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Correction and Reassessment of Intrinsic Viscosity and The Universal CalibrationMethod - A Bridge from Polymers to Small Molecules

Shyhchang S. Huang IAdvanced Technology Group, BFGoodrichBreckn, ilIe, OH 44141 '_

Abstract: The current definition of intrinsic viscosity ([riD, and its application in the universalcalibration method, applies to polymeric materials with molecular weight (MW) >>1,000.However, there have been numerous discussions regarding the [11]of low MW oligomers.Difficulties can be encountered at very low MW, because the [1"1]in some cases becomesnegative, which in theory is impossible. A new model is proposed in this paper to refine thecalculation of [rl]. The new model is based on the fact that [ri] is a polymer volume expansioncoefficient; i.e., [11]times the polymer density is the expansion ratio of the hydrodynamicvolume in solution to the polymer volume in the solid state. In the calculation of [rl], as inmeasuring the density of a low-density material in air, the volume of solvent should be added.For a polymer with MW greater than 10,000, this correction can be neglected. However, thecorrection becomes important for small oligomers. With this correction, the Mark-Houwinkcurve of n-alkane and polystyrene in the low MW region become linear. The universalcalibration curve, calculated with corrected [rl], is also linear across the range from polymer tosmall molecules.

INTRODUCTION

In recent years, there has been increasing interest in producing low molecular weight(MW) polymers and oligomers with applications as plasticizers, detergents, food additives,lubricant additives, copolymer precursors, et.aL. Thus the determination of accurate and/orabsolute MW distribution and MW averages becomes increasingly more important for thepurposes of quality assurance, research and development, patent applications, and productlitigation. However, characterization of low-molecular weight polymers is difficult for bothfundamental and practical reasons. One example is that several parameters for polymers, suchas refractive index and density, do not remain constant in the low oligomer range.

Size-exclusion chromatography (SEC) is by far the most common method for MWdistribution studies of polymers and oligomers. The retention volume in SEC is based on thepolymer size in the mobile phase. The MW is calibrated with a set of known standards or isdetermined directly using an in-line MW detector, such as a light scattering photometer,viscosity detector, or mass spectrometer. In 1967 Benoit et.al, introduced a universalcalibration method in which the retention of a molecule in SEC is related to its hydrodynamicvolume. This can be calculated by multiplying the MW by the intrinsic viscosity ([rl]) t_].The [rl] of a polymer is related to its MW by the equation:

[rfl = kM" (1)where M is the MW of polymer, k and a are Mark-Houwink constants. Therefore, a universalcalibration curve for an SEC column set can be established using a set of polymer standardshaving a wide MW range with known [1"1]values. The true MW distribution of other

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the idea of segment concentration and plotted [_]]_ versus MW _/2. This method appeared tobe particularly useful in the low and medium MW range. A linear relationship of polystyrenein benzene was found down to MW 2,500 t2). To correct the deviation of low MW material,Dondos and Skordilis proposed an error term related to the exponent a of the Mark-Houwink-Sakurada equation c3). Yamakawa and his colleagues attempted to use the exclusion-volumeeffect and an empirical parameter to treat the negative [rl] of low oligomers, ta-s)but was notsuccessful. Recently, Chance and his colleagues also investigated the [rl] of low MWoligomers. The again found that universal calibration method failed in the low oligomerregion t6).

In this paper the problem is looked at from a different perspective, and a new model isproposed to correct the [rl] of polymers and oligomers. The corrected [1"1]is defined as "{rI}"(using curly rather than square brackets). After correction, the log {rI} versus log(MW) forlow MW oligomers of n-alkane and polystyrene is linear, indicating that {rl } will not becomenegative. The universal calibration curve calculated with {rl} can smoothly connects highpolymers with small molecules.

