Supplementary Information
Influence of Surfactants on the Rheology and Stability of Crystallizing Fatty Acid
Pastes
Prachi Thareja1, Anne Golematis2, Carrie B. Street1, Norman J. Wagner1
1
Department of Chemical Engineering
Center for Molecular Engineering and Thermodynamics, University of Delaware
2
DuPont Analytical Sciences, Wilmington, DE
Martin S. Vethamuthu3, Kevin D. Hermanson3 and K.P. Ananthapadmanabhan3
3
Unilever Research and Development, Trumbull, CT
Solubility Calculations
The experimental solubility of PA in phase behavior samples containing 6 wt% surfactant can be compared
to the theory developed in Tzocheva et al. [1].
Symbol List:
Bn
CAPB
CT
CMCM,sat
HCn
ix
iSDS
iTego
iPA
iH2O
k0, k1
KA,mic
KS,mic
Mx
MWx
slope coefficient
cocamidopropyl betaine
total concentration of (all kinds of) surfactant contained in the solution
critical micellization concentration of the mixed surfactant solution
designation for fatty acid with n number of carbon atoms in the fatty acid hydrocarbon
chain
input weight percent of model system component (SDS, Tego, PA, H2O)
input weight percent of SDS
input weight percent of cocamidopropyl betaine
input weight percent of palmitic acid
input weight percent of water
fit parameters for linear function of “energy gain from the transfer of a fatty acid molecule
from a fatty acid crystal into the water phase, as a monomer”[1]
micellization constant related to the work for transferring a fatty acid molecule from the
solution into a micelle
micellization constant related to the work for transferring a monomer of S from the
solution into the micelle
molarity of component x
molecular weight of model system component x (SDS, Tego, PA, H2O)
1
n
q0, q1
SA
SA(20)
SLES
T
x
yA,sat
zA
number of carbon atoms in the fatty acid hydrocarbon chain
fit parameters for linear function of “energy gain from the transfer of a fatty acid molecule
from a fatty acid crystal into the water phase, as a monomer”[1]
fatty acid solubility in pure water
fatty acid solubility in pure water at 20°C
sodium laurylethersulfate
temperature
a model system component (SDS, Tego, PA, H2O)
molar fraction of the fatty acid in the mixed micelles
input mole fraction of fatty acid
β
ρs
ρSDS
ρTego
ρPA
ρH2O
γA,sat
γS,sat
interaction parameter from theory of regular solutions
density of experimental phase behavior sample
density of SDS
density of cocamidopropyl betaine
density of palmitic acid
density of water
activity coefficient of fatty acid in the micelles at saturation
activity coefficient of surfactant in the micelles at saturation
2
Calculating boundary between micelles/(micelles + crystals):
The calculations utilized in Tzocheva et al. [1] to calculate the phase diagram boundary between micelles
and micelles plus crystals have been used here. The equation numbers from this paper are given in
parentheses.
Rearranging a system mass balance gives (Tzocheva Equation 29):
 CMCM ,sat 
SA
z A  1 
 y A,sat 
CT
CT


(1)
The critical micellization concentration of the mixed solution is given by (Tzocheva Equation 28):
CMCM ,sat   A,sat yA,sat KA,mic   S ,sat yS ,sat KS ,mic (2)
yS ,sat  1  yA,sat
(3)
The activity coefficients can be calculated using (Tzocheva Equation 27) (with y A  y A, sat ):
 A, sat  exp   (1  y A, sat )2 
(4)
 S ,sat  exp   ( yA,sat )2 
The expression for β, developed via equilibrium thermodynamics and regular solution theory, is given by
(Tzocheva Equation 24):

