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A lattice model for dynamics in a mixed polymer-diluent glass
556
Journal of Non-Crystalline Solids 131-133 (1991) 556-562 North-Holland
Section 2.5. Glassy polymers
A lattice model for dynamics in a mixed polymer-diluent glass A.A. Jones, P.T. Inglefield, Y. Liu, A.K. R o y a n d B.J. C a u l e y Department of Chemistry, Clark University, Worcester, MA 01610, USA
A lattice model is developed to interpret N M R lineshape and relaxation data on polymer-diluent systems. The lattice is used to count nearest-neighbor contacts which are assumed to influence local dynamics. When applied to the polymer chain, the presence of one diluent-polymer repeat unit contact is considered to suppress sub-glass transition motion through improved packing. Polymer repeat units in contact with more than one diluent molecule on the lattice are presumed to have enhanced mobility. Polymer repeat units surrounded by other polymer repeat units are thought to have no change in their sub-glass transition dynamics. When applied to the diluent, isotropic rotational motion of diluent surrounded by polymer is considered to commence at the apparent thermal glass transition. For those diluent molecules in contact with other diluent molecules on the lattice, sub-glass transition rotational motion occurs at a temperature determined by the intrinsic mobility of the diluent. The motions of the chain and the diluent are reflected in the modulus and are traditionally discussed in terms of plasticization and antiplasticization.
1. Introduction Molecular motions of the polymer chain in polymeric glasses well below the thermal glass transition influence bulk properties. Sub-glass t r a n s i t i o n motions are geometrically restricted and can be divided into two general categories. One class of these motions involves anisotropic rotations such as methyl-group rotation or phenylgroup rotation [1] while the other class involves librational motion [2]. Solid state N M R has proved useful in elucidating a variety of details concerning the rate and amplitude of both categories of motions, One of the best studied systems is bisphenol A polycarbonate (BPA-PC) shown in fig. 1. Both methyl- and phenylene-group rotations occur in glassy BPA-PC, with the phenylene motion linked to a significant mechanical loss [3-14]. The phenylene group reorients by jumping between two minima separated by approximately 180 = and this motion has been named ~r-flips [3]. The rate of occurrence of ,~-flips is inhomogeneous: i.e., different phenylene groups in different spatial locations in the glass execute ~r-flips at different rates. The fractional exponential correlation function
POLYMER
.-BPA-PC
--
~)~"-
--
=,"~ 7~ ! / ' ~
'r--~ ="~
. . . i.
'j?---
o--
OlLUENTS o ~
/ e _II_ o _ _
c 4H
DBP
c__o~ %"6
TOP
Fig.
I'~'~
I PO
3 1. Chemical structures and abbreviations for the polymer
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
repeat unit and the two diluents.
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
with an exponent in the range of 0.15 to 0.20 has been used to characterize the distribution of rates [6,8-9]. The addition of a number of low molecular weight liquids to BPA-PC yields glasses with a single, lower thermal glass transition temperature, and a much reduced low-temperature mechanical loss peak [15]. Solid state N M R studies have shown that the suppression of the mechanical loss coincides with the suppression of the occurrence of ~r-flips [16]. The reduced mechanical loss upon addition of a low molecular weight diluent also leads to a higher modulus than is observed for the pure polymer [15]. The classical term for these phenomena associated with the addition of diluent is antiplasticization. Plasticization or a lowering of the modulus also occurs in BPA-PC at higher concentrations [15]; also, in general the modulus depends on a variety of factors besides concentration such as temperature, the structure of the diluent and the structure of the polymer, Solid state N M R experiments have also shown that the suppression of ~r-flips is associated with an apparent broadening of the inhomogeneous distribution of correlation times [16].
Tip (ms)
In an attempt to provide an interpretational framework for the observations just reviewed, a lattice model is presented below to count polymer-diluent and diluent-diluent contacts. The resuits of a number of recent solid state N M R experiments on BPA-PC plus diluent which motivated the development of this lattice description are summarized [17-18]. The lattice model correlates relaxation behavior of both the polymer and the diluent which is required in order to understand the mechanical response of the multicomponent glass.
2. Polymer dynamics in the presence of diluent The first experimental results to be summarized are on BPA-PC plus di-n-butyl phthalate (DBP). This is a convenient system for consideration by a lattice model since the polymer repeat unit and the diluent are of the same size and thus either one or the other can occupy a lattice site. As mentioned in the introduction, deuterium lineshape studies on phenyl deuterated BPA-PC indicate a suppression of the ~r-flips and a broadening of the apparent distribution of flip rates [16]. The
22 21 20 19 18 17 16-
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-~20-10~
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o'
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TC Fig. 2. Proton Tlo versus temperature for pure BPA-PC d 6 and for BPA-PC d 6 plus 5 wt% perdeutero DBP. The solid line is a smoothed representation of the data for pure polymer and the dashed line for polymer plus diluent. The shaded region indicates suppression of the motion of some of the phenylene groups by the presence of the diluent DBP.
