J. Non-Equilib. Thermodyn. 2016; 41 (1):3–11
Research Article
Ricardas V. Ralys*, Alexander A. Uspenskiy and Alexander A. Slobodov Deriving properties of low-volatile substances from isothermal evaporation curves
DOI: 10.1515/jnet-2015-0030 Received May 31, 2015; revised October 8, 2015; accepted October 19, 2015
Abstract: Mass ux occurring when a substance evaporates from an open surface is proportional to its satu- rated vapor pressure at a given temperature. The proportionality coecient that relates this ux to the vapor pressure shows how far a system is from equilibrium and is called the accommodation coecient. Under vac- uum, when a system deviates from equilibrium to the greatest extent possible, the accommodation coecient equals unity. Under nite pressure, however, the accommodation coecient is no longer equal to unity, and in fact, it is much less than unity. In this article, we consider the isothermal evaporation or sublimation of low- volatile individual substances under conditions of thermogravimetric analysis, when the external pressure of the purging gas is equal to the atmospheric pressure and the purging gas rate varies. When properly treated, the dependence of sample mass over time provides us with various information on the properties of the exam- ined compound, such as saturated vapor pressure, diusion coecient, and density of the condensed (liquid or solid) phase at the temperature of experiment. We propose here the model describing the accommodation coecient as a function of both substance properties and experimental conditions. This model gives the - nal expression for evaporation rate, and thus for mass dependence over time, with approximation parameters resulting in the properties being sought.
Keywords: Saturated vapor pressure, thermogravimetry, evaporation rate, diusion
1 Introduction
Measurement of vapor pressures of low-volatile substances is not an easy task, and assessing this property usually involves both static and dynamic methods [1]. Static methods, such as manometry [2] or ebulliometry [3], require the sample that is being studied to reach its equilibrium state rst. These methods give reliable vapor pressure data, but they are highly demanding in terms of substance quantity, purity, labor, and time. Dynamic methods, on the contrary, do not need to equilibrate the condensed and vapor phases rst. In these methods, vapor pressure is derived from the measured mass ux, and basically, this group includes the tran- spiration, Knudsen [4], and Langmuir methods [5]. Of these, only the transpiration method uses carrier gas to vaporize sample, with the overall mass change at a given experimental time relating to its saturated vapor pressure via the ideal gas law [3, 6, 7]. The Knudsen and Langmuir methods explore mass change under evap- oration into vacuum, with the area of evaporation being the main dierence between them. In the Knudsen method, vapors of a substance escape from a small orice, causing either measurable mass reduction over time or some measurable force like in torsion eusion [8]. In contrast to the Knudsen model, in the Langmuir method, sample vaporizes from an open surface into vacuum. Dynamic methods have high reliability, but their drawback is the need to maintain a sophisticated experimental setup, which involves mass detection and high and tight vacuum.
*Corresponding author: Ricardas V. Ralys: Department of Information Technologies in the Fuel and Energy Complex, ITMO University, 49 Kronverksky Pr., St. Petersburg 197101, Russia, e-mail: [email protected] Alexander A. Uspenskiy, Alexander A. Slobodov: Department of Information Technologies in the Fuel and Energy Complex, ITMO University, 49 Kronverksky Pr., St. Petersburg 197101, Russia, e-mail: [email protected], [email protected] 4 Ë R. V. Ralys, A. A. Uspenskiy and A. A. Slobodov, Properties of low-volatile substances
With the availability of highly accurate modern thermogravimetric analyzers allowing direct measure- ment of mass change with acceptable precision, the Langmuir method has been adapted to the conditions of such equipment. However, the basic assumption of this method is no longer valid. Under vacuum, the vaporized sample fully escapes the surface, and this is no longer observed when evaporating into a purging gas used in thermogravimetry. The equation that describes the evaporation rate in terms of saturated vapor pressure is known as the Langmuir equation:
