A Dynamic Analysis Model for the Diffusion Coefficient in High

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A Dynamic Analysis Model for the Diffusion Coefficient in High-Viscosity Polymer Solution Lile Cai, Junjing Lu, Zhengming Gao, and Ziqi Cai Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b03966 • Publication Date (Web): 24 Oct 2018 Downloaded from http://pubs.acs.org on November 4, 2018

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Industrial & Engineering Chemistry Research

A Dynamic Analysis Model for the Diffusion Coefficient in High-Viscosity Polymer Solution Lile Cai a, Junjing Lub, Zhengming Gao a*, Ziqi Cai a* a State Key Laboratory of Chemical Resource Engineering, School of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China b China Nuclear Power Engineering Co., Ltd. Beijing,100840, China * Corresponding author. Tel.: +8610 64418267; fax: +8610 64449862. E-mail address: [email protected] (Ziqi Cai), [email protected] (Zhengming Gao).

KEYWORDS Diffusion, Viscous polymers, Dynamic model, Molecular simulation, Mass transfer

ABSTRACT An accurate mass transfer model is required to calculate the diffusion coefficient in polymers to describe the mass transfer of volatiles in process intensification, especially in the devolatilization units. However, the diffusion coefficient actually is variate rather than a fixed value due to the changing concentration and viscosity of polymer in the process of devolatilization. In this work, removal rates of the volatiles in liquid film were obtained by the intelligent gravity analysis (IGA), and a discrete mathematical model was established to compute the instantaneous dynamic diffusion coefficients. Based on a large number of experimental data, a correlation of diffusion coefficients was developed by dimensionless regression analysis with temperature,

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concentration, viscosity, molar mass, molecular polarity etc. It can well describe the diffusion process in polymers. Furthermore, molecular simulation was conducted to explore the proper method on bonding method and force field in the calculation of diffusion coefficient.

Highlights 1.Accurate mass transfer model with variate diffusion coefficient was developed.

2.Instantaneous diffusion model was built by differential equation discretization.

3.Dimensionless model correlates various influence factors on diffusion coefficient in polymer.

4.Molecular simulation provides proper way to predict diffusion coefficient in polymer.


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1. Introduction In the synthesis of polymer, there always exists some unexpected components in the product, such as the monomer, solvent, even the by-products. These components, which are called volatiles, negatively affect the properties and performance of the product, even do harm to the environment and users' health. Therefore, the removal or recovery of the residual components from the polymer is essential. Up to now, many methods and apparatuses have been proposed for such a removal process, and the method used for the separation between product and volatile is usually called devolatilization. Some classical apparatuses are applied to the devolatilization process, including the thin-film evaporators1, the falling film reactors2,3, and the single-screw4 and twin-screw extruders5. In these apparatus, polymer melt or solution commonly is high-viscosity liquid with poor fluidity, which may lead to the low removal efficiency. As a result, some new intensification technologies were used in the devolatilization such as the rotating packed bed6, the high speed disperser7 and the ultrasound enhanced devolatilization8. Mass transfer is particularly important in the research of process intensification such as chemical reaction and separation process in the rotating packed bed or high speed disperser9. A precise mass transfer model should be set up to describe the mass transfer in intensification process. Devolatilization of polymer solution mainly involves three steps10: migration of volatile from the polymer solution to the continuous gas-phase interface, gasification on the interface, and then removed by the vacuum system. The first step is the rate-determining step which immensely affects the rate of the whole process, and it is always related to the gas-liquid mass transfer, where three fundamental theories, surface renewal kinetics (Eq. 1), penetration theory (Eq. 2) and diffusion theory (Eq. 3), have been proposed.



𝑘𝑐 = √𝑆𝐷𝐴𝐵

(1)

𝐷

𝑁𝐴 = √ 𝜋𝐴𝐵 (𝐶 − 𝐶𝑒 ) 𝑡 𝑁𝐴 = 𝐷𝐴𝐵

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𝜕 𝜕𝐶 ( ) 𝜕𝑥 𝜕𝑥

(2) (3)

Mass transfer is an important phenomenon in the process of polymer synthesis and impurity removal. Mass transfer models for reactors basically were built based on these mass transfer theories. In the design of devolatilization apparatus, an accurate mass transfer model is usually required in order to describe the behavior of devolatilization apparatus and to predict the influence factor on the apparatus. Roberts 11and Biesenaberger 12 et al presented mass transfer models for reactors based on the penetration theory and the molecular diffusion Table 1 shows some typical models of the devolatilization in various apparatus. Diffusion coefficient of solvent in the polymer solution is a vital variable to describe mass transfer process in reactors of process intensification accoding to the models in Table 1, especially in the synthesis reaction and devolatilization units. There are several test methods in determining the diffusion coefficient in the literatures. Chromatography is a commonly used method with the advantages of accuracy in the infinite dilution, though it is limited in the determination of diffusion coefficient in high viscous solvent or polymers. Tait

17

measured the diffusion coefficient of

benzene, ethyl benzene, propyl benzene in polyvinyl chloride (PVC) and PMMA by gas-liquid chromatography. Arnould 18 measured the infinite dilute diffusion coefficients of alkanes and olefins in PMMA and polyvinyl acetate (PVAC) by antiphase gas chromatography. Iwai Y

19

measured the solubility of m-xylene and nonane vapors in polystyrene and ethylbenzene, and in polybutadiene with a sorption apparatus.


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Gravimetric analysis is another common method. Francoeur 20 measured the average diffusion coefficient of solvent hexane and monomer 5-ethylidene-2-norbornadiene(ENB) in EthylenePropylene-Diene Monomer (EPDM) by the gravimetric analysis, and the diffusion coefficient was used in a mass transfer model in describing a continuous flow stirred tanks. Mccall DW

21

measured the diffusion coefficients of n-hexane, cyclohexane, octane and decane in polyethylene by means of gravimetric adsorption and desorption. Generally, an average diffusion coefficient was determined in the entire diffusion process in the previous research 22. However, the viscosity of the polymer solution and concentration of volatiles, as important factors affecting the diffusion coefficient of the volatiles, are constantly changing in the devolatilization, which is the same as the situation in the gravimetric analysis. In this case, the average diffusion coefficient applied to mass transfer model will lead to a large deviation, especially with high concentration of volatile in the polymer solutions. In this paper, we are going to propose a model in calculating and describing the change of the diffusion coefficient during the devolatilization, by means of the gravimetric method and to explore the influence of environmental conditions and material properties on the volatiles diffusion coefficient. This work includes the experimental determination, model description and molecular simulation, based on the gravity analysis, Fick's law of diffusion and the Einstein equation, respectively. First, a modified numerical analysis method was developed to compute the instantaneous diffusion coefficient based on the thermogravimetric experiment under various conditions including the concentration of the volatiles, solution viscosity, ambient temperature, etc. Then a dimensionless correlation for the volatiles diffusion coefficient in the polymers was built based on thermodynamics, Vrentas-Duda diffusion model and the theory of molecular



polarity23. Furthermore, the molecular simulation is carried on in order to describe the volatiles diffusion phenomena in the polymers by optimizing the force field and the non-bonded method24.

