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The solution viscosity of polybenzyls
European Polymer Journal. 1970, Vol. 6, 7P- I553-1558. Pergamon Press. Printed in England.
THE SOLUTION VISCOSITY OF POLYBENZYLS DEREK B. V. PARKER a n d J. BARRY MYERS Department of Polymer Science and Technology, Brunel University, London, W.3, EnNand
( Recei~'ed 18 February 1970) Abstract--The viscosities of solutions of polybenzyls have been studied according to the view that polybenzyl molecules are highly branched and roughly spherical in shape. A relation between molecular weight and solution viscosity is suggested for these and similar polymers, based on Happel's theory of the viscosity of suspensions of r i n d spheres.
EVER since the early work of Bezzi, cl) attempts have been made to determine, or at least to compare, the molecular weights of samples of polybenzyl by solution ~iscosity methods. However, there is no agreement as to how the viscosity data for the polymer are to be converted into values of molecular weight. There are at least two factors which render the conventional Mark-Houwink treatment inapplicable to polybenzyls. First, these polymers are highly branched, being more or less globular in shape, and quite unlike the flexible chains for which the Mark-Houwink equation was given theoretical justification. The degree of branching and the relative stiffness of the chains probably render the correction for branching due to Zimm and Stockmaye¢ 2~ inapplicable. Second, the soluble polybenzyls are always low polymers with molecular weights rarely exceeding 5000. Thus solutions of relatively high concentration must be used to obtain accurate values of viscosity number. It was therefore considered that one of the theories applying to the viscosity of suspensions of spheres might be applicable to solutions of polybenzyls. The classical theo~' is that due to Einstein/3~ as developed by Simha, Guth, Gold, Vand and others. (4~ This treatment produces relationship of the type: ~?, = I -- A~p '-- Bq) 2 ~- Ctp 3 -~- . . . . . . . where ~o is the volume fraction of the suspended spheres. The constant A is almost universally given the value of 2.5, but there is much less agreement on the value of the coefficient of q~-'. Quoted values range from 7.349 (~¢~ to 16.4. (~a~ Little consideration has been given to the coefficients of higher powers of ~o. The relationship may be written as: ~sp
-- A
-E B q o - S
.......
and if, by analogy with the usual Mark-Houwink procedure, V,p/cpis plotted against % whatever the slope of the curve may be, the intercept should be 2.5. Another treatment of the problem is due to Happel cs~ who derived the equation: 72, = 1 -4:- 5.5 g~ where g is a function of ~o which varies only slowly with cp at low values but which increases more rapidly for higher volume fractions. As ~p --> 0, g --~ 1, hence if ~p/~ is 1553
1554
DEREK B. V. PARKER and J. BARRY MYERS
plotted against 9~, the intercept is 5-5, while the slope is initially ve~" low. The function g is such that log -q~ is virtually linear with ~ up to the point where ~ = 0.3. Other theories have been put forward which also relate log ~, with 9~. These various treatments have been reviewed briefly. (6~ They all appear to agree on one important particular, viz. that the viscosity o f suspensions of spheres, while dependent on the total volume fraction of the particles, is independent of their diameter. If polybenzyl molecules were quite rigid, therefore, it would be expected that the viscosity of their solutions would be independent of their molecular weight. Previously published results (~ do not confirm this. There are indications that the viscosity of suspensions o f spheres is dependent on particle size distribution, especially at high volume fractions; there is reason to believe that the polybenzyls usually have a narrow distribution o f molecular weights.
EXPERIMENTAL Samples of polybenzyl were prepared by the action of stannic chloride on benzyl chloride as described by Valentine and Winter. ctb~ As indicated by Parker et al., ~°~ polybenz).Is with molecular weights greater than about 3000 can only be obtained by the use of large proportions of Lewis acid catalyst. Four samples were prepared, and these were purified by precipitation from dioxane. Impurities of low molecular weight were removed by heating in v a c u o for 10 hr at 150°. The molecular weights of the samples were determined by vapour pressure "osmometry" using a Mechrolab instrument, with benzene as solvent and anthracene as standard.