dm M = −αp S S { , dt V 2πRT ( ) dm S where dt is the evaporation rate (kg/s), α is the accommodation coecient, p is the saturated vapor pres- 2 sure (Pa), SV is the surface of phase boundary area (m ), M is the molar mass of( ) the substance (kg/mol), R is the universal gas constant (J/(mol K)), and T is the sample temperature (K). When vaporizing into vacuum, the accommodation coecient equals unity, i.e. no molecules return to the surface from which they escaped. However, under nite external pressure, most of the escaping molecules return, and this makes the accom- modation coecient much less than unity, and this parameter has 10 5 to 10 4 orders of magnitude (see, e.g., [9–14]). − − Such a large departure from the initial assumption of α = 1 requires that the value of the accommodation coecient be determined rst before deriving the vapor pressure from mass loss measurements. To do so, we need to carry out a series of experiments with calibration substances for which the saturated pressures at temperature of interest are available. This need imposes certain limitations: (1) calibration substances should be structurally similar to those being studied, and (2) experimental conditions should be identical when vaporizing the calibration substances and the samples being studied. Both conditions are not always satised due to evident lack of vapor pressure data for low-volatile compounds. To overcome these obstacles, attempts have been made to create models that explicitly describe the de- pendence of the accommodation coecient or even evaporation rate on substance properties and experimen- tal conditions [15–17]. These models consider evaporation in a thermogravimetric analyzer as a diusion- limited process. When applied to experimental data, the equations proposed in these works give good agree- ment with observed evaporation rates. However, these models do not explain the experimentally observed dependence of evaporation rates on purging gas rates [18–20]. To explain the observed evaporation phenomena, e.g. low accommodation coecient and its depen- dence on purge gas rate, we need to construct a model by taking into account both substance properties and experimental conditions. The model sought is supposed to give explicit expressions for saturated vapor pressure as well.
2 Initial assumptions and model construction
Before deriving the equations of interest, consider under which assumptions they are valid. The following conditions should be met for the model to be valid: 1. Purge gas ux is parallel to the walls of the crucible with the sample. 2. No association or dissociation of the molecules occurs in the vapor phase. 3. Purging gas is neutral to the sample under investigation; thus, neither chemical interactions between them nor gas sorption into the sample takes place. 4. No chemical or phase transformations are observed in the sample during evaporation experiments. 5. The thermal resistivity in the sample is negligible; thus, the sample has the same temperature over its volume. This assumption is readily satised when working with thermogravimetry because substance quantities are small and the heat supply in the apparatus is sucient. 6. The hydrodynamic regimen is laminar; thus, there is stable diusion region over the sample surface. This assumption is also satised when working with thermogravimetry. R. V. Ralys, A. A. Uspenskiy and A. A. Slobodov, Properties of low-volatile substances Ë 5
7. Evaporation, on the one hand, is suciently fast in order to be detected at a reasonable experimental time, and on the other hand, is not very high in order to prevent the surface from undercooling. 8. The concentration of the vapor at the top of diusion layer is non-zero. 9. The amount of the substance diused through the stagnant gas layer is equal to the amount that the purge gas takes away. When the above validity criteria are fullled, the following equations describing the evaporation process are applicable: 1 j = αc S v̄, (2.1) T 4 D ( ) j = (c S − c ), (2.2) T x W V( ̇ ) jT = cW , (2.3) SV 2 where equation (2.1) describes the overall mass ux jT (kg/(m s)) in terms of the Langmuir equation, which involves the accommodation coecient α, saturated vapor concentration c S (kg/m3), and average molecules velocity v̄ (m/s); equation (2.2) describes the mass ux occurring in the diusion( ) layer in terms of the rst Fick’s law, which involves the diusion coecient D (m2/s), diusion path x (m), and concentration at the 3 layer top boundary cW (kg/m ); and equation (2.3) describes the mass ux resulting from convection of 2 diused vapor in the purge gas, with cross-sectional surface SV (m ) and volume rate of the purge gas V̇ (m3/s). The average molecular velocity is considered at equilibrium state of vapor, and thus its value is de- ned by Maxwell–Boltzmann distribution and equals v̄ = x8RT/(πM). The saturated vapor concentration equals c S = p S M/(RT).
