The physical properties of polyisocyanates in solution

European PolymerJournal, 1971, Vol. 7, pp. 805--826.PersamonPress. Printed in England.

THE PHYSICAL PROPERTIES OF POLYISOCYANATES IN SOLUTION B. R. JENNINGS* and B. L. BROWN~ Physics Department, Queen Elizabeth College (University of London), Campden Hill Road, London, W.8., England

(Received 28 duly 1970) Abstract--Electric birefringence measurements have been made on four samples of poly-n-butyl isocyanate in three solvents. The value of ~/'w/~Ys for three of the samples was 1- 2. A description of the apparatus and of its use with both alternating and pulsed direct current electric fields is Oven. The method has been used to obtain molecular relaxation times and dipole moments. These data have been analysed, together with the results of dielectric experiments reported in the literature, in terms of both rigid helices and weakly bending helices. The results indicate that flexibility is encountered for molecular weights in excess of 5 × 104. This is somewhat lower than had previously been reported from other relaxation experiments. Below this molecular weight, the observed molecular relaxation times, which were associated with an end-over-end rotation due to the helices having a permanent dipole moment along their helical axis, were related to a number average molecular weig,,ht. Data were consistent with a helix monomer projection length of 1-3 A and a diameter of 11 A. In the range 5 × 104 < .~?s < l0 s, the data have been interpreted in terms of Hearst's recent theory for weakly bending rods. These results were consistent with the worm-like chain model, used to interpret the data for molecular weights in excess of 100,000. It was shown that the assignment of a specific persistence length was unreliable even for molecular weights as high as 5 × l0 s. The study demonstrates the merits and sensitivity of the electric birefringence method. It is the first time that the "double frequency" component of the birefringence, which is generated when alternating fields are applied to the solutions, has been detected and used to determine molecular parameters for a synthetic polymer. INTRODUCTION THE POLYISOCYANATEShave attracted attention in recent years because o f their apparently helical conformation. M a n y physical techniques have been used to study suitable solutions o f these polymers in efforts both to confirm the helical structure and to evaluate the relevant molecular parameters. Currently,-the helical conform a t i o n is n o t disputed. Conflicting dimensions and characteristics o f the helices have been reported however. There are at least two sources o f uncertainty which directly influence the interpretation o f experimental data. Firstly, the molecules have been shown by a n u m b e r o f workers, (1-7) to be highly extended and stiff in solution. Viscosity, (1-3) dielectric dispersion, (4-~) flow birefringence, (~) light-scattering, (2"s) electric field light scattering, (s) viscoelastic relaxation (s) and sedimentation (1) have all been used to demonstrate this. Before interpreting the experimental data in terms o f a rigid helical structure, care must be taken to examine the limits o f applicability o f this model. In general, a rigid-rod m a y be assumed for low molecular weight material whilst the K r a t k y - P o r o d (9) worm-like chain is used as the molecular weight is progressively increased. O f c o n c e r n here is the question as to the onset o f molecular flexibility with increasing molecular weight o r helix length. W h a t is the highest molecular weight * Present address: Brunel University, Uxbridge, Middlesex, England. I" Present address: Polymer Institute, University of Akron, Akron, Ohio, U.S.A. 805

806

B.R. JENNINGS and B. L. BROWN

material that one can study and yet interpret the data with confidence in terms of the theories for rigid rods ? Using polydisperse samples of poly butyl-isocyanate (PBIC), Yu et aL ~4~ claimed that the molecules were rigid up to a molecular weight (M) of at least 3 × 106. Later experiments by Bur and Roberts tS~ on fractionated samples of this same polymer, indicated that the molecules were flexible with M an order of magnitude smaller than the earlier t4~ result. However, their results were inconsistent as their hydrodynamic data (ref. (5), Fig. 4) indicated rigid rod molecules up to 28 × 104 weight average molecular weight (ATtw), whilst the dipole moments (ref. (5), Fig. 7) were in conformity with a departure from rod-like behaviour at 3~rw= 5.1 × 104. On the other hand, Tsvetkov et aL c6~ showed results (ref. (6), Fig. 8) which had been extrapolated so as to give the impression that the molecules were flexible for molecular weights below 104. Secondly, in the determination of such characteristics of the helix as Lo (the projected length of a monomer onto the helix axis) and b (the helix diameter), monodisperse samples should be studied. Where this is impracticable, the nature of the weighting of the experimental data towards any constituent species must be understood. Apart from the uncertainty of assigning theories for rigid particles to data on flexible molecules mentioned above, polydispersity averaging is probably the most influential factor in the diversity of results hitherto published for Lo. Values of 0.4 A (ref. (7)), 0.5/~ (ref. (4)), 1.1 A (refs. 5, 6)), I. 8-2.1/~ (ref. (3)) and 2.0 ,~, (ref. (7)) have all been reported for the polyisocyanates in solution. Moreover, Shmueli et aL ~°~ obtained a result of 1.94 A from X-ray diffraction studies on PBIC films. The parameter Lo has been evaluated in the majority of studies through the determination of the relaxation time (r) involved in the end-over-end rotation of the rigid or pseudo-rigid helices. This is also the case in the present work. The problem reduces to that of assigning the correct polydispersity average to 1-. With increasing molecular weight and flexibility, the helix deviates markedly from a rigid structure and approaches the extreme of a random coil. The intermediate region has generally been described in terms of the persistence length (q) of the KratkyPorod tg~ worm-like chain. The persistence length is defined as the projection of the flexible chain which is assumed to be infinitely long, along the direction of the first bond. Assessments of q for polyisocyanates in solution have been as diverse as the results for Lo. The following values have been reported, 440 A (ref. (6), i f L o = l " 1 A), 500-600 A (ref. (3)), 400-900 ,~ (ref. (5), by inference from their graphs) and 1300/~ (ref. (7)). In the present study, measurements have been made of the induced birefringence of PBIC solutions using both pulsed and alternating electric fields. By combining our experimental data with information currently available in the literature, we have been able consistently to resolve each of the forementioned uncertainties. In particular, the relatively unknown theory of Hearst ta~J for the end-over-end relaxation of weakly bending rods has been used to indicate the flexibility of the polyisocyanates. The theory is eminently suitable for electric birefringence and dielectric dispersion data. It is consistent with theories for rods and worm-like chains at the limits of its application. Finally, the study demonstrates the versatility of the electric birefringence method. It is quick and sensitive and it enables both geometric and electric molecular parameters to be determined. The data are readily obtained with such precision that slight molecular flexibility can be demonstrated.

