A Functional Approach to Solubility Parameter Computations

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A Functional Approach to Solubility Parameter Computations Jason S Howell, Miranda Renae Roesing, and David Sean Boucher J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b01537 • Publication Date (Web): 10 Apr 2017 Downloaded from http://pubs.acs.org on April 17, 2017

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The Journal of Physical Chemistry

A Functional Approach to Solubility Parameter Computations Jason Howel11, Miranda Roesing,2 David Boucher*2,a 1. Department of Mathematics, College of Charleston, 66 George St., Charleston, SC, USA, 29424. 2. Department of Chemistry and Biochemistry, College of Charleston, 66 George St., Charleston, SC, USA, 29424. a) E-mail address: [email protected], tel. 843-953-6493

ABSTRACT The determination of solubility parameters for solutes represents a challenging mathematical problem of locating the central tendency of solvent affinity based on a limited set of data taken from experimental observations. At present, the most commonly used methods for computing solubility parameters of a solute require a binary classification of solvent affinity for the solute and employ a spherical/ellipsoidal compatibility region in the three-dimensional Hansen solubility parameter space.

Utilizing a binary classification requires an arbitrary solubility

threshold, and an ellipsoidal fitting model imposes a symmetry on the intermolecular forces that is rarely reflected by the experimental data. To overcome these issues, an approach that makes use of accurate solubility data to describe a three-dimensional solubility function, f, is introduced.

The principles of the approach are discussed in detail and the procedures for 1 ACS Paragon Plus Environment


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constructing the solubility function and computing solubility parameters are described. An example using PCBM solubility data available in the literature demonstrates the new method. Lastly, a method that employs f as a predictor of solubility in arbitrary solvents with a proposed measure of reliability is presented.

1. INTRODUCTION Determining the solubility characteristics of small molecules, nanoparticles, and polymers is an important endeavor in many areas, such as bio-therapeutics, pharmaceuticals, materials sciences, petrochemistry, cosmetics, textiles, and the formulations and coatings industry. For instance, one of the most promising hallmarks of polymer-based materials is low-cost, largescale, solution processable manufacturing. A primary requirement for this enterprise is knowledge of the solvents that ensure the solubility of individual components and the mutual solubility and miscibility of constituents in material blends, e.g., bulk heterojunctions. Only after this hurdle is cleared can problems associated with the impact of solvents on the solid-state morphology, material efficiencies, and device performance can be addressed. Solubility parameters, which are a basic physicochemical property of a substance, have played a critical role in the screening and selection of solvents.1-7 Additionally, there are numerous reports in the literature that highlight and discuss the correspondence between solubility parameter theory and several other well-established theories of polymer thermodynamics.8-15 However, as the development of materials with much more complex chemical functionalities and architectures continues to rapidly advance, procedures for more accurate and reliable methods of judicious solvent selection will prove beneficial for a host of practical applications. With respect to more fundamental science, improved correlations between solubility parameter methods and thermodynamic and molecular theories will contribute to our 2 ACS Paragon Plus Environment

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understanding of significant physicochemical system properties, equilibria, and the development of quantitative structure-property relationships. For a common solvent, solubility parameters are measured quantities that can be obtained via well-established experimental methods involving correlations with other physical quantities of the solvent, such as the enthalpy of vaporization, boiling point, refractive index, dipole moment, surface tension, and internal pressure.1-5, 7 However, for a solute these parameters must typically be estimated using an experimental procedure involving several solvents with known parameters. While this approach has been applied in a multitude of polymer and material settings, current methods for estimating the solubility parameters of a solute suffer from theoretical and computational deficiencies that have led to many questions and investigations that seek improvement in solubility parameter theory.13-14, 16-28 In this work a new approach to estimating the solubility parameters of a solute is presented. The approach is straightforward, flexible, mathematically oriented, and addresses several deficiencies present in current methods for estimating solubility parameters of a solute. By introducing a solubility function that describes the behavior of the solute in the space of possible solvents, and taking the solubility parameters of the solute as a measure of the central tendency of the solubility function, the functional approach gives a more natural representation of solutesolvent interactions. This approach also yields a means for predicting solvent affinity with some degree of reliability, and analysis of the solubility function may lead to a greater understanding of the physicochemical interactions that govern solvency. 1.1 Hildebrand and Hansen Solubility Theory

