application of rheological techniques for investigations of polymer branched structures

Ž .Fluid Phase Equilibria 152 1998 327–336

Application of rheological techniques for investigations of polymerbranched structures 1

Z. Dobkowski )

Industrial Chemistry Research Institute, 8 Rydygiera Str., 01-793 Warsaw, Poland

Received 25 February 1997; accepted 7 January 1998

Abstract

Rheological properties, such as the zero shear rate viscosity h and the fluidity difference Dw between the0

Newtonian and non-Newtonian conditions, have been applied for investigations of branched structures ofŽ .polymers. Branching parameters that characterise the long chain branching LCB of macromolecules have been

Ž .determined using the multivariable power function MVP for the dependence of h and Dw on molecular0Ž . Ž .weight M , molecular weight distribution MWD and LCB. In particular, the exponent b of the MVP1

Ž . wŽ .b1 b2 b3 xfunction written as log DwPh s log Bq log h Pg Pq PG enables distinguishing linear and branched˙0 0

polymer structures. Experimental results for PDMS, PP, and PC have been discussed. Literature data for thesepolymers, as well as for PIB and PMMA, have also been shown for comparison of MVP linear master

Ž .dependencies. It has been found that the exponent b is equal to 0.76–0.79 approximately for linear polymers,1

and it is lower than 0.76 for branched ones. The lower the value of b , the higher the amount of branches. The1

quantitative dependence of b on branching degrees can be found, e.g., for PC b s0.30q0.47G. The MVP1 1

linear master dependencies are parallel for each type of polymer considered. It is assumed that their shift can bedependent on some specific constant for a polymer material. The investigations to find such a polymer materialconstant should be continued, and more experimental data are needed, in particular for polymer branchedstructures. q 1998 Elsevier Science B.V. All rights reserved.

Keywords: Rheological properties of polymers; Fluidity difference; Polymer structure parameters; Branching degree;Multivariable power function

1. Introduction

Rheology—a science on deformation and flow of materials—is particularly predestined forinvestigations of polymer materials that exhibit viscoelastic behaviour under a given external force.

) Corresponding author. Fax: q48-22-633-82-95.1 Paper presented at the International Conference on Applied Physical Chemistry, Warsaw, 13–15 November 1996.

0378-3812r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0378-3812 98 00187-3

( )Z. DobkowskirFluid Phase Equilibria 152 1998 327–336328

The ability to flow of polymers is a crucial property that determines not only the polymer preparationprocess, but also the processing method of a given polymer material and its method of forming, as

Ž .well as for applications of final polymer products. The measurement of the melt flow index MFI , orŽ . Ž .the melt flow rate MFR , has commonly been used and it is still used for a preliminary assessment

w x Žof polymer flow properties 1 . It is applied as a simple and quick test at one point on the shear.stress–shear rate flow curve for the control of polymer production, as well as a measure for

Ždistinguishing various polymer grades. Usually, the dependence of polymer flow properties i.e.,. Žrheological properties, such as the viscosity or the shear modulus on the shear rate or on the

.frequency is measured. Hence, it is observed that rheological properties depend, among others, on theŽ .structure of macromolecules, i.e., on the molecular weight M that corresponds to the macro-

Ž . Ž .molecule chain length, molecular weight distribution MWD , and long chain branching LCB ofw x Ž .macromolecules 2–5 . Recently, the multivariable power function MVP has been applied for the

zero shear rate viscosity h and for the fluidity difference Dw to describe simultaneously their0w xdependence on M, MWD and LCB 3,6,7 . Therefore, the inverse problem, i.e., the estimation of M,

MWD and LCB parameters from rheological properties can be solved. Thus, the weight-averageŽ .molecular weight, M , as well as branching degrees or indices of branching as measures of LCBww xcan, in principle, be determined, or at least estimated 8 . In this paper, randomly branched structures

of polydisperse polymers are considered. The MVP function has been applied for the zero shear rateviscosity h and for the fluidity difference Dw to determine branching parameters characterising LCB0

of polymers. Thus, the linear and branched polymer structures can qualitatively be distinguished. Thepossibility for quantitative determinations of the branching degree has also been described. Examples