The Physical Properties of Polyisocyanates in Solution

807

THEORY AND METHOD (i) Introduction Many materials become doubly refracting when subjected to a strong electric field. In such an eventuality, the material acts as a uniaxial crystal with the optic axis parallel to the direction of the applied field. This inducement to birefringence was first reported for borosilicate glass by Kerr," 27 by whose name the effect is generally known. The birefringence (An) is expressed as a function of the field strength (E), the wavelength (Ao) of the light in vacuo and the refractive index (n) of the material by the equation An = B . ) t o . E 2 (1) where B is the Kerr constant. Macromolecules in solution are free to orientate and hence align under the influence of the electric field whence they manifest their inherent or induced optical anisotropy by causing the overall solution to become birefringent. In this case, the birefringence will depend upon the number of solute molecules present. A more meaningful parameter is the specific Kerr constant (K,p), for which the concentration (c as weight per volume) has been divided out. Then An = K~,, ec n E 2 with

K,, = (a/c) a___o. vn

(2)

Whereas the observation of molecular orientation is possible because of the optical anisotropy of the molecules, the origin of the orientation is the interaction of the applied field with any permanent dipole moment (~) of a molecule or with any induced dipole momentwhich results from a difference(g®l-gez) in the volume electric polarizabilities gel and g,2 associated with the major and minor axes, referred to by subscripts 1 and 2 respectively. Moreover, orientation in the electric field is opposed by brownian forces. It is convenient therefore to express the dipole moments in the factors I~2

P = T---ka ~ and Q =

v(gel -- g,2) kT

(3)

where k, T and v are Boltzmann's constant, the absolute temperature and the volume of each solute molecule respectively. In this work, both alternating and pulsed electric fields were used. Each has its advantages. (ii) Birefringence in pulsed d.c. fields In this case, the birefringence can be considered in three temporary regions. The instantaneous application of the step-like d.c. voltage will not be accompanied by an instantaneous rise in the birefringence, because of the finite time required by the molecules to orientate. The birefringence thus follows an exponential growth in which

B.R. JENNINGS and B. L. BROWN

808

both the permanent and the induced dipole moments participate. At any time t after the field is applied, the equation tl 3) An = 2X {P q- Q - 1.5 P exp ( - t/r) zr [0.5 P - Q ]

exp ( - at/r)}

(4)

holds. Here

rrcgE2 X = -(gol -- go2) 15n where (gol-go2) represents the optical anisotropy of the macromolecules and r = ½D with D their rotary diffusion constant. Ultimately, the birefringence will reach a steady maximum value, given by the equation [

Q2

19

An =- Ano = 2x P q- Q q - ' ~ q- ~'~.QP -

1

]

.PZ .

(5)

When the pulse terminates, the molecular array will revert to a random situation as brownian forces disorientate the molecules into their minimum state of potential energy. The birefringence will follow an exponential decay, the time constant or exponent of which will depend upon the molecular geometry and solvent viscosity alone. In this case An = Ano exp (-- 3t/r)

(6)

where Ano is the steady state birefringence given in Eqn.(5). Equation (6) indicates a straightforward means of determining the molecular relaxation time, whilst Eqns. (4) and (5) suggest a means of evaluating P and Q if only their relative contributions can be evaluated. This is difficult using pulsed fields. The method of finding the ratio r = P/Q by comparing the areas above the rise and below the fall of the transient curve as suggested by Yoshioka and Watanabe c14) is generally unreliable as it is experimentally associated with very large errors. (iii) Birefringence in alternating fields When a continuous alternating field of the form E = Eo sin wt, with angular frequency o,, is applied to the solutions, the birefringence becomes An = X {S -- A cos (2oJt -- ~)}

(7)

where S=

P (1 + ~o2 r 2)

Q2 A =

+Q,

p2 + 2PQ

]

(I + 4 co2 r219) "}- (1 --J-4 ¢o2 ~-z/9) (1 + ~o2 ~.2)

and ff is a phase angle. This equation is obtained by suitable combination and manipulation of Peterlin and Stuart's ~15) equations, so as to account for the general case of molecules having both permanent and induced dipole moments. There are some interesting features of Eqn. (7) to which we shall refer later. Firstly, the birefringence consists of two discrete contributions. One of these is time independent and is characterized by the factor S (for "steady"). The other fluctuates at twice the applied field

The Physical Properties of Polyisocyanatesin Solution

809

frequency and is governed by A (for "alternating"). These must be measured in different ways. The steady component offers a simple method of differentiating between permanent and induced dipoles in terms of the frequency dependence of the birefringence. Simultaneously w is found from any frequency dispersion in An. In this respect, the method resembles the dielectric dispersion technique. It gives the same relaxation time. It should be noted that Eqns. (4)-(7) were derived for rigid ellipsoidal molecules. However, the determination of molecular relaxation times using the pulse decay transient (Eqn. (6)), or from the frequency dispersion of the birefringence implied in Eqn. (7), is valid whether the molecules be rigid or not. (iv) Apparatus Diagrammatic representations are given in Figs. 1 and 2 of the apparatus using pulsed and alternating electric fields respectively. The optical system is common to both. In outline, this consisted of a high pressure mercury arc, run from a highly stabilized d.c. supply, as light source. The light passed through a series of collimating lenses and slits before falling onto a Glan-Thompson prism which rendered the transmitted light plane polarized. Between this polarizer and a similar analyser which was set in the crossed mode, the light traversed a cell and a quarter-wave plate. The latter was set with a privileged direction in the azimuth of the polarizer. The cell consisted of a centre body of "Fluon" (PTFE) and two stainless steel end pieces (Fig. 3). The Fluon body was milled out so that two stainless steel electrodes could be mounted 0.2 cm apart. The exposed electrode face dimensions were of 2.5 cm square. These electrodes were terminated on opposite sides of the body in screw connections. The whole centre body was securely screwed to the end pieces. These held microscope slide glass end windows which were cemented to the steel end pices with "Boscoprene" cement Aperture

Slit I

Polori=er

[O"'r*"rl

Cell

"

Q.W.P. Analyser Photomultiplier

1

I Oscilloscope

F]o. 1. Diagrammatic representation of the pulsed field birefringence apparatus. Q.W.P. indicates a

quarter-wave-plate.

