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From thermodynamics to kinetics: Theoretical study of CO2 dissolving in poly (lactic acid) melt
Accepted Manuscript From thermodynamics to kinetics: Theoretical study of CO2 dissolving in poly (lactic acid) melt
Kesong Yu, Hongfu Zhou, Xiangdong Wang, Zhongjie Du, Jianguo Mi PII: DOI: Reference:
S0167-7322(18)34348-4 https://doi.org/10.1016/j.molliq.2019.02.005 MOLLIQ 10405
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
22 August 2018 1 December 2018 1 February 2019
Please cite this article as: K. Yu, H. Zhou, X. Wang, et al., From thermodynamics to kinetics: Theoretical study of CO2 dissolving in poly (lactic acid) melt, Journal of Molecular Liquids, https://doi.org/10.1016/j.molliq.2019.02.005
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ACCEPTED MANUSCRIPT From Thermodynamics to Kinetics: Theoretical Study of CO2 Dissolving in Poly (lactic acid) Melt
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School of Materials and Mechanical Engineering, Beijing Technology and Business University, Beijing, China
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Kesong Yua,b, Hongfu zhoua, Xiangdong Wanga*, Zhongjie Dub, and Jianguo Mi b*
State Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology, Beijing,
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China
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Abstract: Under the framework of dynamic density functional theory, the dissolution and diffusion behaviors
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of CO2
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ACCEPTED MANUSCRIPT 1. Introduction Poly (lactic acid) (PLA) is an aliphatic thermoplastic polyester with the biodegradable attribute and comparable mechanical properties [1,2]. PLA foam has the potential to replace traditional petroleum-based polymer foams in a wide array of applications [3-5]. Moreover, PLA has relatively high CO2 solubility, which provides a good prerequisite for developing nanocellular cells with uniform cell morphology to further enhance
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its impact resistance, toughness, and thermal and sound insulation [6-8]. It is known that the cell morphology is
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generally governed by the following thermodynamic and kinetic properties: CO2 dissolution, PLA swelling,
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CO2-PLA interfacial tension, CO2 diffusion, and PLA viscosity. In order to confirm the condition to produce
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nanocellular PLA foam, these fundamental issues should be clearly identified in advance. It is not easy to handle and ensure the dissolution equilibrium of CO2 in highly PLA melt in experiment, thus
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the available experimental data on CO2 solubility are restricted in a narrow range of temperature and pressure [9-11]. Meanwhile, the swelling ratio of PLA depends on how much CO2 dissolves in PLA [12], and the
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corresponding interfacial structure and tension of CO2-PLA has to be regulated due to PLA plasticization
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[13,14]. The swelling and interfacial properties are more difficult to measure via experimental approaches. A reliable thermodynamic model, capable of accurately describing CO2 solubility, the swelling ratio of glassy
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PLA, and the interfacial structure and property between two phases over a wide range of temperature and
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pressure, is an essential prerequisite for developing a good transport model.
The diffusion of CO2 in PLA is a complex process as
well. It can be influenced by concentration, pressure, temperature, PLA viscosity, external force and so on. For these reasons, the measurement of diffusivity of supercritical CO2 in molten polymer has always been a challenging research topic for the foam processing [15]. T 16,17] Theoretical prediction is an alternative approach to reduce the workload of experiment and to provide more efficient data aimed at predicting the cell size and density of foaming products.
ACCEPTED MANUSCRIPT A multitude of equations of state were applied to calculate and predict gas solubility in polymer systems [1822]. Through the proper choice of specific model parameters regressed from experimental data, these equations are capable of semi-quantitatively evaluating the dissolving behavior. However, the physical meaning of these parameters is obscure, because
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restrict the usefulness of equations as the predictive tools.
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Another important type of equations was derived from the statistical associating fluid theory (SAFT) [23],
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which was widely applicated to describe the dissolution behaviors of gases in macromolecular systems [24-28]. Based on the statistical mechanics, the theory has advantageous over those semi-empirical equations by
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providing molecular-level interpretation. Although SAFT has yielded encouraged results, its further
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improvement is difficult, since its semi-quantitative parameters are often confined to a certain region. Two different empirical equations of state, the Sanchez-Lacombe [29] and Simha-Somocynsky [30] types,
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were used to calculate the swelling property of polymer melts that was induced by gas permeation. These
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equations are also semi-quantitatively and the model parameters rely completely on the available specific pressure–volume–temperature data. It has been recognized that the equations are difficult to predict the swelling
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properties given the extreme shortage of these fundamental data. This failure is a result of the inability to
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accurately predict the chemical potentials of the gas and polymer melt arising from the asymmetric interactions among CO2-CO2, CO2-polymer and polymer-polymer.