Ž . w xare given for polycarbonate PC , taking published data into account 3,6–11 , as well as forŽ . Ž . w xpolydimethylsiloxane PDMS and polypropylene PP this work . It has been found that the MVP

linear master dependencies are parallel for each type of polymer considered. Hence, it is supposedthat their shift can be dependent on some specific constant for a given polymer material. Theinvestigations to find such a polymer material constant should be continued.

2. Theory

The MVP function has been proposed for describing the dependence of polymer properties onw xparameters of their molecular characteristics 3,6 . The following parameters have been considered:

Ž .a the average molecular weights M , where x can be n, v, w or z for the number-, viscosity-,xŽ .weight- or z-average molecular weights; b the polydispersity degree qsM rM as a measure ofw n

Ž .the molecular weight distribution MWD , where q is the ratio of weight-to-number averageŽ .molecular weights; and c the branching degree, e.g., G as the ratio of intrinsic viscosities of

w xbranched and linear macromolecules, cf. definitions in Refs. 3,7–9 . Several polymer properties havew xbeen considered, including rheological properties, such as the zero shear rate melt viscosity h 7 ,0

where

h sKPM a1 Pq a2 PGa3 1Ž .0 x

w xthe melt flow index 9 , and the fluidity ws1rh or the fluidity difference Dw between the

( )Z. DobkowskirFluid Phase Equilibria 152 1998 327–336 329

w xNewtonian and non-Newtonian conditions 3,7,10,11 , where Dws1rhy1rh . Taking Dw into0w xaccount, the linear master dependence in the form of MVP function has been found 3,10,11 , i.e.,

b1 b b2 3log DwPh s log Bq log h Pg Pq PG 2Ž . Ž .Ž .˙0 0

where g is the shear rate. The shear rate g can be replaced by the frequency v in the case of dynamic˙ ˙w xmeasurements, according to the Cox–Merz rule 12 . The exponents a , a and a , and b , b and b1 2 3 1 2 3

can be found from respective regression equations using the experimental data from rheologicalmeasurements.

Moreover, combining MVP functions for the zero shear rate viscosity, for the intrinsic viscosity ofa solution and for the glass transition temperature, a semi-empirical dependence of branching degreeŽ . Ž .g on the branching functionality f and average number of branch points per macromoleculeŽ . w xn was found 3,13,14 :b x f

kg s 0.5 fy1 n q1 3Ž . Ž .f x b x f

where the exponent is given by

ksya r bPa 4Ž . Ž .1s 3t

where, in turn, a is the exponent of the Mark–Houwink equation for a given polymer–solvent1s

system at a given temperature, and a is the branching exponent of the MVP function for the glass3t

transition temperature. The magnitude b is the exponent in the relationship between the branchingŽ . Ž .degrees defined by the intrinsic viscosities G and by the mean square radii of gyration g , i.e., in

the relationship

Gsg b 5Ž .Ž .The exponent b can be found from rheological measurements for h , according to Eq. 1 . Hence,0

w xbsa ra , cf. Refs. 3,6,7 .1 3Ž . w xUsing Eq. 3 , the theoretical equations of Zimm and Stockmayer 15 can be compared with

w xexperimental data 3,13 , as well as the influence of solvent on branching parameters can qualitativelyw xbe assessed 3,16 .

3. Experimental

Ž .The following polymer materials have been considered: bisphenol A polycarbonate PC of variousŽ .grades from various producers; polydimethylsiloxane PDMS , certified reference material aCF368,Ž . Ždelivered by Rheometrics, Germany; polypropylene PP , Malen 330, copolymer about 15% of

.ethylene , delivered by ‘Petrochemia’, Płock, Poland.w xExperimental data on PC have been taken from the quoted papers 3,7,10,11,13,16 . The capillary

viscometer of Instron has been used for measurements of melt viscosity h vs. shear rate g .˙Results for PDMS and PP have been developed for this paper from recent measurements. The

Ž .RDS-2 rheospectrometer Rheometrics has been used for measurements of viscosities, shear moduliw xand tan d , as a function of frequency v, cf. Ref. 8 .