810

B . R . J E N N I N G S and B. L. BROWN

Cell

Q.W.P

Anolyser

Photomultiplier

Tronsformer ( optionoI ) TT

Phase

Frequency

(

~c~-,o

o,,,0o,

J

Oscilloscope with differential omplifier

Oscillator FIG. 2. Diagram of the birefringence apparatus for use with alternating fields. With the switches at positions (i) and (ii) the "steady" and "alternating" components respectively of the birefringence are recorded.

I--B

/

Filling vent ~ _

*..

; ~de

~

"

.... ~ . ~ -

@

,

on BB'

/'

,,ioo,/ ~ ~to,n,o.,e, Electrodes

. Window

Sectic on AA

Copper

P.T.F.E.

FIG. 3. Sample holder. The central body is of polytetrafluorethylene and the end pieces and electrodes of steel.

The Physical Properties of Polyisoeyanatesin Solution

811

number (2762). This is manufactured by Bostik Ltd., Leicester, England, and was found to be very resistant to attacks by the organic solvents used in this study. The cell held approximately 7 ml of solution which could be introduced by syringe through an appropriate opening. The complete cell assembly was simpler than that described previouslyt16) as neither external monitoring of the solvent conditions was required, nor were the solutions very conducting. No light was transmitted by this system prior to the application of the electric field to the solution. When the field was switched on, light penetrated the analyser and was incident on a photomultiplier, the output of which was handled in one of three ways. Using pulsed fields, the photomultiplier signal was amplified, displayed as a single shot event on an oscilloscope and photographed (Fig. I). With alternating electric fields, the "steady" and "alternating" components were measured independently. With the switches of Fig. 2 set to positions (i), the photomultiplier signal passed directly to the Tektronix type 585 oscilloscope. Unfortunately, owing to the imperfection of the polarizing devices and, to a lesser degree, of the cell windows, some residual light reached the photomultiplier when the polarizing prisms were crossed and no field was applied. The difference measurement between this condition and the photomultiplier response when a field was applied to the solution was most readily performed using a differential plug-in amplifier type 1A5 in conjunction with the oscilloscope. Owing to the quadratic nature of the detection ~17~ system, this difference measurement was proportional to the square of the "steady" component of Eqn. (7). The "alternating" component was isolated from the signal by setting the switches at positions (ii) and using the lock-in amplifier or phase sensitive detector. In this case, the reference channel was used. This consisted of the frequency doubler and phase shift network to bring the reference signal to the same phase and frequency as the required component in the photomultiplier signal. Using this arrangement, signals some 45 dB down in the noise level could be measured. 4- 102 tO 104 V +800V

,oov -

I

_,oov

--

_

"

-450 v .-__-#'-

4",-L

"f f

J---

I

FIc. 4. Circuit diagram of the pulse former. For principal components, s¢¢ text.

812

B.R. JENNINGS and B. L. BROWN

Using a suitable transformer, alternating fields of up to 2 kV, corresponding to 10 kV cm-~ across the solutions, were generated by a high power oscillator amplifier type 254 of Airmec (Racal) Ltd., of Reading, England. This generator is capable of producing up to 800 V at 0.19 amps throughout the frequency range of 30 HZ to 30 kHz. Pulses of up to 10 kV amplitude and of duration approaching 25/~s were obtained from a special "hard-valve" generator (see Fig. 4). As pulse generators of this rating are not commercially available, a brief description is as follows. A positive input pulse applied to the common emitter stage turns "on" the transistor (T) which in turn cuts "off" the two A2226 valves (V~ and II2). A 600 V pulse is then generated across the common anode load. This pulse is a.c. coupled to the "White" cathode follower stage which consists of two more A2226 valves (1"3 and 1"4) across a potential difference of 1-2 kV. This cathode follower stage acts as drive for the Cl148 output valve (Vs), of which the control grid rises to 200 V during the duration of the applied input pulse. The current taken by this grid is of the order of 0.5 A, whilst the final output pulses may be as much as 20 A at 10 kV for 25/zs. This power is stored in the capacitor C. No colour filter was used in the incident light beam in order to realize the greatest light intensity. The equivalent wavelength of 500 nm was used for Ao. EXPERIMENTAL AND RESULTS Dr. L. J. Fetters of the University of Akron kindly provided the samples of PBIC in the molecular weight range of 2 × 104 < ATw < 3 × 10 s, together with details of their weight and number average molecular weights from light scattering and osmotic pressure measurements respectively. Three samples (B, C and D) were relatively monodisperse in that the ratio/~,,/37s was about 1.0 to 1.2. The other sample (A) was more polydisperse with this ratio equal to 2.5. Sample B was No. 21 of refs. (3) and (5). A, C and D were classified as I, 6A and 41 by Dr. Fetters. Most measurements were made in benzene although tetrahydrofuran and chloroform also were used as solvents. Dissolution was generally achieved by continuous stirring for a few hours at 40 °. All solvents were of reagent grade. Solutions were filtered through appropriate Millipore solvent resistant filters of 1.2/zm pore diameter. All systems were prepared at various concentrations so that the data could be extrapolated to zero concentration. (i) Pulsed electric fields Pulsed transients were obtained for each system. They did not exhibit any unusual behaviour. For each system, the birefringence was dependent on the square of the field strength (Eqn. (1)). In each case the rotary diffusion constant was obtained from the decay transient using the simulation technique described recently. (18) The results, given in column 7 of Table 1, correspond to infinite dilution except where otherwise stated. All transients were non-symmetrical; that is, the decay exponential was not simply the inverse of the rise exponential (Fig. 5). This implies that the helices had a permanent dipole moment as it is evident from Eqns. (4) and (6) that the trace would have been symmetrical had P = 0. Furthermore, it has been shown tzg) that a predominant permanent dipole moment along a transverse or minor axis of the helix

6.0

2.0

C

D

2.0

5"0

12

12

)~/N (104)

Zero

Zero

Zero

0"25

Concentration (lO-2g.m1-1) (" )~

Cell6

CoH6

Cell6

C4HsO CHCIs CeHe

Solvent

PBIC

0.42 (4-0.04)

3"3 (4-0.3)

30.0 (-I-3.3)

17.8 (-4-1-7) 20.0 (-I-2.0) 25-0 (±2"5)

~-(ps)

SAMPLES

120

15

(±13)

(4-2)

1.66 (4-3.3)

2.8 (-4-0.3) 2.5 (-1-0.3) 2"0 (±0"2)

D (10 4 s-1)

253 (±11)

550 (±25)

1330 (4-80)

1060(-4-50) 1090(-I-55) 1070(-4-50)

Ls (A)

Values of O. 55, O. 56 and O"65 cP, have been used as solvent viscosities for tetrahydrofuran, carbon tetrachloride and benzene respectively in the Burgers equation, which gives the equivalent rod length (L~).