When( ) equating( ) the right-hand parts of (2.2) and (2.3), we easily obtain the expression for the concentra- tion at the top diusion layer boundary:
DSV S cW = c (2.4) DSV + Vẋ ( ) Taking into account (2.3) and (2.4), we write for the mass ux the following expression: ̇ ̇ V DSV S DV S jT = c = c . (2.5) SV DSV + Vẋ DSV + Vẋ ( ) ( ) Comparing (2.5) with the Langmuir equation (2.1), we easily obtain the function that describes the accom- modation coecient: 1 DV̇ 4 DV̇ αc S v̄ = c S â⇒ α = . 4 DSV + Vẋ v̄ DSV + Vẋ ( ) ( ) For the evaporation rate r, after introducing an auxiliary parameter ξ = SV and reconciling the value of the V average molecule velocity, we get ̇ S S dm 2πM D p M RT DSV p M r ≡ − = { SV { = . (2.6) dt RT Dξ + x (RT) 2πM Dξ + x (RT) Basically, we did not assume that the liquid or solid boundary is xed. On the contrary, upon evaporation, there can be a signicant change in its position relative to the top edge of the sample’s crucible. To take this change into account, let us consider the diusion path as the following function:
x(t) = x0 + h(t), (2.7) where h(t) is the height between the sample surface and the top crucible edge (m), and x0 is the diusion path at initial time of an isotherm (m). The height is equal to the dierence in the initial VS0 and current sample volumes VS(t), divided by the cross-sectional area SV : V − V (t) h(t) = S0 S , SV 6 Ë R. V. Ralys, A. A. Uspenskiy and A. A. Slobodov, Properties of low-volatile substances where volumes are in m3 and the area is in m2. The change in height with time is dened by the condensed 3 phase density ρS (kg/m ), according to the following equation: dh(t) 1 dV (t) 1 1 dm r = − S = − = . dt SV dt SV ρS dt ρS SV Assuming that the initial diusion path is constant, we may write
dx p S M D = (2.8) dt ρ(S RT) Dξ + x for the change in the diusion path. In integral form, (2.8) becomes x2 + ξx + C = Kt, 2D S where C is the integration constant and parameter K = p M/(ρS RT) is the auxiliary parameter describing the ratio between vapor and condensed phase densities. Taking( ) into account that x(t = 0) = x0, we derive the nal equation for the diusion path, which is always positive:
2 x(t) = y(Dξ + x0) + 2DKt − Dξ. (2.9)
Embedding (2.9) into (2.6) gives the following relationship for the evaporation rate:
DS p S M r = V . (2.10) 2 (RT) y(Dξ + x0) + 2DKt
This way, we obtain an equation that describes the dependence of mass on time by plain integration of (2.10):
2 m(t) = m0 − SV ρSy(Dξ + x0) + 2DKt − (Dξ + x0), (2.11) where m0 is the initial mass of the sample (kg). Applying (2.11) to mass signal measurement at constant temperature and at dierent purge gas rates results in saturated vapor pressure, diusion coecient of the vaporized substance in purge gas used, and density of the liquid or solid.