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
558
suppression and broadening are both associated with an increased population of slow flippers, To further characterize the distribution of flip rates, proton T1~ were measured as a function of temperature on partially deuterated BPA-PC [17]. The methyl groups of BPA-PC d 6 a r e deuterated while the phenylene groups remain protonated, The proton T1, values are primarily determined by the ,~-flip rates leading to the broad minimum shown in fig. 2 which can be mapped onto the mechanical loss maximum through the use of a fractional exponential correlation function [6,8]. The breadth of the minimum reflects the distribution of ~T-flip rates. The T~p experiment has a characteristic frequency of 44 kHz and the minimum occurs at about - 2 0 ° C while the mechanical experiment has a characteristic frequency of 1 Hz and the maximum occurs at a b o u t - 1 0 0 ° C , On addition of 5 wt% perdeutero DBP, the minimum narrows and is shifted to higher temper-
ature as is also shown in fig. 2. The deuterated diluent is invisible in this N M R experiment so that the TI~ minimum still reflects only the ~r-flip rates. Close examination shows the change in the minimum to be caused by selective suppression of the low-temperature side of the minimum which corresponds to the faster moving phenylene groups. This is shown as a shaded region in fig. 2. In terms of fitting the minimum with a fractional exponential correlation function, the exponent stays constant while the apparent energy increases from 49 to 71 k J / m o l [17]. An examination of the corresponding mechanical loss peak or dielectric loss peaks in related systems does not show a suppression by elimination of the low-temperature side. However, diluent contributions to loss are not easily separated from polymer contributions, as can be done by isotopic labelling in N M R experiments, making simple comparisons difficult. How can the narrower minimum be reconciled
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Too Fig. 3. Same as fig. 2 except for pure p o l y m e r and p o l y m e r plus 10 wt% diluent. N o t e b o t h s u p p r e s s i o n i n d i c a t e d b y s h a d i n g a n d the presence of a new lower t e m p e r a t u r e m i n i m u m at - 90 o C.
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
559
to the broader distribution of flip rates seen in lineshape studies? Phenylene groups whose flip rates are significantly suppressed do not contrib-
F3 =
4pd 3,
(ld)
F4 =
d4"
(le)
ute to the T1, minimum and do not undergo relaxation until the glass transition is reached. If these very slow rates were combined with those remaining in the minimum, the resulting distribution would appear quite broad. The presence of this slow population is indeed visible in the deuteron lineshape, The T1, minimum located at - 2 0 ° C continues to narrow as diluent concentration is increased from 5 to 10-25 wt%. However, an entirely new feature also appears: a minimum located at - 1 0 0 ° C as shown in fig. 3. This lower temperature minimum which develops at higher concentration must correspond to phenylene groups which move more rapidly than those associated with the original minimum in the pure polymer, These new details concerning changes in the distribution of rates of phenylene-group motion upon addition of diluent provide the impetus for
Several assumptions and refinements need to be made to relate these fractional populations to observed relaxation behavior. First, the fractional population, F0, would correspond to the phenyl-
the development of the lattice model. The set-up of polymer and diluent on a lattice is much like the beginning of Flory-Huggins theory. A lattice with six nearest neighbors is considered and either a repeat unit or a diluent occupies a site. Because of chain connectivity, two sites around a polymer unit are occupied by covalently bonded repeat units. There are four remaining nearest-neighbor sites which may be either diluent molecules or other repeat units. Polymer repeat units will be classified according to the number of nearest-neighbor sites which are occupied by diluent. If p is the probability a site is occupied by a polymer repeat unit and the lattice is filled on the basis of random statistics, the fraction of sites, F0, which are polymer units surrounded by other polymer units is F0 = p4.
(la)
If d is the fraction of sites occupied by diluent, the fractions of repeat units with one, two, three and four diluent nearest neighbors are given by E 1 = 4p3d, F 2 = 6p2d 2,
(lb) (lc)
ene groups contributing to the normal relaxation minima or loss peak. The fraction of polymer units with one diluent neighbor is associated with the extent of suppression. The first diluent neighbor is assumed to improve significantly packing around the reference polymer unit and therefore reduces mobility. The rate of motion is assumed to become so slow that relaxation of this unit no longer occurs as part of the low-temperature peak but is shifted to the glass transition. Polymer repeat units with more than one diluent neighbor are assumed to have enhanced mobility and their relaxation is shifted to lower temperatures relative to the original low-temperature loss peak. The lower temperature results from the greater inherent mobility of diluent molecules, as indicated by the glass transition temperature for the diluent as a pure liquid. The clusters of diluent molecules thus promote 'soft' regions in the glass. To a first approximation, the fractional population of repeat units in contact with two or more diluents is the sum of F 2, F 3 and F 4. According to the model, these repeat units account for the lower temperature minimum observed at - 1 0 0 ° C for DBP in BPA-PC. A slight correction to the fractional populations calculated thus far is needed. Suppression of the flip motion is proposed for repeat units in contact with a single diluent and enhanced mobility is proposed for repeat units in contact with two or more diluent moieties. This calculation should be refined, since our intention is to associate suppression with an isolated diluent molecule. If a repeat unit is in contact with a single diluent molecule but that diluent is in contact with other diluent molecules, the reference repeat unit should be considered to have enhanced mobility rather than suppressed mobility. To estimate this effect, the fractional population of diluent in clusters is required. This calculation is related to the initial stages of percolation theory [19] and the number
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
560
1.0
......
molecule is an isolated diluent molecule will be called the extent of suppression, S:
/"
0.a
/"
S = F xp~ = 4pSd.