13"3

B

30

A A A

"I j~

JV/w (104)

Sample

TABLE 1. RELAXATION TIMES FOR

t~

OO

o_.

2"

814

B.R. JENNINGS and B. L. BROWN

, l

l

1 J

\ I

I

I

.

.

.

I .

1 .

.

1 I .

.

t .

"L l

I

FIG. 5. A typical pulse-fieldtransient. Data for sample A in benzene, c=0" 25 g. dl-1. Pulse amplitude of 24 kV. cm- 1 and duration of 25 t~s. would correspond to a negative value of P and hence of the birefringence. This was not the case. Earlier suggestions t4,5.7,s~ that the polyisocyanate helices are polar with t~ directed predominantly along the helix axis are hereby confirmed. If it were assumed that the PBIC helix was both rigid and rod-like for molecular weights below 3 × 105 as suggested by Bur and Roberts, tS) and that the helix diameter was 11 A (refs. (7, 20)), then the helix length could be evaluated using the formula ;~21.2~) 1

D = 2-~

3kT

Wr/oL3(In 2p

y).

(8)

Here ~7ois the solvent viscosity and p the ratio of the length (L) to the diameter (d) of the helices. Burgers t2~ used the factor y ----0.8, whilst according to Broersma, c22~ = 1.57 -- 7 [(In 2 p ) - i _ 0.28] 2. Alternatives to Eqn. (8) have been proposed, c23'24) Broersma's equation is theoretically more precise as it makes allowance for the shape of the ends of the molecule on the orienting torque suffered by the rod. As it differs so little from the alternatives as to allow no experimental differentiation between them, t6~ we have used the simpler Burgers equation. Using Eqn. (8) by successive approximation, L has been evaluated and included in Table 1. Discussion of the applicability of this equation, and evaluation from it of the length (Lo) of the projection of the monomer length on the helix axis, is discussed below. It is gratifying to note the consistence of L for sample A using each of the three solvents. The molecular weight of the monomer was 99. The steady state maximum of the pulsed transients was analysed to give the specific Kerr constants listed in Table 2. Before Eqn. (2) could be applied to the experimental traces, the magnitude of the light intensity reaching the photomultiplier, which was related to the height of the photographic trace, had to be calibrated in terms of absolute birefringence. Using the apparatus in its quadratic mode of detection, t17) this was performed absolutely in terms of the theory of crossed polarizers. Although commonly used, the procedure has been described recently with clarity by Jerrard et al. t2s) A simpler method was also used by the present authors. This involved the comparison of the trace height with that obtained using pure nitrobenzene, for which the Kerr constant B is 386 × 10-7 e s u t26~ at 20 °. The two procedures gave concordant results.

.m ill

12-0

5.0

2.0

B

C

D

(-4- 3)

(4-17)

3-9 (--0.5)

27

140

B/C (10-4 esu. ml. g - ' )

158

423

1081

p (D)

0.79

0- 84

0.89

~,/R. (D)

Assuming t7 ~ 1.09

0.79

0.70

0.81

(D)

IN BENZENE

~,1~.

PBIC

179

481

1224

~ (D)

0.90

0.95

1.00

~,11~. (D)

Assuming v = 0" 8

0.90

0.79

0.91

u/IV,, (D)

Dipole moments (~,) in Debye, calculated with Lo ---- 1 • 3 A and helical diameter of 11 .~. Errors in ~, of order of 30 per cent. Units of ~ are cm 3 g - 1.

-~N (10 4)

Sample

T A B L E 2. K E R R CONSTANT ANALYSIS FOR

9o

o ~"

2"

816

B.R. JENNINGS and B. L. BROWN

(ii) Alternating electric fields The major value of using alternating electric fields is that the presence and direction of any molecular permanent dipole moment may be readily inferred from the frequency dependence of the "steady" component of the birefringence. The smaller the molecular size, the higher the frequency at which end-over-end rotation will be observed in terms of a birefringence dispersion. Only sample B was studied by this method in order to check the results obtained using pulsed fields. Measurements were made with benzene as the solvent. In this case, the analyser was set away from the crossed position in order that the photomultiplier response was directly proportional to the solution birefringence. This is the linear detection setting. (17) The positive sign of the birefringence was determined by interposing the calibration cell containing introbenzene, which gives a positive birefringence and noting that the PBIC solution further increased the photomultiplier response. I0

oe /

\

m° 0 " 6

0.4

0.2

0

I

I

I

5

IO

15

Frequency,

KHz

FIG. 6. "Steady" component frequency dependencefor sample B in benzene; ¢ = 0.25 g. dl- ~,

E=8000 V.cm -x. Figure 6 shows the frequency dependence of the "steady" component of the birefringence for sample B. It is noted that a dispersion is seen which, from Eqn. (7), reveals the presence of a permanent dipole moment. The positive sign of the birefringence shows this to be along the helix axis as mentioned in the previous section. From Eqn. (7) it is seen that the change in the magnitude of the birefringence is half completed for a particular field frequency (o~c)where A n t - An~o - ½ = (1 + ,oo 2 ~ 2 ) - 1 Ano -- An~o

(9)

The Physical Properties of Polyisocyanatesin Solution [

817

"

/. ,/.l

k ,/./L

V VVV::VV VV

o .. °.