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The concept of the solubility parameter was first proposed by Hildebrand,16,

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18, 29

which

defined a single solubility parameter δT in terms of the total cohesive energy density of a solvent E/Vm, where E is the energy of vaporization and Vm is the molar volume of the solvent, δT =

E . Vm

(1)

The parameter δT is a measure of the energy required to separate solvent molecules from one another and provided a simple qualitative measure of solubility behavior based on the principle of “like-dissolves-like.” Hansen improved the theory by partitioning the total internal cohesive solvent forces (δT) into dispersion (δD), permanent dipole (δP), and hydrogen bonding (δH) components.1-4,

7, 30

In Hansen theory, (1) the δD parameter accounts for non-specific

intermolecular interactions related to dispersion forces,

(2) polar interactions attributed to

permanent dipole-permanent dipole forces are registered by the δP parameter, and (3) the δH, or “hydrogen-bonding," parameter is left to account for all of the remaining specific intermolecular interactions such as localized interactions involving specific orbitals, charge-transfer interactions, π-π electron interactions, acid-base interactions, hydrogen bonding, and other complex forming interactions. The solubility parameters satisfy the relationship,

δT = δD2 + δP2 + δH2

(2).

1.2 Hansen Spheres and Hansen Solubility Parameters The three parameters δD, δP, and δH form a three-dimensional space in which scaled values of each of the parameters define an orthogonal set of axes. The guiding principle of Hansen solubility theory is that the set of “good” solvents for a particular solute reside in a sphere (ellipsoid for unscaled axes, will be henceforth referred to as the Hansen Sphere) in the δDδPδHspace, and “poor” solvents reside outside the sphere.3

To employ the Hansen theory for finding


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the solubility parameters of a solute, experimental observations of how well the solute dissolves in several solvents with known δD,δP, and δH values are conducted. Subsequently the solvents are classified as either “good” or “poor” based upon qualitative or quantitative measures, and the Hansen Sphere should be constructed so that as many of the good solvents lie in the sphere and poor solvents lie outside the sphere as possible. Under this construction, the Hansen Solubility Parameters (HSP) of the solute are taken to be the center of the sphere “of best fit.” In other words, the solubility region of the solute in the scaled δDδPδH-space is spherical and consists of all points/solvents (δD,δP,δH) the δDδPδH-space satisfying,

a (δD – δD ) + b(δP – δP ) + c(δH – δH ) ≤ R02 (3) 2

2

2

where R0 is the radius of the sphere, (δD , δP , δH ) are the coordinates of the center of the sphere (and thus are taken to be the HSP of the solute), and a, b, and c are scaling factors. The solubility distance, Ra, between a solvent located at coordinates (δD,δP,δH) and the solute is simply

Ra = a(δD – δD ) + b(δP – δP ) + c(δH – δH ) 2

2

2

(4)

and the solvent is considered to be “good” provided the relative energy distance

RED = Ra / R0 < 1 .

(5)

While the Hansen Sphere/HSP theory originated in the 1960s, it has undergone little refinement despite several deficiencies that stem from the initial development of solubility parameter theory. Most endeavors have been aimed at developing or defining new solubility parameters, or improving the approach used to compute solubility parameters of a solute.13-15, 2128, 31-33

With respect to the latter issue, most recent efforts still preserve the conventional

practices of binary solvent classification and spherical fitting algorithms, and are primarily



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directed at the development of more sophisticated numerical algorithms for computing the center and radius of the Hansen sphere, including optimization strategies, genetic algorithms, nonlinear programming approaches, and even artificial neural networks.25-28, 34 Further innovations in computing Hansen Spheres/HSPs are likely, but in the overall perspective of trying to find the “central tendency” of the solute-solvent interactions, none of the aforementioned approaches address the following issues inherent in the Hansen Sphere/HSP approach: •

Nonuniqueness/Ill-posedness: It is a well-known mathematical fact that, given a set of points in three-dimensional space, there are infinitely many spheres/ellipsoids that contain those points. Additionally, the mathematical problem of finding a sphere that encloses one group of points while excluding another group of points is seldom a well-posed problem. Thus all methods to find a Hansen Sphere are not deterministic and typically rely on finding the sphere of “best fit” subject to varying conditions or constraints. This means that an entire manifold of HSPs can be derived from a single set of experimental data.

