-\lambda}{e}^{-{\lambda}_{k}{t}_{d}},$$$${\lambda}_{k}=\lambda \left(1+{\left(\frac{(2k+1)\pi {L}_{D}}{L}\right)}^{2}\right),$$^{3}] are the thickness and the volume of the specimen, ${C}_{A}$ [Bq/m^{3}] is the initial activity concentration of the RNG in the media, and ${L}_{D}$ [m] is the diffusion length of the RNG in the polymer. In this model, the only two parameters are the partition coefficient K and the diffusion length ${L}_{D}$. The latter is by definition related to the diffusion coefficient D: ${L}_{D}=\sqrt{D/\lambda}$. Thus, if the two parameters K and ${L}_{D}$ (or D) are known, the transport of the RNG in/through a polymer membrane could be quantitatively described. It must be noted that Equation (2) is derived for the more general case of transient radon distribution in the sample and is valid for arbitrary sorption and desorption times. The only restrictions to Equation (2) are the plate shape of the specimens and the exponentially decreasing ambient activity concentration (In [29], Equation (1) is also solved for constant ambient activity concentration and for cylindrical specimens).

#### 2.2. Method for Estimation of K and ${L}_{D}$

#### 2.3. Measurement of the Absorbed Activity

^{222}Rn) passing through the Makrofol DE. However, the polymer foils used in the present work are thin, which leads to very fast desorption of radon from the specimen to the air in the empty LS-vial. This could lead to a change in the counting efficiency and a loss of radon from the vial that could not be followed and corrected for. This is why the approach was modified: the LS-vials are fully filled with distilled water, and the polymer specimen is immersed in it. When the polymer foil is immersed in the water, some of the radon absorbed in the foil is released in the water until equilibrium between the radon concentration in the two media is reached. The equilibrium is determined by the partition coefficient at the border water–polymer. During the process of redistribution of the activity, the Cherenkov counting efficiency changes as the Cherenkov effect depends on the refraction index of the media. Once equilibrium is reached, the efficiency is constant (as it is shown further in this work), and it could be used to determine the activity in the sample.

- These approaches allow precise timing—when the foil is closed in the vial, the activity is “trapped” in the vial, thus it could be attributed to the exact moment of desorption within 1–2 s.
- There is a small (for the Cherenkov) or even no (for the LS) activity leakage from the vials (see further in Section 3.1). Thus, if the samples have to be measured later or for a longer time, the activity will be sufficient for a longer time and precise long measurements can be performed.
- As the activity is “trapped” in the vial, there is no need for temperature control during the measurement. In the case of gamma-spectrometry and external gross counting, the samples have to be kept at the studied temperature; otherwise, the desorption will be compromised. This is inconvenient or even unachievable in the case of a temperature that differs with more than 5–10 °C from the normal room temperature.

## 3. Experiments

^{60}Co (ORTEC, Oak Ridge, TN, USA) and an LS-analyzer RackBeta 1219 (Wallac, Turku, Finland). For the measurements of the activity concentration of radon in air during the exposure a reference monitor AlphaGUARD RnTn Pro (Saphymo, Frankfurt, Germany) was used.

#### 3.1. Estimation of the Counting Efficiencies

^{222}Rn,

^{218}Po,

^{214}Po and the 2 betas of

^{214}Pb,

^{214}Bi) are emitted per one decay of radon.

^{3}) was used. Glass vials were fully filled with this water, unexposed foils were placed in the vials, and the vials were closed tightly. Two vials with water without foils were also prepared for comparison. Then, all the vials were periodically measured on the LS counter in order to follow the signal change in time. The duration of a single measurement was 10 minutes, and the whole follow-up experiment continued for about one week. The vials were also measured at the HPGe detector (2–3 measurements of each vial with a few hours duration) in order to estimate the activity in the vials, thus to estimate the counting efficiency of the LS counter. For the gamma–spectrometry analysis, the 295 keV and 352 keV gamma-lines of