4. Results and discussion

4.1. PC

ŽDiscussion of results on PC revealed that the following branching parameters or indices of. w xbranching can be found 3,7–11,13,16 .

Ž . Ž .1 The exponent b in the relationship between the branching degrees, Eq. 5 , has been foundŽ . w xfrom Eq. 1 . Hence, bs1.31 for randomly branched PC, cf. Refs. 3,7 .

Ž . Ž .2 The exponent b in Eq. 2 was found to be an index of branching. It is equal to approximately1

0.76–0.77 for linear samples, and its lower values b -0.76 are indicating the presence of branched1w xmacromolecules in measured samples 7 , see Fig. 1. It should be noted that the individual straight

lines were obtained for the samples of specified branching degrees, independently of their combinedh –M –q characteristics.0 x

Thus, qualitatively, the lower the value of b , the higher the amount of branches. For quantitative1

determination of a branching degree, the function of b vs. defined branching degree, e.g., vs. G, has1w xto be known. Example, for PC, the following equation was proposed 7,10

b s0.30q0.47G 6Ž .1

It should be noted that for linear polymers, the exponent corresponding to b has been found1w xtheoretically by Graessley 17 for the exponential MWD function with the parameter Zs1, where

b s0.767. Thus, the values of b higher than 0.77"0.01 can be ascribed to experimental1 1

inaccuracy.Ž .For unknown polymer samples, Eq. 2 can be written as

b1log DwPh s log K q log h Pg 7Ž . Ž .Ž .˙0 B 0

where K is the constant, K sBPq b2 PGb3. Hence, the exponent b and the constant K can beB B 1 B

found.Ž . Ž .3 The exponent k in Eq. 3 , ksy1.30, has been found for PC. Then, the theoretical equation

w xof Zimm and Stockmayer 15 for PC functionality fs3 has been compared with experimental data

Ž . Ž .Fig. 1. Linear characteristics of PC samples: DwPh s f gPh . 1 Linear PC samples of various values of h , M and q;˙0 0 0 xŽ . Ž . Ž . Ž . Ž .Gs1. 2 to 5 Branched PC samples of various values of h , M , q and G: 2 Gs0.93; 3 Gs0.83–0.085; 40 x

Ž .Gs0.80–0.82; 5 Gs0.63.


Fig. 2. Dependence of the average branching degree g on the average number of branch points per macromolecule nf x b x fŽ .for PC. Branching functionality f s3. Solid line: experimental data, Eq. 3 ; dashed line: the theoretical dependence of

w xZimm and Stockmayer 15 for f s3.

w x3,13 , cf. Fig. 2. The discrepancy is explained by the fact that the theoretical equations of Zimm andŽStockmayer have been derived with the assumption of the statistics of Flory i.e., for bs1.5, while

.experimental values are usually lower, even down to 0.5 for a certain type of branching for a thetaŽ .solvent i.e., for a s0.5 , not taking end-group volume effects into account. Experimental values for1s

PC were bs1.31 and a s0.82, and end-group volume effects were taken by the exponent a into1s 3t

account. Moreover, the influence of solvent on branching parameters can qualitatively be assessedw x w xusing the exponent k 3,16 . In particular, it has been found earlier, cf. Ref. 16 , that the exponent k

Ž . Ž .in Eq. 3 , see also Eq. 4 , is independent of the solvent quality, while the exponent b depends on aŽsolvent quality and the value of b for a good solvent is higher than that for a theta solvent for more

w x.detailed discussion, see Ref. 16 .

