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Heterogeneous structures in layers of grafted polymer chains
1028
T. M. B1RSHTEIN and A. K. KARAYRV
12. M. G. VITOVSKAYA, I. N. SHTENNIKOVA, E . P. ASTAPENKO and T. V. PEKER, Vysokomol. soyed. AI7: 1161, 1975 (Translated in Polymer Sci. U.S.S.R. 17: 5, 1334, 1975) 13. L. N. ANDREYEVA, A. A. GORBUNOV, S. A. DIDENKO, Ye. V. KORNEYEVA, P. P. LAVRENKO, N. A. PLATE and V. P. SHIBAYEV, Vysokomol. soyed. BIS: 209, 1973 (Not translated in Polymer Sci. U.S.S.R.) 14. V. N. TSVETKOV, V. Ye. ESKIN and S . Ya. FRENKEL', Struktura makromolekul v rastvor¢ (Structure of Macromolecules in Solutions). Moscow, 1964 15. J. E. HEARST and W. H. STOCKMAYER, J. Chem. Phys. 37: 1425, 1962 16. ft. E. HEARST, J. Chem. Phys. 38: 1963, 1962 17. J. C. KIRWOOD and ft. RISEMAN, J. Chem. Phys. 16: 565, 1948 18. H. YAMAKAWA and G. FUJH, Macromolecules 6: 407, 1973 19. R. SIMHA, J. Phys. Chem., 13: 188, 1945 20. G. BOHDANECKY, Macromolecules 16: 1483, 1983
PolymerScience U.S.S.R. Vol. 30, No. 5, pp. 1028-1037, 1988 Printed in Poland
0032-3950/88 $10.00+.00 1989 Pergamon Press plc
HETEROGENEOUS STRUCTURES IN LAYERS OF GRAFTED POLYMER CHAINS* T. M. BIRSHTEIN a n d A . K. KARAYEV Hetero-organic Compounds Institute, U.S.S.R. Academy of Sciences
(Received 29 November 1986) A method for analyzing the structure of a layer of polymer chains grafted to a surface has been developed without an assumption that the layer is homogeneous. The layer was simulated by a system of sublayers of constant density. Numerical calculations were performed by computer for 1-5 and 10 sublayers under all the regimes of a temperature-concentration diagram of state for the layer. It was found that the free energy minimum corresponds to a monotonic diminution in the layer density as the distance from the grafting plane increases. Distribution functions for the units according to the layer height are displaced towards the outer boundary of the layer and grow wider as the unit number increases. When heterogeneity is taken into account it does not affect the conclusion previously reached concerning segregation of two opposing layers. A SYSTEM o f p o l y m e r chains g r a f t e d t o an i m p e r m e a b l e surface ( m a t r i x ) a n d i m m e r s e d in a solvent is u s e d in [1] as a basis f o r a t h e o r e t i c a l d e s c r i p t i o n o f a w h o l e series o f p o l y m e r systems such as, f o r instance, branclaed p o l y m e r s [2, 3], micelles [4] a n d supercrystals [5, 6] o f b l o c k c o p o l y m e r s , a n d a m o r p h o u s - c r y s t a l l i n e p o l y m e r s . O n t h e basis o f scaling ideas [7] t h e o r e t i c a l studies o f t h e s t r u c t u r e o f a layer o f chains g r a f t e d t o a p l a n e m a t r i x [8-10] s t a r t with a n a s s u m p t i o n t h a t t h e l a y e r is h o m o * Vysokomol. soyed. A30: No. 5, 1001-1008, 1988.
Heterogeneous structures in layers of grafted polymer chains
1029
geneous, i.e. there is a constant number of chain units in the basic thickness of the layer (excluding the drop in the number of units close to the grafting surface and in the vicinity of the outer boundary of the layer [9]), and it is assumed that there is a uniform degree of extension of all the chains. On the Other hand, a marked heterogeneity of the layer has been revealed by the data obtained in several experimental and theoretical investigations, including the results of a computer simulation of a layer of relatively short chains. The present paper relates to a simple model developed by us on the basis of a scaling approach with a view to providing a method for analyzing the structure of a layer of long grafted chains in the absence of any assumption that the layer is homogeneous. The data obtained show that the number of chain units decreases significantly as the distance from the grafting plane increases. The model and the method. Let us take a layer of chains grafted to a common plane surface and immersed in a solvent at z =(T-0)/0t> 0. To characterize the rigidity of the grafted chains we take the segmental asymmetry parameter p~> 1, and the number of chain units is sufficiently large, N>>I. The length of a unit (a spherically symmetrical element of the chain)we will take in our case to be a unit length. Now let a be the mean area of grafting per chain; 1 <~a <<.R 2, where R is the mean length of a free chain in solution. The structure of the layer of grafted chains is found by minimizing the free energy for a chain in the layer, expressed as the sum of /tF= AFc + zIFe
(l)
where AFc is the chain units concentration contribution determined at z~>0 by the thermodynamical unfavourableness of collisions involving two or three units in the layer, and AF, is the elastic component taking account of stretching of chains in the layer. Taking the layer to be a semidilute polymer solution of constant concentration c = N/trH,
(2)
where H is the homogeneous layer height, we have [8-10] 4 F c = kl NcXr2-XP3Y
(3)
AF, = k2 N - IH2cyr-Yp- 2+ 3y
(4)
The exponents x and y depend on the type of regime to which the semidilute solution is subjected: x = 5/4, 1 and 2; y = 1/4, 0 and 0 for regimes II+, II,,: and II0 respectively; kx and k2 are numerical coefficients. Minimizing eqn. (1) in respect to H one obtains [8-10] equilibrium values for the height and the free energy of the homogeneous layer (index (1)) x--y
2 -x+y
1
H ( t ) = K 1 N t r 2+x-Y'c2+X-Y p2+X-Y -2x
2(2-x+y)
3
(5)
x
A F ( I ) = K 2 N a 2 * ~ - y z 2+~,-y p Y-2~-~-'-~,
(6)
1030
T.M. BmSHTEiNand A. K. KARAYEV 1