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THEORY

What is Intrinsic Viscosity

The intrinsic viscosity, [rl], or dilute solution viscosity, is measured using a capillaryviscometer, where:

[rl] = ( %/c (2)in which, rlspis called specific viscosity where:

rl,p = rl,-1 =t/t 0- 1 (3)where Tit is the relative viscosity (t/t0), t is the afflux time required for a specified volume ofpolymer solution to flow through a capillary tube, and to is the corresponding afflux time ofthe solvent, rl_p(equation 3) is the excess viscosity contributed by polymer, i.e., in additionto the viscosity from the solvent. Therefore, the [rl] is mainly a measurement of the "change"due to the polymer when it is dissolved in solvent. As early as 1930, Staudinger and Heuer r_attempted to relate this viscosity number to the molecular weight of the polymer. In 1938,Mark is) demonstrated this relationship empirically as equation 1, commonly called Mark-Houwink equation, or Mark-Houwink-Sakurada equation. Flory also investigated therelationship between [rl], MW, and the molecular size in solution tg]. However, the physicalsignificance of [rl] is still not well understood.

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V

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Since the universal calibration method was introduced, its applicability has beenmethou ,proven for high polymers. According to the theory of the universal calibration ,tlj

Vha = [filM (4)where Vhd is the hydrodynamic volume of polymer in solution. For the polymer in the solidstate,

V_d = M/p, (5)or M = Vsdp (6)where V_ and p are the volume and density of polymer in solid phase, respectively.Combining equations 4 and 6:

Vhd = [rl]V,dp, (7)therefore,

[11] = (V_ / Vsd)(1/p) (8)or,

[]lip _'_ Vhd / V_I (9)

In other words, the intrinsic viscosity is actually a comparison of the hydrodynamicvolume of a polymer in solution to its original volume in the solid state. The product of theintrinsic viscosity and the density is the expansion ratio. Table 1 shows data for polystyrene.The density of polystyrene is approximately 1.0 g/mL and the [11]of polystyrene (MW 3x105)in tetrahydrofuran is approximately 100 mL/g. Therefore, the polymer chain expands 100times in solution. Likewise, the expansion ratio of polystyrene (MW 8x106) is approximately1,000. Because the expansion ratio increases with MW very rapidly, the high MW polymerchains are easily deformed and degraded under shear. The higher the MW, the more severethe deformation as well as degradation.

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Correction of Intrinsic Viscosity tiI

For high molecular weight materials for which the universal calibration methodapplies, the above relationship will also apply. However, for low molecular weightoligomers, this relationship breaks down. At very low MW, [1"1]becomes negative. This isincorrect, since neither Vhd or V_dCan be negative.

Why is the intrinsic viscosity negative? Let us recall equations 2 and 3 for thecalculation of the intrinsic viscosity. The [rl] becomes negative when the t for a lowoligomer or small molecule is less than t0. As discussed previously, the intrinsic viscosity isa measurement of volume change. When the hydrodynamic volume of a polymer is muchgreater than the volume of a solvent molecule, t will be greater than to. However, when thehydrodynamic volume of a small oligomer is the same as the solvent molecule, there will beno net increase in the viscosity. The t will be the same as t0, and the apparent [1"1]becomeszero. When the hydrodynamic volume of the oligomer is smaller than the solvent molecule,the intrinsic viscosity becomes negative. This is similar to measuring the density of amaterial in air. If the density of the material is much greater than air, the calculated densitywill be close to the correct value, when measuring the density of a gas that is lighter thanair, the directly calculated "density" becomes negative. The way to correctly calculate the

330........... ._._._ ..... :--'.,,_'__:._...z_ - , ._ ..___ - _.._. _,.._- -._,_

density of a gas is to add the weight of air of the same volume to the measured weight of thegas. Likewise, to "correct" the [rl] of a low oligomer one may add the hydrodynamic volumeof the solvent molecule to the apparent hydrodynamic volume of the oligomer, [rl]Mp.

According to this hypothesis, we can define a "corrected intrinsic viscosity", {r]},which takes into account the volume of the solvent molecule:

Vhd -- {rl}Mp = [rl]Mp + V,v* (10)in which Vsv* is the hydrodynamic volume of the solvent molecule. The "*" is added

because the V_vneeds to be adjusted. During the calculation of [_], rl_, is divided by theconcentration of the polymer or oligomer, in units of weight per volume. Therefore, the [rl]is based on the polymer (or oligomer) density. Every term in equation 10 needs to bemultiplied by its own density:

Therefore, {1]} = [1]] +Ms,/MI, p p (12)

For high polymers with MW > lxl04, the MJMppp term is negligible. However,for low oligomers, this correction becomes significant.