2. Experiments, Models and Simulation 2.1 Experiments Gravimetric analysis was applied to measure the diffusion coefficient for volatiles in the polymers in this research. Liquid sheet of the polymer containing volatiles was paved equably on a plaque which was placed on a scale. Partial pressure of the volatiles was controlled as 0 Pa with nitrogen contacting the upper surface of the plaque slowly (Figure 1). The computer collected the mass data of the liquid sheet in real time in every 2 s. When preparing the liquid sheet for the gravimetric experiments, approximately 6.8 g liquid material containing volatiles and polymer was spread in a sheet with the thickness of ~1 mm in average. The liquid sheet was placed in a tray on a scale within a temperature-controlled chamber, as shown in Figure 1. This gravimetric testing apparatus contains a computer control system that regulated experimental conditions and collected the temperature, pressure, and the mass data. Each experiment was repeated at least twice under the same condition. In the experiments, continuous change of mass data for the liquid sheet was collected by a control system, and the instantaneous diffusion coefficient of volatiles can be calculated by a mathematical model based on the experimental data. Then several systems containing various volatiles and polymers were selected to explore the influence of variable parameters on diffusion coefficient. The volatile matters including pentane, hexane, heptane, octane, methyl alcohol, benzene, and polymers such as Polydimethyl-siloxane (PDMS), polyacrylic-acid (PAA) and polyiso-butylene (PIB) were selected. The experimental variables affecting the diffusion


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coefficient included concentration the of volatile, viscosity of the polymer, ambient temperature, degree of polymerization(DP) and some other properties of volatile components such as molecular weight, molecular volume, solubility and molecular polarity. 2.2 Mathematical model In the solution consisting of volatiles and polymer, polymer can be considered as a stagnated component because of its high viscosity and poor liquidity25. Diffusion coefficient for volatiles in polymer was determined by the concentration analysis model based on the gravimetric test. Here the concentration analysis was set up according to the one-dimensional model of Fick's diffusion law:

𝑁𝐴 =

𝜕 𝜕𝐶 (𝐷𝐴𝐵 ) 𝜕𝑥 𝜕𝑥

(4)

with the following boundary and initial conditions 𝜕𝐶 | =0 𝜕𝑡 𝑥=0

(4𝑎)

𝐶|𝑥=𝐿 = 0

(4𝑏)

𝐶|𝑡=0 = 𝐶0

(4𝑐)

where 𝐶 is the mass concentration (g/g) of the volatile solvent in the polymer, DAB is the mutual diffusion coefficient of volatile component and polymer, L(m) is the thickness of the liquid sheet, t (s) is the time of diffusion, and x(m) is the vertical coordinate in liquid sheet. Equation 4 was resolved to the integral formula Eq. 5, and an average diffusion coefficient of volatiles DAB in the diffusion process can be calculated by this equation in the literatures20,21: ∞ 𝑚𝑡 − 𝑚0 8 1 𝐷𝐴𝐵 (2𝑛 + 1)2 𝜋 2 𝑡 = 1− 2∑ exp(− ) 2 𝑚𝑒𝑞 − 𝑚0 𝜋 4𝐿2 𝑛=0 (2𝑛 + 1)


(5)


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where 𝑚𝑒𝑞 , 𝑚0 ，𝑚𝑡 are the equilibrium mass and initial mass and the mass of the liquid sheet at t moment, respectively. The average diffusion coefficient from the initial to the equilibrium state can be calculated from Eq. 5 based on gravity analysis. However, this average constant diffusion coefficient is not accurate enough to describe the mass transfer in the process of diffusion in considering that the viscosity of the polymer solution increases rapidly with the decrease of volatiles concentration, which is shown in Figure 2 for the heptane-PDMS (various DP) system as an example. The viscosity of polymer solution also varies with the length of the chain temperature

24

26

and

during polymerization. So there should be some modifications for calculating

method in order to obtain the instantaneous diffusion coefficients at moments. Partial differential Eq. 4 was discretized by time and the liquid sheet thickness, which was divided into many layers from x=0 to x=L (shown in Figure 3). In Figure 3, the liquid sheet was divided into N layers with equal thickness, so the thickness of each layer is: 𝛿𝑥 =

𝑚𝑠 𝑁

/(1 − 𝐶𝑖 )/𝜌𝑖 /𝐴

(6)

where 𝑚𝑠 is the instantaneous mass of the solution; N is the quantities of the control volume (layers of liquid sheet); 𝐶𝑖 is the volatiles mass fraction in the ith layer; 𝜌𝑖 (g · cm−1 )is the corresponding density of the solution in the ith layer; A is the surface area of the liquid film exposed to the ambient. It is assumed that the concentration of volatile is uniform within each single control volume, so the property of each control volume is determined by the control point P in the very middle of the corresponding layer. Therefore, Figure 3 is simplified to a one-dimensional situation as shown in Figure 4, where the control point P is an example node with a left point W and a right point E.