2

,d

~./

d

./

~/

FIG. 7. Photographs of the double frequency (alternating) birefringent component, generated using alternating fields at frequencies of (1) 2000 I-lz (2) 1000 Hz. The original single frequency fields are shown for comparison. Here the subscripts refer to the field frequencies. Thus ~ = we- 1 and D = = f t . In this case, f~ = 5 kHz whence • -----31.8 (4-9- 5) t~s and D -----1.57 (4-0"47) × 104s- 1. This is to be compared with the result of 1 "66 (4-0" 17) x 10"s -1 given in Table 1. All these results confirm the data from the pulsed field experiments. One additional piece of information is indicated. The high frequency residue (An,o) of Fig. 6 shows that any induced moment, signified through Q, is negligible in comparison with P. Figure 7 is a photograph of the alternating component of the birefringence. This is, to the authors' knowledge, the first occasion on which this component has been detected for a polymer solution. Thurston and Bowling('s) have measured it for a tobacco mosaic virus solution and various colloidal suspensions where the solute and hence the birefringence were greater. Using the result that Q is negligible for PBIC in benzene, the factor A in Eqn. (7) becomes p2 A~=o =

(1 + 4 ~o2 r2/9) (1 + 0,2 r 2)

whence the frequency dependence of Fig. 8 should follow the equation An Ano

--

[(1 q- 4 oJ2 r2/9) (1 -t- ca2 ¢2)1-~.

(10)

In Fig. 8, the experimental data have been plotted alongside theoretical curves for various D. It is seen that the data are in good agreement with the pulsed and steady component measurements. The theoretical curve for ¢ = 31.2 t~s or D = 1 "6 × 104s-x can be seen to satisfy the data.

818

B.R. JENNINGS and B. L. BROWN I'0

\\ 0'8

0"15

0.4

\

\

\\\\\x~

0.2

0

I 5

I I0 Frequency,

15

I

KHz

Flo. 8. Alternatingcomponent frequency dependence for sample B in benzene; c = 0"25 g.di -~, E = 8000 V.cm -t. Curves (a), (b) arid (c) represent 0.8, 1.6 and 2"4.10"s -t. DISCUSSION (i) Relaxation times f o r helical state Various criteria have been used to indicate the departure of the polyisocyanate helices from the rigid conformation as the polymer molecular weight and length increase. The ratios [@'a)*]/Nw of the radius of gyration (S) with degree of polymerization (N) and [(~w)2//~] of molecular dipole moment ~ ) with N, or the relaxation time alone have all been used. Conflicting results have been obtained as mentioned in the introduction. The difficulty of differentiating between the effects of molecular flexibility and polydispersity may be a contributory factor here. Yu et aL (') suggested that Ln and hence r were related to a molecular weight characterized by ( ~ ~w)*. Bur and Robertg 5> plotted their dielectric relaxation times as a linear function of JlTw, using relatively monodisperse samples of low (i.e. less than 3 × 10 5) molecular weight. Graphs of z vs. M have displayed regions with slopes of 2-7 as expected <4,s~ for rigid rods. The value of Lo obtained from these graphs using Eqn. (8), have been different (Lo = 0.5/~, in ref. (4) and 1. I A in ref. (5). In Fig. 9 we have plotted data for z as functions of both J14Mand Jl~w. Horizontal lines represent the spread in molecular weights. On this graph we have included the data of Bur and Roberts tS) which are appropriate to this molecular weight range. Corresponding number and weight average molecular weights were found in the literature t3~ for their samples. Figure 9 immediately reveals three points. Firstly, the birefringence relaxation times are in excellent agreement with ~- obtained by Bur and Roberts
The Physical Properties of Polyisocyanatesin Solution

/

-2

• O-Me~ []

819

o "Mw

F

-3

•

Ol.7 I-3

./~--o F3

o4.6

~-~ 4-.I "5

t,,

-"

-6

-T

~4'4

/1"0

,4

I ~

I G

log M Fro. 9. Dependence of relaxation time (r) on molecular weight. Horizontal lines represent spread from ~N tO '~]W. Numbers indicate ratio ~'w/~'M. Squar~ and circles represent this work and that of Bur and Roberts~s~respectively. • = 1/2D.

unambiguously on the n u m b e r average molecular weight of the sample. Thirdly, the d a t a cannot b e expressed as satisfactorily by either a weight average (`.s.6) or an intermediate (weight.number) ~ average (~) dependence on M. It is noted with interest that previous electric birefringence experiments c29) on polydisperse helical poly benzyl glutamate solutions gave relaxation times which, when interpreted in terms of Eqn. (8), gave rod lengths corresponding to less than a weight average length. Thus, although our experimental data agree with those of Bur and Roberts, we differ with these workers in the analysis of their relaxation data in terms of the weight averaging of their parameters. Graphs such as Fig. 9 have been used to indicate molecular rigidity. A linear section of approximately 2.7 as slope justifies the use of Burgers ~21)type equations. It is usual to analyse the points along the suitable linear range of the graph of • vs. M in terms of Eqn. (8) (see Table 1). This may be performed collectively using the slope and position of the graph. Furthermore, it has been demonstrated above that LB ----(~s) Lo.

(11)

B. R. JENNINGS and B. L. BROWN

820

IO';'r •

This work

•

Bur and Roberts

106

g ,z 10 5

o o

Monomer projection (a) corresponds to (b) corresponds to (c) corresponds to

I

0

: I0 4

3

1

(Lo) 2.0 1.3.~ 1.0.~

~ I0 5

Molecular weight, no.overage

FIG. 10. Variation of rotary diffusion constant (D) with M. Full lines for rigid cylinders, broken line for weakly bending rod with Lo = 1"3 A, q = 1000 A. Using the value b = 11 A, ¢7.2°) theoretical graphs have been drawn in Fig. 10 for a variety of Lo. Here again, we note three things. Firstly, as expected, Lo only marginally affects the slope of the lines; it rather causes a shift of axis. Secondly, over a restricted molecular weight range, from I(P
The Physical Properties of Polyi~zyanatcs in Solution