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For calculation of gas diffusion in polymer melt,
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In this work, we try to engage the dissolution and diffusion of CO2 molecules in PLA melt using the dynamic
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density functional theory (DDFT) [36]. Given the reliable free-energy function, the theory allows us to quantitatively describe the dynamic density evolution under nonequilibrium condition. In particular, DDFT returns to DFT when the nonlocal density distribution not changes with time. In formulation of the free-energy expression for addressing the CO2-PLA binary system, we integrate the interactions of CO2-CO2, CO2-PLA, and PLA-PLA to reflect the contribution of enthalpy, and the chain connectivity and bond-angle bending energy to account for the contribution of entropy. Unlike previous investigations, where the solubility of polymer in CO2 and the swellings of polymer were overlooked, the present theory simultaneously provides the coexistent curves of CO2 and PLA-CO2 phases and the corresponding swelling properties. During the kinetic diffusion, a
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introduce the idea of including the nonlinear coupling in
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expressing the transport coefficients on the basis of clear thermodynamic evidences.
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Fig. 1. Schematic diagram of repetitive units in a coarse-grained PLA chain. The black, red, and blue balls
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denote carbon, oxygen, and hydrogen atoms, respectively.
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2. Theoretical section
connected
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In the theoretical approach, CO2 is modeled as spherical molecule, and PLA chains are constituted with s. The atomic structure of lactic acid unit in a PLA chain is shown in Fig. 1.
The Lennard-Jones parameters for CO2, CO 3.66 Å, CO / kB 235.56 K, are taken directly from literature [37]. 2
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The force field parameters for the coarse-grained beads are chose from MARTINI force field [38,39] with
p 4.7
Å
p / kB 481.12
In the CO2(1)-PLA(2) binary system, the DDFT equation derived from Langevin equation can be generalized as the following time-dependent free-energy functional form [40,41]
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F [ 1 (r, t ), 2 (r, t )] D (r, t ) (r, t )
where (r,t ) is time-evolution density in real space, D is diffusion coefficient, represents a CO2 molecule or PLA monomer, and F[ 1 (r, t ), 2 (r, t )]
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the free-energy
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F[ 1 (r, t ), 2 (r, t )]
F [ 1 (r, t ), 2 (r, t )] kBT dr (r )[ln (r , t ) 1] F ex [ 1 (r, t ), 2 (r, t )]
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where k B is Boltzmann constant, and the first term on the right hand side denotes the free-energy of ideal-gas.
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The excess term F ex [ 1 (r, t ), 2 (r, t )] includes the contributions of hard-sphere repulsion, dispersive attraction, and chain connectivity and configuration over the ideal gas state.
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The fundamental measure theory [42] is commonly used to describe the contribution of hard-sphere repulsion,
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which is expressed as
F hs [ 1 (r, t ), 2 (r, t )] kBT hs [ni (r )]dr ,
(3)
where hs [ni (r )]
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fundamental geometric measures
free-energy density of component , which is connected to the averaged hs [ni (r )]
,
F att [ 1 (r, t ), 2 (r, t )] k BT (r ) a1[ (r )] a 2 [ (r )] dr
(4)
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The detailed expr
a[ (r)]
The free-energy contribution arising from the chain connectivity [46] and configuration of PLA is expressed as
(5)
where y ' (r1 , r2 ) is the cavity correlation function [46] between
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1 ( r1 r2 ' ) ' 1 y (r1 , r2 ) 2 (r2 ) 2 (r, t ) dr1 (r, t ) ln dr2 ' 2 2 4 ( ) 2 1 , 1 m 1 ( r1 r2 ' ) 1 dr1 (r, t ) ln dr2 g 0 2 (r2 ) 2 4 ( ' ) 2 2 =1 m
chain
and , and '
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' =( + ' ) / 2 . The second term on the right hand side belongs to the
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contribution of chain configuration given by the bond-angle bending formula, and g 0 is triplet function [47]. After introducing CO2 into the melted PLA for enough time, the CO2 phase and CO2-PLA mixed phase reach
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equilibrium at given temperature and pressure. Without consideration of the nonlocal density fluctuation, the
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free-energy expression is simplified to its local type, and can be easily applied to the phase equilibrium
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S 0 / [ L (1 xCO2 )]
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calculation for the binary system to derive the solubilities of CO2 in PLA and PLA in CO2. Accordingly, t
xCO
2
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where 0 is the density of pure polymer, L is the total density of polymer phase after dissolving CO2, and .