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For example, the integral formula of Eq. 4 from W to E in the control volume from time t to time t+Δt： 𝑒

𝑡+∆𝑡 𝜕𝐶

∫𝑤 ∫𝑡

𝑡+∆𝑡

( )𝑑𝑡𝑑𝑥 = ∫𝑡 𝜕𝑡

𝑒 𝜕

𝜕𝐶

∫𝑤 𝜕𝑥 (𝐷𝐴𝐵 𝜕𝑥 ) 𝑑𝑥𝑑𝑡

(7)

And the analytical solution for Eq. 7 is:

δ𝑥,𝑃 (𝐶𝑃1 − 𝐶𝑃0 ) = [

𝐷∆𝑡 (𝐶𝐸0 −𝐶𝑃0 )

(δ0𝑥,𝑃 +δ0𝑥,𝐸) /2

−

0 𝐷∆𝑡 (𝐶𝑃0 −𝐶𝑊 )

(δ0𝑥,𝑊 +δ0𝑥,𝑃) /2

] ∆𝑡

(8)

where δx is the distance between adjacent two control points, 𝐶 is the mass fraction of volatile， 𝐶𝑝1 is the mass fraction of volatile component at t+Δt moment at point P, 𝐶𝑝0 is mass fraction of volatile component at t moment. 𝐷∆𝑡 is diffusion coefficient of volatiles in polymer at the instantaneous ∆𝑡 time.Δt was given according to the sampling interval of balance. The discretization of Eq. 8： 0 𝐶𝑃1 = 𝑎𝐸 𝐶𝐸0 + 𝑎𝑊 𝐶𝑊 + [𝑎𝑝 − (𝑎𝐸 + 𝑎𝑊 )]𝐶𝑃𝑂

Where:

𝑎𝑃 =

𝛿𝑥,𝑃 ∆𝑡

, 𝑎𝐸 =

2𝐷𝐴𝐵 δ0𝑥,𝑃 +δ0𝑥,𝐸

, 𝑎𝑊 =

2𝐷𝐴𝐵 δ0𝑥,𝑊 +δ0𝑥,𝑃

(9) (10)

As Figure 4 shows, Point 1 and point N are in the control volume adjacent to the boundary, and Eq. 8 was modified according to the boundary conditions 4a and 4b: 𝑎1 𝐶11 = 𝑎2 𝐶20 + (𝑎1 − 𝑎2 )𝐶10

(11)

0 𝑎𝑁 𝐶𝑁1 = 𝑎𝑁−1 𝐶𝑁−1 + (𝑎𝑁 − 𝑎𝑁−1 )𝐶𝑁0

(12)

Concentration distribution in each layer can be calculated with the model (Eq. 7 to 12) by computer programming. Diffusion coefficients were determined when the experiment data coincided with the computation fitting results in some period of time.



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2.3 Simulation Diffusion coefficients of the small molecules can be determined based on the law of Einstein or Green-Kubo for molecular simulation27. Einstein equation is based on the Mean Square Displacement (MSD) of molecules and Green-Kubo is based on the Velocity Autocorrelation Function (VACF). Actually, the MSD of small molecule is easy to obtain, while the integration error of VSCF is relatively larger. According to Einstein equation, Eq. 13 shows the correlation between diffusion coefficients and MSD of a small molecules in polymer relative to its original position 28:

D

1 d 2 lim r (t )  r (0) t  6 dt

(13)

Where r(t), r (0) are the displacement of a small molecule between moment t and the initial moment t0 respectively, D is the self-diffusion coefficient of the molecule. Mutual diffusion coefficient of the molecule and polymer can be determined according to the free volume theory. Here Material Studio was used to establish the molecular model. Molecular-dynamics (MD) simulation by momentum scaling method was used to calculate the self-diffusivity of volatiles in polymers. In this paper, three-dimensional periodic cubic microstructures were built by Amorphous Cell module to eliminate the boundary effect 29 as shown in Figure 5. Then the periodic cells were optimized in structure and energy by NPT method (MD simulation by constant atomic number, pressure and temperature) so that the three-dimensional periodic cubic structure cells with the same physical property as experimental sample were selected to calculate NVT (constant atomic number, volume and temperature) molecular dynamics simulation and MSD of small molecules in polymer was set up by NVT simulation. Method of temperature control in molecular dynamics simulation is Andersen method30, and the pressure control is Berendsen method31. The initial speed of atoms is Boltzmann distributed 10 Environment ACS Paragon Plus

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and the total simulation time is 500 ps to 1000 ps based on the results of convergence. In addition, force field which describes the interaction among molecules plays a crucial role in the accuracy of the MSD results. The force field mainly include COMPASS and PCFF in polymer system 32. COMPASS field uses the method of molecular dynamics to modify the nonbonding parameters by the thermophysical properties of liquid or crystalline particles. PCFF is the force field suitable for the polymer systems in the force field of CFF series.

3. Experimental Results Verification tests were carried out to make sure that the height-diameter ratio of the liquid sheet adequately matched the one-dimension diffusion. Experimental results of different sheet diameter under the same condition including the volatile concentration, degree of polymerization and temperature are shown in Figure 6(a). Volatilization-rate for a liquid sheet of diameter 97 mm was almost coincident with those of 156 mm and 180 mm, and it is verified the height-diameter ratio of the liquid sheet about 0.01m is small enough for the one-dimension diffusion. In addition, 15% wt. N-heptane/PDMS (DP=620) and 5% wt. N-heptane/PDMS (DP=620) mixtures at 298 K were used as examples to verify the repeatability of experiments shown in Figure 6(b), and it shows a good repeatability. Orthogonal experiments were performed to explore the influencing factors of diffusion coefficient, such as the temperature, concentration, degree of polymerization and volatile components. The curvature of the volatility curve shown in Figure6 is to some extent a reflection of the size for diffusion coefficient. Therefore, it can be seen that during the process of devolatilization, diffusion rate related to coefficient DAB is always changing with time. Concentration distribution in liquid film was calculated with the mathematical model (Eq.7 to Eq.12) by computer



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programming. The concentration distribution at different vertical positions was shown in Figure 7(a). The average concentration of global liquid film was obtained by integral of distribution. Then diffusion coefficients were determined when the experiment data coincided with the calculated global concentration in some corresponding period of time as shown in Figure 7(b). The variational diffusion coefficient D△t and the corresponding viscosity of the mixture in volatilization is listed in Table S1 with N-heptane and PDMS system as an example. According to Vrentas-Duda free volume diffusion model33, the diffusion coefficient is relevant to many variables such as the temperature, mass fraction of the volatile, glass transition temperature et. al. Dimensionless analysis is an effective way to explain the internal relations between variables. Therefore, a series of interrelated factors were also selected to correlate the diffusion model based on the free-volume diffusion theory in this work. Jiang34 reported that a higher temperature leads to activating the molecular motion. More intense motion of the polymer chain results in a larger molecules diffusion. In this research, the diffusion coefficient with temperature follows the Arrhenius law. N-heptane/PDMS (DP=620) mixture with initial a mass fraction 15% was used as an example, and homologous experimental results for alkane and PDMS system are shown in Figure 8. 𝐸𝑃