15--
/
L3
2
g~ C tC-E N o
a-
> ~..~"~'~"
_
~ l o p p e l
~L
0
l
0 05
1
0 I0
0-15
Volume fraction,
FIG. 1. Plot of viscosity number vs. volume fraction for four samples of polybenzyl. The Happel function is similarly plotted.
The Solution Viscosity of Polybenzyls
i555
Solutions (concentrations ranging from 10 to 150 kg m-3) of the polybenzyl samples were made in toluene, and the viscosity ratios were determined for each, using a suspended level capillaw viscometer, at 2Y, 3Y, and 45:. The density of each solution was also compared with that of toluene, in order that the volume fraction of polybenzyl could be determined tbr each solution. RESULTS The results for viscosity determinations at 25: are displayed graphically in two ways. Figure 1 shows ~sp/P against % while Fig. 2 is a plot o f log to ('0/'70) against ~. On both charts, the curve representing the Happel relationship is also plotted. The molecular weights o f the samples, as determined by the vapour pressure osmometer, are recorded in Table 1. TABLE
1
Happel function
I
2- 06
2.40
Mean slope of log (,7/~1o):6curve "Expansion factor", i.e. ratio of mean slope for polymer samples to that for Happel function Molecular weight of polymer samples
2" 56
3"07
1'24
1"49
1"79
2340
3600
4700
c_~
.2 2
E x:: z
g o
2
"{appel
005
0 i0
3' 68
1880
o
o u
IV
1'17
/
'3 -:
~
Polymer sample II III
015
Volume froction,
FIG. 2. Plot of logarithm of viscosity ratio vs. volume fraction for four samples of polybenz3"l. The Happel function is similarly plotted.
1556
DEREK B. V. PARKER and J. BARRY MYERS
The effect of temperature variation on ~7, was very small. Over the range studied (25°-45 °) the variation was substantially linear; for all four polymers the change in '7, for a solution of concentration 50 kg m -3 was approximately --0- 15 per cent per degree rise in temperature. DISCUSSION Reference to Fig. 1 shows that when dealing with the polybenzyls with very, low molecular weights, experimental error is of such magnitude that extrapolation to 9~ --> 0 is difficult. This difficulty is exacerbated by the fact that in the equation: ~7~_.£= A '-- Bqv ~- C~-" -+- . . . . q~ the coefficients of powers of q~above unity are not negligible, i.e. the lines are far from straight. It is of interest to note that the intercepts all have values slightly in excess of 5.5 which is that for the Happel function. This may be expressed as:
4~ w3
V, -- 1 ~o
-=5.5
~
10 - - Ti_t~84,2/3
10(1 -- ~v1°'3) -- 25cp(1 -- ~ / 3 )
and the curve representing this has also been plotted against ~oin Fig. I. (It is of interest that this curve has a minimum of 5.44 at about ~o= 0.009.) The experimental curves shown in Fig. 1 are all above the Happel curve, and exhibit greater curvature. They may be regarded as comprising a family which tends towards the Happel curve as the molecular weight tends to zero. The values of the intercepts as q0 --> 0 are, however, uncertain, and differ relatively slightly with molecular weight. In Fig. 2, the Happel function is plotted as loglo r/, against ~. This is a line through the origin with only very slight curvature. The experimental results also produce points lying on a family of nearly straight lines passing through the oriNn. The slopes of these curves increase with molecular weight, the Happel curve forming a limit as molecular weight tends to zero.
20--
~ /
oo ._~
I-5 --
x '"
lZ
I jl 1 1 i0 ~
f
[ I000
J I 2000
I 3000
Molecular
! 4000
i 5000
~ bS,S
weight
F[o. 3. Plot of "Expansion factor" vs. molecular weight of polybertzyis.