4.2. PDMS

The values of the zero shear rate viscosity h have been calculated from experimental dependence0

of h vs. v at 303 K. The Rhios 4.0 Rheocalc program of Rheometrics has been used. Then, theŽ .reduced values of DwPh vs. vPh have been plotted according to Eq. 7 , Fig. 3. The linear0 0

dependence has been obtained with the slope b s0.76 as for linear polymer samples. Literature data1w xfor linear and branched PDMS samples 18,19 have been plotted vs. gh , as shown also in Fig. 3 for˙ 0

comparison. Thus, it has been found that the exponent b is equal to about 0.76 for the linear1Ž w xsamples. It is lower for the branched PDMS sample the sample BO in Ref. 18 , in Fig. 3 denoted by

. Žq points , and b s0.69 in this case. The slope for another sample considered as branched one the1w x .sample BG in Ref. 18 , of low branching degree, in Fig. 3 denoted by = points is b s0.77, cf.1

Table 1. It suggests that the structure of macromolecules in this sample is linear. It seems possiblethat branched structures of PDMS macromolecules in this case are not significant andror they arehighly regular and such macromolecules could be considered as linear ones of specific but notdisclosed structural unit.

w xIt has also been observed that the Cox–Merz rule 12 was satisfied for PDMS samples, i.e., theindependent variables gh or vh may be applied equivalently for the plots in Fig. 3. Thus,˙ 0 0

experimental data obtained in various modes of rheological techniques, e.g., using capillary or


Ž . Ž .Fig. 3. Linear characteristics of PDMS samples: DwPh s f gPh or vPh . PDMS standard, according to Fig. 3 I ;˙0 0 0w x Ž . Ž . w x Ž .PDMS according to Ref. 18 : linear v, `, B and branched q, = ; PDMS according to Ref. 19 \, ^, ^ .

rotational viscometers, are comparable. Moreover, one straight line has been obtained for linearPDMS samples of various h , i.e., of various M, at different temperatures. It conforms to the earlier0

observation for linear PC samples, see Fig. 1.

4.3. PP

The same procedure done on PDMS has been applied for PP samples. Thus, the samples of MalenŽ .330 PP were measured at different temperatures, i.e., in the range of 453 to 493 K 180–2208C . The

Ž .reduced values of Dwh vs. vh have been plotted according to Eq. 7 , and one common straight0 0

line has been obtained, independently of temperature of measurements. The slope b about 0.54 has1

been found in this case. It suggests that the investigated Malen 330 PP samples exhibit a branchedstructure of macromolecules. Unfortunately, numerical values of the branching degree were notdetermined.

4.4. Comparison of the linear master dependence for Õarious polymers

Results for described polymer samples have been compared with literature experimental data forŽ .various polymers. The data can be treated according to the MVP function, Eq. 2 , if the values of

branching degrees are known for branched samples. Thus, the set of data for linear and branchedsamples made it possible to calculate the exponents b , b and b and the constant B. This has been1 2 3

w x Ž .the case for PC 10,20 , see Table 1. For unknown samples, the data were treated according to Eq. 7


Table 1Linear master dependence for selected polymers

cŽ .Polymer Sample Reference Mode Temperature 8C b B1

aŽ . w xPP = A PP1, lin 22 v 175 0.70 8.91 ey4Ž .B PP2, br 0.59 3.16 ey3Ž .C PP3, br 0.54 1.12 ey2

w x Ž .C Malen this work v 180–220 0.54"0.01 1.19 ey2w x Ž .C NPL this work v 210 0.59 1.01 ey2w xPDMS A lin 18,19 g 21r23 0.76 3.55 ey4˙w xA lin this work v 30 0.76 3.48 ey4

b w x Ž .B br 18 g 23 0.767 5.62 ey4˙w x Ž .C br 18 g 23 0.686 1.78 ey3˙w xPIB lin 19 g 21 0.78 1.26 ey4˙w xPMMA Lucryl, lin 21 g , v 190–230 0.76 1.00 ey4˙w x w xPC lin 10,20 not given 300 0.756 1.78 ey5

br 0.686br 0.675

w xlin 7 g 280 0.79 2.51 ey5˙w xlin 10 g 280 0.77 3.55 ey5˙

br 0.70br 0.66br 0.60

a w xThe sample is considered in Ref. 22 as linear one, however, the exponent b s0.70 suggests a distinct branching effect1

since b -0.76.1b w xThe sample is considered in Ref. 18 as branched one, however, the exponent b s0.767 suggests that the sample is linear.1