The fraction o f repeat units that are in contact with a microscopic cluster of diluents through a single diluent molecule is /'1(1 _pS), and this situation will lead to enhanced mobility of the reference unit. These repeat units should be combined with the fractions F 2, F 3, and F4 to yield a fraction with enhanced mobility, E:
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Fig. 4. Plots of Fo, S, and E versus diluent concentration, d. F0 (solid line) indicates the fraction of the polymer repeat units surrounded by polymer which contributes to the - 2 0 °C minimum; S (dotted line) is the fraction of polymer repeat units in contact with one diluent leading to a suppression of the sub-glass transition motions; E (dashed line) is the fraction of polymer repeat units in contact with diluent clusters leading to the appearance of a m i n i m u m at - 9 0 ° C at higher diluent concentrations. Associatedmathematicalexpressions are given
in the text. of clusters of various sizes is not easily calculated. However, only the fractional population of isolated diluent molecules, G 1, is required here and that quantity is simply given by Ga = p 6 . (2) Now the probability that a repeat unit is in contact with one diluent molecule and that diluent
To summarize the factors influencing polymer chain dynamics, the intensity of the low-temperature loss peak is given by F0, the extent of suppression is given by S, and the intensity of a new lOSS peak caused by contact with mobile diluent groupings is given by E. These quantities are given plotted in fig. 4 a s a function of diluent c o n c e n t r a tion, d. These calculations assume the glass transition temperature remains higher than the position of the low temperature loss peak of the pure polymer.
3. Diluent dynamics in polymeric glasses Thus far changes in the mobility of polymer repeat units with diluent concentration have been considered but similar changes in diluent mobility are also observed. To monitor diluent dynamics
+
=
?
j SLOW C O M P O N E N T
_ FAST
COMPONENT
COMPOSITE
Fig. 5. Partially collapsed chemical shift anisotropy lineshape. The points are experimental observations and the lines are simulations. The simulation consists of a narrow component corresponding to the fast population of diluent and a broad component corresponding to the slow population. The two components are combined to simulate the observed lineshape.
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
independent of the polymer matrix, phosphorous31 N M R of phosphate ester diluents has proved quite informative [18,20]. The system-for consideration here will be a trioctyl phosphate (TOP) in BPA-PC (structures shown in fig. 1). The 31p chemical shift anisotropy line shape for 15 wt% TOP shows considerable collapse at temperatures well below the glass transition indicating the presence of considerable diluent motion. The lineshape collapse has been simulated by combining two populations with two rates referred to as the fast and slow components. Figure 5 shows a typical spectrum in the collapse region and the fast and slow contributions to the total lineshape. At temperatures just below the thermal glass transition, the fast component is undergoing rapid, isotropic Brownian rotational diffusion on the N M R timescale while the slow component is undergoing rather little reorientation on the same timescale. This bimodal character used to describe the lineshape can readily be meshed with the lattice model. The fast population is associated
.....................
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........
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.
,
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•
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i
0. a
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Fig. 6. Plots of G 1 and 1 - G 1 versus concentration of diluent, d. G 1 (solid line) is the fraction of diluent molecules surrounded by polymer in the lattice model and 1 - G 1 (dotted line) is the fraction of diluent molecules in contact with other diluent molecules. Mathematical expressions for these quantities are given in the text.
561
with diluent molecules in contact with other diluent molecules while the slow population is associated with diluent molecules surrounded by polymer units. Quantitatively, the slow population is simply given by G 1 and the fast population is given by 1 - G1. A plot of these two quantities as a function of diluent concentration is shown in fig. 6. To apply this approach to TOP in BPA-PC, the size of the diluent moiety relative to the repeat unit must be considered. The polymer repeat unit is used to define the size of the lattice unit and based on molecular weight, the TOP is considered to occupy two lattice units which leads to G~ =pS. At the concentration of 15 wt%, the slow population is predicted to be 38%, in good agreement with observation.
4. Discussion The lattice model assumes the diluent and polymer repeat units are randomly distributed on the available sites and focuses on nearest-neighbor contacts to interpret molecular level dynamics. At the low concentrations of diluent considered so far, the thermal glass transition is associated with motion of repeat units in contact with zero or one diluent molecule plus the motion of isolated diluent molecules. Polymer units in contact with diluent clusters and the clusters themselves are associated with motion which occurs below the thermal glass transition and is controlled by the intrinsic mobility of the diluent which could be characterized by the glass transition temperature of the pure diluent. At low concentration of diluent, the range of the motion of these more mobile regions associated with diluent clusters would extend only over a few lattice sites and thus would not constitute a second glass transition. As the concentration of diluent is raised beyond the regime of the N M R studies considered here, the range of motion associated with the more mobile regions would grow and the apparent thermal glass transition temperature might jump to a value c l o s e r t o t h e i n t r i n s i c d i l u e n t , Tg. T h e r e a r e i n d i c a tions of such a jump in the apparent glass transi-
tion temperature with concentration in other perimental studies [21,22].
