Multicomponent Polymer Systems

23 Kinetics of Aggregation and Dimensions of Supramolecular Structure in Noncrystalline Block Copolymers M. HOFFMANN, G. KAMPF, H. KROMER, and G. PAMPUS Farbenfabriken Bayer AG, 509 Leverkusen, Germany

Styrene-butadiene copolymers were investigated for relationships between molecular structure, phase separation, and microheterogeneous structure. Phase separation starts at slightly greater concentrations than in mixtures of homopolymers. During solvent evaporation, aggregates shrink to unfavorable dimensions. By annealing above their T , they become more uniform in size and shape, achieve sharper boundaries, and rearrange to well ordered arrays. In solutions containing 10 ). 5

8

3

358

MULTICOMPONENT POLYMER SYSTEMS

Q/ I 10

3

1 2

1

1 4

1 1 1—i—l—i 6 W

1 2

4

1

1 4

Figure 4. Critical concentrations, g , of phase separation for block copolymers and polymer mixtures in benzene solutions at 25°C as functions of the polystyrene molecular weight g5

Detection methods: • x-ray small-angle scattering A depolarization of scattered light X viscosity # macroscopic phase separation Further details of the phase separation of polymer mixtures were obtained by analyzing the high styrene phase sedimented at a weight concentration g of benzene (Table I I ) . In accordance with general experience with immiscible polymers, the styrene phase contains very little polybutadiene. In the same manner, only very little polystyrene can then be dissolved i n the polybutadiene phase. The aggregates of the block polymers are certainly also free B

Table II.

Polystyrene MW

Polystyrene in Polymer wt%

5500 12000 32000

29 29 29

Analyst: of Styrene Phase

%

Polybutadiene in Swollen Precipitate, %

Benzene in Swollen Precipitate wt%

Precipitate vol%

45 62 78

0.13 0.17

43 59 68

22 25 20

SB,

23.

Aggregation and Supramolecular Structure

HOFFMANN ET AL.

359

of the other kind of blocks. Thus, for example, styrene aggregates cannot be stained with O s 0 , and calculations of = 1.05 . Finally, the aggregates in the presence of benzene, as well as the precipitate of Table II, will probably contain benzene. If solvents are used which do not possess a high dissolving power for both kinds of blocks (high second virial coefficients of osmotic pressure), phase separation occurs at considerably lower concentrations, and the solvent content of the aggregates is lower than that of the matrix. 4

s

Kinetics of

ag

s

Aggregation

The speed of aggregation depends on the solvent. If it is a precipitant for one sort of block and if this block comprises a sufficiently large part of the molecule, aggregates of this kind of block are formed even in dilute solutions. P 42 in n-hexane at 60°C, for instance, forms genuine solutions with a viscosity of 0.56 dl/gram and a light scattering value which yields the correct molecular weight (M = 82000). If such solutions (c = 0.3 gram/liter) are cooled quickly, a strong turbidity occurs within tenths of a second and remains constant afterwards, giving a molecular weight of 2.55 X 10 . The osmometric molecular weight is 160,000 and the intrinsic viscosity 0.53 dl/gram. The aggregates thus form very quickly in dilute solutions and have a broad size distribution. Light scattering shows that their sphere diameter at c = 0.3 gram/liter is 720 A , which corresponds well to double the thickness of a single molecular coil in hexane (h = 280 A , h$ 85 A ) . w

6

B

When the concentrations are smaller, the capability of forming aggregates decreases; for example at c = 0.1 gram/liter to M = 1.7 X 10 and at c = 0.03 gram/liter to M = 0.5 X 10 . These degrees of aggregation remain constant for several days and seem to be equilibrium values. In better solvents—e.g., in cyclohexane at 25 °C—there is no aggregation, but only a decrease in the interaction between the polymer and the solvent, osmotically measured as d(p /c)/dc, with increasing styrene content. The interaction decreases, as expected, to a greater extent than corresponds to the styrene content and has dropped, for 30% styrene, to half the original value. It is possible in such or better solvents to produce moderately concentrated solutions, without any aggregation (see Figure 3). In such cases only the critical concentration (see Figure 4) and the rate of evaporation or cooling determine the aggregation. Molecular mobility does not play a major role for we found in experiments with polymer mixtures that turbidity occurred on cooling even at high concentrations (at least 50% ) within seconds. Such speeds of aggregation are understandable. For example, molecules with M — 10 , 6