It seems possible that branched structures of PDMS macromolecules in this case are highly regular and such macromoleculescould be considered as linear ones of specific but not disclosed structural unit. Or the branching degree can be considered as

w xinsignificant, cf. Ref. 18 .c Ž .The values in parentheses correspond to the values of K since the branching degree was not known, cf. Eq. 7 .BŽ .= Capital letters A, B, C denote respective straight lines in Fig. 4.

Žto determine the exponent b and the constant B in the case of linear samples providing the1w x.exponent b s0, as it has been found for PC 7 . In the case of branched samples, where the2

numerical values for the branching degrees were not known, the value of K can only be found.B

Nevertheless, the samples can be classified qualitatively in respect of their branching, see Table 1 forPP and PDMS. Thus, for samples of unknown branching degree, it has been assumed that effects ofmolecular weight polydispersity and branching can be included into the constant K sBPq b2 PGb3,B

Ž .cf. Eq. 7 .The results of calculations are shown in Table 1 and in Fig. 4. The data in Fig. 4 have been plotted

Ž . Ž . wŽ . b1x wŽ . b1x Žaccording to Eq. 7 , where Ys log DwPh and Xs log h Pg or Xs log h Pv since the˙0 0 0.use of g and v has been found to be equivalent . Thus, parallel straight lines have been obtained for˙

all polymers here considered. Hence, for a given polymer, one master line should be obtained forlinear and branched samples giving the constant B. It may be supposed that the master lines forpolymer types can be shifted into one common universal master line. Such shift can be dependent onsome specific parameter, presumably structural one, for a given polymer material. More experimentaldata, however, are needed, in particular, numerical values for branching degrees to calculate the


Fig. 4. The linear master dependence for selected polymers. Straight lines denoted as in Table 1.

values of constant B. The investigations to find such a polymer material structural parameter and auniversal constant should be continued.

5. Conclusions

The discussed approach, using various techniques of rheological measurements for the viscosity ofpolymer materials and the multivariable power function for data treatment, can be applied fordistinguishing linear and branched structures of macromolecules and for estimation of the extent ofbranching. The comparison of theoretical and experimental data can also be performed for branchedpolymers.

Branching parameters as exponents of respective MVP functions can be found from experimentalŽ .data. Thus, it has been found that the exponent b is equal to 0.76–0.79 approximately for linear1

polymers and it is lower than 0.76 for branched ones. The lower the value of b , the higher the1

amount of branches. The quantitative dependence of b on branching degrees can be found.1

It has also been found that the MVP linear master dependencies are parallel for each type ofpolymer considered. Hence, it is supposed that their shift can be dependent on some specific structural


parameter, constant for a given polymer material. The investigations to find such a polymer materialconstant should be continued. However, more experimental data, in particular, numerical values forbranching degrees are needed.

6. List of symbols

Ž .a , a , a constants, exponents of MVP function for h s f M ,q,G1 2 3 0 x

b the exponent for the relationship between G and g branching degreesŽ .b , b , b constants, exponents of MVP function for DwPh s f gPh , q, G˙1 2 3 0 0

Ž .B the constant in the linear master dependence, Eq. 2f the branching functionality, i.e., the number of end groups per macromoleculeg the branching degree defined as the ratio of the radii of gyration of branched

and linear macromoleculesŽ . Ž .g the x-average branching degree g for a given branching functionality ff x

G the branching degree defined as the ratio of intrinsic viscosities of branched andlinear macromolecules