3 Experimental data treatment
Given a set of curves describing the experimental dependence of the sample’s mass over time under isother- mal conditions at dierent purge gas rates, we may extract the values of dierent substance properties. If we introduce auxiliary variables such as
m0 − m(t) 2 η ≡ , a0 ≡ (Dξ + x0) ⋅ ρS , a1 ≡ 2KDρS , SV then (2.11) is reduced to 2 η = ya0 + a1t − a0. This form can be easily linearized to yield the expression
2 η = −2a0η + a1t. (3.1)
The measurement of sample mass change over time at dierent purge gas rates and its approximation using (3.1) results in a set of parameters (a0, a1) dened at these varied purge gas rates. From this set of parameters, we readily obtain the saturated vapor pressure using the following relationships:
S c 2 S ∂a0 S ∂a0 a1 = 2 ρS D = 2c DρS , DρS = â⇒ a1 = 2c , ρ(S) ∂ξ T ∂ξ T ( ) ( ) R. V. Ralys, A. A. Uspenskiy and A. A. Slobodov, Properties of low-volatile substances Ë 7 and 1 RT a p S = 1 . (3.2) 2 M (∂a0/∂ξ)T ( ) To derive not only the saturated vapor pressure but the diusion coecient and liquid or solid density as well, we need to transform the mass of the sample as a function of time to the dependence of time needed to reach the same diusion path under dierent purge gas rates. To do so, we rearrange (2.7) into the following relationship: x(t) − x0 = h0 − h(t), (3.3) where h0 is the initial height of the sample (m) (this variable is made equal to the sample crucible during the start of experiment). If we divide the current sample height by its initial height, another auxiliary variable appears upon this action – the degree of evaporation φ:
h(t) φ(t) ≡ . (3.4) h0 Under isothermal conditions, the degree of evaporation may be expressed in terms of mass values:
m(t) φ(t) = . (3.5) m0 Equations (3.3)–(3.5) allow us to write an equation for the change in diusion path with time, which is ob- tained directly from experimental data:
∆x(t) ≡ x(t) − x0 = h0(1 − φ(t)). (3.6)
From (3.6), it follows that the same diusion paths are reached when same degrees of evaporation are reached, under the condition of same initial sample height. Rearranging (2.9) yields the expression
2 2 x(t) − x0 + 2Dξ(x(t) − x0) = 2KDt.
Finally, we obtain the time to reach the needed degree of evaporation as a function of change in the diusion path: ξ x (∆x)2 t = ∆x ⋅ + 0 + . K KD 2KD If we introduce, for the sake of convenience, the additional variables x 1 1 c ≡ 0 , c ≡ , b ≡ 0 KD 1 K 2 2KD and consider the change in diusion path as an independent variable, we obtain the function
2 t = (c0 + c1ξ)∆x + b2(∆x) . (3.7)
When we consider the same degree of evaporation at dierent purge gas rates, we readily dene the auxiliary coecients from (3.7), which results in the diusion coecient and condensed phase density, taking into account variables (a0, a1) from (3.1):
2 ∂a 2 D = 0 , (3.8) a1c1 ∂ξ T a1c1 ρs = . (3.9) 2(∂a0/∂ξ)T By having a set of evaporation curves measured at the same temperature but with dierent purge gas rates, we can derive the desired properties (saturated vapor pressure, diusion coecient, and condensed phase density) by applying (3.2), (3.8), and (3.9), respectively, where auxiliary parameters result from data approx- imation using (3.1) and (3.7). 8 Ë R. V. Ralys, A. A. Uspenskiy and A. A. Slobodov, Properties of low-volatile substances
4 Experimental test of the proposed evaporation model
4.1 Experimental setup and materials used
As a thermogravimetric analyzer, we used the Netzsch Libra 209 F1 TGA apparatus with corundum crucibles without lid. The crucibles have a height of 3 mm and diameter of 6 mm. The purging gas was dry nitrogen of high purity (99.999%). The calibration was performed according to the guides provided by Netzsch, and it included calibration by temperature and mass. Temperature calibration was carried out using highly pure metals (In, Sn, Bi, Zn, Al, Ag, and Au) and by measuring their melting points using a heating rate of 5 K/min at ve purge gas rates, 20, 80, 135, 190, and 250 mL/min. Mass calibration was performed using an internal standard embedded into the apparatus, also at the above-mentioned purge gas rates. Temperature uncer- tainty did not exceed 0.1 K, and mass uncertainty did not exceed 2 × 10 10 kg after calibration.
Testing the proposed model required stable low-volatile substances− with well-known saturated vapor pressures in a large temperature interval. We needed to ensure that the model is correct in both liquid and solid samples. As substances satisfying these requirements, we selected di-n-butylphthalate (CAS RN 84742, liquid sample) and benzoic acid (CAS RN 65850, solid sample). Both samples were of 99.5% purity, with further purication done. We distilled the liquid and sublimated the solid under vacuum. After purication, we tested the samples using gas chromatography to obtain a purity greater than 99.9%.