where Kx = [kt x / k 2 ( 2 - h ) l 2+x-y and K2--klK~X+k2 K2-y are numerical coefficients. The concentration of units in the layer is given by equations (2) and (5). Let us now generalize the scheme under discussion. To allow for possible heterogeneity of the layer we will subdivide it into a fixed number n=2, 3 ... of plane sublayers (steps) of thickness hi, where i= 1, 2, ... n, and the layers can be numbered (as to distance) from the matrix. Taking account of the marked linear stretching of chains in the layer let us subdivide all the grafted chains into n types, such that chains of the jth type (their fraction being 0~j) are located in the firstj sublayers and end in the jth sublayer. Let flji be the number of units for chains of thejth type that are located in the ith layer ( i ~ j = 1, 2 ..... n). Thus the distribution of chain ends according to layers is given by the vector ~=(~q, ~2, ..., ~ ) , (7) and the distribution of units for chains of each type is given by the rows of the triangular matrix
l/s,, o
li
0 )
n
with the normalizations ~ ~l= 1 and ~, flsi= 1. i=1
i=1
The proposed model is a discrete variant of an analytical method of investigation of a layer of grafted chains developed by Semenov [11] involving an approximation of a self-consistent field on condition that the unit concentration is constant. In this case only the elastic component AFe was minimized on the basis of distributions of chain ends and the stretching of the chains (in a discrete model based on at and fl). In the present instance a concentration distribution in the layer is allowed for, and dFc is likewise taken into account. Moreover in the case of the discrete model it is possible not only to analyze a self-consistent field approximation (regimes II 0 and IIms), but also to go beyond the limits of this approximation (regime II+). We would point out that generally speaking the total free energy of the layer also includes another factor, namely a surface component determined by deficiencies in contacts at the interface between sublayers in the discrete model. However, simple calculations show that the contribution of this surface term decreases as the number of sublayers increases and is substantially less than the value of AF in all cases, including that of transition to a continued limit. Free energy of the graduated layer. The total free energy AF (") of the layer may be expressed as the sum of free energies of the sublayers AFt A F (n) = ~ AF i
(9)
i=1
We will treat each sublayer as being a homogeneous layer of grafted chains with an effective grafting area that increases with the increase in i. Let us say that the number
Heterogeneous structures in layers of grafted polymer chains
1031
of units and the sublayer thicknesses are such that they may be accomodated within the framework of the scheme outlined above. For this to happen it is essential that the thickness of a sublayer should be significantly greater than the density correlation radius. As in eqn. (1), values of ziF i will now be expressed as the sum of the concentration and the elastic components, calculating for a chain grafted to the matrix (i.e. calculating for an area a). Generalizing expressions (3) and (4) we obtain for the ith ~ublayer (zlFc)~ = (zlF(,1)) ( N i ) N - l(ei/c)~
(LIF,)i = (AF[, 1>) (Ni- 1) N (ci/c) V(hffH) 2, where
(10) (1 I)
tl
( m i ) N - 1 =Ai = ~ o~jflji
(12)
j=i tl
(13)
( N [ 1) N = B i = ~ajlfl.i, j=l
ci = (Ni)/ahl
(14)
A further problem is that of minimizing the total free energy d F (') in respect to all values of {~j}, {fli~} and {h+}. This is most simply achieved by minimizing in respect to the set of {h~} values, since each of these determines solely the free energy of the ith sublayer. Equilibrium values of ht are obtained from the condition
ZF,
[(aFt),
= 0
Now with the aid of equations (3) and (4) as well as (10)-(14), and assuming that the numbers of units in all the sublayers correspond to the same regime of the semidilute solution (II+, IIm: or II0) we obtain the height, the free energy and the concentration of units in the ith sublayer 1
hi=H(1)Ai(AiBi)
2+~-y
(15)
x
/IF i =AF(I)Ai(Ai Bi)2 +x-v
(16)
1
ct = cCU(Ai Bl) 2 +x-y,
(17)
where H (1), AF (1) and c CI) are the height, the free energy and the concentration of units in the homogeneous (graduated, n = 1) layer under the regime in question, determined by eqns. (2), (5) and (6). It is evident from equations (9) and (15)-(17) that all the characteristics of sublayers and of the layer as a whole are distinguishable solely by the numerical coefficients of Ai and B i (eqns. (12)and (13)) from the characteristics of the homogeneous layer. This ineans that a minimization of.the total free energy of the layer may be reduced to a minimization (based on sets of {~i} and {flji} values, taking differing values of n) of
1032
T. M. ~ s n r v l s
and A. K. KAltA'ttv
the total coefficient X
O f ~ A~(AtB~) 2+~-y |=1
Minimization of the nonlinear function ~ for many variables (a nonlinear programming problem) was carried out by a "sliding tolerance" method, using a Flexi program [12], for n=2, 3, 4, 5, 10, under regimes II+, IIml and ]I0.
1"5I~~j.~"~.
y ~.1/y ( .
•
!
+
Z
03
1.0
1 I !
I l I I
g'5
l t L
t 1 I
=
I
I
I
I
I I
J
I
I
Z lln
w 5 4 ,7 i FIo. 1. Relative characteristics of the system yt=)ly(l~ vs. the reciprocal of the number of sublayers 1/n for regimes II+ (1), II=t (2), and IIo (3): a= JF(")MF(I}; b ffi c~")/c~~); c-H(n)/H ~1~. Figure 1 shows the relative values of free energy of the layer AF(")/JF (1) vs. the preset number of sublayers. It can be seen that under all the regimes the free energy decreases with the increase in n, such that a free energy minimum corresponds to a heterogeneous layer with a distribution of stretched chain ends in the entire thickness n
of the layer: Figure 1 also shows plots ofn vs. the total layer height H(')/H°)= ~, h~/H (t)
Heterogeneous structures in layers of grafted polymer chains
1033
and the concentration in the first sublayer c]")/c~ . It can be seen that H (") and c]") increase with increasing n, and at n = 10 the values may be said to describe the heterogeneous layer structure satisfactorily, b Figure 2 shows the found values of components of [~j} and of the triangular matrix [flji} when n = 10 for regime IIe. The straight lines parallel to the abscissa have been drawn at intervals of flj~, Bj2 . . . . . flJi. The curves connect ~ji values to a fixed value of i.