Ž .ksya r bPa the exponent of semi-empirical equation for g , where a is the exponent of1s 3t f x 1s

the extended Mark–Houwink equation of a given polymer–solvent system at agiven temperature, a is the branching exponent of the MVP function for the3t

glass transition temperatureŽ .K the constant for the h s f M ,q,G dependence0 x

Ž . b2 b3K the constant in Eq. 7 , K sBPq PGB B

LCB the long chain branchingM the molecular weightM the average molecular weights M , where x can be n, v, w or z for thex x

number-, viscosity-, weight- or z-average molecular weightsMVP the multivariable power functionMWD the molecular weight distribution

Ž .n the x-average number of branch points per macromolecule n for a givenb x f bŽ .branching functionality f

qsM rM the polydispersity degreew n

g the shear rate, sy1˙h the viscosity, Pa sh the zero shear rate viscosity, Pa s0

Ž .y1ws1rh the fluidity, Pa sDws1rhy1rh the fluidity difference between the Newtonian and non-Newtonian conditions,0

Ž .y1Pa sv the frequency, sy1

References

w x1 J.F. Petersen, Praktische Rheometrie von Kunststoffschmelzen, in: Praktische Rheologie der Kunststoffe, VDI-Verlag,Dusseldorf, 1978, p. 45.

w x2 J.D. Ferry, Viscoelastic Properties of Polymers, 3rd edn., Wiley, New York, 1980.


w x Ž .3 Z. Dobkowski, General Dependence of Polymer Properties on their Molecular Characteristics in Polish , Prace nauk.,ser. Chemia, z. 43. Warsaw University of Technology Publications, Warsaw, 1988.

w x4 H.A. Barnes, J.F. Hutton, K. Walters, An Introduction to Rheology, Elsevier, Amsterdam, 1989.w x5 C.L. Rohn, Analytical Polymer Rheology, Hanser, Munich, 1995.w x Ž .6 Z. Dobkowski, Eur. Polym. J. 17 1981 1131.w x Ž .7 Z. Dobkowski, Eur. Polym. J. 18 1982 1051.w x Ž .8 Z. Dobkowski, M. Zielonka, Polimery 42 1997 321.w x Ž .9 Z. Dobkowski, Rheol. Acta 25 1986 195.

w x Ž .10 Z. Dobkowski, Polym. Bull. 19 1988 165.w x Ž .11 Z. Dobkowski, The linearization of master curves for polymer condensed systems, in: C. Gallegos Ed. , Progress and

Trends in Rheology, IV, Steinkopff Verlag, Darmstadt, 1994, p. 365.w x Ž .12 W.P. Cox, E.H. Merz, J. Polym. Sci. 28 1958 619.w x Ž .13 Z. Dobkowski, J. Appl. Polym. Sci. 28 1983 3105.w x14 Z. Dobkowski, Determination of long chain branching of polymers by multitechnique procedures, in: J. Mitchell, Jr.

Ž .Ed. , Applied Polymer Analysis and Characterization, Hanser Publishers, Munich, 1987, p. 341.w x Ž .15 B.H. Zimm, W.H. Stockmayer, J. Chem. Phys. 17 1949 1301.w x Ž .16 Z. Dobkowski, J. Appl. Polym. Sci. 30 1985 355.w x Ž .17 W.W. Graessley, J. Chem. Phys. 47 1967 1942.w x Ž .18 N. El Kissi, J.M. Piau, P. Attane, G. Turrel, Rheol. Acta 32 1993 293.w x Ž .19 N. Ohl, W. Gleissle, J. Rheol. 37 1993 381.w x20 K. Idel, D. Freitag, W. Nouvertne, Ger. Offen. 2718466, Bayer, 2 November 1978.w x21 M. Lecomte, IUPAC WP 4.2.1 Report, Shell Research, Louvain-la-Neuve, August 1993.w x Ž .22 R. Hingmann, B.L. Marczinke, J. Rheol. 38 1994 573.

application of rheological techniques for investigations of polymer branched structures

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