ex-
562
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
This lattice model is quite simplistic and is unlikely to be applicable under all circumstances, For instance, the dynamics of diluent molecules in all clusters of two or more molecules are described
[21 J. Hirschinger, H. Miura, K.H. Gardner, and A.D. English, Macromolecules 23 (1990) 2153. [31 H.W. Spiess, Colloid Polym. Sci. 261 (1983) 193. [4] H.W. Spiess, Advances in Polymer Science, Vol. 66
by one rate which most certainly must be an approximation. However, this picture based on local contacts on a lattice appears to be a suitable basis for the interpretation of a variety of relaxation experiments in mixed glasses.
[51 P.T. Inglefield, A.A. Jones, R.P. Lubianez and J.F. O'Gara, Macromolecules 14 (1981)288. [61 A.A. Jones, J.F. O'Gaxa, P.T. Inglefield, J.T. Bendler, A.F. Yee and K.L. Ngai, Macromolecules 16 (1983) 658. [71 P.T. Inglefield, R.M. Amici, J.F. O'Gara, C.-C. Hung and
5. Conclusions I n a mixed diluent-polymer glass, the lattice
model relates changes in relaxation of both the polymer and the diluent to nearest-neighbor contacts between the two components. At low c o n centration of diluent, suppression of sub-Tg polym e r motion is linked to improved packing caused by isolated diluent molecules. At higher c o n centrations, diluent-diluent clusters predicted on the basis of random statistics produce local points of high mobility both for the diluent and for adjacent polymer repeat units. The competitive effects of suppression by isolated diluent and enhanced mobility by diluent clusters leads to cornplicated mechanical and dielectric response as a function of concentration, intrinsic mobility of the diluent, and the nature of sub-Tg polymer motion in pure polymer, This research was carried out with support from National Science Foundation G r a n t N M R 9001678.
References [1] A.A. Jones, in: Molecular Dynamics in Restricted Geometries, eds. J. Klafter and M.J. Drake (Wiley, New York, 1989).
(Springer, Berlin, 1985) p. 23.
A.A. Jones, Macromolecules 16 (1983) 1552. [8] J.F. O'Gara, A.A. Jones, C.-C. Hung and P.T. Inglefield, Macromolecules 18 (1986) 1117. [9] A.K. Roy, A.A. Jones and P.T. Inglefield, Macromolecules 19 (1986) 1356.
[10] J.J. Connolly, P.T. lnglefield and A.A. Jones, J. Chem.
Phys. 86 (1987) 6602. [11] J. Schaefer, E.O. Stejskal and R. Buchdahl, Macromolecules 10 (1977) 384. [12] J. Schaefer, E.O. Stejskal, R.A. McKay and W.T. Dixon, Macromolecules 17 (1984) 1479. [13] T.R. Steger, J. Schaefer, E.O. Stejskal and R.A. McKay, Macromolecules 13 (1980) 1127. [14] J. Schaefer, E.O. Stejskal, D. Perchak, J. Skolnik and R. Yaris, Macromolecules 18 (1985) 368. [15] L.A. Belflore, P.M. Henrichs, D.J. Massa, N. Zumbulyadis, W.P. Rothwell and S.L. Cooper, Macromolecules 16 (1983) 1744. I161 E.W. Fischer, G.P. Hellman, H.W. Spiess, S.F. Horth, U.E. Carius and M. Wherle, Makromol. Chem. Suppl. 12
(1985) 189. [17] Y. Liu, A.K. Roy, A.A. Jones, P.T. Inglefield and P. Ogden, Macromolecules 23 (1990) 968. [181 B.J. Cauley, C. Cipriani, K. Ellis, A.K. Roy, A.A. Jones, P.T. Inglefield, B.J. McKinley and R.P. Kambour, Macromolecules 24 (1991) 403. [19] D. Stauffer, Introduction to Percolation Theory (Taylor and Francis, Philadelphia, PA, 1985) p. 24. [201 R.P. Kambour, J.M. Kelly, B.J. McKinley, B.J. Cauley, P.T. Inglefield and A.A. Jones, Macromolecules 21 (1988) 2937. [21] K. Adachi, M. Hattori and Y. lshida, J. Polym. Sci. Polym. Phys. Ed. 15 (1977) 693. [22] D.J. Plazek, E. Riande, H. Markovitz and N. Raghupathi, J. Polym. Sci. Polym. Phys. Ed. 17 (1979) 2189.
