João G D Preturlan

Loïc Favergeon
email: favergeon@emse.fr

Laetitia Vieille

Sara Quiligotti

Comprehensive thermodynamic study of the calcium sulfate-water vapor system. Part 2: Physical modeling of adsorption phenomena

Keywords: adsorption, thermodynamics, phase equilibrium, calcium sulfate

The calcium sulfate-water vapor system is extremely relevant for a vast number scientific and industrial applications. Although the vast number of studies about this chemical system, some critical thermodynamic aspects seem to have been left aside. In this context, in the first part of this study, we investigated the thermodynamic equilibrium between two calcium sulfate polymorphs (AIII-CaSO % and CaSO % ⋅ 0.5 H , O) and quantified the water vapor adsorption as a function of the temperature and water vapor partial pressure. In this second part, we employ a rigorous modeling approach to determine the thermodynamic nature of the water adsorption phenomena on these two polymorphs. We developed macroscopic solution models (ideal and non-ideal) for modeling monolayer adsorption on AIII-CaSO % . This allowed the calculation of the energies of adsorption for this phenomenon, evidencing a physisorption mechanism. For the β-CaSO % ⋅ 0.5 H , O, we interpreted the water adsorption using a multilayer adsorption model. For both materials, we showed that nitrogen adsorption data was not sufficient to represent their entire surface areas and porosity profiles compared to the water vapor sorption capacity of these materials.

Introduction

In the first part of this study, we reported an experimental study about the interaction between calcium sulfate polymorphs soluble anhydrite (AIII-CaSO % ) and calcium sulfate β-hemihydrate (β-CaSO % ⋅ 0.5H , O), both obtained from the dehydration of synthetic gypsum (CaSO % ⋅ 2H , O), and water vapor at atmospheric pressure. The equilibrium conditions between AIII-CaSO % and β-CaSO % ⋅ 0.5H , O in the presence of water vapor were confirmed and compared to available data in the literature. This equilibrium was also confirmed to be univariant. By thermogravimetric analysis under controlled temperature (T) and water vapor partial pressure (P 1 2 3 ), we were also able to observe that compounds of the form CaSO % ⋅ εH , O would show two bivariant zones as recalled in Figure 1. These two bivariant zones, a lower one (for 0 ≤ ε ≤ 0.15) and an upper one (for ε ≥ 0.5), were interpreted as caused by water adsorption on AIII-CaSO % and β-CaSO % ⋅ 0.5H , O, respectively. We observed that the extent of the water adsorption phenomena can be reasonably considerable when compared to the crystalline water content of both phases depending on the P 1 2 3

values. The lack of knowledge regarding these variations in water content could create uncertainties when manipulating or quantifying calcium sulfate based materials.

This second part of this study aims to analyze the nature of the water adsorption on each previously studied phase, AIII-CaSO % and β-CaSO % ⋅ 0.5H , O, and to propose a physical interpretation for this phenomenon. For this, we plotted adsorption isotherms corresponding to each bivariant zone in Figure 1, and we compared the experimental data with thermodynamic models for monolayer and multilayer adsorption. (Shown in the first part of this study). [START_REF] Preturlan | Thermodynamic Equilibrium between Calcium Sulfate and Water Vapor. Part 1: Experimental Study and Phase Equilibria[END_REF]

Thermodynamic models

Before analyzing the water adsorption phenomenon responsible for the bivariant behavior observed in the upper and lower bivariant zones in Figure 1, it is convenient to recall some principles. First of all, we consider the model for water vapor adsorption for which the gas molecules adsorb locally on identical points on the surface (adsorption sites) with no mobility within the adsorbed phase. That is, once the adsorbed molecule is fixed on an adsorption site, the surface of the solid becomes a network of free and occupied adsorption sites. At maximum surface coverage, all the adsorption sites are occupied forming a monomolecular layer. Everett also adds that considering the number of available adsorption sites to be independent of the temperature was observed as a satisfactory procedure to represent experimental data. [START_REF] Everett | The Thermodynamics of Adsorption. Part II.-Thermodynamics of Monolayers on Solids[END_REF][START_REF] Everett | The Thermodynamics of Adsorption. Part III.-Analysis and Discussion of Experimental Data[END_REF] In this context, adsorption behaves like a surface solution of free and occupied adsorption sites for which the equilibrium equation writes [START_REF] Soustelle | Thermodynamics Surfaces and Capillary Systems[END_REF] S + G (;) ⇌ SG (>?@) , (1) where S represents the free adsorption sites, G (;) the adsorbate in the surrounding gas phase, and SG (>?@) the occupied adsorption sites. This equilibrium can be treated by a macroscopic solution model and the law of mass action for the equilibrium in (1 writes as follows

K CD = x GH γ GH x G γ G P (2)
where x J and γ J represent the molar fraction and the activity coefficient of the species i. The parameter P takes account of the pressure, and it is given by the ratio P = P H P°⁄ , where P H is the gas G partial pressure and P° is standard reference pressure (taken as the unit pressure).

This expression is usually simplified by the introduction of the surface coverage fraction θ. This parameter is given by the ratio between the concentration of occupied adsorption sites and the total concentration of adsorption sites. Therefore, the molar fractions in (2 can be expressed as x GH = θ and x G = 1 -θ. Furthermore, for a first approximation, we consider the surface solution to be ideal. This approximation is often interpreted as the case for which there is no interaction between occupied adsorption sites. However, to be precise, this approximation assumes that interactions between occupied and free adsorption sites are identical to the interaction between two occupied adsorption sites. As a consequence, the activity coefficients are γ GH = γ G = 1 and (2 simplifies to the following expression where the pressure term P is homographic to the surface coverage

θ = K P P 1 + K P P (3)
This expression is similar to the well-known Langmuir isotherm equation used to interpret type I isotherms according to the classification of Brunauer, Deming, and Deming and Teller, normally referred to as the BDDT or Brunauer classification, which was standardized by the IUPAC. [START_REF] Rouquerol | Adsorption by Powder and Porous Solids: Principles, Methodology and Applications[END_REF] For this reason, we replace the general equilibrium constant K CD by K P . The adsorption equilibrium constant K P can then be written as follows

K P (T) = K Q exp T- E >?@ RT W (4)
where R stands for the gas constant, K Q represents a pre-exponential constant and E >?@ accounts for the energy of adsorption. As an alternative, Everett considers this equilibrium constant as depending on the change in enthalpy and entropy of adsorption. [START_REF] Everett | The Thermodynamics of Adsorption. Part II.-Thermodynamics of Monolayers on Solids[END_REF][START_REF] Rouquerol | Adsorption by Powder and Porous Solids: Principles, Methodology and Applications[END_REF] For this case, the energy of adsorption would correspond to the change in enthalpy of adsorption Δ > Q H, and the parameter K Q would be a function of the change in entropy of adsorption Δ > Q S. According to this approximation, these two parameters change with the surface coverage except for an assumption of an ideal surface solution and identical adsorption sites. [START_REF] Ruthven | Principles of Adsorption and Adsorption Processes[END_REF] Considering that surface coverage θ can also be defined as the ratio between the adsorbed molar quantity n > per moles of solid and the monolayer capacity n Z per moles of solid, (3 can be rewritten as

n > n Z = K P P 1 + K P P (5)
which is usually employed in its linear form

P n > = 1 K P n Z + P n Z (6)
These forms are useful to treat adsorption data because they allow the quick estimation of the energy of adsorption by using (4 from the fitted values of K P (T). The surface area S [ of the solid in m , g ⁄ is also easily assessed from the fitted values of monolayer capacity n Z and by considering the cross-sectional area σ of the adsorbing molecule according to the following expression

S [ = n Z σN `a M @ (7)
where N `a is the Avogadro's number and M @ the molar mass of the solid.