Hydrodynamic Volume of Small Molecules

If the above correction is correct, then equation 12 should also be applicable tosmall molecules, and the universal calibration curve of polymers and oligomers calculatedusing that equation should be extendable to small molecules. However, the intrinsicviscosities of small molecules are normally very low and are difficult to measure accurately.Small molecules that are approximately spherical will not expand when dissolved in solution,and the hydrodynamic volume can be estimated:

Vhd = M / p (13)

EXPERIMENTAL

Tetrahydrofuran (THF), stabilized with 250ppm butylated hydroxytoluene (BHT),and toluene (EM Science, Gibbstown, NJ), were used as the mobile phases in SEC. Butylatedhydroxyltoluene (BHT), 1,2,4-trichlorobenzene (TCB), p-xylene, N,N-dimethylacetamide(DMAc), and isopropanol were from Aldrich Chemical Co. (Milwaukee, WI). Chloroform, o-dichlorobenzene(ODCB), and chlorobenzene were from Burdick and Jackson/Baxter(Muskegon, MI). Cyclohexane and 2-butanone(MEK) were OmniSolv grade from EM Science(Gibbstown, NJ). Acetone, acetonitrile, and methanol were all Optima grade from FisherScientific (Pittsburgh, PA).

The polystyrene standards of MW 750 and 2630 were purchased from PressureChemicals (Pittsburgh, PA) and TosoHaas (Montgomeryville, PA), respectively. Apolystyrene standard kit EasiCal, PS-2 from Polymer Laboratories, Inc. (Shropshire, UK) wasused for the universal calibration curve.

The density, or specific gravity, of polystyrene was measured using a pycnometer,No. 1620, 25 mL (Coming, Inc., Coming, NY). The universal calibration curve was studied

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using a Waters 590 LC pump, a Micromeritics Model 725 autosampler, a Waters Columnoven, and a Waters 410 DRI detector. The viscosity of small molecules, positive or negative,was measured using a gel permeation chromatograph/viscometer 150CV from Waters(Milford, MA). A PLgel 5lma guard and a 51arn 100_ column (Polymer Laboratories, Inc.,Shropshire, UK) in series was used in both studies.

RESULTS / DISCUSSION

n-Alkane in Tetrahydrofuran

The molecular weights and densities of the n-alkanes listed in Table 2 are from theCRC Handbook of Chemistry and Physics t12J.The intrinsic viscosities, [_], of n-alkanes intetrahydrofuran are from reference 6. The corrected intrinsic viscosities, {11}, are calculatedwith equation 12 and listed in Table 2. The [11]and {11}are plotted against molecular weightas shown in Figure 1. The Figure shows the viscosity data down to decane. Below decane,the [rl] is reduced dramatically. For n-alkanes smaller than octane, the [rl] becomes negativein toluene (Figure 2). However, after correction, the log{rl } values are approximately linear

versus log MW, meaning the {1"1}of smaller n-alkanes will not be negative. The Mark- iHouwink constant a (the slope of the linear curve) is 0.84, which is much higher than 0.5 aspredicted for most low MW polymers t.33. This indicates that the n-alkane molecules in this

configuration .MW range probably still remain in a random coil [ta]

Styrene Oligomer.st

In order to calculate the corrected intrinsic viscosities of styrene oligomers, the idensities of these oligomers are needed. It is very difficult to directly measure the density ofeach oligomer, because a large amount of material is required. Therefore, these wereestimated using the densities of the unimer (1-phenylhexane) and two mixtures of oligomers.First, the relation of density versus reciprocal of molecular weight, as shown in Figure 3using n-alkane as an example, is linear. Next, the densities of unimer was taken from ahandbook tgl as 0.861. The densities of the styrene oligomers of molecular weight 750 and2,630 were measured with a pycnometer to be 1.028 and 1.070, respectively. The densities ofpolystyrene oligomers were obtained from the linear curve in Figure 4 are are listed in Table3.