𝐷𝐴𝐵 = 𝐷0 𝑒 −𝑅𝑇

(14)

where Ep is the Diffusion Activation Energy(J·mol-1), and it is the potential barrier when atoms overcome their constraints to jump, which is related to the structures of the polymer and the small molecule. Diffusion Activation Energy of each mixture system can be acquired by the law of Arrhenius as shown in Figure 8. The slope of curve for ln D-1/T is just the value of Ep/R, in which R is the ideal gas constant(J·mol-1·K-1). Arrhenius curve for diffusion activation energy of


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different volatiles-polymer systems is shown in Figure 9 based on Thermogravimetric Analysis (TGA). DAB is mainly influenced by temperature, so Eq. 14 was treated as the basis equation of researching influence factor on diffusion coefficient, with pre-exponential factor D0 containing the other influence factors. During the volatilization, the viscosity and concentration of the volatiles in the polymer are always changing, and it will lead to the change of diffusion coefficient for the volatiles correspondingly. N-heptane/PDMS mixture (DP=620) with an initial concentration 15% wt. at 298K was taken as an example, and the changing diffusion coefficient with mass fraction during the volatilization is shown in Figure10 as the calculation results of Eqs. 6-13. The correlation between the diffusion coefficient and the mass fraction of volatiles was obtained based on Eq. 15:

𝐷𝐴𝐵 =

D′ 0

×

𝐸 − 𝑃 𝑅𝑇 𝑒

𝑤∗ =

(𝑤 ∗ )0.925 𝑤𝐴

𝑤𝑚𝑖𝑥

(15) (16)

where D′0 is the pre-exponential factor,𝑤𝐴 and 𝑤𝑚𝑖𝑥 are the mass of volatile in polymer and the mass of the mixture respectively. Eq. 15 includes factors of temperature and mass fraction of volatile, and both of them affect the viscosity and density of the system, which has influence on the diffusion coefficient of polymer system as shown in Eq.17 and Eq.18 by experimental measurement. A and B are constants. Dimensionless analysis was used to summarize the effects of above variables on diffusion coefficient in this work. Since Schmitt Number Sc describes the matter traveling through a two-phase interface which is mainly included in the double-film theory



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and permeation theory, it is introduced to modified the Eq. 15 in a dimensionless way shown in Eq. 20. 𝜇 = 𝐴(𝑤 ∗ )𝑎 (𝑇 ∗ )𝑏

(17)

𝜌 = 𝐵(𝑤 ∗ )𝑐 (𝑇 ∗ )𝑑

(18)

𝑇 𝑇𝑔

(19)

𝜇 𝜌𝐷𝐴𝐵

(20)

𝑇∗ = 𝑆𝐶 =

where Tg is the glass transition temperature (K)of the polymer. Substituted Eqs. 17-20 into Eq.15, and the dimensionless model is:

𝐷𝐴𝐵 =

𝐸 A(𝑤 ∗ )𝑏 (𝑇 ∗ )𝑐 − 𝑃 ′ 𝑅𝑇 = D × 𝑒 (𝑤 ∗ )0.925 0 B(𝑤 ∗ )𝑐 (𝑇 ∗ )𝑑 𝑆𝐶

(21)

Where the short form of Eq.21 is: 𝐸𝑝

𝐷1 𝑆𝑐 𝑒 −𝑅·𝑇 (𝑇 ∗ )𝑑 ( 𝑤 ∗ )𝑒 = 1

(22)

Diffusion coefficient is also associated with the structure and the polymerization degree of the polymer, and the diffusion coefficient for the volatiles (alkanes) and PDMS with different degree of polymerization are shown in Figure 11.

Except for the polymerization degree, some other internal factors of materials including the diffusion activation energy, molecular polarizability and molecular mass, molecular volume are also

considered.

Dimensionless

variables

𝑀∗ 𝐷𝑝

, 𝑉∗ ,

and

self-defined

interaction

parameters 𝛹 including molecular polarizability and solubility parameter which affect intermolecular forces between volatiles and polymer were introduced to correlation (Eq. 22).


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Based on the experimental data on volatile diffusion coefficients, Eq. 23 which correlates above mentioned variables is obtained.

34.5 ∙ 𝑆𝑐 × 𝑒

−

𝐸𝑝 𝑅·𝑇

2.478

×

(𝑇 ∗ )4.74

×(𝑤

∗ )0.882

𝑀∗ ×( ) 𝐷𝑝

× (𝑉 ∗ )−4.71 × 𝛹 −0.28 = 1 (23)

𝑀∗ =

𝑀𝐴 𝑀𝐵

𝛹=(

𝑉∗ =

𝑉𝐴 𝑉𝐵

𝑉𝐴 ∙(𝛿𝐴 −𝛿𝐵 )2 𝑅𝑇

)

(24) (25)

where Ep (J/mol) is the diffusion activation energy which the atoms must overcome when jumping from one balance position to another, T (K) is the temperature of polymer and Tg (K) is the glass transition temperature, 𝑤𝐴 and 𝑤𝑚𝑖𝑥 are the mass (g) of the small volatile molecule in polymer and the mass (g) of the mixture respectively. MA, MB (g/mol) are the molecular molar weight of the small volatile molecule and monomer of polymer. VA, VB (cm3/mol) are the molecular molar volume of small volatile molecule and monomer of polymer. Dp is the degree of polymerization of polymer. δA and δB (J/cm3)0.5 in Eq. 25 for interaction parameters 𝛹 are the solubility parameter (SP) of volatile molecule and polymer. Line A in Figure 12 showed the result of correlation (Eq.23) for diffusion coefficient of heptane in PDMS system during the process of heptane removal. Line B is the average value of diffusion coefficient calculated by Eq. 5. N-heptane/PDMS (DP=620) mixture with an initial Nheptane concentration 15% wt. at 298K was used as an example here. Equation 23 on diffusion coefficient DAB was substituted into the Fick’s diffusion law (Eq. 4) to verify the accuracy of correlation Eq.23 and it is shown in Figure 13. N-heptane concentration



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over time in volatilization process was almost consistent with experimental result as shown, while there was a larger deviation by using the average diffusion coefficient. In Figure 14, the diffusion coefficients calculated by Eq.23 were compared with the experimental results under different conditions and systems, and the relative deviation was within ±10%. 4.