10
1t
1"0
t,t, I, f',I,,f',","-10
fl
i O ~/,:~/:/m
FIG. 2 FIG. 3 Fro. 2. Values of components of the vector {~,} (eqn. (7)) and of the triangular matrix {Bu} (eqn. (8)) with n= 10 for regime Iio. FiG. 3. Density distribution histograms for units in the layer with n= 10 and flattened curves for regimes II, (1), Ilms (2) and II0 (3), Marked off along the axes are the values referring to characteristics of the homogeneous layer under the same regime:the fractured lines refer to the case of the homogeneous layer (n = 1).
Structure of the heterogeneous layer. Figure 3 shows the histograms of density distribution in the layer at n = 10 under the differing regimes. It can be seen that under all the regimes the density decreases with increasing distance h from the grafting plane. The c(h) relations are described by a convex curve such that in a considerable portion of the layer, adjoining the grafting plane, the density is practically a constant. The relative width of density distribution ztc=(c2-~-2)!/2/~ is small, and amounts to •c =0.34, 0.36 and 0.25 for regimes II +, IImy and II 0 respectively. Using the values of {aq} and {fljl} one may now characterize the conformations of chains in the layer in some detail. Figure 4 shows the histograms of distribution for units
1034
T, M, Bmsrlr~tN and A, K. KAR~t~nJv
A F(n)/,~tdO 6"0
it
-
5"0 W(h
q'8 0,3
r'-'-
O,f
I
?-kJ
:' '
I
V
I
0"5
11
,1
_._3.
i i
t'8. T ..
/'0 h/H u)
i I-Z
I 1"8
I 2.u
H/H (0
FIG. 4 F1o. 5 Fro. 4. Histograms of the distribution of units in the layer when n= 10. Unit numbers: N/4 (1, 1'), N]2 (2), 3N/4 (3), N (4, 4'). 1-4-regime II,; I', 4'-regime II+. FIG. 5. Plots of the relative free energy for two interacting layers of grafted chains vs. distance between the grafting planes: 1,/'-homogeneous layers, n= 1; 2, 2'-layers comprising n=2 sublayers; 3, 3'-layers comprising n = 3 sublayers; 1-3-segregated layers; 1'-3'-overlapping of outer sublayers. numbering N/4, N/2, 3N/4 and N (reading off the grafting plane) in the layer. For all the regimes there is a shift towards larger h and a significant widening of the function o f units distribution as the number of units increases (see Fig. 4). For instance, for regime II,,y the widths of distribution z l h = h i ( - ~2)1/~/~ are 0.13, 0-20, 0.27 and 0.30 for units numbers N/4, N[2, 3N/4 and N respectively. It therefore appears that in the case of a plane layer of grafted chains immersed in a solvent all the exponential laws previously detected for conditions of homogeneity of a layer are satisfied. This being so, I) the chain ends are distributed along the length of the layer; 2) this length is longer than for the case where one pustulates an equal extension for all the chains and assumes that the layer is homogeneous; 3) all the chains are stretched to a nonuniform eztent, and the stretching decreases as distance from the matrix increases; 4) the number of units likewise decreases with increasing distance from the
Heterogeneous structures in layers of grafted polymer chains
1035
matrix (no account is taken of density reductions in the immediate vicinity of the matrix on a blobe-size scale. Alter analyzing these results one may explain them from a simple physical standpoint. Let us take as the initial state a homogeneous layer of grafted uniformly stretched chains. A major contribution to free energy of the layer is made by the elastic component AFt. It is clear that AFt will be reduced if one introduces a function of chain distribution according to length, such that some of the chains will be stretched less than in a homogeneous layer. In this case one may allow for even a slight increase in stretching for a certain number of the chains. In regard to the concentration component AFt one may say that a necessary condition for reduction in this component will be a decrease in the mean density of units in the heterogeneous layer, i.e. an increase in the limiting thickness of the layer. As was pointed out above, this condition is compatiblc with a requirement that AFt has to be reduced. However, it is not a sutficienl condition. ,Since/JF¢ is a nonlinear ftmction of the nuanber of units it increases with the increase in the concentration gradient for the layer, (and) with the increase in its hctcrogeneity. The degree of heterogeneity is highest in the case of uniform stretching of each chain, and decreases in the case of nonuniform stretching i.e. where the degree of ~tretch decreases with increasing distance from the matrix along the chain contour with an accompanying rise in ~4F,. A compromise between the AF components is obtainable by density equalization though there will still be heterogeneity. This being so we have a fulfilment of the condition AF~")/AF~)=AF~")/AF~)=.4F~"3/AF~<1, i.e. not only the total free energy of the layer AF ~"), but equally its components AI'~"~ and Jk~." prove to be lower than for the homogeneous layer. The results of this investigation have been briefly outlined in [13], and are in good agreement with those obtained by other methods in regard to layers immersed in large amounts of a solvent. Khalatur used the Monte-Carlo method to model a layer of PE chains (containing up to 400 units) that were grafted to a surface. He showed that the density of the layer remains approximately constant within an interval of about 1/3-1/2 of the total layer thickness, after which the density decreases markedly. A similar result was reported in [15] by authors who modelled the conformation of a chain in a medium field stemming from other chains. Cantor and Dill modelled a layer of short chains [16], and showed that functions of units distribution in the layer are displaced towards the outer boundary of the layer, and widen as the unit number increases. In view of the results of the present investigation this conclusion could now be extended to cover long grafted chains as well. Dual layers of grafted chains. A homogeneous plane layer of grafted chains was used as a model in a theoretical study of a number of more complex structures. On a previous occasion [17] we investigated the interaction of two opposing homogeneous layers, and a segregation effect was observed lbr these layers: when the opposing grafting planes neared one another and arrived at a distance less than twice the individual layer height it was found that a compacting or a "squeezing up" of each layer took place such that no interpenetration of the layers could occur. The resu!ts of study of the heterogeneit~¢ of the layer call for an investigation of the
1036
T.M..BIRSHTEIN and A. K. KARAYEV
.... ....... +t , - - - = - 4 ~ ~ . . . . . . 1-4-1 4~___~, ;L_Ulr___~.r__." . .
i!