1
1  y 
A, sat
2


SA
ln 

y K
 A,sat A,mic 
(5)
The linear equation used to fit the molar fraction of fatty acid in mixed micelles versus the number of
carbon atoms in the fatty acid hydrocarbon chain is given by (Tzocheva Equation 23 (for n = 14, 16, 18)):
ln  y A,sat   q0  q1n
(6)
From Tzocheva Table 5, at 25°C, for HCn in SLES: q0 = 2.41, q1 = -0.303; for HCn in CAPB: q0 = 2.69, q1 =
0.322.
KA,mic versus the number of carbon atoms in the fatty acid hydrocarbon chain is given by (Tzocheva
Equation 20):
ln K A,mic  k0  k1n
(7)
From Tzocheva Table 5, at 25°C, for HCn in SLES: k0 = 3.71, k1 = -1.15; for HCn in CAPB: k0 = 3.43, k1 = -1.13
3
n = 16 for palmitic acid
Tzocheva et al. used the following equation to interpolate literature data for log SA versus T (Tzocheva
Equation 15):
log S A  log S A(20)  Bn T  20 
(8)
where the equation for SA(20) is given by Lucassen[2] (Tzocheva Equation 16):
log S A(20)  2.82  0.65n
(9)
The values of Bn are given by (Tzocheva Equation 17):
Bn  0.00419( n  8)  0.00147
(10)
Using these equations and assuming 6 wt% sodium laurylethersulfate (SLES) as an approximation of 4.5
wt% SDS and 1.5 wt% CAPB, the phase boundary between micelles and micelles plus crystals has been
calculated and will be compared to experimental results as follows.
Information about the model system components is given in Table S1.
SDS
CAPB
PA
H2O
Table S1. Model system component information
Molecular
Molecular Formula Weight (g/mol)
Density (g/ml)
NaC12H25SO4
288.38[3]
> 1.1 at 20°C[3]
C19H38N2O3
342.52[4]
1.043 at 25°C[4]
C16H32O2
256.42[3]
0.852 at 25°C[5]
H2O
18.01528[6]
1[6]
Hydrogen
Density (g/ml)
0.0875
0.116
0.106
0.111
The molar concentration of total surfactant in the phase behavior samples is calculated as follows. The
density of the sample is calculated using a weighted density of each component:
s  iSDS SDS  iTego Tego  iPA PA  iH 2O H 2O (11)
The molarity of each component is given by:
Mx 
ix s
MWx
(12)
The molar concentration of total surfactant is given by:
4
M x  MSDS  MCAPB  M PA
(13)
A portion of the phase diagram of mole fraction of fatty acid (palmitic acid) versus total surfactant
concentration is given in the following figure. The phase boundary between micelles and micelles plus
crystals calculated from Tzocheva et al. [1] is shown as the ‘Theory’ line. The compositions of experimental
phase behavior samples are plotted as points in the figure. From the pictures, the solubility of palmitic acid
in the surfactant solution is consistent with the theory.
micelles + crystals
zA (mole fraction)
0.09
Theory
6 wt% S
0.4 wt% PA
0.3 wt% PA
0.06
micelles
0.25 wt% PA
0.03
0.1 wt% PA
0.0
0.1
0.2
0.3
CT (M)
Figure S1. A portion of the phase diagram of mole fraction of fatty acid (palmitic acid) versus total
surfactant with experimental phase behavior sample compositions and pictures in cross-polarized light.
5
Predicting compositions based on input component amounts and a linear fit to the fraction of solids
content in 12 wt% surfactant system versus input PA
Note that the solid phase includes palmitic acid, surfactant (SDS and Tego), and H2O (or D2O).
Assumptions:
- variable solubility of PA in surfactant solution
- variable amount of surfactant in the liquid portion of the sample
- the solubility of PA in a surfactant solution of a given composition of 3:1 SDS:Tego is the same as the
solubility of PA in a surfactant solution in the presence of additional PA (that is, in the complete model
system formulations)
- the solubility of PA in an aqueous surfactant solution is equal to the solubility of PA in the same
surfactant solution made in D2O
- the total amount of solids is less than the input PA (allowing the use of total input PA crystallized as an
initial guess)
- In the absence of any information to the contrary, we assume that the immobile surfactant has the same
proportion as the surfactant in the dispersed phase. SDS and Tego are assumed to act as an effective
surfactant “S.”
- The slope of the fraction of solids content as a function of input PA in the model system formulations
containing 6 wt% surfactant can be estimated by the slope of the fraction of solids content as a function of
input PA in the model system formulations containing 12 wt% surfactant.
Symbol List:
cx
d1,2
iS,PAx
ix
lx
mS,PAx
mx
n1,2
S
STDNMR
εx
ρx
calculated wt% of x in crystal or immobile form
denominator in density expression
input wt% of x based on PA and S only
input wt% of x based on the total weight
calculated wt% of x in the liquid portion of the formulation
measured wt% of x in crystal or immobile form based on PA and S only
measured wt% of x in crystal or immobile form based on the total weight
numerator in density expression
effective surfactant, ¾ sodium dodecyl sulfate (SDS) and ¼ cocamidopropyl betaine (CAPB)
solids content measured via TDNMR
uncertainty in x
density of x
The following function was used to fit STDNMR (the fraction of solids content) as a function of input PA in the
12 wt% surfactant model system formulations in D2O:
6
S TDNMR  1.65iPA  4.1 (14)
The uncertainties in the slope (1.65±0.04) and intercept (-4.1±0.5) were obtained by using linear fitting in
Origin 8. The x-intercept indicates there is a finite solubility of PA in the surfactant solution. From the
solubility experiments detailed in the supplementary material of our complementary study [7], the
solubility of PA in a solution containing 6 wt% of 3:1 SDS:Tego is approximately 0.3 wt% PA. The line
describing fraction of solids content versus input PA should therefore have an x-intercept at approximately
0.3 wt% PA:
0  1.65*0.3  xintercept
0.5  xintercept
(15)
(16)
The uncertainty of the x-intercept will be taken as 0.1, the uncertainty in the solubility experiments (since
the solubility of PA in a 6 wt% surfactant solution was between 0.3 wt% PA and 0.4 wt% PA) [7].
The uncertainty in the first term of the linear function is given by:
  iPA   0.04 