On the other hand, the nonlocal density distributions of CO2 and PLA at the CO2-PLA interface can be derived by minimizing the excess free-energy
F ex 1 (r, t ), 2 (r, t ) (r, t ) exp (r, t )
(7)
ACCEPTED MANUSCRIPT with an ordinary Picard iteration procedure. Here is the chemical potential of component at bulk state. The iteration is repeated until the average fractional difference over any grid point between the old and new
(r ) is less than 1.0 104 . The CO2-PLA interfacial tension is the excess free-energy at the interface of polymer phase and CO2 phase
/ A [ 1 (r), 2 (r)] ( 1 , 2 ) / A
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(8)
dr (r, t)
(9)
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[ 1 (r), 2 (r)] F[ 1 (r), 2 (r)]
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where A is the interfacial area, [ 1 (r), 2 (r)] is the grand potential, and can be calculated by
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Under the framework of dynamic DDFT, an individual CO2 molecule is chosen as a test particle located at time t 0 . The initial density distribution is numerically specified by a Gaussian distribution
T (r, 0) ( / )3/ 2 exp( r )
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(10)
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where is set as 200. Other CO2 molecules and PLA monomers are distributed around the labeled particle. It is
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assumed that the labeled CO2 is in equilibrium, the density distribution of the surrounding molecules is approximated to their radial distribution functions
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respectively. D0
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During the dynamic calculations, the length and time are nondimensionalized by P and B P 2 / D0 ,
dr 0.01 P D0
dt 105 B
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R2
Do ,0 lim R 2 /(6t )
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t
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After derivation of the time-dependent local density distribution of the test particle, the mean square
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displacement R 2 can be extracted from a simple statistical formula R 2 drT (r, t ) r . As such, the 2
local diffusion coefficient is determined by the Stokes-Einstein relation, D / Do lim R 2 /(6t )
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t
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the extended test-particle method [51], we allow
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one bead from an arbitrarily selected block to be fixed at the origin location, and in a symmetric external field [41]. The mean square
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displacement of a motion chain is calculated from the statistical average of all beads. Similarly, the diffusion
As a result,
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coefficient is calculated by D p / Do lim R 2 /(6t ) , and D0 t
kBT / 6 RD p
R
Dp
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Fig. 2. Comparison of the radial distribution function of PLA given by the theoretical calculation and
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molecular simulation.
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3. Results and discussion
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The macroscopic properties of polymers are always related to their microscopic structure.
1.25 ∙ F ex [ 1 (r, t ), 2 (r, t )]
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F ex [ (r, t )]
The theoretical prediction is generally in accordance with the simulation result, indicating that the theory can grasp the essential features of chain conformation.
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Fig. 3. Solubilities of CO2 in PLA at various temperatures and pressures. x 1 is the mass fraction (wt%) of
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CO2 in PLA melt. The scatter dots are the experimental data.
We then calculate the phase equilibria of CO2-PLA using the free-energy equation without consideration of
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the nonlocal density distribution. Fig. 3 presents the measured and calculated solubilities of CO2 in PLA melt.
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Although PLA is a semi-crystalline polymer, we choose the temperature below the cold crystallization temperature or above the melting temperature of PLA to avoid the generation of crystals, i.e., PLA can be
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regarded as an amorphous polymer. It can be known that CO2 solubility in PLA increases with decreasing
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temperature and increasing pressure The average deviations between calculated and experimental values [9-11] are 8.46% (308 K), 7.58% (333 K), and 9.69% (463 K), respectively. The general agreement between
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theoretical calculations and experimental data has been achieved, which indicates that the theoretical model provides a reasonable free-energy expression for the asymmetric interactions between PLA chains and CO2 molecules. Another advantage of phase equilibrium calculation is to provide the solubilities of PLA in supercritical CO2 phase according to the chemical potential equilibrium in the two phases. As shown in Fig. 4, it is difficult to detect PLA at low pressure due to its high cohesive energy. At middle pressure 50 solubilities of PLA in CO2 can be predicted by theoretical prediction, although the values are
he
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Fig. 4. Solubilities of PLA in CO2 at various temperatures and pressures. y1 is the mass fraction (wt%) of
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PLA in CO2 phase.
According to the definition of polymer expansion in supercritical CO2, the swelling ratios can be calculated
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through the equilibrium densities of PLA before and after the swelling process. From Fig. 5 one can see that the
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swelling of PLA increases with elevated pressure or depressed temperature. When the temperature is constant, the swelling ratio increases monotonously with pressure, and shows a significant slowdown of slop at high pressure. By comparing the swelling curves with the pressure-composition curves, one can find that the swelling ratios are coherently associated with CO2 solubilities. For instance, the slowing enhanced solubility at high pressure results in small changes in PLA density and swelling ratio. In general, the averaged relative deviation is 4.52%, showing a good agreement between the predicted and experimental swelling ratios.
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Fig. 5. Calculated swelling ratios for PLA in the presence of CO2 at elevated pressures. The scatter dots are the
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experimental data.
Fig. 6. Density distributions of CO2 and PLA at the phase interface region at (a) 463 K, 1 MPa; (b) 463 K, 10
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MPa; (c) 333K, 10 MPa.
