LA VULNERACION DE LA MANIFESTACIÓN DE VOLUNTAD EN LA EMISION DE LETRAS DE CAMBIO INCOMPLETAS
4.2. DE LA FALTA DE RIGUROSIDAD EN LA EMISION DE LETRAS DE CAMBIO INCOMPLETAS
4.2.5. Pacto de Completamiento e Integración de la Letra de Cambio Incompleta
Subsequently, we compare obtained activities with experimental data for a concise, but varied, set of polymer-solvent mixtures. Finally, Flory-Huggins χ-interaction parameters are calculated using PAC-MAC and compared with an experimental database containing 779 values. This chapter contains a proof of concept for the treatment of polymers within the PAC-MAC model. The derivation of the expression for the activity coefficients is included as Supporting Information.
Prediction of Polymer-Solvent Miscibility Properties Using PAC-MAC
THEORETICAL BASIS
PAC-MAC is a force field based quasi-chemical method for rapid calculation of multi-component phase diagrams. The theoretical basis is extensively discussed for solvent mixtures in Chapter II and Chapter III. This section contains a basic overview and an explanation of the required modifications to handle polymers within PAC-MAC. The method consists of 3 successive computational steps (surface generation, sampling of pair configurations and solving a system of equations) which are described in three different subsections.
Surface generation
The surface of a solvent molecule is defined by the outermost area formed by spheres around the atomic nuclei. The coordinates of the atomic nuclei are obtained using a molecular energy minimization in vacuum. And the radius of a sphere around atom k is related to the Lennard-Jones distance parameter kk via:
0.62 surf
k kk
R (1)
The molecular surface is subsequently subdivided in, mainly hexagonal and pentagonal, surface panels with an area of about 0.5 Å2 per panel.
Within the PAC-MAC method, a polymer is represented by an energetically minimized linear chain of iso-conformational segments, see Figure 3. A single segment typically consists of a few monomers (by default 2). The big advantage of a linear chain of equal-shaped segments is the possibility to use periodic boundary conditions to reduce computational calculation time. We are aware that a linear polymer chain does not represent the physical configuration of a polymer in solution. However, the shape of a random coil is only visible on the scale of a few Kuhn lengths. On a smaller scale (within the Kuhn length), the assumption to treat neighboring monomers as a linear chain is plausible. Moreover, as shown by C. Loschen and A. Klamt, two different configurations for the PEG polymer yield nearly identical results in COSMO-RS.15 Subsequently, the surface of a polymer is, in accordance with a solvent molecule, defined by the exterior formed by spheres around the atomic nuclei and subdivided in surface panels with an area of about 0.5 Å2 per panel. See Figure 1 for an illustration of the surface generation of PEG (total area: 123.4 Å2/segment).
Chapter IV
Figure 1. Surface generation of PEG. Top: Energetically minimized configuration of a linear polymer with iso-conformational segments (using 2 monomers per segment). Bottom: Corresponding polymer surface in PAC-MAC, divided into 234 surface panels per segment of approximately 0.5 Å2.
Sampling of pair configurations
Using the obtained molecular structures of all different components in the mixture (solvent molecules and polymers), we sample a large diverse set of pair configurations. The sampling procedure is performed for all possible combinations of two adjacent components in the mixture. A total of m pair configurations are sampled for each combination of two components, with m = 5·104 by default. The pair configurations represent a uniform set of all possible orientations between two molecules within their first coordination shell. For details concerning the computational procedure of the formation of pair configurations, we refer to
Chapter II and Chapter III.