w

6

w

osm

4

360


which means D • rji = 10" (35), still diffuse over distances of 300 A even i n a medium of rji « 10 poise i n about 0.5 second. The first stage of aggregation w i l l be reached in most cases very quickly, at speeds approximately proportional to the solvent viscosity. Reaggregation, however, occurs much slower, primarily i n undiluted polymers and at high polymer concentrations, because the styrene content of the aggregates elevates the transition temperature and thus reduces the chain mobility. The reaggregation slows down as it progresses. The two-block polymer P 45 (about 4 0 % styrene) was annealed, starting from a very poor superstructure (see Figure l a ) , until the outer ring, shown i n Figure l b and which is caused by the particle form factor, just became visually recognizable. The annealing time necessary for obtaining this stage of aggregation depends strongly on temperature: 8

3

T, C

t, mm.

110 100 90 80 70

25 180 1800 36000

a

The temperature dependence of the velocity corresponds to the temperature dependence of the viscosity of polystyrene near the glass transition temperature.

Figure 5. Electron micrographs of films of block copolymers which were cast from aliphatic hydrocarbons (C + C ) at 25° C and contrasted at 25°C for 1 hour with OsO vapor 5

6

u

(a) P 41 (17% styrene blocks) (b) P 32 (26% styrene blocks) (c) P 32, specimen prepared by slow evaporation of a benzene solution

23.

HOFFMANN ET AL.


361

The following equation holds approximately: In *(min) = -

15.7 + ^

5

( 4 )

The value of the glass transition temperature thus determined is about 50 °C lower than the value observed in macroscopic specimens of styrene homopolymer of molecular weight equal to that of the styrene blocks. Electron micrographs of the same polymer show more or less pronounced supramolecular structures depending on the preparation, the nature of the solvent, and the rate of evaporation. Fast evaporation from dilute solutions at temperatures far below the softening temperature of the aggregates results in a supramolecular structure we shall call the starting structure. Irregular aggregates with diffuse boundaries and broad distributions of the shapes, sizes, and distances are characteristic features of the starting structure (Figure 5 ) . If these samples are annealed at temperatures around 100°C, a supramolecular structure is obtained after some hours which is much more regular than the starting structure. It no longer changes on continued annealing, and we call it therefore the end structure. Figure 6 shows some examples. During annealing and reaggregation, structures can be observed which are still faulty (Figure 7 ) . These faults, which also include diffuse phase boundaries, are richer in free energy than their surroundings and facilitate reaggregation. According to Figure 7b it is estimated that in annealed samples the diffuse interface of the aggregates is not thicker than about 10-20 A . It is remarkable that the degree of perfection of the end structure of an annealed block copolymer depends on the quality of the starting structure and thus, for example, on the solvent (3, 17, 18, 30, 53, 70). If solutions of methylene chloride and acetone are sprayed into a vacuum, even intensive annealing is far from resulting in an end structure which is as good as that obtained when the polymer solution (xylene) is evaporated slowly at elevated temperatures ( 1 2 0 ° C ) . Thus, after a certain time of annealing, reaggregation comes to a standstill, independent of the degree of order that has been reached. Obviously, the displacement of butadiene chains out of the styrene aggregates takes place more quickly than the building up of long range order. Because of the gradually increasing inhibition of chain mobility, the well ordered equilibrium structure cannot always be reached. However, the mean dimensions of the supramolecular structure after tempering do not depend on the nature of the solvent or the evaporation technique and thus are typical for each polymer. This is contrary to some assumptions in the literature (37, 38). Literature data (2, 3, 4, 21, 30) on the kinetics of reaggregation are in satisfactory agreement with our experiments. The regeneration of the mechanical properties of strongly deformed SBS