After the phase equilibria have been achieved, the interfacial density distributions of CO2 and PLA can be determined by the density functional method, and the results are shown in Fig. 6. From the three subFig.s, one can immediately see the enrichments of CO2 in the interfacial region. The decrease of PLA density leads to reduced short-range repulsion of CO2-PLA, which is beneficial for the enhancement of CO2 concentration. At low pressure, the stacking of CO2 at the interface is pretty low (Fig. 6a). When the pressure is high enough, the
ACCEPTED MANUSCRIPT CO2 stack in the interface region becomes notably due to high solubility, since the attractions between PLA and CO2 are much stronger than that among CO2 molecules. As for influence of temperature, it could be seen from the subFig.s 6b and 6c that the density of CO2 at the interface increases at lower temperature, because low
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temperature can result in stronger asymmetric attractions.
Fig. 7. Variation of interfacial tension with respect to temperature and pressure. The scatter dots are the
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experimental data.
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According to the interfacial density distributions under different temperature and pressure effects, the interfacial tensions can be easily calculated and presented in Fig. 7. The approximate agreement of experimental
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values [55,56] and theoretical predictions manifests that interfacial structure analysis is reliable. When the temperature is constant, the interfacial tension decreases as pressure increases at a different rate. An increase of pressure results in high CO2 solubility, polymer swelling, and the reduced free volume, therefore leads to low interfacial tension. It is also shown that the lower temperature sample has a higher interfacial tension. Under low pressure, solubility also declines, and the influence of temperature becomes more obvious. At high pressure, however, solubility is large enough and the influence of dissolved CO2 becomes trivial. Therefore, the differential of interfacial tension between different temperatures decreases. Figure 7 also indicates that the predicted and experimental values are in good consistence.
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Fig. 8. Nonlocal density relaxation process of the test CO2 molecule.
t 0.1 B , the density distribution is described as r, t 0 / g(r ) . T
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initial stage
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Fig. 8 shows the density evolution of the test CO2 molecule at elevated time during diffusion process. In the
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t 5.0 B
t 50.0 B
and approaches to its bulk density
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As time goes by, the density peaks gradually decrease
The corresponding free-energy evolution of the test CO2 molecule is given in Fig. 9. One sees a monotone decline over the whole diffusion process. In it obvious that the declining free-energy provides the driving force for diffusion. Nevertheless, the slopes of these curves are not constant. At the intermediate region, those second
ACCEPTED MANUSCRIPT derivatives on the three curves change from minus to positive, and then return to minus. This phenomenon means that the diffusion rate of CO2 slows down in this region. Due to the frequent collisions with its neighboring CO2 molecules and PLA chains, the test molecule is confined to a small region but this restriction is not strong enough. Consequently, it escapes from the confinement. It can also be seen that, the temperature
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has a perceivable influence on free-energy. As temperature decreases, the concentration of CO2 increases, and
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the time cost to cross the confinement increases, resulting in the dropdown of driving force at low temperature.
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Fig. 9.
According to the time-dependent density evolutions, the diffusion coefficients can be easily derived and the results are shown in Fig. 10. At high pressure, more CO2 molecules are dissolved into the PLA melt, thus the collision frequency increases, leading to slower diffusivity. Under the pressure of 20 MPa, the diffusivity at 453 K is almost two orders of magnitude higher than the value at 323 K. It is unsurprised sine the free-energy driving force has been depressed at low temperature. One can also see that the predicted diffusion coefficients
ACCEPTED MANUSCRIPT are overestimated when compared with the corresponding experimental data [57,58]. It could be that there exist some entanglements and crystal regions in PLA during experimental test at low temperature. This effect has been ignored in theoretical study. Because CO2 molecules can hardly permeate through the entanglements and crystals, the actual diffusion has been retarded to some extent. In addition, the calculated diffusivity at high
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temperature deviates obviously from
. The scatted dots denote experimental data
Fig. 11 depicts the calculated viscosities of PLA at 383K and 453 K. It can be seen that the viscosity of PLA is reduced rapidly with increasing pressure, especially before the supercritical state of CO2. When PLA melt absorbs CO2 molecules, the chains rearrange themselves towards a new equilibrium conformation. For small CO2 concentration equivalent to low pressure, the plasticizing effect is reflected in the rapid rising swelling
ACCEPTED MANUSCRIPT ratio as shown in Fig. 5. Thus effect leads to enhanced ease of hole formation and chain motion. After that, the variation tendency of the curve becomes planus. Although the plasticizing is prominent at high pressure, the slope gradient of increasing swelling ratio decreases. Fig. 11 also shows that the predicted viscosities deviate from the experimental values with about one order of magnitude, indicating that description of chain dynamic
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given by the theoretical model is only qualitatively reasonable due to some crude approximations, such as chain
. The scatted dots
denote experimental data.
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block definition and regardlessness of hydrodynamic interactions given by dissolved CO2 molecules.