821

is also noted that when displayed in this manner the relaxation times indicate molecular flexibility at the same molecular weight (5" 1 × 104) as Bur and Roberts (s) reported from their dipole moment values. Using a weight average molecular weight, they had the anomaly of predicting 5.1 × 104 from dipole moments and yet 28 x 104 from relaxation times. Their discrepancy is hereby resolved. A monomer projection distance of 1.3 A is in close agreement with the conclusions of other workers. Two sets of authors t5'6) favour 1.1 A. Furthermore, it is noted that Yu et al. (4) reported Lo = 0.5 A using the polydispersity weighting La = (N,, fi/n)* Lo with polydisperse samples having the ratio llTw/MN in the range of 3.2-4.3. If, as a rough estimate, a value of 4 is taken for ~ w / ~ s , and the number average weighting of LB as indicated in the present work is accepted, then the data of Yu et al. (4) would indicate that L o = 1 . 0 A, a result that is consistent with one of those authors' later study (5) on monodisperse material where Lo = 1.1 A was cited. This gives added support to our suggestion that a number average weighting should be ascribed to and La. In an attempt to account for the curvature of Fig. 10 the experimental data has been interpreted using the little known theory of Hearst. (11) This purports to account for the rotary diffusion constant of weakly bending rods. The basic equation may be written in the form

irMa

-,--,M(4.5 L.M2p___~-- 10.2 + 4b/a )]

D - - -k-T ' p a [ 7 0 (3.LM/2pb)--4.92 + 4 b / a + - ~

(12) where p=M/L, the mass per unit length of the molecule. Equation (12) hold only when x [=(L/q)] ~ 1, where L is the overall extended length of the rod and q is the persistence length of the worm-like chain model. (9) The parameters a and b are identical with those used in theories of the frictional properties of large molecules, (a°.al) where the molecules are considered as an assemblage of a number of frictional elements with effective Stokes diameter a, separated by a distance b. Hearst and Stockmayer (31) have shown that rod molecules are best represented by a series of touching spheres, in which case a -----b. Furthermore, they state that a (----b) is equal to the diameter of the rod to a close approximation. The present authors have therefore used a ----b = 10 A (refs. (7, 20)) in Eqn. 02) for polyisocyanates. The weakly bending rod theory reduces to the same form as that for rods when the molecules become sufficiently small or q becomes infinite. This is shown in Fig. 10 where the broken line represents this theory for L o = l .3 A and q = 1000 A. At low molecular weights, the broken line is coincident with the Burgers theory for Lo = 1.3 A. The particular value ofq = 1000 A was obtained by trial and error to find the best fitting curve. It satisfies the data extremely well. Theoretical curves for the weakly bending rod model but with other values of q similarly approached the Burgers line asymptotically at low molecular weights, but gave varying degrees of curvature at high molecular weights. This was to be expected as the molecular flexibility, which was characterized through q, became more noticeable with increasing M. Dev et al. (~) have rightly pointed out that as the truly rigid polymer situation is encountered at a relatively low molecular weight, one must be careful to assign a correct value to b before determining Lo. Thus, as the molecular length is reduced, the ratio L/b ~ p assumes increasing importance in Eqn. (8). However, a variation in the

822

B.R. JENNINGS and B. L. BROWN

choice of b between the limits 8 ~< b(A) ~< 12.5 corresponds to percentage variations in Lo = 1 "3 fit of only 6 and 3 per cent for material with -~N = 10" and l0 s respectively. For a molecular weight just below the onset of flexibility (about 5 × 104=Jl7~ from Fig. 10) the determination of Lo -----1.3 (4- 0-1) A is unlikely to be seriously affected. (ii) The worm-like chain mode/ Above 5 × 10" number average molecular weight, PBIC is flexible and deviation from the Burgers equation increases with increasing M. Although the weakly bending rod model fits the data in Fig. 10, at very high M the worm-like chain and ultimately the true random coil models must be used to express the molecular shape. Unfortunately, we did not have samples of very high molecular weight. However, Bur and Roberts measured ~-for molecular weights up to 10v and, because of the good agreement between our data and theirs at low molecular weights, we make a reappraisal of their data in terms of our current interpretation. As the molecular flexibility arises from increased helical length, values for b ( = 10 A) and Lo (----1.3 A) remain unchanged. For the worm-like chain, the rotary diffusion constant is given by the equation (~1) D = ~o q'~'i

0.126

+ 0.318

03)

-- 0-37 + 0.16

and is valid only when x[ =(L/q)] ~ 1. When x is infinitely large, the worm-like chain becomes synonymous with the random coil. I0 3

,(o)~

I0 2

e-

go

:6 t o

L i n e s c orre,, persistence

n,.

(a) I000, (t (d) 200, (el I

3 10 6

IO s Molecular

1 I0 ? weight,

I0 e

no.overage

Fla. I1. High molecular weight continuation of Fig. 10. Full lines represent the flexible worm-like chain model.

The PhysicalPropertiesof Polyisocyanatesin Solution

823

Bur and Roberts' high molecular weight data are plotted in Fig. 11. The molecular weight range of the abscissa complements that of Fig. 10. Theoretical curves are also given in Fig. 11 for PBIC with Lo -----1" 3 A and various values ofq. Three points emerge from this graph. Firstly, as might be expected, the lowest molecular weight data correspond to q = 1000A as predicted by the weakly bending rod theory. This theory therefore appears to be compatible with the worm-like chain model from which it was deduced,c11)Secondly, the data cannot be satisfied by a single value of q, at least in the range 10s <3~n < 107. Thirdly, as the molecular flexibility increases with increasing molecular weight, the data approach a curve for q = 400 (:[: 50) asymptotically. A wide variety of values for q for polyisocyanates exist in the literature, including 1300 A (ref. (7)), 500-600 A (ref. (3)) and 440 A (ref. (6)). The deductions of the previous paragraph suggest that, subject to the accuracy of Bur and Roberts' dielectric relaxation times, the original data from which the values of q were deduced by the various authors, were not necessarily in discord. The molecular weight of the predominant material used in the evaluation of q might be very relevant here. Furthermore, the present display of the data in Fig. 11 accounts for the lack of success in the fitting of theoretical curves for single combinations of Lo with q to the experimental datata.s~ at various molecular weights. When a sufficiently high M is reached, the data of Bur and Roberts indicate a probable constancy in q of 400 (-4- 50A). A single value of q can only be upheld if Lo is allowed to vary, This implies the most unlikely tendency of the PBIC helix to change pitch with increasing flexibility. The apparent variation in q for PBIC is evident not solely from the analysis of the molecular relaxation times. In a private communication, Dr. Fetters has kindly pointed out that a similar conclusion can be drawn from an analysis of the dipole moments published by Bur and Roberts. According to Dr. Fetters, Fig. 8 of ref. (5) can also be well represented by the worm-like chain model throughout the studied molecular weight range, ifq be allowed to vary with M; a larger q being consistent with smaller M as mentioned in this study. These two observations may indicate either the inadequacy of the worm-like chain model for the polyisocyanate system or the unsatisfactory mathematical representation of the model through the relevant equations used. The former suggestion is reminiscent of the comments of Dev et al. that the concept of the worm-like chain must be applied with caution to systems of low molecular weight where the persistence length is comparable to the extended molecular length. Finally, it is interesting to draw attention to the work of Moha et al. ta2~ on poly benzyl glutamate in helix promoting solvents. Fig. 4 of their work shows the tendency ofq to vary with molecular weight and also approach a high molecular weight asymptote characterized by q in the range of 300--400 A. Although this variation in q was not specifically commented on by these authors, they did notice a wide variety of values from 300 A to 925 ~, for their own work and that of other authors. (iii) Kerr constants Once calibrated in terms of birefringence, the steady maxima of the transient photomultiplier response could be interpreted using Eqns. (1), (2) and (5). It is important to note that Eqn. (5) is only applicable to the equilibrium conditions encountered with pulses of long enough duration. Table 2 summarizes the data. Sample

