Esso Research and Engineering Co., Brit. Patent 798,542 (July 23, 1958). Hollyday, W. C., Jr., Munsell, M. W. (to Esso Research and Engineering Co.), U. S. Patent 3,076,791 (Feb. 5, 1963). Moyer, R. C . (to Pure Oil C o . ) , U. S. Patent 3,143,877 (Aug. 11, 1964). Rogers, D. T., et al., “Mechanism of Engine Sludge Formation and Additives Action,” SAE Golden Anniversary Fuels and Lubricants Meeting, Philadelphia, Pa., November 1955. Rohm & Haas G.m.b.H., Brit. Patent 937,734 (Sept. 25, 1963).
Shell International Research Maatschappiy N V , Brit. Patent 973,393 (Oct. 28, 1964). Spindt, R. S.,et al., “Nitrogen Oxides, Combustion, and Engine Deposits,” SAE Golden Anniversary Fuels and Lubricants Meeting, Philadelphia. Pa., Sovember 1955. Verley, G. M. (to Sinclair Research), U. S.Patent 3,044,860 (July 17, 1962). RECEIVED for review April 29, 1968 ACCEPTED August 23, 1968 Division of Petrochemistry, 155th Meeting, ACS, San Francisco, Calif., April 1968. Permission to publish granted by the British Petroleum Co., Ltd.
GEL PERMEATION CHROMATOGRAPHY Calibration Curue from Polydisperse Standards S .
T .
B A L K E ,
A .
E .
H A M I E L E C ,
A N D
B .
P .
LECLAIR
Chemical Engineering Department, McMaster Uniuersity, Hamilton, Ontario, Canada S .
1.
P E A R C E
Shell Oil,Ltd., Montreal, P. Q., Canada A search technique and computer program (Fortran IV) have been developed to determine the calibration curve for a gel permeation chromatograph using broad molecular weight distribution standards. Two average molecular weights (number, weight, or viscosity) for one or two polydisperse standards are required. A Rosenbrock search is used to find the calibration curve constants. This search has been tested for a variety of cases on an IBM 7040 computer. An evaluation of these results is presented.
THEuse
of the gel permeation chromatograph (GPC) for the determination of molecular weight distribution (MWD) of polymers is developing rapidly. T o interpret the GPC chromatogram it is necessary to establish a quantitative relationship between molecular weight and elution volume. The GPC is usually calibrated with narrow MWD standards ( M , / M , 5 1.1) using the elution volume a t the peak position. I t is not valid to use the peak position for broad MWD standards. For many linear homopolymers narrow MWD standards are not readily available, thus presenting the problem of determining the calibration curve with broad MWD standards. Rodriguez et ai. (1966) suggested a graphical procedure, but with little elaboration. Cantow et al. (1967) proposed a procedure for finding the calibration curve given the complete MWD of one broad MWD standard. They are also able to correct their calibration curve for the effect of concentration. More recently, Frank et al. (1968) developed a technique for finding a calibration curve using a number- and a weight-average molecular weight of one broad MWD standard. Their technique involves a graphical approximation which makes it difficult to employ. I n this paper a precise computer technique is presented, tested, and evaluated for several cases. These include searches for linear calibration curves given the following 54
I & E C PRODUCT RESEARCH A N D DEVELOPMENT
data: one broad MWD standard and its number- and weight-average molecular weights; two broad MWD standards, given two number-average molecular weights, two weight-average molecular weights, two viscosity-average molecular weights, and one number- and one weightaverage molecular weight. Searches for nonlinear calibrations are also discussed. Development
Linear Calibration Curve. A linear calibration curve may be expressed as
VOL = C1 - C2 logm(MW) where VOL = elution volume in counts MW = molecular weight of a monodisperse sample C l and C2 = constants to be found by search program
The method involves the injection of one or two broad MWD standards into the GPC under conditions where the effect of correction for imperfect resolution on the molecular weight averages of a polydisperse sample is negligible. [If skewing and symmetrical axial dispersion correction may be estimated, the true calibration curve
where
Table I. Test Samples
Sample No.
M"
M,
M,IMn
7A 108 3A 17 103
48,000 197,400 333,000 64,000 48,000
55,000 250,000 430,000 125,800 107,100
1.14 1.26 1.29 1.95 2.21
may be obtained even. under extreme conditions (Balke and Hamielec, 1968)]. This usually means residence times of about 100 minutes or greater, according to Duerksen and Hamielec (1968). This assumption of negligible correction for imperfect resolution can be checked by injecting other broad MWD standards to test the validity of the GPC calibration curve over a range of molecular weights. I n this program valu'es of C1 and C2 are guessed by the search and the calibration curve is used to convert the GPC chromatogram to a differential MWD, assuming infinite resolution. This differential M W D is then used to calculate the appropriate molecular weight averages (M,,, M,, or M u ) . One of the following objective functions is minimized.