Journal of Non-Crystalline Solids 131-133 (1991) 556-562 North-Holland
Section 2.5. Glassy polymers
A lattice model for dynamics in a mixed polymer-diluent glass A.A. Jones, P.T. Inglefield, Y. Liu, A.K. R o y a n d B.J. C a u l e y Department of Chemistry, Clark University, Worcester, MA 01610, USA
A lattice model is developed to interpret N M R lineshape and relaxation data on polymer-diluent systems. The lattice is used to count nearest-neighbor contacts which are assumed to influence local dynamics. When applied to the polymer chain, the presence of one diluent-polymer repeat unit contact is considered to suppress sub-glass transition motion through improved packing. Polymer repeat units in contact with more than one diluent molecule on the lattice are presumed to have enhanced mobility. Polymer repeat units surrounded by other polymer repeat units are thought to have no change in their sub-glass transition dynamics. When applied to the diluent, isotropic rotational motion of diluent surrounded by polymer is considered to commence at the apparent thermal glass transition. For those diluent molecules in contact with other diluent molecules on the lattice, sub-glass transition rotational motion occurs at a temperature determined by the intrinsic mobility of the diluent. The motions of the chain and the diluent are reflected in the modulus and are traditionally discussed in terms of plasticization and antiplasticization.
1. Introduction Molecular motions of the polymer chain in polymeric glasses well below the thermal glass transition influence bulk properties. Sub-glass t r a n s i t i o n motions are geometrically restricted and can be divided into two general categories. One class of these motions involves anisotropic rotations such as methyl-group rotation or phenylgroup rotation [1] while the other class involves librational motion [2]. Solid state N M R has proved useful in elucidating a variety of details concerning the rate and amplitude of both categories of motions, One of the best studied systems is bisphenol A polycarbonate (BPA-PC) shown in fig. 1. Both methyl- and phenylene-group rotations occur in glassy BPA-PC, with the phenylene motion linked to a significant mechanical loss [3-14]. The phenylene group reorients by jumping between two minima separated by approximately 180 = and this motion has been named ~r-flips [3]. The rate of occurrence of ,~-flips is inhomogeneous: i.e., different phenylene groups in different spatial locations in the glass execute ~r-flips at different rates. The fractional exponential correlation function
POLYMER
.-BPA-PC
--
~)~"-
--
=,"~ 7~ ! / ' ~
'r--~ ="~
. . . i.
'j?---
o--
OlLUENTS o ~
/ e _II_ o _ _
c 4H
DBP
c__o~ %"6
TOP
Fig.
I'~'~
I PO
3 1. Chemical structures and abbreviations for the polymer
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
repeat unit and the two diluents.
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
with an exponent in the range of 0.15 to 0.20 has been used to characterize the distribution of rates [6,8-9]. The addition of a number of low molecular weight liquids to BPA-PC yields glasses with a single, lower thermal glass transition temperature, and a much reduced low-temperature mechanical loss peak [15]. Solid state N M R studies have shown that the suppression of the mechanical loss coincides with the suppression of the occurrence of ~r-flips [16]. The reduced mechanical loss upon addition of a low molecular weight diluent also leads to a higher modulus than is observed for the pure polymer [15]. The classical term for these phenomena associated with the addition of diluent is antiplasticization. Plasticization or a lowering of the modulus also occurs in BPA-PC at higher concentrations [15]; also, in general the modulus depends on a variety of factors besides concentration such as temperature, the structure of the diluent and the structure of the polymer, Solid state N M R experiments have also shown that the suppression of ~r-flips is associated with an apparent broadening of the inhomogeneous distribution of correlation times [16].
Tip (ms)
In an attempt to provide an interpretational framework for the observations just reviewed, a lattice model is presented below to count polymer-diluent and diluent-diluent contacts. The resuits of a number of recent solid state N M R experiments on BPA-PC plus diluent which motivated the development of this lattice description are summarized [17-18]. The lattice model correlates relaxation behavior of both the polymer and the diluent which is required in order to understand the mechanical response of the multicomponent glass.
2. Polymer dynamics in the presence of diluent The first experimental results to be summarized are on BPA-PC plus di-n-butyl phthalate (DBP). This is a convenient system for consideration by a lattice model since the polymer repeat unit and the diluent are of the same size and thus either one or the other can occupy a lattice site. As mentioned in the introduction, deuterium lineshape studies on phenyl deuterated BPA-PC indicate a suppression of the ~r-flips and a broadening of the apparent distribution of flip rates [16]. The
22 21 20 19 18 17 16-
/
15
13 1211 10 ~
9
8765
557
i
-~20-10~
50/0
-80' -d
-40' -~_
o
o'
2;
4;
6~
80' 1~0 ~2'0
TC Fig. 2. Proton Tlo versus temperature for pure BPA-PC d 6 and for BPA-PC d 6 plus 5 wt% perdeutero DBP. The solid line is a smoothed representation of the data for pure polymer and the dashed line for polymer plus diluent. The shaded region indicates suppression of the motion of some of the phenylene groups by the presence of the diluent DBP.