_
"-'If I it '.:ll i'+ 'III I
Z?t (D
FIG. 6. Change in the density profile of opposing layers orl their drawing closer together for the c a s e where n = 2. Solid Iines - segregated layers; fractured lines - overlapping of outer sublayers.
structure of layers when interaction between thcm takes place. To simplify the matter we will limit ourselves to the case where a heterogeneous layer is modelled by two or three sublayers (n = 2 or 3). Let us compare the dependence of free energies of interaction of opposing layers on the distance between gcafting planes for the case where one has segregated layers, and layers whose outer sublayers penetrate one another. Presetting the number of sublayers and conditions of interaction of the opposing layers we carried out a minimization with respect to data sets for {~} and {flj+}, taking each value of distance between the sublayers. The results of the free energy calculations appear ill Fig. 5, where curves 1-3 relate to segregated layers consisting of n= 1-3 sublayers, and curves 1'-3' correspond to layers consisting of n = 1-3 sublayers with overlapped outer sublayers (the number of sublayers being equal to n). It is seen from Fig. 5 that when allowance is made for the heterogeneity of the interacting layers this does not affectthe main result reported in [17]; the most favourable state is that where segregation of opposing layers has come about. Interpenetration of layers leads to a free energy increase, the amount of the gain being proportional to the interpenetration region (as is seen from Fig. 5 the discrepancy between curves with n and n' decreases in the order 1-1', 2-2', 3-3'). Figure 6 shows the equilibrium density profile for segregated opposing layers where n = 2. It is seen that when the layers draw closer together density values in the dual layer are equalized and there is a narrowing of the chain ends distribution function. Translated,by R. J. A. HENDRY REFERENCES 1. T. M. B I R S T H E I N and Ye. B. Z H U L I N A , Konformatsii makromolekul svyazannykh s poverknostyarni razdela (Conformations of Macromolecules Attached to Boundary Surfaces). p. 16, Pushchino, 1983 2. T. M. B I R S H T E I N and E. B. Z H U L I N A , Polymer 25: 1453, 1984
Superslow relaxation processes in amorphous linear polymers
1039
3. T. M. BIRSTHEIN, O. V. BORISOV, Ye. B, ZHULINA, A. R. KHOKHLOV and T. Yu. YURASOVA, Vysokomol. soyed. A29: 1169, 1987 (Translated in Polymer Sci. U.S.S.R. 29: 6, 1293, 1987) 4. Ye. B. ZHULINA and T. M. BIRSTHEIN, Vysokomol. soyed. A27:511, 1985 (Translated in Polymer Sci. U.S.S.R. 27: 3, 570, 1985) 5. T. M. BIRSTHEIN and Ye. B. ZHUL1NA, Wysokomol. soyed. A27: 1613, 1985 (Translated in Polymer Sci. U.S.S.R. 27: 8, 1807, 1985) 6. Ye. B. ZHULINA and T. M. BIRSTHEIN, Vysokomol. soyed. A28: 2589, 1986 (Translated in Polymer Sci. U.S.S.R. 28: 12, 2880, 1986) 7. P. G. de GENNES, Idei skeilinga v fizike polimerov (Scaling Ideas in Polymer Physics), Moscow, 1982 8. S. ALEXANDER, J. Phys., 38: 983, 1977 9. P. G. de GENNES, Macromolecules 13: 1069, 1980 10. T. M. BIRSTHEIN and Ye. B. ZHULINA, Vysokomol. soyed. A25: 1862, 1983 (Translated in Polymer Sci. U.S.S.R. 25: 9, 2165, 1983) 11. A. N. SEMENOV, Zh. Eksperim. i Teoret. lZiziki 88: 1242, 1985 12. D. HIMMELBLAU, Nelineinoye programmirovanie (Nonlinear Programming). Moscow, 1975 13. T. M. BIRSTHEIN, O. V. BORISOV, Ye.. B. ZHULINA and A. K. KARAYEV, IV All-Union Conf. on "Mathermaticl Methods for Investigation of Polymers and Biopolymers". p. 4, Pushchino, 1985 14. P. G. K H A L A T U R , Kolloid. zh. 45: 1171, 1983 15. A. M. SKVORTSOV, A. A. GORBUNOV and I. V. PAVLUSHKOV, All-Union Sympos. on Intermolec. Interaction and Molecular Conformations, p. 69, Pushchino, 1986 16. R. S. CANTOR and K. A. DILL, Macromolecules 17: 384, 1984 17. T. G. BIRSTHEIN and A. K. KARAYEV, Vysokomol. soyed. A29: 1882, 1987 (Translated in Polymer Sci. U.S.S.R. 29: 9, 2066, 1987)
Polymer Science U.$,S.R. Vol. 30, No, 5, pp. 1037-1046, 1988 Printed in Poland
0032,--3950/88 $10.00+ ,00 1989 Pergamon Press plc
SUPERSLOW RELAXATION PROCESSES IN AMORPHOUS LINEAR POLYMERS AND THEIR INTERPRETATION* Yu. G. YANOVSKII, V. N. POKROVSKn, YU. K. KOKORIN, Y'U. N. KARNET a n d L. V. TITKOVA A. V. Topchiyev Institute of Petrochemical Synthesis, U.S.S.R. Academy of Sciences (Received 1 December 1986) Relaxation processes in the region of the flow state (the terminal zone) in 1,4-polybutadiene with a narrow MWD have been investigated by a dynamic method at low frequencies over a wide temperature range. Superslow relaxation processes observed in experiments are discussed in terms of the theory of viscoelasticity of linear polymers. * Vysokomol. soyed. A30: No. 5, 1009-1016, 1988.