The Solution Viscosity of Polybenzyls
1557
Table 1 contains values of an "Expansion factor", i.e. the ratio of the mean slopes of the experimental curves of Fig. 2 to the slope of the Happel curve. These ratios are plotted against molecular weights in Fig. 3. It will be noted that the points lie on a smooth curve, and, though the relationship is not quite linear, the curve could be used to determine molecular weights from viscosity data. It should be noted that Fig. 2 is plotted using apparent volume fraction as abscissa. In fact, as the apparent solute density of polybenzyl in toluene does not vary appreciably with molecular weight, it would suffi.ce to plot log ,/, against concentration. The advantage of this procedure over any based on the limiting viscosity number is that the curve of log rlr against 9~is substantially linear up to quite high values of volume fraction, and hence the slope of the mean straight line is not greatly affected by inaccuracies in the viscosity determinations at low concentrations. This is in contrast with the curves of viscosity number against 9~, where values obtained at high volume fractions are virtually without effect on the accuracy of the extrapolation to limiting viscosity number. As has been pointed out, if the molecules of polymer were rigid spheres, molecular weight would not be expected to influence solution viscosity. The tact that, for polybenzyls, viscosity is not independent of molecular weight must imply that, when surrounded by a good solvent, the polymer molecule is able to expand by bond rotation so that its effective boundary is greater in extent than when the polymer is in the glassy state. Some molecules of solvent will be immobilized in the interstices of the polymer molecule. Study of a diagram of polybenzyl such as that published (v~ makes it clear that such expansion is possible, and evidently the proportional expansion increases as the size of the molecule increases, since our experimental "Exansion factor" increases with molecular weight. It is perhaps significant that the variation of viscosity ratio with temperature is very slight and negative. This would be expected if the molecule is composed of numerous branches which are fully extended except for slight shortening caused by thermal motion. It is possible that this method of treatment of viscosity data may be applicable to other polymers which are highly branched and therefore approximately globular in shape. However, it must be noted that the shape of the curve in Fig. 3 will depend very much on the polymer concerned, since the magnitude of the "Expansion factor" clearly depends on the details of molecular structure.
REFERENCES (1) (a) S. Bezzi, Gazz. chim. ital. 66, 491 (1936). (b) L. Valentine and R. W. Winter, J. chem. Soc. 4768 (1956). (c) D. B. V. Parker, W. G. Davies and K. D. South, J. chem. Soc. (B), 471 (1967). (2) B. H. Zimm and W. H. Stockmayer, J. chem. Phys. 17, 1301 (I949). (3) A. Einstein, Ann. Phys. 19, 289 (1906). (4) (a) E. Guth and R. Simha, Kolloidzeitschrift 74, 266 (1936). (b) E. Guth and O. Gold, Phys. Ret'. 53, 322 (1939). (c) V. Vand, .i". phys. Coll. Chem. 52, 277 (1948). (d) E. Guth, Rubb. Chem. Technol. 23, 635 (1950). (e) R. Sirnha, J. appl. Phys. 23, 1020 (1952). (5) J. Happel, J. appl. Phys. 28, 1288 (1957). (6) F. Horsfall, The t'iscosity of suspensions of rigid spheres, R.A.P.R.A. Research Report No. 175 (1969). (7) D. B. V. Parker, Europ. Polym. J. 5, 93 (1969).
1558
D E R E K B. V. P A R K E R and J. BARRY MYERS
R~sum~--Les viscosit~s de solutions de polybenzyles ont ~t6 6tudi~es en tenant compte de ce que les mol6cules de polybenzyle sont fortement ramifi6es et sensiblement sph~riques. Pour ces polym6rs et ceux qui leurs sont semblabtes, on propose une relation entre la masse mol6culaire et la viscosit6 de la solution, bas8e sur la theorie de Happel sur la viscosit6 des suspensions de spheres rigides.
Sommario---Si ~ studiata la viscosita di soluzioni di polibenzili, dal punto di vista che le molecole di polibenzile sono molto ramificate e di forma approssimativamente sferica. Per questi e simili polimeri, si avanza una teoria per cui esiste una relazione tra peso molecolare e viscosit5, di soluzione, basata sulla teoria di Happel circa la viscositb, di sospensioni di sfere rigide. Zusammenfassung--Die Viskosit~.ten der L{Ssungen yon Polybenzylen warden untersucht im Hinblick darauf, dab Polybenzylmolekfile hoch verzweigt sind und etwa Kugelgestalt besitzen. F/Jr diese und ~ihnliche Polymere wird eine Beziehung zwischen Molekulargewicht und .L6sungs~iskosit~.t vorgeschlagen, die auf der Theorie von Happel tiber die Viskositat yon Suspensionen starrer Kugeln basiert.