Molecular Simulation In this work, a constant temperature Molecular-dynamics (MD) simulation by momentum

scaling method was used to study the self-diffusivity of volatiles in the polymer. Moleculardynamics simulations of a three-dimensional (3D) model of the cubic microstructures were used to study the grain-boundary (GB) diffusion. The cubic-microstructure periodic models were built to optimize the structure of the model and process the MD simulation by NPT module. Suitable calculation parameters were selected by comparing the simulation results on the physical properties of the material with the actual values. Then MSD results of volatiles molecule was set up by the NVT dynamics simulation based on the NPT module. Optimized cubic-microstructure periodic model for volatiles and polymer was calculated by NVT dynamic in 600 ps by recording the trajectory of the molecule in each picosecond. Take the N-heptane/N-hexane/N-pentanePDMS mixture with an initial concentration 10% wt. at 298K as an example, the motion of mean square displacement (MSD) of the volatiles molecules over time is shown in Figure 15. Self-diffusion coefficients of the volatile molecules can be determined by plotting the curve of MSD-time according to the Einstein's formula (Eq. 13). Then the mutual diffusion coefficient DAB can be calculated by Vrentas-duda model33. 𝐷𝐴𝐵 = 𝐷𝐴 (1 − 𝜙𝐴 )2 (1 − 2𝜒𝜙𝐴 )


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where DA and DAB are respectively the self-diffusion coefficients of volatile molecules and mutual diffusion coefficients of volatiles and polymers, m2·s-1; χ is the interaction parameters of the volatiles and the polymer in Vrentas-duda model, 𝜙𝐴 is the volume fraction of the volatile in the polymer. In MD simulation, the force field are mainly PCFF35 and COMPASS36. COMPASS field uses the method of molecular dynamics to modify the non-bonding parameters by the thermos-physical properties of liquid or crystalline particles, and PCFF is suitable for the polymer systems in the force field of CFF series. Table 2 and Table 3 show the average deviation of DAB for N-heptane or N-pentane /PDMS mixture with concentration 10% wt. at 298K between the experimental and simulation results with different force field and non-bond methods. It can be seen that the PCFF field is more suitable for present volatile-polymer system. From E1, E4 and E8, it can be seen that the results from Atombased method have large deviation with the experimental value, and the results of Ewald Nonbond method are closer to the experimental value. Therefore, the PCFF force field, the Ewald non-bonding method and the ultra-fine precision were selected in the simulations. Glass transition temperature of PDMS was set up to verify the accuracy of the MD simulation. Calculating the specific volume of the polymer under different temperature is a common method in acquiring the glass transition temperature in the molecular simulation37. When the specific volume of a material suddenly changes, the corresponding temperature is glass transition temperature of the polymer. Figure16 shows the specific volume of PDMS with temperature. The glass transition temperature of PDMS is approximately 150K, and it is consistent with the results in literature38.



Diffusion coefficients derived from molecular simulation were compared with the experimental value, as shown in Figure 17. Based on the optimized force field and bonding method, the diffusion coefficients of volatiles in another several volatile-polymer systems were simulated, and the results were compared with the values obtained from the correlation (Eq. 23), as shown in Table S2. The results matched well, with a maximum relative error less than 10%. The diffusion activation energy in different systems can be calculated based on the Arrhenius Equation and the values of diffusion activation energy for different systems from Figure 9 are listed in Table S2.

5. Conclusion Accurate determination of the diffusion coefficient is an important subject in the mass transfer. In polymer system with high viscosity, diffusion coefficient is variate rather than a fixed value due to the changing concentration of volatiles and the viscosity of system in process of diffusion devolatilization. The variate was explored by mathematic model of differential equation discretization based on the gravity analysis experiment. The diffusion coefficient of volatiles in polymer during the instantaneous period Δt was relevant to the corresponding concentration, viscosity and temperature etc. Diffusion coefficient obtained from the gravity analysis conforms to the Arrhenius laws. Meanwhile, the mutual relation for the diffusion coefficient and temperature, viscosity, mass fraction was also determined by the gravimetric analysis under various conditions. A dimensionless regression model was used to summarize the effects of variables above and the physical properties of volatiles and polymers such as the intermolecular force, molecular weight and the molecular polarity diversity on diffusion coefficient. The interaction parameter was used to modify the correlation. It is indicated that the dimensionless correlation for diffusion coefficient of volatiles in polymer agrees well with the experimental determination and it is more accurate than the average diffusion coefficient to describe the process of devolatilization. Furthermore, 18 Environment ACS Paragon Plus

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molecular simulation was used to predict the diffusion coefficient for the volatiles in polymer in this work. The results showed that the polymerization degree of the polymer and the volatiles concentration are two independent factors in determining the diffusion coefficients of volatiles. PCFF force field, the Ewald non-bonding method and the ultra-fine precision are optimal method for simulation of diffusion coefficient in polymer, and the results of molecular simulation agree well with experimental results and correlation. NOMENCLATURE A

the total mass transfer area in the reactor, m2

C

mass fraction of volatiles

C0

initial concentration of volatiles, mol·L-1

Ce

equilibrium concentration of volatiles, mol·L-1

Cf

final concentration of volatiles, mol·L-1

c*

saturated concentration of gas in liquid, mol·L-1

D

diffusion coefficient, m2·s-1

D1

pre-exponential factor in Eq. 22

DAB

diffusion coefficient of components A and B, m2·s-1

Dp

polymerization degree of the polymer

Ep

diffusion activation energy, J·mol-1

kc

mass transfer coefficient , m·s-1

M

molecular weight, g·mol-1

m0

initial mass of liquid sheet, g

meq

the equilibrium mass of liquid sheet, g

mt

mass of the liquid sheet at t moment, g

N

quantities of the control volume



NA

molar flux, mol·m-2·s-1

NS

rotational speed of the shafts in reactor, rev/s

qe

amount of solvent removed from polymer per m2, mol·m-2

R

ideal gas constant, J·mol-1·K-1

RD

radius of the disc in table 1, m

r

displacement of a small molecule in polymer.

S

mass transfer area, m2

Sc

Schmitt Number

T

temperature of polymer, K

Tg

glass transition temperature of polymer, K

t

time of devolatilization, s

Δt

data collection interval, s

te

time for which polymer is exposed to vacuum, s

V

molecular volume, cm3·mol-1

VL

liquid volume flow rate, cm3/s.

wmix

mass of mixture, g

wA

mass of component A, g

x

vertical coordinate, m

Ψ

self-defined interaction parameters in Eq. 23

ρ

density of the solution, kg·m-3

μ

viscosity of mixture, Pa·s

χ

interaction parameters of volatiles and polymer in Vrentas-duda model

φA

volume fraction of volatiles

δA, δB solubility parameter of volatiles and polymer, (J·cm-3)0.5

Ω

Surface renewal rate s-1


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AUTHOR INFORMATION Corresponding Authors *E-mail [email protected] (Ziqi Cai). * E-mail [email protected] (Zhengming Gao)

ACKNOWLEDGMENT The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. 21676007 & 21506005).