The {rl } and the hydrodynamic volume, {_ }M, of styrene oligomers are thencalculated based on the [rl] data in reference 6. These are shown in Table 3. The [11]and i{rI} were plotted against molecular weight as shown in Figure 5. As expected, the curve of [[11]drops sharply at the unimer. After correction, the {rl} of all oligomers become I

approximately linear, as well. The Mark-Houwink constant "a" of these five oligomers is0.38, which is considerably lower than 0.5. This may be due to the chain end (n-hexylgroup) effect.

•Universal Calibration Curve

The SEC of a set of polystyrene standards was studied in tetrahydrofuran. Theuniversal calibration curves of polystyrene were constructed using both [filM and {rl}M asshown in Figure 6. The {rl} of polystyrene of molecular weight >2,000 is very close to [rl].

Several small molecules, 1,2,4-trichlorobenzene, o-dichlorobenzene, chlorobenzene,p-xylene, and toluene were also studied. They were chosen because they are aromatic andthus chemically similar to styrene oligomers and polymers. The calculated hydrodynamicvolumes of these small molecules, M/p, are shown in Table 4. Figure 7, an expansion ofFigure 6, shows that the {rl}M curve for polystyrene fits the small molecule data points betterthan the [_]M curve. The data points of the small molecules do not fall on a straight linebecause they are not exactly rigid and spherical.

Intrinsic viscosity of small molecules

There is another way to test to see this correction method is valid. Combiningequations 11 and 13::

Vhd = [rl]lVlv + lVldp v -- lVlv/pp (14)The subscript "p" here also stands for spherical and rigid molecules.

[rl]Mp = M.p/pp - lVl_/pp (15)Or, [rl] -- (1 - 1Vl_lv)/pv (16)Therefor, the apparent intrinsic viscosity, [rl], of a spherical and rigid molecule with MW(instead of MW/p) greater than that of the solvent will be positive. And vice vei'sa.

A series of small molecules that are approximately spherical were injected into aSEC/IV system in both THF and toluene as mobile phases. The positive or negative resultsare listed in Table 4 in the order of molecular volume, i.e., MW/p. It is not surprising to findthat the molecules that have larger molecular volumes do not necessarily have a positiveviscosity, and vice versa. On the other hand, if the molecules are listed simply in the orderof MW, as shown in Table 5, they follow the prediction given in equation 15, except forseveral molecules with molecular size very close to the solvent.

SUMMARY / CONCLUSION

An equation was derived from universal calibration theory that indicates that theintrinsic viscosity divided by density becomes the expansion ratio for the volume of solidpolymer to the hydrodynamic volume of polymer in solution. With this concept a new modelhas been proposed to correct the intrinsic viscosity by adding the volume of solvent to theapparent hydrodynamic volume of the polymer. This correction is negligible for high MWpolymers, but it is significant for small oligomers (MW<I,000). The curve of correctedintrinsic viscosities, {1"1},versus molecular weight (in logarithmic scale) is linear down at lowmolecular weight, since this correction prevents the viscosity of very small molecules frombecoming negative. The corrected universal calibration curve using {1"1}M, instead of [rl]M,

333

can be smoothly extended to the hydrodynamic volume of small molecules, i.e., molecularweight divided by density. Positive/negative viscosity studies on small and sphericalmolecules also support this correction method as being valid.

ACKNOWLEDGEMENT

The author thanks the continuing support and encouragement from the managementof BFGoodrich company. The measurement of the density of polystyrene oligomers byH. Robert Lindesmith and David Gregus is also appreciated.

REFERENCES

(1) Z. Grubusic, P. Rempp, and Benoit; Polymer Letters, 5_,p 753 (1967).(2) A. Dondos and H. Benoit; Polymer, _ p. 1161 (1977).(3) A. Dondos and V. Skordilis; J. Polym. Sci., Phys. _ p. 615-621 (1985).(4) F. Abe, Y. Einaga, and H. Yamakawa; Macromolecules, _ 4423-4428 (1991); andreferences therein.