THE SOLUTION VISCOSITY OF POLYBENZYLS DEREK B. V. PARKER a n d J. BARRY MYERS Department of Polymer Science and Technology, Brunel University, London, W.3, EnNand
( Recei~'ed 18 February 1970) Abstract--The viscosities of solutions of polybenzyls have been studied according to the view that polybenzyl molecules are highly branched and roughly spherical in shape. A relation between molecular weight and solution viscosity is suggested for these and similar polymers, based on Happel's theory of the viscosity of suspensions of r i n d spheres.
EVER since the early work of Bezzi, cl) attempts have been made to determine, or at least to compare, the molecular weights of samples of polybenzyl by solution ~iscosity methods. However, there is no agreement as to how the viscosity data for the polymer are to be converted into values of molecular weight. There are at least two factors which render the conventional Mark-Houwink treatment inapplicable to polybenzyls. First, these polymers are highly branched, being more or less globular in shape, and quite unlike the flexible chains for which the Mark-Houwink equation was given theoretical justification. The degree of branching and the relative stiffness of the chains probably render the correction for branching due to Zimm and Stockmaye¢ 2~ inapplicable. Second, the soluble polybenzyls are always low polymers with molecular weights rarely exceeding 5000. Thus solutions of relatively high concentration must be used to obtain accurate values of viscosity number. It was therefore considered that one of the theories applying to the viscosity of suspensions of spheres might be applicable to solutions of polybenzyls. The classical theo~' is that due to Einstein/3~ as developed by Simha, Guth, Gold, Vand and others. (4~ This treatment produces relationship of the type: ~?, = I -- A~p '-- Bq) 2 ~- Ctp 3 -~- . . . . . . . where ~o is the volume fraction of the suspended spheres. The constant A is almost universally given the value of 2.5, but there is much less agreement on the value of the coefficient of q~-'. Quoted values range from 7.349 (~¢~ to 16.4. (~a~ Little consideration has been given to the coefficients of higher powers of ~o. The relationship may be written as: ~sp
-- A
-E B q o - S
.......
and if, by analogy with the usual Mark-Houwink procedure, V,p/cpis plotted against % whatever the slope of the curve may be, the intercept should be 2.5. Another treatment of the problem is due to Happel cs~ who derived the equation: 72, = 1 -4:- 5.5 g~ where g is a function of ~o which varies only slowly with cp at low values but which increases more rapidly for higher volume fractions. As ~p --> 0, g --~ 1, hence if ~p/~ is 1553
1554
DEREK B. V. PARKER and J. BARRY MYERS
plotted against 9~, the intercept is 5-5, while the slope is initially ve~" low. The function g is such that log -q~ is virtually linear with ~ up to the point where ~ = 0.3. Other theories have been put forward which also relate log ~, with 9~. These various treatments have been reviewed briefly. (6~ They all appear to agree on one important particular, viz. that the viscosity o f suspensions of spheres, while dependent on the total volume fraction of the particles, is independent of their diameter. If polybenzyl molecules were quite rigid, therefore, it would be expected that the viscosity of their solutions would be independent of their molecular weight. Previously published results (~ do not confirm this. There are indications that the viscosity of suspensions o f spheres is dependent on particle size distribution, especially at high volume fractions; there is reason to believe that the polybenzyls usually have a narrow distribution o f molecular weights.
EXPERIMENTAL Samples of polybenzyl were prepared by the action of stannic chloride on benzyl chloride as described by Valentine and Winter. ctb~ As indicated by Parker et al., ~°~ polybenz).Is with molecular weights greater than about 3000 can only be obtained by the use of large proportions of Lewis acid catalyst. Four samples were prepared, and these were purified by precipitation from dioxane. Impurities of low molecular weight were removed by heating in v a c u o for 10 hr at 150°. The molecular weights of the samples were determined by vapour pressure "osmometry" using a Mechrolab instrument, with benzene as solvent and anthracene as standard.
15--
/
L3
2
g~ C tC-E N o
a-
> ~..~"~'~"
_
~ l o p p e l
~L
0
l
0 05
1
0 I0
0-15
Volume fraction,
FIG. 1. Plot of viscosity number vs. volume fraction for four samples of polybenzyl. The Happel function is similarly plotted.