T. M. B1RSHTEIN and A. K. KARAYRV
12. M. G. VITOVSKAYA, I. N. SHTENNIKOVA, E . P. ASTAPENKO and T. V. PEKER, Vysokomol. soyed. AI7: 1161, 1975 (Translated in Polymer Sci. U.S.S.R. 17: 5, 1334, 1975) 13. L. N. ANDREYEVA, A. A. GORBUNOV, S. A. DIDENKO, Ye. V. KORNEYEVA, P. P. LAVRENKO, N. A. PLATE and V. P. SHIBAYEV, Vysokomol. soyed. BIS: 209, 1973 (Not translated in Polymer Sci. U.S.S.R.) 14. V. N. TSVETKOV, V. Ye. ESKIN and S . Ya. FRENKEL', Struktura makromolekul v rastvor¢ (Structure of Macromolecules in Solutions). Moscow, 1964 15. J. E. HEARST and W. H. STOCKMAYER, J. Chem. Phys. 37: 1425, 1962 16. ft. E. HEARST, J. Chem. Phys. 38: 1963, 1962 17. J. C. KIRWOOD and ft. RISEMAN, J. Chem. Phys. 16: 565, 1948 18. H. YAMAKAWA and G. FUJH, Macromolecules 6: 407, 1973 19. R. SIMHA, J. Phys. Chem., 13: 188, 1945 20. G. BOHDANECKY, Macromolecules 16: 1483, 1983
PolymerScience U.S.S.R. Vol. 30, No. 5, pp. 1028-1037, 1988 Printed in Poland
0032-3950/88 $10.00+.00 1989 Pergamon Press plc
HETEROGENEOUS STRUCTURES IN LAYERS OF GRAFTED POLYMER CHAINS* T. M. BIRSHTEIN a n d A . K. KARAYEV Hetero-organic Compounds Institute, U.S.S.R. Academy of Sciences
(Received 29 November 1986) A method for analyzing the structure of a layer of polymer chains grafted to a surface has been developed without an assumption that the layer is homogeneous. The layer was simulated by a system of sublayers of constant density. Numerical calculations were performed by computer for 1-5 and 10 sublayers under all the regimes of a temperature-concentration diagram of state for the layer. It was found that the free energy minimum corresponds to a monotonic diminution in the layer density as the distance from the grafting plane increases. Distribution functions for the units according to the layer height are displaced towards the outer boundary of the layer and grow wider as the unit number increases. When heterogeneity is taken into account it does not affect the conclusion previously reached concerning segregation of two opposing layers. A SYSTEM o f p o l y m e r chains g r a f t e d t o an i m p e r m e a b l e surface ( m a t r i x ) a n d i m m e r s e d in a solvent is u s e d in [1] as a basis f o r a t h e o r e t i c a l d e s c r i p t i o n o f a w h o l e series o f p o l y m e r systems such as, f o r instance, branclaed p o l y m e r s [2, 3], micelles [4] a n d supercrystals [5, 6] o f b l o c k c o p o l y m e r s , a n d a m o r p h o u s - c r y s t a l l i n e p o l y m e r s . O n t h e basis o f scaling ideas [7] t h e o r e t i c a l studies o f t h e s t r u c t u r e o f a layer o f chains g r a f t e d t o a p l a n e m a t r i x [8-10] s t a r t with a n a s s u m p t i o n t h a t t h e l a y e r is h o m o * Vysokomol. soyed. A30: No. 5, 1001-1008, 1988.
Heterogeneous structures in layers of grafted polymer chains
1029
geneous, i.e. there is a constant number of chain units in the basic thickness of the layer (excluding the drop in the number of units close to the grafting surface and in the vicinity of the outer boundary of the layer [9]), and it is assumed that there is a uniform degree of extension of all the chains. On the Other hand, a marked heterogeneity of the layer has been revealed by the data obtained in several experimental and theoretical investigations, including the results of a computer simulation of a layer of relatively short chains. The present paper relates to a simple model developed by us on the basis of a scaling approach with a view to providing a method for analyzing the structure of a layer of long grafted chains in the absence of any assumption that the layer is homogeneous. The data obtained show that the number of chain units decreases significantly as the distance from the grafting plane increases. The model and the method. Let us take a layer of chains grafted to a common plane surface and immersed in a solvent at z =(T-0)/0t> 0. To characterize the rigidity of the grafted chains we take the segmental asymmetry parameter p~> 1, and the number of chain units is sufficiently large, N>>I. The length of a unit (a spherically symmetrical element of the chain)we will take in our case to be a unit length. Now let a be the mean area of grafting per chain; 1 <~a <<.R 2, where R is the mean length of a free chain in solution. The structure of the layer of grafted chains is found by minimizing the free energy for a chain in the layer, expressed as the sum of /tF= AFc + zIFe
(l)
where AFc is the chain units concentration contribution determined at z~>0 by the thermodynamical unfavourableness of collisions involving two or three units in the layer, and AF, is the elastic component taking account of stretching of chains in the layer. Taking the layer to be a semidilute polymer solution of constant concentration c = N/trH,
(2)
where H is the homogeneous layer height, we have [8-10] 4 F c = kl NcXr2-XP3Y
(3)
AF, = k2 N - IH2cyr-Yp- 2+ 3y
(4)
The exponents x and y depend on the type of regime to which the semidilute solution is subjected: x = 5/4, 1 and 2; y = 1/4, 0 and 0 for regimes II+, II,,: and II0 respectively; kx and k2 are numerical coefficients. Minimizing eqn. (1) in respect to H one obtains [8-10] equilibrium values for the height and the free energy of the homogeneous layer (index (1)) x--y
2 -x+y
1
H ( t ) = K 1 N t r 2+x-Y'c2+X-Y p2+X-Y -2x
2(2-x+y)
3
(5)
x
A F ( I ) = K 2 N a 2 * ~ - y z 2+~,-y p Y-2~-~-'-~,
(6)
1030
T.M. BmSHTEiNand A. K. KARAYEV 1