SUPPORTING INFORMATION Figure Captions (Supporting Information paragraph) Figure 1 Schematic diagram of gravimetric testing apparatus Figure 2 Curve of solution viscosity with volatiles decreasing Figure 3 Boundary conditions and control volume in liquid film Figure 4 The schematic diagram of one-dimensional calculation model Figure 5 Three-dimensional Amorphous Cell model for volatiles-polymer system (a. pentane, heptane in PDMS

b. heptane in PIB

c. methanol in PAA)

Figure 6 Volatilization rate with time in contrastive diameter film (a) and replicated experiments (b) Figure 7 Concentration distribution at different vertical positions (a) Comparison of model calculation with experiment results (b) Figure 8 Logarithm of Diffusion coefficients under different 1/T of Heptane-PDMS Figure 9 Diffusion coefficient with temperature in different systems Figure 10 Diffusion coefficient with the concentration of volatiles decreasing



Figure 11 Diffusion coefficients of different volatiles and polymer(different DP) Figure 12 Diffusion coefficient with the decreasing concentration of volatiles Figure 13 Change of volatiles concentration for modle calculation and constant D with experiment in volatilization process Figure 14 Error of correlation model and experiment result Figure 15 The MSD track over time for different alkanes in PDMS Figure 16 The specific volume of PDMS by molecular simulation with temperature Figure 17 The diffusion coefficient with different factors for experiment and simulation of Heptane-PDMS Table Captions Table 1. Mass transfer models in the literatures Table 2. Deviation for DAB of N-heptane/PDMS by simulation and experiment Table 3. Deviation for DAB of N-pentane /PDMS by simulation and experiment

Supporting Information for Publication Table S1. Diffusion coefficients changing with mass fraction of volatiles (N-heptane) in PDMS Table S2. Error checking of the model in other devolatilization systems

REFERENCES (1) (2) (3) (4) (5)

Freese, H. L.; Glover, W. B. Mechanically Agitated Thin-Film Evaporators. Chem. Eng. Prog. 1979, 75 (1), 52–58. Schaal, F.; Schilling, K.; Hasse, H. Separation Efficiency of Thin-Film Evaporators: Experiments with Water-Ethylene Glycol and Methanol-Water and Stage-Based Modeling. Chem. Eng. Process. Process Intensif. 2008, 47 (2), 209–214. Xi, Z.; Zhao, L.; Liu, Z. New Falling Film Reactor for Melt Polycondensation Process. Macromol. Symp. 2007, 259 (1), 10–16. Hasson A, Gerhart R J. Method for removing volatile substances from polyphenylene ether resin blends: EP, CA 2030055 A1. 1991. Khanin D, Scarola L S. Process for devolatilization: US, US 5932691 A. 1999.


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Wang, D. G.; Guo, F.; Chen, J. F.; Liu, H.; Zhang, Z. T. Preparation of Nano Aluminium Trihydroxide by High Gravity Reactive Precipitation. Chem. Eng. J. 2006, 121 (2), 109– 114. Nie, A.; Gao, Z.; Xue, L.; Cai, Z.; Evans, G. M.; Eaglesham, A. Micromixing Performance and the Modeling of a Confined Impinging Jet Reactor/High Speed Disperser. Chem. Eng. Sci. 2018, 184, 14–24. Rege, S. U.; Yang, R. T.; Cain, C. A. Desorption by Ultrasound: Phenol on Activated Carbon and Polymeric Resin. Aiche J. 1998, 44 (7), 1519–1528. Ouyang, Y.; Xiang, Y.; Gao, X.-Y.; Li, W.-L.; Zou, H.-K.; Chu, G.-W.; Chen, J.-F. Micromixing Efficiency in a Rotating Packed Bed with Non-Newtonian Fluid. Chem. Eng. J. 2018. Albalak, R. J.; Tadmor, Z.; Talmon, Y. Scanning Electron Microscopy Studies of Polymer Melt Devolatilization. Aiche J. 1987, 33 (5), 234–235. Roberts, G. W. A Surface Renewal Model for the Drying of Polymers during Screw Extrusion. Aiche J. 1970, 16 (5), 878–882. Biesenberger, J. A. Polymer Devolatilization: Theory of Equipment. Polym. Eng. Sci. 2010, 20 (15), 1015–1022. Woyuan, L. I.; Wei, W. U.; Zou, H.; Chu, G.; Lei, S.; Chen, J. A Mass Transfer Model for Devolatilization of Highly Viscous Media in Rotating Packed Bed. Chinese Journal of Chemical Engineering 2010, 18 (2), 194–201. Secor, R. M. A Mass Transfer Model for a Twin‐screw Extruder. Polym. Eng. Sci. 1986, 26 (9), 647–652. Meeuwse, M.; Lempers, S.; Schaaf, J. Van Der; Schouten, J. C. Liquid−Solid Mass Transfer and Reaction in a Rotor−Stator Spinning Disc Reactor. Ind. Eng. Chem. Res. 2010, 49 (21), 10751–10757. Danckwerts, P. V. B1 – Significance of Liquid-Film Coefficients in Gas Absorption. Insights Into Chem. Eng. 1981, 43 (6), 51–75. Tait, P. J. T.; Abushihada, A. M. The Use of a Gas Chromatographic Technique for the Study of Diffusion in Polymers. J. Chromatogr. Sci. 1979, 17 (4), 219–223. Arnould, D.; Laurence, R. L. Size Effects on Solvent Diffusion in Polymers. Ind.eng.chem.res 1992, 31 (1), 218–228. Iwai, Y.; Arai, Y. Measurement and Prediction of Solubilities of Hydrocarbon Vapors in Molten Polymers. J. Chem. Eng. Japan 2006, 22 (2), 155–161. Francoeur, A. J. B.; Karimi, H.; Mcauley, K. B.; D’Agnillo, L. A Model for the Devolatilization of EPDM Rubber in a Series of Steam Stripping Vessels. Aiche J. 2014, 60 (7), 2596–2606. Mccall, D. W.; Douglass, D. C.; Anderson, E. W. Diffusion in Ethylene Polymers. IV. J. Chem. Phys. 1959, 30 (3), 771–773. Cocchi, G.; Angelis, M. G. De; Doghieri, F. Solubility and Diffusivity of Liquids for Food and Pharmaceutical Applications in Crosslinked Polydimethylsiloxane (PDMS) Films: II. Experimental Data on Mixtures. J. Memb. Sci. 2015, 492, 612–619. Sacide, A.; Larry, D. J. Modeling of Multicomponent Drying Polymer Films. Aiche J. 2010, 45 (4), 896–905. Hillman, A. R.; Igor Efimov, A.; Ryder, K. S. Time-Scale- and Temperature-Dependent Mechanical Properties of Viscoelastic Poly(3,4-Ethylenedioxythiophene) Films. J. Am. Chem. Soc. 2005, 127 (47), 16611–16620.



