Supporting Information
Ternary phase diagram and fiber morphology for
nonsolvent/DMAc/polyamic-acid systems
Chaoqing Yin, Jie Dong, Zhentao Li, Zixin Zhang, Qinghua Zhang*
State Key Laboratory for Modification of Chemical Fibers and Polymer Materials, College of
Materials Science and Engineering, Donghua University, Shanghai 201620, People’s Republic of China
*Corresponding author: Q. Zhang, Tel/Fax: 0086-21-67792854. E-mail: qhzhang@dhu.edu.cn
THEORY
1. Thermodynamics of ternary systems
Tompa[1] is the first person who applied Flory-Huggins model to analyze nonsolvent/solvent/polymer
ternary systems. According to this theory, Gibbs free energy of mixing, ∆Gm, can be expressed as follows:
The subscripts refer to nonsolvent (1), solvent (2) and polymer (3), ni and i are the number of moles
and the volume fraction of component i, respectively. R and T are the gas constant and absolute temperature,
respectively. gij is a concentration-dependent binary interaction parameter between components i and j.
However, in most ternary systems, independence of concentration is taken into consideration for the
thermodynamic evaluations of nonsolvent/polymer and solvent/polymer interaction parameters, g13 and g23.
Therefore, in the following equations, χ13 and χ23 are used to replace g13 and g23. The solvent-nonsolvent
parameter, g12, is assumed to be a function of 2, with 2= 2/( 1+ 2) by Pouchlý et al.[2].
1.1. Binodal curve
The chemical potential of a species in a mixture can be defined as the slope of the Gibbs free energy of
the system with respect to a change in the number of moles of just that species. Thus, the chemical potential
of component i can be expressed as follows:
Based on the expression of ∆Gm (Eq. 1) and the definition of chemical potential (Eq. 2), the following
three equations are derived for the chemical potentials of the components in the mixture:
Where, ∆μi is the difference between the chemical potential of component i and pure state, and vi is the
molar volume of component i.
The condition for liquid-liquid equilibrium is:
Where, the subscripts denote dilute phase (A) and concentrated phase (B), respectively. According to mass
conservation law, it obeys the following equation:
Tie lines connect the pair of equilibrium compositions in the polymer-rich phase (1,A, 2,A, 3,A) and
polymer-poor phase (1,B, 2,B, 3,B). It should be noted that only binary interaction parameters are
considered. To specify the tie line compositions, these six unknowns can be obtained. Substituting Eqs.
(3)-(5) in Eq. (6) provide three equations, and the other two equations are given by Eq. (7). There are five
relations among the six unknown compositions, the polymer composition in the polymer-poor phase, 3,B,
is set as an independent variable, and the Newton-Raphson method can be used to solve the remaining
system.
1.2. Spinodal curve
A curve that separates a meta-stable region from an unstable region in the coexistence region of a
binary fluid is the so-called spinodal curve, which is thermodynamically defined as:
The equation for the spinodal is:
Where
, therefore, the relationship for ∆Gm obeys the following:
Also, the components of ternary system obey the volume conservation law (Eq. 13). When the
interaction parameters are available, substitution of Eqs. (10)-(12) in Eq. (9) along with Eq. (13) results in
two equations with three variables, which can be solved numerical by choosing one of the variables (in our
case 3) as the independent variable. This is done by using the same numerical procedure as for the binodal.
1.3. Critical point
The critical point where the binodal and spinodal curves intersect can be expressed as[1]:
The critical point composition meets the following equation, and it can be solved by the similar
procedure used for spinodal:
2. Evaluation of the binary interaction parameters
2.1. Nonsolvent/solvent interaction parameter
The nonsolvent-solvent interaction parameter, g12, is determined from the excess Gibbs free energy
(∆GE) data using the following equations[3]:
Where ni and i are the number of moles and the volume fraction of component i, respectively. By Eq. (17),
g12 can be calculated as a function of 2, and ∆GE can be directly obtained from experimental data. But a
few solvent/nonsolvent systems have experimental data on ∆GE available. ∆GE can be obtainde from
activity coefficients data, which can be calculated based on Universal Quasichemical Functional Group
Activity Coefficient (UNIFAC) model[4-8].
The original UNIFAC model, which combines the functional group concept with a model for activity
coefficients based on an extension of the quasi-chemical theory of liquid mixtures, was proposed by
Fredenslund et al. in 1975.[8] The activity coefficient is expressed as a function of composition and
temperature, it is calculated as a sum of the combinatorial and residual parts:
ln γi = ln γi (c) + ln γi (R)
(18)
Expressions of combinatorial part are as follows:
ln γi (c) = ln
ϕi
θi
ϕi
+ 5q i ln + li − ∑ xj lj
xi
ϕi
xi
(19)
j
li = 5(ri − q i ) − (ri − 1)
(20)
Where subscripts i and j denote the components, k is the group, i and θi are the volume fraction and
area fraction of component i, respectively, and xi is the mole fraction of component i. Parameters of ri and
qi are relative to molecular van der Waals volumes and molecular surface areas, respectively, and they are
calculated as the sum of the group volume and group area parameter, Rk and Qk, as Eq. (22) shows.