362


Figure 6. Electron micrographs of films of some block copolymers cast from various solvents and contrasted with OsOt vapor after annealing (a) P 41, 17% styrene blocks, BS, 1 hour 100°C, hydrocarbon (b) P 41, 17% styrene blocks, BS, 1 hour 110°C, methyl ethyl ketone (c) P 32, 26% styrene blocks, BS, 1 hour 120°C, benzene (d) P 46, 30% styrene blocks, BS, 1 hour 100°C, benzene (e) P 53, 22% styrene blocks, S/BS, 6 hours 100°C, benzene (f) P 54, 32% styrene blocks, S/BS, 1 hour 100°C, benzene

23.

HOFFMANN ET AL.

Figure 7.


363

Electron micrographs of intermediate stages of supramolecular order

(a) P 42, 15% styrene blocks, BS, 1 hour 100°C, hydrocarbon, short rods and thicker spheres (b) P 41, 17% styrene blocks, BS, 1 hour 120°C, hydrocarbon, different ways of packing cylinders (c) P 42, 15% styrene blocks, BS, 1 hour 100°C, hydrocarbon, spheres arranged along lines (d) P 32, 26% styrene blocks, BS, 25'C, benzene, branched thread-like aggregates which tend to form layers

polymers at room temperature w i l l probably, however, also depend on other mechanisms (recombination of radicals). Aggregate Shapes and Long-Range Order of the Aggregates According to our observations and to literature data (4, 19, 22, 24, 47, 49, 50, 54, 56, 63, 64, 65), the following aggregate shapes occur i n the end structures: spheres, ellipsoids, long, substantially cylindrical rods in

364


various regular arrangements, as well as layers, and the corresponding shapes of the matrix. The arrangement of the aggregates is a statistical one only in mixtures of block polymers with butadienes. In most cases, however, we find long range order. W i t h spheres and frequently also with rods, arrangements of the centers are found like in the corresponding closest packings. The aggregates do not touch one another because the butadiene chains belonging to every styrene aggregate form a shell around the aggregate and enforce a minimum distance. Such packings have a surprisingly precise long range order going far beyond the length of extended molecules (see Figure 6a). The aggregates are obviously so uniform in their forms and sizes that they build up a lattice of far-reaching order. Such regular aggregates can be formed only by polymers of very uniform molecular structure. The long range order thus is disturbed greatly if polymers are mixed, even similar polymers ( F i g ure 8). If many aggregate sizes are measured in Figures 8b and 8c, the resulting size distribution exhibits two maxima, corresponding to the values of the components of the mixture. Together with the irregular distribution of the styrene domains in Figure 8b, this proves that in such mixtures a separation into the components occurs, primarily on annealing. It is remarkable that this demixing in the system shown in Figure 8c is no

Figure 8.

Electron micrographs of mixtures (1:1) of two-block copolymers cast from benzene

(a) P 42 (15% styrene blocks, M = 83000) + P 32 (26% styrene blocks, M = 70000), not annealed (b) P 42 + P 32, annealed for 1 hour at 100°C (c) P 41 (17% styrene blocks, M = 48000) + P 42, annealed for 1 hour at 100°C

23.

HOFFMANN ET AL.