•

Inconsistency with experimental observations: When the sphere problem is illposed, the Hansen Sphere will contain one or more poor solvents, exclude one or more good solvents, or possibly both, which disagrees with experimental observations.26, 33-39

•

Inconsistency of center/radius computations: As a variety of methods for fitting the Hansen Sphere have been utilized, different types of algorithms can arrive at (sometimes dramatic) different HSP/ܴ଴ results for the same set of observed data.25-26, 28, 36, 40

For example, independent calculations by Vebber et al.28 and Gharagheizi25


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for the δD,δP, and δH parameters of polyethersulfone (PES) vary individually by as much as 20%, and there is a 24% discrepancy in the computed radius of each Hansen sphere despite using the exact same data set. •

Isotropy:

The Hansen Sphere/HSP theory assumes that the intermolecular

interactions and thermodynamics related to the dispersive, polar, and hydrogenbonding parameters equally influence the solubility behavior. The hypothesis of isotropic solvent behavior in these three directions has been questioned by several authors and has been repeatedly demonstrated to be erroneous.9,

36-37, 41-42

Rapid

changes in solvent affinity can occur differently in different directions and there may be two solvents whose coordinates are the same distance from the HSP of the solute but one is a good solvent while the other is poor. •

Prediction of nonphysical behavior: It is physically impossible for any of the δD,δP, and δH values to be negative. However, it is very common for Hansen theory to predict a HSP whose values are smaller than the radius, R0, implying that part of the solubility region lies outside the realm of physical possibility.28, 31, 33-35, 38-40, 43-46

•

Scaling factors: Through experimentation, Hansen empirically determined that the correct scaling factors in eq 3 should be a = 4, b = 1, and c = 1Several subsequent studies have questioned the validity of these values and have proposed alternate values.36, 47

•

Binary classification of solvents: While experimental procedures employed to gather the solubility data vary, in many cases the dissolved concentration of the solute is found. This quantitative data must then be converted to a binary classification of good/poor, which neglects the magnitude of the solvent's affinity for the solute.



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Often the concentration threshold is set arbitrarily, and some studies indicate that a single data set can produce widely varying HSP/R0 results for different threshold values.43-44 1.3 Convex Solubility Parameters Convex Solubility Parameters (CSP) were introduced to address several issues with the Hansen Sphere/HSP theory.41 Based on the principle that an additive (convex) combination of good solvents should also be a good solvent, the solubility region is taken to be the convex hull of all experimentally-observed good solvents. This construction does not assume isotropic solvent behavior and guarantees that (a) the solubility region is unique and a well-posed mathematical problem, (b) all of the solubility region is physically reasonable, and (c) independent of the scaling of the axes. The CSP is taken to be the center of mass of the convex solubility region, treating it as a solid with uniform density. As the center of mass of a convex body is always contained within the body, this guarantees that the CSP is physically reasonable and represents an additive combination of experimentally observed good solvents. Figure 1 gives an example of the difference between HSP and CSP calculations. The polymer in question is lignin and the experimental solubility (good/poor) was determined using a test bed of 82 solvents, 16 of which were deemed as good solvents.28, 41 Note that the calculated HSP of (21.7, 14.2, 16.9) of lignin is significantly removed from all good solvents - in fact, the minimum Ra from the lignin HSP to any of the 16 good solvents is 9.7 MPa1/2, while the computed CSP of (17.8, 11.0, 14.4) is within 9.3 MPa1/2 of 94% (15/16) good solvents. The convex solubility region also better reflects the varying influence that the parameters δD,δP, and

δH have one the solubility of lignin.