824

B.R. JENNINGS and B. L. BROWN

A has not been included as the instrument had not been calibrated at that stage o f the study. In the evaluation of the Kerr constants, 500 nm and I. 5 were used for ~o and n respectively. It has been shown in this work and elsewhere that the molecules have negligible induced dipole moment and that the permanent dipole moment is predominantly along the helical axis. In order to determine the magnitude of this moment, the optical anisotropy (gol-go2) and the partial specific volume (9) must be known. Tsvetkov et aL ¢7~ measured the former from flow birefringence experiments in toluene, citing values of 6-6 and 7-6 × 1025cm 3 for the monomer optical anisotropy. Using the average o f 7 × 1025 cm 3 and assuming it to be suitable for benzene solutions, then

V(gol - - g o 2 )

=

7 X 1025 cm a

where V is ,r r 2/-'0, the equivalent monomer volume, and r is the helix radius o f 5.5 A. Two values of fl can be found in the literature. Fetters and Yu ¢2°) determined 1.09 cm a g-~ for polyisocyanates in toluene, whilst both Burchard cl) and Tsvetkov et al. ~7~ used ~ = 0.8 cm a g - t for PBIC in benzene or tetrahydrofuran. This large difference is unlikely to be due to the different solvents alone. In Table 2, molecular and monomer dipole moments have been evaluated using each value of 9. The table also includes the dipole moment per monomer calculated from both number and weight average degrees o f polymerization. Two inferences are drawn from these results. Firstly, from the relaxation data o f this and previous c5) work, the helix is expected to be rigid for M < 5 x l0 s. Molecular flexibility, which accompanies increased molecular weight, is characterized by a decrease in ~ / N ) . Table 2 probably indicates that the dipole moment deduced from Kerr constants is more closely related to a weight than a number average parameter. Use of ~N not only gives poorer constancy o f the monomer dipole moment for these samples, but also an apparent increase of this factor with M. The choice o f a weight average is in concord with Bur and Roberts ¢5~ for dielectric measurements on PBIC and Boeckel e t a / . (29) for Kerr effect measurements on helical polybenzyl glutamate. The possible assignment of a weight averaging to t~ is not to be confused with the earlier selection o f a number average for r. Secondly, dielectric measurements ts) have indicated that ~ / N ) = 1 . 1 3 Debye. Considering the uncertainty in the value of (go 1 -- go2) used above, this agrees well with the results in Table 2. Furthermore, it would indicate that for PBIC in benzene 0.8 cm a g - I is probably closer to e than 1.09 cm a g - L This conclusion awaits experimental verification. Acknowledgements--Tbe authors thank Dr. Fetters for the kind provision of samples. The Imperial

Chemical Industries Ltd. and the Central Research Fund of this University are both especiallythanked for funds which enabled the apparatus to be built. One of us (B.L.B.) thanks this College for the award of a student demonstratorship whilst both of us thank Mr. R. Webb for designing and building the pulse generator and Professor Burge of this department for his encouragement and the provision of facilities.

(1) (2) (3) (4) (5) (6) (7)

REFERENCES W. Burchard, Makromolek. Chem. 67, 182 (1963). N. S. Schneider, S. Furusaki and R. W. Lenz, J. Polym. Sci. A3, 133 (1965). L. J. Fetters and H. Yu, Macromolecules (in press). H. Yu, A. J. Bur and L. J. Fetters, J. chem. Phys. 44, 2568 (1966). A. J. Bur and D. E. Roberts, J. chem. Phys. 51, 406 (1969). S. B. Dev, R. Y. Lochhead and A. M. North, Disc. Faraday Soc. 49, 244 (1970). V. N. Tsvetkov, Y. I. Ryumtsev, I. N. Shtermikova and G. I. Okhrimenko, Vysokomol. soyed. 8, 1466 (1966).

The Physioal Properties of Polyisocyanates in Solution (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)

825

H. Plummer and B. R. Jennings, Europ. Polym. J. 6, 171 (1970). O. Kratky and G. Pored, Rec. Tray. Chim. 68, 1106 (1949). V. Shmueli, W. Traub and K. Rosenheck, J. Polym. Sci. A2, 7, 515 (1969). J. E. Hearst, Y. chem. Phys. 38, 1002 (1963). J. Kerr, Phil. Mag. SO, 337, 446 (1875). H. Benoit, Ann. Phys. 6, 561 (1951). K. Yoshioka and H. Watanabe, Nippon, Kagaku Zasshi. 84, 626 (1963). A. Peterlin and H. A. Stuart, Hand.und Jahrbuch Chemischen Physik, Vol. 8, lB. Backer & Frier, Leipzig (1943). B. R. Jennings, B. L. Brown and H. Plnmmer, J. Coll. Int. Sci. 32, 606 (1970). C. T. O'Konski, Encyclopedia o f Polymer Science and Technology, Vol. 9, p. 551. Interscience (1966). B. L. Brown and B. R. Jennings, J. Phys. (E), Scient. Instrum. 3, 195 (1970). I. Tinoco, J. Am. chem. Sac. 77, 4486 (1955). L. J. Fetters and H. Yu, A.C.S. Polymer Preprints 7, 443 (1966). J. M. Burgers, Verhandel. Koninkl, Ned. Akad. Welenschap. ,4fdel. Natuurk. Sac. 1 No. 4. 16, 113