Aggregation and Svpramolecular Structure

365

longer caused by a different chemical composition of the components but only by their different molecular weights. A further form of the arrangement of aggregates is considerably important from the standpoint of mechanical properties. Thread-shaped aggregates, especially the pearl strings occurring on incomplete annealing, often touch one another if the sty rent; content is sufficiently high, and thus form a hard network inside a soft matrix (33). When deformations are strong, this network, with its very long relaxation time, is presumably broken. The polymer, which in cases of little deformations is elastic and has a high modulus of elasticity, can therefore be deformed plastically at higher strains and remains soft after such deformations. Anomalous behavior of the butadiene-st) rene block copolymers may thus be understood (33). A substantial reduction of the glass transition temperature of the aggregates by the stress applied (6, 7, 31, 39, 51, 68) need not be assumed in order to explain the stress-strain behavior and the non-Newtonian viscosity (33) of the two-block polymers. Size and Shape of the Aggregates as a Function of Molecular Structure Table III presents data on the molecular structure, and Table I V gives mean values from several measurements of the supramolecular structure of these polymers. Surprisingly, every polymer appeared in several end structures. The two-block polymer P 41, for example, shows in 50 preparations (films) spotty aggregations just as often as streaky ones. The energy difference between the two shapes does not seem to be great. It seems that, in general, more than one characteristic supramolecular structure may exist in a given block copolymer. Moreover the diameters D (and necessarily the distances A ) of the two kinds of aggregates are distinctly and reproducibly different. Both shapes are found to have narrow size distributions—e.g., in cur two-block polymers mostly mean deviations of d=13% of the mean value. The two aggregate thicknesses thus depend upon molecular weight in a different way. Interpreting the aggregate sizes will not be as easy as has hitherto been tried (4, 6, 7,9,21,22,31,32,39,47, 49, 50, 51, 54, 63, 64, 68). In our search for quantitative relationships between the aggregate dimensions and the molecular structure, we first discuss the conformation of the sequences in the aggregates. If the sequences were densely folded or pressed together to compact nodules (69) without substantial mutual penetration, the aggregate thicknesses should—since no butadiene sequences are allowed to be present in the aggregate—not be much bigger than double the edge length:

366


Table III. Preparation 87 5 88 93 41 42 94 39 95 30 32 96 31 46 45 97 98 99 40 38 44 69 u 52 53 55 54 90 91 92 68

Type B B BS BS BS BS BS BS BS BS BS BS BS BS BS BS BS BS B/S" B/S S/B< S/B< BS/B S/BS S/BS S/BS BS/BS BS/BS S/BS/BS/BS S B Star 4

Styrene,

%

0 0 10 10 20 20 20 20 30 30 30 30 30 37 46 50 80 80 20 20 20 20 28 27 40 40 50 20 20 20

Molecular

Styrene Block,

%

0 0 7.5 6.5 17.5 15.1 17.5 17.7 25.5 24.7 26.0 (25.5) 24.0 30 41 41 68 68 17.8 22.6 19 19 (22) 22.2 (32) (32) 42 18 18 18

T)T»1

0.74 1.08 1.79 1.82 0.68 1.06 1.16 0.80 0.46 0.63 0.80 1.03 1.48 0.58 0.83 0.90 0.54 0.50 0.94 0.65 0.42 0.64 0.79 0.86 0.57 0.72 0.61 0.76 0.76 1.51

° Integral distribution of molecular weights. The stroke indicates a new charging of monomer. 6

of cubic nodules. In P 41 there is 2d = 63 A , while experimentally for spheres D — 212 A (x-ray) and for long cylinders D = 176 A is found. Similar differences between 2d and D are found with all preparations. Consequently, compressed coils do not exist. The extended chain is another conformation which has been discussed often and proved to exist—e.g., i n linear polyethylene crystallized at high pressures. It has also been assumed for amorphous polymers i n the form of strands of parallel molecules [kink model ( 5 7 ) ] . The maximum length L (x-ray) of a sequence of vinyl monomers of basic moM lecular weight M is -=-=- • 2.52 A . Straight strands of molecules can be K

z

0

23.

HOFFMANN ET AL.


367

Structure

10% — —

50%

90%

— —

— —

5000

10000

15000

14000

13000 13000 14000 21000

26000

21000 8500 23000

36000 10500 41000

46000 13500 47000

13000 16000

5200 6200 8000 — 3500 5000 c d

17000 14000 8200 9700 13000 3000 5700 10000

9500 11500 17000 11000 9000 14000

M

— —

12000 12000 10000 15000 17000 11000 12000 45000 20000 25000 36000 13000 36000 32000 56000 52000 13000 15000 4400 7200 16000 11000 9000 10000 14000 5000

39000 67000 147000 163000 38000 69000 78000 47000 22000 35000 50000 72000 121000 35000 54000 48000 26000 24000 59000 32000 19000 33000 24000 52000 32000 46000 20000 22000