FMAX2 = (MW(1) - T E M P 2 ) 2+ (MN(1) TEMP1)'
(1)
FMAX2 = (MN(1:I - T E M P l ) ' + (MN(2) TEMP1)'
(2a)
FMAX2 = (MW(1) - T E M P 2 ) *+ (MW(2) TEMP2)' (2b) FMAX2 = (VMW(1) - T E M P 5 ) ' + (VMW(2) TEMP5)' (212) FMAX2 = (MW(1) - TEMP2)"
(MiX(2) TEMP1)'
(2d)
FMAX2 = objective function MW(1) = experimental weight-average molecular weight for sample 1 MN(1) = experimental number-average molecular weight for sample 1 VMW (1) = experimental viscosity-average molecular weight for sample 1 T E M P 1 = calculated number-average molecular weight T E M P 2 = calculated weight-average molecular weight T E M P 5 = calculated viscosity-average molecular weight The computer program uses a Rosenbrock search as described by Wilde (1964). The search is restricted to the desired range of possible C1 and C2 by a "high failure" technique. Nonlinear Calibration Curve. A nonlinear calibration curve might be expressed as
The search program can be easily modified to search for N constants given N bits of information. This involves N / 2 to N broad MWD standards. An alternative approach might involve the search for straight-line segments of the nonlinear calibration using the existing program. No attempt has been made here to evaluate this procedure. Effective Calibration Curves. An effective calibration curve may be obtained where correction for imperfect resolution is not negligible. This involves using the calibration curve constants obtained by search on the imperfectly resolved chromatogram of a broad standard only for unknowns with chromatograms similar to that of the
Table II. Input Data for Program Test
Search Parameters
Initial Estimate
~
Run SampleNo. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
3A 7A 7A 108 103 103 103 103 103 103 103 7A 3A 3A
M, 333,000 48,000 48,000 197,000
First Sample MI'
ML
M"
Second Sample Ma
17
64,430
125,800
3A 3A 3A 17 17 17 17 17 17 17 3A 17 17
333,000 333,000 333,000
Iststep, M,
430,000
107,100 107,100 107,100 107,100 107,100 107,100 107.100 53,000" 333,000 333,000
Sample No.
125,800 125,800 125,800 125,800 125,800 125,800 125,800 413,000" 125,800 125,800
CC
.
0.005 0.005 0.005 0.002 0.005 0.005 0.1 0.005 0.005 0.005 0.05 0.05 0.005 0.005 0.005
Accel, A
Deccel, BE
2.5 1.5 1.5 2.5 1.5 2.5 2.5 2.5 1.5 2.5 2.5 3.5 1.5 1.5 1.5
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
~
CI
C2
20 20 50 50 20 80 50 50 50 20 20 20 20 50 20
15 15 6 6 15 5 6 6 6 15 15 15 15 6 15
' a of 0.70 in Mark-Houwink-Sakurada relation used.
VOL. 8 NO. 1 M A R C H 1 9 6 9
55
Table 111. Results of Program Test
output Cl c2 (actual (actual value value No. of Run No. = 37.55) = 3.62) iterations
1
37.54
3.62
166
2
37.56
3.62
409
3
37.52
3.61
209
4
41.34
4.32
195
5
37.43
3.60
348
6
72.31
9.42
198
7
47.71
5.59
136
8
49.79
6.00
58
9
49.79
6.00
219
10
42.11
4.50
250
11
40.66
4.22
219
12
36.74
3.47
260
13
37.53
3.62
470
14
37.35
3.58
234
15
37.33
3.59
403
APP~X time 2 min. 28 sec. 4 min. 8 sec. 3 min. 36 sec. 3 min. 27 sec. 4 min. 0 sec. 4 min. 21 sec. 4 min. 44 sec. 5 min. 11 sec. 5 min. 1 2 sec. 6 min. 15 sec. 4 min. 6 sec. 5 min. 6 sec. 5 min. 30 sec. 3 min. 39 sec. 5 min. 16 sec.
Remarks
standard. The effective calibration curve constants automatically accomplish necessary resolution correction. Another possibility is to use a series of broad standards over the range of elution volumes desired and combine the linear segments so obtained to obtain a complete effective calibration curve. A nonlinear calibration curve search would provide even more flexibility in the above procedures. None of the above are evaluated in this paper. Evaluation
Slowb
Very slowb
Very slowb Very slow*
Very slow6 Very slowb Lax criterion" Slow6
"Search criterion was 20 rather than 0.1. bSearch criterion of objective function variation (Y1 - Y2) = 0.1 not reached.