Two calculations are performed for each sampled pair configuration: the calculation of the surface panels which are covered by the adjacent molecule and the calculation of the total intermolecular energy. A surface panel is covered by the adjacent molecule if a vector at the center of the panel and perpendicular to the convex hull of the molecule pierces the surface of the adjacent molecule. A maximum distance, at which intersection of the adjacent molecule is possible, is not taken into account. However, if the adjacent molecule is a polymer, then only the surface of the nearest segment is taken to avoid intersection of the polymer at very large distances. The calculation of the intermolecular energy wi is performed using the OPLS-AA force field24. In the OPLS-AA force field, the intermolecular energy consists of Van der Waals and electrostatic interactions. Hydrogen bonding effects are included within the partial charges of the atoms. In the case of pair configurations containing polymers, only the interaction with the nearest segment is taken in the energy calculation, to be consistent with the calculation of the occupied surface panels. Figure 2 shows the occupied surface panels and intermolecular energy of a single pair configuration for all 4 combinations of adjacent components within a binary mixture of PEG and ethanol. Note that only pair configurations between two molecules within their first coordination shell are sampled in the PAC- MAC method, so three-body interactions and long-range interactions with molecules in a second or higher coordination shell are not taken into account.
Prediction of Polymer-Solvent Miscibility Properties Using PAC-MAC
Figure 2. Examples of sampled pair configurations with occupied surface panels (black colored) and intermolecular energy for ethanol-ethanol (1), ethanol-PEG (2), PEG-ethanol (3) and PEG-PEG (4).
Solving system of equations
Using the set of occupied surface panels and calculated intermolecular energies, the probabilities of occurrence IJ
i
x of each sampled pair configuration i
1..m between molecules I and J is calculated. Moreover, unoccupied fractions ,vac j I
x are calculated for all surface panels j
1..LI
of molecule I. The unoccupied fraction, vac j I
x refers to the probability that surface panel j is not covered by neighboring molecules. The pair probabilities and unoccupied fractions are calculated by minimizing the following expression for the mixing free energy:
mix id fv comb vac int
F F F F F F (2)
In which, for a binary mixture containing molecules A and B (respectively with mole fraction xA and xB), the 5 free energy terms are given by:
, ln id I I I A B B F x x Nk T
(3) , ln fv fv I I I A B B I F x Nk T x
(4)Chapter IV , , 1 1 ln 2 IJ comb m IJ i i I A B J A B i B I J x m F z x Nk T y y
(5) , , , 1 ln I I vac L vac j j I vac I j I I vac I A B j B i I A x F x x Nk T A x
(6) , , 1 1 2 IJ int m IJ i i I A B J A B i B B w F z x Nk T k T
(7)A description of all used variables within Equations (3)-(7) is given in the List of Symbols.
In comparison with the system of equations presented in Chapter III, only the Staverman-Guggenheim correction term25, 26 is replaced by the Elbro free volume correction term, Equation (4)14. Elbro et al. designed their expression specifically for polymer solutions to incorporate free volume contributions. A combinatorial effect which is known to be important within polymer solutions27 and is missing in the Staverman-Guggenheim expression. Our definition for the free volume fraction of component A is given by:
, A A A fv A I I I I A B x v v x v v
(8) With vI and vI being respectively the molar volume and the molar hard-core volume of component I.
Minimization of the free energy, Equation (2), is performed subject to the constraint that the covered fraction of surface panel j by neighboring molecules equals the occupied fraction of panel j:
, , 1 , 1 1 1 2 2 1.. , , m m IJ IJ JI JI vac i ij i ij I j I I J A B i J A B i I z x o z x o x x x j L I A B
(9)Minimization of the free energy results in the following expression for the activity of component I:
, , , 1 , , , 1 ln ln 1 (1 ) (1 ) , I I fv Lfv I I mix vac mix
I I j j I
j I L
I pure vac pure
j j I j a x x x I A B
(10)Prediction of Polymer-Solvent Miscibility Properties Using PAC-MAC Where jI are Lagrange multipliers.
For details concerning the derivation towards the activities, Equation (10), we refer to the Supporting Information.
