Effect of Shear on Self-Assembling Block Copolymers and Phase

Chapter 15

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Effect of Shear on Self-Assembling Block Copolymers and Phase-Separating Polymer Blends M. Muthukumar Polymer Science and Engineering Department and Materials Research Laboratory, University of Massachusetts, Amherst, MA 01002

The theoretical efforts to understand the effect of shear on block copolymers and polymer blends are briefly reviewed. The known analytical results are only perturbative, and the status of theory is at a primitive stage in comparison with the recent advances made in the experimental front. Much more theoretical and computational work is necessary to fully understand the rich phenomenological experimental results. The effect of shear on the relative stability and features of different mor phologies formed by block coplymers and on the spinodal decomposition of polymeric mixtures has been an active research area. Although the exper imental work has produced rich phenomenological results of tremendous intrigue, theoretical progress has been rather modest. Whatever little we know based on theory is reviewed here and much more theoretical and computational work is required to even begin to understand the various phenomenological results. Although the fact that the composition fluc tuations are suppressed by shear is obvious, the calculation of the extent of this effect is technically very complicated. Only the linearized theory has been studied so far, thus restricting the applicability of theoretical predictions to the general experimental conditions. Even the verification of predictions of linear theory by experiment is not yet available for the appropriate conditions. The general theoretical framework is based on the pioneering paper (i) by Onuki and Kawasaki. The results presented below are essentially contained in the original theory of Onuki and Kawasaki. Formulation We consider a volume element r in an A - Β diblock or A - Β - A triblock copolymer or in a polymer blend (A + B) under shear. Based on the 0097-6156/95/0597-0220$12.00/0 © 1995 American Chemical Society

Nakatani and Dadmun; Flow-Induced Structure in Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1995.

15. MUTHUKUMAR

Effect of Shear on Self-Assembling Block Copolymers

221

conservation of mass and momentum, the equation of motion for the local volume fraction 4>{v,t) of component A at time t obeys (1) ^

= -V· J-V-V0 + /

£-„V»v-(v,g)

x +

(la)

(nx.

(16)

Here J is the local current of the A component, J = 3 +J +J . d

The diffusive

e

(2)

c

flux

caused by the chemical potential is given by

where Λ is the Onsager coefficient, keT is the Boltzman constant times the temperature, and F is the free energy functional appropriate to a par ticular polymer system. The elastic flux J corresponds to the transport of material due to gradients in strain inside the system. J is the convective flux due to the imposed flow field. We take the shear flow of the form, e

c

u(r)=72/e

(4)

x

where u(r) is the average velocity at r, e is the unit vector along x-axis and 7 is the shear rate, ν is the deviation of the local velocity from its average u. η is the shear viscosity of the system. (· · denotes taking the transverse part. / and f are noises. In general, the free energy F is a Taylor series in the order parameter φ(τ). Usually, the series is truncated at the quartic term. Substituting the functional derivative ÔF/δφ in equations l a and l b , we get a coupled set of nonlinear equations. While this set of equations should be solved numerically to obtain the results given by the continuity equations, we (28) have so far considered only the linear regime in view of the analytical tractability. If we are interested in only the linear terms of φ(τ) in equation l a then equations l a and l b become decoupled with ν replaced by u. Making this linear approximation and assuming further that the ve locity gradients are weak due to high viscosities of the polymeric material and that the elastic contribution to the free energy is negligible, we obtain x

0

θφ

A

SF

.

θφ

r

2

where only the term of φ is kept in F. Nakatani and Dadmun; Flow-Induced Structure in Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1995.

,. c

222

FLOW-INDUCED STRUCTURE IN POLYMERS

\J dv f άτ'φ{τ)Τ (τ-τ')φ{τ')

kT

+ >-.

2

(6)

B

Assuming further that the Onsager coefficient A is local, and using the fluctuation-dissipation theorem for the noise / , J k

k ( t )

(0) + 2 L [* dse'^-'M

k\s)

0

(48)

Jo where /(k, t) = 2 L

Γ ds[n k (s) - Kk (β)] Λ> 2

0

2

(49)

4

2

For -γ* > 1, fc (i) is approximated by 2

2

k (t) = (k + 7 i f c ) + k\ y

(50)

x

Using the change of variables,

and 2

3(1, τ) = * / ( * )

(51)

k

we obtain 2

3ι(τ) = 3 ι ) ( 0 ) β ^ > + 2

F

(52)

F

Γdul (u)e ^- ^

( τ

±

Jo where 2

F ( l , r) = 2 [ du [l (u) - Kl\ (u)] Jo ±

2

2

l (u) = l ±

±

+ 2ul l

x y

2

2

+ ul

(53)

x

2

with l\ = ξ + l . z

Results. We now proceed to look at the limiting behaviors of I. Since F ( l , r) is complicated, we ignore the cross term 2ul l in equa tion 53. The asymptotic results obtained below are independent of this approximation. Now, x y