4. Conclusion We have presented a DDFT approach to quantitatively predict the CO2 solubilities and diffusivities, PLA swelling ratios and viscosities, and CO2-PLA interfacial tensions. Before its application, the theoretical approach has been fully tested by the computational radial distribution of PLA, some experimental solubilities, interfacial tensions, and diffusion coefficients. It has been shown that the predictions of solubilities, swelling
ACCEPTED MANUSCRIPT ratios, interfacial tensions, diffusivities, and viscosities are generally reasonable. Therefore, we conclude that such theoretical approach has some advantages over previous thermodynamic or dynamic equations: (i) the solubilities of CO2 in PLA and PLA in CO2 as well as the swelling ratios of PLA can be determined simultaneously by the phase equilibrium calculations for the binary system; (ii) the local and nonlocal density distributions can be derived by the same theory under equilibrium condition; (iii) the time-dependent diffusion
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process is performed strictly on the density and free-energy evolution, such that the kinetic process is coherently
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associated with the thermodynamic state, and the driven force can be clearly identified from the free-energy
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variation. Therefore, this work provides a comprehensive model for in-depth understanding the role of various factors in influencing PLA foaming that could find important applications for the control of cell density and
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morphology.
Acknowledgements
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This work has been supported by the Natural Science Foundation of Beijing (2162012), the National Science
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Foundation of China (51673004) , and the 2017 Special Commercial Projects (19005757053).
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Kesong Yu, Hongfu Zhou, Xiangdong Wang, Zhongjie Du, Jianguo Mi PII: DOI: Reference:
S0167-7322(18)34348-4 https://doi.org/10.1016/j.molliq.2019.02.005 MOLLIQ 10405
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
22 August 2018 1 December 2018 1 February 2019
Please cite this article as: K. Yu, H. Zhou, X. Wang, et al., From thermodynamics to kinetics: Theoretical study of CO2 dissolving in poly (lactic acid) melt, Journal of Molecular Liquids, https://doi.org/10.1016/j.molliq.2019.02.005
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ACCEPTED MANUSCRIPT From Thermodynamics to Kinetics: Theoretical Study of CO2 Dissolving in Poly (lactic acid) Melt
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School of Materials and Mechanical Engineering, Beijing Technology and Business University, Beijing, China
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Kesong Yua,b, Hongfu zhoua, Xiangdong Wanga*, Zhongjie Dub, and Jianguo Mi b*
State Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology, Beijing,
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China
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Abstract: Under the framework of dynamic density functional theory, the dissolution and diffusion behaviors
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of CO2
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ACCEPTED MANUSCRIPT 1. Introduction Poly (lactic acid) (PLA) is an aliphatic thermoplastic polyester with the biodegradable attribute and comparable mechanical properties [1,2]. PLA foam has the potential to replace traditional petroleum-based polymer foams in a wide array of applications [3-5]. Moreover, PLA has relatively high CO2 solubility, which provides a good prerequisite for developing nanocellular cells with uniform cell morphology to further enhance
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its impact resistance, toughness, and thermal and sound insulation [6-8]. It is known that the cell morphology is
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generally governed by the following thermodynamic and kinetic properties: CO2 dissolution, PLA swelling,
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CO2-PLA interfacial tension, CO2 diffusion, and PLA viscosity. In order to confirm the condition to produce
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nanocellular PLA foam, these fundamental issues should be clearly identified in advance. It is not easy to handle and ensure the dissolution equilibrium of CO2 in highly PLA melt in experiment, thus
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the available experimental data on CO2 solubility are restricted in a narrow range of temperature and pressure [9-11]. Meanwhile, the swelling ratio of PLA depends on how much CO2 dissolves in PLA [12], and the
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corresponding interfacial structure and tension of CO2-PLA has to be regulated due to PLA plasticization
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[13,14]. The swelling and interfacial properties are more difficult to measure via experimental approaches. A reliable thermodynamic model, capable of accurately describing CO2 solubility, the swelling ratio of glassy
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PLA, and the interfacial structure and property between two phases over a wide range of temperature and
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pressure, is an essential prerequisite for developing a good transport model.
The diffusion of CO2 in PLA is a complex process as
well. It can be influenced by concentration, pressure, temperature, PLA viscosity, external force and so on. For these reasons, the measurement of diffusivity of supercritical CO2 in molten polymer has always been a challenging research topic for the foam processing [15]. T 16,17] Theoretical prediction is an alternative approach to reduce the workload of experiment and to provide more efficient data aimed at predicting the cell size and density of foaming products.
ACCEPTED MANUSCRIPT A multitude of equations of state were applied to calculate and predict gas solubility in polymer systems [1822]. Through the proper choice of specific model parameters regressed from experimental data, these equations are capable of semi-quantitatively evaluating the dissolving behavior. However, the physical meaning of these parameters is obscure, because
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restrict the usefulness of equations as the predictive tools.