^{214}Pb were used. The experiment was carried-out contrariwise—unexposed foil in water with activity, instead of exposed foil in distilled water, in order to ensure better counting statistics, thus, to be more sensitive to slight changes in the signal due to the redistribution of radon between the water and the foils. The follow-up measurements at the LS-counter show that the signal of all samples, except those with Makrofol foils, decreases purely exponentially with the same (statistically) effective half-life as the signal of the distilled water samples (see, for example, Figure 1a). The average value of the effective half-life is 3.728(36) d, which is slightly lower than the radon half-life of 3.8232(8) d [35]. It was also observed that, in the first 60-70 h of the follow-up, the signal from the samples with the two types of Makrofol foils increases, reaches a maximum and then starts to decrease and, after 60–70 h, the decrease becomes exponential with the same above-mentioned effective half-life (see Figure 1a). The initial increase of the signal could be explained by the absorption of radon in the Makrofol foils—these foils absorb a significant part of the radon from the water. Due to their higher refraction index ∼1.6 [36] (compared to that of the water 1.33), they have higher efficiency for Cherenkov light emission (the higher the refraction index is, the lower is the threshold energy for the beta-particles to produce Cherenkov effect). Additionally, the Makrofol material possesses some (poor) scintillation properties [36], which also might lead to increasing the counting efficiency.

^{−1}] and the gamma-spectroscopically measured activity in the sample $A(t)$:

#### 3.2. Estimation of K and ${L}_{D}$

^{222}Rn source (≈100 kBq, ≈200 mL), a peristaltic pump, and another “control” drexel (700 mL) (see Figure 3). The radon activity was promptly introduced in the system by opening all valves and turning on the pump at 2 L/min flow-rate for 5 min. After that, all valves were closed and the foils were exposed for 2–3 days. Thus, the exposure activity concentration in these three experiments was of the order of tens of MBq/m

^{3}. Such high activity concentration was needed to ensure good counting statistics for the follow-up of the foils. During the exposure, each drexel was placed in a bigger hermetic vessel, and the radon concentration in the bigger vessels was measured by the AlphaGUARD. This was done in order to check for radon leakage from the drexels. In all experiments, the leakage from the drexels was found to be less than 1% of the radon activity in the drexel. During the exposure, the bigger vessel (with the “exposure” drexel inside) was placed in a thermostat [37] and the exposure temperature was kept stable within 1 °C. The “control” drexel was used for estimation of the activity concentration during the exposure: Because the exposure activity concentration was above the measurement range of the AlphaGUARD, the activity from the “control” drexel was diluted in a larger vessel with a well-known volume. Thus, the activity concentration in the larger vessel was lowered to the measurement range of the AlphaGUARD—it was measured, and the initial (exposure) activity concentration was calculated based on this measurement and the volume ratio of the drexel and the larger vessel. The exposure data are summarized in Table 2.

## 4. Results

- The uncertainties of the individual points of the desorption follow-up. We aim to achieve relative uncertainty of the net counting rate comparable to or better than that of the counting efficiency (see Table 1), i.e., a few percent;
- The change (decrease) of the absorbed activity due to the desorption. The model curve (see Equation (4)) is a sum of several exponents in which the quantities K and ${L}_{D}$ are parameters. In order to achieve a better estimate of the parameters, it is important to observe greater differences in the activity in the sample, i.e., to follow the desorption for a longer time. However, this leads to a decrease in the counting rate and an increase in its statistical uncertainty.