where Kx = [kt x / k 2 ( 2 - h ) l 2+x-y and K2--klK~X+k2 K2-y are numerical coefficients. The concentration of units in the layer is given by equations (2) and (5). Let us now generalize the scheme under discussion. To allow for possible heterogeneity of the layer we will subdivide it into a fixed number n=2, 3 ... of plane sublayers (steps) of thickness hi, where i= 1, 2, ... n, and the layers can be numbered (as to distance) from the matrix. Taking account of the marked linear stretching of chains in the layer let us subdivide all the grafted chains into n types, such that chains of the jth type (their fraction being 0~j) are located in the firstj sublayers and end in the jth sublayer. Let flji be the number of units for chains of thejth type that are located in the ith layer ( i ~ j = 1, 2 ..... n). Thus the distribution of chain ends according to layers is given by the vector ~=(~q, ~2, ..., ~ ) , (7) and the distribution of units for chains of each type is given by the rows of the triangular matrix
l/s,, o
li
0 )
n
with the normalizations ~ ~l= 1 and ~, flsi= 1. i=1
i=1
The proposed model is a discrete variant of an analytical method of investigation of a layer of grafted chains developed by Semenov [11] involving an approximation of a self-consistent field on condition that the unit concentration is constant. In this case only the elastic component AFe was minimized on the basis of distributions of chain ends and the stretching of the chains (in a discrete model based on at and fl). In the present instance a concentration distribution in the layer is allowed for, and dFc is likewise taken into account. Moreover in the case of the discrete model it is possible not only to analyze a self-consistent field approximation (regimes II 0 and IIms), but also to go beyond the limits of this approximation (regime II+). We would point out that generally speaking the total free energy of the layer also includes another factor, namely a surface component determined by deficiencies in contacts at the interface between sublayers in the discrete model. However, simple calculations show that the contribution of this surface term decreases as the number of sublayers increases and is substantially less than the value of AF in all cases, including that of transition to a continued limit. Free energy of the graduated layer. The total free energy AF (") of the layer may be expressed as the sum of free energies of the sublayers AFt A F (n) = ~ AF i
(9)
i=1
We will treat each sublayer as being a homogeneous layer of grafted chains with an effective grafting area that increases with the increase in i. Let us say that the number
Heterogeneous structures in layers of grafted polymer chains
1031
of units and the sublayer thicknesses are such that they may be accomodated within the framework of the scheme outlined above. For this to happen it is essential that the thickness of a sublayer should be significantly greater than the density correlation radius. As in eqn. (1), values of ziF i will now be expressed as the sum of the concentration and the elastic components, calculating for a chain grafted to the matrix (i.e. calculating for an area a). Generalizing expressions (3) and (4) we obtain for the ith ~ublayer (zlFc)~ = (zlF(,1)) ( N i ) N - l(ei/c)~
(LIF,)i = (AF[, 1>) (Ni- 1) N (ci/c) V(hffH) 2, where
(10) (1 I)
tl
( m i ) N - 1 =Ai = ~ o~jflji
(12)
j=i tl
(13)
( N [ 1) N = B i = ~ajlfl.i, j=l
ci = (Ni)/ahl
(14)
A further problem is that of minimizing the total free energy d F (') in respect to all values of {~j}, {fli~} and {h+}. This is most simply achieved by minimizing in respect to the set of {h~} values, since each of these determines solely the free energy of the ith sublayer. Equilibrium values of ht are obtained from the condition
ZF,
[(aFt),
= 0
Now with the aid of equations (3) and (4) as well as (10)-(14), and assuming that the numbers of units in all the sublayers correspond to the same regime of the semidilute solution (II+, IIm: or II0) we obtain the height, the free energy and the concentration of units in the ith sublayer 1
hi=H(1)Ai(AiBi)
2+~-y
(15)
x
/IF i =AF(I)Ai(Ai Bi)2 +x-v
(16)
1
ct = cCU(Ai Bl) 2 +x-y,
(17)
where H (1), AF (1) and c CI) are the height, the free energy and the concentration of units in the homogeneous (graduated, n = 1) layer under the regime in question, determined by eqns. (2), (5) and (6). It is evident from equations (9) and (15)-(17) that all the characteristics of sublayers and of the layer as a whole are distinguishable solely by the numerical coefficients of Ai and B i (eqns. (12)and (13)) from the characteristics of the homogeneous layer. This ineans that a minimization of.the total free energy of the layer may be reduced to a minimization (based on sets of {~i} and {flji} values, taking differing values of n) of
1032
T. M. ~ s n r v l s
and A. K. KAltA'ttv
the total coefficient X
O f ~ A~(AtB~) 2+~-y |=1
Minimization of the nonlinear function ~ for many variables (a nonlinear programming problem) was carried out by a "sliding tolerance" method, using a Flexi program [12], for n=2, 3, 4, 5, 10, under regimes II+, IIml and ]I0.
1"5I~~j.~"~.
y ~.1/y ( .
•
!
+
Z
03
1.0
1 I !
I l I I
g'5
l t L
t 1 I
=
I
I
I
I
I I
J
I
I
Z lln
w 5 4 ,7 i FIo. 1. Relative characteristics of the system yt=)ly(l~ vs. the reciprocal of the number of sublayers 1/n for regimes II+ (1), II=t (2), and IIo (3): a= JF(")MF(I}; b ffi c~")/c~~); c-H(n)/H ~1~. Figure 1 shows the relative values of free energy of the layer AF(")/JF (1) vs. the preset number of sublayers. It can be seen that under all the regimes the free energy decreases with the increase in n, such that a free energy minimum corresponds to a heterogeneous layer with a distribution of stretched chain ends in the entire thickness n
of the layer: Figure 1 also shows plots ofn vs. the total layer height H(')/H°)= ~, h~/H (t)
Heterogeneous structures in layers of grafted polymer chains
1033
and the concentration in the first sublayer c]")/c~ . It can be seen that H (") and c]") increase with increasing n, and at n = 10 the values may be said to describe the heterogeneous layer structure satisfactorily, b Figure 2 shows the found values of components of [~j} and of the triangular matrix [flji} when n = 10 for regime IIe. The straight lines parallel to the abscissa have been drawn at intervals of flj~, Bj2 . . . . . flJi. The curves connect ~ji values to a fixed value of i.