4.2 Test procedure
The test procedure was set as simple as possible. The aim was to ensure that (2.11), which describes the variation of mass under isothermal condition and varied purge gas rates, holds true. To do so, we evalu- ated parameters (a0, a1) by the approximation of the mass signals acquired at ve purge gas rates (20, 80, 135, 190, and 250 mL/min) at several temperatures (353.15, 383.15, and 413.15 K for di-n-butylphthalate; 323.15, 343.15, and 383.15 K for benzoic acid). According to the model proposed, parameter a1 should be independent of the purge gas rate, whereas parameter a0 should display linear dependence on the reciprocal purge gas rate, or equivalently, it should be linearly dependent on parameter ξ, which is the reciprocal linear gas velocity. To keep parameter a0 linear on the reciprocal linear gas velocity, we maintained the same initial sample mass within the 1 × 10 8-kg tolerance. As soon as benzoic acid turned into powder, we needed to get a uniform sublimating surface.− To do this, we fused the sample and then cooled it to the required temperature. We recorded the isothermal mass signal within 3.6 × 104 s to attain signicant mass changes.
4.3 Test results
Upon performing the above-described test procedure, we obtained the results, which show that the model proposed is valid under the studied experimental conditions (see the typical mass signals in Figure 1). The dependence of parameters (a0, a1) on purge gas rate is depicted in Figures 2 and 3. From Figure 2, it fol- lows that there is the strong linear dependence of the a parameter on the reciprocal linear gas velocity ξ. In Figure 3, we see that there is no dependence of parameter a1 on the ξ variable. R. V. Ralys, A. A. Uspenskiy and A. A. Slobodov, Properties of low-volatile substances Ë 9
Figure 1. Evaporation curves for benzoic acid at T 343.15 K under dierent purge gas rates. =
Figure 2. Dependence of parameter a on the reciprocal linear Figure 3. Values of the parameter a1 at dierent purge gas gas velocity ξ for di-n-butylphthalate at T 383.15 K. rates for di-n-butylphthalate at T 383.15 K.
= = 5 Conclusions
In this work, we have developed a model that describes the evaporation process of low-volatile pure sub- stances in isothermal thermogravimetric analysis. We have shown that the model takes into account both substance properties and experimental conditions when the evaporation rate is computed. If we perform evaporation measurements at several purge gas rates, we can easily derive the saturated vapor pressure, dif- fusion coecient, and density of solid or liquid sample using the model. 10 Ë R. V. Ralys, A. A. Uspenskiy and A. A. Slobodov, Properties of low-volatile substances
T = 353.15 K T = 383.15 K T = 413.15 K 6 6 6 V(N2) a0 a1 × 10 a0 a1 × 10 a0 a1 × 10 (mL/min) (kg/m2) (kg/(sm2)) (kg/m2) (kg/(sm2)) (kg/m2) (kg/(sm2)) 20 1.333 1.032 1.311 10.14 1.277 72.37 80 1.082 1.038 1.057 10.13 1.023 72.41 135 1.045 1.034 1.020 10.17 0.985 72.38 190 1.030 1.031 1.005 10.17 0.971 72.36 250 1.022 1.032 0.996 10.14 0.962 72.36
Table 1. Values of the approximation parameters in (2.11) for di-n-butylphthalate at three dierent temperatures.
T = 323.15 K T = 343.15 K T = 383.15 K 6 6 6 V(N2) a0 a1 × 10 a0 a1 × 10 a0 a1 × 10 (mL/min) (kg/m2) (kg/(sm2)) (kg/m2) (kg/(sm2)) (kg/m2) (kg/(sm2)) 20 1.581 1.317 1.583 9.46 1.587 260.6 80 1.382 1.314 1.379 9.47 1.374 260.8 135 1.353 1.314 1.349 9.49 1.342 259.2 190 1.341 1.316 1.337 9.48 1.329 261.0 250 1.335 1.316 1.330 9.50 1.322 260.2
Table 2. Values of the approximation parameters in (2.11) for benzoic acid at three dierent temperatures.
Tables 1 and 2 contain the data on parameters (a0, a1) at three dierent temperatures for di-n-butyl- phthalate and benzoic acid. These data as well as Figures 2 and 3 show that the model we propose in this work adequately describes the process of evaporation of low-volatile substances under the conditions of thermo- gravimetric experiment.
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