11000

194000

d

39000 67000 159000 175000 48000 83000 95000 58000 32000 49000 70000 97000 157000 48000 90000 80000 82000 76000 72000 44000 23500 40000 63000 71000 50000 66000 68000 54000 49000 238000

Polymerized with addition of ether. Star with four branches of M = 48500.

arranged only i n layers without any packing difficulties. It is obvious that spherical or cylindrical aggregates cannot be built up from completely extended styrene sequences because the aggregates do not contain butadiene and must have the same density everywhere. Moreover, the experimental D and D are far smaller than 2 L and even smaller than L . It must therefore be assumed that such molecular strands, if they occur at all, are kinked or curved. Before drawing further conclusions regarding the conformation, we have; to take into account that i n such aggregates the conformation requirements of both kinds of blocks must be fulfilled. The relations between aggregate size and block length may be expected to be especially simple if two-block polymers of the same composition but of different K

z

368


Table IV.

Supramolecular

Preparation 88 93 41

ho,s 79 80 71

ho,B 490 530 220

9s 0.073 0.063 0.168

Aggregate Composition St St St

42

90

310

0.146

St

94 39 95 30 32

100 79 79 90 106

345 250 161 211 260

0.168 0.170 0.244 0.237 0.250

St St St St St

96 31 46

120 150 83

320 435 200

0.244 0.229 0.284

St St St

45

150

270

0.385

St

97

140

255

0.40

St

98 99 40 38 44 69 u 52 53

195 187 83 90 45 60 93 75

160 150 283 200 140 190 170 265

0.70 0.70 0.171 0.216 0.181 0.181 (0.21) 0.212

Bu Bu St St St St St St

69

200

0.30

St

55

Bu 54

71

250

0.30

St

90

87

153

0.395

Bu St

91 92

~65

162 ~140

0.172 0.172

St St

68

75

380

0.172

St

° h : end-to-end distance in athermic solutions. >h of the opposite interfaces the energy per chain w i l l also increase because the blocks are fixed to at least one interface and thus forced to form less probable conformations, elongated i n the direction of the normal to the interfaces, in order to fill space between the interfaces completely with matter. For these reasons and contrary to some opinions expressed recently (23, 37,38, 46, 52), a minimum of free energy exists for each set of blocks at some definite distance of the interfaces which is approximately equal to the coil size. If the aggregates are not layers, there is a broad distribution of distances between the interfaces i n different directions so that the average of (q — 1) must be used in Equation 9. The distance between opposite interfaces may be somewhat larger than p • h or q • h because of the overlapping of coils (see below). Considering the ways in which coils with Gaussian distributions of segments can overlap, we estimate the optimum distance of interfaces i n the following way. If two coils approach each other until the concentration of segments is approximately constant everywhere between their centers of gravity, the distance of their centers of gravity is about 0.55 h . The concentration m

K

z

8

0

0

0

2

0j8

0>B

0

372


of segments decreases i n outward direction and, at about 0.35 ho, is only one-third of the maximum value between the centers of gravity. Thus 0.55 h + 2 • 0.35 h = 1.25 1% may be considered as the "diameter of the aggregate/' This derivation neglects the fact that h is increased i f one end of the coil is fixed within an interface (8, 15, 16, 72). O n the other hand, such a dilatation of a coil may be diminished by the necessity to fill the space between interfaces. Our experiments show (see below) that 1.2 h is a reasonable value for the actual end-to-end distance between plane interfaces. I n the case of cylinders one has to discuss the distribution of segments over the cross section. The segments of six coils arranged at the corners of a regular hexagon with suitable dimensions w i l l overlap i n such a way that the concentration of segments is nearly constant inside the hexagon and a small distance to the outside. Such an arrangement is similar to the circular cross section of a cylinder and has a "diameter" of about 1.5 h. A n arrangement of coils similar to a sphere may be built u p using 3.6 = 18 coils and has a "diameter" of about 1.8 h. F o r these reasons, the following relations are expected to hold as a first approximation: 0