^{2}/s] (also shown in Table 3). The diffusion coefficients, the partition coefficients, and the permeabilities versus temperature are also shown in Figure 5, Figure 6 and Figure 7 (the diffusion lengths are not shown as ${L}_{D}=\sqrt{D/\lambda}$). It is seen that their temperature dependences could be described analytically for the studied temperature interval (5–31 °C). The parameters of the linear fits shown in Figure 5, Figure 6 and Figure 7 are summarized in Table 4. That allows for estimating the values of the quantities for a given temperature in that interval and thus to model the absorption and transport of radon in the polymers.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LDPE | Low-Density Polyethylene |

LDPE-A | Low-Density Polyethylene with Anti-slip coating |

HDPE | High-Density Polyethylene |

PE | Polyethylene |

PP | Polypropylene |

LS | Liquid Scintillation |

HPGe | High-Purity Germanium |

TDCR | Triple to Double Coincidence Ratio |

RNG | Radioactive Noble Gas |

SLP | Short-Lived Progeny |

CD | Compact Disc |

SI | International System of Units (from French: Système International (d’unités)); |

“radon” | short for the ^{222}Rn isotope |

“thoron” | short for the ^{220}Rn isotope |

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**Figure 1.**Signal follow-up (in semi-logarithmic scale) of several samples measured at the liquid scintillation counter in Cherenkov-counting mode: (

**a**) unexposed polymer foils immersed in water with radon activity and (

**b**) a Makrofol N foil exposed to radon immersed in distilled water. The points are the experimental data (the uncertainties—not shown, are within the size of the symbols), the solid line is a linear fit of the data, and the dashed line is extrapolation of the fit for better visualization. The signals decrease linearly in semi-log. scale (i.e., exponentially) and the slopes are very close.

**Figure 2.**Cherenkov-counting efficiencies as a function of time for the two types Makrofol foils immersed in water. The uncertainties (not shown) at the level of 1$\sigma $ are about 5% for the points from the first experiment and about 3% for the points from the second experiment.

**Figure 3.**A scheme of the exposure system. In the beginning of the exposure, the activity of the radon source was promptly introduced in the system by the pump. Then, the valves “V” were closed, and the system was disconnected.

**Figure 4.**Experimental data (points) and theoretical curve fits (solid lines) of the desorption follow-up of radon from (

**a**) High-density polyethylene and (

**b**) Makrofol DE foils for the estimation of the partition coefficient and diffusion length at different temperatures. To fit the same scale, the activity data of Makrofol DE at 10 °C are multiplied by 10, as the radon activity concentration in this experiment was one order of magnitude lower than in the other three. The uncertainties are at the level of 1$\sigma $. The embedded smaller graph presents the same data in semi-log scale—it is seen that, in the early desorption the dependences are nonlinear in semi-log scale, i.e., they are sums of several exponents rather than single exponents.

**Figure 5.**Temperature dependence of the diffusion coefficients of the studied materials. Note that the dependence is ln(D) vs. T. The points are the experimental data and the solid lines are linear fits of the data. The uncertainties are at the level of 1$\sigma $.

**Figure 6.**Temperature dependence of the partition coefficients of the studied materials. Note that the dependence is ln(K) vs. T. The points are the experimental data and the solid lines are linear fits of the data. The uncertainties are at the level of 1$\sigma $.

**Figure 7.**Temperature dependence of the permeabilities of the studied materials. Note that the dependence is ln(P) vs. T. The points are the experimental data and the solid lines are linear fits of the data. The uncertainties are at the level of 1$\sigma $.

**Table 1.**Counting efficiencies for: polymer foils in distilled water counted in Cherenkov mode (1–6), distilled water counted in Cherenkov mode (7) and Makrofol N foil dissolved in toluene based liquid scintillation cocktail (LSC) (8). The counting efficiencies are given after reaching equilibrium distribution between radon concentration in the two phases (polymer–water) and/or equilibrium of radon and its short-lived progeny, i.e., these are steady-state counting efficiencies. For (1–4, 7,8), equilibrium is reached after 3–5 h and for (5,6) equilibrium is reached after 60–70 h.