I n this evaluation samples of high and low polydispersity were used (Table I). Maximum computer time permitted per run was approximately 6 minutes. Previous programs using a Hooke and Jeeves pattern search had required 7 to 30 minutes to reach an optimum. Tables I1 and I11 show that, in general, convergence to the correct optimum (&0.7%) occurred after about 200 iterations within approximately 3% minutes using an initial step size of 0.005, an acceleration factor of 2.5 or 1.5, and a deceleration factor of -1. I n the only exceptional case slow convergence occurred when two very broad molecular weight samples, each of known but of almost equal weight-average molecular weights, were used. These samples were examined in runs 6 to 13. In none of these instances was the search criterion for compfetion, Y1 - Y2 = 0.1, reached although values as low as Y1 - Y2 = 2.0 were obtained a t values far from the correct C1 and C2-that is, the criterion was almost satisfied far from the optimum of these runs. I t was evident from examination of the calculated molecular weights that the objective function was not satisfied. This situation was attributed to the automatic step size reduction in the search, which usually resulted in very rapid convergence. Attempts to remedy this situation by changing the three search parameters, the search criterion
Table IV. Input Data
Polydisperse PVC sample 1 GPC response
Counts Height Counts Height
23.0 0.0
23.5 0.6
24.0 1.9
24.5 4.8
25.0 11.9
25.5 24.2
26.0 43.9
26.5 66.1
27.0 74.2
27.5 111.1
28.0 118.8
28.5 117.8
29.0 108.8
29.5 90.9
30.0 70.8
30.5 51.6
31.0 38.7
31.5 27.2
32.0 18.0
32.5 11.5
33.0 7.9
33.5 4.7
34.0 2.9
34.5 1.6
35.0 0.3
35.5 0.0
Average molecular weight M,"
= 67,800
M.b = 28,200 Search parameters CC = 0.005 A = 1.5 BE = -1
First guess of C1 = 20 First guess of C2 = 15 Output data C1 = 48.87 C2 = 4.36 Time = 2 min. 52 sec. No. of iterations = 305 a
56
Measured by light scattering. Measured by osmometry.
I & E C PRODUCT RESEARCH A N D DEVELOPMENT
Table V. Verification of GPC Calibration Curve
P vc Sample No.
(MJos (Osmometry)
2 3 4
38,000 49,000 31,000 60,000
5
(MnJcPc[ ( M n ) c P C- (Mn)as] GPC ( M d O S 43,000 50,000 37,000 57,000
+13.2 +2.1 +19.4 -5.0
for convergence and particularly the initial guesses of C1 and C2, met with considerable success. Run 12, although slow, was very near convergence when the time limit expired. Using the values of C 1 and C2 obtained with the search technique and PVC sample 1 (Table IV) the GPC results calculated assuming infinite resolution (Table V) differ in M , from -5.0 to +19.4’% compared with osmometer values, and in M, from -19.9 to -1.070, compared to SOFICA light scattering measurements. Considering the sources of uncertainty in all three methods of measurement as well as the fact that the search technique was used only for sample 1 (no attempt was made to optimize a C1 and C2 for the range of molecular weights of interest), these results were comidered satisfactory. Recommendations for Use
crf Search Program
Use search parameter values of
CC = 0.005 A = 2.5 BB = -1 Use a t least two different sets of initial guesses of C1 and C2 to test each case. Use a search criterion of Y1 - Y2 = 0.1 and a computer time limit of 6 minutes. Adjust initial guesses of C1 and C2 to speed convergence. Altering search parameters, or criterion for convergence, are less effective solutions to the problem.
(MY)LS
(Light Scattering) 128,000 148,000 99,000 176,000
( M ~ ) [ ( ~M s I~c p c~- ( M ~ L s ]loo (GPCI (Mi, 115,000 133,000 98,000 141,000
-10.2 -10.2 -1.0 -19.9
technique has been developed and used to determine a linear calibration curve for a wide number of cases (Tables I1 and 111). The program may be very easily modified to search for parameters of a nonlinear calibration curve. Results of program evaluation and specific recommendations for its use are presented. A source program listing is available on request. Acknowledgment
The authors appreciate financial support from the Chinook Chemicals Corp., Ltd., and the National Research Council of Canada. Data on PVC from Imperial Oil Enterprises, Sarnia, Ontario, and the original Rosenbrock search program from John D. Sheel of McMaster University are also gratefully acknowledged. literature Cited
Balke, S. T., Hamielec, A. E., Reprints, Sixth International Seminar, Gel Permeation Chromatography, Miami Beach, Fla., 1968. Cantow, M. J. R., Porter, R. S., Johnson, J. F., J . Polymer Sci. 5 , Part A-1, 1391 (1967). Duerksen, J. H., Hamielec, A. E., J . Polymer Sci. 6, Part C, 83 (1968). Frank, F. C., Ward, I. M., Williams, T., Reprints, Fifth International Seminar, Gel Permeation Chromatography, London, 1968. Rodriguez, F., Kulakowski, R., Clark, 0. K., IND. ENG. RES.DEVELOP. 5,121 (1966). CHEM.PROD. Wilde, D. J., “Optimum Seeking Methods,” p. 151, Prentice-Hall, Englewood Cliffs, N . J., 1964.
Conclusions
A program to determine a linear calibration curve from polymer samples of known weight number- or viscosityaverage molecular we:ights utilizing a Rosenbrock search
RECEIVED for review May 20, 1968 ACCEPTED November 4, 1968
VOL. 8 NO. 1 M A R C H 1969
57