 
 (17)
 iPA   1.65 
2
 FirstTerm _ S
TDNMR
 1.65* iPA
2
The uncertainty in the second term is given by:
 SecondTerm _ S
TDNMR
 0.1 (18)
The uncertainty in STDNMR is given by:
S
TDNMR
2
2
  FirstTerm
  SecondTerm
_ STDNMR
_ STDNMR
(19)
For the samples where the total amount of solids is less than the input PA, the linear function can be used
to calculate the fraction of solids content for a given input PA.
The maximum amount of PA that can crystallize is equal to the input PA (iPA). Therefore, iPA can be used as
a first guess for the amount of crystalline PA (cPA). Subtracting iPA from the estimated total amount of solids
gives the amount of immobile surfactant (cS):
cS  csolid  iPA (20)
εc,S (the uncertainty in the amount of immobile surfactant) is given by:
7
 c   c2
S
solid
  i2PA
(21)
The amount of surfactant remaining in the liquid portion of the formulation (lS) can be calculated by
subtracting the amount of immobile surfactant (cS) of the sample from the input amount of surfactant:
lS  iS  cS
(22)
εl,S (the uncertainty in the amount of surfactant in the liquid portion of the formulation) is given by:
 l   i2   c2
S
S
(23)
S
The surfactant solubilizes a finite amount of PA. From the Solubility experiments section in the
supplementary material from our complementary study[7], assuming that the amount of surfactant in the
liquid portion of the formulation follows the same trend as the solubility experiments samples (is in the
solubility experiments is equal to ls in the formulations), the amount of PA in the liquid portion of the
formulation (lPA) is given by:
lPA   0.0051  0.0007  * ls2   0.022  0.008  * ls
(24)
The uncertainty in the first term is given by:
 0.0007    ls    ls 

    
 0.0051   ls   ls 
2
2
 FirstTerm _ l  0.0051* ls * ls
PA
2
(25)
The uncertainty in the second term is given by:
 0.008    ls 