10
1t
1"0
t,t, I, f',I,,f',","-10
fl
i O ~/,:~/:/m
FIG. 2 FIG. 3 Fro. 2. Values of components of the vector {~,} (eqn. (7)) and of the triangular matrix {Bu} (eqn. (8)) with n= 10 for regime Iio. FiG. 3. Density distribution histograms for units in the layer with n= 10 and flattened curves for regimes II, (1), Ilms (2) and II0 (3), Marked off along the axes are the values referring to characteristics of the homogeneous layer under the same regime:the fractured lines refer to the case of the homogeneous layer (n = 1).
Structure of the heterogeneous layer. Figure 3 shows the histograms of density distribution in the layer at n = 10 under the differing regimes. It can be seen that under all the regimes the density decreases with increasing distance h from the grafting plane. The c(h) relations are described by a convex curve such that in a considerable portion of the layer, adjoining the grafting plane, the density is practically a constant. The relative width of density distribution ztc=(c2-~-2)!/2/~ is small, and amounts to •c =0.34, 0.36 and 0.25 for regimes II +, IImy and II 0 respectively. Using the values of {aq} and {fljl} one may now characterize the conformations of chains in the layer in some detail. Figure 4 shows the histograms of distribution for units
1034
T, M, Bmsrlr~tN and A, K. KAR~t~nJv
A F(n)/,~tdO 6"0
it
-
5"0 W(h
q'8 0,3
r'-'-
O,f
I
?-kJ
:' '
I
V
I
0"5
11
,1
_._3.
i i
t'8. T ..
/'0 h/H u)
i I-Z
I 1"8
I 2.u
H/H (0
FIG. 4 F1o. 5 Fro. 4. Histograms of the distribution of units in the layer when n= 10. Unit numbers: N/4 (1, 1'), N]2 (2), 3N/4 (3), N (4, 4'). 1-4-regime II,; I', 4'-regime II+. FIG. 5. Plots of the relative free energy for two interacting layers of grafted chains vs. distance between the grafting planes: 1,/'-homogeneous layers, n= 1; 2, 2'-layers comprising n=2 sublayers; 3, 3'-layers comprising n = 3 sublayers; 1-3-segregated layers; 1'-3'-overlapping of outer sublayers. numbering N/4, N/2, 3N/4 and N (reading off the grafting plane) in the layer. For all the regimes there is a shift towards larger h and a significant widening of the function o f units distribution as the number of units increases (see Fig. 4). For instance, for regime II,,y the widths of distribution z l h = h i ( - ~2)1/~/~ are 0.13, 0-20, 0.27 and 0.30 for units numbers N/4, N[2, 3N/4 and N respectively. It therefore appears that in the case of a plane layer of grafted chains immersed in a solvent all the exponential laws previously detected for conditions of homogeneity of a layer are satisfied. This being so, I) the chain ends are distributed along the length of the layer; 2) this length is longer than for the case where one pustulates an equal extension for all the chains and assumes that the layer is homogeneous; 3) all the chains are stretched to a nonuniform eztent, and the stretching decreases as distance from the matrix increases; 4) the number of units likewise decreases with increasing distance from the
Heterogeneous structures in layers of grafted polymer chains
1035
matrix (no account is taken of density reductions in the immediate vicinity of the matrix on a blobe-size scale. Alter analyzing these results one may explain them from a simple physical standpoint. Let us take as the initial state a homogeneous layer of grafted uniformly stretched chains. A major contribution to free energy of the layer is made by the elastic component AFt. It is clear that AFt will be reduced if one introduces a function of chain distribution according to length, such that some of the chains will be stretched less than in a homogeneous layer. In this case one may allow for even a slight increase in stretching for a certain number of the chains. In regard to the concentration component AFt one may say that a necessary condition for reduction in this component will be a decrease in the mean density of units in the heterogeneous layer, i.e. an increase in the limiting thickness of the layer. As was pointed out above, this condition is compatiblc with a requirement that AFt has to be reduced. However, it is not a sutficienl condition. ,Since/JF¢ is a nonlinear ftmction of the nuanber of units it increases with the increase in the concentration gradient for the layer, (and) with the increase in its hctcrogeneity. The degree of heterogeneity is highest in the case of uniform stretching of each chain, and decreases in the case of nonuniform stretching i.e. where the degree of ~tretch decreases with increasing distance from the matrix along the chain contour with an accompanying rise in ~4F,. A compromise between the AF components is obtainable by density equalization though there will still be heterogeneity. This being so we have a fulfilment of the condition AF~")/AF~)=AF~")/AF~)=.4F~"3/AF~<1, i.e. not only the total free energy of the layer AF ~"), but equally its components AI'~"~ and Jk~." prove to be lower than for the homogeneous layer. The results of this investigation have been briefly outlined in [13], and are in good agreement with those obtained by other methods in regard to layers immersed in large amounts of a solvent. Khalatur used the Monte-Carlo method to model a layer of PE chains (containing up to 400 units) that were grafted to a surface. He showed that the density of the layer remains approximately constant within an interval of about 1/3-1/2 of the total layer thickness, after which the density decreases markedly. A similar result was reported in [15] by authors who modelled the conformation of a chain in a medium field stemming from other chains. Cantor and Dill modelled a layer of short chains [16], and showed that functions of units distribution in the layer are displaced towards the outer boundary of the layer, and widen as the unit number increases. In view of the results of the present investigation this conclusion could now be extended to cover long grafted chains as well. Dual layers of grafted chains. A homogeneous plane layer of grafted chains was used as a model in a theoretical study of a number of more complex structures. On a previous occasion [17] we investigated the interaction of two opposing homogeneous layers, and a segregation effect was observed lbr these layers: when the opposing grafting planes neared one another and arrived at a distance less than twice the individual layer height it was found that a compacting or a "squeezing up" of each layer took place such that no interpenetration of the layers could occur. The resu!ts of study of the heterogeneit~¢ of the layer call for an investigation of the
1036
T.M..BIRSHTEIN and A. K. KARAYEV
.... ....... +t , - - - = - 4 ~ ~ . . . . . . 1-4-1 4~___~, ;L_Ulr___~.r__." . .
i!