(5) K. Horita, N. Sawatari, T. Yoshizaki, Y. Einaga, and H. Yamakawa; Macromolecules,4455 A,_63 (1995); and references therein.(6) R. R. Chance, S. P. Baniukiewcz, D. Mintz, and G. Ver Strate; Int. J. PoIym. Anal. &Characterization 1_,1 (1995).(7) H. Staudinger and W. Heuer; Chem. Ber. 63, 222 (1930).(8) H. Mark; in "Techniques of Polymer Characterization" Chapter 6, Butterworth, London,1959.

(9) P. Flory; in "Principles of Polymer Chemistry", Chapter XIV, Cornell University Press,Ithaca, NY (1953).(10) M. Haney, J. Armonas, and L. Rosen; in "Detection and Data Analysis in Size ExclusionChromatography", Ed. by T. Provder, Chapter 7, ACS Symposium Series 352 (1987).(11) W. Roff, J. Scott, and J. Pacitti; in "Handbook of Common Polymers", Section 4, CRCPress, Cleveland, OH (1971).(12) CRC Handbook of Chemistry and Physics, 67th ed., by CRC Press, Boca Raton, FL(1986-1987).(13) U. Bianchi, M. Dalpiaz, and E. Patrone; Makromol. Chem., 8_9.0,112 (1964)(14) M. Bohdaneck_ and J. Kov_: "Viscosity of Polymer Solutions", Elsevier, Amsterdam, p.87 (1982).

Table 1.Polymer ExpansionRatioof PolystyreneinTHF

Mw [_1]' ExpansionRatio(mi/gm) [zl]W

9,350 8.50 965,000 34.0 36

275,000 95A 1001,890,000 376.5 3958,420,000 1034. 1084

a. Calculatedusingthe equation:Log M = 0.0128 + 0.712 • Log[q], (Reference 10)

b. Densityof polystyreneis 1.04 - 1.05 (Reference 11)

Table 2.CorrectedIntrinsicViscositiesof n-Alkanes

Carbon (g/mole) (g/ml) (dl/g) _ (ml/g) (dl/g) (dl/mde) (dl/mole)

10 142.28 0.7300 0.0063 103.0 0.700 0.0133 1_92 0_0

12 17034 0.7497 0.0106 1275 0.566 0.0163 2.777 1,806

14 198.39 0.7628 0.0137 1513 0.477 0.0185 3.670 2.718

16 226.45 0.7733 0.0173 175.1 0A12 0.0214 4J345 3.q18

18 254.50 0.7768 0.0195 197.7 0365 0.0232 5.904 4.963

36 t 506.98 0.8168 0.0381 414.1 0.174 0.0398 20.18 19.32

1. See_ 1Z2. SeeRefemrce6.3. M,c(of)=7Zll,_124. (q}=_ * M,,/(M_pvJ,Eq,am12

Table 3.Corrected IntrinsicViscositiesof Styrene OligomersinTHF

Oagomer (g/mole)(g/ml) ((g/g) Mpop (ml/g) (dl/g) (all/mole)((g/mole)

Un_ner 162.27 0_61 0.0117 139.7 0.517 0.0169 2.75 1.90

Dimer 266.42 0_46 0.0181 252.0 0.287 0.0210 5.59 4.82

Tdmer 370.57 0.984 0.0219 364.6 0.198 0.0239 8.86 8.12

Tetramer 474.72 1.006 0.0238 477.6 0.151 0.0253 12.01 11.30

Pentamer578.87 1.020 0.0265 590.4 0.122 0.0277 16.03 15.34

t. SeeReference9.2. SeeRefenmce12.3. Mx(of)=72.11,Refenmce94 zquauont2.

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Table 4.