2

F(l, τ) ~ 2(l -Κΐ\)τ ±

2

3

+ \ΐ {\-2Kl\)τ χ

5

- ^Kl*r


(54)

229

230


1) . It follows from equation 54 that fluctuations with l± > l/y/K cannot be enhanced. Therefore we may assume that Zj_ < 1. Now, for longer times, r) goes essentially as - r . 5

2) . For l = 0,

r) grows indefinitely.

x

3) . For l > 1, l± < 1: approximated as

The second integral in equation 52 can be

x

F

2

£ll(u)e- ^c£duu ll

1

exp(^

5 U

)

1

(55)

^(έ) ^ ^-^·^] 5

where T's are gamma functions and τ ' = § Ζ^τ . Substituting equation 4

5

55 into eauation 52 yields for r > Ιχ ^ 2

2

3(1 τ) = ς τ - + · · · .

(56)

)

Therefore the intensity decays with time instead of reaching a constant value as claimed by IOK. 4) . ZJL < 1, l < 1: approximated as x

£

F(1 u)

2

dull («) e~" '

=

I

Now the second integral in equation 52 can be

3

2 I l \~i j ~ dpp exp ( | | l | " * p x

i , Γ * exp

x

(±

I

i

I" ) 1

x

.

6

|j>)

(57)

Substituting equation 57 into equation 52 yields for τ > 1/ \ l \ x

5

3(1,

r) = V 2 7 I

l | ~ i exp x

-

L

_ |£ )

(58)

T

5

so that the scattered intensity decays exponentially with τ . This result also is in disagreement with the theoretical prediction of IOK but in agree ment with the results of Nakatani et al (16).


15.

MUTHUKUMAR

Effect of Shear on Self-Assembling Block Copolymers

Conclusions The status of theoretical predictions on the effects of flow on polymeric systems is still in a primitive state. Even in the simplest cases, the prob lem is technically and computationally intense although the methodology is clear. There are many issues which need to be addressed by theoretical and computational methods before we can successfully predict the phase behavior of polymers under flow. Some of this issues are the following: 1. Calculation of nonlinear effects even for the mean field case. 2. Numerical solution of coupled equations between the order param eter and velocity field. 3. Derivation of correct coupled equations including the contributions from elasticity of the various components. 4. Incorporation of topological constraints, as associated with bridges and loops of A — Β — A triblock copolymers in self-assembled morphologies for example, and coupling with external flow fields. 5. Generalization of the coupled equations when there are many order parameters. Acknowledgments I am grateful to the polymer blend group at NIST for making critical comments on this problem. Acknowledgment is made to the NSF grant DMR-9221146001 and to the Materials Research Laboratory at the Uni versity of Massachusetts. Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Onuki, Α.; Kawasaki, K . Ann. Phys. 1979, 121, pp. 456. Frederickson, G.H.; J. Chem. Phys. 1986, 85, pp. 5306. Onuki, Α.; J. Chem. Phys. 1987, 87, pp. 3692. Cates, M.E.; Milner, S.T. Phys. Rev. Lett. 1989, 62, pp. 1856. Marques, C.M.; Cates, M . E . J. Phys. (Paris) 1990, 51, pp. 1733. Frederickson, G.H.; Larson, R.G. J. Chem. Phys. 1987, 86, pp. 1553. Douglas, J. Macromolecules. 1992, 25, pp. 1462. Imaeda, T.; Onuki, Α.; Kawasaki, K . Prof. Theor. Phys. 1984, 71, pp. 16. Leibler, L. Macromolecules. 1980, 13, pp. 1602. Ohta, T.; Kawasaki, K . Macromolecules. 1986, 19, pp. 2621. Muthukumar, M . Macromolecules 1993, 26, pp. 5259. Melenkevitz, J.; Muthukumar, M . Macromolecules. 1991, 24, pp. 4199.


231

232


13. Lescancec, R.L.; Muthumkumar, M . Macromolecules. 1993, 26, pp. 3908. 14. Brazovskii, S.A. Sov. Phys. JETP 1979, 91, pp. 7228. 15. Frederickson, G.H.; Helfand, E. J. Chem. Phys. 1987, 87, pp. 697. 16. Hobbie, E.K.; Hair, D.W.; Nakatami, A.I.; Han, C.C. Phys. Rev. Lett. 1982, 69, pp. 1951. RECEIVED February 15, 1995


Effect of Shear on Self-Assembling Block Copolymers and Phase

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