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Another important type of equations was derived from the statistical associating fluid theory (SAFT) [23],
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which was widely applicated to describe the dissolution behaviors of gases in macromolecular systems [24-28]. Based on the statistical mechanics, the theory has advantageous over those semi-empirical equations by
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providing molecular-level interpretation. Although SAFT has yielded encouraged results, its further
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improvement is difficult, since its semi-quantitative parameters are often confined to a certain region. Two different empirical equations of state, the Sanchez-Lacombe [29] and Simha-Somocynsky [30] types,
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were used to calculate the swelling property of polymer melts that was induced by gas permeation. These
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equations are also semi-quantitatively and the model parameters rely completely on the available specific pressure–volume–temperature data. It has been recognized that the equations are difficult to predict the swelling
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properties given the extreme shortage of these fundamental data. This failure is a result of the inability to
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accurately predict the chemical potentials of the gas and polymer melt arising from the asymmetric interactions among CO2-CO2, CO2-polymer and polymer-polymer.
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For calculation of gas diffusion in polymer melt,
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In this work, we try to engage the dissolution and diffusion of CO2 molecules in PLA melt using the dynamic
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density functional theory (DDFT) [36]. Given the reliable free-energy function, the theory allows us to quantitatively describe the dynamic density evolution under nonequilibrium condition. In particular, DDFT returns to DFT when the nonlocal density distribution not changes with time. In formulation of the free-energy expression for addressing the CO2-PLA binary system, we integrate the interactions of CO2-CO2, CO2-PLA, and PLA-PLA to reflect the contribution of enthalpy, and the chain connectivity and bond-angle bending energy to account for the contribution of entropy. Unlike previous investigations, where the solubility of polymer in CO2 and the swellings of polymer were overlooked, the present theory simultaneously provides the coexistent curves of CO2 and PLA-CO2 phases and the corresponding swelling properties. During the kinetic diffusion, a
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introduce the idea of including the nonlinear coupling in
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expressing the transport coefficients on the basis of clear thermodynamic evidences.
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Fig. 1. Schematic diagram of repetitive units in a coarse-grained PLA chain. The black, red, and blue balls
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denote carbon, oxygen, and hydrogen atoms, respectively.
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2. Theoretical section
connected
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In the theoretical approach, CO2 is modeled as spherical molecule, and PLA chains are constituted with s. The atomic structure of lactic acid unit in a PLA chain is shown in Fig. 1.
The Lennard-Jones parameters for CO2, CO 3.66 Å, CO / kB 235.56 K, are taken directly from literature [37]. 2
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The force field parameters for the coarse-grained beads are chose from MARTINI force field [38,39] with
p 4.7
Å
p / kB 481.12
In the CO2(1)-PLA(2) binary system, the DDFT equation derived from Langevin equation can be generalized as the following time-dependent free-energy functional form [40,41]
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F [ 1 (r, t ), 2 (r, t )] D (r, t ) (r, t )
where (r,t ) is time-evolution density in real space, D is diffusion coefficient, represents a CO2 molecule or PLA monomer, and F[ 1 (r, t ), 2 (r, t )]
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the free-energy
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F[ 1 (r, t ), 2 (r, t )]
F [ 1 (r, t ), 2 (r, t )] kBT dr (r )[ln (r , t ) 1] F ex [ 1 (r, t ), 2 (r, t )]
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where k B is Boltzmann constant, and the first term on the right hand side denotes the free-energy of ideal-gas.
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The excess term F ex [ 1 (r, t ), 2 (r, t )] includes the contributions of hard-sphere repulsion, dispersive attraction, and chain connectivity and configuration over the ideal gas state.
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The fundamental measure theory [42] is commonly used to describe the contribution of hard-sphere repulsion,
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which is expressed as
F hs [ 1 (r, t ), 2 (r, t )] kBT hs [ni (r )]dr ,
(3)
where hs [ni (r )]
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fundamental geometric measures
free-energy density of component , which is connected to the averaged hs [ni (r )]
,
F att [ 1 (r, t ), 2 (r, t )] k BT (r ) a1[ (r )] a 2 [ (r )] dr
(4)
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The detailed expr
a[ (r)]
The free-energy contribution arising from the chain connectivity [46] and configuration of PLA is expressed as
(5)
where y ' (r1 , r2 ) is the cavity correlation function [46] between
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1 ( r1 r2 ' ) ' 1 y (r1 , r2 ) 2 (r2 ) 2 (r, t ) dr1 (r, t ) ln dr2 ' 2 2 4 ( ) 2 1 , 1 m 1 ( r1 r2 ' ) 1 dr1 (r, t ) ln dr2 g 0 2 (r2 ) 2 4 ( ' ) 2 2 =1 m
chain
and , and '
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' =( + ' ) / 2 . The second term on the right hand side belongs to the
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contribution of chain configuration given by the bond-angle bending formula, and g 0 is triplet function [47]. After introducing CO2 into the melted PLA for enough time, the CO2 phase and CO2-PLA mixed phase reach
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equilibrium at given temperature and pressure. Without consideration of the nonlocal density fluctuation, the
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free-energy expression is simplified to its local type, and can be easily applied to the phase equilibrium
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S 0 / [ L (1 xCO2 )]
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calculation for the binary system to derive the solubilities of CO2 in PLA and PLA in CO2. Accordingly, t
xCO
2
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where 0 is the density of pure polymer, L is the total density of polymer phase after dissolving CO2, and .