θi =
x i qi
∑j xj qj
xi ri
∑j xj rj
(21)
q i = ∑k νk (i) Q k
(22)
ϕi =
ri = ∑k νk (i) R k
Where νk (i) is the number of groups of type k in molecule i. The group parameters Rk and Qk are
available in literatures.
Expressions of residual parts are as follows:
ln γi (R) = ∑ νk (i) [ln Γk − ln Γk (i) ]
(23)
k
Where Γk is the group residual activity coefficient, and Γk(i) is the residual activity coefficient of group
k in a reference solution containing only molecules of type i.
ln Γk = Q k [1 − ln (∑ θm ψmk ) − ∑
m
ln Γk
(i)
= Q k [1 − ln (∑ θm
m
(i)
ψmk ) − ∑
m
Qm Xm
θm =
∑n Q n Xn
Xm =
∑i νm (i) xi
∑i ∑k νk (i) xi
θm ψkm
]
∑n θn ψnm
m
θm
(i)
=
X m (i)
θm (i) ψkm
∑n θn (i) ψnm
Q m Xm (i)
∑n Q n Xn (i)
νm (i)
=
∑k νk (i)
(24)
]
(25)
(26)
(27)
ψmn = exp (−
amn
)
T
(28)
Xm is the fraction of group m in the mixture. ψmn is decided by amn, characterizing the interaction
between groups n and m. Ethanol was divided into three groups: -CH3, -CH2 and -OH; ethylene glycol was
divided into four groups: 2×-CH2 and 2×-OH, DMAc was divided into two groups: -CH3 and –CON(CH3)2.
For each group-group interaction, there are two parameter: amn≠anm, and they are also available in
literatures, as listed in Table S-1.
1
2
3
4
5
Table S-1. The interaction parameters anm between groups
n
H2O
-CH3
-CH2
-OH
-CON(CH3)2
m
H2O
0
300
300
-229.1
835.6
-CH3
1318
0
0
986.5
390.9
-CH2
1318
0
0
986.5
390.9
-OH
353.5
156.4
156.4
0
-382.7
-CON(CH3)2
-509.3
27.97
27.97
394.8
0
2.2. Solvent/polymer and nonsolvent/polymer interaction parameters
Solvent/polymer interaction parameter (χ23) can be determined by measuring the activity of the solvent
via a variety of techniques including light scattering[9], osmotic pressure[10], gas-liquid equilibrium[11]
and viscometric method[12]. The nonsolvent/polymer interaction parameter (χ13) can be evaluated via
simple equilibrium swelling method[13]. However, these methods are not quite suitable for systems in our
study due to the low vapor pressure of DMAc, high volatility of ethanol and easy degradation of PAA in
water. Therefore, it is a good way to calculate χ23 and χ13 via Hansen solubility parameter theory[14] (Eq.
29).
χ12 = α
V1
2
[(δd1 − δd2 )2 + 0.25(δp1 − δp2 ) + 0.25(δh1 − δh2 )2 ]
RT
(29)
Where δd, δp and δh are the dispersion force, polar force and hydrogen bond components of the solubility
parameter, respectively. In most cases, the results derived from an optimum value of 0.6 are better than an
assumed value of one[15].
The δd, δp and δh values of solvent and nonsolvent can be directly obtained in literatures[14] by
Lydersen (1955), and the corresponding values for polymers can be derived by Hoy’s method[16].
According to the structure of PAA (Scheme 1), it can be divided into the following groups, and the values
of the group contributions are available in literatures, as listed in Table S-2.
Table S-2. Values of group contributions
Vi/(cm3/mol)
△Ti（p）
59.5
13.2
0.019
173
63
7.2
0.013
2
1131
895
28.3
0.073
2
565
415
23.3
0.045
N
0.7
125
125
12.6
0.014
NH
0.7
368
368
11.0
0.031
O
0.3
235
216
6.5
0.018
Groups
Numbers
CH
13.3
249
C
11.4
CONH
COOH
The solubility parameters of DMAc, ethanol, water, EG and PAA, which are based on the results of
Hoy’s method for estimation, are listed in Table S-3.
Table S-3. The solubility parameters of DMAc, ethanol, water, ethylene glycol and PAA
Component
δd/(MPa1/2)
δp/(MPa1/2)
δh/(MPa1/2)
δ/(MPa1/2)
V(ml/mol)
DMAc
16.8
11.5
10.2
22.8
92.5
water
15.5
16.0
42.3
47.8
18.0
ethanol
15.8
8.8
19.4
26.5
58.5
EG
17.0
11.0
26.0
33.0
55.8
PAA
15.4
14.8
13.0
25.0
--
3. Evaluation of the binary interaction parameters
To better illustrate the relationship between experimental and calculated data in the ternary systems,
the
experimental
and
calculated
compositions
for
PAA/DMAc/H2O,
PAA/DMAc/EG
and
PAA/DMAc/ethanol systems are listed in Table S-4.
Table S-4. The experimental and calculated compositions for the ternary systems.
