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A Quantitative Approach to Polymer Solubility ParametersMiranda Roesing and David Boucher, Department of Chemistry and Biochemistry, College of Charleston

Jason Howell, Department of Mathematics, College of Charleston

Solubility Parameters

• Solubility parameters are three intermolecular forcesthat govern solubility in Hansen’s model

•”

d

Dispersion forces: the attraction between allcompounds based on instantanious changes inelectron density

•”

p

Dipole-Dipole Forces: the attraction/repulsionbased on permanent dipole

•”

h

Hydrogen Bonding: the attractive forces based onhydrogen bonding

• Objective: move beyond binary classification ofsolvents to an approach that more accuratelyrepresents quantitative solvent behavior

• Method: gather solubility data and construct asolubility function f (”

d

, ”

p

, ”

h

) that represents theconcentration of the polymer when dissolved in asolvent with parameters (”

d

, ”

p

, ”

h

)

Introduction

• Solution: solute dissolved in solvent• Based on the concept of “like dissolves like” - e.g., polar

solvents dissolve polar solutes• Additive combinations of good solvents should also be

good solvents• Hildebrand model: solubility is based on

thermodynamics• Hansen model: solubility is based on intermolecular

forces

Hansen Solubility Parameters [1]

• Difference between solubility parameters of solvent andsolute give “solubility distance” R

a

:R

2a

= 4(”d,1 ≠ ”

d,2)2 + (”p,1 ≠ ”

p,2)2 + (”h,1 ≠ ”

h,2)2

• HSP taken to be center of sphere• Better solvents are closer to center• Issues:

• Spheres can be largely extrapolated - overestimates solubilityregion

• Portions of sphere outside physical parameter range• Hydrogen Bonding is a sort of “catch all” for every force not

listed above• Good solvents may lie outside sphere and poor solvents may be

inside sphere

Convex Solubility Parameters [2]

• Convex hull: smallest region in space that contains agiven set of points and all line segments between thosepoints

• Solubility region of polymer defined by the convex hullQ of all good solvents x1, . . . , x

t

:

Q =8<

:x

����� x =tX

i=1⁄

i

x

i

,

tX

i=1⁄

i

= 1, ⁄

i

Ø 0, i = 1, . . . , t

9=

;

• CSP taken to be center of mass of convex solubilityregion (assuming uniform density)

• Determining if a solvent is inside the hull can be doneby solving a linear feasibility problem:

find q

such that q =tX

i=1⁄

i

x

i

,

tX

i=1⁄

i

= 1, ⁄

i

Ø 0, i = 1, . . . , t

Motivation

Existing approaches for computing solubility parameters of polymers employ a binary classification of solvents - eithera solvent is “good” or “bad.” Determination of good/bad solvents is usually done experimentally via a “trained eye”or setting an arbitrary cutoff/threshold of polymer concentration in the solvent. Both HSP and CSP approaches treatall good solvents equally, regardless of how much of the polymer they can actually dissolve.

This work is the first step towards computation of solubility parameters that incorporates quantitative behavior ofsolvents and more accurately reflects the location of the polymer in the solubility parameter space.

Gathering Solubility Data for P3HT

• Start with 29 pure solvents• Poly(3-hexylthiophene) (P3HT) polymer added to solvent• Use centrifugation to isolate saturated supernatant solution• Measure UV/Vis absorbance spectra, A vs. ⁄

• Use Beer’s Law, A = Á¸C to obtain P3HT concentration C

Most Effective SolventsSolvent ”

d

”

p

”

h

Conc. (mg/mL)Chloroform 17.8 3.1 5.7 38.002-chlorotoluene 19.0 4.9 2.3 18.00Toluene 18.0 1.4 2.0 7.54Tetrahydrofuran 16.8 5.7 8.0 6.11p-xylene 17.6 1.0 3.1 3.34Dichloromethane 18.2 6.3 6.1 0.82Cyclohexylbenzene 18.7 0.0 1.0 0.602-chlorophenol 20.3 5.5 13.9 0.47

Definition of Solubility Function f and Functional Solubility Parameter (FSP)

• �: region in parameter space formed by convex hull of all solvents tested•

f : � æ R3, f (x) = f (”d

, ”

p

, ”

h

) = y concentration of P3HT in solvent with parameters (”d

, ”

p

, ”

h

)•

f is constructed using continuous piecewise linear interpolant of experimental data via first-order Lagrangeinterpolating polynomials {„

i

}, i = 1, . . . , N on Delaunay triangulation T of �: pure solvent coordinates at node x

i

withconcentration y

i

:

„

i

(xj

) =8<

:y

i

if i = j

0 if i ”= j

, f(x) =NX

i=1„

i

(x)

• Evaluation of f at any point x in � is simply the proportional average of the 4 vertices of the tetrahedron containing x

• FSP (”̄d

, ”̄

p

, ”̄

h

) computed as center of mass of � treated as a solid with density f :

V =ZZZ

�f d�, ”̄

d

= 1V

ZZZ”

d

f d�, ”̄

p

= 1V

ZZZ”

p

f d�, ”̄

h

= 1V

ZZZ”

h

f d�• Integrals computed numerically via Gaussian quadrature rules at points x

q

with weights w

q

, q = 1, . . . , m:ZZZ

�f d� =

X

KœT

✓ZZZ

K

f dK

◆¥

X

KœT

0

@mX

q=1w

q

f (xq

)1

A =X

KœT

0

@mX

q=1w

q

0

@nX

i=1„

i

(xq

)1

A

1

A

P3HT Results

Method Cutoff (mg/mL) ”

d

”

p

”

h

HSP [3] 5 17.8 5.6 4.02 18.3 4.3 5.2

0.5 18.4 3.7 6.3CSP 5 17.9 3.8 4.5

2 17.8 3.6 4.50.5 18.0 3.4 4.20.1 18.6 4.0 6.3

FSP 17.8 4.6 5.2

Visualization of f

Conclusion

• The functional solubility approach employs quantitativesolubility data to produce a more accurate solubilityparameter.

• No need to set arbitrary concentration threshold.• Construction of solubility function f allows for

prediction of solute concentration for arbitrary solventwith parameters in �.

• Computation of FSP is accomplished through numericalintegration of linear polynomials.

• Concept is easily generalized to include moreparameters.

References

[1] C. M. Hansen. Hansen Solubility Parameters: A User’s Handbook, Second Edition, CRC Press, 2007.

[2] J. S. Howell, B. O. Stephens, and D. S. Boucher. Convex solubility parameters for polymers. J. Polym.Sci. Part B: Polym. Phys., 53(16), 2015, 1089-1097.

[3] F. Machui, S. Langner, X. Zhu, S. Abbott, C. J. Brabec. Determination of the P3HT:PCBM solubilityparameters via a binary solvent gradient method: Impact of solubility on the photovoltaic performance,Solar Energy Materials and Solar Cells, 100, 2012, 138-146.

Acknowledgements

The authors acknowledge the financial support of the College of Charleston Office of UndergraduateResearch and Creative Activities (MAYS Grant No. MA2015-006) and the Howard Hughes MedicalInstitute, Pre-College and Undergraduate Science Education Program (HHMI Grant No. 52006290and 52007537).

Contact Information

Email: roesingmr@g.cofc.edu, boucherds@cofc.edu, howelljs@cofc.edu