Nevertheless, it is important to highlight that the physical meaning of the parameter n Z has to be considered with a degree of caution. This parameter was classically considered as a limiting adsorption uptake quantity due to a monolayer filling mechanism for type I isotherms. However, Rouquerol et al. [START_REF] Rouquerol | Adsorption by Powder and Porous Solids: Principles, Methodology and Applications[END_REF] and Gregg and Sing. [START_REF] Gregg | Specific Surface and Porosity[END_REF] discuss that in this context this value should preferentially be considered as a saturation capacity of the adsorbent due mostly to a micropore filling mechanism within pores that are accessible by the molecules of the adsorbate rather than the classic monolayer filling mechanism.

To address the issue of non-ideality of the surface solution into account and evaluate the validity of this hypothesis for the experimental data, we need a macroscopic solution model allowing the mathematical representation of the activity coefficients of surface solution species S and SG (>?@) in (2. When the shift from ideal behavior is not large, it is reasonable to consider the approximation of regular solutions. Hence, the activity coefficients of each component of the surface solution can be expressed by the second-order Margules activity model as follows [START_REF] Soustelle | Thermodynamics Surfaces and Capillary Systems[END_REF][START_REF] Authelin | Thermodynamics of Non-Stoichiometric Pharmaceutical Hydrates[END_REF][START_REF] Okhrimenko | Thermodynamic Study of MgSO4-H2O System Dehydration at Low Pressure in View of Heat Storage[END_REF] ln γ G = B(T)x GH , = B(T)θ ,

ln γ GH = B(T)x G , = B(T)(1 -θ) , (8)
where B(T) is a parameter that depends on the temperature and the intermolecular lateral interactions within the solution. Using a microscopic approach from statistical physics, the coefficient B(T) of regular solutions is typically represented by 4

B(T) = α T ( 10
)
where α is a temperature and composition dependent interaction term.

Substituting (8 and (9 in (2 gives the following expression:

K fG P = θ 1 -θ exp[B(T)(1 -2θ)] (11)
where K CD was replaced by K fG (for the regular solution approximation) and the surface coverage is also given by θ = i j i k

. This equation is also known as the Fowler-Guggenheim adsorption isotherm model. For this model, if B = 0, we obtain the Langmuir adsorption model. For non-ideal solutions, the parameter B needs to be less than 2, otherwise, if B ≥ 2, there is phase separation or no solution is formed at equilibrium (demixing case). Figure 2 The choice for the Margules activity coefficient model to express the non-ideality of the surface solution came from the use of the regular solution approach. As previously mentioned, this approach allows a relatively simple way of treating non-ideality of a solution when the behavior does not differ much from the ideal one. However, there are many other more sophisticated and complicated activity coefficient models for liquid-vapor equilibrium (Redlich-Kister, Van Laar, NRTL, etc.) that can be used depending on the complexity of the system. [START_REF] Prausnitz | Molecular Thermodynamics of Fluid-Phase Equilibria[END_REF] Activity coefficients have also been expressed by temperature independent polynomial factors. [START_REF] Carey | Equilibrium in the Clinoptilolite-H2O System[END_REF][START_REF] Fialips | Hydration-Dehydration Behavior and Thermodynamics of Chabazite[END_REF] Both models presented by (3 and (11 correspond to localized adsorption for which we consider the hypothesis of theoretical monolayer adsorption. These models are not adapted to represent adsorption isotherm profiles as for type II and type IV isotherms, which are normally interpreted as being produced by a multilayer adsorption mechanism for non-porous and mesoporous materials, respectively. For isotherms with these profiles, the Brunauer-Emmett-Teller (BET) theory for multilayer adsorption can then be employed to fit the adsorption curve and derive the BET area of the sample for each temperature condition. [START_REF] Brunauer | Adsorption of Gases in Multimolecular Layers[END_REF] The BET equation is then given by the following expression

1 n > (1/P n -1) = 1 n Z C opq + C opq -1 n Z C opq P n ( 12
)
where P n stands for the relative pressure (or relative humidity RH) and is given by the ratio P H P H @>r (T) ⁄ (P H @>r (T) stands for the saturation pressure of the adsorbing gas for the isotherm temperature T). The coefficient C opq is the BET parameter, and it is defined as an energetic term as follows

C opq = exp s -tE >?@,u -E v w RT x (13)
where E >?@,u is the energy of adsorption of the first layer of adsorbed gas molecules, and E v is the liquefaction energy for this adsorbed gas.

In rare cases, the adsorption isotherms profile is the product of only one type of adsorption phenomenon. That is, usually, adsorption isotherms are in fact an association of isotherms of different types. For example, type II or type IV can commonly be associated with type I isotherms.

This association means mean that pores with different size ranges are present within the analyzed sample. For instance, micropores could be present along with mesopores and macropores. For these cases, we observe the appearance of a classic "knee" in the isotherm profile indicating the filling of micropores (type I) before a change in concavity due to multilayer adsorption (type II or IV). The BET theory is typically applied for purely type II or type IV isotherms for the relative pressure range of 0.05 ≤ P n ≤ 0.30 and normally reaches its limitations for microporous materials. This would restrain the use of this theory to estimate the BET area of association of isotherms. However, some authors argued that this method can still give representative information about the pores and surface of the sample. Mooney et al. argued about the use of the BET theory to model combined type I and type II water vapor adsorption isotherms as long as the usual requirements (relative pressure ranges) are fulfilled. [START_REF] Mooney | Adsorption of Water Vapor by Montmorillonite. I. Heat of Desorption and Application of BET Theory[END_REF][START_REF] Mooney | Adsorption of Water Vapor by Montmorillonite. II. Effect of Exchangeable Ions and Lattice Swelling as Measured by X-Ray Diffraction[END_REF] According to the criteria published by Llewellyn et al., we are still able to consider the BET area as a "BET strong retention capacity" or "saturation capacity" of the material if two criteria are respected: (i) the value of C opq must be positive for the range of points chosen for the calculation and (ii) the term n > tP 1 2 3 @>r -P 1 2 3 w must continuously increase (self-consistency criterion). [START_REF] Llewellyn | Is the BET Equation Applicable to Microporous Adsorbents? Stud[END_REF] The described methods were applied to interpret the adsorption phenomenon for AIII-CaSO % and β-CaSO % ⋅ 0.5H , O. We chose isotherms for six temperature values in each bivariant zone in Figure 1 to analyze the nature of this phenomenon and have a broad description of each zone instead of working with adsorption at only one temperature as most of the studies aiming only at calculating specific surface areas. Figure 3 shows the linear transform of the adsorbed quantity values and the fitted curves obtained by using (6. These linear transforms were used to fit the values of n Z and K P shown in Table 1 and Figure 4. Overall, the localized monolayer ideal adsorption model represents the experimental data points and their trends with more dispersion between experimental and fitted values for higher temperatures. To better evaluate the overall validity of the assumptions taken into account by this model, we analyzed the values of the fitting parameters n Z and K CD and the adsorption isotherms plotted in Figure 5. S opq (N , ) = (14 ± 1) m , /g (C opq = 63). This value is almost 5 times inferior to the average value obtained from water vapor adsorption data. This difference can be associated with the fact that nitrogen might not access to the entire pore network because of its larger molecular size when compared to water vapor. Nitrogen molecules are considered to have a cross-sectional area of 0.162 nm , , which is reasonably more significant than the water molecules cross-sectional area of 0.125 nm , . Therefore, water molecules would more easily gain access to micropores with reduced dimensions (such as the supermicropores with a pore diameter between 2 and 0.7 nm) in contrast to nitrogen molecules. [START_REF] Lastoskie | Pore Size Distribution Analysis of Microporous Carbons: A Density Functional Theory Approach[END_REF] This different behavior of the material in the presence of water compared to the nitrogen is also sustainable considering the t-plot curve for the nitrogen adsorption data. The nitrogen adsorption results are indeed characteristic of non-porous materials and, as a consequence, they do not show the same textural properties as the water vapor adsorption data, which indicate the behavior of a microporous material. Furthermore, this kind of discrepancy in adsorption results obtained by different gases has been observed several times in the literature.