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
558
suppression and broadening are both associated with an increased population of slow flippers, To further characterize the distribution of flip rates, proton T1~ were measured as a function of temperature on partially deuterated BPA-PC [17]. The methyl groups of BPA-PC d 6 a r e deuterated while the phenylene groups remain protonated, The proton T1, values are primarily determined by the ,~-flip rates leading to the broad minimum shown in fig. 2 which can be mapped onto the mechanical loss maximum through the use of a fractional exponential correlation function [6,8]. The breadth of the minimum reflects the distribution of ~T-flip rates. The T~p experiment has a characteristic frequency of 44 kHz and the minimum occurs at about - 2 0 ° C while the mechanical experiment has a characteristic frequency of 1 Hz and the maximum occurs at a b o u t - 1 0 0 ° C , On addition of 5 wt% perdeutero DBP, the minimum narrows and is shifted to higher temper-
ature as is also shown in fig. 2. The deuterated diluent is invisible in this N M R experiment so that the TI~ minimum still reflects only the ~r-flip rates. Close examination shows the change in the minimum to be caused by selective suppression of the low-temperature side of the minimum which corresponds to the faster moving phenylene groups. This is shown as a shaded region in fig. 2. In terms of fitting the minimum with a fractional exponential correlation function, the exponent stays constant while the apparent energy increases from 49 to 71 k J / m o l [17]. An examination of the corresponding mechanical loss peak or dielectric loss peaks in related systems does not show a suppression by elimination of the low-temperature side. However, diluent contributions to loss are not easily separated from polymer contributions, as can be done by isotopic labelling in N M R experiments, making simple comparisons difficult. How can the narrower minimum be reconciled
23-22-
I*/,
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T1 p
11~
(ms) 12 10' 11 -
5 'I -120
10%
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I
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-60
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I
I
I
I
i
I
-20
0
20
40
60
80
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Too Fig. 3. Same as fig. 2 except for pure p o l y m e r and p o l y m e r plus 10 wt% diluent. N o t e b o t h s u p p r e s s i o n i n d i c a t e d b y s h a d i n g a n d the presence of a new lower t e m p e r a t u r e m i n i m u m at - 90 o C.
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
559
to the broader distribution of flip rates seen in lineshape studies? Phenylene groups whose flip rates are significantly suppressed do not contrib-
F3 =
4pd 3,
(ld)
F4 =
d4"
(le)
ute to the T1, minimum and do not undergo relaxation until the glass transition is reached. If these very slow rates were combined with those remaining in the minimum, the resulting distribution would appear quite broad. The presence of this slow population is indeed visible in the deuteron lineshape, The T1, minimum located at - 2 0 ° C continues to narrow as diluent concentration is increased from 5 to 10-25 wt%. However, an entirely new feature also appears: a minimum located at - 1 0 0 ° C as shown in fig. 3. This lower temperature minimum which develops at higher concentration must correspond to phenylene groups which move more rapidly than those associated with the original minimum in the pure polymer, These new details concerning changes in the distribution of rates of phenylene-group motion upon addition of diluent provide the impetus for
Several assumptions and refinements need to be made to relate these fractional populations to observed relaxation behavior. First, the fractional population, F0, would correspond to the phenyl-
the development of the lattice model. The set-up of polymer and diluent on a lattice is much like the beginning of Flory-Huggins theory. A lattice with six nearest neighbors is considered and either a repeat unit or a diluent occupies a site. Because of chain connectivity, two sites around a polymer unit are occupied by covalently bonded repeat units. There are four remaining nearest-neighbor sites which may be either diluent molecules or other repeat units. Polymer repeat units will be classified according to the number of nearest-neighbor sites which are occupied by diluent. If p is the probability a site is occupied by a polymer repeat unit and the lattice is filled on the basis of random statistics, the fraction of sites, F0, which are polymer units surrounded by other polymer units is F0 = p4.
(la)
If d is the fraction of sites occupied by diluent, the fractions of repeat units with one, two, three and four diluent nearest neighbors are given by E 1 = 4p3d, F 2 = 6p2d 2,
(lb) (lc)
ene groups contributing to the normal relaxation minima or loss peak. The fraction of polymer units with one diluent neighbor is associated with the extent of suppression. The first diluent neighbor is assumed to improve significantly packing around the reference polymer unit and therefore reduces mobility. The rate of motion is assumed to become so slow that relaxation of this unit no longer occurs as part of the low-temperature peak but is shifted to the glass transition. Polymer repeat units with more than one diluent neighbor are assumed to have enhanced mobility and their relaxation is shifted to lower temperatures relative to the original low-temperature loss peak. The lower temperature results from the greater inherent mobility of diluent molecules, as indicated by the glass transition temperature for the diluent as a pure liquid. The clusters of diluent molecules thus promote 'soft' regions in the glass. To a first approximation, the fractional population of repeat units in contact with two or more diluents is the sum of F 2, F 3 and F 4. According to the model, these repeat units account for the lower temperature minimum observed at - 1 0 0 ° C for DBP in BPA-PC. A slight correction to the fractional populations calculated thus far is needed. Suppression of the flip motion is proposed for repeat units in contact with a single diluent and enhanced mobility is proposed for repeat units in contact with two or more diluent moieties. This calculation should be refined, since our intention is to associate suppression with an isolated diluent molecule. If a repeat unit is in contact with a single diluent molecule but that diluent is in contact with other diluent molecules, the reference repeat unit should be considered to have enhanced mobility rather than suppressed mobility. To estimate this effect, the fractional population of diluent in clusters is required. This calculation is related to the initial stages of percolation theory [19] and the number
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
560
1.0
......
molecule is an isolated diluent molecule will be called the extent of suppression, S:
/"
0.a
/"
S = F xp~ = 4pSd.