The Solution Viscosity of Polybenzyls
i555
Solutions (concentrations ranging from 10 to 150 kg m-3) of the polybenzyl samples were made in toluene, and the viscosity ratios were determined for each, using a suspended level capillaw viscometer, at 2Y, 3Y, and 45:. The density of each solution was also compared with that of toluene, in order that the volume fraction of polybenzyl could be determined tbr each solution. RESULTS The results for viscosity determinations at 25: are displayed graphically in two ways. Figure 1 shows ~sp/P against % while Fig. 2 is a plot o f log to ('0/'70) against ~. On both charts, the curve representing the Happel relationship is also plotted. The molecular weights o f the samples, as determined by the vapour pressure osmometer, are recorded in Table 1. TABLE
1
Happel function
I
2- 06
2.40
Mean slope of log (,7/~1o):6curve "Expansion factor", i.e. ratio of mean slope for polymer samples to that for Happel function Molecular weight of polymer samples
2" 56
3"07
1'24
1"49
1"79
2340
3600
4700
c_~
.2 2
E x:: z
g o
2
"{appel
005
0 i0
3' 68
1880
o
o u
IV
1'17
/
'3 -:
~
Polymer sample II III
015
Volume froction,
FIG. 2. Plot of logarithm of viscosity ratio vs. volume fraction for four samples of polybenz3"l. The Happel function is similarly plotted.
1556
DEREK B. V. PARKER and J. BARRY MYERS
The effect of temperature variation on ~7, was very small. Over the range studied (25°-45 °) the variation was substantially linear; for all four polymers the change in '7, for a solution of concentration 50 kg m -3 was approximately --0- 15 per cent per degree rise in temperature. DISCUSSION Reference to Fig. 1 shows that when dealing with the polybenzyls with very, low molecular weights, experimental error is of such magnitude that extrapolation to 9~ --> 0 is difficult. This difficulty is exacerbated by the fact that in the equation: ~7~_.£= A '-- Bqv ~- C~-" -+- . . . . q~ the coefficients of powers of q~above unity are not negligible, i.e. the lines are far from straight. It is of interest to note that the intercepts all have values slightly in excess of 5.5 which is that for the Happel function. This may be expressed as:
4~ w3
V, -- 1 ~o
-=5.5
~
10 - - Ti_t~84,2/3
10(1 -- ~v1°'3) -- 25cp(1 -- ~ / 3 )
and the curve representing this has also been plotted against ~oin Fig. I. (It is of interest that this curve has a minimum of 5.44 at about ~o= 0.009.) The experimental curves shown in Fig. 1 are all above the Happel curve, and exhibit greater curvature. They may be regarded as comprising a family which tends towards the Happel curve as the molecular weight tends to zero. The values of the intercepts as q0 --> 0 are, however, uncertain, and differ relatively slightly with molecular weight. In Fig. 2, the Happel function is plotted as loglo r/, against ~. This is a line through the origin with only very slight curvature. The experimental results also produce points lying on a family of nearly straight lines passing through the oriNn. The slopes of these curves increase with molecular weight, the Happel curve forming a limit as molecular weight tends to zero.
20--
~ /
oo ._~
I-5 --
x '"
lZ
I jl 1 1 i0 ~
f
[ I000
J I 2000
I 3000
Molecular
! 4000
i 5000
~ bS,S
weight
F[o. 3. Plot of "Expansion factor" vs. molecular weight of polybertzyis.