0

0

0

D

K

= 1.8 VKK ; D = 1.5 p h z

z

0

; D = 1.2 p h s

s

(10)

0

Analogously, the preferred average distance between opposite interfaces across the matrix should be about 1.2 qh . The experimental results of Table I V show for the case of layered aggregation that the value of p is approximately 1.07 and q about 0.96. Neither A W nor the limitation (8, 15,16, 72) of the conformation space seem to influence substantially the values of p and q i n this case. O u r experiments do not confirm the assumption D = 2h but give the approximate value D « 1.2 h for layers. The two energy terms related to the styrene blocks and the butadiene blocks, respectively, have minima at different values of D ( A — D is proportional to D for constant ). The sum of both terms shows an energy minimum at such a value of D that both p and q differ from 1. A minimum of free energy is obtained without considering the influence of the interfacial energy A W . W e have therefore tried to explain the existence of an equiUbrium diameter without taking into account A W . As the effective values of the distances between interfaces cannot easily be determined for the matrix (except layered structures), we developed the following approach to overcome this difficulty (for symbols used see text referring to Equations 5-9). Let us first discuss the structure of an isolated aggregate or micella: It consists of an inner domain, the "nucleus" (e.g., styrene blocks) and a surrounding "shell" (e.g., butadiene blocks). Such a shell of thickness 0>B

8

8

0

8

0

8

23.

373


HOFFMANN ET AL.

1.2 • q • h and volume V n may contain a maximum number of N heii blocks of one kind and the nucleus (volume V uci) up to N i blocks of the other kind. For isolated spherical ( K ) domains we find: 8 h e

0)B

8

t

n

/iVsheiA

\iVnucl )

=

Vsheii -9B*M

S

(D

=

K

+ 2 - 1.2 - q

K

^nuel ' 9S* M

K

D

B

9B • M

S

•h y 0tB

-

DK

Z

#

Z

K

, 2 » 1,2 «

=

n u c

• ft A _ ! 1 . 0

*^ s

( \ n

W i t h Equations 5, 6, and 7 one obtains

(fe). - (0 «•* • £ (*-^f J - '} • *r^5S +

21

The analogous derivation for cylindrical ( Z ) aggregates gives:

(fe), - {[•*• • s ( ^ ) ' l ' - >} • +

«*>

For p = 1 and q = 1 the ratio N ii/N ieu8 is substantially greater than & A (ratio of numbers of B blocks to S blocks in the molecule). This means that the butadiene chains attached to the surface of a styrene nucleus fill the space determined by their coil sizes to only a very small degree. The butadiene shells of adjacent aggregates therefore penetrate each other to such an extent that the shells and thus the whole space between the styrene nuclei are completely filled with matter. (In undiluted block copolymers, isolated aggregates with non-overlapping shells would have anomalous conformations and so an increased free energy.) The overlapping can formally be taken into account by introducing overlap factors, F and F , which are a measure of the penetration of the shells and influence the values p and q of Equation 9: nuc

8he

K

z

\ N ^ )

K

~

s '[ N ^ )

Z

=

F Z

'S

( 1 3 )

If F and F are regarded as constant values, q/p depends only on the volume fraction but not on the molecular weight (neglecting A W as a first approximation). This is confirmed by experimental data (Table I V ) . Moreover, Table I V shows that F and F , calculated by Equation 12 from experimental p-values (1.2 q = 1), are nearly constant for different . F appears to be about 9 and F about 4. The experimental values of F and F can be interpreted by simple geometric considerations. A cross section of a hexagonal cylinder packing with the volume fraction = 0.25 shows, for example, that the shells K

z

8

z

K

8

K

z

K

8

z

374


of three neighboring cylinders overlap entirely i n the space between them. The shells of the three next-but-one cylinders also protrude partly into this field and thus increase the mean overlap factor F from 3 to approximately 4. F does not depend greatly upon h /D and , so that the value F = 4 may be expected to be a suitable first approximation for all hexagonal cylinder packings. If nearly Gaussian-shaped segment density distributions are assumed around the centers of the butadiene coils, such overlapping of the shells gives a nearly constant over-all concentration of matter everywhere between the aggregate nuclei. For a closest packing of spherically symmetric particles consisting of nucleus and shell, the value of the overlap factor, F = 9, can be shown to be plausible in an analogous manner. Deviations from these values of F and F should increase the free energy in Equation 9. W i t h the aid of the conditions z