Chapter IV
RESULTS AND DISCUSSION
Within the calculation of the Elbro free volume correction term, Equation (4), two molecular volumes are required: the molar volume vI and the molar hard-core volume vI
. In principle, Bondi’s van der Waals volume represents the hard-core
volume.14 However, we decided to replace the required atomic Bondi radii by atomic radii HS
k
R . The atomic radii HS k
R are related to the Lennard-Jones distance parameter kk in order to be consistent with surf
k R in Equation (1): 0.56 HS k kk R (11)
The scaling factor of 0.5·21/6 = 0.56 relates
kk
with half the distance at which a Lennard-Jones potential reaches its minimum. The molar volume, on the other hand, should be obtained from experimental data. However, using a QSPR estimation28 or a group contribution method29, very accurate prediction of the molar volume can be obtained without the necessity of having experimental data available. In accordance with the calculation of the molecular surface, using Equation (1), and the calculation of the hard-core volume, using Equation (11), we decided to define the molecular volume by spheres around the atomic nuclei of which the radii Vm
k
R are equal to the Lennard-Jones distance parameter kk multiplied by a scaling factor
Vm
s :
Vm Vm
k kk
R s (12)
A comparison with experimental data shows that a scaling factor of Vm
s = 0.66 results in the best fit of molar volumes at 298 K for both solvent molecules as well as the repeat unit of amorphous atactic homopolymers (see Figure 3). The scaling factor sVm is, as expected, bigger than the scaling factor of 0.56 for HS
k
R in Equation (11) because the molar volume is bigger than the molar hard-core volume. Note that
Vm
s is optimized at 298 K; a temperature dependence of the scaling factors is not yet taken into account.
Prediction of Polymer-Solvent Miscibility Properties Using PAC-MAC
Figure 3. Molar volume calculated within the PAC-MAC model in comparison with experimental values at 298 K, for both solvent molecules (blue) and homopolymers (red). A single scaling factor of Vm
s = 0.66 is required within the PAC-MAC model.
The experimental data of the solvent molecules is extracted from the online available chemical properties of products supplied by Alfa Aesar30. The data related to polymers is extracted from the CROW Polymer Database31.
Solvent activities as a function of solvent fraction are calculated using the PAC- MAC model for 4 different polymer-solvent mixtures, see Figure 4. A comparison with experimental data is provided in every case.
Chapter IV
Figure 4. Obtained activities as a function of weight fraction xw or volume fraction : the PAC-MAC model in comparison with experimental data. Experimental data is obtained from: Booth and Devoy32 (PEO – benzene), Brown et al.33 (PP – diethyl ketone), Allen et al.34 (PP – diethyl ketone) and Wohlfarth35 (other mixtures).
As shown in Figure 4, the PAC-MAC model not always accurately predicts solvent activities in polymer-solvent mixtures. The calculated activity of chlorobenzene in a mixture with polyethylene is slightly overestimated. However, an observed root mean squared error (RMSE) of 0.053 is much smaller than the calculated deviation in a mixture of poly(ethylene oxide) with aromatic compounds benzene or toluene. PAC-MAC incorrectly predicts phase separation of poly(ethylene oxide) (PEO) in both benzene and toluene. PEO, dissolved in various solvents, shows characteristic cluster formation36. The origin of the observed clustering in PEO solutions is under debate. Possible reasons are crystallization of the PEO groups37, a polymer-rich phase due to a phase transition at low temperature38 and impurities in the solvent39. Several assumptions are made within the PAC-MAC model in order to minimize the calculation time. The most important assumptions are: treating polymers as linear chains of equal-shaped segments, neglecting interactions beyond the first coordination shell and neglecting many-body interactions. Therefore, the predictive capacity of PAC-MAC is expected to be
Prediction of Polymer-Solvent Miscibility Properties Using PAC-MAC limited for mixtures in which cluster formation or crystallization phenomena are involved.