The fraction o f repeat units that are in contact with a microscopic cluster of diluents through a single diluent molecule is /'1(1 _pS), and this situation will lead to enhanced mobility of the reference unit. These repeat units should be combined with the fractions F 2, F 3, and F4 to yield a fraction with enhanced mobility, E:
// o= 0.6.
\
/,"
, , , / , ~ o
0.4.
o.2
. . . . . ........ .. . . . . . . . . . . . .
..;4,
/ , f"'fS/"
0.0 !" . O.
(3)
........
E = FI ( 1 - pS ) + Fz + F3 + F4,
. . . . . . 0.2
0.4
E 0.6
o.a
= 4p3d(1
_ p 5 ) + 6p2d2 + 4 p d 3 + d 4.
(4)
1.0
Concentration
Fig. 4. Plots of Fo, S, and E versus diluent concentration, d. F0 (solid line) indicates the fraction of the polymer repeat units surrounded by polymer which contributes to the - 2 0 °C minimum; S (dotted line) is the fraction of polymer repeat units in contact with one diluent leading to a suppression of the sub-glass transition motions; E (dashed line) is the fraction of polymer repeat units in contact with diluent clusters leading to the appearance of a m i n i m u m at - 9 0 ° C at higher diluent concentrations. Associatedmathematicalexpressions are given
in the text. of clusters of various sizes is not easily calculated. However, only the fractional population of isolated diluent molecules, G 1, is required here and that quantity is simply given by Ga = p 6 . (2) Now the probability that a repeat unit is in contact with one diluent molecule and that diluent
To summarize the factors influencing polymer chain dynamics, the intensity of the low-temperature loss peak is given by F0, the extent of suppression is given by S, and the intensity of a new lOSS peak caused by contact with mobile diluent groupings is given by E. These quantities are given plotted in fig. 4 a s a function of diluent c o n c e n t r a tion, d. These calculations assume the glass transition temperature remains higher than the position of the low temperature loss peak of the pure polymer.
3. Diluent dynamics in polymeric glasses Thus far changes in the mobility of polymer repeat units with diluent concentration have been considered but similar changes in diluent mobility are also observed. To monitor diluent dynamics
+
=
?
j SLOW C O M P O N E N T
_ FAST
COMPONENT
COMPOSITE
Fig. 5. Partially collapsed chemical shift anisotropy lineshape. The points are experimental observations and the lines are simulations. The simulation consists of a narrow component corresponding to the fast population of diluent and a broad component corresponding to the slow population. The two components are combined to simulate the observed lineshape.
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
independent of the polymer matrix, phosphorous31 N M R of phosphate ester diluents has proved quite informative [18,20]. The system-for consideration here will be a trioctyl phosphate (TOP) in BPA-PC (structures shown in fig. 1). The 31p chemical shift anisotropy line shape for 15 wt% TOP shows considerable collapse at temperatures well below the glass transition indicating the presence of considerable diluent motion. The lineshape collapse has been simulated by combining two populations with two rates referred to as the fast and slow components. Figure 5 shows a typical spectrum in the collapse region and the fast and slow contributions to the total lineshape. At temperatures just below the thermal glass transition, the fast component is undergoing rapid, isotropic Brownian rotational diffusion on the N M R timescale while the slow component is undergoing rather little reorientation on the same timescale. This bimodal character used to describe the lineshape can readily be meshed with the lattice model. The fast population is associated
.....................
1.0
........................... l l ~[ l ~ ~ I
0.o
........
0.6
/
_ "~ 0.4
/
~-
0.2 0.0
•
•0
.
,
0.2
•
•
, 0.4 0.6 Concentration ,
i
0. a
1.0
Fig. 6. Plots of G 1 and 1 - G 1 versus concentration of diluent, d. G 1 (solid line) is the fraction of diluent molecules surrounded by polymer in the lattice model and 1 - G 1 (dotted line) is the fraction of diluent molecules in contact with other diluent molecules. Mathematical expressions for these quantities are given in the text.
561
with diluent molecules in contact with other diluent molecules while the slow population is associated with diluent molecules surrounded by polymer units. Quantitatively, the slow population is simply given by G 1 and the fast population is given by 1 - G1. A plot of these two quantities as a function of diluent concentration is shown in fig. 6. To apply this approach to TOP in BPA-PC, the size of the diluent moiety relative to the repeat unit must be considered. The polymer repeat unit is used to define the size of the lattice unit and based on molecular weight, the TOP is considered to occupy two lattice units which leads to G~ =pS. At the concentration of 15 wt%, the slow population is predicted to be 38%, in good agreement with observation.