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Athinodoros, B.; Inglefield, P. T.; Jones, A. A.; Wen, W. A Nuclear Magnetic Resonance Study of Toluene-Polyisobutylene Solutions. 2. Vrentas-Duda Theory. J. Polym. Sci. Part B Polym. Phys. 1995, 33 (10), 1505–1514. Carroll, B.; Bocharova, V.; Carrillo, J. M. Y.; Kisliuk, A.; Cheng, S.; Yamamoto, U.; Schweizer, K. S.; Sumpter, B. G.; Sokolov, A. P. Diffusion of Sticky Nanoparticles in a Polymer Melt: Crossover from Suppressed to Enhanced Transport. Macromolecules 2018, 51 (6). Seifert, U. Generalized Einstein or Green-Kubo Relations for Active Biomolecular Transport. Phys. Rev. Lett. 2010, 104 (13), 138101. Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids / M.P. Allen, D.J. Tildesley. 1988, No. 88. Kucukpinar, E.; Doruker, P. Molecular Simulations of Small Gas Diffusion and Solubility in Copolymers of Styrene. Polymer (Guildf). 2003, 44 (12), 3607–3620. Andersen, H. C. Molecular Dynamics Simulations at Constant Pressure and/or Temperature. J. Chem. Phys. 1980, 72 (4), 2384–2393. Berendsen, H. J. C.; Postma, J. P. M.; Gunsteren, W. F. Van; Dinola, A.; Haak, J. R. Molecular Dynamics with Coupling to an External Bath. J. Chem. Phys. 1984, 81 (8), 3684– 3690. YaoLi; DeyiKong; JiangZhen; JuyanXu. A Base Transport Model for Ultra-Thin-Base SiGe HBT. Int. J. Electron. 2000, 87 (11), 1281–1287. Hong, S. U. Prediction of Polymer/Solvent Diffusion Behavior Using Free-Volume Theory. Ind. Eng. Chem. Res. 1995, 34 (7), 2536–2544. Jiang, W. H. Infinite Dilution Diffusion Coefficients of -Hexane, -Heptane and -Octane in Polyisobutylene by Inverse Gas Chromatographic Measurements. Eur. Polym. J. 2001, 37 (8), 1705–1712. Sun, H.; Mumby, S. J.; Maple, J. R.; Hagler, A. T. An Ab Initio CFF93 All-Atom Force Field for Polycarbonates. J. Am. Chem. Soc. 1994, 116 (7), 2978–2987. Sun, H. COMPASS: An Ab Initio Force-Field Optimized for Condensed-Phase ApplicationsOverview with Details on Alkane and Benzene Compounds. J. Phys. Chem. B 1998, 102 (38), 7338–7364. Buchholz, J.; Paul, W.; Varnik, F.; Binder, K. Cooling Rate Dependence of the Glass Transition Temperature of Polymer Melts: Molecular Dynamics Study. J. Chem. Phys. 2002, 117 (15), 7364–7372. Onyon, P. F. Polymer Handbook. Nature 1972, 238 (5358), 56.


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Figure 1

Figure 1 Schematic diagram of gravimetric testing apparatus

ACS Paragon Plus Environment


Figure 2 12 11 10

Viscosity/Pas

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 45

Volatiles： n-heptane Polymer： PDMS

DP=250 DP=850 DP=1050

9 8 7 6 5 4 3 2 1 20

15

10

5

0

Volatiles Residual Concentration %

Figure 2

Curve of solution viscosity with volatiles decreasing


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Figure 3

𝑪|𝒙=𝑳 = 𝟎

𝝏𝑪 | =𝟎 𝝏𝒕 𝒙=𝟎 Figure 3

Boundary conditions and control volume in liquid film



Figure 4

Figure 4

The schematic diagram of one-dimensional calculation model


Page 28 of 45

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Figure 5

a. pentane, heptane in PDMS Figure 5

b. heptane in PIB

c. methanol in PAA

Three-dimensional Amorphous Cell model for volatiles-polymer system



Figure 6

7.0

97mm 180mm 150mm

Material： N-heptane/PDMS Concentration： 10% wt. 0.8 Temperature： 298K

1.0

Mixture mass/g

Volatilization Mass Fraction

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.6 0.4 0.2

15% wt. N-heptane/PDMS-sample1 15% wt. N-heptane/PDMS-sample2 5% wt. N-heptane/PDMS-sample2 5% wt. N-heptane/PDMS-sample1

6.5

6.0

0.0 0

2000

4000

6000

8000

10000

12000

5.5

0

500

1000

1500

2000

Time/s

Time/s

(a)

(b)

Figure 6 Volatilization rate with time in contrastive diameter film (a)and replicated experiments (b)

Time/s


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Figure 7

a Figure 7

b

Concentration distribution at different vertical positions (a) Comparison of model calculation with experiment results (b)



Figure 8

-20.7

Initial concentration: 15% wt Polymer： PDMS（ DP=620) Volatiles： n-heptane

-20.8

Ln D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-20.9

-21.0

-21.1

0.0030

0.0031

0.0032

0.0033

0.0034

-1 1/T ( K )

Figure 8

Logarithm of Diffusion coefficients under different 1/T of Heptane-PDMS


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Figure 9

-21

Ln D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-22

-23

Pentane-PDMS Octane-PDMS

Hexane-PDMS Methanol-PDMS

Benzene-PDMS

Heptane+PIB

1/298

1/303

1/313

Heptane-PDMS CH2Cl2-PDMS Methanol+PAA

1/323

1/333

1/T (1/K)

Figure 9 Diffusion coefficient with temperature in different systems



Figure 10

Systems： N-heptane-PDMS Temperature： 298K Initial concentration:10% wt.