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Figure 1. Hansen Sphere/HSP and convex hull/CSP for lignin based on binary classification of 82 (16 good/66 poor) solvents.28, 41 (a) Three-dimensional view and (b) projection onto the δDδPplane. While the convex hull/CSP approach addresses almost all of the issues inherent to the Hansen Sphere/HSP approach, it still does not account for varying degrees of solvent affinity (i.e., all good solvents are weighted equally) and in the presence of quantitative solubility data, an arbitrary good/bad solvent threshold must be set. As long as the threshold is greater than 0 mg/mL, useful data regarding the moderate affinity of some solvents is essentially discarded in the process of computing the CSP.

This deficiency is addressed with the new functional

approach as described in the Methods section.



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1.4 Other Approaches to Computing Solubility Parameters In studying the dispersions of single-walled nanotubes in several solvents, Coleman et al. presented an approach to computing solubility parameters that used the average of all coordinates of good solvents weighted by the dissolved concentrations.31-32 While this approach emphasizes the importance of relative solvent affinity in the solubility parameter calculations, the method is susceptible to influence by the number of observations in particular regions of the solubility space. This will be discussed more in the Methods section.

2. METHODS 2.1 The Functional Approach The method presented here frames the solubility parameter problem in a more mathematical context by considering the solubility of a solute in different solvents to behave in a function-like manner - at every point in the physically feasible parameter space, the solute should dissolve into a solvent located at that point with some (possibly zero) concentration in mg/mL. This extends the main ideas behind the convex hull/CSP approach by incorporating quantitative solubility data.

The new approach to computing solubility parameters, termed

functional solubility parameters (FSP), utilizes such a function defined on a subset of the

δDδPδH--space whose output gives the dissolved concentration of the solute (in mg/mL) in a solvent with coordinates (δD, δP, δH). This function is assumed to be continuous everywhere it is defined, and the FSP is then taken to be the center of mass of the parameter space when employing the function as a weighting function (density) - the “central tendency” of the function's behavior. The solubility function can be approximated using observed concentrations of the solute in a small or moderate number of solvents. The domain of the approximate function


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is taken to be the convex hull of all solvents with affinity for the solute, and the approximate FSP can be calculated using numerical integration. This approach thus overcomes the limitations of the HSP and CSP approaches and gives a solubility parameter much more consistent with experimentally observed data. It should be noted that if a binary classification for solubility is used, then the functional solubility approach reduces to the convex hull approach. An example that illustrates the main differences between the HSP, CSP, and the new FSP approaches is presented below. For ease of presentation, the example is limited to a hypothetical two-parameter (δa,δb) system. Suppose a polymer has been dissolved in several solvents, each with different solubility parameters (coordinates) in the parameter space. The location of the solvents along with their concentration of the dissolved polymer (given in mg/mL) are given in Figure 2a. To use either the HSP or CSP approach for computing solubility parameters, it becomes necessary to bin the solvents into two categories - good solvents and poor solvents. Such cutoffs may be established via a natural separation in the concentrations, or perhaps with a target concentration in mind, however they may often be chosen arbitrarily. Given the data in Figure 2, a solubility threshold of 1.0 mg/mL divides the solvents into two groups: four good solvents and six poor solvents.



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Figure 2. Solvents with dissolved polymer concentrations in parameter space (a). Domain Ω of the solubility function f given by the Delaunay triangulation of the convex hull of all points with nonzero concentration (b). Contour plot of f, the continuous piecewise linear interpolant of the solubility data (c). For solvents with a concentration of at least 1.0 mg/mL, Hansen solubility region (dotted circle), convex solubility region (dashed quadrilateral), and computed HSP, CSP, and FSP (d).