_
"-'If I it '.:ll i'+ 'III I
Z?t (D
FIG. 6. Change in the density profile of opposing layers orl their drawing closer together for the c a s e where n = 2. Solid Iines - segregated layers; fractured lines - overlapping of outer sublayers.
structure of layers when interaction between thcm takes place. To simplify the matter we will limit ourselves to the case where a heterogeneous layer is modelled by two or three sublayers (n = 2 or 3). Let us compare the dependence of free energies of interaction of opposing layers on the distance between gcafting planes for the case where one has segregated layers, and layers whose outer sublayers penetrate one another. Presetting the number of sublayers and conditions of interaction of the opposing layers we carried out a minimization with respect to data sets for {~} and {flj+}, taking each value of distance between the sublayers. The results of the free energy calculations appear ill Fig. 5, where curves 1-3 relate to segregated layers consisting of n= 1-3 sublayers, and curves 1'-3' correspond to layers consisting of n = 1-3 sublayers with overlapped outer sublayers (the number of sublayers being equal to n). It is seen from Fig. 5 that when allowance is made for the heterogeneity of the interacting layers this does not affectthe main result reported in [17]; the most favourable state is that where segregation of opposing layers has come about. Interpenetration of layers leads to a free energy increase, the amount of the gain being proportional to the interpenetration region (as is seen from Fig. 5 the discrepancy between curves with n and n' decreases in the order 1-1', 2-2', 3-3'). Figure 6 shows the equilibrium density profile for segregated opposing layers where n = 2. It is seen that when the layers draw closer together density values in the dual layer are equalized and there is a narrowing of the chain ends distribution function. Translated,by R. J. A. HENDRY REFERENCES 1. T. M. B I R S T H E I N and Ye. B. Z H U L I N A , Konformatsii makromolekul svyazannykh s poverknostyarni razdela (Conformations of Macromolecules Attached to Boundary Surfaces). p. 16, Pushchino, 1983 2. T. M. B I R S H T E I N and E. B. Z H U L I N A , Polymer 25: 1453, 1984
Superslow relaxation processes in amorphous linear polymers
1039
3. T. M. BIRSTHEIN, O. V. BORISOV, Ye. B, ZHULINA, A. R. KHOKHLOV and T. Yu. YURASOVA, Vysokomol. soyed. A29: 1169, 1987 (Translated in Polymer Sci. U.S.S.R. 29: 6, 1293, 1987) 4. Ye. B. ZHULINA and T. M. BIRSTHEIN, Vysokomol. soyed. A27:511, 1985 (Translated in Polymer Sci. U.S.S.R. 27: 3, 570, 1985) 5. T. M. BIRSTHEIN and Ye. B. ZHUL1NA, Wysokomol. soyed. A27: 1613, 1985 (Translated in Polymer Sci. U.S.S.R. 27: 8, 1807, 1985) 6. Ye. B. ZHULINA and T. M. BIRSTHEIN, Vysokomol. soyed. A28: 2589, 1986 (Translated in Polymer Sci. U.S.S.R. 28: 12, 2880, 1986) 7. P. G. de GENNES, Idei skeilinga v fizike polimerov (Scaling Ideas in Polymer Physics), Moscow, 1982 8. S. ALEXANDER, J. Phys., 38: 983, 1977 9. P. G. de GENNES, Macromolecules 13: 1069, 1980 10. T. M. BIRSTHEIN and Ye. B. ZHULINA, Vysokomol. soyed. A25: 1862, 1983 (Translated in Polymer Sci. U.S.S.R. 25: 9, 2165, 1983) 11. A. N. SEMENOV, Zh. Eksperim. i Teoret. lZiziki 88: 1242, 1985 12. D. HIMMELBLAU, Nelineinoye programmirovanie (Nonlinear Programming). Moscow, 1975 13. T. M. BIRSTHEIN, O. V. BORISOV, Ye.. B. ZHULINA and A. K. KARAYEV, IV All-Union Conf. on "Mathermaticl Methods for Investigation of Polymers and Biopolymers". p. 4, Pushchino, 1985 14. P. G. K H A L A T U R , Kolloid. zh. 45: 1171, 1983 15. A. M. SKVORTSOV, A. A. GORBUNOV and I. V. PAVLUSHKOV, All-Union Sympos. on Intermolec. Interaction and Molecular Conformations, p. 69, Pushchino, 1986 16. R. S. CANTOR and K. A. DILL, Macromolecules 17: 384, 1984 17. T. G. BIRSTHEIN and A. K. KARAYEV, Vysokomol. soyed. A29: 1882, 1987 (Translated in Polymer Sci. U.S.S.R. 29: 9, 2066, 1987)
Polymer Science U.$,S.R. Vol. 30, No, 5, pp. 1037-1046, 1988 Printed in Poland
0032,--3950/88 $10.00+ ,00 1989 Pergamon Press plc
SUPERSLOW RELAXATION PROCESSES IN AMORPHOUS LINEAR POLYMERS AND THEIR INTERPRETATION* Yu. G. YANOVSKII, V. N. POKROVSKn, YU. K. KOKORIN, Y'U. N. KARNET a n d L. V. TITKOVA A. V. Topchiyev Institute of Petrochemical Synthesis, U.S.S.R. Academy of Sciences (Received 1 December 1986) Relaxation processes in the region of the flow state (the terminal zone) in 1,4-polybutadiene with a narrow MWD have been investigated by a dynamic method at low frequencies over a wide temperature range. Superslow relaxation processes observed in experiments are discussed in terms of the theory of viscoelasticity of linear polymers. * Vysokomol. soyed. A30: No. 5, 1009-1016, 1988.