Application of thermodynamic methods and discussion

Other authors such as Hagymassy et al. [START_REF] Hagymassy | Pore Structure Analysis by Water Vapor Adsorption[END_REF] , Mooney et al. [START_REF] Mooney | Adsorption of Water Vapor by Montmorillonite. I. Heat of Desorption and Application of BET Theory[END_REF][START_REF] Mooney | Adsorption of Water Vapor by Montmorillonite. II. Effect of Exchangeable Ions and Lattice Swelling as Measured by X-Ray Diffraction[END_REF] , Médout-Marère et al. [START_REF] Médout-Marère | Thermodynamic Analysis of the Immersion of a Swelling Clay[END_REF] , and Salles et al. [START_REF] Salles | Hydration Sequence of Swelling Clays: Evolutions of Specific Surface Area and Hydration Energy[END_REF] have shown that porous materials such as concrete, cement, silica, and clays show more important sorption of water vapor than nitrogen. Gregg and Willing have also shown that during the decomposition of calcium sulfate dihydrate, different gases (oxygen, nitrogen, CCl % , water, etc.) are adsorbed to different extents for the same material. [START_REF] Gregg | The Dehydration of Gypsum. Part I. A Study of the Thermal Decomposition of Gypsum by the Adsorption of Oxygen and of Nitrogen at -183[END_REF][START_REF] Gregg | The Dehydration of Gypsum. Part II. A Study of the Thermal Dehydration of Qypsum by Means of the Adsorption of Carbon Tetra-Chloride Vapour at 25°[END_REF][START_REF] Gregg | The Dehydration of Gypsum. Part III. A Study of the[END_REF] As a consequence, the analysis of the surface area and the porosity by nitrogen adsorption at 77K, techniques that are typically employed and mastered for materials characterization, showed not to be sufficiently adapted to understand the water adsorption capabilities of AIII-CaSO % . As a complement to the observations in this study, to verify the full porosity range water molecules have access to, it would be interesting to perform a full adsorption isotherm for water vapor. That is, analyzing the water adsorption capacity of the materials for a large P 1 2 3 range, from almost completely dry conditions to near saturation pressure. However, this was not technically feasible using our experimental apparatus (which is limited to water vapor partial pressures up to 60 hPa) as detailed in the first part of this study. [START_REF] Preturlan | Thermodynamic Equilibrium between Calcium Sulfate and Water Vapor. Part 1: Experimental Study and Phase Equilibria[END_REF] Moreover, as phase transition towards CaSO % ⋅ 0.5H , O can take place as the P 1 2 3 increases, which would limit a full adsorption spectrum of the AIII-CaSO % . Other interesting experiments to carry out to elucidate the different accessibility of gases in the pore network of AIII-CaSO % would be performing systematic adsorption experiments with different gases showing diverse cross-sectional areas and polarity.

adsorption could be that the partial pressure of water vapor can increase locally in pores and thus induce capillary condensation of water. This phenomenon could produce local hydration reactions near the surface of the solid and eventually locally hydrate the sample of AIII-CaSO % to CaSO % ⋅ 0.5H , O at pore walls increasing the retention capacity of water of the solid. This local effect would be complicated to measure experimentally, but the estimation of pore saturating pressures (based on pore dimensions) could be compared to the values of P 1 2 3 and T for phase transition to verify the plausibility of this phenomenon as further work. The second fitting parameter K P is presented in the form of a van't Hoff plot in Figure 4(b). The energy of adsorption was estimated as E >?@ = -16.5 kJ ⋅ mol ‡u . As expected for an adsorption process, E >?@ < 0 (evidencing an exothermic process). Moreover, the low value of |E >?@ | < 20 kJ ⋅ mol ‡u confirms the hypothesis of a physisorption process as previously considered to develop this model. The overall quality of fit presented in the van't Hoff plot in Figure 4(b) shows a linear trend with some dispersion. This can be caused by experimental effects combined with, as previously mentioned, non-ideal effects within the surface solution.

Finally, we chose to use the maximum value of monolayer coverage in Table 1 (i.e., n Z = 0.141 mol H , O mol CaSO % ⁄ ) to plot the adsorption isotherms and compare these curves as shown in Figure 5. The calculated curves represent the overall trend for each temperature, but with some shifts. These shifts are evidently produced by the difference between the used monolayer capacity value and the obtained fitted values for each temperature. To improve the quality of the fit for each curve, we evaluated the hypothesis of a non-ideal surface solution. is compared to the experimental points (blue squares).

To address the issue of non-ideality of the surface solution into account and evaluate the validity of this hypothesis for interpreting the experimental data, we employed a macroscopic solution model to express the activity coefficients of its components. When the shift from ideal behavior is not considerably large (as observed in Figure 4 and Figure 5), it is reasonable to consider the approximation of regular solutions. We employed then the mathematical expression shown by (11, which is based on the Margules activity model. To evaluate this model for the presented adsorption isotherms, we considered the number of adsorption sites to be constant and took the maximum value of monolayer coverage, n Z = 0.141 mol H , O mol CaSO % ⁄ , calculated for the ideal surface solution case. Figure 6 shows the comparison of the calculated surface coverage data plotted with the fitted curves. Table 2 shows the fitting parameters K fG and B and also the calculated values of α according to (10. Parameters α and K fG are graphically represented in Figure 7.