The Hansen “Sphere” (circle in two dimensions) is obtained by finding the circle of best fit (in this case simply the bounding circle of the good solvents of minimum radius), and the HSP is taken to be the center of the circle (details of this and subsequent computations can be found in the Supplemental Information). The circle is shown in Figure 2d with a dotted line and the HSP is indicated. Noting that the good solvents lie on or very close to the boundary of the solubility region while a “poor” solvent is contained within the region, this example illustrates that it is possible 12 ACS Paragon Plus Environment

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for the computed HSP to actually be closer to a poor solvent than any good solvent. Again, it should be emphasized that different algorithms may produce different Hansen spheres, depending on the statistical measures used to quantify “best fit.” The convex solubility region is constructed by taking the convex hull of the four good solvents, creating the dashed quadrilateral in Figure 2d. The CSP is taken to be the center of mass of the convex hull, treating it as a solid with constant density. While this construction of the solubility region always guarantees that the entirety of the region is physically reasonable, and that the CSP will always be contained within the solubility region, as with the Hansen approach, the relative efficacy of the good solvents is ignored. It is clear that the CSP may be closer to solvents with lower dissolved concentrations, as the CSP is closer to solvents with dissolved concentrations of 1.3 mg/mL and 1.1 mg/mL than it is to solvents with dissolved concentrations of 1.6 mg/mL and 3.2 mg/mL. The functional solubility parameter (FSP) method proceeds by taking the solubility region Ω to be the convex hull of all solvents with affinity for the solute and subsequently partitioning Ω into simplices (triangles in 2-D, tetrahedra in 3-D), as shown in Figure 2b. The solubility function f is taken to be the continuous piecewise linear interpolant of the dissolved polymer concentration data, as shown in Figure 2c. The functional solubility parameters (FSP) are taken to be the center of mass of Ω weighted by f - essentially treating f as a function that describes the “density” of the solvent affinity in Ω. This natural representation of the solubility behavior of the solute ensures that all observed solvents with any affinity for the solute are taken into account, and the better solvents weigh more heavily in the solubility parameter calculation. This approach has several advantages to both the Hansen Sphere/HSP and Convex Solubility Region/CSP methods. The primary advantage is that all (good) solvents are not treated equally,



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instead solvents with the highest affinity for the solute are weighted more heavily in the calculation of the central tendency of solvent behavior. Several other advantages and insights into the physicochemical behavior of the solvent-solute interactions include: •

The use of a function (only a single output for any given input) is the most mathematically accurate representation of the nature of solvent-solute interactions, and is implied by the reproducibility of experimental results, i.e., one should expect the dissolved concentration of the solute in any given solvent to always be the same.

•

Requiring continuity of the solubility function accurately represents the true nature of solubility of a solute in a collection of nearby solvents. Mathematically, continuity implies that small (infinitesimal) changes in input (solubility parameters of the solvent) should result in small changes in the output (dissolved concentration of the solute). In other words, a very small change in one of the solubility parameters of the solvent should not result in a discontinuous jump in the solubility of the solute. Continuity does not preclude steep changes in solubility behavior over an interval, it merely guarantees that the solubility does not jump up or down in a discontinuous fashion. This is in contrast to both the Hansen Sphere and the Convex Solubility region, which presume “good” solvent affinity while in the region and truncates to “bad” solvent affinity outside of the region.

•

As the domain Ω of the solubility function f is the convex hull of all solvents demonstrating any affinity for the solute (obtained via experimentation), it can be used to estimate the dissolved concentration of the solute in any solvent found inside

Ω. This aspect of the FSP approach gives much greater information than either the


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Hansen or Convex approaches, which merely indicate if a solvent should be “good” or “poor” based on its coordinates in the parameter space. •

The sensitivity of the solute to the different solubility parameters δD,δP, δH is implicitly encoded in the solubility function f via partial derivatives of f with respect to the each parameter. Indeed, the gradient, ∇f, at any point gives the direction of maximum increase of solvent affinity. The continuous piecewise linear construction of f means that ∇f is easy to compute on any tetrahedron in the triangulation of the solubility region Ω.

We remark that the solubility function f described above is only an approximation (the continuous piecewise linear interpolant) to the exact function that represents the true solubility of the solute at any point in the parameter space. More on the quality and reliability of f as an approximation to the true solubility function will be discussed in the context of the application of the method to find the solubility parameters of PCBM. The method for constructing the solubility function and computing the functional solubility parameters of the solute requires the following steps: 1. Gather sufficiently many experimental data points via dissolving the solute in a variety of solvents. 2. Construct the solubility function f by first defining the domain Ω of f and then interpolating the experimental data with continuous piecewise linear basis functions. 3. Compute the functional solubility parameters by finding the center of mass of Ω treated as a solid object with density f via numerical integration. Several of the computational procedures that will be discussed in steps 2 and 3 above are utilized in multiple mathematical contexts. Indeed, the numerical solution of partial differential 15 ACS Paragon Plus Environment


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equations via the finite element method generally requires all of these algorithms with the objective of approximating the solution to a partial differential equation.48 Thus many software packages that implement the finite element method can be utilized to perform some of the computations required for finding the FSP of a solute. The mathematical and computational details of the construction of the solubility function and the computation of the FSP are provided in the Supplementary Information.