No | Sample | Counting Efficiency |
---|---|---|

1 | PP in water | 0.380(12) |

2 | LDPE in water | 0.371(12) |

3 | LDPE-A in water | 0.400(14) |

4 | HDPE in water | 0.407(13) |

5 | Makrofol N in water | 1.168(36) |

6 | Makrofol DE in water | 0.883(29) |

7 | distilled water | 0.376(12) |

8 | Makrofol N in LSC | 4.946(29) |

**Table 2.**Exposure conditions of the four experiments for estimation of the partition coefficients and diffusion lengths of radon in polymer foils: initial activity concentration of radon ${C}_{A}$ [MBq/m

^{3}], exposure duration (sorption time) ${t}_{s}$ [h], temperature T [°C], and the average thickness L [µm] of the stack of polymer foils of the given type. The uncertainties are at the level of 1$\sigma $. The uncertainties of the thickness include the instrumental uncertainty of the micrometer and the standard deviation of the thickness of the stack of the polymers. “N/A” means that polymers of that type are not used in the given experiment.

${\mathit{C}}_{\mathit{A}}$ [MBq/m^{3}] | ${\mathit{t}}_{\mathit{s}}$ [h] | T [∘C] | L [µm] | |||||
---|---|---|---|---|---|---|---|---|

PP | LDPE | LDPE-A | HDPE | Makrofol N | Makrofol DE | |||

52.4(36) | 46.23 | 21(1) | 31.4(11) | 74.0(28) | 97.0(37) | 123.8(18) | 42.1(11) | 50.6(12) |

49.5 (31) | 52.03 | 5(1) | 31.1(10) | 74.1(24) | 92.0(24) | 123.8(30) | 41.9(11) | 50.0(10) |

31.4 (20) | 48.17 | 31(1) | 29.7(11) | 76.7(39) | 89.6(11) | 120.3(12) | 42.0(11) | 50.2(11) |

1.442(75) | 69.43 | 10(1) | N/A | N/A | N/A | N/A | 41.6(11) | 50.7(11) |

**Table 3.**Partition coefficients polymer–air, diffusion lengths, diffusion coefficients and permeabilities of radon for the studied polymers at different temperatures. The temperature was kept constant within 1 °C. All uncertainties are at the level of 1$\sigma $. For comparison, values obtained in previous studies are given.

PP | LDPE | LDPE-A | HDPE | Makrofol N | Makrofol DE | CD/Makrofol ^{a} | |
---|---|---|---|---|---|---|---|

T [°C] | Partition Coefficient K | ||||||

5 | 6.13(55) | 4.18(39) | 4.05(42) | 3.63(33) | 211(16) | 77.5(67) | 21.5(43) |

10 | – | – | – | – | 183(12) | 72.8(58) | 24.3(36) |

21 | 3.69(38) | 3.66(38) | 3.13(41) | 2.51(22) | 103.3(79) | 34.6(30) | 26.2(19) |

31 | 3.25(43) | 3.70(43) | 2.96(30) | 2.44(21) | 70.2(51) | 27.8(24) | 22.9(10) |

20 | 2.17(14) ^{b}2.40(22) ^{b} | 2.21(13) ^{b} | 112(12) ^{c} | 27.6(16) ^{b} | |||

T [°C] | Diffusion Length ${L}_{D}$ [µm] | ||||||

5 | 67.6(51) | 605(30) | 646(36) | 460(19) | 18.0(10) | 20.8(10) | 42.2(16) |

10 | – | – | – | – | 23.9(10) | 26.8(10) | 42.8(11) |

21 | 198(10) | 1210(64) | 1204(85) | 880(22) | 36.2(10) | 43.3(13) | 53.8(5) |

31 | 300(15) | 1880(140) | 1722(54) | 1252(23) | 52.1(15) | 62.9(16) | 75.5(8) |

20 | 1463(33) ^{b}1437(94) ^{b} | 721(9) ^{b} | 38.9(13) ^{c} | 50.8(10) ^{b} | |||