  
 0.022   ls 
2
 SecondTerm _ l  0.022* ls
PA
2
(26)
The uncertainty in lPA is given by:
l   2
PA
FirstTerm _ lPA
2
  SecondTerm
_l
(27)
PA
Now, a better estimate of PA in the solid portion (cPA) of the sample is given by:
cPA  iPA  lPA (28)
εc,PA (the uncertainty in amount of crystalline PA) is given by:
8
c   2   2
PA
iPA
lPA
(29)
The amount of immobile surfactant is now better estimated by:
cS  csolid  cPA
(30)
This iteration is repeated until cS, lS, lPA, cPA do not change.
The values for the model system formulations in D2O are given in the following table:
Table S2. Calculated compositions of 12 wt% and 6 wt% surfactant model system formulations in D2O
12 wt% surfactant
6 wt% surfactant
(lPA+ls)/(cP
(lPA+ls)/(cPA
cS,
cPA,
ls,
lPA,
cS,
cPA,
ls,
lPA,
iPA,
+cS),
A+cS),
wt% wt%
wt%
wt%
wt%
wt%
wt%
wt%
wt%
wt%
wt%
7
1±1
6.2±0.1
11±1 0.8±0.1 1.6±0.3 4.1±0.3 6.91±0.03 1.9±0.3 0.09±0.03 0.18±0.03
8 1.8±0.9 7.3±0.1 10.2±0.9 0.8±0.1 1.2±0.2 4.8±0.3 7.94±0.01 1.2±0.3 0.06±0.01 0.10±0.03
9 2.4±0.9 8.3±0.1 9.6±0.9 0.7±0.1 1.0±0.1 5.4±0.4 8.97±0.01 0.6±0.4 0.03±0.01 0.05±0.03
10 3.0±0.9 9.4±0.1 9.0±0.9 0.6±0.1 0.8±0.1
6
10
0
0
0
11 3.6±0.8 10.5±0.1 8.4±0.8 0.6±0.1 0.64±0.08 6.6±0.5 11.03±0.01 -0.6±0.5 -0.03±0.01 -0.04±0.03
12 4.2±0.8 11.52±0.09 7.8±0.8 0.48±0.09 0.53±0.06 7.2±0.5 12.06±0.02 -1.2±0.5 -0.06±0.02 -0.07±0.03
13 4.8±0.8 12.57±0.08 7.2±0.8 0.43±0.08 0.44±0.05 7.9±0.5 13.09±0.03 -1.9±0.5 -0.09±0.03 -0.09±0.03
14 5.4±0.8 13.63±0.07 6.6±0.8 0.37±0.07 0.37±0.05 8.5±0.6 14.12±0.05 -2.5±0.6 -0.12±0.05 -0.12±0.03
15 6.0±0.8 14.68±0.07 6.0±0.8 0.32±0.07 0.31±0.04 9.1±0.6 15.15±0.08 -3.1±0.6 -0.15±0.08 -0.13±0.03
16 6.6±0.8 15.73±0.06 5.4±0.8 0.27±0.06 0.26±0.04 9.7±0.7 16.2±0.1 -3.7±0.7 -0.2±0.1 -0.15±0.03
17 7.2±0.9 16.78±0.05 4.8±0.9 0.22±0.05 0.21±0.04 10.3±0.7 17.2±0.2 -4.3±0.7 -0.2±0.2 -0.17±0.03
18 7.8±0.9 17.82±0.05 4.2±0.9 0.18±0.05 0.17±0.04 11.0±0.8 18.3±0.2 -5.0±0.8 -0.3±0.2 -0.18±0.03
19 8.4±0.9 18.85±0.04 3.6±0.9 0.15±0.04 0.14±0.03 11.6±0.8 19.3±0.3 -5.6±0.8 -0.3±0.3 -0.19±0.03
20 9±1 19.89±0.04 3±1 0.11±0.04 0.11±0.03 12.2±0.9 20.3±0.3 -6.2±0.9 -0.3±0.3 -0.20±0.03
The following function was used to fit STDNMR (the fraction of solids content) as a function of input PA in the
12 wt% surfactant model system formulations in H2O:
S TDNMR  1.61iPA  4.0 (31)
The uncertainties in the slope (1.61±0.08) and intercept (-4±1) were obtained by using linear fitting in
Origin 8. The x-intercept indicates there is a finite solubility of PA in the surfactant solution. From the
solubility experiments detailed in the supplementary material of our complementary study [7], the
solubility of PA in a solution containing 6 wt% of 3:1 SDS:Tego is approximately 0.3 wt% PA. The line
describing fraction of solids content versus input PA should therefore have an x-intercept at approximately
0.3 wt% PA:
9
0  1.61*0.3  xintercept
0.5  xintercept
(32)
(33)
The uncertainty of the x-intercept will be taken as 0.1, the uncertainty in the solubility experiments (since
the solubility of PA in a 6 wt% surfactant solution was between 0.3 wt% PA and 0.4 wt% PA) [7].
The uncertainty in the first term of the linear function is given by:
  iPA   0.08 