2.2 Gathering Solubility Data In practice, a comprehensive set of solvents with disparate solubility parameters should be used to obtain a meaningful set of solubility data. In a typical experiment, the polymer solubility is determined by adding a known amount of polymer to a specific volume of solvent. Each solution is vigorously stirred for several hours or days at the desired temperature to ensure complete dissolution. The solutions are visibly monitored and additional polymer may be added as needed to achieve the solubility limit. For many polymers the preparation of the solutions can be quite tedious because care must be taken to avoid gelation. The final solutions are usually centrifuged or filtered to remove excess, undissolved polymer. The solubility behavior in each solvent can then be examined using a convenient spectroscopic or analytical technique that is well-suited for the polymer under consideration, or, within the context of the binary classification, each solution may simply be visually inspected by a trained observer and the solvent categorized as “good” (high solubility) or “poor” (low solubility).

The functional

approach requires accurate measurements of the dissolved concentration of polymer in each solvent; thus, visual inspection is not an appropriate method for assessing solubility behavior. The data gathered from the experimental procedure for FSP computations consists of a list of


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solvents xi, i = 1,..., n (n > 4), each with coordinates xi = (δD,i,δP,i,δH,i) in the parameter space, and the quantitative dissolved concentration yi of the solute in solvent xi.

2.3 Comparison with the Weighted Average Method As mentioned in the Background section, Coleman et al. determined that a better approach to computing the solubility parameters of graphene dispersions was to utilize a “weighted average” approach, which incorporates the dissolved concentrations into the computation of the solubility parameters via the formula,

∑ c (δ ) = ∑c i

δj

j i

i

, (6)

i

i

Where j ∈ {D, P, H} and ci is the dissolved concentration of the polymer in the ith solvent.31-32 Essentially, this approach treats the solubility data as a system of point masses where the mass of each point is given by, and the solubility parameter is taken to be the center of mass of this system of point masses. While this approach does succeed in weighting solvents with higher affinity over those with lower affinity, the result of these computations are susceptible to influence by the number of observations (solvents) in different regions of the parameter space.

A large number of

moderately-weighted data points (point masses) in a small region of space can “pull” the center of mass away from data points in other regions with larger weights. In other words, given an outcome derived from a particular set of observations, additional observations that do not add any “new” information should have minimal, if any, effect on the outcome. This problem can manifest in solubility parameter computations as many of the



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commonly used solvents available for empirical testing can cluster in small regions of the parameter space. A straightforward example will illustrate this. In Figure 3a, suppose you have a simplified two-parameter system with four point masses at the vertices of a square, with one of the vertices having a dissolved solute concentration of 20 mg/mL (mass of 20), while the other three vertices have a concentration of 10 mg/mL.

Figure 3. Illustration of a weighted average approach vs. functional approach. The center of mass of a system of four point masses (a) is significantly different than the center of mass of a system of six point masses (b) while the center of mass computed using the functional approach is the same to two decimal places in (c) and (d).