On the other hand, the nonlocal density distributions of CO2 and PLA at the CO2-PLA interface can be derived by minimizing the excess free-energy
F ex 1 (r, t ), 2 (r, t ) (r, t ) exp (r, t )
(7)
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(r ) is less than 1.0 104 . The CO2-PLA interfacial tension is the excess free-energy at the interface of polymer phase and CO2 phase
/ A [ 1 (r), 2 (r)] ( 1 , 2 ) / A
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(8)
dr (r, t)
(9)
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[ 1 (r), 2 (r)] F[ 1 (r), 2 (r)]
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where A is the interfacial area, [ 1 (r), 2 (r)] is the grand potential, and can be calculated by
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Under the framework of dynamic DDFT, an individual CO2 molecule is chosen as a test particle located at time t 0 . The initial density distribution is numerically specified by a Gaussian distribution
T (r, 0) ( / )3/ 2 exp( r )
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(10)
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where is set as 200. Other CO2 molecules and PLA monomers are distributed around the labeled particle. It is
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assumed that the labeled CO2 is in equilibrium, the density distribution of the surrounding molecules is approximated to their radial distribution functions
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respectively. D0
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During the dynamic calculations, the length and time are nondimensionalized by P and B P 2 / D0 ,
dr 0.01 P D0
dt 105 B
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R2
Do ,0 lim R 2 /(6t )
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t
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After derivation of the time-dependent local density distribution of the test particle, the mean square
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displacement R 2 can be extracted from a simple statistical formula R 2 drT (r, t ) r . As such, the 2
local diffusion coefficient is determined by the Stokes-Einstein relation, D / Do lim R 2 /(6t )
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t
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the extended test-particle method [51], we allow
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one bead from an arbitrarily selected block to be fixed at the origin location, and in a symmetric external field [41]. The mean square
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displacement of a motion chain is calculated from the statistical average of all beads. Similarly, the diffusion
As a result,
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coefficient is calculated by D p / Do lim R 2 /(6t ) , and D0 t
kBT / 6 RD p
R
Dp
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Fig. 2. Comparison of the radial distribution function of PLA given by the theoretical calculation and
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molecular simulation.
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3. Results and discussion
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The macroscopic properties of polymers are always related to their microscopic structure.
1.25 ∙ F ex [ 1 (r, t ), 2 (r, t )]
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F ex [ (r, t )]
The theoretical prediction is generally in accordance with the simulation result, indicating that the theory can grasp the essential features of chain conformation.
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Fig. 3. Solubilities of CO2 in PLA at various temperatures and pressures. x 1 is the mass fraction (wt%) of
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CO2 in PLA melt. The scatter dots are the experimental data.
We then calculate the phase equilibria of CO2-PLA using the free-energy equation without consideration of
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the nonlocal density distribution. Fig. 3 presents the measured and calculated solubilities of CO2 in PLA melt.
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Although PLA is a semi-crystalline polymer, we choose the temperature below the cold crystallization temperature or above the melting temperature of PLA to avoid the generation of crystals, i.e., PLA can be
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regarded as an amorphous polymer. It can be known that CO2 solubility in PLA increases with decreasing
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temperature and increasing pressure The average deviations between calculated and experimental values [9-11] are 8.46% (308 K), 7.58% (333 K), and 9.69% (463 K), respectively. The general agreement between
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theoretical calculations and experimental data has been achieved, which indicates that the theoretical model provides a reasonable free-energy expression for the asymmetric interactions between PLA chains and CO2 molecules. Another advantage of phase equilibrium calculation is to provide the solubilities of PLA in supercritical CO2 phase according to the chemical potential equilibrium in the two phases. As shown in Fig. 4, it is difficult to detect PLA at low pressure due to its high cohesive energy. At middle pressure 50 solubilities of PLA in CO2 can be predicted by theoretical prediction, although the values are
he
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Fig. 4. Solubilities of PLA in CO2 at various temperatures and pressures. y1 is the mass fraction (wt%) of
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PLA in CO2 phase.
According to the definition of polymer expansion in supercritical CO2, the swelling ratios can be calculated
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through the equilibrium densities of PLA before and after the swelling process. From Fig. 5 one can see that the
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swelling of PLA increases with elevated pressure or depressed temperature. When the temperature is constant, the swelling ratio increases monotonously with pressure, and shows a significant slowdown of slop at high pressure. By comparing the swelling curves with the pressure-composition curves, one can find that the swelling ratios are coherently associated with CO2 solubilities. For instance, the slowing enhanced solubility at high pressure results in small changes in PLA density and swelling ratio. In general, the averaged relative deviation is 4.52%, showing a good agreement between the predicted and experimental swelling ratios.