GPCV V'scosity + I - in THF & Toluene

Molecule MW_1) Densit_1) MWld. in Tol.(2) inTHF_BHT 220.36 0.901 244.8 + +TCB 181.46 1.454 124.8 + +p-Xylene 106.17 0.866 122.6 + +ODCB 147.01 1.308 112.6 + +cy-Hexane 84.16 0.779 108.1 +Toluene 92.14 0.867 106.3 +Cl-benzene 112.56 1.106 101.8 + +DMAc 87.12 0.932 93.5 + +MEK 72.11 0.805 89.6 - -THF 72.11 0.888 81.?. -Chlo_o;+orm 119.38 1.489 80.?. + +i-PrOH 60.09 0.785 76.5 - +Acetone 58.08 0.791 73.5 - -AcCN 41.05 0.782 52.5 - -MeOH 32.04 0.791 40.5 - -

"|'""iT ...... T'!

(1)SeeR._msn_e12(2)OmaCtwmaa_=p_=xdtions,m ==pedmsnt=e=_, Wmr IS0CVwmu=ed

Table 5.

GPCV V'scosity + I- in THF & Toluene

Molecule : MW(1) Dense) MWld. in1"o1.(2) inTHF(=)BHT i 220.36 0.901 244.8 + +TCB i 181.40 1.454 124.8 + +ODC8 147.01 1.306 112.6 + +Chlo+-o;'orm 119.38 1.489 80.2 + +Cl-benzene 112.56 1.108 101.8 + +p-Xylene 108.17 0.866 122.6 + +Toluene 92.14 0.867 106.3 +DMAc 87.12 0.932 93.5 + +cy-Hexane 84.16 0.779 108.1 +MEK 72.11 0.805 89.6THF 72.11 0.888 81.2i-PrOH 60.09 0.785 76.5 - +A,_to_e 58.08 0.791 73.5AcCN 41.os 0.782 52.5 -MeOH ......3...2:...0(l....0.791 40.5 -(1)seeRa_ren_12(2)l_d _ _,-._nL m ==;x.t._ _ w==.1socvw=s.=ed

s O,=[Ivl II=t_

c= _._.,,'--IIP

c,_ _r _.o "_"° e"c /

0.5 //100 200 300 400 500

MolecularWeight

FIGURE1 Intrinsicviscositiesversusmolecularweightof n-alkanes.The data are fromTable 2.

_4.M

FIGURE 2 Viscosity chromatogram of n-alkanes. Instrument: Waters

t50CV; Column: PLgel 5p, Guard + 1001_; Mobile phase: toluene, 0.5ml/min.

ii 337

I;

° 0.9

0.8 -_

o.,,0.7 _

0.65 '_

0.60 0.002 0.004 0.006 0.008 0.01 0.012 0.014

1/MwHGURE 3 Density of n-alkanes versus the reciprocal of molecular weight.The data are from the CRC Handbook of Chemistry and Physics.

, 0.9

1. (xl04)

FI[G_ 4 Dcusity of styreneoligom¢_ versusthe tecip[X_L[ of mo|ecu]m"

weight. The data of standard samples (I): see text. iI

_-a22 i

J

MC S"

/ U_r1

100 200 300 400 500

MolecularWeight

FIGURE 5 Intmsic viscosities versus molecular weight of styrene oligomers.The data are from Table 3. O: [q], intrinsic viscosity; R: {q}, conected [q].

,3;3.8, -.,, '_._u_,._ ..... _,.: L.._L,,_,_h_ .... - - -.......

1000 , PS" + M/d,sm.mol.

loo "',.

_" 10"hl

10 11 12 13 14 15 16 17 18 19 20

RetentionTime(Minutes)

FIGURE 6 Universal calibration curve bY.polystyrenestandardsandsmall molecules. Instrument:a Waters _'LC pump, a Micro:_riticsModel 725 autosampler,a Waters Column oven, and a Waters411+ODR]detector; Column: PLgel 5Ix,Guard+ 100A; Mobile phase:telrahydrofuranwith 250 ppm BHT, 0.5 ml/min.

• [IV]*M,PS... _ <˜Ð�€�PS_, * M/d,smtool._

10

1 (IBz13 14 15 16 17 18 19

RetenUonTime(IVgnutes)

FIGURE 7 Expansion od Figure 6 at low molecularweight area.

339. .,._,,_....... _ ....... _....,..._