Initially, we observe that the fitted curves in Figure 6 correspond more adequately to the experimental points for each temperature value compared to the previous model. Therefore, the consideration of an additional fitting parameter that takes account of the non-ideality of the surface solution is suitable for the considered system. This parameter considers the interaction energies between adsorption sites not be equivalent. The results are presented as a function of P (the ratio between the water vapor partial pressure P 1 2 3

and the reference unit pressure P ‰ ). The localized monolayer adsorption model combined with the regular solution approach (dashed-line fitted curves) is compared to the experimental points (blue squares). shows the van't Hoff plot for K fG . We observe that the experimental points show less dispersion than for the ideal surface solution approach. We can also estimate the energy of adsorption value as E >?@ = -18.0 kJ ⋅ mol ‡u . The negative and low value of this energy is following previous hypotheses of exothermic processes and physisorption. Besides, this value is not very different compared to the one obtained previously evidencing the small shift in behavior compared to the ideal surface solution hypothesis. Finally, we recall that the chosen regular solution model implies phase separation for B(T) > 2. Hence, the fitted values of B(T) shown in Table 1 confirm the thermodynamic feasibility of the surface solution. Overall, the hypothesis of the non-ideal surface solution represented by the regular solution hypothesis improves the quality of the fitted curves and fitted parameters compared to the ideal surface solution approach. The behavior of water adsorption on AIII-CaSO % is well described by the hypothesis of monolayer adsorption where the heterogeneities within the interactions of the surface solution components play a part in this phenomenon. Depending on the objective of the analysis, a predictive equation giving the adsorbed quantity can be proposed to represent the experimental data in the given range respecting the experimental errors. To analyze this observed behavior, we first consider the group of curves for which the profiles show a change in concavity, i.e., from 45℃ to 70℃. Even though we are limited in terms of the experimental pressure range (5 to 60 hPa), these isotherms seem to be typical of non-porous or mesoporous materials, corresponding to type II and IV isotherms, respectively. The BET theory for multilayer adsorption can then be employed to analyze the behavior of this material and derive the water vapor BET area S opq (H , O) of the sample for each temperature condition. To apply this equation, we reasoned similarly as for water vapor adsorption on AIII-CaSO % . The overall quantity of water ε measured in the upper bivariant region was considered to have two contributions: one by crystalline water content and the other by the adsorbed water content. For the CaSO % ⋅ 0.5H , O, the crystalline water content is equal to 0.5 mol H , O mol CaSO % ⁄ . Therefore, the adsorbed water content can be expressed as n > = ε -0.5. We then plotted these experimental values of the adsorbed water content n > as a function of the relative pressure P n as shown in Figure 8(b). As expected, we observe that with increasing temperature we reduce the relative pressure interval for the isotherm (because the saturation pressure value increases systematically). For this reason, for the lower temperature values, from 45℃ to 70℃, we manage to apply the BET equation (cf. (12).

However, for the higher values of temperature, 80℃ and 90℃, the relative pressure interval is too reduced to consolidate the type II or IV behavior of the isotherm and apply the BET equation accordingly. Figure S4 in the supporting information section shows the BET plots for temperature values from 45℃ to 70℃ from which we obtained the parameters shown in Table 3. Figure S5 shows the self-consistency plots for this same temperature interval, and they respect the criteria recalled in the methods section. [START_REF] Llewellyn | Is the BET Equation Applicable to Microporous Adsorbents? Stud[END_REF] Using these parameters we could trace the BET adsorption isotherms from 45℃ to 70℃ as shown in Figure 8 Overall, we observe that the BET equation represents well the experimental data for the temperature values from 45℃ to 70℃. However, we observe a shift between the model and the experimental points for low relative pressure values. We can observe that the experimental curves present a sharp change in concavity for these low values of P n (or a "knee", as usually referred to in the literature) at the beginning of the isotherm, and these profiles are normally associated with the phenomenon of micropore filling. This effect is more evident for the temperatures from 50℃

to 70℃. This "knee" frequently represents that these isotherms are, in fact, an association of type II or type IV along with a type I isotherm. Usually, the BET theory reaches its physical limits for microporous materials. However, as the two criteria published by Llewellyn et al. [START_REF] Llewellyn | Is the BET Equation Applicable to Microporous Adsorbents? Stud[END_REF] were verified, the n Z parameter and the calculated BET area do present a physical meaning and are suitable to represent the water retention capacity of the sample for each given temperature value.

We also observe that the calculated value for the monolayer capacity n Z and the BET area values increase with temperature. It ranges from S opq (H , O) = 46 m , g ⁄ for 45℃ to S opq (H , O) = 61 m , g ⁄ for 70℃. This would mean that for higher temperatures, more water is required to completely fill the pores of the sample. This effect is may be that at higher temperatures the pores of the material are more available to water adsorption compared to lower temperatures. We also observed that the C opq parameter decreases with increasing temperature. This would represent that the BET theory less adequately represents the behavior of the system at these temperatures.

Finally, the non-ideality of the adsorbed layer (or surface solution) can also be compensated by these fitting parameters.

Finally, we notice a similar effect when comparing the water vapor and nitrogen adsorption phenomena, like the one observed for AIII-CaSO % . Figure S3 in the supporting information document shows the results for one nitrogen adsorption and desorption isotherm obtained at 77K

for one sample of β-hemihydrate. This estimated BET area of the material was S opq (N , ) = (7.3 ± 1.6) m , /g (C opq > 100). Therefore, we observed that the values of S opq (H , O) in Table 3 are much superior to the nitrogen BET area S opq (N , ). The t-plot for the hemihydrate shown in Figure S3 also the weak retention of nitrogen by micropores when compared to the water vapor adsorption. Therefore, the nitrogen adsorption data supports a non-porous material in contrast to water adsorption data that evidences the presence of micropores. These micropores would be preferentially accessible by molecules smaller than nitrogen, such as water. In other words, the water molecules seem to have access to a more extended surface than nitrogen molecules. Therefore, similar discussions as proposed for the adsorption on AIII-CaSO % are also convenient for the β-CaSO % ⋅ 0.5H , O. For example, the broadly used nitrogen adsorption techniques seem to be insufficient to represent the whole extent of the β-CaSO % ⋅ 0.5H , O textural properties. To obtain more reliable and complete data about the surface area and the pore network of this material, it would also be interesting to perform complete water vapor adsorption isotherms for the given temperatures with relative water vapor pressure (or relative humidity) values up to near saturation values. Using different adsorbing gases could also increase the comprehension of the textural properties of β-hemihydrate.

Furthermore, we observe that the specific surface area values estimated by water adsorption for β-hemihydrate (between 46 and 61 m , g ⁄ ) are sensibly inferior to the ones calculated for AIII-CaSO % (between 60 and 78 m , g ⁄ ). This difference may be produced by the fact that the hydration reaction decreases the number of open pores for adsorption due to the increase in the crystalline water content. This also may be caused by the structural change taking place during the phase transition between these two polymorphs.

The main differences between the considerations for adsorption on AIII-CaSO % compared to the lower temperature isotherms obtained for β-CaSO % ⋅ 0.5H , O reside in the adsorption mechanism.

Water vapor adsorption seems to proceed with a multilayer adsorption mechanism with the likely presence of micropores.

Finally, other mathematical models for adsorption on microporous-mesoporous materials can be found in the literature. However, they present little gain in physical interpretation compared to the BET model. For temperatures of 80℃ and 90℃, we observe that due to restricted relative pressure range we could only obtain the beginning of their isotherms. This made it difficult to describe the general profile of the adsorption isotherms for these conditions.

Conclusions

This was the second part of a two-part study about the thermodynamic aspects of the thermodynamic equilibrium between water vapor and two calcium sulfate polymorphs, AIII-CaSO % and β-CaSO % ⋅ 0.5H , O, obtained from the dehydration of gypsum. The first part consisted in the experimental determination of the equilibrium conditions between these two compounds and the quantification of the water adsorption phenomenon. [START_REF] Preturlan | Thermodynamic Equilibrium between Calcium Sulfate and Water Vapor. Part 1: Experimental Study and Phase Equilibria[END_REF] The second part aim was to propose physical interpretation for the observed adsorption phenomena by means of thermodynamic modeling of the experimental adsorption isotherms.