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Fig. 5. Calculated swelling ratios for PLA in the presence of CO2 at elevated pressures. The scatter dots are the
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experimental data.
Fig. 6. Density distributions of CO2 and PLA at the phase interface region at (a) 463 K, 1 MPa; (b) 463 K, 10
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MPa; (c) 333K, 10 MPa.
After the phase equilibria have been achieved, the interfacial density distributions of CO2 and PLA can be determined by the density functional method, and the results are shown in Fig. 6. From the three subFig.s, one can immediately see the enrichments of CO2 in the interfacial region. The decrease of PLA density leads to reduced short-range repulsion of CO2-PLA, which is beneficial for the enhancement of CO2 concentration. At low pressure, the stacking of CO2 at the interface is pretty low (Fig. 6a). When the pressure is high enough, the
ACCEPTED MANUSCRIPT CO2 stack in the interface region becomes notably due to high solubility, since the attractions between PLA and CO2 are much stronger than that among CO2 molecules. As for influence of temperature, it could be seen from the subFig.s 6b and 6c that the density of CO2 at the interface increases at lower temperature, because low
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temperature can result in stronger asymmetric attractions.
Fig. 7. Variation of interfacial tension with respect to temperature and pressure. The scatter dots are the
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experimental data.
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According to the interfacial density distributions under different temperature and pressure effects, the interfacial tensions can be easily calculated and presented in Fig. 7. The approximate agreement of experimental
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values [55,56] and theoretical predictions manifests that interfacial structure analysis is reliable. When the temperature is constant, the interfacial tension decreases as pressure increases at a different rate. An increase of pressure results in high CO2 solubility, polymer swelling, and the reduced free volume, therefore leads to low interfacial tension. It is also shown that the lower temperature sample has a higher interfacial tension. Under low pressure, solubility also declines, and the influence of temperature becomes more obvious. At high pressure, however, solubility is large enough and the influence of dissolved CO2 becomes trivial. Therefore, the differential of interfacial tension between different temperatures decreases. Figure 7 also indicates that the predicted and experimental values are in good consistence.
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Fig. 8. Nonlocal density relaxation process of the test CO2 molecule.
t 0.1 B , the density distribution is described as r, t 0 / g(r ) . T
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initial stage
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Fig. 8 shows the density evolution of the test CO2 molecule at elevated time during diffusion process. In the
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t 5.0 B
t 50.0 B
and approaches to its bulk density
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As time goes by, the density peaks gradually decrease
The corresponding free-energy evolution of the test CO2 molecule is given in Fig. 9. One sees a monotone decline over the whole diffusion process. In it obvious that the declining free-energy provides the driving force for diffusion. Nevertheless, the slopes of these curves are not constant. At the intermediate region, those second
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the time cost to cross the confinement increases, resulting in the dropdown of driving force at low temperature.
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Fig. 9.
According to the time-dependent density evolutions, the diffusion coefficients can be easily derived and the results are shown in Fig. 10. At high pressure, more CO2 molecules are dissolved into the PLA melt, thus the collision frequency increases, leading to slower diffusivity. Under the pressure of 20 MPa, the diffusivity at 453 K is almost two orders of magnitude higher than the value at 323 K. It is unsurprised sine the free-energy driving force has been depressed at low temperature. One can also see that the predicted diffusion coefficients
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temperature deviates obviously from
. The scatted dots denote experimental data
Fig. 11 depicts the calculated viscosities of PLA at 383K and 453 K. It can be seen that the viscosity of PLA is reduced rapidly with increasing pressure, especially before the supercritical state of CO2. When PLA melt absorbs CO2 molecules, the chains rearrange themselves towards a new equilibrium conformation. For small CO2 concentration equivalent to low pressure, the plasticizing effect is reflected in the rapid rising swelling
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given by the theoretical model is only qualitatively reasonable due to some crude approximations, such as chain
. The scatted dots
denote experimental data.
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block definition and regardlessness of hydrodynamic interactions given by dissolved CO2 molecules.
4. Conclusion We have presented a DDFT approach to quantitatively predict the CO2 solubilities and diffusivities, PLA swelling ratios and viscosities, and CO2-PLA interfacial tensions. Before its application, the theoretical approach has been fully tested by the computational radial distribution of PLA, some experimental solubilities, interfacial tensions, and diffusion coefficients. It has been shown that the predictions of solubilities, swelling
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process is performed strictly on the density and free-energy evolution, such that the kinetic process is coherently
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associated with the thermodynamic state, and the driven force can be clearly identified from the free-energy
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variation. Therefore, this work provides a comprehensive model for in-depth understanding the role of various factors in influencing PLA foaming that could find important applications for the control of cell density and
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morphology.
Acknowledgements
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This work has been supported by the Natural Science Foundation of Beijing (2162012), the National Science
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Foundation of China (51673004) , and the 2017 Special Commercial Projects (19005757053).
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