A weighted average/point mass approach would compute the center of mass of this system as (0.6, 0.6), as the larger mass naturally contributes more toward the overall mass of the system. (details of this and subsequent computations can be found in the Supplementary Information). 18 ACS Paragon Plus Environment

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This outcome also makes sense in treating the concentrations as masses, wherein one would expect the solubility parameter of a polymer with this solubility data to be closer to the solvent located at (1,1) than the other three solvents as it exhibits a higher affinity for the solute. In Figure 3b, two points with concentration 10 mg/mL have been added to the system. This moves the weighted average of the system to (0.5, 0.5), a natural result when considering the data as a system of point masses. However, when considering the coordinates and weights as solubility data, the two new observations lend no new information - the dissolved concentrations that have been added at those locations are exactly what one would expect under normal assumptions (e.g., a solvent posited halfway between two solvents with concentrations of 10 mg/mL should also have a concentration of 10 mg/mL). Therefore, the two new observations should not have any influence on solubility parameter computations - the new observations are not enough information to move the solubility parameter away from the solvent with highest affinity, and certainly not enough information to reduce both coordinates by over 16%. However, note that in Figure 3c and Figure 3d, the function ݂ that interpolates the mass/concentration data is mostly the same. Treating the square domain ߗ as a lamina with variable density ݂ , and computing its center of mass gives the same solubility parameter (to two decimal places) of (0.54, 0.54) for both cases. The conclusion that can be drawn from this example is that utilizing a weighted average/point mass approach to solubility parameter computations can skew the result toward regions in the parameter space with a large number of observations, without taking into full account the quantitative value of solvents with high affinity and fewer neighbors.

This

disadvantage is overcome by instead treating the solubility region as a continuum in the parameter space and the concentration data as the density of a solid occupying the solubility



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region.

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Then center of mass of the solid with variable density will be a more consistent

measurement of where the “center” of a polymer's solubility behavior lies.

3. RESULTS AND DISCUSSION 3.1. Computation of FSP for PCBM PCBM, [6,6]-phenyl-C61-butyric acid methyl ester, is a fullerene derivative that has emerged as a benchmark electron acceptor in organic photovoltaic (OPV) materials. Typically, PCBM is combined with an electron donor polymer moiety in solution and the mixture is processed to form functional optoelectronic and photovoltaic materials. Thus, the solubility properties of PCBM are vital for the processing and optimization of these materials. Solubility data for PCBM, which was taken from Machui et al.,43 is presented in Table 1. For the purposes of this study, all concentrations in Table 1 reported as less than 0.1 mg/mL were taken to be 0 mg/mL, and the solubility in bromoform was taken to be 165 mg/mL.


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Table 1. PCBM solubility data from Machui et al.43 Solvent Heteroaromatics tetrahydrothiophene tetrahydrofuran Halogenated aromatics bromobenzene chlorobenzene 1,2-dichlorobenzene 2-chlorotoluene 1,2,4-trichlorobenzene 3-chlorotoluene Halogenated alkanes chloroform bromoform Arenes o-xylene styrene tetrahydronaphthalene toluene α-methylstyrene benzene quinoline Green solvents β-pinene α-pinene limonene oleic acid Other aromatics 2-chlorophenol benzaldehyde benzyl benzoate aniline Alcohols cyclohexanol 1-butanol 2-propanol benzyl alcohol methanol ethanol Polar solvents cyclohexanone acetone dimethylsulfoxide water Other cyclohexane

δD[MPa1/2 ]

δP[MPa1/2 ]

δH[MPa1/2 ]

C[mg/mL]

18.6 16.8

6.7 5.7

6 8

6.7 1.8

19.2 19 18.3 19 20.2 18.9

5.5 4.3 7.7 4.9 4.2 3.9

4.1 2 2.8 2.3 3.2 2.9

30.8 59.5 42.1 65.3 81.4 48.1

17.8 20

3.1 5

5.7 7

28.8 >165

17.8 18.6 19.6 18 18.5 18.4 20.5

1 1 2 1.4 2.4 0 5.6

3.1 4.1 2.9 2 2.4 2 5.7

22.1 29 114.8 15.6 40.7 16 54.8

17.1 17.3 17.2 16

3 2.4 1.8 2.8

2.7 3.1 4.3 6.2

2.8 1.6 7.4 1.8

19 19.4 20 20.1

5.5 7.4 5.1 5.8

13.9 5.3 5.2 11.2

112 20.8 8.8 16.5

17.4 16 15.8 18.4 14.7 15.8

4.1 5.7 6.1 6.3 12.3 8.8

13.5 15.8 16.4 13.7 22.3 19.4

3.1

A Functional Approach to Solubility Parameter Computations

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