T [°C] | Diffusion Coefficient D [10^{−14} m^{2}/s] | ||||||

5 | 0.96(14) | 76.9(77) | 87.4(97) | 44.3(37) | 0.0677(79) | 0.0911(84) | |

10 | – | – | – | – | 0.120(10) | 0.151(11) | |

21 | 8.20(85) | 307(33) | 304(43) | 162(8) | 0.275(15) | 0.394(25) | |

31 | 18.9(19) | 739(111) | 623(39) | 329(12) | 0.570(32) | 0.831(43) | |

T [°C] | Permeability P [10^{−13} m^{2}/s] | ||||||

5 | 0.59(10) | 32.1(44) | 35.4(54) | 16.1(20) | 1.43(20) | 0.706(89) | |

10 | – | – | – | – | 2.20(24) | 1.10(12) | |

21 | 3.03(44) | 113(17) | 95.1(18) | 40.7(41) | 2.84(27) | 1.36(15) | |

31 | 6.1(10) | 273(52) | 184(22) | 80.4(75) | 4.00(37) | 2.31(23) |

- PP – Polypropylene, LDPE – Low-Density Polyethylene, LDPE-A – Low-Density Polyethylene with Anti-slip coating, HDPE – High-Density Polyethylene.

$\mathbf{ln}(\mathit{D})={\mathit{a}}_{\mathit{D}}+{\mathit{b}}_{\mathit{D}}\mathit{T}$ Figure 5 | $\mathbf{ln}(\mathit{K})={\mathit{a}}_{\mathit{K}}+{\mathit{b}}_{\mathit{K}}\mathit{T}$ Figure 6 | $\mathbf{ln}(\mathit{P})={\mathit{a}}_{\mathit{P}}+{\mathit{b}}_{\mathit{P}}\mathit{T}$ Figure 7 | ||||
---|---|---|---|---|---|---|

Polymer | ${\mathit{a}}_{\mathit{D}}$ | ${\mathit{b}}_{\mathit{D}}$ | ${\mathit{a}}_{\mathit{K}}$ | ${\mathit{b}}_{\mathit{K}}$ | ${\mathit{a}}_{\mathit{P}}$ | ${\mathit{b}}_{\mathit{P}}$ |

PP | −32.76(35) | 0.1159(51) | 1.93(11) | −0.0262(59) | −30.87(23) | 0.092(10) |

LDPE | −28.33(16) | 0.0869(80) | 1.45(11) | −0.0053(56) | −26.88(19) | 0.0815(96) |

LDPE−A | −28.13(16) | 0.0755(64) | 1.45(12) | −0.0123(56) | −26.69(19) | 0.0635(82) |

HDPE | −28.81(13) | 0.0771(55) | 1.33(12) | −0.0158(56) | −27.49(16) | 0.0619(68) |

Makrofol N | −35.22(12) | 0.0791(57) | 5.603(82) | −0.0441(43) | −29.61(13) | 0.0347(59) |

Makrofol DE | −35.00(11) | 0.0844(54) | 4.62(14) | −0.0443(73) | −30.39(14) | 0.0410(69) |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Georgiev, S.; Mitev, K.; Dutsov, C.; Boshkova, T.; Dimitrova, I. Partition Coefficients and Diffusion Lengths of ^{222}Rn in Some Polymers at Different Temperatures. *Int. J. Environ. Res. Public Health* **2019**, *16*, 4523.
https://doi.org/10.3390/ijerph16224523

**AMA Style**

Georgiev S, Mitev K, Dutsov C, Boshkova T, Dimitrova I. Partition Coefficients and Diffusion Lengths of ^{222}Rn in Some Polymers at Different Temperatures. *International Journal of Environmental Research and Public Health*. 2019; 16(22):4523.
https://doi.org/10.3390/ijerph16224523

**Chicago/Turabian Style**

Georgiev, Strahil, Krasimir Mitev, Chavdar Dutsov, Tatiana Boshkova, and Ivelina Dimitrova. 2019. "Partition Coefficients and Diffusion Lengths of ^{222}Rn in Some Polymers at Different Temperatures" *International Journal of Environmental Research and Public Health* 16, no. 22: 4523.
https://doi.org/10.3390/ijerph16224523