For AIII-CaSO % , we proposed models of localized monolayer adsorption and the adsorbed layer was considered as a solution between free and occupied adsorption sites. Initially, we assumed ideal behavior of the surface solution and employed the Langmuir adsorption model to fit it. This model was not sufficient to represent the integrality of the adsorption isotherms and the fitted parameters presented some dispersion. Afterwards, the fitting was improved by the assumption of non-ideal behavior of the surface solution and the introduction of a parameter to take account of the interactions between the solution components. For both methods the values of the energy of adsorption were considerably low and confirmed an initial hypothesis of physisorption.

For β-CaSO % ⋅ 0.5H , O, the water adsorption was modeled assuming multilayer adsorption. The adsorption isotherm profiles for temperature from 45℃ to 70℃ were not homographic as the ones for AIII-CaSO % and we employed the BET theory to analyze them. We observed shifts between the BET theory and the experimental points for low relative pressure ranges and attributed these effects to the presence of micropores in the matrix of the material. Isotherms for temperatures of 80℃ and 90℃ were left out of the modeling because they did not present a sufficiently large range of relative pressure to assess their characteristics.

For both materials, we observed an important effect concerning the differences between specific surface and profile of the adsorption isotherm for water vapor and nitrogen. The adsorption profiles obtained for water adsorption were typical of microporous materials. However, nitrogen adsorption data obtained at 77 K showed a profile for non-porous materials.

Figure 1 .

 1 Figure 1. Isobars showing the overall water content ε of the calcium sulfate compounds as a

  shows a representation of adsorption isotherms for these different values of the parameter B. For B ≥ 2 we observe that the same pressure P 1 2 3 show more than one θ value. This behavior has no physical meaning and indicates the case of solution demixing, as highlighted in zones D1 and D2.[START_REF] Authelin | Thermodynamics of Non-Stoichiometric Pharmaceutical Hydrates[END_REF][START_REF] Okhrimenko | Thermodynamic Study of MgSO4-H2O System Dehydration at Low Pressure in View of Heat Storage[END_REF][START_REF] Butt | Physics and Chemicstry of Interfaces[END_REF][START_REF] Soustelle | Thermodynamic Modeling of Solid Phases[END_REF]

Figure 2 .

 2 Figure 2. Water adsorption isotherms showing the evolution of the isotherm profile according to

3. 1 .

 1 Thermodynamic modeling of water adsorption on soluble anhydrite. The profiles of the adsorption isotherms for the AIII-CaSO % corresponding to the lower bivariant zone (0 < ε < 0.15) in Figure1are shown in Figure5for six temperature values. These values were chosen to provide a relatively broad description of this bivariant zone. The isotherm profiles are analogous to homographic curves presenting a saturation plateau. This corresponds to type I isotherms according to the BDDT classification. Therefore, we initially evaluated the hypothesis of localized monolayer adsorption with ideal behavior. This case is represented by the Langmuir adsorption model shown by (3. The quantity of water ε is considered to correspond entirely to adsorbed water because the sample is constituted of the anhydrous crystalline phase AIII-CaSO % . Therefore, we take ε = n > for this data set.

Figure 3 .

 3 Figure 3. Langmuir linear transformation of the adsorption isotherms compared to the

Figure 4 .

 4 Figure 4. Fitting parameters n Z and K P for the localized monolayer ideal adsorption model. (a)

Figure 5 .

 5 Figure 5. Isotherms for the adsorbed quantity of water n > on AIII-CaSO % for temperatures between

Figure 6 .

 6 Figure 6. Adsorption isotherms for phase AIII-CaSO % for temperatures between 125℃ to 250℃.

Figure 7 (

 7 Figure 7(a) shows that the interaction parameter α decreases with increasing temperature. This

Figure 7 (

 7 Figure 7(b) shows the van't Hoff plot for K fG . We observe that the experimental points show

Figure 7 .

 7 Figure 7. Fitting parameters α and K fG for the localized monolayer non-ideal adsorption model:

Figure 8 (

 8 a) shows the isotherms representing the overall water contents for temperature values between 45℃ and 90℃ in the upper bivariant zone in Figure1. Analogously as the previous section, we used six temperatures values to run through the different possible behaviors of the water vapor adsorption within the bivariant zone for which ε ≥ 0.5. Two main kinds of profiles can be distinguished. At first, the curves observed for temperature values from 45℃ to 70℃ present a change in concavity and recall the profiles for type II or type IV isotherms according to the BDDT classification. On the other hand, the curves for 80℃ and 90℃ do not present the same change in concavity and recall a homographic profile of type I (with a saturation plateau).

  (a) and Figure 8(b) by the dashed lines. For the temperature values of 80℃ and 90℃, the presented dotted lines in Figure 8(a) are only a guide for the eyes to evidence the different observed profile.

Figure 8 .

 8 Figure 8. Isotherms for the (a) overall quantity of water of β-CaSO % ⋅ 0.5H , O and (b) adsorbed

Table 1 .

 1 Fitting parameters for the localized monolayer ideal adsorption hypothesis (LangmuirThe fitting parameter n Z is listed in Table1and plotted in Figure4(a) as a function of the temperature. We observe a sensible decreasing trend for this parameter with stabilization between 200℃ and 250℃. As previously mentioned, this parameter is usually assumed to be constant or independent of the temperature. Therefore, the variation observed for this parameter can perhaps be associated with the non-ideality of the surface solution (presence of interactions between molecules) or due to surface heterogeneities. The non-ideality effect is further evaluated. The monolayer capacity can be used to estimate specific surface area values ranging from S [ (H

	adsorption).
	T (℃)	K P	n Z (mol H , O mol CaSO % ⁄	) S [ (H , O) (g m , ⁄ )
	125	70.6	0.141 ± 0.007	78 ± 4
	150	62.5	0.125 ± 0.006	69 ± 3
	175	44.7	0.122 ± 0.006	68 ± 3
	200	44.4	0.109 ± 0.005	60 ± 3
	225	33.8	0.110 ± 0.005	61 ± 3
	250	23.7	0.116 ± 0.005	64 ± 3

, O) = (78 ± 4) m , /g to S [ (H , O) = (60 ± 3) m , /g (considering an average cross-sectional area for the water molecule

[START_REF] Mcclellan | Cross-Sectional Areas of Molecules Adsorbed on Solid Surfaces[END_REF]

of σ = 0.125 nm , and molar mass of anhydrous CaSO % as 136.14 g mol ⁄ ). We can estimate an average surface area of S [ (H , O) ƒƒƒƒƒƒƒƒƒƒƒ = (66 ± 5) m , /g. These surface area measurements can be compared to measurements obtained by nitrogen adsorption data and calculations performed using the BET theory. Figure

S2

in the supporting information file shows the results of nitrogen adsorption and desorption isotherms obtained at 77K for one sample after dehydration at 200℃. As the isotherm presents a type II profile with an almost negligible hysteresis loop, we calculate the nitrogen BET area of the material as corresponding to

Table 2 .

 2 Fitting parameters for non-ideal surface solution (regular solution approach)

	T (℃)	K fG	B	α
	125	64.3 2.31 ⋅ 10 ‡u	8040
	150	42.4 9.75 ⋅ 10 ‡OE	6360
	175	30.9 6.13 ⋅ 10 ‡OE	5410
	200	24.4 8.96 ⋅ 10 ‡%	4880
	225	20.7 3.46 ⋅ 10 ‡,	4650
	250	17.2 9.03 ⋅ 10 ‡,	4280
	3.2

. Thermodynamic modeling of water adsorption on

β-hemihydrate.