z

0>B

8

z

K

K

z

9 4 1

F F 1.2 • q K z

(14)

and Equations 12 and 13 the quantities p and p and thus, according to Equation 10, the diameters may be calculated if the volume fraction of the aggregates and the structural type of the molecule (s styrene blocks and b butadiene blocks) are known. The data in Table V show good agreement between experimental and calculated p values and confirm the validity of the assumptions used —i.e., A W « 0 and 1.2 • q 1. It should be remembered that the average experimental values of F and F may reflect to some extent an influence of A W =7^ 0 or 1.2 • q ^ 1 on p. Nevertheless, the introduction of F already leads to a minimum of A G for p ^ l . Table V also includes values for q which is a. measure of the deformation of the butadiene sequences along the line connecting the centers of aggregate nuclei. K

z

8

K

z

mim

- D = 1.2 q ,

A

K

K

min

K

• h

0tB

;A

Z

- D = 1.2 g z

m i n

, • h B z

Ql

(15)

Thus q as well as the average q are proportional to p for a constant value of . If i n the case of spherical aggregates 1.2 • q , which is always smaller than 1, becomes smaller than 0.5, the two-block polymers tend to avoid this strong deformation of the butadiene sequences by forming cylindrical aggregates. Both particle shapes are observed i n cases when q is calculated to be approximately 0.5 to 0.7 both for spheres and cylinders. Equation 9 gives the excess free energy and may be simplified by expressing q and D in terms of p. Neglecting A G we find a value of 12 min

8

min

miD

m

23.


HOFFMANN ET AL.

375

Table V . Comparison of Experimental and Calculated Data"

Type

fait

BS

0.07

Aggregate Composition Shape St St St St St St St St St St St Bu Bu St St St St

0.17 0.21 0.25 0.39 0.29 BSB SBS (SB)

0.21 0.21 0.30 0.17

4

K Z K Z K Z K Z K Z S K Z Z Z Z K

D h„ exp.

Fexp

2.8

11.3

2.7

8.7

2.5

9.5

2.0

3.9

1.9 1.2

4.5

1.2 1.5 2.2 2.1 3.3

3.2 6.3 6.9 13.4

D h„ cole.

1.2 q calc.

3.07 1.90 2.65 1.92 2.56 1.94 2.51 1.96 2.38 2.07 1.2 1.35 1.09 1.30 2.90 2.98 2.9

0.61 0.82 0.50 0.74 0.46 0.72 0.43 0.70 0.32 0.62 1.2 0.40 0.68 0.72 0.72 0.67 0.63

min

° Abbreviations same as in Table IV. D is the diameter of the aggregates formed by the indicated sequences.

erg/cm for the interfacial energy A W in the case of P 41. This value is certainly determined mainly b y the influence of F on p. W e propose to neglect A W and to use Equations 12 to 14 only. The experimental values of p are greater than 1, so that a considerable deformation of the styrene coils must be expected. O n the other hand, the most probable form of a statistical coil does not seem to be spherically symmetric (44). Rather, the coil shape resembles an ellipsoid, the maximum length being 1.25 h and the short axes about 0.45 and 0.71 h. Arranging such coils i n an analogous manner as described above, we estimate the diameters to be 1.5 h (layer), 2.0 h (cylinder), and 2.4 h (sphere). Moreover, Equations 12, 13, and 14 do not depend on the variables h and h themselves, but only on the ratio h /h . It can therefore not yet be concluded from the agreement between calculation and experiment that the assumption h = h is correct. It is possible that i n undiluted polymers and their concentrated solutions the coil diameters h are greater b y a factor independent of the molecular weight and of the chemical nature than the values measured in athermic diluted solutions. A parallel arrangement of the chains for lengths

Multicomponent Polymer Systems

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