The Solution Viscosity of Polybenzyls
1557
Table 1 contains values of an "Expansion factor", i.e. the ratio of the mean slopes of the experimental curves of Fig. 2 to the slope of the Happel curve. These ratios are plotted against molecular weights in Fig. 3. It will be noted that the points lie on a smooth curve, and, though the relationship is not quite linear, the curve could be used to determine molecular weights from viscosity data. It should be noted that Fig. 2 is plotted using apparent volume fraction as abscissa. In fact, as the apparent solute density of polybenzyl in toluene does not vary appreciably with molecular weight, it would suffi.ce to plot log ,/, against concentration. The advantage of this procedure over any based on the limiting viscosity number is that the curve of log rlr against 9~is substantially linear up to quite high values of volume fraction, and hence the slope of the mean straight line is not greatly affected by inaccuracies in the viscosity determinations at low concentrations. This is in contrast with the curves of viscosity number against 9~, where values obtained at high volume fractions are virtually without effect on the accuracy of the extrapolation to limiting viscosity number. As has been pointed out, if the molecules of polymer were rigid spheres, molecular weight would not be expected to influence solution viscosity. The tact that, for polybenzyls, viscosity is not independent of molecular weight must imply that, when surrounded by a good solvent, the polymer molecule is able to expand by bond rotation so that its effective boundary is greater in extent than when the polymer is in the glassy state. Some molecules of solvent will be immobilized in the interstices of the polymer molecule. Study of a diagram of polybenzyl such as that published (v~ makes it clear that such expansion is possible, and evidently the proportional expansion increases as the size of the molecule increases, since our experimental "Exansion factor" increases with molecular weight. It is perhaps significant that the variation of viscosity ratio with temperature is very slight and negative. This would be expected if the molecule is composed of numerous branches which are fully extended except for slight shortening caused by thermal motion. It is possible that this method of treatment of viscosity data may be applicable to other polymers which are highly branched and therefore approximately globular in shape. However, it must be noted that the shape of the curve in Fig. 3 will depend very much on the polymer concerned, since the magnitude of the "Expansion factor" clearly depends on the details of molecular structure.
REFERENCES (1) (a) S. Bezzi, Gazz. chim. ital. 66, 491 (1936). (b) L. Valentine and R. W. Winter, J. chem. Soc. 4768 (1956). (c) D. B. V. Parker, W. G. Davies and K. D. South, J. chem. Soc. (B), 471 (1967). (2) B. H. Zimm and W. H. Stockmayer, J. chem. Phys. 17, 1301 (I949). (3) A. Einstein, Ann. Phys. 19, 289 (1906). (4) (a) E. Guth and R. Simha, Kolloidzeitschrift 74, 266 (1936). (b) E. Guth and O. Gold, Phys. Ret'. 53, 322 (1939). (c) V. Vand, .i". phys. Coll. Chem. 52, 277 (1948). (d) E. Guth, Rubb. Chem. Technol. 23, 635 (1950). (e) R. Sirnha, J. appl. Phys. 23, 1020 (1952). (5) J. Happel, J. appl. Phys. 28, 1288 (1957). (6) F. Horsfall, The t'iscosity of suspensions of rigid spheres, R.A.P.R.A. Research Report No. 175 (1969). (7) D. B. V. Parker, Europ. Polym. J. 5, 93 (1969).
1558
D E R E K B. V. P A R K E R and J. BARRY MYERS
R~sum~--Les viscosit~s de solutions de polybenzyles ont ~t6 6tudi~es en tenant compte de ce que les mol6cules de polybenzyle sont fortement ramifi6es et sensiblement sph~riques. Pour ces polym6rs et ceux qui leurs sont semblabtes, on propose une relation entre la masse mol6culaire et la viscosit6 de la solution, bas8e sur la theorie de Happel sur la viscosit6 des suspensions de spheres rigides.
Sommario---Si ~ studiata la viscosita di soluzioni di polibenzili, dal punto di vista che le molecole di polibenzile sono molto ramificate e di forma approssimativamente sferica. Per questi e simili polimeri, si avanza una teoria per cui esiste una relazione tra peso molecolare e viscosit5, di soluzione, basata sulla teoria di Happel circa la viscositb, di sospensioni di sfere rigide. Zusammenfassung--Die Viskosit~.ten der L{Ssungen yon Polybenzylen warden untersucht im Hinblick darauf, dab Polybenzylmolekfile hoch verzweigt sind und etwa Kugelgestalt besitzen. F/Jr diese und ~ihnliche Polymere wird eine Beziehung zwischen Molekulargewicht und .L6sungs~iskosit~.t vorgeschlagen, die auf der Theorie von Happel tiber die Viskositat yon Suspensionen starrer Kugeln basiert.