Table 3 .

 3 Parameters obtained from the application of the BET model for isotherms from 45℃ to

	70℃
	T (℃) C opq n Z (mol H , O mol CaSO % ⁄	) S opq (H , O) (m , g ⁄ )
	45	141	8.40 ⋅ 10 ‡,	46
	50	144	8.30 ⋅ 10 ‡,	45
	60	110	9.40 ⋅ 10 ‡,	51
	70	77.5	1.11 ⋅ 10 ‡u	61

  Furthermore, the surface area values calculated from water adsorption data (either S [ (H , O) for AIII-CaSO % or S opq (H , O) for β-CaSO % ⋅ 0.5H , O) were several times larger than the values S opq (N , ) obtained P H Partial pressure of adsorbing gas G P° Reference pressure (normally taken as the unit pressure) R Universal gas constant (= 8.314 J ⋅ mol ‡u ⋅ K ‡u ) Specific surface area (m , ⋅ g ‡u ) S opq (N , ) BET area calculated from nitrogen adsorption data (m , ⋅ g ‡u ) S opq (H , O) BET area calculated from water vapor adsorption data (m , ⋅ g ‡u )

	θ	Surface coverage fraction
	K P	Langmuir equilibrium constant (ideal solution)
	K P,Q	Pre-exponential constant for Langmuir adsorption isotherm model
	E >?@	Adsorption energy (J ⋅ mol ‡u ⋅ K ‡u )
	n >	Adsorbed quantity (mol H , O mol CaSO % ⁄	)
	n Z	Monolayer capacity (mol H , O mol CaSO % ⁄	)
	S [
	N `a	Avogadro's number
	M @	Molar mass (g ⋅ mol ‡u )
	B	Margules interaction parameter
	α	Interaction parameter
	α Q	Pre-exponential factor
	Γ	Thermal coefficient (kJ ⋅ mol ‡u )
	K fG	Equilibrium constant from regular solution approximation (non-ideal adsorption)
	P n	Relative pressure
	P H @>r (T)	Saturation pressure of gas G
	C opq	BET constant
	GREEK LETTERS
	ε	Overall water content (mol H , O mol CaSO % ⁄	)
	σ	Cross-sectional area of an adsorbing gas molecule (nm , )

molecules have reduced dimensions and may access smaller pores when compared to nitrogen molecules. Other effects of capillary condensation could also play a role in this phenomenon.

ASSOCIATED CONTENT

The following files are available free of charge.

Additional characterization data for the initial sample and for its dehydration product at 200℃ is provided in Support_Preturlan_JGD.PDF

AUTHOR INFORMATION

Corresponding

Table of contents graphics

Multilayer model Monolayer model


Mascia Bedendo
email: mbedendo@audencia.com

Linus Siming

Incentivizing organ donation through a non-monetary posthumous award

Since 2013, The Order of St John Award for Organ Donation is offered to the families of deceased solid

INTRODUCTION

In 2013, the Order of St John together with the National Health Service (NHS) Blood and Transplant introduced The Order of St John Award for Organ Donation in the United Kingdom. This non-monetary award, which takes the form of a pin badge and a scroll, is given to the families of deceased solid organ donors as recognition of the generosity of the donor at public award ceremonies. An additional explicitly stated purpose of the award is to raise awareness of the importance of solid organ donation and inspire others to follow in the honored donors' footsteps [START_REF] John | Order of St John ceremonies, frequently asked questions and answers[END_REF]. To the best of our knowledge, this award represents the first nationwide official recognition for deceased organ donors in any country. Previous awards have either been ad hoc in nature or granted by individual hospitals [START_REF] Gardiner | The Order of St John Award for organ donation[END_REF].

We use the introduction of the award to study the incentivizing effects of posthumous recognition on organ donation. There is ample theoretical ground for believing that awards can spur solid organ donation among both living and deceased potential donors. [START_REF] Niederle | Philanthropically funded heroism awards for kidney donors?[END_REF] survey a representative sample of adults in the United States to collect approval rates of various policies that reward living kidney donors. They find that a policy that rewards a kidney donor with a medal of heroism and a monetary prize (provided by a private foundation rather than taxpayers) has the significantly highest approval. [START_REF] Woodfine | New thinking about honor and deceased organ donation[END_REF] argue that an increased level of nonmonetary public recognition of the family members of deceased organ donors in the United States could inspire others in the community to consent to organ donation.

To estimate to what extent posthumous non-monetary public recognition initiatives can incentivize solid organ donation, we use a difference-in-differences approach around the introduction of The Order of St John Award for Organ Donation. We have natural treatment and control groups since this award is given only to commemorate those who donated solid organs--e.g. heart, kidneys, liver, lungs or pancreas--whereas tissue donation and eye-only donation (cornea) are not recognized under this scheme. While cornea and tissue transplantations can have a dramatic effect on a patient's quality of life, they are not considered lifesaving [START_REF] Gaum | Tissue and corneal donation and transplantation in the UK[END_REF] and, as such, were not included in this award scheme. Accordingly, in our analysis, the treatment group consists of deceased donors of solid organs and the control group consists of deceased donors of cornea who did not donate any solid organ. Our difference-in-differences analysis suggests that the award has produced two important effects: An increase in the number of deceased solid organ donors and an increase in the willingness to donate solid organs among the population that has registered as donors.

Our results contribute to the literature on successful policy incentives put in place to increase the supply of transplantable organs. [START_REF] Stoler | Incentivizing organ donor registrations with organ allocation priority[END_REF] include a recent overview of such policies, which for example include systems where individuals are considered organ donors unless they actively opt out [START_REF] Horvat | Informing the debate: rates of kidney transplantation in nations with presumed consent[END_REF], priority schemes for registered organ donors (Kessler andRoth, 2012, 2014), financial compensation to the donor or the donor's family [START_REF] Friedman | Payment for living organ donation should be legalised[END_REF], and organ exchange [START_REF] Roth | Kidney exchange[END_REF]. Understanding which policies can spur organ donation is of course important beyond the academic discourse, as there is a global shortage of organs [START_REF] Stoler | Incentivizing organ donor registrations with organ allocation priority[END_REF]. Our study also contributes to the literature on the incentivizing effects of official non-monetary awards. A growing number of empirical papers have documented that a bestowing body can push individuals to outperform along desired dimensions through awards. For example, [START_REF] Ager | Killer incentives: Relative position, performance and risk-taking among German fighter pilots[END_REF] find that bravery awards can trigger fighter pilots to take more risks in combat while [START_REF] Raff | Knighthoods, damehoods and CEO behaviour[END_REF] show that governments can use knighthoods to induce business leaders to uphold an excessive number of employees. This study contributes to the literature on honors and awards by being the first to show that even posthumous awards can have an incentivizing effect in increasing the output targeted by the bestowing body.