 

 iPA   1.61 
2
 FirstTerm _ S
TDNMR
 1.61* iPA
2
(34)
The uncertainty in the second term is given by:
 SecondTerm _ S
TDNMR
1
(35)
The uncertainty in STDNMR is given by:
S
TDNMR
2
2
  FirstTerm
  SecondTerm
_ STDNMR
_ STDNMR
(36)
For the samples where the total amount of solids is less than the input PA, the linear fit function can be
used to calculate the fraction of solids content for a given input PA.
cS+cH2O can be compared to the amount of immobilized surfactant calculated for the D2O system:
Table S3. Predicted compositions of 12 wt% surfactant model system in H2O
cS+cH2O,
cS, wt%,
wt%,
iPA,
from
from
D2O
wt%
H2O
system
system
7
4.1±0.3
3.8±0.6
11
6.6±0.5
6.2±0.9
12
7.2±0.5
7±1
13
7.9±0.5
7±1
18
11.0±0.8
10±1
The difference in the solids content from the D2O system and the H2O system provides an indication of the
amount of immobile H2O.
10
Density Calculation:
From the supplementary material in the complementary study [7], the density of the effective surfactant S
is 1.018 g/ml. Using an assumption of ideal mixing, the density of the solids is given by:
mS  mPA n1

mS mPA d1

 solid 
S
(37)
 PA
The uncertainty in the numerator is given by:
n 
1

  
2
mS /  S
mPA /  PA

2
(38)
The uncertainty in the denominator (assuming no uncertainty in the density of each component) is given
by:
d 
1
    
2
mS
2
(39)
mPA
The uncertainty in the solids density is given by:
n  n   d 
 1  1   1 
d1  n1   d1 
2

crystal
2
(40)
The density of the liquid phase, also assuming ideal mixing, is given by:
liquid 
lS  lPA
n
 2
lS
l
d2
 PA
S
(41)
 PA
The uncertainty in the numerator is given by:
n 
2
    
2
lS
lPA
2
(42)
The uncertainty in the denominator (assuming no uncertainty in the density of each component) is given
by:
11
d 
2

  
2
lS /  S
lPA /  PA

2
(43)
The uncertainty in the density of the liquid is given by:
n  n   d 
 2  2   2 
d2  n2   d2 
2