(1938). (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32)

S. Broersma, J. chem. Phys. 32, 1626 (1960). F. Perrin, I. phys. Rad. $, 497 (1934). J. G. Kirkwood and P. L. Auer, J. chem. Phys. 19, 281 (1951). H. G. Jerrard, C. L. Riddiford and P. Ingram, J. Phys. (E), Scient. lnstrum. 2, 761 (1969). J. R. Partington. In Advanced Treatise on Physical Chemistry, p. 279. Longmans, London (1953). J. Badoz, J.phys. Paul. 17, Suppl. 11,143A (1956). G. B. Thurston and D. I. Bowling J. Coll. Int. Sci. 30, 34 (1969). G. Boeckel, J. C. G-enzling,G. Weill and H. Benoit, J. Chim.phys. 59, 999 (1962). J. G. Kirkwood, J. Polym. Sci. 12, 1 (1954). J. E. Hearst and W. H. Stockmayer, J. chem. Phys. 37, 1425 (1962). P. Moha, G. Weill and H. Benoit, J. Chim. phys. 61, 1240 (1964).

R/n~a6--Dos mosuros de bir6fringence 61ectdque ont 6t6 effectu6es sur quatre 6chantillons do poly (isocyanate do n-butyle) dans trois solvants. Le rapport .~Vfr//~TfNvaut 1,2 pour trois des 6chantflions. On d~crit l ' a p p m ~ et son utilisationavec dos champs 61ectriquoscr66s par un courant alteraatff et par un courant continu puls6. La m6thodo a 6t6 utilis6e pour obtenir los temps de relaxation et los moments dipolaires moi6culairos. Los r6sultats ant 6t6 analys6s, de m~me qua ceux obtenus par des mosuros di61ectdquos d6crits dana la litt6ratureen envisageantdos h61icosrigides et des h61icosfaiblementttexiblos. Les r6sultats indiquent qua la flexibilit6 apparait pour dos masses mol6culairos sup6deuros i 5 x 104. Cette valeur ost quelque peu inf6fieure hce qui r6sultait d'autres exp6riencos de relaxation. Pour des masses mol6culaires inf6rieures, los temps de relaxation mol6culaire observ6s, attfibu6s/t una rotation d'ensemble do la rnol&'ule due au fait qua los h61icos poss6dent un moment dipolaire permanent le long de leur axe h61icoldal, sent refi6s/l la masse mol6culaire moyenne on nombre. Cos donn6os sent coh6rentes avec une Iongueur de 1,3 A pour la projection de rh6fice du motif monom6rique d'un diam6tre de 10 ,~. Dans Iv domaine 5 × 104 < MN < 10s, los donn6es ant 6t6 interpr6t6es salon la th6orie r6cente do Hearst sur los batonnets faiblement flexibles. Cos r6sultats sent en accord ave¢ le mod61e de chaine "en serlaentin", utilis6 pour interpr6ter los donn6os dans los cas ou la masse mol6culaire est sup6fieure/t 100.000. On a montr6 que la d6termination d'une longueur sp6cifique perrnanante n'ost pas justifiable m6me pour dos masses mol6cula/ros aussi importantos qua 5 x 10s. Carte 6tude met an valeur los m6ritos et la sensibilit6 de la m6thodo de bir6fringence 61ectrique. C'ost la premi6re fois qua le caract6re "fr6quence double" de la bir6ffingence, qui ost produite g6n6ndement quand dos solutions sent soumisos/t des champs 61ectfiquos alternatifs, a 6t6 raise on 6vidence et utilis6e pour d6terminer des param6tres mol6culaires d'un polym6re synth6tique.

826

B.R. JENNINGS and B. L. BROWN

Zu,~mmenfassung--Elektrische Doppelbrechungsmossungen wurden mit vier Proben yon Poly-nButyl Isocyanat in drei LOsungsmitteln gomacht. Der Weft yon ~rw/~r~v war fftr drvi der Proben 1,2. Eine Boschreibung des Apparats und seiner Verwendung sow#hi fhr Wechselstrom- wie f/Jr pulsierende Gleichstromfelder wird gegeben. Die Method# ist verwandt word#n, um molekularo Relaxationszciten und Dipolmomente zu vrhalten. Diese Angaben sind zusammen mit den, in der Literatur berichteten Ergobnissen dielek-trischer Experimente sow#hi in Form yon starr~n Spiralen und auch leicht biegenden Spiralen analysiert worden. Die Ergebniss# zeigen, dass Flexibilitht ffir Molekulargewichte von mehr als 5 × 104 angetroffen wird. Dies ist etwas niedriger, als yon anderen Relaxationsexperimenten frfiher berichtet wurde. Die beobachteten molekularen Rolaxationszeiton, welch# mit einer Endv-tiber Endedrohung infolge oines pormanenten Dipolmomentes der Spiralen entlang ihrvr Spiralenachse vorbunden wurden, wurden unter diesem Molokulargewicht auf vine Durchschnittsmolekulargewichtszahl bezogen. Daten standen mit einer Monomer Spiralenprojektionslange yon 1,3 A und einem Durchmvsser yon 11 Aim Einldang. Im Bervich 5 x 104 < ~1'~ < 105 sind die Daten in Form der neuen Hearst Thoorie fib" schwach biogonde Stabe gedeutet worden. Diesv Ergebnisse standen mit dem wurmartigvn Kottenmodell in Einklang, das zur Deutung der Daten ffir Molekulargvwichte ~bor 100 000 benutzt wurde. Es wurde gvzoigt, dass selbst ffir so grosso Molekulargowichte wie 5 x 105 die Festlegung einor spezifischen Persistenzlange unzuverlassig war. Die Untersuchung zeigt die Vorteile und die Empfindlichkeit der ©loktrischenDoppelbrvchungsmethode. Es ist das vrste Mal, class der "Doppelfrvquenz" Bestandteil der Doppelbrvchung ermittelt wurde, wvlcher er'zeugt wird, wvnn Wechselfolder bvi den L6sungvn vorwandt word#n, und dass er benutzt wurde, um die molekularen Parameter ffir einen synthotischvn Polymer zu vrmitteln.