INSTITUTIONAL BACKGROUND

Currently, organ donation in the United Kingdom is conducted through an opt-in system.1 An individual who wishes to donate her organs can register with the NHS Organ Donor Register. The family members of a registered donor will be consulted after the person's death to ensure that she had not changed her mind after registration and to give their final consent to the organ donation. The United Kingdom organ donation legislation states that consent lies with the deceased, but, in practice, relatives' wishes are respected if they do not consent to the organ donation [START_REF] Vincent | Consent for organ donation[END_REF]. The family's consent is also sought in case the deceased was not registered as a donor. According to the NHS Blood and Transplant Organ Donation and The award, which is depicted in Figure 1, is wearable in the form of a pin and is offered by invitation to all families in the United Kingdom whose relative donated a solid organ after death. The award is open to people of all faiths or no faith. The concept of the award was modeled on The Elizabeth Cross [START_REF] Gardiner | The Order of St John Award for organ donation[END_REF], which is a posthumous award given to the next of kin of members of the British Armed Forces killed in action. The Order of St John Award for Organ Donation, together with a scroll, is awarded posthumously to donors and accepted on their behalf by relatives at an official regional ceremony. In 2014, 65% of the families of deceased solid organ donors accepted the offer to participate to the ceremony and receive the award. Those ceremonies have generally attracted a significant level of media publicity.3

METHOD

To analyze if a non-monetary posthumous award can incentivize organ donation we gather data on deceased solid organ and cornea-only donors in the United Kingdom over the years 2010 to 2017 from the annual issues of the NHS Blood and Transplant

Organ Donation and Transplantation Activity Reports. 4 The treatment group consists of deceased donors where at least one solid organ was recovered for the purpose of transplantation and the control group consists of deceased donors where only corneas were recovered for the purpose of transplantation. Some of the donors in the treatment group may have also donated their corneas, but none of the donors in the control group donated a solid organ. We therefore refer to the treatment (control) group as solid organ (cornea-only) donors. We use the data to estimate the following model of deceased donors:

Y t,i = Post award + Treatment group + d(Treatment group × Post award) + e t,i (1)
where Y t,i is the natural logarithm of deceased donors during year t in group i. The two groups i are the treatment and control groups. For the treatment (control) group Y is the log of the number of deceased solid organ (cornea-only) donors from 1 April to 31

March in each year. The first year in which the award is in place is the one ending on 31

March 2014. The period-specific dummy variable Post award measures the common shocks to the two groups in the pre-award and post-award periods. It is an indicator variable that takes the value of one for each year that the award system is in place (2014, 2015, 2016, and 2017), and zero otherwise (2010, 2011, 2012, and 2013). Thus,

Post award controls for nation-wide shocks and trends that shape the dependent variables over time in the United Kingdom, such as the overall number of deceased individuals, general medical progress or general health-care policy changes. 5 The dummy variable Treatment group, which equals one for solid organ donors and zero for cornea-only donors, accounts for time-invariant differences in the number of donations due to group-specific factors. By including these indicator variables in the model, all of the common time variation and the cross-sectional variation across groups are removed.

The interaction Treatment group × Post award is an indicator variable that takes the value of one for the treatment group for each year that the award system is in place (i.e., after year 2013), and zero otherwise. Thus, the coefficient d is the difference-indifferences estimator that measures the change in the outcome variable stemming from the introduction of The Order of St John Award for Organ Donation. This estimator shows the changes in the number of solid organ donors following the introduction of the award, controlling for changes in the dependent variable among cornea-only donors, which are unaffected by the award.

RESULTS AND DISCUSSION

Impact on solid organ donations

5 For example, in addition to the recommendation of granting appropriate recognition to individual donors, the Organ Donation Taskforce issued 12 suggestions (Organ Donation Taskforce, 2008) regarding funding, organizational improvements, data monitoring, training of donation teams, increasing awareness among medical staff and the general public. While several of the suggestions were implemented, none of them focused solely on solid organ or cornea donation. Therefore, we expect their effects to be captured by this period dummy.

In Figure 2 we plot the number of deceased solid organ and cornea-only donors by year.

The upper panel shows the raw number of deceased donors while the lower panel shows normalized values with year 2010 as the start year. We first note that the number of solid organ donors is lower than the number of cornea-only donors throughout the sample period. Importantly, while the trends up to the introduction are similar for both groups, we note a diverging pattern in the number of donors after the introduction of the award, as only solid organ donors continue to increase from that point onwards. An important assumption of specification ( 1) is that the error term e t,i is uncorrelated with the time period and the group indicator variables. Essentially, the trend in terms of number of solid organ donors should not differ from the trend in terms of number of cornea-only donors before the award is introduced. The plot in Figure 2 suggests that this is indeed the case, which confirms the appropriateness of using deceased corneaonly donors as a control group.

The first column of Table 1 shows the estimates of the difference-in-differences model As mentioned in Section 2, Wales operates under a presumed consent system since December 2015. To ensure that the results are not driven by this change, we re-estimate regression (1) after removing all observations that relate to Wales and report the estimates in the second column of Table 1. 6 The estimated coefficient for the interaction term Treatment group × Post award of 0.226 translates into an increase of 25% in deceased solid organ donors after the introduction of the award compared to the change in cornea-only donors. The coefficient is significant at the 5% level. We conclude that the 2015 change in the Welsh organ donation system cannot explain away our main findings.

In order to argue that the introduction of the award has played a significant role in explaining the relative increase in solid organ donations, we have to rule out alternative channels. The most plausible competing explanation is medical progress in solid organ retrieval. Compared to solid organs, cornea retrieval for transplantation is considerably easier due to the longer post-mortem time span over which the organ can be recovered (hours or days for corneas versus minutes for solid organs), a less complicated recovery procedure and a longer life span of the organ between retrieval and transplant (see Guam et al., 2012;[START_REF] Osband | Extraction time of kidneys from deceased donors and impact on outcomes[END_REF]. Under the hypothesis that, upon either brain or circulatory death, individuals are screened for potential donation of any type of organ (solid or cornea), medical progress in solid organ retrieval in the Post award period would likely translate into a redistribution of donors from cornea-only (control group) to solid organs (treatment group, which includes donors of solid organs and cornea).

To assess whether this could drive our findings, we retrieve information on the donation process of solid organs from the annual NHS Blood and Transplant Organ Donation and Transplantation Activity Reports. The first step of the process is the identification, made by clinicians, of eligible donors, i.e. patients for whom death was confirmed following neurological tests (for brain death) or who had treatment withdrawn (for circulatory death) and had no absolute medical contraindications to solid organ donation. Once eligible donors are identified, specialized clinical staff is expected to approach the family of the deceased to ask for consent to organ donation. If family consent is granted, the retrieval of solid organs can take place. Table 2 shows the average percentage of success at each step of the donation process over the periods before (Pre award) and after (Post award) the introduction of the award, i.e. the proportion of eligible donors over audited deaths, the proportion of eligible donors whose family was approached for donation, the family consent rate, and the percentage of cases where solid organs were successfully retrieved following family consent.7

If medical progress played a crucial role in driving up the number of solid organ donors

Post award, we would expect the proportion of eligible donors over audited deaths and, most importantly, the percentage of successful donations over donors for which consent was granted, to increase substantially in the second period compared to the first.

However, we record a modest increase (from 14.3% to 15.8%) in the first indicator and a slight decrease (from 68.7% to 68.1%) in the second. Instead, we observe an important increase (from 56.6% to 60.4%) in the family consent rate to solid organ donation, which is the only step in the process directly affected by the award scheme.

Unfortunately, separate consent rates for solid organ and cornea-only donations are not available, hence we cannot perform a difference-in-differences analysis of consent rates around the introduction of the award. Given that there is no indication from Table 2 that medical advances in the retrieval of solid organs may have significantly contributed to narrowing the gap with respect to cornea retrieval, the relative increase in solid organ donations compared to cornea-only donations likely stems from an increase in the consent rate for the former compared to the latter, which is consistent with the incentives provided by the award.