CPAA (wt%)
Composition ratio（PAA:DMAc:H2O）
Experimental
Calculated
Deviations
0.5
0.50: 85.89: 13.61
0.50: 86.30: 13.20
0.00: 0.41: -0.41
1.5
1.50: 85.70: 12.80
1.50: 86.13: 12.37
0.00: 0.43: -0.43
2.0
2.00: 85.40: 12.60
2.00: 85.80: 12.20
0.00: 0.40: -0.40
2.5
2.50: 85.22: 12.28
2.50: 85.37: 12.13
0.00: 0.15: -0.15
3.0
3.00: 84.90: 12.10
3.00: 85.09: 11.91
0.00: 0.19: -0.19
CPAA (wt%)
Composition ratio（PAA:DMAc:ethanol）
Experimental
Calculated
Deviations
0.5
0.50: 67.23: 32.27
0.50: 67.75: 31.75
0.00: 0.52: -0.52
1.5
1.50: 67.02: 31.48
1.50: 67.28: 31.22
0.00: 0.26: -0.26
2.0
2.00: 66.63: 31.37
2.00: 66.91 :31.09
0.00: 0.28: -0.28
2.5
2.50: 66.27: 31.23
2.50: 66.53 :30.97
0.00: 0.26: -0.26
3.0
3.00: 65.79: 31.21
3.00: 66.22: 30.78
0.00: 0.43: -0.43
CPAA (wt%)
Composition ratio（PAA:DMAc:EG）
Experimental
Calculated
Deviations
0.5
0.50: 87.30: 12.20
0.50: 87.51: 11.99
0.00: 0.21: -0.21
1.5
1.50: 86.73: 11.77
1.50: 87.15: 11.35
0.00: 0.42: -0.42
2.0
2.00: 86.30: 11.70
2.00: 86.71: 11.29
0.00: 0.41: -0.41
2.5
2.50: 86.29: 11.21
2.50: 86.48: 11.02
0.00: 0.19: -0.19
3.0
3.00: 85.90: 11.10
3.00: 86.09: 10.91
0.00: 0.19: -0.19
Reference
1. Tompa H (1956) Polymer solutions. Butterworths Scientific Publications, London
2. Pouchly J (1989) Sorption equilibria in ternary-systems-polymer mixed-solvent. Pure Appl Chem
61 (6):1085-1095
3. Mulder M (1991) Basic principles of membrane technology second edition. Dordrecht: Kluwer
Academic Publishers, Netherland
4. Kuramochi H, Noritomi H, Hoshino D, Nagahama K (1996) Measurements of solubilities of two
amino acids in water and prediction by the UNIFAC model. Biotechnol Prog 12 (3):371-379
5. Kan AT, Tomson MB (1996) UNIFAC prediction of aqueous and nonaqueous solubilities of
chemicals with environmental interest. Environ Sci Technol 30 (4):1369-1376
6. Fu YH, Orbey H, Sandler SI (1996) Prediction of vapor-liquid equilibria of associating mixtures
with UNIFAC models that include association. Ind Eng Chem Res 35 (12):4656-4666
7. Albuquerque LC, Simoes AN, Ventura CM, Goncalves RC, Macedo EA (1996) Infinite dilution
activity coefficients predicted from UNIFAC model. new experimental data for the solvolytic reactions of
2-chloro-2-methylpropane
in
methanol/ethanol,
methanol/2-methoxyethanol,
and
ethanol/2-methoxyethanol. Ind Eng Chem Res 35 (10):3759-3762
8. Lei ZG, Zhang JG, Li QS, Chen BH (2009) UNIFAC Model for Ionic Liquids. Ind Eng Chem Res
48 (5):2697-2704
9. Zeman L, Tkacik G (1988) Thermodynamic analysis of a membrane-forming system water
N-methyl-2-pyrrolidone polyethersulfone. J Membrane Sci 36:119-140
10. Fuoss RM, Mead DJ (1943) Osmotic pressures of polyvinyl chloride solutions by a dynamic
method. J Phys Chem 47 (1):59-70
11. Vainberg YV, Mokrushina LV, Balashova IM (1999) Infinite-dilution activity coefficients of
solvents in solvent-polymer systems from inverse gas-liquid chromatography. Theor Found Chem Eng 33
(4):313-318
12. Qian JW, Rudin A (1992) Prediction of thermodynamic properties of polymer-solutions. Eur
Polym J 28 (7):725-732
13.
Tan
L,
Pan
D,
Pan
N
(2008)
Thermodynamic
study
of
a
water-dimethylformamide-polyacrylonitrile ternary system. J Appl Polym Sci 110 (6):3439-3447
14. Hansen CM (2007) Hansen solubility parameters: a user's handbook. CRC Press, Boca Raton
15. Lindvig T, Michelsen ML, Kontogeorgis GM (2002) A Flory-Huggins model based on the Hansen
solubility parameters. Fluid Phase Equilibr 203 (1-2):247-260
16. Krevelen DWV, Nijenhuis KT (2009) Properties of polymers: their correlation with chemical
structure; their numerical estimation and prediction from additive group contributions. Fourth, completely
revised edition edn. Elsevier, New York