4. Discussion The lattice model assumes the diluent and polymer repeat units are randomly distributed on the available sites and focuses on nearest-neighbor contacts to interpret molecular level dynamics. At the low concentrations of diluent considered so far, the thermal glass transition is associated with motion of repeat units in contact with zero or one diluent molecule plus the motion of isolated diluent molecules. Polymer units in contact with diluent clusters and the clusters themselves are associated with motion which occurs below the thermal glass transition and is controlled by the intrinsic mobility of the diluent which could be characterized by the glass transition temperature of the pure diluent. At low concentration of diluent, the range of the motion of these more mobile regions associated with diluent clusters would extend only over a few lattice sites and thus would not constitute a second glass transition. As the concentration of diluent is raised beyond the regime of the N M R studies considered here, the range of motion associated with the more mobile regions would grow and the apparent thermal glass transition temperature might jump to a value c l o s e r t o t h e i n t r i n s i c d i l u e n t , Tg. T h e r e a r e i n d i c a tions of such a jump in the apparent glass transi-
tion temperature with concentration in other perimental studies [21,22].
ex-
562
A.A. Jones et al. / Dynamics in a mixed polymer-diluent glass
This lattice model is quite simplistic and is unlikely to be applicable under all circumstances, For instance, the dynamics of diluent molecules in all clusters of two or more molecules are described
[21 J. Hirschinger, H. Miura, K.H. Gardner, and A.D. English, Macromolecules 23 (1990) 2153. [31 H.W. Spiess, Colloid Polym. Sci. 261 (1983) 193. [4] H.W. Spiess, Advances in Polymer Science, Vol. 66
by one rate which most certainly must be an approximation. However, this picture based on local contacts on a lattice appears to be a suitable basis for the interpretation of a variety of relaxation experiments in mixed glasses.
[51 P.T. Inglefield, A.A. Jones, R.P. Lubianez and J.F. O'Gara, Macromolecules 14 (1981)288. [61 A.A. Jones, J.F. O'Gaxa, P.T. Inglefield, J.T. Bendler, A.F. Yee and K.L. Ngai, Macromolecules 16 (1983) 658. [71 P.T. Inglefield, R.M. Amici, J.F. O'Gara, C.-C. Hung and
5. Conclusions I n a mixed diluent-polymer glass, the lattice
model relates changes in relaxation of both the polymer and the diluent to nearest-neighbor contacts between the two components. At low c o n centration of diluent, suppression of sub-Tg polym e r motion is linked to improved packing caused by isolated diluent molecules. At higher c o n centrations, diluent-diluent clusters predicted on the basis of random statistics produce local points of high mobility both for the diluent and for adjacent polymer repeat units. The competitive effects of suppression by isolated diluent and enhanced mobility by diluent clusters leads to cornplicated mechanical and dielectric response as a function of concentration, intrinsic mobility of the diluent, and the nature of sub-Tg polymer motion in pure polymer, This research was carried out with support from National Science Foundation G r a n t N M R 9001678.
References [1] A.A. Jones, in: Molecular Dynamics in Restricted Geometries, eds. J. Klafter and M.J. Drake (Wiley, New York, 1989).
(Springer, Berlin, 1985) p. 23.
A.A. Jones, Macromolecules 16 (1983) 1552. [8] J.F. O'Gara, A.A. Jones, C.-C. Hung and P.T. Inglefield, Macromolecules 18 (1986) 1117. [9] A.K. Roy, A.A. Jones and P.T. Inglefield, Macromolecules 19 (1986) 1356.
[10] J.J. Connolly, P.T. lnglefield and A.A. Jones, J. Chem.
Phys. 86 (1987) 6602. [11] J. Schaefer, E.O. Stejskal and R. Buchdahl, Macromolecules 10 (1977) 384. [12] J. Schaefer, E.O. Stejskal, R.A. McKay and W.T. Dixon, Macromolecules 17 (1984) 1479. [13] T.R. Steger, J. Schaefer, E.O. Stejskal and R.A. McKay, Macromolecules 13 (1980) 1127. [14] J. Schaefer, E.O. Stejskal, D. Perchak, J. Skolnik and R. Yaris, Macromolecules 18 (1985) 368. [15] L.A. Belflore, P.M. Henrichs, D.J. Massa, N. Zumbulyadis, W.P. Rothwell and S.L. Cooper, Macromolecules 16 (1983) 1744. I161 E.W. Fischer, G.P. Hellman, H.W. Spiess, S.F. Horth, U.E. Carius and M. Wherle, Makromol. Chem. Suppl. 12
(1985) 189. [17] Y. Liu, A.K. Roy, A.A. Jones, P.T. Inglefield and P. Ogden, Macromolecules 23 (1990) 968. [181 B.J. Cauley, C. Cipriani, K. Ellis, A.K. Roy, A.A. Jones, P.T. Inglefield, B.J. McKinley and R.P. Kambour, Macromolecules 24 (1991) 403. [19] D. Stauffer, Introduction to Percolation Theory (Taylor and Francis, Philadelphia, PA, 1985) p. 24. [201 R.P. Kambour, J.M. Kelly, B.J. McKinley, B.J. Cauley, P.T. Inglefield and A.A. Jones, Macromolecules 21 (1988) 2937. [21] K. Adachi, M. Hattori and Y. lshida, J. Polym. Sci. Polym. Phys. Ed. 15 (1977) 693. [22] D.J. Plazek, E. Riande, H. Markovitz and N. Raghupathi, J. Polym. Sci. Polym. Phys. Ed. 17 (1979) 2189.