8

Viscosity Diffusion coefficient 7

5

6 5 4

4

Viscosity/Pa·s

6

D×1010m2/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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3

3

2 2

0.00

0.02

0.04

0.06

0.08

0.10

Mass Fraction of Volatiles

Figure 10 Diffusion coefficient with the concentration of volatiles decreasing


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Figure 11 8.0

Systems： alkanes and PDMS Initial concentration： 10% wt. Temperature： 298K

7.5 7.0

7

D×1010m2/s

6.5

D×1010m2/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


6.0 5.5 5.0 4.5 4.0

Systems： N-heptane-PDMS Initial concentration： 10% wt. Temperature： 298K

6

5

4

3.5 3.0

3

2.5 2.0

4

5

6

7

8

9

200

400

Carbon atom number of volatiles

600

800

1000

1200

DP

Figure 11 Diffusion coefficients of different volatiles and polymer(different DP)



Figure 12

6

Average value Model value

D/×10-10m2/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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A 4

B 2 0.0

0.2

0.4

0.6

0.8

1.0

Removal ratio

Figure 12 Diffusion coefficient with the decreasing concentration of volatiles


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Figure 13

numerical calculation of modle experiment data of gravimetric analysis numerical calculation of constant D

0.1

wA/wmix Kg/Kg

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.075

0.05

0.025

0

2000

4000

6000

8000

Time/ s Figure 13 Change of volatiles concentration for modle calculation and constant D with experiment in volatilization process



Figure 14 7

Modle calculation Experiment results

6

10 2

D ×10 m /s-Modle

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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5 4 3 2 1 0

0

1

2

3 10 2

4

5

6

7

D ×10 m /s-Experiment

Figure 14 Error of correlation model and experiment result


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Figure 15

500

N-heptane Initial concentration: 10% wt. N-hexane Temperature: 298K N-pentane

450 400 350

MSD/2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


300 250 200 150 100 50 0 100

200

300

400

500

600

Time/ ps

Figure 15 The MSD track over time for different alkanes in PDMS



Figure 16

1.10

Specific volume of PDMS 1.05

3

Specific volume (cm /g)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1.00

0.95

0.90 0

50

100

150

200

250

300

350

400

Temperature (K)

Figure 16

The specific volume of PDMS by molecular simulation with temperature


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Figure 17 7

10 2 m /s

6

6

D×10

10 2 m /s

Experiment Simulation

D×10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5

Experiment Simulation

4

2

4

300

310

320

330

0.00

0.05

0.10

Mass Concentration/kg/kg

T (K)

(a) D with Temperature (b) D with mass fraction of volatiles Figure 17 The diffusion coefficient with different factors for experiment and simulation ofHeptane-PDMS


0.15


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Table 1 Table 1. Mass transfer models in the literatures Author

Model

Roberts11

𝐷𝑡𝑒 2 𝑞𝑒 = 2𝐶 ( ) 𝜋

Theoretical basis

Application

Penetration theory Diffusion theory

screw extruder

1

W. Li 13

2 𝐴

𝐶𝑓 − 𝐶0 = 𝛿

𝑓

Secor14

Meeuwse.15

Danckwerts16

𝐴𝑓

𝐷𝑡

√

𝜋

(𝐶𝑒 − 𝐶0 ) 𝑍/𝑎

𝐶𝑓 𝐴 4𝐷𝑓𝑁𝑆 = (1 − √ ) 𝐶0 𝑉𝐿 𝜋

4 2𝐷Ω𝑅𝐷 𝑘𝐶 = √ 5 𝜋𝑑𝑏 𝑐∗ − a 𝑁𝐴 = 1 1 + b√𝐷 𝑘𝑐

penetration theory diffusion theory

rotating packed bed

penetration theory diffusion theory

twin-screw extruder

surface renewal kinetics diffusion theory

spinning disc reactor

two-film theory

Evaporator

penetration theory


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Table 2

Table 2 Deviation for DAB of N-heptane/PDMS by simulation and experiment Case

Force field

Non-bond method

Quality

E1

pcff

Atom based

Fine

E2

pcff

Group based

E3

pcff

E5

D-exp

D-sim

Relative error of D

4.52

4.98

+10.18%

Coarse

4.52

4.86

+8.20%

Group based

Medium

4.52

4.71

+4.20%

pcff

Group based

Ultra-fine

4.52

4.63

+2.43%

E6

pcff

Ewald

Coarse

4.52

4.436

-1.85%

E7

pcff

Ewald

Medium

4.52

4.43

-1.96%

E8

pcff

Ewald

Fine

4.52

4.41

-2.43%

E9

pcff

Ewald

Ultra-fine

4.52

4.44

-1.77%

E10

COMPASS

Group based

Fine

4.52

4.78

+5.76%

E11

COMPASS

Ewald

Fine

4.52

4.73

+4.64%



Page 44 of 45

Table 3 Table 3 Deviation for DAB of N- pentane /PDMS by simulation and experiment Case

Force field

Non-bond method

Quality

Relative error of D

E12

pcff

Atom based

Fine

+4.21%

E13

pcff

Group based

Fine

+3.19%

E14

pcff

Ewald

Coarse

-2.05%

E15

pcff

Ewald

Medium

-3.02%

E16

pcff

Ewald

Fine

-3.23%

E17

pcff

Ewald

Ultra-fine

-3.13%

E18

COMPASS

Group based

Fine

+7.44%

E19

COMPASS

Ewald

Fine

+5.34%


Page 45 of 45

8

5

Ep (J/mol)

7 6 5

4

4

-21

Ln D

Diffusion coefficient Viscosity

Viscosity/Pa·s

6

D×1010m2/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


-22

3

3

-23

2 2

0.00

0.02

0.04

0.06

0.08

Mass Fraction of Volatiles 𝐸𝑝

𝐴 ∙ 𝑆𝑐 × 𝑒−𝑅·𝑇

∗ 𝐵

× 𝑇

0.10

Pentane-PDMS Octane-PDMS Benzene-PDMS 1/298

∗ 𝐶

× 𝑤

1/303

𝑀∗ × 𝐷𝑝

Hexane-PDMS Methanol-PDMS Heptane+PIB

𝐷

1/313

Heptane-PDMS CH2Cl2-PDMS Methanol+PAA

1/T

1/323

∗ 𝐸

× 𝑉

1/333

= 𝛹𝐹

Volatile Solute in Polymers


A Dynamic Analysis Model for the Diffusion Coefficient in High

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