liquid
2
(44)
Table S4. 6% surfactant model system formulation components density
Input Density of
Density of
PA,
solids,
Liquid, g/ml
wt%
g/ml
7
0.9±0.3
1.0±0.2
11
0.9±0.4
1.0±0.4
12
0.9±0.5
1.0±0.5
15
0.9±0.6
1.0±0.3
18
0.9±0.8
1.0±0.2
The density of the solids is lower than that of the liquid, and the calculation shows that the solids will, to
within the calculated uncertainty, have a tendency to cream or remain neutrally buoyant.
From the above calculations (results given in Table S2), the maximum amount of PA that can be
accommodated by 6 wt% surfactant is approximately 9 wt%. In model system formulations with the
amount of PA exceeding approximately 9 wt%, the remainder of the PA crystallizes and floats on the top of
the formulation.
Neglecting hydration (since it does not matter if the water floats or sinks) and assuming that S completely
solidifies, the density of the solids containing surfactant is higher than the density of the excess PA crystals
(0.90±0.08 g/ml versus 0.852 g/ml). This can explain why the unstable formulations have some solids that
float and some solids that sink.
12
DSC data for 6 wt% surfactant model system formulations
0.0
7% Palmitic Acid Melting
0.0
Watts / Gram
-0.4
-0.6
-0.2
Watts / Gram
t= 30min
t= 1hr
t= 1hr 30min
t= 2hrs
t= 2hrs 30min
t= 3hrs
t=19hrs
-0.2
8% Palmitic Acid Melting
-0.8
-0.4
t=30min
t=74min
t=96min
t=140min
t=5hrs
t=8hrs
-0.6
-0.8
20
25
30
35
40
45
50
55
60
65
70
75
20
30
40
Temperature (C)
0.0
70
-0.2
-0.4
-0.6
t=30min
t=74min
t=96min
t=2hr30min
-0.8
20
25
30
35
40
45
50
55
60
65
70
Watts/ Gram
Watts / Gram
60
10% Palmitic Acid Melting
0.0
9% Palmitic Acid Melting
-0.2
-0.4
-0.6
t=30min
t=3hrs
t=5hrs
t=12hrs
-0.8
75
20
25
30
35
Temperature (C)
40
45
50
55
60
65
70
75
Temperature (C)
12%Palmitic Acid Melting
0.0
0.0
15% Palmitic Acid Melting
-0.2
Watts/ Gram
-0.2
Watts/ Gram
50
Temperature (C)
-0.4
-0.6
t= 40min
t= 1hr 20min
t= 3hrs
t= 6hrs
t= 12hrs
-0.8
20
25
30
35
40
45
-0.4
-0.6
t= 30min
t=74min
t=96min
t-140min
t=6hrs
-0.8
50
55
Temperature (C)
60
65
70
75
20
25
30
35
40
45
50
55
60
65
70
75
Temperature (C)
13
0.0
18% Palmitic Acid Melting
Watts/Gram
-0.2
-0.4
t= 30min
t=74min
t=96min
t-140min
-0.6
-0.8
20
25
30
35
40
45
50
55
60
65
70
75
Temperature (C)
SAXS data for 6 wt% surfactant and 12 wt% surfactant model system formulations
12 wt% surfactant
6 wt% surfactant
60
50
d (Å)
40
30
20
10
0
7
10
18
PA content (wt%)
Table S5. SAXS peaks and spacings
12 wt% surfactant 6 wt% surfactant
PA content
q (Å-1)
d (Å)
q (Å-1)
d (Å)
(wt%)
7
0.128
48.91
0.153
41.00
7
0.173
36.36
0.175
35.95
7
0.340
18.46
0.349
18.01
7
0.523
12.02
0.523
12.02
7
1.358
4.63
1.358
4.63
7
1.429
4.40
1.430
4.40
7
1.516
4.14
1.511
4.16
7
1.615
3.89
1.618
3.89
7
1.692
3.71
1.691
3.72
14
10
10
10
10
10
10
10
10
10
0.146
0.171
0.347
0.518
1.352
1.424
1.508
1.615
1.687
43.01
36.78
18.12
12.13
4.65
4.41
4.17
3.89
3.73
0.147
0.173
0.349
0.521
1.356
1.425
1.508
1.606
1.680
42.82
36.22
18.01
12.07
4.64
4.41
4.17
3.91
3.74
18
18
18
18
18
18
18
18
18
0.139
0.171
0.338
0.519
1.350
1.424
1.509
1.618
1.688
45.24
36.78
18.60
12.11
4.66
4.41
4.17
3.89
3.72
0.151
0.173
0.349
0.498
1.357
1.427
1.510
1.609
1.684
41.71
36.22
18.01
12.62
4.63
4.40
4.16
3.91
3.73
15
16000
16000
14000
14000
12000
12000
10000
10000
I (a.u.)
I (a.u.)
6 wt% surfactant
8000
8000
6000
6000
4000
4000
2000
0
0.0
7 wt% PA
t=19hr
0.5
1.0
1.5
2.0
2.5
-1
10 wt% PA
t=20hr
2000
0
0.0
0.5
1.0
1.5
2.0
2.5
-1
q (Å )
q (Å )
18000
16000
14000
I (a.u.)
12000
10000
8000
6000
4000
2000
0
0.0
18 wt% PA
t=2hr30min
0.5
1.0
1.5
2.0
2.5
-1
q (Å )
16
12 wt% surfactant
30000
12000
25000
8000
I (a.u.)
I (a.u.)
20000
15000
4000
10000
7 wt% PA
t=5.5hr
5000
0.0
0.5
1.0
1.5
2.0
2.5
0
0.0
-1
10 wt% PA
t=24hr
0.5
1.0
1.5
2.0
2.5
-1
q (Å )
q (Å )
18000
16000
14000
12000
I (a.u.)
10000
8000
6000
4000
2000
18 wt% PA
t=3hr
0
-2000
0.0
0.5
1.0
1.5
2.0
2.5
-1
q (Å )
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[4]
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[5]
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[6]
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[7]
Street C.B., Yarovoy Y., Wagner N.J., M.S. Vethamuthu, K.D. Hermanson, K.P.
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17