Impact on organ donor registrations

We further examine the incentivizing role of the Order of St John Award for Organ Donation by investigating if the willingness--when alive--to opt in the NHS Organ Donor Register to donate solid organs has been affected by the award. As described in Section 2, organ donation in the United Kingdom is currently conducted through an optin system. However, a number of those who opt in are restricted donors, i.e. unwilling to donate one or more specific organs. Naturally, any donor unwilling to donate one or more solid organs has a reduced probability of receiving the award. If the award spurs the willingness to donate solid organs, we would expect a reduction in the number of restrictions that relate to solid organs among the registered donors, compared to those that relate to corneas.

We gather data on the restrictions towards solid organ and cornea donations in the United Kingdom over the years 2010 to 2017 from the annual issues of the NHS Blood and Transplant Organ Donation and Transplantation Activity Reports. The average percentage of unrestricted donors (out of all individuals registered in the National Organ Register) is 88%. When we look closer at the unwillingness to donate specific organs, some interesting patterns emerge. Figure 3 plots the percentage of donors who are not prepared to donate a specific organ, for each year in our sample period. We observe two patterns when we look at the trends before and after the introduction of the award:

While there is a slightly increasing trend in donors who specifically opt out of cornea donations, the trends for donors who refuse to donate one of the solid organs are mostly downward sloping. In Table 3, we formally test these trends. We employ the same estimation strategy as in (1) in a set of repeated difference-in-differences regressions where the control group is always the natural logarithm of all living registrants not prepared to donate their corneas, but where the treatment group is, in turn, the natural logarithm of all living registrants not prepared to donate their kidneys, pancreas, heart, lungs, and liver, respectively. As before, all data relate to the year from 1 April to 31

March, standard errors are corrected for heteroscedasticity, period-and group-specific dummy variables are employed, and the key variable of interest is the interaction between the treatment group and the period after the award was introduced.

We see from Table 3 that the coefficient of the interaction term, Treatment group × Post award, is negative and statistically significant for pancreas, heart and liver, suggesting that the unwillingness to donate any of these solid organs has decreased compared to the unwillingness to donate corneas after the introduction of the award. The interaction coefficient for kidneys and lungs is also negative, but not statistically significant. We interpret the findings reported in Table 3 as an indication that the introduction of the award overall spurred an increased willingness to register to become solid organ donors.

The effect is however weaker compared to the direct impact estimated on actual solid organ donations.

CONCLUSION

Using the 2013 introduction of The Order of St John Award for Organ Donation in the United Kingdom, we are able to make an important contribution to the literature on policies that can incentivize organ donation. We show that the introduction of the award has led to an increase in the number of deceased solid organ donors and, to a lesser extent, in the willingness to register as solid organ donors.

Our results suggest that it is possible to incentivize organ donation through schemes of official recognition of the deceased donor. Consequently, policy makers outside the United Kingdom may want to consider establishing official awards for organ donation in a spirit similar to The Order of St John Award for Organ Donation as a means, among others, of reducing the current shortage of transplantable solid organs. Notes. This table presents the results of a difference-in-differences regression of the (log) number of deceased organ donors over the sample period 2010 to 2017. Post award is an indicator variable that takes the value of one for each year that the award system is in place (2014, 2015, 2016, and 2017), and zero otherwise (2010, 2011, 2012, and 2013). Notes. This table presents the results of difference-in-differences regressions of the (log) number of registered donors unwilling to donate a particular organ over the sample period 2010 to 2017. Post award is an indicator variable that takes the value of one for each year that the award system is in place (2014, 2015, 2016, and 2017), and zero otherwise (2010, 2011, 2012, and 2013). The respective Treatment group is indicated in the heading of each column. In all regressions, the control group is the (log) number of all registrants that are not prepared to donate their corneas. Robust standard errors in parentheses. ***, ** and * indicate statistical significance at the 1%, 5% and 10% levels, respectively.

( 1 )

 1 . The coefficient of the interaction term Treatment group × Post award represents the increase in the logarithmic outcome variable for a change in the indicator variable from zero to one. Hence, the estimated coefficient of 0.222 translates into a change of 100 × (exp 0.222 -1), or an increase of 25% in deceased solid organ donors over the years 2014 to 2017 with respect to the change in cornea-only donors. The coefficient is significant at the 5% level. This estimate suggests that the award has a large positive impact on the supply of solid organs from deceased donors.

Figure 1 .

 1 Figure 1. The Order of St John Award for Organ Donation

Figure 2 .

 2 Figure 2. Deceased Donors per Year

Figure 3 .

 3 Figure 3. Unwillingness of Registered Donors to Donate Specific Organs

  Treatment group is an indicator variable equal to one for solid organ donors and zero for cornea-only donors. Results in the first (second) column include the full sample (exclude data from Wales). Robust standard errors in parentheses. ***, ** and * indicate statistical significance at the 1%, 5% and 10% levels, respectively.

Table 1 . Estimated Impact of Award on Deceased Donors

 1

	Natural logarithm of deceased donors
	Full sample	Excluding Wales

Table 3 . Estimated Impact of Award on Unwillingness to Donate

 3

		Kidneys	Pancreas	Heart	Lungs	Liver
	Treatment group × Post award	-0.079	-0.152**	-0.132*	-0.118	-0.143*
		(0.079)	(0.072)	(0.071)	(0.079)	(0.079)
	Treatment group	-2.512***	-1.242***	-1.225***	-1.334***	-1.819***
		(0.061)	(0.057)	(0.056)	(0.061)	(0.061)
	Post award	0.258***	0.258***	0.258***	0.258***	0.258***
		(0.064)	(0.064)	(0.064)	(0.064)	(0.064)
	Constant	7.496***	7.496***	7.496***	7.496***	7.496***
		(0.053)	(0.053)	(0.053)	(0.053)	(0.053)
	Observations	16	16	16	16	16
	R-squared	0.997	0.991	0.991	0.991	0.995

Since December

2015, Wales operates under a presumed consent system (soft opt-out) where deceased adults are deemed to have consented to organ donation unless they had positively opted out before their death. We will account for this in our empirical analysis. Changes to an opt-out system are currently (October 2018) under discussion in both the British and Scottish Parliaments.2 Available from http://www.odt.nhs.uk/statistics-and-reports/annual-activity-report/.

Information from http://odt.nhs.uk/pdf/advisory_group_papers/NODC/Order_of_St_John_Award_for_ organ_Donation_Report.pdf accessed on 7 December, 2017.

Available from http://www.odt.nhs.uk/statistics-and-reports/annual-activity-report/ accessed on 7 December, 2017. Data on soft tissue other than cornea are not available.

The number of deceased Welsh cornea-only donors is missing from the activity report of 2017. The NHS Blood and Transplant kindly provided us with the number of corneas donated in that year by type of donor and geographical area, from which we were able to estimate the missing observation.

The average over the Pre award period is computed over three years(2011, 2012 and 2013) as the procedure for recording information on the donation process changed in 2010, and data for that year are not directly comparable.

organ donors in the United Kingdom to honor the donors and inspire others to donate. We evaluate the effects of this award using a difference-in-differences approach that builds on the fact that solid organ donors are eligible for the award, while cornea-only donors are not. We find that the introduction of the award led to an increase in the number of deceased solid organ donors, as well as in the general willingness to register as solid organ donors when alive. Organ donation; incentives for organ donation; posthumous

 2014, 2015, 2016, and 2017.