The activity of 1,2-dimethylbenzene in a mixture with poly(vinyl methyl ether) at 373.15 K is modeled perfectly by PAC-MAC. However, a temperature dependence of the solubility, probably caused by thermal expansion of the solvent and related dissimilarity in free volume27, is not observed. Since temperature independent scaling factors are used for the calculation of both the molar hard-core volume and the molar volume, PAC-MAC predicts a constant free volume fraction with changing temperature.
Besides the errors attributed to assumptions made within the model, also the experimental data might lead to deviations. Two similar samples of amorphous polypropylene solvated in diethyl ketone show significant different activities of the solvent33, 34. The difference is caused by a minor but different degree of crystallinity within the two samples according to Allen et al.34
A comparison of the PAC-MAC method with the competitive COSMO-RS method is given in Figure 5 for 2 binary mixtures.
Figure 5. Obtained activities as a function of weight fraction xw: the PAC-MAC
model in comparison with COSMO-RS15 and experimental data40.
COSMO-RS shows similar accuracy as PAC-MAC based on the mixtures in Figure 5. We cannot draw any statistical conclusion based on a comparison involving only 2 polymer-solvent mixtures. However, since COSMO-RS also does not include clustering effects and contains the same Elbro free volume correction, it is likely that similar results are observed.
In the limit of polymer length towards infinity, lim
p s p
v v v = 0, the Flory-
Huggins χ-interaction parameter is related to the solvent activity, as, via:
2
ln( )as ln(s)pp (13)
The accuracy of the PAC-MAC method is tested by comparing the obtained Flory- Huggins χ-interaction parameter, calculated using Equation (13), with experimental
Chapter IV
values taken from the Polymer Handbook3. The comparison for 779 data points is given in Figure 6. The dataset is divided in 5 different categories, related to the polymer in the mixture: cellulose (16 points), polyalkenes (216 points), polystyrenes (312 points), polyacrylates (176 points) and polyglycols (59 points).
Figure 6. Scatterplot of 2
, PAC-MAC versus experimental.
Figure 6 shows correlation between the calculated and experimental values. Especially polymer solutions containing polystyrenes or polyacrylates correlate properly with experimental data: the obtained correlation coefficients are respectively 0.925 and 0.946 (see Figure 7). However, a total observed RMSE of 0.593 is not very accurate. As shown in Figure 4, and caused by clustering effects36, PAC-MAC overestimates the χ-interaction parameter for mixtures containing polyglycol/polyoxide. Moreover, significant outliers are obtained for hydroxypropyl cellulose – acetone (exp.: 0.43, PAC-MAC: -2.53) and hydroxypropyl cellulose – tetrahydrofuran (exp.: 0.33, PAC-MAC: -2.19), revealing the limitations of PAC- MAC for mixtures containing polymers with large repeat units. A large repeat unit leads to a large spread in the calculated intermolecular pair energies because many atoms are contributing to the Van der Waals and electrostatic interaction energy. As a consequence, a few sampled pair configurations, with a very low intermolecular energy, will have a very dominant contribution to the mixing properties resulting in an imprecise estimation of the χ-interaction parameter. Of course, also the accuracy of the experimental data must be taken into account; the standard deviations of the experimental values of the substituted cellulose solutions are unknown and may not be negligible. Finally, the accuracy of PAC-MAC is always limited to the accuracy of the force fields. Monte Carlo simulations show that the error in mix
F caused by the force field can be significant, up to 1.0 kBT.17 On the other hand, the use of force field parameters provides a lot of handles to optimize. In the spirit of group
Prediction of Polymer-Solvent Miscibility Properties Using PAC-MAC contribution methods5, the observed RMSE will be greatly reduced if many parameters are available to adjust. Based on Figure 6, PAC-MAC is a quick tool for obtaining a rough indication of the χ-interaction parameter.
Figure 7. Scatterplot of 2
, PAC-MAC versus experimental, for solutions containing polystyrenes (left) and polyacrylates (right).
Chapter IV